Research Article Optimal Ascent Guidance for Air-Breathing...

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Hindawi Publishing Corporation Mathematical Problems in Engineering Volume 2013, Article ID 313197, 11 pages http://dx.doi.org/10.1155/2013/313197 Research Article Optimal Ascent Guidance for Air-Breathing Launch Vehicle Based on Optimal Trajectory Correction Xuefang Lu, Yongji Wang, and Lei Liu Key Laboratory of Ministry of Education for Image Processing and Intelligent Control, School of Automation, Huazhong University of Science and Technology, Wuhan 430074, China Correspondence should be addressed to Yongji Wang; [email protected] Received 14 October 2013; Revised 28 October 2013; Accepted 28 October 2013 Academic Editor: Hui Zhang Copyright © 2013 Xuefang Lu et al. is is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. An optimal guidance algorithm for air-breathing launch vehicle is proposed based on optimal trajectory correction. e optimal trajectory correction problem is a nonlinear optimal feedback control problem with state inequality constraints which results in a nonlinear and nondifferentiable two-point boundary value problem (TPBVP). It is difficult to solve TPBVP on-board. To reduce the on-board calculation cost, the proposed guidance algorithm corrects the reference trajectory in every guidance cycle to satisfy the optimality condition of the optimal feedback control problem. By linearizing the optimality condition, the linear TPBVP is obtained for the optimal trajectory correction. e solution of the linear TPBVP is obtained by solving linear equations through the Simpson rule. Considering the solution of the linear TPBVP as the searching direction for the correction values, the updating step size is generated by linear search. Smooth approximation is applied to the inequality constraints for the nondifferentiable Hamiltonian. e sufficient condition for the global convergence of the algorithm is given in this paper. Finally, simulation results show the effectiveness of the proposed algorithm. 1. Introduction e development of space technology has given rise to the expectation that launchers will become low cost and fully reusable. e launch vehicle with hypersonic air-breathing propulsion is considered to reduce the cost of payloads taken to the Earth’s orbit. e air-breathing launch has inherent features that make it a candidate for future space transportation [1]. e impulse of air-breathing propulsion, which is approximately 3000 s [2], is significantly higher than that of a rocket (360 s). Air-breathing propulsion brings high impulse as well as strong nonlinear, coupling aerodynamic and thrust. e traditional design category for ascent guidance is to drive an optimal nominal trajectory off-board. e guidance problem is then transformed into a tracking problem for the designed optimal nominal trajectory. e design of a trajectory could be formulated as a global optimization task [3]. e methods for numerical optimization of con- tinuous dynamic systems could be termed “Hamiltonian” (indirect method) and “Transcription” (direct method) [4]. For the optimization problem with inequality constraints, smoothing approximation was considered in [5, 6]. A filled function approach for nonsmooth constrained global opti- mization was presented in [7]. Linear-quadratic optimization was implemented to optimal control in [8]. Direct and indirect optimization methods were implemented to the off- board trajectory optimization problem in previous literature. e numerical algorithms of trajectory optimization for vehicles were summarized and systematically analyzed in [9, 10]. A new concept of pseudocontrol sets to solve optimal control problems was proposed in [11]. is approach reduces the calculation cost by combining large-scale linear program- ming algorithms with discretization of the continuous system dynamics on small segments. An algorithm for multiobjective optimization was presented in [12]. Intelligent algorithms can also be used for trajectory optimization problems. e par- ticle swarm optimization (PSO) method was implemented to the space trajectory optimization in [13]. e simulation results showed the effective of PSO in finding the optimal solution to the space trajectory optimization problems, with great numerical accuracy. Approximate numerical methods

Transcript of Research Article Optimal Ascent Guidance for Air-Breathing...

Page 1: Research Article Optimal Ascent Guidance for Air-Breathing ...downloads.hindawi.com/journals/mpe/2013/313197.pdfResearch Article Optimal Ascent Guidance for Air-Breathing Launch Vehicle

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2013 Article ID 313197 11 pageshttpdxdoiorg1011552013313197

Research ArticleOptimal Ascent Guidance for Air-Breathing Launch VehicleBased on Optimal Trajectory Correction

Xuefang Lu Yongji Wang and Lei Liu

Key Laboratory of Ministry of Education for Image Processing and Intelligent Control School of AutomationHuazhong University of Science and Technology Wuhan 430074 China

Correspondence should be addressed to Yongji Wang wangyjchmailhusteducn

Received 14 October 2013 Revised 28 October 2013 Accepted 28 October 2013

Academic Editor Hui Zhang

Copyright copy 2013 Xuefang Lu et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

An optimal guidance algorithm for air-breathing launch vehicle is proposed based on optimal trajectory correction The optimaltrajectory correction problem is a nonlinear optimal feedback control problem with state inequality constraints which results in anonlinear and nondifferentiable two-point boundary value problem (TPBVP) It is difficult to solve TPBVP on-board To reducethe on-board calculation cost the proposed guidance algorithm corrects the reference trajectory in every guidance cycle to satisfythe optimality condition of the optimal feedback control problem By linearizing the optimality condition the linear TPBVP isobtained for the optimal trajectory correction The solution of the linear TPBVP is obtained by solving linear equations throughthe Simpson rule Considering the solution of the linear TPBVP as the searching direction for the correction values the updatingstep size is generated by linear search Smooth approximation is applied to the inequality constraints for the nondifferentiableHamiltonian The sufficient condition for the global convergence of the algorithm is given in this paper Finally simulation resultsshow the effectiveness of the proposed algorithm

1 Introduction

The development of space technology has given rise to theexpectation that launchers will become low cost and fullyreusable The launch vehicle with hypersonic air-breathingpropulsion is considered to reduce the cost of payloadstaken to the Earthrsquos orbit The air-breathing launch hasinherent features that make it a candidate for future spacetransportation [1] The impulse of air-breathing propulsionwhich is approximately 3000 s [2] is significantly higher thanthat of a rocket (360 s) Air-breathing propulsion brings highimpulse as well as strong nonlinear coupling aerodynamicand thrust

The traditional design category for ascent guidance is todrive an optimal nominal trajectory off-board The guidanceproblem is then transformed into a tracking problem forthe designed optimal nominal trajectory The design of atrajectory could be formulated as a global optimizationtask [3] The methods for numerical optimization of con-tinuous dynamic systems could be termed ldquoHamiltonianrdquo(indirect method) and ldquoTranscriptionrdquo (direct method) [4]

For the optimization problem with inequality constraintssmoothing approximation was considered in [5 6] A filledfunction approach for nonsmooth constrained global opti-mization was presented in [7] Linear-quadratic optimizationwas implemented to optimal control in [8] Direct andindirect optimization methods were implemented to the off-board trajectory optimization problem in previous literatureThe numerical algorithms of trajectory optimization forvehicles were summarized and systematically analyzed in[9 10] A new concept of pseudocontrol sets to solve optimalcontrol problemswas proposed in [11]This approach reducesthe calculation cost by combining large-scale linear program-ming algorithmswith discretization of the continuous systemdynamics on small segments An algorithm formultiobjectiveoptimization was presented in [12] Intelligent algorithms canalso be used for trajectory optimization problems The par-ticle swarm optimization (PSO) method was implementedto the space trajectory optimization in [13] The simulationresults showed the effective of PSO in finding the optimalsolution to the space trajectory optimization problems withgreat numerical accuracy Approximate numerical methods

2 Mathematical Problems in Engineering

of optimization were presented for multiorbit noncoplanarorbit transfers of low-thrust spacecraft in [14]

Many control methods were implemented to trajectorytracking guidance and control problem [15] In the previousliterature many researchers focused on the robust controlmethod [16 17] It has obtained successful application in theindustry [18 19] For the networked control system a H-infinity step tracking control method was presented in [20]An adaptive fuzzy robust control for a class of nonlinearsystems was proposed in [21] An adaptive guidance law andoff-board trajectory optimization for air-breathing launchvehicle were addressed in [22] In that paper the optimalcontrol problem was solved using SQP method And theadaptive guidance law was developed using a feedback loopbased on a second-order rate controller for angle of attack Arobust state feedback guidance law was generated in real timeusing the indirect Legendre pseudospectral feedbackmethodin [23] In that paper the guidance problem was convertedinto a trajectory state regulation problem which is a lineartime varying system

However the accuracy of the trajectory tracking methodis low with the disturbance and the modeling error Itlacks the autonomy and adaptability to cope with the non-nominal vehicle and mission conditions needed for futurereusable launch vehicles [24] To improve the performanceand the accuracy of the guidance the optimal guidancemethod generates the prospective trajectory by fast trajectoryoptimization on-board based on current flight state Thisguidance method has become a research focus with theadvance in on-board computation capability Ping Lu usedthe indirect method to pose the trajectory optimizationproblem as a nonlinear two-point boundary value problem(TPBVP) in [25 26] From this model the optimal thrustvector that satisfied the optimality condition was derivedA finite difference method was employed to solve thenonlinear TPBVP through numerical calculations Similarlya fast trajectory optimization for hypersonic air-breathingvehicles was presented in [27] The indirect method wasimplemented for ascent trajectory optimization on-boardconsidering the features of the air-breathing vehicle Ping Lursquoswork is significant for the optimal ascent guidance Howeverthe TPBVP is nonlinear and nondifferentiable for the airbreathing launch vehicle It is difficult to solve the nonlinearand nondifferentiable TPBVP on-board in every guidancecycle Using the directmethod a new guidance concept basedon nonlinear programming (NLP) method was proposedin [24] NLP-based guidance concepts appear advantageousover conventional methods because the on-board guidancealgorithm allows a single algorithm to be implemented fordifferent vehicles and missions The optimal control problemwas parameterized into a nonlinear programming problemthat was solved by the gradient projection algorithm In [28]the reference trajectorywas updated for disturbance by an on-board algorithm that satisfied the real-time requirement Anew real-time guidance method derived from the optimalitycondition was proposed in [29] In that paper the simpleguidance parameters were updated in real time In [30] aguidance method was presented for online launcher ascenttrajectory updating based on neural networks In the paper

the utilization of a neural network approximation was usedonline during the ascent flight with a training processperformed off-line

In this paper we present an optimal guidance algorithmfor air breathing launch vehicle based on optimal trajectorycorrection to reduce the on-board calculation cost Consider-ing the current vehicle state as the initial condition the opti-mal trajectory correction problem is referred to as a nonlinearoptimal control problem with state inequality constraintsFor the real-time requirement of the on-board algorithm thelinear TPBVP is obtained for optimal trajectory correctionby linearizing the optimality condition in this paper TheSimpson rule is applied to transform the linear TPBVPinto linear equations Considering the solution of the linearTPBVP as the searching direction for the correction valuesthe updating step size is generated by linear search Smoothapproximation is applied to the inequality constraints for thenondifferentiable Hamiltonian The sufficient condition forthe global convergence of the algorithm is given in this paperFinally simulation results for different cases of the modelingerror show the effectiveness of the proposed algorithm Insummation the main contributions of this paper are givenas follows

(1) We reduce the on-board calculation cost of theguidance method a lot Comparing with the methodsin previous literatures which solves nonlinear equa-tions in every guidance cycle the proposed guidancemethod in this paper solves linear equations once inevery guidance cycle only

(2) We obtain the relationship between the global conver-gence and the guidance cycle of the online algorithmThe sufficient condition of the global convergence ofthis algorithm is given

The remainder of this paper is organized as followsSection 2 presents the state normalized-energy differentialequations of motion of the vehicle and the optimal controlproblem for optimal ascent guidance Section 3 providesdetails on the optimal guidance algorithm Section 4 presentsan analysis of the global convergence of the proposedalgorithm Section 5 discusses the proposed differentiableapproximation for Hamiltonian Section 6 presents the simu-lations for theGenericHypersonic Vehicle (GHV)model andscramjet engine Section 7 presents the conclusion

2 Problem Formulation

21 Ascent Dynamic The formulation of motion of the air-breathing vehicle in the longitudinal plane is presented in thissection As shown in Figure 1 the thrust 119879 gravitation 119892 andthe aerodynamic lift force119871 and drag force119863 are acting on thevehicle The angle of attack 120572 is the angle between the bodyaxis and the vector of the velocity V The flight path angle 120574 isthe angle between the vector of the velocity V and the groundplane

Mathematical Problems in Engineering 3

Center of mass

V

T

D

L

120572

120574

Figure 1 The forces acting on the vehicle

Themotion of the vehicle in the longitudinal plane can bedescribed as follows

119889119903

119889119905= V sin 120574

119889V119889119905

=119879 cos120572 minus 119863

119898minus 119892 sin 120574

119889120574

119889119905=

1

V[119879 sin120572 + 119871

119898cos120590 minus 119892 cos 120574 + V2

119903cos 120574]

119889119898

119889119905= minus

119879

119868sp

(1)

where 119903 119898 119868sp and 119892 are the altitude weight of the vehicleimpulse and gravity acceleration respectively

The lift force 119871 and the drag force119863 are given as follows

119871 =1

2120588V2119878ref119862119871 (119872119886 120572)

119863 =1

2120588V2119878ref119862119863 (119872119886 120572)

(2)

where 120588 and 119878ref are the air density of the current altitudeand the reference area respectively 119862

119871and 119862

119863are the lift

coefficient and the drag coefficient respectively which are thenonlinear functions of the angle of attack120572 andMachnumber119872119886 For the air-breathing engine the thrust is given by

119879 = 119879 (120588119872119886 120572 120601) (3)

where119879 is the nonlinear function of 120588 120572119872119886 and the throttle

command 120601In [25] the equations were normalized to reduce

the numerical calculation error Variable substitutions areapplied as follows

119877 =119903

1199030

119881 =V

radic11989201199030

119892 =11989201199032

0

1199032 (4)

where 1199030and 119892

0are Earth radius and the ground gravity

acceleration respectively Considering the thrust 119879 andaerodynamic forces as a composition of forces we define thenormalized accelerations 119886

119879and 119886119871as follows

119886119879=

119879 cos120572 minus 119863

1198981198920

119886119871=

119879 sin120572 + 119871

1198981198920

(5)

We define

120591 =119905

radic11990301198920

(6)

The normalized equations of motion are obtained from (1)(4) (5) and (6) as follows

119889119877

119889120591= 119881 sin 120574

119889119881

119889120591= 119886119879minussin 1205741198772

119889120574

119889120591=

1

119881[119886119871+ (1198812minus

1

119877)(

cos 120574119877

)]

119889119898

119889120591= minus

119879radic1199030

119868spradic1198920

(7)

The mission of ascent is to send the vehicle to therequired final state which is denoted as (119877

119891 119881119891 120574119891) Define

normalized-energy 119864 as follows

119864 =1198812

2minus

1

119877 (8)

From (8) we obtain

119889119864

119889120591= 119881119886119879

119889119877

119889119864=sin 120574119886119879

119889119881

119889119864=

1

119886119879119881

[119886119879minussin 1205741198772

]

119889120574

119889119864=

1

1198861198791198812

[119886119871+ (1198812minus

1

119877)(

cos 120574119877

)]

119889119898

119889119864= minus

119879radic1199030

119886119879119881119868spradic119892

0

(9)

The integration interval [1198640 119864119891] is fixed

22 Optimal Feedback Control Problem for Ascent GuidanceIn this section the optimal feedback control problem foroptimal ascent guidance is addressed This problem differsfrom trajectory optimization off-board in that the initialstate of the optimal control problem is obtained from thenavigation system We denote the state and the guidancecommand of the trajectory as

119909 (119864) = [119877 (119864) 119881 (119864) 120574 (119864) 119898 (119864)]119879

119906 (119864) = [120572 (119864) 120601 (119864)]119879

(10)

With the feedback vehicle state 119909119888119896obtained from navigation

system the optimal ascent guidance algorithm is used to

4 Mathematical Problems in Engineering

generate new reference trajectory [119909(119864) 119906(119864)]119879 and output

119906(119864) from the following optimal control problem

min 119869 = 120601 (119909 (119864119891)) + int

119864119891

119864119888119896+1

119892 (119909 119906) 119889119864

st = 119891 (119909 119906)

119909 (119864119888119896) = 119909119888

119896

119882 (119909 (119864119891)) = 0

119862 (119909 119906) le 0

(11)

where 119864119888119896is the normalized energy for current vehicle state

119909119888119896

3 Optimal Ascent Guidance Algorithm

In [25 26] the optimal control problem was solved on-boardin every guidance cycle However it is difficult to solve thenonlinear and nondifferentiable TPBVP on-board for air-breathing launch vehicles In this section we propose an opti-mal guidance algorithmThis guidance algorithmupdates thereference trajectory [119909 119906]

119879 in every guidance cycle to dealwith the unknowndisturbance For the current state119909119888

119896of the

119896th guidance cycle the optimal feedback control problem istransformed into a linear TPBVP problem using optimalitycondition linear approximation The searching direction ofthe correction values 119889

119896is obtained by solving the linear

TPBVP With the searching direction the new referencetrajectory is generated by linear search For the 119896th guidancecycle the trajectory update is performed as follows

[119909119896+1(119864) 119906119896+1

(119864)]119879

= [119909119896(119864) 119906119896(119864)]119879

+ 119897119896119889119879

119896(119864) (12)

The initial reference trajectory [1199090(119864) 119906

0(119864)]119879 is derived

from the off-board from trajectory optimization by directmethodThe flow chart of the algorithm is shown in Figure 2Figure 3 shows the time sequence of algorithm

31 Linear Approximation for the Optimality Condition Inthis section we linearize the optimality condition for opti-mality trajectory updating The optimal control problem (11)without inequality constraints is given as follows

min 119869 = 120601 (119909 (119864119891)) + int

119864119891

119864119888119896+1

119892 (119909 119906) 119889119864

st = 119891 (119909 119906)

119909 (119864119888119896) = 119909119888

119896

119882 (119909 (119864119891)) = 0

(13)

The Hamiltonian of (13) is given as follows

119867 = 119892 (119909 119906) + 120582119879119891 (119909 119906) (14)

Begin

Linear approximation for the

optimality condition the costate

equations the motion equations

Solving the linear TPBVP

Linear search for the step size

Updating the optimal trajectory

End

Figure 2 Flow chart of the optimal ascent guidance algorithm

where 120582 is the costate vector The state equations costateequations and optimality condition for the optimal controlproblem are given by

= 119891 (119909 119906)

= minus(120597119867

120597119909)

119879

120597119867

120597119906= 0

(15)

The initial and transversality conditions are given by

119909 (119864119888119896) = 119909119888

119896

119882 (119909 (119864119891)) = 0

120582 (119864119891) = [

120597120601

120597119909 (119864119891)

]

119879

+ [120603119879 120597119882

120597119909 (119864119891)

]

119879

(16)

We denote the trajectory correction variables as follows

[

[

Δ119909 (119864)

Δ119906 (119864)

Δ120582 (119864)

]

]

= [

[

119909 (119864) minus 119909119896(119864)

119906 (119864) minus 119906119896(119864)

120582 (119864) minus 120582119896(119864)

]

]

(17)

Mathematical Problems in Engineering 5

middot middot middot

Output theguidancecommanduk+1(E)

Output theguidancecommanduk+2(E)

middot middot middot

middot middot middot

The time consuming ofthe online algorithm The guidance cycle

Trajectoryupdating

beginsupdating

Trajectory

ends

Trajectoryupdating

beginsupdating

Trajectory

ends

Trajectoryupdating

beginsupdating

Trajectory

ends

End

Time (120591)kΔ120591 (k + 1)Δ120591 (k + 2)Δ120591

Figure 3 Time sequence of the optimal ascent guidance algorithm

We denote that

119867119896

119906119909=

1205972(119892 + 120582

119879119891)

120597119906120597119909(119909119896 119906119896 120582119896)

119867119896

119906= [

120597119867

120597119906(119909119896 119906119896 120582119896)]

119879

119867119896

119906119906=

1205972(119892 + 120582

119879119891)

1205971199062(119909119896 119906119896 120582119896)

119867119896

119909= [

120597119867

120597119909(119909119896 119906119896 120582119896)]

119879

119867119896

119909119909=

1205972(119892 + 120582

119879119891)

1205971199092(119909119896 119906119896 120582119896)

119891119896

119906=

120597119891

120597119906(119909119896 119906119896)

119867119896

119909119906=

1205972(119892 + 120582

119879119891)

120597119909120597119906(119909119896 119906119896 120582119896)

119891119896

119909=

120597119891

120597119909(119909119896 119906119896)

119892119896

119906= [

120597119892

120597119906(119909119896 119906119896)]

119879

119892119896

119909= [

120597119892

120597119909(119909119896 119906119896)]

119879

(18)

The first-order Taylor expansion of (15) on the old referencetrajectory [119909

119896119906119896

120582119896]119879 is

[120597119867

120597119906]

119879

asymp 119867119896

119906+ 119867119896

119906119909Δ119909 + 119867

119896

119906119906Δ119906 + (119891

119896

119906)119879

Δ120582

asymp minus119867119896

119909minus 119867119896

119909119909Δ119909 minus 119867

119896

119909119906(119909119896 119906119896 120582119896) Δ119906 minus (119891

119896

119909)119879

Δ120582

asymp 119891 (119909119896 119906119896) + 119891119896

119909Δ119909 + 119891

119896

119906Δ119906

(19)

The following equations are established

119867119896

119906+ (119891119896

119906)119879

Δ120582 = 119892119896

119906+ (119891119896

119906)119879

120582

119867119896

119909+ (119891119896

119909)119879

Δ120582 = 119892119896

119909+ (119891119896

119909)119879

120582

(20)

Substituting (20) into (19) yields

= minus119892119896

119909minus 119867119896

119909119909Δ119909 minus 119867

119896

119909119906(119909119896 119906119896 120582119896) Δ119906 minus (119891

119896

119909)119879

120582

Δ = 119891119896

119909Δ119909 + 119891

119896

119906Δ119906 minus

119896+ 119891 (119909

119896 119906119896)

119892119896

119906+ 119867119896

119906119909Δ119909 + 119867

119896

119906119906Δ119906 + (119891

119896

119906)119879

120582 = 0

Δ119909 (119864119888119896) + Δ119909

119896(119864119888119896) = 119909119888

119896

120597119882

120597119909 (119864119891)

Δ119909 (119864119891) +119882(119909

119896(119864119891)) = 0

120582 (119864119891) = [

120597120601

120597119909 (119864119891)

]

119879

+ [120603119879 120597119882

120597119909 (119864119891)

]

119879

(21)

This equation is a linear TPBVP about the variables[Δ119909 Δ119906 120582]

119879 The linear TPBVP is solved using the Simpsonrules in Section 6

32 Backtracking Line Search for the Step Size In this sectionthe step size 119897

119896is determined by backtracking line search

considering the solution of the linear TPBVP as the searchdirection The penalty function for problem (13) is given by

119901119896= 119869 (119909 119906) + 120578

10038171003817100381710038171003817119882(119909119891)10038171003817100381710038171003817+ int

119864119891

119864119888119896

1003817100381710038171003817119891 (119909 119906) minus 1003817100381710038171003817

+1003817100381710038171003817119909 (119864119888119896) minus 119909119888

119896

1003817100381710038171003817

(22)

6 Mathematical Problems in Engineering

where 120578 is the penalty parameter satisfied 120578 gt max |120582| Wedenote

119910 = [119909119896

119906119896

] 119889119896= [

Δ119909

Δ119906] (23)

where [Δ119909 Δ119906 120582]119879 is the solution of the linear TPBVP (21)

The minimizer of penalty function 119901119896(119909 119906) corresponds

to the solution of problem (11) Backtracking line search wasused for large-scale nonlinear optimization problems in [31]The step size 119897

119896is determined by backtracking line search that

satisfies

119897119896= max 119897 isin 119878 119901

119896+1(119910119896+ 119897119889119896)

le 119901119896+1

(119910119896) + 1198971205761120575119901119896+1

(119910119896 119889119896)

119878 = 119897 119897 = 1205762

119898 119898 = 0 1 2

(24)

where 1205761

isin (0 1) 1205762

isin (0 1) and 120575119901119896+1

(119910119896 119889119896) is the

variational of the functional 119901119896+1

(119910) caused by variational119889119896of 119910 on 119910

119896 Considering 119889

119896as the solution of (21)

120575119901119896+1

(119910119896 119889119896) le 0 is established The new state and guidance

command of the reference trajectory are generated as follows

[119909119896+1

119906119896+1

] = [119909119896

119906119896

] + 119897119896[Δ119909

Δ119906] (25)

4 The Global Convergence of the Algorithm

In [32 33] the convergence of the off-board algorithm wasanalyzed But the relationship between the global conver-gence and the guidance cycle 120591 has not been discussed for on-board optimization in the previous literatures In this sectionwe state and prove the global convergence of the on-boardalgorithm proposed in the preceding section The solutionsequence [119909

119896119906119896]119879 generated by the algorithm will converge

to the solution of the feedback optimal control problemUnlike off-board trajectory optimization the initial conditionof the feedback optimal control problem depends on thecurrent state of the vehicle in real time For the 119896th guidancecycle the optimal feedback control problem is described as(13)Thepenalty function119901

119896for this feedback optimal control

problem is given by (22) For the subsequent guidance cyclethe optimal feedback guidance problem is given by

min 119869 = 119882(119909 (119864119891)) + int

119864119891

119864119888119896+1

119892 (119909 119906) 119889119864

st = 119891 (119909 119906)

119909 (119864119888119896+1

) = 119909119888119896+1

119882 (119909 (119864119891)) = 0

(26)

where

119864119888119896+1

= 119864119888119896+ int

120591119896+Δ120591

120591119896

119889119864

119889120591119889120591 (27)

Δ120591 is the guidance cycle and 119909119888119896+1

is the new state of vehiclefrom navigation system at the time 120591

119896+ Δ120591 The penalty

function for optimal problem (24) is given by

119901119896+1

(119909 119906) = 119869 (119909 119906) + 12057810038171003817100381710038171003817119882(119909119896(119864119891))10038171003817100381710038171003817

+ 120578int

119864119891

119864119888119896+1

1003817100381710038171003817119891 (119909 119906) minus 1003817100381710038171003817 119889119864

+ 1205781003817100381710038171003817119909 (119864119888119896+1) minus 119909119888

119896+1

1003817100381710038171003817

(28)

We define the entire penalty function as follows

119901119896(119909 119906) = 119901

119896(119909 119906)

+

119896minus1

sum

119894=0

int

119864119888119894+1

119864119888119894

[1205781003817100381710038171003817119891 (119909119894 119906119894) minus 119894

1003817100381710038171003817 + 119892 (119909119894 119906119894)] 119889119864

(29)

The entire penalty function 119901119896(119909 119906) includes the penalty

function on the whole integration interval [1198640 119864119891]

Theorem 1 If the minimizer of 119901119896(119909119896 119906119896) is exited 119892(119909 119906) ge

0 and Δ120591 is selected and satisfies

120578100381710038171003817100381710038171003817119909119888119896+1

minus 119909119888119896minus int119864119888119896+1

119864119888119896119896119889119864

100381710038171003817100381710038171003817

minus120575119901119896+1

(119910119896 119889119896)

lt 1198971198961205763 (30)

where 1205763isin (0 120576

1) The solution sequence [119909

119896119906119896]119879 generated

from (24) will converge to the solution of the feedback optimalcontrol problem

Proof From (22) (28) (29) and

119909119896(119864119888119896+1

) = 119909119896(119864119888119896) + int

119864119888119896+1

119864119888119896

119896119889119864 (31)

we can obtain that

119901119896+1

(119909119896 119906119896) minus 119901119896(119909119896 119906119896)

= 120578

100381710038171003817100381710038171003817100381710038171003817

119909119896(119864119888119896) minus 119909119888

119896

minus(119909119888119896+1

minus 119909119888119896minus int

119864119888119896+1

119864119888119896

119896119889119864)

100381710038171003817100381710038171003817100381710038171003817

minus 1205781003817100381710038171003817119909119896 (119864119888119896) minus 119909119888

119896

1003817100381710038171003817

(32)

Mathematical Problems in Engineering 7

From (32) we obtain

limΔ120591rarr0

[119901119896+1

(119909119896 119906119896) minus 119901119896(119909119896 119906119896)] = 0

lim119891(119909119896119906119896)minus 119909119896rarr0

[119901119896+1

(119909119896 119906119896) minus 119901119896(119909119896 119906119896)] = 0

(33)

119901119896+1

(119909119896 119906119896) minus 119901119896(119909119896 119906119896)

= 120578

100381710038171003817100381710038171003817100381710038171003817

119909119896(119864119888119896) minus 119909119888

119896+1+ int

119864119888119896+1

119864119888119896

119896119889119864

100381710038171003817100381710038171003817100381710038171003817

minus1003817100381710038171003817119909119896 (119864119888119896) minus 119909119888

119896

1003817100381710038171003817

le 1205781003817100381710038171003817119909119896 (119864119888119896) minus 119909119888

119896

1003817100381710038171003817

+

100381710038171003817100381710038171003817100381710038171003817

119909119888119896+1

minus 119909119888119896minus int

119864119888119896+1

119864119888119896

119896119889119864

100381710038171003817100381710038171003817100381710038171003817

minus1003817100381710038171003817119909119896 (119864119888119896) minus 119909119888

119896

1003817100381710038171003817

= 120578

100381710038171003817100381710038171003817100381710038171003817

119909119888119896+1

minus 119909119888119896minus int

119864119888119896+1

119864119888119896

119896119889119864

100381710038171003817100381710038171003817100381710038171003817

(34)

From conditions (24) and (30) it is obtained that

119901119896+1

(119909119896+1

119906119896+1

)

le 119901119896+1

(119909119896 119906119896) + 1198971198961205761120575119901119896+1

(119889119896)

= 119901119896(119909119896 119906119896) + 1198971198961205761120575119901119896+1

(119889119896)

+ 120578

100381710038171003817100381710038171003817100381710038171003817

119909119896(119864119888119896) minus 119909119888

119896

minus (119909119888119896+1

minus 119909119888119896minus int

119864119888119896+1

119864119888119896

119896119889119864)

100381710038171003817100381710038171003817100381710038171003817

minus1003817100381710038171003817119909119896 (119864119888119896) minus 119909119888

119896

1003817100381710038171003817

le 119901119896(119909119896 119906119896) + 120578

100381710038171003817100381710038171003817100381710038171003817

119909119888119896+1

minus 119909119888119896minus int

119864119888119896+1

119864119888119896

119896119889119864

100381710038171003817100381710038171003817100381710038171003817

+ 1198971198961205761120575119901119896+1

(119910119896 119889119896)

le 119901119896(119909119896 119906119896) + 120578

100381710038171003817100381710038171003817100381710038171003817

119909119888119896+1

minus 119909119888119896minus int

119864119888119896+1

119864119888119896

119896119889119864

100381710038171003817100381710038171003817100381710038171003817

+ 1198971198961205763120575119901119896+1

(119910119896 119889119896)

lt 119901119896(119909119896 119906119896)

(35)

Thus the sequence 119901119896(119909119896 119906119896) 119896 = 1 2 3 is monotoni-

cally decreasing Considering the existence of a minimizer119901119896(119909119896 119906119896) converges Thus the following equation is estab-

lished

lim119896rarrinfin

(1198971198961205763120575119901119896+1

(119910119896 119889119896) minus 1198971198961205761120575119901119896+1

(119910119896 119889119896)) = 0 (36)

From (36) we derive

lim119896rarrinfin

(119897119896120575119901119896+1

(119910119896 119889119896)) = 0 (37)

If 119889119896

= 0 [119909119896

119906119896]119879 is not the minimizer of penalty function

then 120575119901119896+1

(119910119896 119889119896) = 0 Considering that 119897

119896is bounded away

from zero we derive

lim119896rarrinfin

119889119896= 0

lim119896rarrinfin

120575119901119896+1

(119910119896 119889119896) = 0

lim119896rarrinfin

[119896minus 119891 (119909

119896 119906119896)] = 0

lim119896rarrinfin

(119896+ 119867119896

119909) = 0

lim119896rarrinfin

119867119896

119906= 0

(38)

Thus the solution sequence [119909119896

119906119896]119879 generated by the

algorithmwill converge to the solution of the optimal controlproblem

5 Smooth Approximation forInequality Constraints

The nondifferentiable TPBVP from optimal control problem(11) with the inequality constraints brings difficulties for theproposed algorithm which can only solve a smooth problemSmooth approximations for nondifferentiable optimizationproblems have been studied in [5 6 34] In this sectionwe introduce the smooth approximation method for theinequality constraints If the inequality constraint function isabout the state and the guidance command

119862 (119909 119906) le 0 (39)

the Hamiltonian for the optimal control problem (11) withinequality constraints (39) is given by

119867 = 119892 (119909 119906) + 120582119879119891 (119909 119906) + 120583max 119862 (119909 119906) 0 (40)

However the Hamiltonian (40) is a nondifferentiable func-tion which brings difficulty for the proposed algorithmFor the smooth Hamiltonian approximation a differentiablefunction is substituted for max119862(119909 119906) 0 We denote thenondifferentiable function as follows

max 0 119910 = int

119910

minusinfin

119908 (119911) 119889119911 (41)

where 119908(119911) is the step function

119908 (119911) = 1 if 119911 ge 0

0 if 119911 lt 0(42)

Considering 119904(119911 119886) as the approximation function of 119908(119911)the differentiable approximation to max0 119910 is obtained as

119891119901(119910 119886) asymp int

119910

minusinfin

119904 (119911 119886) 119889119911 = 119910 +1

119886In (1 + 119890

minus119886119910) (43)

8 Mathematical Problems in Engineering

where 119904(119911 119886) is Sigmoid function

119904 (119911 119886) =1

1 + 119890minus119886119911 119886 gt 0 (44)

The Properties of 119891119901(119910 119886) 119886 gt 0 are as follows [5]

(1) 119891119901(119910 119886) is 119899-times continuously differentiable for any

positive integer 119899 with 120597119891119901(119910 119886)120597119910 = 1(1 + 119890

minus119886119910)

and 1205972119891119901(119910 119886)120597119910

2= 119886119890minus119886119910

(1 + 119890minus119886119910

)2

(2) 119891119901(119910 119886) is strictly convex and strictly increasing onR

(3) 119891119901(119910 119886) gt max0 119910

(4) lim119886rarrinfin

119891119901(119910 119886) = max0 119910

(5) 119891119901(119910 119886) gt 119891

119901(119910 1198861) for 119886 lt 119886

1 119910 isin R

Substituting (43) into (40) the following differentiableHamiltonian approximation is obtained

= 119892 (119909 119906) + 120582119879119891 (119909 119906) + 120583 [119862 +

1

119886In (1 + 119890

minus119886119862)] (45)

If the inequality constraint function is a 119902th-order statevariable inequality constraint given by

119862 (119909) le 0 (46)

If 119901 times derivatives of 119862(119909) are required before 119906 appearsexplicitly in the result the Hamiltonian with inequalityconstraints (46) is given by

119867 = 119892 (119909 119906) + 120582119879119891 (119909 119906) + 120583119862

(119902) (47)

where

120583 = 0 if 119862 lt 0 (48)

Considering the differentiable approximation to 120583 as follows

120583 = 120583119908 (119862) asymp 1205831

1 + 119890minus119886119862(49)

the following differentiable Hamiltonian approximation isobtained

= 119892 (119909 119906) + 120582119879119891 (119909 119906) + 120583

1

1 + 119890minus119886119862119862(119902) (50)

With the differentiable Hamiltonian approximation theinequality constraints are considered in the proposed guid-ance algorithm

6 Simulation Results

61 Numerical Calculation Based on Linear TPBVPDiscretiza-tion In this section the linear TPVBP is transformed intolinear equations using the Simpson rule which correspondsto the three-point Newton-Cotes quadrature rule The Simp-son rule for differential equation 119889119883119889119864 = 119910(119864) is given by

int

119887

119886

119910 (119864) 119889119864 =119887 minus 119886

6[119910 (119886) + 4119910(

119886 + 119887

2) + 119910 (119887)] (51)

From (51) we obtain

119883(119887) minus 119883 (119886) =119887 minus 119886

6[119910 (119886) + 4119910(

119886 + 119887

2) + 119910 (119887)] (52)

We disperse the normalized energy 119864 into119873 equal segmentsas [1198640

1198641

sdot sdot sdot 119864119873] with 119864

0= 119864119888119896and 119864

119873= 119864119891 The

differential equation 119889119883119889119864 = 119910(119883 119864) can be transformedinto the following equation

119883(119864119894) minus 119883 (119864

119894minus1)

=119864119894minus 119864119894minus1

6[119910 (119883

119894 119864119894)

+ 4119910 (119883119894+ 119883119894minus1

2119864119894+ 119864119894minus1

2)

+ 119910 (119883119894minus1

119864119894minus1

)] 119894 = 1 2 3 119873

(53)

Based on (53) the linear TPBVP (21) can be transformed intolinear equations described by

119860119883 = 119861 (54)

The linear equations can be solved by Gaussian eliminationFinally we can obtain the solution Δ119909(119864) Δ119906(119864) and

120582(119864) of the linear TPBVP (21) from the solution of the linearequations through interpolating

62 GHV Model The proposed method is applied to theGenericHypersonicVehicle (GHV) to verify the effectivenessby flight simulation In order to develop a complete setof aerodynamic coefficients experimental longitudinal andlateral-directional aerodynamics were obtained for the GHVby using six Langley wind tunnels [35] The aerodynamicdatabase of the GHV is shown in [35] The GHV model withscramjet propulsion system is used for the simulation where119878ref = 33473m2 and the full-scale weight119898ini = 110000 kg

According to the characterization of scramjet propulsionsystem described in [2 27] the development of the propul-sion system begins with the following equation

119879 = 05120601119868sp (120601119872119886) 120588V1198920119862119879 (120572119872119886) (55)

where 120601 is the throttle commandwhich varies from 0 to 1Thecoefficient 119862

119879depends on 120572 and119872

119886as follows

119862119879= minus00012119872

1198861205722+ 0008120572 + 14 minus 01119872

119886 (56)

The impulse 119868sp is given by the following equation

119868sp = minus811198722

119886+ 1004119872

119886minus 52 + 100120601 (57)

63 Simulation Results for Modeling and Initial State ErrorConsidering the high cost of the flight experimental testresearchers do flight numerical simulation hardware-in-the-loop simulation and final flight experimental test to verifythe guidance algorithm step by step In this section flight

Mathematical Problems in Engineering 9

Table 1 Initial and required final states of the vehicle

119903ini(m)

Vini(ms)

120574ini(deg)

119903119903119891

(m)V119903119891

(ms)120574119903119891

(deg)20000 1000 0 35000 2000 0

numerical simulation results for the GHV with modelingerror and initial state error are given to show the effectivenessof the proposed guidance algorithm For the modeling errorcaused by the wind tunnel experimental test the aerody-namic coefficients bias is considered in the flight simulationThe simulations end once the final normalized energy 119864

119891

is reached The optimal guidance algorithm updates thetrajectory every guidance cycle which is 03 s The initial andrequired final states of the air-breathing vehicle are shownin Table 1 The number of the discrete nodes 119873 = 10 Thedynamic pressure inequality constraint is given by

119876 =1

2120588V2 le 140000Pa (58)

To assess the capability of the proposed algorithm to dealwith disturbance the aerodynamic coefficients bias and theinitial state error in the simulations are given by followingcases

(1) Δ119862119871= minus01119862

119871 Δ119862119863= +01119862

119863

(2) Δ119862119871= minus01119862

119871 Δ119862119863= +01119862

119863 Δ119903ini = +500m

(3) Δ119862119871= minus01119862

119871 Δ119862119863= +01119862

119863 Δ119903ini = minus500m

(4) Δ119862119871= minus01119862

119871 Δ119862119863= +01119862

119863 Δ119903ini = +1000m

(5) Δ119862119871= minus01119862

119871 Δ119862119863= +01119862

119863 Δ119903ini = minus1000m

The simulation results of the optimal guidance algorithmare shown in Figures 4 5 6 7 and 8 The altitude velocityflight path angle angle of attack and dynamic pressure arerespectively shown in Figures 4 to 8 As expected the finalstate is achieved under aerodynamic bias and initial stateerror Figure 8 shows that the state inequality constraint issatisfied The dynamic pressure is less than 140000 Pa fordifferent cases

Table 2 shows the terminal error and the fuel cost ofthe optimal guidance algorithm The terminal accuracy ishigh for different cases Moreover the fuel cost of theoptimal guidance algorithm is lower than that of the trackingguidance algorithm A greater initial state error will result inmore fuel savings In summarywith lower fuel cost the resultsof the proposed optimal guidance algorithm satisfies all theequality and inequality constraints The simulation resultsshow the great potential for the final flight experimental test

7 Conclusion

In this paper we present an optimal guidance algorithmfor air-breathing launch vehicle based on optimal trajectorycorrection The proposed guidance algorithm corrects thereference trajectory in every guidance cycle to satisfy the opti-mality condition of the optimal feedback control problem

0 14012010080604020

18000

21000

24000

27000

30000

33000

Alti

tude

(m)

Time (s)

Case 1Case 2Case 3

Case 4Case 5

Figure 4 The altitude of the simulation results

1200

1400

1600

1800

2000

2200Ve

loci

ty (m

s)

Time (s)0 14012010080604020

Case 1Case 2Case 3

Case 4Case 5

Figure 5 The velocity of the simulation results

By linearizing the optimality condition the linear TPBVPis obtained for optimal trajectory correction The solutionof the linear TPBVP is derived using the Simpson rule Thenew trajectory is generated by a linear search for the stepsize of the solution Smooth Hamiltonian approximation isimplemented to the inequality constraints The sufficiencycondition for the global convergence of the guidance algo-rithm is given in this paper

Finally simulations for 5 different cases of modeling andinitial state error are presented Compared with the trackingguidance method the simulation results prove the low fuelcost and high precision of the proposed optimal guidancemethod

10 Mathematical Problems in Engineering

Table 2 The simulation results for optimal guidance algorithm

Case Δ119903119891(m) ΔV

119891(ms) Δ120574

119891(deg) Fuel cost (kg) Fuel cost (tracking method) (kg)

1 minus2820 0136 020 457219 4574622 859 minus003 minus015 458314 4628393 minus1508 007 020 458474 4605664 10816 047 minus023 460292 4776005 minus3243 019 minus016 460133 468894

0

2

4

6

8

Time (s)

Flig

ht p

ath

angl

e (D

EG)

minus2

minus4

0 14012010080604020

Case 1Case 2Case 3

Case 4Case 5

Figure 6 The flight path angle of the simulation results

Ang

le o

f atta

ck (D

EG)

0

1

2

3

4

minus1

minus2

minus3

minus4

minus5

minus6

Time (s)0 14012010080604020

Case 1Case 2Case 3

Case 4Case 5

Figure 7 The angle of attack of the simulation results

Acknowledgments

This work was supported in part by the National NatureScience Foundation of China nos 61203081 and 61174079Doctoral Fund of Ministry of Education of China

20000

40000

60000

80000

100000

120000

140000

Dyn

amic

pre

ssur

e (Pa

)

Time (s)0 14012010080604020

Case 1Case 2Case 3

Case 4Case 5

Figure 8 The dynamic pressure of the simulation results

no 20120142120091 andPrecisionmanufacturing technologyand equipment for metal parts no 2012DFG70640

References

[1] H J Henry Jr and C H McLellan ldquoAir-breathing launch vehi-cle for earth-orbit shuttle New technology and developmentapproachrdquo Journal of Aircraft vol 8 no 5 pp 381ndash387 1971

[2] J E CorbanA J Calise andGA Flandro ldquoReal-time guidancealgorithm for aerospace plane optimal ascent to low earth orbitrdquoin Proceedings of the American Control Conference pp 2475ndash2481 Pittsburgh PA USA June 1989

[3] C H Yam D D Lorenzo and D Izzo ldquoLow-thrust trajectorydesign as a constrained global optimization problemrdquo Journalof Aerospace Engineering G vol 225 no 11 pp 1243ndash1251 2011

[4] B A Conway ldquoA survey of methods available for the numericaloptimization of continuous dynamic systemsrdquo Journal of Opti-mization Theory and Applications vol 152 no 2 pp 271ndash3062012

[5] C H Chen and O L Mangasarian ldquoSmoothing methodsfor convex inequalities and linear complementarity problemsrdquoMathematical Programming vol 71 no 1 pp 51ndash69 1995

[6] X Chen ldquoSmoothing methods for nonsmooth nonconvexminimizationrdquo Mathematical Programming vol 134 no 1 pp71ndash99 2012

[7] W Wang Y Shang and Y Zhang ldquoA filled function approachfor nonsmooth constrained global optimizationrdquoMathematical

Mathematical Problems in Engineering 11

Problems in Engineering vol 2010 Article ID 310391 9 pages2010

[8] L I Rozonoer ldquoLinear-quadratic optimization and some gen-eral hypotheses on optimal controlrdquoMathematical Problems inEngineering vol 5 no 4 pp 275ndash289 1999

[9] J T Betts ldquoSurvey of numerical methods for trajectory opti-mizationrdquo Journal of Guidance Control and Dynamics vol 21no 2 pp 193ndash207 1998

[10] G Q Huang Y P Lu and Y Nan ldquoA survey of numericalalgorithms for trajectory optimization of flight vehiclesrdquo ScienceChina-Technological Sciences vol 55 no 9 pp 2538ndash2560 2012

[11] Y Ulybyshev ldquoDiscrete pseudocontrol sets for optimal controlproblemsrdquo Journal of Guidance Control and Dynamics vol 33no 4 pp 1133ndash1142 2010

[12] M Vasile and F Zuiani ldquoMulti-agent collaborative search anagent-based memetic multi-objective optimization algorithmapplied to space trajectory designrdquo Journal of Aerospace Engi-neering G vol 225 no 11 pp 1211ndash1227 2011

[13] M Pontani and B A Conway ldquoParticle swarm optimizationapplied to space trajectoriesrdquo Journal of Guidance Control andDynamics vol 33 no 5 pp 1429ndash1441 2010

[14] Y P Ulybyshev ldquoOptimization of low-thrust orbit transfers withconstraintsrdquo Cosmic Research vol 50 no 5 pp 376ndash390 2012

[15] M Chen R Mei and B Jiang ldquoSliding mode control for a classof uncertainMIMOnonlinear systemswith application to near-space vehiclesrdquoMathematical Problems in Engineering vol 2013Article ID 180589 9 pages 2013

[16] H Zhang Y Shi M Q Xu and H T Cui ldquoObserver-basedtracking controller design for networked predictive controlsystems with uncertain markov delaysrdquo in Proceedings of theAmerican Control Conference pp 5682ndash5687 2012

[17] H Zhang and Y Shi ldquoRobust H-infinity sliding-mode controlfor markovian jump systems subject to intermittent observa-tions and partially known transition probabilitiesrdquo Systems ampControl Letters vol 62 no 12 pp 1114ndash1124 2013

[18] Z Shuai H Zhang J Wang J Li and M Ouyang ldquoCombinedAFS andDYC control of four-wheel-independent-drive electricvehicles over CAN network with time-varying delaysrdquo IEEETransactions on Vehicular Technology no 99 p 1 2013

[19] H Zhang Y Shi and A Saadat Mehr ldquoRobust static outputfeedback control and remote PID design for networked motorsystemsrdquo IEEETransactions on Industrial Electronics vol 58 no12 pp 5396ndash5405 2011

[20] H Zhang Y Shi and M X Liu ldquoH-infinity step trackingcontrol for networked discrete-time nonlinear systems withintegral and predictive actionsrdquo IEEE Transactions on IndustrialInformatics vol 9 no 1 pp 337ndash345 2013

[21] X J Wang and S P Wang ldquoAdaptive fuzzy robust controlfor a class of nonlinear systems via small gain theoremrdquoMathematical Problems in Engineering vol 2013 Article ID201432 11 pages 2013

[22] V Brinda S Dasgupta and M Lal ldquoTrajectory optimizationand guidance of an air breathing hypersonic vehiclerdquo in Pro-ceedings of the 14th AIAAAHI International Space Planes andHypersonics Systems Technologies Conference vol 2 pp 856ndash869 November 2006

[23] B Tian and Q Zong ldquoOptimal guidance for reentry vehiclesbased on indirect legendre pseudospectral methodrdquoActa Astro-nautica vol 68 no 7-8 pp 1176ndash1184 2011

[24] M H Graszliglin J Telaar and UM Schottle ldquoAscent and reentryguidance concept based on NLP-methodsrdquo Acta Astronauticavol 55 no 3ndash9 pp 461ndash471 2004

[25] P Lu and B Pan ldquoHighly constrained optimal launch ascentguidancerdquo Journal of Guidance Control and Dynamics vol 33no 2 pp 404ndash414 2010

[26] P Lu H Sun and B Tsai ldquoClosed-loop endo-atmosphericascent guidancerdquo in Proceedings of the AIAA Guidance Navi-gation and Control Conference Monterey Calif USA 2002

[27] J Oscar Murillo and P Lu ldquoFast ascent trajectory optimizationfor hypersonic air-breathing vehiclesrdquo in Proceedings of theAIAA Guidance Navigation and Control Conference pp 5ndash8Toronto Canada 2010

[28] M Marrdonny and M Mobed ldquoA guidance algorithm forlaunch to equatorial orbitrdquo Aircraft Engineering and AerospaceTechnology vol 81 no 2 pp 137ndash148 2009

[29] T Yamamoto and J Kawaguchi ldquoA new real-time guidancestrategy for aerodynamic ascent flightrdquo Acta Astronautica vol61 no 11-12 pp 965ndash977 2007

[30] C Filici andR S Sanchez Pena ldquoOnline guidance updates usingneural networksrdquo Acta Astronautica vol 66 no 3-4 pp 477ndash485 2010

[31] P T Boggs A J Kearsley and J W Tolle ldquoA global convergenceanalysis of an algorithm for large-scale nonlinear optimizationproblemsrdquo SIAM Journal on Optimization vol 9 no 4 pp 833ndash862 1999

[32] S Schlenkrich and A Walther ldquoGlobal convergence of quasi-Newton methods based on adjoint Broyden updatesrdquo AppliedNumerical Mathematics vol 59 no 5 pp 1120ndash1136 2009

[33] S Schlenkrich A Griewank and A Walther ldquoOn the localconvergence of adjoint Broyden methodsrdquo Mathematical Pro-gramming vol 121 no 2 pp 221ndash247 2010

[34] S Lian and L Zhang ldquoA simple smooth exact penalty functionfor smooth optimization problemrdquo Journal of Systems Science ampComplexity vol 25 no 3 pp 521ndash528 2012

[35] S Keshmiri R Colgren and M Mirmirani ldquoDevelopment ofan aerodynamic database for a generic hypersonic air vehiclerdquoin Proceedings of the AIAA Guidance Navigation and ControlConference pp 3978ndash3998 San Francisco Calif USA August2005

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Complex AnalysisJournal of

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Page 2: Research Article Optimal Ascent Guidance for Air-Breathing ...downloads.hindawi.com/journals/mpe/2013/313197.pdfResearch Article Optimal Ascent Guidance for Air-Breathing Launch Vehicle

2 Mathematical Problems in Engineering

of optimization were presented for multiorbit noncoplanarorbit transfers of low-thrust spacecraft in [14]

Many control methods were implemented to trajectorytracking guidance and control problem [15] In the previousliterature many researchers focused on the robust controlmethod [16 17] It has obtained successful application in theindustry [18 19] For the networked control system a H-infinity step tracking control method was presented in [20]An adaptive fuzzy robust control for a class of nonlinearsystems was proposed in [21] An adaptive guidance law andoff-board trajectory optimization for air-breathing launchvehicle were addressed in [22] In that paper the optimalcontrol problem was solved using SQP method And theadaptive guidance law was developed using a feedback loopbased on a second-order rate controller for angle of attack Arobust state feedback guidance law was generated in real timeusing the indirect Legendre pseudospectral feedbackmethodin [23] In that paper the guidance problem was convertedinto a trajectory state regulation problem which is a lineartime varying system

However the accuracy of the trajectory tracking methodis low with the disturbance and the modeling error Itlacks the autonomy and adaptability to cope with the non-nominal vehicle and mission conditions needed for futurereusable launch vehicles [24] To improve the performanceand the accuracy of the guidance the optimal guidancemethod generates the prospective trajectory by fast trajectoryoptimization on-board based on current flight state Thisguidance method has become a research focus with theadvance in on-board computation capability Ping Lu usedthe indirect method to pose the trajectory optimizationproblem as a nonlinear two-point boundary value problem(TPBVP) in [25 26] From this model the optimal thrustvector that satisfied the optimality condition was derivedA finite difference method was employed to solve thenonlinear TPBVP through numerical calculations Similarlya fast trajectory optimization for hypersonic air-breathingvehicles was presented in [27] The indirect method wasimplemented for ascent trajectory optimization on-boardconsidering the features of the air-breathing vehicle Ping Lursquoswork is significant for the optimal ascent guidance Howeverthe TPBVP is nonlinear and nondifferentiable for the airbreathing launch vehicle It is difficult to solve the nonlinearand nondifferentiable TPBVP on-board in every guidancecycle Using the directmethod a new guidance concept basedon nonlinear programming (NLP) method was proposedin [24] NLP-based guidance concepts appear advantageousover conventional methods because the on-board guidancealgorithm allows a single algorithm to be implemented fordifferent vehicles and missions The optimal control problemwas parameterized into a nonlinear programming problemthat was solved by the gradient projection algorithm In [28]the reference trajectorywas updated for disturbance by an on-board algorithm that satisfied the real-time requirement Anew real-time guidance method derived from the optimalitycondition was proposed in [29] In that paper the simpleguidance parameters were updated in real time In [30] aguidance method was presented for online launcher ascenttrajectory updating based on neural networks In the paper

the utilization of a neural network approximation was usedonline during the ascent flight with a training processperformed off-line

In this paper we present an optimal guidance algorithmfor air breathing launch vehicle based on optimal trajectorycorrection to reduce the on-board calculation cost Consider-ing the current vehicle state as the initial condition the opti-mal trajectory correction problem is referred to as a nonlinearoptimal control problem with state inequality constraintsFor the real-time requirement of the on-board algorithm thelinear TPBVP is obtained for optimal trajectory correctionby linearizing the optimality condition in this paper TheSimpson rule is applied to transform the linear TPBVPinto linear equations Considering the solution of the linearTPBVP as the searching direction for the correction valuesthe updating step size is generated by linear search Smoothapproximation is applied to the inequality constraints for thenondifferentiable Hamiltonian The sufficient condition forthe global convergence of the algorithm is given in this paperFinally simulation results for different cases of the modelingerror show the effectiveness of the proposed algorithm Insummation the main contributions of this paper are givenas follows

(1) We reduce the on-board calculation cost of theguidance method a lot Comparing with the methodsin previous literatures which solves nonlinear equa-tions in every guidance cycle the proposed guidancemethod in this paper solves linear equations once inevery guidance cycle only

(2) We obtain the relationship between the global conver-gence and the guidance cycle of the online algorithmThe sufficient condition of the global convergence ofthis algorithm is given

The remainder of this paper is organized as followsSection 2 presents the state normalized-energy differentialequations of motion of the vehicle and the optimal controlproblem for optimal ascent guidance Section 3 providesdetails on the optimal guidance algorithm Section 4 presentsan analysis of the global convergence of the proposedalgorithm Section 5 discusses the proposed differentiableapproximation for Hamiltonian Section 6 presents the simu-lations for theGenericHypersonic Vehicle (GHV)model andscramjet engine Section 7 presents the conclusion

2 Problem Formulation

21 Ascent Dynamic The formulation of motion of the air-breathing vehicle in the longitudinal plane is presented in thissection As shown in Figure 1 the thrust 119879 gravitation 119892 andthe aerodynamic lift force119871 and drag force119863 are acting on thevehicle The angle of attack 120572 is the angle between the bodyaxis and the vector of the velocity V The flight path angle 120574 isthe angle between the vector of the velocity V and the groundplane

Mathematical Problems in Engineering 3

Center of mass

V

T

D

L

120572

120574

Figure 1 The forces acting on the vehicle

Themotion of the vehicle in the longitudinal plane can bedescribed as follows

119889119903

119889119905= V sin 120574

119889V119889119905

=119879 cos120572 minus 119863

119898minus 119892 sin 120574

119889120574

119889119905=

1

V[119879 sin120572 + 119871

119898cos120590 minus 119892 cos 120574 + V2

119903cos 120574]

119889119898

119889119905= minus

119879

119868sp

(1)

where 119903 119898 119868sp and 119892 are the altitude weight of the vehicleimpulse and gravity acceleration respectively

The lift force 119871 and the drag force119863 are given as follows

119871 =1

2120588V2119878ref119862119871 (119872119886 120572)

119863 =1

2120588V2119878ref119862119863 (119872119886 120572)

(2)

where 120588 and 119878ref are the air density of the current altitudeand the reference area respectively 119862

119871and 119862

119863are the lift

coefficient and the drag coefficient respectively which are thenonlinear functions of the angle of attack120572 andMachnumber119872119886 For the air-breathing engine the thrust is given by

119879 = 119879 (120588119872119886 120572 120601) (3)

where119879 is the nonlinear function of 120588 120572119872119886 and the throttle

command 120601In [25] the equations were normalized to reduce

the numerical calculation error Variable substitutions areapplied as follows

119877 =119903

1199030

119881 =V

radic11989201199030

119892 =11989201199032

0

1199032 (4)

where 1199030and 119892

0are Earth radius and the ground gravity

acceleration respectively Considering the thrust 119879 andaerodynamic forces as a composition of forces we define thenormalized accelerations 119886

119879and 119886119871as follows

119886119879=

119879 cos120572 minus 119863

1198981198920

119886119871=

119879 sin120572 + 119871

1198981198920

(5)

We define

120591 =119905

radic11990301198920

(6)

The normalized equations of motion are obtained from (1)(4) (5) and (6) as follows

119889119877

119889120591= 119881 sin 120574

119889119881

119889120591= 119886119879minussin 1205741198772

119889120574

119889120591=

1

119881[119886119871+ (1198812minus

1

119877)(

cos 120574119877

)]

119889119898

119889120591= minus

119879radic1199030

119868spradic1198920

(7)

The mission of ascent is to send the vehicle to therequired final state which is denoted as (119877

119891 119881119891 120574119891) Define

normalized-energy 119864 as follows

119864 =1198812

2minus

1

119877 (8)

From (8) we obtain

119889119864

119889120591= 119881119886119879

119889119877

119889119864=sin 120574119886119879

119889119881

119889119864=

1

119886119879119881

[119886119879minussin 1205741198772

]

119889120574

119889119864=

1

1198861198791198812

[119886119871+ (1198812minus

1

119877)(

cos 120574119877

)]

119889119898

119889119864= minus

119879radic1199030

119886119879119881119868spradic119892

0

(9)

The integration interval [1198640 119864119891] is fixed

22 Optimal Feedback Control Problem for Ascent GuidanceIn this section the optimal feedback control problem foroptimal ascent guidance is addressed This problem differsfrom trajectory optimization off-board in that the initialstate of the optimal control problem is obtained from thenavigation system We denote the state and the guidancecommand of the trajectory as

119909 (119864) = [119877 (119864) 119881 (119864) 120574 (119864) 119898 (119864)]119879

119906 (119864) = [120572 (119864) 120601 (119864)]119879

(10)

With the feedback vehicle state 119909119888119896obtained from navigation

system the optimal ascent guidance algorithm is used to

4 Mathematical Problems in Engineering

generate new reference trajectory [119909(119864) 119906(119864)]119879 and output

119906(119864) from the following optimal control problem

min 119869 = 120601 (119909 (119864119891)) + int

119864119891

119864119888119896+1

119892 (119909 119906) 119889119864

st = 119891 (119909 119906)

119909 (119864119888119896) = 119909119888

119896

119882 (119909 (119864119891)) = 0

119862 (119909 119906) le 0

(11)

where 119864119888119896is the normalized energy for current vehicle state

119909119888119896

3 Optimal Ascent Guidance Algorithm

In [25 26] the optimal control problem was solved on-boardin every guidance cycle However it is difficult to solve thenonlinear and nondifferentiable TPBVP on-board for air-breathing launch vehicles In this section we propose an opti-mal guidance algorithmThis guidance algorithmupdates thereference trajectory [119909 119906]

119879 in every guidance cycle to dealwith the unknowndisturbance For the current state119909119888

119896of the

119896th guidance cycle the optimal feedback control problem istransformed into a linear TPBVP problem using optimalitycondition linear approximation The searching direction ofthe correction values 119889

119896is obtained by solving the linear

TPBVP With the searching direction the new referencetrajectory is generated by linear search For the 119896th guidancecycle the trajectory update is performed as follows

[119909119896+1(119864) 119906119896+1

(119864)]119879

= [119909119896(119864) 119906119896(119864)]119879

+ 119897119896119889119879

119896(119864) (12)

The initial reference trajectory [1199090(119864) 119906

0(119864)]119879 is derived

from the off-board from trajectory optimization by directmethodThe flow chart of the algorithm is shown in Figure 2Figure 3 shows the time sequence of algorithm

31 Linear Approximation for the Optimality Condition Inthis section we linearize the optimality condition for opti-mality trajectory updating The optimal control problem (11)without inequality constraints is given as follows

min 119869 = 120601 (119909 (119864119891)) + int

119864119891

119864119888119896+1

119892 (119909 119906) 119889119864

st = 119891 (119909 119906)

119909 (119864119888119896) = 119909119888

119896

119882 (119909 (119864119891)) = 0

(13)

The Hamiltonian of (13) is given as follows

119867 = 119892 (119909 119906) + 120582119879119891 (119909 119906) (14)

Begin

Linear approximation for the

optimality condition the costate

equations the motion equations

Solving the linear TPBVP

Linear search for the step size

Updating the optimal trajectory

End

Figure 2 Flow chart of the optimal ascent guidance algorithm

where 120582 is the costate vector The state equations costateequations and optimality condition for the optimal controlproblem are given by

= 119891 (119909 119906)

= minus(120597119867

120597119909)

119879

120597119867

120597119906= 0

(15)

The initial and transversality conditions are given by

119909 (119864119888119896) = 119909119888

119896

119882 (119909 (119864119891)) = 0

120582 (119864119891) = [

120597120601

120597119909 (119864119891)

]

119879

+ [120603119879 120597119882

120597119909 (119864119891)

]

119879

(16)

We denote the trajectory correction variables as follows

[

[

Δ119909 (119864)

Δ119906 (119864)

Δ120582 (119864)

]

]

= [

[

119909 (119864) minus 119909119896(119864)

119906 (119864) minus 119906119896(119864)

120582 (119864) minus 120582119896(119864)

]

]

(17)

Mathematical Problems in Engineering 5

middot middot middot

Output theguidancecommanduk+1(E)

Output theguidancecommanduk+2(E)

middot middot middot

middot middot middot

The time consuming ofthe online algorithm The guidance cycle

Trajectoryupdating

beginsupdating

Trajectory

ends

Trajectoryupdating

beginsupdating

Trajectory

ends

Trajectoryupdating

beginsupdating

Trajectory

ends

End

Time (120591)kΔ120591 (k + 1)Δ120591 (k + 2)Δ120591

Figure 3 Time sequence of the optimal ascent guidance algorithm

We denote that

119867119896

119906119909=

1205972(119892 + 120582

119879119891)

120597119906120597119909(119909119896 119906119896 120582119896)

119867119896

119906= [

120597119867

120597119906(119909119896 119906119896 120582119896)]

119879

119867119896

119906119906=

1205972(119892 + 120582

119879119891)

1205971199062(119909119896 119906119896 120582119896)

119867119896

119909= [

120597119867

120597119909(119909119896 119906119896 120582119896)]

119879

119867119896

119909119909=

1205972(119892 + 120582

119879119891)

1205971199092(119909119896 119906119896 120582119896)

119891119896

119906=

120597119891

120597119906(119909119896 119906119896)

119867119896

119909119906=

1205972(119892 + 120582

119879119891)

120597119909120597119906(119909119896 119906119896 120582119896)

119891119896

119909=

120597119891

120597119909(119909119896 119906119896)

119892119896

119906= [

120597119892

120597119906(119909119896 119906119896)]

119879

119892119896

119909= [

120597119892

120597119909(119909119896 119906119896)]

119879

(18)

The first-order Taylor expansion of (15) on the old referencetrajectory [119909

119896119906119896

120582119896]119879 is

[120597119867

120597119906]

119879

asymp 119867119896

119906+ 119867119896

119906119909Δ119909 + 119867

119896

119906119906Δ119906 + (119891

119896

119906)119879

Δ120582

asymp minus119867119896

119909minus 119867119896

119909119909Δ119909 minus 119867

119896

119909119906(119909119896 119906119896 120582119896) Δ119906 minus (119891

119896

119909)119879

Δ120582

asymp 119891 (119909119896 119906119896) + 119891119896

119909Δ119909 + 119891

119896

119906Δ119906

(19)

The following equations are established

119867119896

119906+ (119891119896

119906)119879

Δ120582 = 119892119896

119906+ (119891119896

119906)119879

120582

119867119896

119909+ (119891119896

119909)119879

Δ120582 = 119892119896

119909+ (119891119896

119909)119879

120582

(20)

Substituting (20) into (19) yields

= minus119892119896

119909minus 119867119896

119909119909Δ119909 minus 119867

119896

119909119906(119909119896 119906119896 120582119896) Δ119906 minus (119891

119896

119909)119879

120582

Δ = 119891119896

119909Δ119909 + 119891

119896

119906Δ119906 minus

119896+ 119891 (119909

119896 119906119896)

119892119896

119906+ 119867119896

119906119909Δ119909 + 119867

119896

119906119906Δ119906 + (119891

119896

119906)119879

120582 = 0

Δ119909 (119864119888119896) + Δ119909

119896(119864119888119896) = 119909119888

119896

120597119882

120597119909 (119864119891)

Δ119909 (119864119891) +119882(119909

119896(119864119891)) = 0

120582 (119864119891) = [

120597120601

120597119909 (119864119891)

]

119879

+ [120603119879 120597119882

120597119909 (119864119891)

]

119879

(21)

This equation is a linear TPBVP about the variables[Δ119909 Δ119906 120582]

119879 The linear TPBVP is solved using the Simpsonrules in Section 6

32 Backtracking Line Search for the Step Size In this sectionthe step size 119897

119896is determined by backtracking line search

considering the solution of the linear TPBVP as the searchdirection The penalty function for problem (13) is given by

119901119896= 119869 (119909 119906) + 120578

10038171003817100381710038171003817119882(119909119891)10038171003817100381710038171003817+ int

119864119891

119864119888119896

1003817100381710038171003817119891 (119909 119906) minus 1003817100381710038171003817

+1003817100381710038171003817119909 (119864119888119896) minus 119909119888

119896

1003817100381710038171003817

(22)

6 Mathematical Problems in Engineering

where 120578 is the penalty parameter satisfied 120578 gt max |120582| Wedenote

119910 = [119909119896

119906119896

] 119889119896= [

Δ119909

Δ119906] (23)

where [Δ119909 Δ119906 120582]119879 is the solution of the linear TPBVP (21)

The minimizer of penalty function 119901119896(119909 119906) corresponds

to the solution of problem (11) Backtracking line search wasused for large-scale nonlinear optimization problems in [31]The step size 119897

119896is determined by backtracking line search that

satisfies

119897119896= max 119897 isin 119878 119901

119896+1(119910119896+ 119897119889119896)

le 119901119896+1

(119910119896) + 1198971205761120575119901119896+1

(119910119896 119889119896)

119878 = 119897 119897 = 1205762

119898 119898 = 0 1 2

(24)

where 1205761

isin (0 1) 1205762

isin (0 1) and 120575119901119896+1

(119910119896 119889119896) is the

variational of the functional 119901119896+1

(119910) caused by variational119889119896of 119910 on 119910

119896 Considering 119889

119896as the solution of (21)

120575119901119896+1

(119910119896 119889119896) le 0 is established The new state and guidance

command of the reference trajectory are generated as follows

[119909119896+1

119906119896+1

] = [119909119896

119906119896

] + 119897119896[Δ119909

Δ119906] (25)

4 The Global Convergence of the Algorithm

In [32 33] the convergence of the off-board algorithm wasanalyzed But the relationship between the global conver-gence and the guidance cycle 120591 has not been discussed for on-board optimization in the previous literatures In this sectionwe state and prove the global convergence of the on-boardalgorithm proposed in the preceding section The solutionsequence [119909

119896119906119896]119879 generated by the algorithm will converge

to the solution of the feedback optimal control problemUnlike off-board trajectory optimization the initial conditionof the feedback optimal control problem depends on thecurrent state of the vehicle in real time For the 119896th guidancecycle the optimal feedback control problem is described as(13)Thepenalty function119901

119896for this feedback optimal control

problem is given by (22) For the subsequent guidance cyclethe optimal feedback guidance problem is given by

min 119869 = 119882(119909 (119864119891)) + int

119864119891

119864119888119896+1

119892 (119909 119906) 119889119864

st = 119891 (119909 119906)

119909 (119864119888119896+1

) = 119909119888119896+1

119882 (119909 (119864119891)) = 0

(26)

where

119864119888119896+1

= 119864119888119896+ int

120591119896+Δ120591

120591119896

119889119864

119889120591119889120591 (27)

Δ120591 is the guidance cycle and 119909119888119896+1

is the new state of vehiclefrom navigation system at the time 120591

119896+ Δ120591 The penalty

function for optimal problem (24) is given by

119901119896+1

(119909 119906) = 119869 (119909 119906) + 12057810038171003817100381710038171003817119882(119909119896(119864119891))10038171003817100381710038171003817

+ 120578int

119864119891

119864119888119896+1

1003817100381710038171003817119891 (119909 119906) minus 1003817100381710038171003817 119889119864

+ 1205781003817100381710038171003817119909 (119864119888119896+1) minus 119909119888

119896+1

1003817100381710038171003817

(28)

We define the entire penalty function as follows

119901119896(119909 119906) = 119901

119896(119909 119906)

+

119896minus1

sum

119894=0

int

119864119888119894+1

119864119888119894

[1205781003817100381710038171003817119891 (119909119894 119906119894) minus 119894

1003817100381710038171003817 + 119892 (119909119894 119906119894)] 119889119864

(29)

The entire penalty function 119901119896(119909 119906) includes the penalty

function on the whole integration interval [1198640 119864119891]

Theorem 1 If the minimizer of 119901119896(119909119896 119906119896) is exited 119892(119909 119906) ge

0 and Δ120591 is selected and satisfies

120578100381710038171003817100381710038171003817119909119888119896+1

minus 119909119888119896minus int119864119888119896+1

119864119888119896119896119889119864

100381710038171003817100381710038171003817

minus120575119901119896+1

(119910119896 119889119896)

lt 1198971198961205763 (30)

where 1205763isin (0 120576

1) The solution sequence [119909

119896119906119896]119879 generated

from (24) will converge to the solution of the feedback optimalcontrol problem

Proof From (22) (28) (29) and

119909119896(119864119888119896+1

) = 119909119896(119864119888119896) + int

119864119888119896+1

119864119888119896

119896119889119864 (31)

we can obtain that

119901119896+1

(119909119896 119906119896) minus 119901119896(119909119896 119906119896)

= 120578

100381710038171003817100381710038171003817100381710038171003817

119909119896(119864119888119896) minus 119909119888

119896

minus(119909119888119896+1

minus 119909119888119896minus int

119864119888119896+1

119864119888119896

119896119889119864)

100381710038171003817100381710038171003817100381710038171003817

minus 1205781003817100381710038171003817119909119896 (119864119888119896) minus 119909119888

119896

1003817100381710038171003817

(32)

Mathematical Problems in Engineering 7

From (32) we obtain

limΔ120591rarr0

[119901119896+1

(119909119896 119906119896) minus 119901119896(119909119896 119906119896)] = 0

lim119891(119909119896119906119896)minus 119909119896rarr0

[119901119896+1

(119909119896 119906119896) minus 119901119896(119909119896 119906119896)] = 0

(33)

119901119896+1

(119909119896 119906119896) minus 119901119896(119909119896 119906119896)

= 120578

100381710038171003817100381710038171003817100381710038171003817

119909119896(119864119888119896) minus 119909119888

119896+1+ int

119864119888119896+1

119864119888119896

119896119889119864

100381710038171003817100381710038171003817100381710038171003817

minus1003817100381710038171003817119909119896 (119864119888119896) minus 119909119888

119896

1003817100381710038171003817

le 1205781003817100381710038171003817119909119896 (119864119888119896) minus 119909119888

119896

1003817100381710038171003817

+

100381710038171003817100381710038171003817100381710038171003817

119909119888119896+1

minus 119909119888119896minus int

119864119888119896+1

119864119888119896

119896119889119864

100381710038171003817100381710038171003817100381710038171003817

minus1003817100381710038171003817119909119896 (119864119888119896) minus 119909119888

119896

1003817100381710038171003817

= 120578

100381710038171003817100381710038171003817100381710038171003817

119909119888119896+1

minus 119909119888119896minus int

119864119888119896+1

119864119888119896

119896119889119864

100381710038171003817100381710038171003817100381710038171003817

(34)

From conditions (24) and (30) it is obtained that

119901119896+1

(119909119896+1

119906119896+1

)

le 119901119896+1

(119909119896 119906119896) + 1198971198961205761120575119901119896+1

(119889119896)

= 119901119896(119909119896 119906119896) + 1198971198961205761120575119901119896+1

(119889119896)

+ 120578

100381710038171003817100381710038171003817100381710038171003817

119909119896(119864119888119896) minus 119909119888

119896

minus (119909119888119896+1

minus 119909119888119896minus int

119864119888119896+1

119864119888119896

119896119889119864)

100381710038171003817100381710038171003817100381710038171003817

minus1003817100381710038171003817119909119896 (119864119888119896) minus 119909119888

119896

1003817100381710038171003817

le 119901119896(119909119896 119906119896) + 120578

100381710038171003817100381710038171003817100381710038171003817

119909119888119896+1

minus 119909119888119896minus int

119864119888119896+1

119864119888119896

119896119889119864

100381710038171003817100381710038171003817100381710038171003817

+ 1198971198961205761120575119901119896+1

(119910119896 119889119896)

le 119901119896(119909119896 119906119896) + 120578

100381710038171003817100381710038171003817100381710038171003817

119909119888119896+1

minus 119909119888119896minus int

119864119888119896+1

119864119888119896

119896119889119864

100381710038171003817100381710038171003817100381710038171003817

+ 1198971198961205763120575119901119896+1

(119910119896 119889119896)

lt 119901119896(119909119896 119906119896)

(35)

Thus the sequence 119901119896(119909119896 119906119896) 119896 = 1 2 3 is monotoni-

cally decreasing Considering the existence of a minimizer119901119896(119909119896 119906119896) converges Thus the following equation is estab-

lished

lim119896rarrinfin

(1198971198961205763120575119901119896+1

(119910119896 119889119896) minus 1198971198961205761120575119901119896+1

(119910119896 119889119896)) = 0 (36)

From (36) we derive

lim119896rarrinfin

(119897119896120575119901119896+1

(119910119896 119889119896)) = 0 (37)

If 119889119896

= 0 [119909119896

119906119896]119879 is not the minimizer of penalty function

then 120575119901119896+1

(119910119896 119889119896) = 0 Considering that 119897

119896is bounded away

from zero we derive

lim119896rarrinfin

119889119896= 0

lim119896rarrinfin

120575119901119896+1

(119910119896 119889119896) = 0

lim119896rarrinfin

[119896minus 119891 (119909

119896 119906119896)] = 0

lim119896rarrinfin

(119896+ 119867119896

119909) = 0

lim119896rarrinfin

119867119896

119906= 0

(38)

Thus the solution sequence [119909119896

119906119896]119879 generated by the

algorithmwill converge to the solution of the optimal controlproblem

5 Smooth Approximation forInequality Constraints

The nondifferentiable TPBVP from optimal control problem(11) with the inequality constraints brings difficulties for theproposed algorithm which can only solve a smooth problemSmooth approximations for nondifferentiable optimizationproblems have been studied in [5 6 34] In this sectionwe introduce the smooth approximation method for theinequality constraints If the inequality constraint function isabout the state and the guidance command

119862 (119909 119906) le 0 (39)

the Hamiltonian for the optimal control problem (11) withinequality constraints (39) is given by

119867 = 119892 (119909 119906) + 120582119879119891 (119909 119906) + 120583max 119862 (119909 119906) 0 (40)

However the Hamiltonian (40) is a nondifferentiable func-tion which brings difficulty for the proposed algorithmFor the smooth Hamiltonian approximation a differentiablefunction is substituted for max119862(119909 119906) 0 We denote thenondifferentiable function as follows

max 0 119910 = int

119910

minusinfin

119908 (119911) 119889119911 (41)

where 119908(119911) is the step function

119908 (119911) = 1 if 119911 ge 0

0 if 119911 lt 0(42)

Considering 119904(119911 119886) as the approximation function of 119908(119911)the differentiable approximation to max0 119910 is obtained as

119891119901(119910 119886) asymp int

119910

minusinfin

119904 (119911 119886) 119889119911 = 119910 +1

119886In (1 + 119890

minus119886119910) (43)

8 Mathematical Problems in Engineering

where 119904(119911 119886) is Sigmoid function

119904 (119911 119886) =1

1 + 119890minus119886119911 119886 gt 0 (44)

The Properties of 119891119901(119910 119886) 119886 gt 0 are as follows [5]

(1) 119891119901(119910 119886) is 119899-times continuously differentiable for any

positive integer 119899 with 120597119891119901(119910 119886)120597119910 = 1(1 + 119890

minus119886119910)

and 1205972119891119901(119910 119886)120597119910

2= 119886119890minus119886119910

(1 + 119890minus119886119910

)2

(2) 119891119901(119910 119886) is strictly convex and strictly increasing onR

(3) 119891119901(119910 119886) gt max0 119910

(4) lim119886rarrinfin

119891119901(119910 119886) = max0 119910

(5) 119891119901(119910 119886) gt 119891

119901(119910 1198861) for 119886 lt 119886

1 119910 isin R

Substituting (43) into (40) the following differentiableHamiltonian approximation is obtained

= 119892 (119909 119906) + 120582119879119891 (119909 119906) + 120583 [119862 +

1

119886In (1 + 119890

minus119886119862)] (45)

If the inequality constraint function is a 119902th-order statevariable inequality constraint given by

119862 (119909) le 0 (46)

If 119901 times derivatives of 119862(119909) are required before 119906 appearsexplicitly in the result the Hamiltonian with inequalityconstraints (46) is given by

119867 = 119892 (119909 119906) + 120582119879119891 (119909 119906) + 120583119862

(119902) (47)

where

120583 = 0 if 119862 lt 0 (48)

Considering the differentiable approximation to 120583 as follows

120583 = 120583119908 (119862) asymp 1205831

1 + 119890minus119886119862(49)

the following differentiable Hamiltonian approximation isobtained

= 119892 (119909 119906) + 120582119879119891 (119909 119906) + 120583

1

1 + 119890minus119886119862119862(119902) (50)

With the differentiable Hamiltonian approximation theinequality constraints are considered in the proposed guid-ance algorithm

6 Simulation Results

61 Numerical Calculation Based on Linear TPBVPDiscretiza-tion In this section the linear TPVBP is transformed intolinear equations using the Simpson rule which correspondsto the three-point Newton-Cotes quadrature rule The Simp-son rule for differential equation 119889119883119889119864 = 119910(119864) is given by

int

119887

119886

119910 (119864) 119889119864 =119887 minus 119886

6[119910 (119886) + 4119910(

119886 + 119887

2) + 119910 (119887)] (51)

From (51) we obtain

119883(119887) minus 119883 (119886) =119887 minus 119886

6[119910 (119886) + 4119910(

119886 + 119887

2) + 119910 (119887)] (52)

We disperse the normalized energy 119864 into119873 equal segmentsas [1198640

1198641

sdot sdot sdot 119864119873] with 119864

0= 119864119888119896and 119864

119873= 119864119891 The

differential equation 119889119883119889119864 = 119910(119883 119864) can be transformedinto the following equation

119883(119864119894) minus 119883 (119864

119894minus1)

=119864119894minus 119864119894minus1

6[119910 (119883

119894 119864119894)

+ 4119910 (119883119894+ 119883119894minus1

2119864119894+ 119864119894minus1

2)

+ 119910 (119883119894minus1

119864119894minus1

)] 119894 = 1 2 3 119873

(53)

Based on (53) the linear TPBVP (21) can be transformed intolinear equations described by

119860119883 = 119861 (54)

The linear equations can be solved by Gaussian eliminationFinally we can obtain the solution Δ119909(119864) Δ119906(119864) and

120582(119864) of the linear TPBVP (21) from the solution of the linearequations through interpolating

62 GHV Model The proposed method is applied to theGenericHypersonicVehicle (GHV) to verify the effectivenessby flight simulation In order to develop a complete setof aerodynamic coefficients experimental longitudinal andlateral-directional aerodynamics were obtained for the GHVby using six Langley wind tunnels [35] The aerodynamicdatabase of the GHV is shown in [35] The GHV model withscramjet propulsion system is used for the simulation where119878ref = 33473m2 and the full-scale weight119898ini = 110000 kg

According to the characterization of scramjet propulsionsystem described in [2 27] the development of the propul-sion system begins with the following equation

119879 = 05120601119868sp (120601119872119886) 120588V1198920119862119879 (120572119872119886) (55)

where 120601 is the throttle commandwhich varies from 0 to 1Thecoefficient 119862

119879depends on 120572 and119872

119886as follows

119862119879= minus00012119872

1198861205722+ 0008120572 + 14 minus 01119872

119886 (56)

The impulse 119868sp is given by the following equation

119868sp = minus811198722

119886+ 1004119872

119886minus 52 + 100120601 (57)

63 Simulation Results for Modeling and Initial State ErrorConsidering the high cost of the flight experimental testresearchers do flight numerical simulation hardware-in-the-loop simulation and final flight experimental test to verifythe guidance algorithm step by step In this section flight

Mathematical Problems in Engineering 9

Table 1 Initial and required final states of the vehicle

119903ini(m)

Vini(ms)

120574ini(deg)

119903119903119891

(m)V119903119891

(ms)120574119903119891

(deg)20000 1000 0 35000 2000 0

numerical simulation results for the GHV with modelingerror and initial state error are given to show the effectivenessof the proposed guidance algorithm For the modeling errorcaused by the wind tunnel experimental test the aerody-namic coefficients bias is considered in the flight simulationThe simulations end once the final normalized energy 119864

119891

is reached The optimal guidance algorithm updates thetrajectory every guidance cycle which is 03 s The initial andrequired final states of the air-breathing vehicle are shownin Table 1 The number of the discrete nodes 119873 = 10 Thedynamic pressure inequality constraint is given by

119876 =1

2120588V2 le 140000Pa (58)

To assess the capability of the proposed algorithm to dealwith disturbance the aerodynamic coefficients bias and theinitial state error in the simulations are given by followingcases

(1) Δ119862119871= minus01119862

119871 Δ119862119863= +01119862

119863

(2) Δ119862119871= minus01119862

119871 Δ119862119863= +01119862

119863 Δ119903ini = +500m

(3) Δ119862119871= minus01119862

119871 Δ119862119863= +01119862

119863 Δ119903ini = minus500m

(4) Δ119862119871= minus01119862

119871 Δ119862119863= +01119862

119863 Δ119903ini = +1000m

(5) Δ119862119871= minus01119862

119871 Δ119862119863= +01119862

119863 Δ119903ini = minus1000m

The simulation results of the optimal guidance algorithmare shown in Figures 4 5 6 7 and 8 The altitude velocityflight path angle angle of attack and dynamic pressure arerespectively shown in Figures 4 to 8 As expected the finalstate is achieved under aerodynamic bias and initial stateerror Figure 8 shows that the state inequality constraint issatisfied The dynamic pressure is less than 140000 Pa fordifferent cases

Table 2 shows the terminal error and the fuel cost ofthe optimal guidance algorithm The terminal accuracy ishigh for different cases Moreover the fuel cost of theoptimal guidance algorithm is lower than that of the trackingguidance algorithm A greater initial state error will result inmore fuel savings In summarywith lower fuel cost the resultsof the proposed optimal guidance algorithm satisfies all theequality and inequality constraints The simulation resultsshow the great potential for the final flight experimental test

7 Conclusion

In this paper we present an optimal guidance algorithmfor air-breathing launch vehicle based on optimal trajectorycorrection The proposed guidance algorithm corrects thereference trajectory in every guidance cycle to satisfy the opti-mality condition of the optimal feedback control problem

0 14012010080604020

18000

21000

24000

27000

30000

33000

Alti

tude

(m)

Time (s)

Case 1Case 2Case 3

Case 4Case 5

Figure 4 The altitude of the simulation results

1200

1400

1600

1800

2000

2200Ve

loci

ty (m

s)

Time (s)0 14012010080604020

Case 1Case 2Case 3

Case 4Case 5

Figure 5 The velocity of the simulation results

By linearizing the optimality condition the linear TPBVPis obtained for optimal trajectory correction The solutionof the linear TPBVP is derived using the Simpson rule Thenew trajectory is generated by a linear search for the stepsize of the solution Smooth Hamiltonian approximation isimplemented to the inequality constraints The sufficiencycondition for the global convergence of the guidance algo-rithm is given in this paper

Finally simulations for 5 different cases of modeling andinitial state error are presented Compared with the trackingguidance method the simulation results prove the low fuelcost and high precision of the proposed optimal guidancemethod

10 Mathematical Problems in Engineering

Table 2 The simulation results for optimal guidance algorithm

Case Δ119903119891(m) ΔV

119891(ms) Δ120574

119891(deg) Fuel cost (kg) Fuel cost (tracking method) (kg)

1 minus2820 0136 020 457219 4574622 859 minus003 minus015 458314 4628393 minus1508 007 020 458474 4605664 10816 047 minus023 460292 4776005 minus3243 019 minus016 460133 468894

0

2

4

6

8

Time (s)

Flig

ht p

ath

angl

e (D

EG)

minus2

minus4

0 14012010080604020

Case 1Case 2Case 3

Case 4Case 5

Figure 6 The flight path angle of the simulation results

Ang

le o

f atta

ck (D

EG)

0

1

2

3

4

minus1

minus2

minus3

minus4

minus5

minus6

Time (s)0 14012010080604020

Case 1Case 2Case 3

Case 4Case 5

Figure 7 The angle of attack of the simulation results

Acknowledgments

This work was supported in part by the National NatureScience Foundation of China nos 61203081 and 61174079Doctoral Fund of Ministry of Education of China

20000

40000

60000

80000

100000

120000

140000

Dyn

amic

pre

ssur

e (Pa

)

Time (s)0 14012010080604020

Case 1Case 2Case 3

Case 4Case 5

Figure 8 The dynamic pressure of the simulation results

no 20120142120091 andPrecisionmanufacturing technologyand equipment for metal parts no 2012DFG70640

References

[1] H J Henry Jr and C H McLellan ldquoAir-breathing launch vehi-cle for earth-orbit shuttle New technology and developmentapproachrdquo Journal of Aircraft vol 8 no 5 pp 381ndash387 1971

[2] J E CorbanA J Calise andGA Flandro ldquoReal-time guidancealgorithm for aerospace plane optimal ascent to low earth orbitrdquoin Proceedings of the American Control Conference pp 2475ndash2481 Pittsburgh PA USA June 1989

[3] C H Yam D D Lorenzo and D Izzo ldquoLow-thrust trajectorydesign as a constrained global optimization problemrdquo Journalof Aerospace Engineering G vol 225 no 11 pp 1243ndash1251 2011

[4] B A Conway ldquoA survey of methods available for the numericaloptimization of continuous dynamic systemsrdquo Journal of Opti-mization Theory and Applications vol 152 no 2 pp 271ndash3062012

[5] C H Chen and O L Mangasarian ldquoSmoothing methodsfor convex inequalities and linear complementarity problemsrdquoMathematical Programming vol 71 no 1 pp 51ndash69 1995

[6] X Chen ldquoSmoothing methods for nonsmooth nonconvexminimizationrdquo Mathematical Programming vol 134 no 1 pp71ndash99 2012

[7] W Wang Y Shang and Y Zhang ldquoA filled function approachfor nonsmooth constrained global optimizationrdquoMathematical

Mathematical Problems in Engineering 11

Problems in Engineering vol 2010 Article ID 310391 9 pages2010

[8] L I Rozonoer ldquoLinear-quadratic optimization and some gen-eral hypotheses on optimal controlrdquoMathematical Problems inEngineering vol 5 no 4 pp 275ndash289 1999

[9] J T Betts ldquoSurvey of numerical methods for trajectory opti-mizationrdquo Journal of Guidance Control and Dynamics vol 21no 2 pp 193ndash207 1998

[10] G Q Huang Y P Lu and Y Nan ldquoA survey of numericalalgorithms for trajectory optimization of flight vehiclesrdquo ScienceChina-Technological Sciences vol 55 no 9 pp 2538ndash2560 2012

[11] Y Ulybyshev ldquoDiscrete pseudocontrol sets for optimal controlproblemsrdquo Journal of Guidance Control and Dynamics vol 33no 4 pp 1133ndash1142 2010

[12] M Vasile and F Zuiani ldquoMulti-agent collaborative search anagent-based memetic multi-objective optimization algorithmapplied to space trajectory designrdquo Journal of Aerospace Engi-neering G vol 225 no 11 pp 1211ndash1227 2011

[13] M Pontani and B A Conway ldquoParticle swarm optimizationapplied to space trajectoriesrdquo Journal of Guidance Control andDynamics vol 33 no 5 pp 1429ndash1441 2010

[14] Y P Ulybyshev ldquoOptimization of low-thrust orbit transfers withconstraintsrdquo Cosmic Research vol 50 no 5 pp 376ndash390 2012

[15] M Chen R Mei and B Jiang ldquoSliding mode control for a classof uncertainMIMOnonlinear systemswith application to near-space vehiclesrdquoMathematical Problems in Engineering vol 2013Article ID 180589 9 pages 2013

[16] H Zhang Y Shi M Q Xu and H T Cui ldquoObserver-basedtracking controller design for networked predictive controlsystems with uncertain markov delaysrdquo in Proceedings of theAmerican Control Conference pp 5682ndash5687 2012

[17] H Zhang and Y Shi ldquoRobust H-infinity sliding-mode controlfor markovian jump systems subject to intermittent observa-tions and partially known transition probabilitiesrdquo Systems ampControl Letters vol 62 no 12 pp 1114ndash1124 2013

[18] Z Shuai H Zhang J Wang J Li and M Ouyang ldquoCombinedAFS andDYC control of four-wheel-independent-drive electricvehicles over CAN network with time-varying delaysrdquo IEEETransactions on Vehicular Technology no 99 p 1 2013

[19] H Zhang Y Shi and A Saadat Mehr ldquoRobust static outputfeedback control and remote PID design for networked motorsystemsrdquo IEEETransactions on Industrial Electronics vol 58 no12 pp 5396ndash5405 2011

[20] H Zhang Y Shi and M X Liu ldquoH-infinity step trackingcontrol for networked discrete-time nonlinear systems withintegral and predictive actionsrdquo IEEE Transactions on IndustrialInformatics vol 9 no 1 pp 337ndash345 2013

[21] X J Wang and S P Wang ldquoAdaptive fuzzy robust controlfor a class of nonlinear systems via small gain theoremrdquoMathematical Problems in Engineering vol 2013 Article ID201432 11 pages 2013

[22] V Brinda S Dasgupta and M Lal ldquoTrajectory optimizationand guidance of an air breathing hypersonic vehiclerdquo in Pro-ceedings of the 14th AIAAAHI International Space Planes andHypersonics Systems Technologies Conference vol 2 pp 856ndash869 November 2006

[23] B Tian and Q Zong ldquoOptimal guidance for reentry vehiclesbased on indirect legendre pseudospectral methodrdquoActa Astro-nautica vol 68 no 7-8 pp 1176ndash1184 2011

[24] M H Graszliglin J Telaar and UM Schottle ldquoAscent and reentryguidance concept based on NLP-methodsrdquo Acta Astronauticavol 55 no 3ndash9 pp 461ndash471 2004

[25] P Lu and B Pan ldquoHighly constrained optimal launch ascentguidancerdquo Journal of Guidance Control and Dynamics vol 33no 2 pp 404ndash414 2010

[26] P Lu H Sun and B Tsai ldquoClosed-loop endo-atmosphericascent guidancerdquo in Proceedings of the AIAA Guidance Navi-gation and Control Conference Monterey Calif USA 2002

[27] J Oscar Murillo and P Lu ldquoFast ascent trajectory optimizationfor hypersonic air-breathing vehiclesrdquo in Proceedings of theAIAA Guidance Navigation and Control Conference pp 5ndash8Toronto Canada 2010

[28] M Marrdonny and M Mobed ldquoA guidance algorithm forlaunch to equatorial orbitrdquo Aircraft Engineering and AerospaceTechnology vol 81 no 2 pp 137ndash148 2009

[29] T Yamamoto and J Kawaguchi ldquoA new real-time guidancestrategy for aerodynamic ascent flightrdquo Acta Astronautica vol61 no 11-12 pp 965ndash977 2007

[30] C Filici andR S Sanchez Pena ldquoOnline guidance updates usingneural networksrdquo Acta Astronautica vol 66 no 3-4 pp 477ndash485 2010

[31] P T Boggs A J Kearsley and J W Tolle ldquoA global convergenceanalysis of an algorithm for large-scale nonlinear optimizationproblemsrdquo SIAM Journal on Optimization vol 9 no 4 pp 833ndash862 1999

[32] S Schlenkrich and A Walther ldquoGlobal convergence of quasi-Newton methods based on adjoint Broyden updatesrdquo AppliedNumerical Mathematics vol 59 no 5 pp 1120ndash1136 2009

[33] S Schlenkrich A Griewank and A Walther ldquoOn the localconvergence of adjoint Broyden methodsrdquo Mathematical Pro-gramming vol 121 no 2 pp 221ndash247 2010

[34] S Lian and L Zhang ldquoA simple smooth exact penalty functionfor smooth optimization problemrdquo Journal of Systems Science ampComplexity vol 25 no 3 pp 521ndash528 2012

[35] S Keshmiri R Colgren and M Mirmirani ldquoDevelopment ofan aerodynamic database for a generic hypersonic air vehiclerdquoin Proceedings of the AIAA Guidance Navigation and ControlConference pp 3978ndash3998 San Francisco Calif USA August2005

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Journal of

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Mathematical PhysicsAdvances in

Complex AnalysisJournal of

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OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Journal of

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society

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Decision SciencesAdvances in

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Page 3: Research Article Optimal Ascent Guidance for Air-Breathing ...downloads.hindawi.com/journals/mpe/2013/313197.pdfResearch Article Optimal Ascent Guidance for Air-Breathing Launch Vehicle

Mathematical Problems in Engineering 3

Center of mass

V

T

D

L

120572

120574

Figure 1 The forces acting on the vehicle

Themotion of the vehicle in the longitudinal plane can bedescribed as follows

119889119903

119889119905= V sin 120574

119889V119889119905

=119879 cos120572 minus 119863

119898minus 119892 sin 120574

119889120574

119889119905=

1

V[119879 sin120572 + 119871

119898cos120590 minus 119892 cos 120574 + V2

119903cos 120574]

119889119898

119889119905= minus

119879

119868sp

(1)

where 119903 119898 119868sp and 119892 are the altitude weight of the vehicleimpulse and gravity acceleration respectively

The lift force 119871 and the drag force119863 are given as follows

119871 =1

2120588V2119878ref119862119871 (119872119886 120572)

119863 =1

2120588V2119878ref119862119863 (119872119886 120572)

(2)

where 120588 and 119878ref are the air density of the current altitudeand the reference area respectively 119862

119871and 119862

119863are the lift

coefficient and the drag coefficient respectively which are thenonlinear functions of the angle of attack120572 andMachnumber119872119886 For the air-breathing engine the thrust is given by

119879 = 119879 (120588119872119886 120572 120601) (3)

where119879 is the nonlinear function of 120588 120572119872119886 and the throttle

command 120601In [25] the equations were normalized to reduce

the numerical calculation error Variable substitutions areapplied as follows

119877 =119903

1199030

119881 =V

radic11989201199030

119892 =11989201199032

0

1199032 (4)

where 1199030and 119892

0are Earth radius and the ground gravity

acceleration respectively Considering the thrust 119879 andaerodynamic forces as a composition of forces we define thenormalized accelerations 119886

119879and 119886119871as follows

119886119879=

119879 cos120572 minus 119863

1198981198920

119886119871=

119879 sin120572 + 119871

1198981198920

(5)

We define

120591 =119905

radic11990301198920

(6)

The normalized equations of motion are obtained from (1)(4) (5) and (6) as follows

119889119877

119889120591= 119881 sin 120574

119889119881

119889120591= 119886119879minussin 1205741198772

119889120574

119889120591=

1

119881[119886119871+ (1198812minus

1

119877)(

cos 120574119877

)]

119889119898

119889120591= minus

119879radic1199030

119868spradic1198920

(7)

The mission of ascent is to send the vehicle to therequired final state which is denoted as (119877

119891 119881119891 120574119891) Define

normalized-energy 119864 as follows

119864 =1198812

2minus

1

119877 (8)

From (8) we obtain

119889119864

119889120591= 119881119886119879

119889119877

119889119864=sin 120574119886119879

119889119881

119889119864=

1

119886119879119881

[119886119879minussin 1205741198772

]

119889120574

119889119864=

1

1198861198791198812

[119886119871+ (1198812minus

1

119877)(

cos 120574119877

)]

119889119898

119889119864= minus

119879radic1199030

119886119879119881119868spradic119892

0

(9)

The integration interval [1198640 119864119891] is fixed

22 Optimal Feedback Control Problem for Ascent GuidanceIn this section the optimal feedback control problem foroptimal ascent guidance is addressed This problem differsfrom trajectory optimization off-board in that the initialstate of the optimal control problem is obtained from thenavigation system We denote the state and the guidancecommand of the trajectory as

119909 (119864) = [119877 (119864) 119881 (119864) 120574 (119864) 119898 (119864)]119879

119906 (119864) = [120572 (119864) 120601 (119864)]119879

(10)

With the feedback vehicle state 119909119888119896obtained from navigation

system the optimal ascent guidance algorithm is used to

4 Mathematical Problems in Engineering

generate new reference trajectory [119909(119864) 119906(119864)]119879 and output

119906(119864) from the following optimal control problem

min 119869 = 120601 (119909 (119864119891)) + int

119864119891

119864119888119896+1

119892 (119909 119906) 119889119864

st = 119891 (119909 119906)

119909 (119864119888119896) = 119909119888

119896

119882 (119909 (119864119891)) = 0

119862 (119909 119906) le 0

(11)

where 119864119888119896is the normalized energy for current vehicle state

119909119888119896

3 Optimal Ascent Guidance Algorithm

In [25 26] the optimal control problem was solved on-boardin every guidance cycle However it is difficult to solve thenonlinear and nondifferentiable TPBVP on-board for air-breathing launch vehicles In this section we propose an opti-mal guidance algorithmThis guidance algorithmupdates thereference trajectory [119909 119906]

119879 in every guidance cycle to dealwith the unknowndisturbance For the current state119909119888

119896of the

119896th guidance cycle the optimal feedback control problem istransformed into a linear TPBVP problem using optimalitycondition linear approximation The searching direction ofthe correction values 119889

119896is obtained by solving the linear

TPBVP With the searching direction the new referencetrajectory is generated by linear search For the 119896th guidancecycle the trajectory update is performed as follows

[119909119896+1(119864) 119906119896+1

(119864)]119879

= [119909119896(119864) 119906119896(119864)]119879

+ 119897119896119889119879

119896(119864) (12)

The initial reference trajectory [1199090(119864) 119906

0(119864)]119879 is derived

from the off-board from trajectory optimization by directmethodThe flow chart of the algorithm is shown in Figure 2Figure 3 shows the time sequence of algorithm

31 Linear Approximation for the Optimality Condition Inthis section we linearize the optimality condition for opti-mality trajectory updating The optimal control problem (11)without inequality constraints is given as follows

min 119869 = 120601 (119909 (119864119891)) + int

119864119891

119864119888119896+1

119892 (119909 119906) 119889119864

st = 119891 (119909 119906)

119909 (119864119888119896) = 119909119888

119896

119882 (119909 (119864119891)) = 0

(13)

The Hamiltonian of (13) is given as follows

119867 = 119892 (119909 119906) + 120582119879119891 (119909 119906) (14)

Begin

Linear approximation for the

optimality condition the costate

equations the motion equations

Solving the linear TPBVP

Linear search for the step size

Updating the optimal trajectory

End

Figure 2 Flow chart of the optimal ascent guidance algorithm

where 120582 is the costate vector The state equations costateequations and optimality condition for the optimal controlproblem are given by

= 119891 (119909 119906)

= minus(120597119867

120597119909)

119879

120597119867

120597119906= 0

(15)

The initial and transversality conditions are given by

119909 (119864119888119896) = 119909119888

119896

119882 (119909 (119864119891)) = 0

120582 (119864119891) = [

120597120601

120597119909 (119864119891)

]

119879

+ [120603119879 120597119882

120597119909 (119864119891)

]

119879

(16)

We denote the trajectory correction variables as follows

[

[

Δ119909 (119864)

Δ119906 (119864)

Δ120582 (119864)

]

]

= [

[

119909 (119864) minus 119909119896(119864)

119906 (119864) minus 119906119896(119864)

120582 (119864) minus 120582119896(119864)

]

]

(17)

Mathematical Problems in Engineering 5

middot middot middot

Output theguidancecommanduk+1(E)

Output theguidancecommanduk+2(E)

middot middot middot

middot middot middot

The time consuming ofthe online algorithm The guidance cycle

Trajectoryupdating

beginsupdating

Trajectory

ends

Trajectoryupdating

beginsupdating

Trajectory

ends

Trajectoryupdating

beginsupdating

Trajectory

ends

End

Time (120591)kΔ120591 (k + 1)Δ120591 (k + 2)Δ120591

Figure 3 Time sequence of the optimal ascent guidance algorithm

We denote that

119867119896

119906119909=

1205972(119892 + 120582

119879119891)

120597119906120597119909(119909119896 119906119896 120582119896)

119867119896

119906= [

120597119867

120597119906(119909119896 119906119896 120582119896)]

119879

119867119896

119906119906=

1205972(119892 + 120582

119879119891)

1205971199062(119909119896 119906119896 120582119896)

119867119896

119909= [

120597119867

120597119909(119909119896 119906119896 120582119896)]

119879

119867119896

119909119909=

1205972(119892 + 120582

119879119891)

1205971199092(119909119896 119906119896 120582119896)

119891119896

119906=

120597119891

120597119906(119909119896 119906119896)

119867119896

119909119906=

1205972(119892 + 120582

119879119891)

120597119909120597119906(119909119896 119906119896 120582119896)

119891119896

119909=

120597119891

120597119909(119909119896 119906119896)

119892119896

119906= [

120597119892

120597119906(119909119896 119906119896)]

119879

119892119896

119909= [

120597119892

120597119909(119909119896 119906119896)]

119879

(18)

The first-order Taylor expansion of (15) on the old referencetrajectory [119909

119896119906119896

120582119896]119879 is

[120597119867

120597119906]

119879

asymp 119867119896

119906+ 119867119896

119906119909Δ119909 + 119867

119896

119906119906Δ119906 + (119891

119896

119906)119879

Δ120582

asymp minus119867119896

119909minus 119867119896

119909119909Δ119909 minus 119867

119896

119909119906(119909119896 119906119896 120582119896) Δ119906 minus (119891

119896

119909)119879

Δ120582

asymp 119891 (119909119896 119906119896) + 119891119896

119909Δ119909 + 119891

119896

119906Δ119906

(19)

The following equations are established

119867119896

119906+ (119891119896

119906)119879

Δ120582 = 119892119896

119906+ (119891119896

119906)119879

120582

119867119896

119909+ (119891119896

119909)119879

Δ120582 = 119892119896

119909+ (119891119896

119909)119879

120582

(20)

Substituting (20) into (19) yields

= minus119892119896

119909minus 119867119896

119909119909Δ119909 minus 119867

119896

119909119906(119909119896 119906119896 120582119896) Δ119906 minus (119891

119896

119909)119879

120582

Δ = 119891119896

119909Δ119909 + 119891

119896

119906Δ119906 minus

119896+ 119891 (119909

119896 119906119896)

119892119896

119906+ 119867119896

119906119909Δ119909 + 119867

119896

119906119906Δ119906 + (119891

119896

119906)119879

120582 = 0

Δ119909 (119864119888119896) + Δ119909

119896(119864119888119896) = 119909119888

119896

120597119882

120597119909 (119864119891)

Δ119909 (119864119891) +119882(119909

119896(119864119891)) = 0

120582 (119864119891) = [

120597120601

120597119909 (119864119891)

]

119879

+ [120603119879 120597119882

120597119909 (119864119891)

]

119879

(21)

This equation is a linear TPBVP about the variables[Δ119909 Δ119906 120582]

119879 The linear TPBVP is solved using the Simpsonrules in Section 6

32 Backtracking Line Search for the Step Size In this sectionthe step size 119897

119896is determined by backtracking line search

considering the solution of the linear TPBVP as the searchdirection The penalty function for problem (13) is given by

119901119896= 119869 (119909 119906) + 120578

10038171003817100381710038171003817119882(119909119891)10038171003817100381710038171003817+ int

119864119891

119864119888119896

1003817100381710038171003817119891 (119909 119906) minus 1003817100381710038171003817

+1003817100381710038171003817119909 (119864119888119896) minus 119909119888

119896

1003817100381710038171003817

(22)

6 Mathematical Problems in Engineering

where 120578 is the penalty parameter satisfied 120578 gt max |120582| Wedenote

119910 = [119909119896

119906119896

] 119889119896= [

Δ119909

Δ119906] (23)

where [Δ119909 Δ119906 120582]119879 is the solution of the linear TPBVP (21)

The minimizer of penalty function 119901119896(119909 119906) corresponds

to the solution of problem (11) Backtracking line search wasused for large-scale nonlinear optimization problems in [31]The step size 119897

119896is determined by backtracking line search that

satisfies

119897119896= max 119897 isin 119878 119901

119896+1(119910119896+ 119897119889119896)

le 119901119896+1

(119910119896) + 1198971205761120575119901119896+1

(119910119896 119889119896)

119878 = 119897 119897 = 1205762

119898 119898 = 0 1 2

(24)

where 1205761

isin (0 1) 1205762

isin (0 1) and 120575119901119896+1

(119910119896 119889119896) is the

variational of the functional 119901119896+1

(119910) caused by variational119889119896of 119910 on 119910

119896 Considering 119889

119896as the solution of (21)

120575119901119896+1

(119910119896 119889119896) le 0 is established The new state and guidance

command of the reference trajectory are generated as follows

[119909119896+1

119906119896+1

] = [119909119896

119906119896

] + 119897119896[Δ119909

Δ119906] (25)

4 The Global Convergence of the Algorithm

In [32 33] the convergence of the off-board algorithm wasanalyzed But the relationship between the global conver-gence and the guidance cycle 120591 has not been discussed for on-board optimization in the previous literatures In this sectionwe state and prove the global convergence of the on-boardalgorithm proposed in the preceding section The solutionsequence [119909

119896119906119896]119879 generated by the algorithm will converge

to the solution of the feedback optimal control problemUnlike off-board trajectory optimization the initial conditionof the feedback optimal control problem depends on thecurrent state of the vehicle in real time For the 119896th guidancecycle the optimal feedback control problem is described as(13)Thepenalty function119901

119896for this feedback optimal control

problem is given by (22) For the subsequent guidance cyclethe optimal feedback guidance problem is given by

min 119869 = 119882(119909 (119864119891)) + int

119864119891

119864119888119896+1

119892 (119909 119906) 119889119864

st = 119891 (119909 119906)

119909 (119864119888119896+1

) = 119909119888119896+1

119882 (119909 (119864119891)) = 0

(26)

where

119864119888119896+1

= 119864119888119896+ int

120591119896+Δ120591

120591119896

119889119864

119889120591119889120591 (27)

Δ120591 is the guidance cycle and 119909119888119896+1

is the new state of vehiclefrom navigation system at the time 120591

119896+ Δ120591 The penalty

function for optimal problem (24) is given by

119901119896+1

(119909 119906) = 119869 (119909 119906) + 12057810038171003817100381710038171003817119882(119909119896(119864119891))10038171003817100381710038171003817

+ 120578int

119864119891

119864119888119896+1

1003817100381710038171003817119891 (119909 119906) minus 1003817100381710038171003817 119889119864

+ 1205781003817100381710038171003817119909 (119864119888119896+1) minus 119909119888

119896+1

1003817100381710038171003817

(28)

We define the entire penalty function as follows

119901119896(119909 119906) = 119901

119896(119909 119906)

+

119896minus1

sum

119894=0

int

119864119888119894+1

119864119888119894

[1205781003817100381710038171003817119891 (119909119894 119906119894) minus 119894

1003817100381710038171003817 + 119892 (119909119894 119906119894)] 119889119864

(29)

The entire penalty function 119901119896(119909 119906) includes the penalty

function on the whole integration interval [1198640 119864119891]

Theorem 1 If the minimizer of 119901119896(119909119896 119906119896) is exited 119892(119909 119906) ge

0 and Δ120591 is selected and satisfies

120578100381710038171003817100381710038171003817119909119888119896+1

minus 119909119888119896minus int119864119888119896+1

119864119888119896119896119889119864

100381710038171003817100381710038171003817

minus120575119901119896+1

(119910119896 119889119896)

lt 1198971198961205763 (30)

where 1205763isin (0 120576

1) The solution sequence [119909

119896119906119896]119879 generated

from (24) will converge to the solution of the feedback optimalcontrol problem

Proof From (22) (28) (29) and

119909119896(119864119888119896+1

) = 119909119896(119864119888119896) + int

119864119888119896+1

119864119888119896

119896119889119864 (31)

we can obtain that

119901119896+1

(119909119896 119906119896) minus 119901119896(119909119896 119906119896)

= 120578

100381710038171003817100381710038171003817100381710038171003817

119909119896(119864119888119896) minus 119909119888

119896

minus(119909119888119896+1

minus 119909119888119896minus int

119864119888119896+1

119864119888119896

119896119889119864)

100381710038171003817100381710038171003817100381710038171003817

minus 1205781003817100381710038171003817119909119896 (119864119888119896) minus 119909119888

119896

1003817100381710038171003817

(32)

Mathematical Problems in Engineering 7

From (32) we obtain

limΔ120591rarr0

[119901119896+1

(119909119896 119906119896) minus 119901119896(119909119896 119906119896)] = 0

lim119891(119909119896119906119896)minus 119909119896rarr0

[119901119896+1

(119909119896 119906119896) minus 119901119896(119909119896 119906119896)] = 0

(33)

119901119896+1

(119909119896 119906119896) minus 119901119896(119909119896 119906119896)

= 120578

100381710038171003817100381710038171003817100381710038171003817

119909119896(119864119888119896) minus 119909119888

119896+1+ int

119864119888119896+1

119864119888119896

119896119889119864

100381710038171003817100381710038171003817100381710038171003817

minus1003817100381710038171003817119909119896 (119864119888119896) minus 119909119888

119896

1003817100381710038171003817

le 1205781003817100381710038171003817119909119896 (119864119888119896) minus 119909119888

119896

1003817100381710038171003817

+

100381710038171003817100381710038171003817100381710038171003817

119909119888119896+1

minus 119909119888119896minus int

119864119888119896+1

119864119888119896

119896119889119864

100381710038171003817100381710038171003817100381710038171003817

minus1003817100381710038171003817119909119896 (119864119888119896) minus 119909119888

119896

1003817100381710038171003817

= 120578

100381710038171003817100381710038171003817100381710038171003817

119909119888119896+1

minus 119909119888119896minus int

119864119888119896+1

119864119888119896

119896119889119864

100381710038171003817100381710038171003817100381710038171003817

(34)

From conditions (24) and (30) it is obtained that

119901119896+1

(119909119896+1

119906119896+1

)

le 119901119896+1

(119909119896 119906119896) + 1198971198961205761120575119901119896+1

(119889119896)

= 119901119896(119909119896 119906119896) + 1198971198961205761120575119901119896+1

(119889119896)

+ 120578

100381710038171003817100381710038171003817100381710038171003817

119909119896(119864119888119896) minus 119909119888

119896

minus (119909119888119896+1

minus 119909119888119896minus int

119864119888119896+1

119864119888119896

119896119889119864)

100381710038171003817100381710038171003817100381710038171003817

minus1003817100381710038171003817119909119896 (119864119888119896) minus 119909119888

119896

1003817100381710038171003817

le 119901119896(119909119896 119906119896) + 120578

100381710038171003817100381710038171003817100381710038171003817

119909119888119896+1

minus 119909119888119896minus int

119864119888119896+1

119864119888119896

119896119889119864

100381710038171003817100381710038171003817100381710038171003817

+ 1198971198961205761120575119901119896+1

(119910119896 119889119896)

le 119901119896(119909119896 119906119896) + 120578

100381710038171003817100381710038171003817100381710038171003817

119909119888119896+1

minus 119909119888119896minus int

119864119888119896+1

119864119888119896

119896119889119864

100381710038171003817100381710038171003817100381710038171003817

+ 1198971198961205763120575119901119896+1

(119910119896 119889119896)

lt 119901119896(119909119896 119906119896)

(35)

Thus the sequence 119901119896(119909119896 119906119896) 119896 = 1 2 3 is monotoni-

cally decreasing Considering the existence of a minimizer119901119896(119909119896 119906119896) converges Thus the following equation is estab-

lished

lim119896rarrinfin

(1198971198961205763120575119901119896+1

(119910119896 119889119896) minus 1198971198961205761120575119901119896+1

(119910119896 119889119896)) = 0 (36)

From (36) we derive

lim119896rarrinfin

(119897119896120575119901119896+1

(119910119896 119889119896)) = 0 (37)

If 119889119896

= 0 [119909119896

119906119896]119879 is not the minimizer of penalty function

then 120575119901119896+1

(119910119896 119889119896) = 0 Considering that 119897

119896is bounded away

from zero we derive

lim119896rarrinfin

119889119896= 0

lim119896rarrinfin

120575119901119896+1

(119910119896 119889119896) = 0

lim119896rarrinfin

[119896minus 119891 (119909

119896 119906119896)] = 0

lim119896rarrinfin

(119896+ 119867119896

119909) = 0

lim119896rarrinfin

119867119896

119906= 0

(38)

Thus the solution sequence [119909119896

119906119896]119879 generated by the

algorithmwill converge to the solution of the optimal controlproblem

5 Smooth Approximation forInequality Constraints

The nondifferentiable TPBVP from optimal control problem(11) with the inequality constraints brings difficulties for theproposed algorithm which can only solve a smooth problemSmooth approximations for nondifferentiable optimizationproblems have been studied in [5 6 34] In this sectionwe introduce the smooth approximation method for theinequality constraints If the inequality constraint function isabout the state and the guidance command

119862 (119909 119906) le 0 (39)

the Hamiltonian for the optimal control problem (11) withinequality constraints (39) is given by

119867 = 119892 (119909 119906) + 120582119879119891 (119909 119906) + 120583max 119862 (119909 119906) 0 (40)

However the Hamiltonian (40) is a nondifferentiable func-tion which brings difficulty for the proposed algorithmFor the smooth Hamiltonian approximation a differentiablefunction is substituted for max119862(119909 119906) 0 We denote thenondifferentiable function as follows

max 0 119910 = int

119910

minusinfin

119908 (119911) 119889119911 (41)

where 119908(119911) is the step function

119908 (119911) = 1 if 119911 ge 0

0 if 119911 lt 0(42)

Considering 119904(119911 119886) as the approximation function of 119908(119911)the differentiable approximation to max0 119910 is obtained as

119891119901(119910 119886) asymp int

119910

minusinfin

119904 (119911 119886) 119889119911 = 119910 +1

119886In (1 + 119890

minus119886119910) (43)

8 Mathematical Problems in Engineering

where 119904(119911 119886) is Sigmoid function

119904 (119911 119886) =1

1 + 119890minus119886119911 119886 gt 0 (44)

The Properties of 119891119901(119910 119886) 119886 gt 0 are as follows [5]

(1) 119891119901(119910 119886) is 119899-times continuously differentiable for any

positive integer 119899 with 120597119891119901(119910 119886)120597119910 = 1(1 + 119890

minus119886119910)

and 1205972119891119901(119910 119886)120597119910

2= 119886119890minus119886119910

(1 + 119890minus119886119910

)2

(2) 119891119901(119910 119886) is strictly convex and strictly increasing onR

(3) 119891119901(119910 119886) gt max0 119910

(4) lim119886rarrinfin

119891119901(119910 119886) = max0 119910

(5) 119891119901(119910 119886) gt 119891

119901(119910 1198861) for 119886 lt 119886

1 119910 isin R

Substituting (43) into (40) the following differentiableHamiltonian approximation is obtained

= 119892 (119909 119906) + 120582119879119891 (119909 119906) + 120583 [119862 +

1

119886In (1 + 119890

minus119886119862)] (45)

If the inequality constraint function is a 119902th-order statevariable inequality constraint given by

119862 (119909) le 0 (46)

If 119901 times derivatives of 119862(119909) are required before 119906 appearsexplicitly in the result the Hamiltonian with inequalityconstraints (46) is given by

119867 = 119892 (119909 119906) + 120582119879119891 (119909 119906) + 120583119862

(119902) (47)

where

120583 = 0 if 119862 lt 0 (48)

Considering the differentiable approximation to 120583 as follows

120583 = 120583119908 (119862) asymp 1205831

1 + 119890minus119886119862(49)

the following differentiable Hamiltonian approximation isobtained

= 119892 (119909 119906) + 120582119879119891 (119909 119906) + 120583

1

1 + 119890minus119886119862119862(119902) (50)

With the differentiable Hamiltonian approximation theinequality constraints are considered in the proposed guid-ance algorithm

6 Simulation Results

61 Numerical Calculation Based on Linear TPBVPDiscretiza-tion In this section the linear TPVBP is transformed intolinear equations using the Simpson rule which correspondsto the three-point Newton-Cotes quadrature rule The Simp-son rule for differential equation 119889119883119889119864 = 119910(119864) is given by

int

119887

119886

119910 (119864) 119889119864 =119887 minus 119886

6[119910 (119886) + 4119910(

119886 + 119887

2) + 119910 (119887)] (51)

From (51) we obtain

119883(119887) minus 119883 (119886) =119887 minus 119886

6[119910 (119886) + 4119910(

119886 + 119887

2) + 119910 (119887)] (52)

We disperse the normalized energy 119864 into119873 equal segmentsas [1198640

1198641

sdot sdot sdot 119864119873] with 119864

0= 119864119888119896and 119864

119873= 119864119891 The

differential equation 119889119883119889119864 = 119910(119883 119864) can be transformedinto the following equation

119883(119864119894) minus 119883 (119864

119894minus1)

=119864119894minus 119864119894minus1

6[119910 (119883

119894 119864119894)

+ 4119910 (119883119894+ 119883119894minus1

2119864119894+ 119864119894minus1

2)

+ 119910 (119883119894minus1

119864119894minus1

)] 119894 = 1 2 3 119873

(53)

Based on (53) the linear TPBVP (21) can be transformed intolinear equations described by

119860119883 = 119861 (54)

The linear equations can be solved by Gaussian eliminationFinally we can obtain the solution Δ119909(119864) Δ119906(119864) and

120582(119864) of the linear TPBVP (21) from the solution of the linearequations through interpolating

62 GHV Model The proposed method is applied to theGenericHypersonicVehicle (GHV) to verify the effectivenessby flight simulation In order to develop a complete setof aerodynamic coefficients experimental longitudinal andlateral-directional aerodynamics were obtained for the GHVby using six Langley wind tunnels [35] The aerodynamicdatabase of the GHV is shown in [35] The GHV model withscramjet propulsion system is used for the simulation where119878ref = 33473m2 and the full-scale weight119898ini = 110000 kg

According to the characterization of scramjet propulsionsystem described in [2 27] the development of the propul-sion system begins with the following equation

119879 = 05120601119868sp (120601119872119886) 120588V1198920119862119879 (120572119872119886) (55)

where 120601 is the throttle commandwhich varies from 0 to 1Thecoefficient 119862

119879depends on 120572 and119872

119886as follows

119862119879= minus00012119872

1198861205722+ 0008120572 + 14 minus 01119872

119886 (56)

The impulse 119868sp is given by the following equation

119868sp = minus811198722

119886+ 1004119872

119886minus 52 + 100120601 (57)

63 Simulation Results for Modeling and Initial State ErrorConsidering the high cost of the flight experimental testresearchers do flight numerical simulation hardware-in-the-loop simulation and final flight experimental test to verifythe guidance algorithm step by step In this section flight

Mathematical Problems in Engineering 9

Table 1 Initial and required final states of the vehicle

119903ini(m)

Vini(ms)

120574ini(deg)

119903119903119891

(m)V119903119891

(ms)120574119903119891

(deg)20000 1000 0 35000 2000 0

numerical simulation results for the GHV with modelingerror and initial state error are given to show the effectivenessof the proposed guidance algorithm For the modeling errorcaused by the wind tunnel experimental test the aerody-namic coefficients bias is considered in the flight simulationThe simulations end once the final normalized energy 119864

119891

is reached The optimal guidance algorithm updates thetrajectory every guidance cycle which is 03 s The initial andrequired final states of the air-breathing vehicle are shownin Table 1 The number of the discrete nodes 119873 = 10 Thedynamic pressure inequality constraint is given by

119876 =1

2120588V2 le 140000Pa (58)

To assess the capability of the proposed algorithm to dealwith disturbance the aerodynamic coefficients bias and theinitial state error in the simulations are given by followingcases

(1) Δ119862119871= minus01119862

119871 Δ119862119863= +01119862

119863

(2) Δ119862119871= minus01119862

119871 Δ119862119863= +01119862

119863 Δ119903ini = +500m

(3) Δ119862119871= minus01119862

119871 Δ119862119863= +01119862

119863 Δ119903ini = minus500m

(4) Δ119862119871= minus01119862

119871 Δ119862119863= +01119862

119863 Δ119903ini = +1000m

(5) Δ119862119871= minus01119862

119871 Δ119862119863= +01119862

119863 Δ119903ini = minus1000m

The simulation results of the optimal guidance algorithmare shown in Figures 4 5 6 7 and 8 The altitude velocityflight path angle angle of attack and dynamic pressure arerespectively shown in Figures 4 to 8 As expected the finalstate is achieved under aerodynamic bias and initial stateerror Figure 8 shows that the state inequality constraint issatisfied The dynamic pressure is less than 140000 Pa fordifferent cases

Table 2 shows the terminal error and the fuel cost ofthe optimal guidance algorithm The terminal accuracy ishigh for different cases Moreover the fuel cost of theoptimal guidance algorithm is lower than that of the trackingguidance algorithm A greater initial state error will result inmore fuel savings In summarywith lower fuel cost the resultsof the proposed optimal guidance algorithm satisfies all theequality and inequality constraints The simulation resultsshow the great potential for the final flight experimental test

7 Conclusion

In this paper we present an optimal guidance algorithmfor air-breathing launch vehicle based on optimal trajectorycorrection The proposed guidance algorithm corrects thereference trajectory in every guidance cycle to satisfy the opti-mality condition of the optimal feedback control problem

0 14012010080604020

18000

21000

24000

27000

30000

33000

Alti

tude

(m)

Time (s)

Case 1Case 2Case 3

Case 4Case 5

Figure 4 The altitude of the simulation results

1200

1400

1600

1800

2000

2200Ve

loci

ty (m

s)

Time (s)0 14012010080604020

Case 1Case 2Case 3

Case 4Case 5

Figure 5 The velocity of the simulation results

By linearizing the optimality condition the linear TPBVPis obtained for optimal trajectory correction The solutionof the linear TPBVP is derived using the Simpson rule Thenew trajectory is generated by a linear search for the stepsize of the solution Smooth Hamiltonian approximation isimplemented to the inequality constraints The sufficiencycondition for the global convergence of the guidance algo-rithm is given in this paper

Finally simulations for 5 different cases of modeling andinitial state error are presented Compared with the trackingguidance method the simulation results prove the low fuelcost and high precision of the proposed optimal guidancemethod

10 Mathematical Problems in Engineering

Table 2 The simulation results for optimal guidance algorithm

Case Δ119903119891(m) ΔV

119891(ms) Δ120574

119891(deg) Fuel cost (kg) Fuel cost (tracking method) (kg)

1 minus2820 0136 020 457219 4574622 859 minus003 minus015 458314 4628393 minus1508 007 020 458474 4605664 10816 047 minus023 460292 4776005 minus3243 019 minus016 460133 468894

0

2

4

6

8

Time (s)

Flig

ht p

ath

angl

e (D

EG)

minus2

minus4

0 14012010080604020

Case 1Case 2Case 3

Case 4Case 5

Figure 6 The flight path angle of the simulation results

Ang

le o

f atta

ck (D

EG)

0

1

2

3

4

minus1

minus2

minus3

minus4

minus5

minus6

Time (s)0 14012010080604020

Case 1Case 2Case 3

Case 4Case 5

Figure 7 The angle of attack of the simulation results

Acknowledgments

This work was supported in part by the National NatureScience Foundation of China nos 61203081 and 61174079Doctoral Fund of Ministry of Education of China

20000

40000

60000

80000

100000

120000

140000

Dyn

amic

pre

ssur

e (Pa

)

Time (s)0 14012010080604020

Case 1Case 2Case 3

Case 4Case 5

Figure 8 The dynamic pressure of the simulation results

no 20120142120091 andPrecisionmanufacturing technologyand equipment for metal parts no 2012DFG70640

References

[1] H J Henry Jr and C H McLellan ldquoAir-breathing launch vehi-cle for earth-orbit shuttle New technology and developmentapproachrdquo Journal of Aircraft vol 8 no 5 pp 381ndash387 1971

[2] J E CorbanA J Calise andGA Flandro ldquoReal-time guidancealgorithm for aerospace plane optimal ascent to low earth orbitrdquoin Proceedings of the American Control Conference pp 2475ndash2481 Pittsburgh PA USA June 1989

[3] C H Yam D D Lorenzo and D Izzo ldquoLow-thrust trajectorydesign as a constrained global optimization problemrdquo Journalof Aerospace Engineering G vol 225 no 11 pp 1243ndash1251 2011

[4] B A Conway ldquoA survey of methods available for the numericaloptimization of continuous dynamic systemsrdquo Journal of Opti-mization Theory and Applications vol 152 no 2 pp 271ndash3062012

[5] C H Chen and O L Mangasarian ldquoSmoothing methodsfor convex inequalities and linear complementarity problemsrdquoMathematical Programming vol 71 no 1 pp 51ndash69 1995

[6] X Chen ldquoSmoothing methods for nonsmooth nonconvexminimizationrdquo Mathematical Programming vol 134 no 1 pp71ndash99 2012

[7] W Wang Y Shang and Y Zhang ldquoA filled function approachfor nonsmooth constrained global optimizationrdquoMathematical

Mathematical Problems in Engineering 11

Problems in Engineering vol 2010 Article ID 310391 9 pages2010

[8] L I Rozonoer ldquoLinear-quadratic optimization and some gen-eral hypotheses on optimal controlrdquoMathematical Problems inEngineering vol 5 no 4 pp 275ndash289 1999

[9] J T Betts ldquoSurvey of numerical methods for trajectory opti-mizationrdquo Journal of Guidance Control and Dynamics vol 21no 2 pp 193ndash207 1998

[10] G Q Huang Y P Lu and Y Nan ldquoA survey of numericalalgorithms for trajectory optimization of flight vehiclesrdquo ScienceChina-Technological Sciences vol 55 no 9 pp 2538ndash2560 2012

[11] Y Ulybyshev ldquoDiscrete pseudocontrol sets for optimal controlproblemsrdquo Journal of Guidance Control and Dynamics vol 33no 4 pp 1133ndash1142 2010

[12] M Vasile and F Zuiani ldquoMulti-agent collaborative search anagent-based memetic multi-objective optimization algorithmapplied to space trajectory designrdquo Journal of Aerospace Engi-neering G vol 225 no 11 pp 1211ndash1227 2011

[13] M Pontani and B A Conway ldquoParticle swarm optimizationapplied to space trajectoriesrdquo Journal of Guidance Control andDynamics vol 33 no 5 pp 1429ndash1441 2010

[14] Y P Ulybyshev ldquoOptimization of low-thrust orbit transfers withconstraintsrdquo Cosmic Research vol 50 no 5 pp 376ndash390 2012

[15] M Chen R Mei and B Jiang ldquoSliding mode control for a classof uncertainMIMOnonlinear systemswith application to near-space vehiclesrdquoMathematical Problems in Engineering vol 2013Article ID 180589 9 pages 2013

[16] H Zhang Y Shi M Q Xu and H T Cui ldquoObserver-basedtracking controller design for networked predictive controlsystems with uncertain markov delaysrdquo in Proceedings of theAmerican Control Conference pp 5682ndash5687 2012

[17] H Zhang and Y Shi ldquoRobust H-infinity sliding-mode controlfor markovian jump systems subject to intermittent observa-tions and partially known transition probabilitiesrdquo Systems ampControl Letters vol 62 no 12 pp 1114ndash1124 2013

[18] Z Shuai H Zhang J Wang J Li and M Ouyang ldquoCombinedAFS andDYC control of four-wheel-independent-drive electricvehicles over CAN network with time-varying delaysrdquo IEEETransactions on Vehicular Technology no 99 p 1 2013

[19] H Zhang Y Shi and A Saadat Mehr ldquoRobust static outputfeedback control and remote PID design for networked motorsystemsrdquo IEEETransactions on Industrial Electronics vol 58 no12 pp 5396ndash5405 2011

[20] H Zhang Y Shi and M X Liu ldquoH-infinity step trackingcontrol for networked discrete-time nonlinear systems withintegral and predictive actionsrdquo IEEE Transactions on IndustrialInformatics vol 9 no 1 pp 337ndash345 2013

[21] X J Wang and S P Wang ldquoAdaptive fuzzy robust controlfor a class of nonlinear systems via small gain theoremrdquoMathematical Problems in Engineering vol 2013 Article ID201432 11 pages 2013

[22] V Brinda S Dasgupta and M Lal ldquoTrajectory optimizationand guidance of an air breathing hypersonic vehiclerdquo in Pro-ceedings of the 14th AIAAAHI International Space Planes andHypersonics Systems Technologies Conference vol 2 pp 856ndash869 November 2006

[23] B Tian and Q Zong ldquoOptimal guidance for reentry vehiclesbased on indirect legendre pseudospectral methodrdquoActa Astro-nautica vol 68 no 7-8 pp 1176ndash1184 2011

[24] M H Graszliglin J Telaar and UM Schottle ldquoAscent and reentryguidance concept based on NLP-methodsrdquo Acta Astronauticavol 55 no 3ndash9 pp 461ndash471 2004

[25] P Lu and B Pan ldquoHighly constrained optimal launch ascentguidancerdquo Journal of Guidance Control and Dynamics vol 33no 2 pp 404ndash414 2010

[26] P Lu H Sun and B Tsai ldquoClosed-loop endo-atmosphericascent guidancerdquo in Proceedings of the AIAA Guidance Navi-gation and Control Conference Monterey Calif USA 2002

[27] J Oscar Murillo and P Lu ldquoFast ascent trajectory optimizationfor hypersonic air-breathing vehiclesrdquo in Proceedings of theAIAA Guidance Navigation and Control Conference pp 5ndash8Toronto Canada 2010

[28] M Marrdonny and M Mobed ldquoA guidance algorithm forlaunch to equatorial orbitrdquo Aircraft Engineering and AerospaceTechnology vol 81 no 2 pp 137ndash148 2009

[29] T Yamamoto and J Kawaguchi ldquoA new real-time guidancestrategy for aerodynamic ascent flightrdquo Acta Astronautica vol61 no 11-12 pp 965ndash977 2007

[30] C Filici andR S Sanchez Pena ldquoOnline guidance updates usingneural networksrdquo Acta Astronautica vol 66 no 3-4 pp 477ndash485 2010

[31] P T Boggs A J Kearsley and J W Tolle ldquoA global convergenceanalysis of an algorithm for large-scale nonlinear optimizationproblemsrdquo SIAM Journal on Optimization vol 9 no 4 pp 833ndash862 1999

[32] S Schlenkrich and A Walther ldquoGlobal convergence of quasi-Newton methods based on adjoint Broyden updatesrdquo AppliedNumerical Mathematics vol 59 no 5 pp 1120ndash1136 2009

[33] S Schlenkrich A Griewank and A Walther ldquoOn the localconvergence of adjoint Broyden methodsrdquo Mathematical Pro-gramming vol 121 no 2 pp 221ndash247 2010

[34] S Lian and L Zhang ldquoA simple smooth exact penalty functionfor smooth optimization problemrdquo Journal of Systems Science ampComplexity vol 25 no 3 pp 521ndash528 2012

[35] S Keshmiri R Colgren and M Mirmirani ldquoDevelopment ofan aerodynamic database for a generic hypersonic air vehiclerdquoin Proceedings of the AIAA Guidance Navigation and ControlConference pp 3978ndash3998 San Francisco Calif USA August2005

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Page 4: Research Article Optimal Ascent Guidance for Air-Breathing ...downloads.hindawi.com/journals/mpe/2013/313197.pdfResearch Article Optimal Ascent Guidance for Air-Breathing Launch Vehicle

4 Mathematical Problems in Engineering

generate new reference trajectory [119909(119864) 119906(119864)]119879 and output

119906(119864) from the following optimal control problem

min 119869 = 120601 (119909 (119864119891)) + int

119864119891

119864119888119896+1

119892 (119909 119906) 119889119864

st = 119891 (119909 119906)

119909 (119864119888119896) = 119909119888

119896

119882 (119909 (119864119891)) = 0

119862 (119909 119906) le 0

(11)

where 119864119888119896is the normalized energy for current vehicle state

119909119888119896

3 Optimal Ascent Guidance Algorithm

In [25 26] the optimal control problem was solved on-boardin every guidance cycle However it is difficult to solve thenonlinear and nondifferentiable TPBVP on-board for air-breathing launch vehicles In this section we propose an opti-mal guidance algorithmThis guidance algorithmupdates thereference trajectory [119909 119906]

119879 in every guidance cycle to dealwith the unknowndisturbance For the current state119909119888

119896of the

119896th guidance cycle the optimal feedback control problem istransformed into a linear TPBVP problem using optimalitycondition linear approximation The searching direction ofthe correction values 119889

119896is obtained by solving the linear

TPBVP With the searching direction the new referencetrajectory is generated by linear search For the 119896th guidancecycle the trajectory update is performed as follows

[119909119896+1(119864) 119906119896+1

(119864)]119879

= [119909119896(119864) 119906119896(119864)]119879

+ 119897119896119889119879

119896(119864) (12)

The initial reference trajectory [1199090(119864) 119906

0(119864)]119879 is derived

from the off-board from trajectory optimization by directmethodThe flow chart of the algorithm is shown in Figure 2Figure 3 shows the time sequence of algorithm

31 Linear Approximation for the Optimality Condition Inthis section we linearize the optimality condition for opti-mality trajectory updating The optimal control problem (11)without inequality constraints is given as follows

min 119869 = 120601 (119909 (119864119891)) + int

119864119891

119864119888119896+1

119892 (119909 119906) 119889119864

st = 119891 (119909 119906)

119909 (119864119888119896) = 119909119888

119896

119882 (119909 (119864119891)) = 0

(13)

The Hamiltonian of (13) is given as follows

119867 = 119892 (119909 119906) + 120582119879119891 (119909 119906) (14)

Begin

Linear approximation for the

optimality condition the costate

equations the motion equations

Solving the linear TPBVP

Linear search for the step size

Updating the optimal trajectory

End

Figure 2 Flow chart of the optimal ascent guidance algorithm

where 120582 is the costate vector The state equations costateequations and optimality condition for the optimal controlproblem are given by

= 119891 (119909 119906)

= minus(120597119867

120597119909)

119879

120597119867

120597119906= 0

(15)

The initial and transversality conditions are given by

119909 (119864119888119896) = 119909119888

119896

119882 (119909 (119864119891)) = 0

120582 (119864119891) = [

120597120601

120597119909 (119864119891)

]

119879

+ [120603119879 120597119882

120597119909 (119864119891)

]

119879

(16)

We denote the trajectory correction variables as follows

[

[

Δ119909 (119864)

Δ119906 (119864)

Δ120582 (119864)

]

]

= [

[

119909 (119864) minus 119909119896(119864)

119906 (119864) minus 119906119896(119864)

120582 (119864) minus 120582119896(119864)

]

]

(17)

Mathematical Problems in Engineering 5

middot middot middot

Output theguidancecommanduk+1(E)

Output theguidancecommanduk+2(E)

middot middot middot

middot middot middot

The time consuming ofthe online algorithm The guidance cycle

Trajectoryupdating

beginsupdating

Trajectory

ends

Trajectoryupdating

beginsupdating

Trajectory

ends

Trajectoryupdating

beginsupdating

Trajectory

ends

End

Time (120591)kΔ120591 (k + 1)Δ120591 (k + 2)Δ120591

Figure 3 Time sequence of the optimal ascent guidance algorithm

We denote that

119867119896

119906119909=

1205972(119892 + 120582

119879119891)

120597119906120597119909(119909119896 119906119896 120582119896)

119867119896

119906= [

120597119867

120597119906(119909119896 119906119896 120582119896)]

119879

119867119896

119906119906=

1205972(119892 + 120582

119879119891)

1205971199062(119909119896 119906119896 120582119896)

119867119896

119909= [

120597119867

120597119909(119909119896 119906119896 120582119896)]

119879

119867119896

119909119909=

1205972(119892 + 120582

119879119891)

1205971199092(119909119896 119906119896 120582119896)

119891119896

119906=

120597119891

120597119906(119909119896 119906119896)

119867119896

119909119906=

1205972(119892 + 120582

119879119891)

120597119909120597119906(119909119896 119906119896 120582119896)

119891119896

119909=

120597119891

120597119909(119909119896 119906119896)

119892119896

119906= [

120597119892

120597119906(119909119896 119906119896)]

119879

119892119896

119909= [

120597119892

120597119909(119909119896 119906119896)]

119879

(18)

The first-order Taylor expansion of (15) on the old referencetrajectory [119909

119896119906119896

120582119896]119879 is

[120597119867

120597119906]

119879

asymp 119867119896

119906+ 119867119896

119906119909Δ119909 + 119867

119896

119906119906Δ119906 + (119891

119896

119906)119879

Δ120582

asymp minus119867119896

119909minus 119867119896

119909119909Δ119909 minus 119867

119896

119909119906(119909119896 119906119896 120582119896) Δ119906 minus (119891

119896

119909)119879

Δ120582

asymp 119891 (119909119896 119906119896) + 119891119896

119909Δ119909 + 119891

119896

119906Δ119906

(19)

The following equations are established

119867119896

119906+ (119891119896

119906)119879

Δ120582 = 119892119896

119906+ (119891119896

119906)119879

120582

119867119896

119909+ (119891119896

119909)119879

Δ120582 = 119892119896

119909+ (119891119896

119909)119879

120582

(20)

Substituting (20) into (19) yields

= minus119892119896

119909minus 119867119896

119909119909Δ119909 minus 119867

119896

119909119906(119909119896 119906119896 120582119896) Δ119906 minus (119891

119896

119909)119879

120582

Δ = 119891119896

119909Δ119909 + 119891

119896

119906Δ119906 minus

119896+ 119891 (119909

119896 119906119896)

119892119896

119906+ 119867119896

119906119909Δ119909 + 119867

119896

119906119906Δ119906 + (119891

119896

119906)119879

120582 = 0

Δ119909 (119864119888119896) + Δ119909

119896(119864119888119896) = 119909119888

119896

120597119882

120597119909 (119864119891)

Δ119909 (119864119891) +119882(119909

119896(119864119891)) = 0

120582 (119864119891) = [

120597120601

120597119909 (119864119891)

]

119879

+ [120603119879 120597119882

120597119909 (119864119891)

]

119879

(21)

This equation is a linear TPBVP about the variables[Δ119909 Δ119906 120582]

119879 The linear TPBVP is solved using the Simpsonrules in Section 6

32 Backtracking Line Search for the Step Size In this sectionthe step size 119897

119896is determined by backtracking line search

considering the solution of the linear TPBVP as the searchdirection The penalty function for problem (13) is given by

119901119896= 119869 (119909 119906) + 120578

10038171003817100381710038171003817119882(119909119891)10038171003817100381710038171003817+ int

119864119891

119864119888119896

1003817100381710038171003817119891 (119909 119906) minus 1003817100381710038171003817

+1003817100381710038171003817119909 (119864119888119896) minus 119909119888

119896

1003817100381710038171003817

(22)

6 Mathematical Problems in Engineering

where 120578 is the penalty parameter satisfied 120578 gt max |120582| Wedenote

119910 = [119909119896

119906119896

] 119889119896= [

Δ119909

Δ119906] (23)

where [Δ119909 Δ119906 120582]119879 is the solution of the linear TPBVP (21)

The minimizer of penalty function 119901119896(119909 119906) corresponds

to the solution of problem (11) Backtracking line search wasused for large-scale nonlinear optimization problems in [31]The step size 119897

119896is determined by backtracking line search that

satisfies

119897119896= max 119897 isin 119878 119901

119896+1(119910119896+ 119897119889119896)

le 119901119896+1

(119910119896) + 1198971205761120575119901119896+1

(119910119896 119889119896)

119878 = 119897 119897 = 1205762

119898 119898 = 0 1 2

(24)

where 1205761

isin (0 1) 1205762

isin (0 1) and 120575119901119896+1

(119910119896 119889119896) is the

variational of the functional 119901119896+1

(119910) caused by variational119889119896of 119910 on 119910

119896 Considering 119889

119896as the solution of (21)

120575119901119896+1

(119910119896 119889119896) le 0 is established The new state and guidance

command of the reference trajectory are generated as follows

[119909119896+1

119906119896+1

] = [119909119896

119906119896

] + 119897119896[Δ119909

Δ119906] (25)

4 The Global Convergence of the Algorithm

In [32 33] the convergence of the off-board algorithm wasanalyzed But the relationship between the global conver-gence and the guidance cycle 120591 has not been discussed for on-board optimization in the previous literatures In this sectionwe state and prove the global convergence of the on-boardalgorithm proposed in the preceding section The solutionsequence [119909

119896119906119896]119879 generated by the algorithm will converge

to the solution of the feedback optimal control problemUnlike off-board trajectory optimization the initial conditionof the feedback optimal control problem depends on thecurrent state of the vehicle in real time For the 119896th guidancecycle the optimal feedback control problem is described as(13)Thepenalty function119901

119896for this feedback optimal control

problem is given by (22) For the subsequent guidance cyclethe optimal feedback guidance problem is given by

min 119869 = 119882(119909 (119864119891)) + int

119864119891

119864119888119896+1

119892 (119909 119906) 119889119864

st = 119891 (119909 119906)

119909 (119864119888119896+1

) = 119909119888119896+1

119882 (119909 (119864119891)) = 0

(26)

where

119864119888119896+1

= 119864119888119896+ int

120591119896+Δ120591

120591119896

119889119864

119889120591119889120591 (27)

Δ120591 is the guidance cycle and 119909119888119896+1

is the new state of vehiclefrom navigation system at the time 120591

119896+ Δ120591 The penalty

function for optimal problem (24) is given by

119901119896+1

(119909 119906) = 119869 (119909 119906) + 12057810038171003817100381710038171003817119882(119909119896(119864119891))10038171003817100381710038171003817

+ 120578int

119864119891

119864119888119896+1

1003817100381710038171003817119891 (119909 119906) minus 1003817100381710038171003817 119889119864

+ 1205781003817100381710038171003817119909 (119864119888119896+1) minus 119909119888

119896+1

1003817100381710038171003817

(28)

We define the entire penalty function as follows

119901119896(119909 119906) = 119901

119896(119909 119906)

+

119896minus1

sum

119894=0

int

119864119888119894+1

119864119888119894

[1205781003817100381710038171003817119891 (119909119894 119906119894) minus 119894

1003817100381710038171003817 + 119892 (119909119894 119906119894)] 119889119864

(29)

The entire penalty function 119901119896(119909 119906) includes the penalty

function on the whole integration interval [1198640 119864119891]

Theorem 1 If the minimizer of 119901119896(119909119896 119906119896) is exited 119892(119909 119906) ge

0 and Δ120591 is selected and satisfies

120578100381710038171003817100381710038171003817119909119888119896+1

minus 119909119888119896minus int119864119888119896+1

119864119888119896119896119889119864

100381710038171003817100381710038171003817

minus120575119901119896+1

(119910119896 119889119896)

lt 1198971198961205763 (30)

where 1205763isin (0 120576

1) The solution sequence [119909

119896119906119896]119879 generated

from (24) will converge to the solution of the feedback optimalcontrol problem

Proof From (22) (28) (29) and

119909119896(119864119888119896+1

) = 119909119896(119864119888119896) + int

119864119888119896+1

119864119888119896

119896119889119864 (31)

we can obtain that

119901119896+1

(119909119896 119906119896) minus 119901119896(119909119896 119906119896)

= 120578

100381710038171003817100381710038171003817100381710038171003817

119909119896(119864119888119896) minus 119909119888

119896

minus(119909119888119896+1

minus 119909119888119896minus int

119864119888119896+1

119864119888119896

119896119889119864)

100381710038171003817100381710038171003817100381710038171003817

minus 1205781003817100381710038171003817119909119896 (119864119888119896) minus 119909119888

119896

1003817100381710038171003817

(32)

Mathematical Problems in Engineering 7

From (32) we obtain

limΔ120591rarr0

[119901119896+1

(119909119896 119906119896) minus 119901119896(119909119896 119906119896)] = 0

lim119891(119909119896119906119896)minus 119909119896rarr0

[119901119896+1

(119909119896 119906119896) minus 119901119896(119909119896 119906119896)] = 0

(33)

119901119896+1

(119909119896 119906119896) minus 119901119896(119909119896 119906119896)

= 120578

100381710038171003817100381710038171003817100381710038171003817

119909119896(119864119888119896) minus 119909119888

119896+1+ int

119864119888119896+1

119864119888119896

119896119889119864

100381710038171003817100381710038171003817100381710038171003817

minus1003817100381710038171003817119909119896 (119864119888119896) minus 119909119888

119896

1003817100381710038171003817

le 1205781003817100381710038171003817119909119896 (119864119888119896) minus 119909119888

119896

1003817100381710038171003817

+

100381710038171003817100381710038171003817100381710038171003817

119909119888119896+1

minus 119909119888119896minus int

119864119888119896+1

119864119888119896

119896119889119864

100381710038171003817100381710038171003817100381710038171003817

minus1003817100381710038171003817119909119896 (119864119888119896) minus 119909119888

119896

1003817100381710038171003817

= 120578

100381710038171003817100381710038171003817100381710038171003817

119909119888119896+1

minus 119909119888119896minus int

119864119888119896+1

119864119888119896

119896119889119864

100381710038171003817100381710038171003817100381710038171003817

(34)

From conditions (24) and (30) it is obtained that

119901119896+1

(119909119896+1

119906119896+1

)

le 119901119896+1

(119909119896 119906119896) + 1198971198961205761120575119901119896+1

(119889119896)

= 119901119896(119909119896 119906119896) + 1198971198961205761120575119901119896+1

(119889119896)

+ 120578

100381710038171003817100381710038171003817100381710038171003817

119909119896(119864119888119896) minus 119909119888

119896

minus (119909119888119896+1

minus 119909119888119896minus int

119864119888119896+1

119864119888119896

119896119889119864)

100381710038171003817100381710038171003817100381710038171003817

minus1003817100381710038171003817119909119896 (119864119888119896) minus 119909119888

119896

1003817100381710038171003817

le 119901119896(119909119896 119906119896) + 120578

100381710038171003817100381710038171003817100381710038171003817

119909119888119896+1

minus 119909119888119896minus int

119864119888119896+1

119864119888119896

119896119889119864

100381710038171003817100381710038171003817100381710038171003817

+ 1198971198961205761120575119901119896+1

(119910119896 119889119896)

le 119901119896(119909119896 119906119896) + 120578

100381710038171003817100381710038171003817100381710038171003817

119909119888119896+1

minus 119909119888119896minus int

119864119888119896+1

119864119888119896

119896119889119864

100381710038171003817100381710038171003817100381710038171003817

+ 1198971198961205763120575119901119896+1

(119910119896 119889119896)

lt 119901119896(119909119896 119906119896)

(35)

Thus the sequence 119901119896(119909119896 119906119896) 119896 = 1 2 3 is monotoni-

cally decreasing Considering the existence of a minimizer119901119896(119909119896 119906119896) converges Thus the following equation is estab-

lished

lim119896rarrinfin

(1198971198961205763120575119901119896+1

(119910119896 119889119896) minus 1198971198961205761120575119901119896+1

(119910119896 119889119896)) = 0 (36)

From (36) we derive

lim119896rarrinfin

(119897119896120575119901119896+1

(119910119896 119889119896)) = 0 (37)

If 119889119896

= 0 [119909119896

119906119896]119879 is not the minimizer of penalty function

then 120575119901119896+1

(119910119896 119889119896) = 0 Considering that 119897

119896is bounded away

from zero we derive

lim119896rarrinfin

119889119896= 0

lim119896rarrinfin

120575119901119896+1

(119910119896 119889119896) = 0

lim119896rarrinfin

[119896minus 119891 (119909

119896 119906119896)] = 0

lim119896rarrinfin

(119896+ 119867119896

119909) = 0

lim119896rarrinfin

119867119896

119906= 0

(38)

Thus the solution sequence [119909119896

119906119896]119879 generated by the

algorithmwill converge to the solution of the optimal controlproblem

5 Smooth Approximation forInequality Constraints

The nondifferentiable TPBVP from optimal control problem(11) with the inequality constraints brings difficulties for theproposed algorithm which can only solve a smooth problemSmooth approximations for nondifferentiable optimizationproblems have been studied in [5 6 34] In this sectionwe introduce the smooth approximation method for theinequality constraints If the inequality constraint function isabout the state and the guidance command

119862 (119909 119906) le 0 (39)

the Hamiltonian for the optimal control problem (11) withinequality constraints (39) is given by

119867 = 119892 (119909 119906) + 120582119879119891 (119909 119906) + 120583max 119862 (119909 119906) 0 (40)

However the Hamiltonian (40) is a nondifferentiable func-tion which brings difficulty for the proposed algorithmFor the smooth Hamiltonian approximation a differentiablefunction is substituted for max119862(119909 119906) 0 We denote thenondifferentiable function as follows

max 0 119910 = int

119910

minusinfin

119908 (119911) 119889119911 (41)

where 119908(119911) is the step function

119908 (119911) = 1 if 119911 ge 0

0 if 119911 lt 0(42)

Considering 119904(119911 119886) as the approximation function of 119908(119911)the differentiable approximation to max0 119910 is obtained as

119891119901(119910 119886) asymp int

119910

minusinfin

119904 (119911 119886) 119889119911 = 119910 +1

119886In (1 + 119890

minus119886119910) (43)

8 Mathematical Problems in Engineering

where 119904(119911 119886) is Sigmoid function

119904 (119911 119886) =1

1 + 119890minus119886119911 119886 gt 0 (44)

The Properties of 119891119901(119910 119886) 119886 gt 0 are as follows [5]

(1) 119891119901(119910 119886) is 119899-times continuously differentiable for any

positive integer 119899 with 120597119891119901(119910 119886)120597119910 = 1(1 + 119890

minus119886119910)

and 1205972119891119901(119910 119886)120597119910

2= 119886119890minus119886119910

(1 + 119890minus119886119910

)2

(2) 119891119901(119910 119886) is strictly convex and strictly increasing onR

(3) 119891119901(119910 119886) gt max0 119910

(4) lim119886rarrinfin

119891119901(119910 119886) = max0 119910

(5) 119891119901(119910 119886) gt 119891

119901(119910 1198861) for 119886 lt 119886

1 119910 isin R

Substituting (43) into (40) the following differentiableHamiltonian approximation is obtained

= 119892 (119909 119906) + 120582119879119891 (119909 119906) + 120583 [119862 +

1

119886In (1 + 119890

minus119886119862)] (45)

If the inequality constraint function is a 119902th-order statevariable inequality constraint given by

119862 (119909) le 0 (46)

If 119901 times derivatives of 119862(119909) are required before 119906 appearsexplicitly in the result the Hamiltonian with inequalityconstraints (46) is given by

119867 = 119892 (119909 119906) + 120582119879119891 (119909 119906) + 120583119862

(119902) (47)

where

120583 = 0 if 119862 lt 0 (48)

Considering the differentiable approximation to 120583 as follows

120583 = 120583119908 (119862) asymp 1205831

1 + 119890minus119886119862(49)

the following differentiable Hamiltonian approximation isobtained

= 119892 (119909 119906) + 120582119879119891 (119909 119906) + 120583

1

1 + 119890minus119886119862119862(119902) (50)

With the differentiable Hamiltonian approximation theinequality constraints are considered in the proposed guid-ance algorithm

6 Simulation Results

61 Numerical Calculation Based on Linear TPBVPDiscretiza-tion In this section the linear TPVBP is transformed intolinear equations using the Simpson rule which correspondsto the three-point Newton-Cotes quadrature rule The Simp-son rule for differential equation 119889119883119889119864 = 119910(119864) is given by

int

119887

119886

119910 (119864) 119889119864 =119887 minus 119886

6[119910 (119886) + 4119910(

119886 + 119887

2) + 119910 (119887)] (51)

From (51) we obtain

119883(119887) minus 119883 (119886) =119887 minus 119886

6[119910 (119886) + 4119910(

119886 + 119887

2) + 119910 (119887)] (52)

We disperse the normalized energy 119864 into119873 equal segmentsas [1198640

1198641

sdot sdot sdot 119864119873] with 119864

0= 119864119888119896and 119864

119873= 119864119891 The

differential equation 119889119883119889119864 = 119910(119883 119864) can be transformedinto the following equation

119883(119864119894) minus 119883 (119864

119894minus1)

=119864119894minus 119864119894minus1

6[119910 (119883

119894 119864119894)

+ 4119910 (119883119894+ 119883119894minus1

2119864119894+ 119864119894minus1

2)

+ 119910 (119883119894minus1

119864119894minus1

)] 119894 = 1 2 3 119873

(53)

Based on (53) the linear TPBVP (21) can be transformed intolinear equations described by

119860119883 = 119861 (54)

The linear equations can be solved by Gaussian eliminationFinally we can obtain the solution Δ119909(119864) Δ119906(119864) and

120582(119864) of the linear TPBVP (21) from the solution of the linearequations through interpolating

62 GHV Model The proposed method is applied to theGenericHypersonicVehicle (GHV) to verify the effectivenessby flight simulation In order to develop a complete setof aerodynamic coefficients experimental longitudinal andlateral-directional aerodynamics were obtained for the GHVby using six Langley wind tunnels [35] The aerodynamicdatabase of the GHV is shown in [35] The GHV model withscramjet propulsion system is used for the simulation where119878ref = 33473m2 and the full-scale weight119898ini = 110000 kg

According to the characterization of scramjet propulsionsystem described in [2 27] the development of the propul-sion system begins with the following equation

119879 = 05120601119868sp (120601119872119886) 120588V1198920119862119879 (120572119872119886) (55)

where 120601 is the throttle commandwhich varies from 0 to 1Thecoefficient 119862

119879depends on 120572 and119872

119886as follows

119862119879= minus00012119872

1198861205722+ 0008120572 + 14 minus 01119872

119886 (56)

The impulse 119868sp is given by the following equation

119868sp = minus811198722

119886+ 1004119872

119886minus 52 + 100120601 (57)

63 Simulation Results for Modeling and Initial State ErrorConsidering the high cost of the flight experimental testresearchers do flight numerical simulation hardware-in-the-loop simulation and final flight experimental test to verifythe guidance algorithm step by step In this section flight

Mathematical Problems in Engineering 9

Table 1 Initial and required final states of the vehicle

119903ini(m)

Vini(ms)

120574ini(deg)

119903119903119891

(m)V119903119891

(ms)120574119903119891

(deg)20000 1000 0 35000 2000 0

numerical simulation results for the GHV with modelingerror and initial state error are given to show the effectivenessof the proposed guidance algorithm For the modeling errorcaused by the wind tunnel experimental test the aerody-namic coefficients bias is considered in the flight simulationThe simulations end once the final normalized energy 119864

119891

is reached The optimal guidance algorithm updates thetrajectory every guidance cycle which is 03 s The initial andrequired final states of the air-breathing vehicle are shownin Table 1 The number of the discrete nodes 119873 = 10 Thedynamic pressure inequality constraint is given by

119876 =1

2120588V2 le 140000Pa (58)

To assess the capability of the proposed algorithm to dealwith disturbance the aerodynamic coefficients bias and theinitial state error in the simulations are given by followingcases

(1) Δ119862119871= minus01119862

119871 Δ119862119863= +01119862

119863

(2) Δ119862119871= minus01119862

119871 Δ119862119863= +01119862

119863 Δ119903ini = +500m

(3) Δ119862119871= minus01119862

119871 Δ119862119863= +01119862

119863 Δ119903ini = minus500m

(4) Δ119862119871= minus01119862

119871 Δ119862119863= +01119862

119863 Δ119903ini = +1000m

(5) Δ119862119871= minus01119862

119871 Δ119862119863= +01119862

119863 Δ119903ini = minus1000m

The simulation results of the optimal guidance algorithmare shown in Figures 4 5 6 7 and 8 The altitude velocityflight path angle angle of attack and dynamic pressure arerespectively shown in Figures 4 to 8 As expected the finalstate is achieved under aerodynamic bias and initial stateerror Figure 8 shows that the state inequality constraint issatisfied The dynamic pressure is less than 140000 Pa fordifferent cases

Table 2 shows the terminal error and the fuel cost ofthe optimal guidance algorithm The terminal accuracy ishigh for different cases Moreover the fuel cost of theoptimal guidance algorithm is lower than that of the trackingguidance algorithm A greater initial state error will result inmore fuel savings In summarywith lower fuel cost the resultsof the proposed optimal guidance algorithm satisfies all theequality and inequality constraints The simulation resultsshow the great potential for the final flight experimental test

7 Conclusion

In this paper we present an optimal guidance algorithmfor air-breathing launch vehicle based on optimal trajectorycorrection The proposed guidance algorithm corrects thereference trajectory in every guidance cycle to satisfy the opti-mality condition of the optimal feedback control problem

0 14012010080604020

18000

21000

24000

27000

30000

33000

Alti

tude

(m)

Time (s)

Case 1Case 2Case 3

Case 4Case 5

Figure 4 The altitude of the simulation results

1200

1400

1600

1800

2000

2200Ve

loci

ty (m

s)

Time (s)0 14012010080604020

Case 1Case 2Case 3

Case 4Case 5

Figure 5 The velocity of the simulation results

By linearizing the optimality condition the linear TPBVPis obtained for optimal trajectory correction The solutionof the linear TPBVP is derived using the Simpson rule Thenew trajectory is generated by a linear search for the stepsize of the solution Smooth Hamiltonian approximation isimplemented to the inequality constraints The sufficiencycondition for the global convergence of the guidance algo-rithm is given in this paper

Finally simulations for 5 different cases of modeling andinitial state error are presented Compared with the trackingguidance method the simulation results prove the low fuelcost and high precision of the proposed optimal guidancemethod

10 Mathematical Problems in Engineering

Table 2 The simulation results for optimal guidance algorithm

Case Δ119903119891(m) ΔV

119891(ms) Δ120574

119891(deg) Fuel cost (kg) Fuel cost (tracking method) (kg)

1 minus2820 0136 020 457219 4574622 859 minus003 minus015 458314 4628393 minus1508 007 020 458474 4605664 10816 047 minus023 460292 4776005 minus3243 019 minus016 460133 468894

0

2

4

6

8

Time (s)

Flig

ht p

ath

angl

e (D

EG)

minus2

minus4

0 14012010080604020

Case 1Case 2Case 3

Case 4Case 5

Figure 6 The flight path angle of the simulation results

Ang

le o

f atta

ck (D

EG)

0

1

2

3

4

minus1

minus2

minus3

minus4

minus5

minus6

Time (s)0 14012010080604020

Case 1Case 2Case 3

Case 4Case 5

Figure 7 The angle of attack of the simulation results

Acknowledgments

This work was supported in part by the National NatureScience Foundation of China nos 61203081 and 61174079Doctoral Fund of Ministry of Education of China

20000

40000

60000

80000

100000

120000

140000

Dyn

amic

pre

ssur

e (Pa

)

Time (s)0 14012010080604020

Case 1Case 2Case 3

Case 4Case 5

Figure 8 The dynamic pressure of the simulation results

no 20120142120091 andPrecisionmanufacturing technologyand equipment for metal parts no 2012DFG70640

References

[1] H J Henry Jr and C H McLellan ldquoAir-breathing launch vehi-cle for earth-orbit shuttle New technology and developmentapproachrdquo Journal of Aircraft vol 8 no 5 pp 381ndash387 1971

[2] J E CorbanA J Calise andGA Flandro ldquoReal-time guidancealgorithm for aerospace plane optimal ascent to low earth orbitrdquoin Proceedings of the American Control Conference pp 2475ndash2481 Pittsburgh PA USA June 1989

[3] C H Yam D D Lorenzo and D Izzo ldquoLow-thrust trajectorydesign as a constrained global optimization problemrdquo Journalof Aerospace Engineering G vol 225 no 11 pp 1243ndash1251 2011

[4] B A Conway ldquoA survey of methods available for the numericaloptimization of continuous dynamic systemsrdquo Journal of Opti-mization Theory and Applications vol 152 no 2 pp 271ndash3062012

[5] C H Chen and O L Mangasarian ldquoSmoothing methodsfor convex inequalities and linear complementarity problemsrdquoMathematical Programming vol 71 no 1 pp 51ndash69 1995

[6] X Chen ldquoSmoothing methods for nonsmooth nonconvexminimizationrdquo Mathematical Programming vol 134 no 1 pp71ndash99 2012

[7] W Wang Y Shang and Y Zhang ldquoA filled function approachfor nonsmooth constrained global optimizationrdquoMathematical

Mathematical Problems in Engineering 11

Problems in Engineering vol 2010 Article ID 310391 9 pages2010

[8] L I Rozonoer ldquoLinear-quadratic optimization and some gen-eral hypotheses on optimal controlrdquoMathematical Problems inEngineering vol 5 no 4 pp 275ndash289 1999

[9] J T Betts ldquoSurvey of numerical methods for trajectory opti-mizationrdquo Journal of Guidance Control and Dynamics vol 21no 2 pp 193ndash207 1998

[10] G Q Huang Y P Lu and Y Nan ldquoA survey of numericalalgorithms for trajectory optimization of flight vehiclesrdquo ScienceChina-Technological Sciences vol 55 no 9 pp 2538ndash2560 2012

[11] Y Ulybyshev ldquoDiscrete pseudocontrol sets for optimal controlproblemsrdquo Journal of Guidance Control and Dynamics vol 33no 4 pp 1133ndash1142 2010

[12] M Vasile and F Zuiani ldquoMulti-agent collaborative search anagent-based memetic multi-objective optimization algorithmapplied to space trajectory designrdquo Journal of Aerospace Engi-neering G vol 225 no 11 pp 1211ndash1227 2011

[13] M Pontani and B A Conway ldquoParticle swarm optimizationapplied to space trajectoriesrdquo Journal of Guidance Control andDynamics vol 33 no 5 pp 1429ndash1441 2010

[14] Y P Ulybyshev ldquoOptimization of low-thrust orbit transfers withconstraintsrdquo Cosmic Research vol 50 no 5 pp 376ndash390 2012

[15] M Chen R Mei and B Jiang ldquoSliding mode control for a classof uncertainMIMOnonlinear systemswith application to near-space vehiclesrdquoMathematical Problems in Engineering vol 2013Article ID 180589 9 pages 2013

[16] H Zhang Y Shi M Q Xu and H T Cui ldquoObserver-basedtracking controller design for networked predictive controlsystems with uncertain markov delaysrdquo in Proceedings of theAmerican Control Conference pp 5682ndash5687 2012

[17] H Zhang and Y Shi ldquoRobust H-infinity sliding-mode controlfor markovian jump systems subject to intermittent observa-tions and partially known transition probabilitiesrdquo Systems ampControl Letters vol 62 no 12 pp 1114ndash1124 2013

[18] Z Shuai H Zhang J Wang J Li and M Ouyang ldquoCombinedAFS andDYC control of four-wheel-independent-drive electricvehicles over CAN network with time-varying delaysrdquo IEEETransactions on Vehicular Technology no 99 p 1 2013

[19] H Zhang Y Shi and A Saadat Mehr ldquoRobust static outputfeedback control and remote PID design for networked motorsystemsrdquo IEEETransactions on Industrial Electronics vol 58 no12 pp 5396ndash5405 2011

[20] H Zhang Y Shi and M X Liu ldquoH-infinity step trackingcontrol for networked discrete-time nonlinear systems withintegral and predictive actionsrdquo IEEE Transactions on IndustrialInformatics vol 9 no 1 pp 337ndash345 2013

[21] X J Wang and S P Wang ldquoAdaptive fuzzy robust controlfor a class of nonlinear systems via small gain theoremrdquoMathematical Problems in Engineering vol 2013 Article ID201432 11 pages 2013

[22] V Brinda S Dasgupta and M Lal ldquoTrajectory optimizationand guidance of an air breathing hypersonic vehiclerdquo in Pro-ceedings of the 14th AIAAAHI International Space Planes andHypersonics Systems Technologies Conference vol 2 pp 856ndash869 November 2006

[23] B Tian and Q Zong ldquoOptimal guidance for reentry vehiclesbased on indirect legendre pseudospectral methodrdquoActa Astro-nautica vol 68 no 7-8 pp 1176ndash1184 2011

[24] M H Graszliglin J Telaar and UM Schottle ldquoAscent and reentryguidance concept based on NLP-methodsrdquo Acta Astronauticavol 55 no 3ndash9 pp 461ndash471 2004

[25] P Lu and B Pan ldquoHighly constrained optimal launch ascentguidancerdquo Journal of Guidance Control and Dynamics vol 33no 2 pp 404ndash414 2010

[26] P Lu H Sun and B Tsai ldquoClosed-loop endo-atmosphericascent guidancerdquo in Proceedings of the AIAA Guidance Navi-gation and Control Conference Monterey Calif USA 2002

[27] J Oscar Murillo and P Lu ldquoFast ascent trajectory optimizationfor hypersonic air-breathing vehiclesrdquo in Proceedings of theAIAA Guidance Navigation and Control Conference pp 5ndash8Toronto Canada 2010

[28] M Marrdonny and M Mobed ldquoA guidance algorithm forlaunch to equatorial orbitrdquo Aircraft Engineering and AerospaceTechnology vol 81 no 2 pp 137ndash148 2009

[29] T Yamamoto and J Kawaguchi ldquoA new real-time guidancestrategy for aerodynamic ascent flightrdquo Acta Astronautica vol61 no 11-12 pp 965ndash977 2007

[30] C Filici andR S Sanchez Pena ldquoOnline guidance updates usingneural networksrdquo Acta Astronautica vol 66 no 3-4 pp 477ndash485 2010

[31] P T Boggs A J Kearsley and J W Tolle ldquoA global convergenceanalysis of an algorithm for large-scale nonlinear optimizationproblemsrdquo SIAM Journal on Optimization vol 9 no 4 pp 833ndash862 1999

[32] S Schlenkrich and A Walther ldquoGlobal convergence of quasi-Newton methods based on adjoint Broyden updatesrdquo AppliedNumerical Mathematics vol 59 no 5 pp 1120ndash1136 2009

[33] S Schlenkrich A Griewank and A Walther ldquoOn the localconvergence of adjoint Broyden methodsrdquo Mathematical Pro-gramming vol 121 no 2 pp 221ndash247 2010

[34] S Lian and L Zhang ldquoA simple smooth exact penalty functionfor smooth optimization problemrdquo Journal of Systems Science ampComplexity vol 25 no 3 pp 521ndash528 2012

[35] S Keshmiri R Colgren and M Mirmirani ldquoDevelopment ofan aerodynamic database for a generic hypersonic air vehiclerdquoin Proceedings of the AIAA Guidance Navigation and ControlConference pp 3978ndash3998 San Francisco Calif USA August2005

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

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Page 5: Research Article Optimal Ascent Guidance for Air-Breathing ...downloads.hindawi.com/journals/mpe/2013/313197.pdfResearch Article Optimal Ascent Guidance for Air-Breathing Launch Vehicle

Mathematical Problems in Engineering 5

middot middot middot

Output theguidancecommanduk+1(E)

Output theguidancecommanduk+2(E)

middot middot middot

middot middot middot

The time consuming ofthe online algorithm The guidance cycle

Trajectoryupdating

beginsupdating

Trajectory

ends

Trajectoryupdating

beginsupdating

Trajectory

ends

Trajectoryupdating

beginsupdating

Trajectory

ends

End

Time (120591)kΔ120591 (k + 1)Δ120591 (k + 2)Δ120591

Figure 3 Time sequence of the optimal ascent guidance algorithm

We denote that

119867119896

119906119909=

1205972(119892 + 120582

119879119891)

120597119906120597119909(119909119896 119906119896 120582119896)

119867119896

119906= [

120597119867

120597119906(119909119896 119906119896 120582119896)]

119879

119867119896

119906119906=

1205972(119892 + 120582

119879119891)

1205971199062(119909119896 119906119896 120582119896)

119867119896

119909= [

120597119867

120597119909(119909119896 119906119896 120582119896)]

119879

119867119896

119909119909=

1205972(119892 + 120582

119879119891)

1205971199092(119909119896 119906119896 120582119896)

119891119896

119906=

120597119891

120597119906(119909119896 119906119896)

119867119896

119909119906=

1205972(119892 + 120582

119879119891)

120597119909120597119906(119909119896 119906119896 120582119896)

119891119896

119909=

120597119891

120597119909(119909119896 119906119896)

119892119896

119906= [

120597119892

120597119906(119909119896 119906119896)]

119879

119892119896

119909= [

120597119892

120597119909(119909119896 119906119896)]

119879

(18)

The first-order Taylor expansion of (15) on the old referencetrajectory [119909

119896119906119896

120582119896]119879 is

[120597119867

120597119906]

119879

asymp 119867119896

119906+ 119867119896

119906119909Δ119909 + 119867

119896

119906119906Δ119906 + (119891

119896

119906)119879

Δ120582

asymp minus119867119896

119909minus 119867119896

119909119909Δ119909 minus 119867

119896

119909119906(119909119896 119906119896 120582119896) Δ119906 minus (119891

119896

119909)119879

Δ120582

asymp 119891 (119909119896 119906119896) + 119891119896

119909Δ119909 + 119891

119896

119906Δ119906

(19)

The following equations are established

119867119896

119906+ (119891119896

119906)119879

Δ120582 = 119892119896

119906+ (119891119896

119906)119879

120582

119867119896

119909+ (119891119896

119909)119879

Δ120582 = 119892119896

119909+ (119891119896

119909)119879

120582

(20)

Substituting (20) into (19) yields

= minus119892119896

119909minus 119867119896

119909119909Δ119909 minus 119867

119896

119909119906(119909119896 119906119896 120582119896) Δ119906 minus (119891

119896

119909)119879

120582

Δ = 119891119896

119909Δ119909 + 119891

119896

119906Δ119906 minus

119896+ 119891 (119909

119896 119906119896)

119892119896

119906+ 119867119896

119906119909Δ119909 + 119867

119896

119906119906Δ119906 + (119891

119896

119906)119879

120582 = 0

Δ119909 (119864119888119896) + Δ119909

119896(119864119888119896) = 119909119888

119896

120597119882

120597119909 (119864119891)

Δ119909 (119864119891) +119882(119909

119896(119864119891)) = 0

120582 (119864119891) = [

120597120601

120597119909 (119864119891)

]

119879

+ [120603119879 120597119882

120597119909 (119864119891)

]

119879

(21)

This equation is a linear TPBVP about the variables[Δ119909 Δ119906 120582]

119879 The linear TPBVP is solved using the Simpsonrules in Section 6

32 Backtracking Line Search for the Step Size In this sectionthe step size 119897

119896is determined by backtracking line search

considering the solution of the linear TPBVP as the searchdirection The penalty function for problem (13) is given by

119901119896= 119869 (119909 119906) + 120578

10038171003817100381710038171003817119882(119909119891)10038171003817100381710038171003817+ int

119864119891

119864119888119896

1003817100381710038171003817119891 (119909 119906) minus 1003817100381710038171003817

+1003817100381710038171003817119909 (119864119888119896) minus 119909119888

119896

1003817100381710038171003817

(22)

6 Mathematical Problems in Engineering

where 120578 is the penalty parameter satisfied 120578 gt max |120582| Wedenote

119910 = [119909119896

119906119896

] 119889119896= [

Δ119909

Δ119906] (23)

where [Δ119909 Δ119906 120582]119879 is the solution of the linear TPBVP (21)

The minimizer of penalty function 119901119896(119909 119906) corresponds

to the solution of problem (11) Backtracking line search wasused for large-scale nonlinear optimization problems in [31]The step size 119897

119896is determined by backtracking line search that

satisfies

119897119896= max 119897 isin 119878 119901

119896+1(119910119896+ 119897119889119896)

le 119901119896+1

(119910119896) + 1198971205761120575119901119896+1

(119910119896 119889119896)

119878 = 119897 119897 = 1205762

119898 119898 = 0 1 2

(24)

where 1205761

isin (0 1) 1205762

isin (0 1) and 120575119901119896+1

(119910119896 119889119896) is the

variational of the functional 119901119896+1

(119910) caused by variational119889119896of 119910 on 119910

119896 Considering 119889

119896as the solution of (21)

120575119901119896+1

(119910119896 119889119896) le 0 is established The new state and guidance

command of the reference trajectory are generated as follows

[119909119896+1

119906119896+1

] = [119909119896

119906119896

] + 119897119896[Δ119909

Δ119906] (25)

4 The Global Convergence of the Algorithm

In [32 33] the convergence of the off-board algorithm wasanalyzed But the relationship between the global conver-gence and the guidance cycle 120591 has not been discussed for on-board optimization in the previous literatures In this sectionwe state and prove the global convergence of the on-boardalgorithm proposed in the preceding section The solutionsequence [119909

119896119906119896]119879 generated by the algorithm will converge

to the solution of the feedback optimal control problemUnlike off-board trajectory optimization the initial conditionof the feedback optimal control problem depends on thecurrent state of the vehicle in real time For the 119896th guidancecycle the optimal feedback control problem is described as(13)Thepenalty function119901

119896for this feedback optimal control

problem is given by (22) For the subsequent guidance cyclethe optimal feedback guidance problem is given by

min 119869 = 119882(119909 (119864119891)) + int

119864119891

119864119888119896+1

119892 (119909 119906) 119889119864

st = 119891 (119909 119906)

119909 (119864119888119896+1

) = 119909119888119896+1

119882 (119909 (119864119891)) = 0

(26)

where

119864119888119896+1

= 119864119888119896+ int

120591119896+Δ120591

120591119896

119889119864

119889120591119889120591 (27)

Δ120591 is the guidance cycle and 119909119888119896+1

is the new state of vehiclefrom navigation system at the time 120591

119896+ Δ120591 The penalty

function for optimal problem (24) is given by

119901119896+1

(119909 119906) = 119869 (119909 119906) + 12057810038171003817100381710038171003817119882(119909119896(119864119891))10038171003817100381710038171003817

+ 120578int

119864119891

119864119888119896+1

1003817100381710038171003817119891 (119909 119906) minus 1003817100381710038171003817 119889119864

+ 1205781003817100381710038171003817119909 (119864119888119896+1) minus 119909119888

119896+1

1003817100381710038171003817

(28)

We define the entire penalty function as follows

119901119896(119909 119906) = 119901

119896(119909 119906)

+

119896minus1

sum

119894=0

int

119864119888119894+1

119864119888119894

[1205781003817100381710038171003817119891 (119909119894 119906119894) minus 119894

1003817100381710038171003817 + 119892 (119909119894 119906119894)] 119889119864

(29)

The entire penalty function 119901119896(119909 119906) includes the penalty

function on the whole integration interval [1198640 119864119891]

Theorem 1 If the minimizer of 119901119896(119909119896 119906119896) is exited 119892(119909 119906) ge

0 and Δ120591 is selected and satisfies

120578100381710038171003817100381710038171003817119909119888119896+1

minus 119909119888119896minus int119864119888119896+1

119864119888119896119896119889119864

100381710038171003817100381710038171003817

minus120575119901119896+1

(119910119896 119889119896)

lt 1198971198961205763 (30)

where 1205763isin (0 120576

1) The solution sequence [119909

119896119906119896]119879 generated

from (24) will converge to the solution of the feedback optimalcontrol problem

Proof From (22) (28) (29) and

119909119896(119864119888119896+1

) = 119909119896(119864119888119896) + int

119864119888119896+1

119864119888119896

119896119889119864 (31)

we can obtain that

119901119896+1

(119909119896 119906119896) minus 119901119896(119909119896 119906119896)

= 120578

100381710038171003817100381710038171003817100381710038171003817

119909119896(119864119888119896) minus 119909119888

119896

minus(119909119888119896+1

minus 119909119888119896minus int

119864119888119896+1

119864119888119896

119896119889119864)

100381710038171003817100381710038171003817100381710038171003817

minus 1205781003817100381710038171003817119909119896 (119864119888119896) minus 119909119888

119896

1003817100381710038171003817

(32)

Mathematical Problems in Engineering 7

From (32) we obtain

limΔ120591rarr0

[119901119896+1

(119909119896 119906119896) minus 119901119896(119909119896 119906119896)] = 0

lim119891(119909119896119906119896)minus 119909119896rarr0

[119901119896+1

(119909119896 119906119896) minus 119901119896(119909119896 119906119896)] = 0

(33)

119901119896+1

(119909119896 119906119896) minus 119901119896(119909119896 119906119896)

= 120578

100381710038171003817100381710038171003817100381710038171003817

119909119896(119864119888119896) minus 119909119888

119896+1+ int

119864119888119896+1

119864119888119896

119896119889119864

100381710038171003817100381710038171003817100381710038171003817

minus1003817100381710038171003817119909119896 (119864119888119896) minus 119909119888

119896

1003817100381710038171003817

le 1205781003817100381710038171003817119909119896 (119864119888119896) minus 119909119888

119896

1003817100381710038171003817

+

100381710038171003817100381710038171003817100381710038171003817

119909119888119896+1

minus 119909119888119896minus int

119864119888119896+1

119864119888119896

119896119889119864

100381710038171003817100381710038171003817100381710038171003817

minus1003817100381710038171003817119909119896 (119864119888119896) minus 119909119888

119896

1003817100381710038171003817

= 120578

100381710038171003817100381710038171003817100381710038171003817

119909119888119896+1

minus 119909119888119896minus int

119864119888119896+1

119864119888119896

119896119889119864

100381710038171003817100381710038171003817100381710038171003817

(34)

From conditions (24) and (30) it is obtained that

119901119896+1

(119909119896+1

119906119896+1

)

le 119901119896+1

(119909119896 119906119896) + 1198971198961205761120575119901119896+1

(119889119896)

= 119901119896(119909119896 119906119896) + 1198971198961205761120575119901119896+1

(119889119896)

+ 120578

100381710038171003817100381710038171003817100381710038171003817

119909119896(119864119888119896) minus 119909119888

119896

minus (119909119888119896+1

minus 119909119888119896minus int

119864119888119896+1

119864119888119896

119896119889119864)

100381710038171003817100381710038171003817100381710038171003817

minus1003817100381710038171003817119909119896 (119864119888119896) minus 119909119888

119896

1003817100381710038171003817

le 119901119896(119909119896 119906119896) + 120578

100381710038171003817100381710038171003817100381710038171003817

119909119888119896+1

minus 119909119888119896minus int

119864119888119896+1

119864119888119896

119896119889119864

100381710038171003817100381710038171003817100381710038171003817

+ 1198971198961205761120575119901119896+1

(119910119896 119889119896)

le 119901119896(119909119896 119906119896) + 120578

100381710038171003817100381710038171003817100381710038171003817

119909119888119896+1

minus 119909119888119896minus int

119864119888119896+1

119864119888119896

119896119889119864

100381710038171003817100381710038171003817100381710038171003817

+ 1198971198961205763120575119901119896+1

(119910119896 119889119896)

lt 119901119896(119909119896 119906119896)

(35)

Thus the sequence 119901119896(119909119896 119906119896) 119896 = 1 2 3 is monotoni-

cally decreasing Considering the existence of a minimizer119901119896(119909119896 119906119896) converges Thus the following equation is estab-

lished

lim119896rarrinfin

(1198971198961205763120575119901119896+1

(119910119896 119889119896) minus 1198971198961205761120575119901119896+1

(119910119896 119889119896)) = 0 (36)

From (36) we derive

lim119896rarrinfin

(119897119896120575119901119896+1

(119910119896 119889119896)) = 0 (37)

If 119889119896

= 0 [119909119896

119906119896]119879 is not the minimizer of penalty function

then 120575119901119896+1

(119910119896 119889119896) = 0 Considering that 119897

119896is bounded away

from zero we derive

lim119896rarrinfin

119889119896= 0

lim119896rarrinfin

120575119901119896+1

(119910119896 119889119896) = 0

lim119896rarrinfin

[119896minus 119891 (119909

119896 119906119896)] = 0

lim119896rarrinfin

(119896+ 119867119896

119909) = 0

lim119896rarrinfin

119867119896

119906= 0

(38)

Thus the solution sequence [119909119896

119906119896]119879 generated by the

algorithmwill converge to the solution of the optimal controlproblem

5 Smooth Approximation forInequality Constraints

The nondifferentiable TPBVP from optimal control problem(11) with the inequality constraints brings difficulties for theproposed algorithm which can only solve a smooth problemSmooth approximations for nondifferentiable optimizationproblems have been studied in [5 6 34] In this sectionwe introduce the smooth approximation method for theinequality constraints If the inequality constraint function isabout the state and the guidance command

119862 (119909 119906) le 0 (39)

the Hamiltonian for the optimal control problem (11) withinequality constraints (39) is given by

119867 = 119892 (119909 119906) + 120582119879119891 (119909 119906) + 120583max 119862 (119909 119906) 0 (40)

However the Hamiltonian (40) is a nondifferentiable func-tion which brings difficulty for the proposed algorithmFor the smooth Hamiltonian approximation a differentiablefunction is substituted for max119862(119909 119906) 0 We denote thenondifferentiable function as follows

max 0 119910 = int

119910

minusinfin

119908 (119911) 119889119911 (41)

where 119908(119911) is the step function

119908 (119911) = 1 if 119911 ge 0

0 if 119911 lt 0(42)

Considering 119904(119911 119886) as the approximation function of 119908(119911)the differentiable approximation to max0 119910 is obtained as

119891119901(119910 119886) asymp int

119910

minusinfin

119904 (119911 119886) 119889119911 = 119910 +1

119886In (1 + 119890

minus119886119910) (43)

8 Mathematical Problems in Engineering

where 119904(119911 119886) is Sigmoid function

119904 (119911 119886) =1

1 + 119890minus119886119911 119886 gt 0 (44)

The Properties of 119891119901(119910 119886) 119886 gt 0 are as follows [5]

(1) 119891119901(119910 119886) is 119899-times continuously differentiable for any

positive integer 119899 with 120597119891119901(119910 119886)120597119910 = 1(1 + 119890

minus119886119910)

and 1205972119891119901(119910 119886)120597119910

2= 119886119890minus119886119910

(1 + 119890minus119886119910

)2

(2) 119891119901(119910 119886) is strictly convex and strictly increasing onR

(3) 119891119901(119910 119886) gt max0 119910

(4) lim119886rarrinfin

119891119901(119910 119886) = max0 119910

(5) 119891119901(119910 119886) gt 119891

119901(119910 1198861) for 119886 lt 119886

1 119910 isin R

Substituting (43) into (40) the following differentiableHamiltonian approximation is obtained

= 119892 (119909 119906) + 120582119879119891 (119909 119906) + 120583 [119862 +

1

119886In (1 + 119890

minus119886119862)] (45)

If the inequality constraint function is a 119902th-order statevariable inequality constraint given by

119862 (119909) le 0 (46)

If 119901 times derivatives of 119862(119909) are required before 119906 appearsexplicitly in the result the Hamiltonian with inequalityconstraints (46) is given by

119867 = 119892 (119909 119906) + 120582119879119891 (119909 119906) + 120583119862

(119902) (47)

where

120583 = 0 if 119862 lt 0 (48)

Considering the differentiable approximation to 120583 as follows

120583 = 120583119908 (119862) asymp 1205831

1 + 119890minus119886119862(49)

the following differentiable Hamiltonian approximation isobtained

= 119892 (119909 119906) + 120582119879119891 (119909 119906) + 120583

1

1 + 119890minus119886119862119862(119902) (50)

With the differentiable Hamiltonian approximation theinequality constraints are considered in the proposed guid-ance algorithm

6 Simulation Results

61 Numerical Calculation Based on Linear TPBVPDiscretiza-tion In this section the linear TPVBP is transformed intolinear equations using the Simpson rule which correspondsto the three-point Newton-Cotes quadrature rule The Simp-son rule for differential equation 119889119883119889119864 = 119910(119864) is given by

int

119887

119886

119910 (119864) 119889119864 =119887 minus 119886

6[119910 (119886) + 4119910(

119886 + 119887

2) + 119910 (119887)] (51)

From (51) we obtain

119883(119887) minus 119883 (119886) =119887 minus 119886

6[119910 (119886) + 4119910(

119886 + 119887

2) + 119910 (119887)] (52)

We disperse the normalized energy 119864 into119873 equal segmentsas [1198640

1198641

sdot sdot sdot 119864119873] with 119864

0= 119864119888119896and 119864

119873= 119864119891 The

differential equation 119889119883119889119864 = 119910(119883 119864) can be transformedinto the following equation

119883(119864119894) minus 119883 (119864

119894minus1)

=119864119894minus 119864119894minus1

6[119910 (119883

119894 119864119894)

+ 4119910 (119883119894+ 119883119894minus1

2119864119894+ 119864119894minus1

2)

+ 119910 (119883119894minus1

119864119894minus1

)] 119894 = 1 2 3 119873

(53)

Based on (53) the linear TPBVP (21) can be transformed intolinear equations described by

119860119883 = 119861 (54)

The linear equations can be solved by Gaussian eliminationFinally we can obtain the solution Δ119909(119864) Δ119906(119864) and

120582(119864) of the linear TPBVP (21) from the solution of the linearequations through interpolating

62 GHV Model The proposed method is applied to theGenericHypersonicVehicle (GHV) to verify the effectivenessby flight simulation In order to develop a complete setof aerodynamic coefficients experimental longitudinal andlateral-directional aerodynamics were obtained for the GHVby using six Langley wind tunnels [35] The aerodynamicdatabase of the GHV is shown in [35] The GHV model withscramjet propulsion system is used for the simulation where119878ref = 33473m2 and the full-scale weight119898ini = 110000 kg

According to the characterization of scramjet propulsionsystem described in [2 27] the development of the propul-sion system begins with the following equation

119879 = 05120601119868sp (120601119872119886) 120588V1198920119862119879 (120572119872119886) (55)

where 120601 is the throttle commandwhich varies from 0 to 1Thecoefficient 119862

119879depends on 120572 and119872

119886as follows

119862119879= minus00012119872

1198861205722+ 0008120572 + 14 minus 01119872

119886 (56)

The impulse 119868sp is given by the following equation

119868sp = minus811198722

119886+ 1004119872

119886minus 52 + 100120601 (57)

63 Simulation Results for Modeling and Initial State ErrorConsidering the high cost of the flight experimental testresearchers do flight numerical simulation hardware-in-the-loop simulation and final flight experimental test to verifythe guidance algorithm step by step In this section flight

Mathematical Problems in Engineering 9

Table 1 Initial and required final states of the vehicle

119903ini(m)

Vini(ms)

120574ini(deg)

119903119903119891

(m)V119903119891

(ms)120574119903119891

(deg)20000 1000 0 35000 2000 0

numerical simulation results for the GHV with modelingerror and initial state error are given to show the effectivenessof the proposed guidance algorithm For the modeling errorcaused by the wind tunnel experimental test the aerody-namic coefficients bias is considered in the flight simulationThe simulations end once the final normalized energy 119864

119891

is reached The optimal guidance algorithm updates thetrajectory every guidance cycle which is 03 s The initial andrequired final states of the air-breathing vehicle are shownin Table 1 The number of the discrete nodes 119873 = 10 Thedynamic pressure inequality constraint is given by

119876 =1

2120588V2 le 140000Pa (58)

To assess the capability of the proposed algorithm to dealwith disturbance the aerodynamic coefficients bias and theinitial state error in the simulations are given by followingcases

(1) Δ119862119871= minus01119862

119871 Δ119862119863= +01119862

119863

(2) Δ119862119871= minus01119862

119871 Δ119862119863= +01119862

119863 Δ119903ini = +500m

(3) Δ119862119871= minus01119862

119871 Δ119862119863= +01119862

119863 Δ119903ini = minus500m

(4) Δ119862119871= minus01119862

119871 Δ119862119863= +01119862

119863 Δ119903ini = +1000m

(5) Δ119862119871= minus01119862

119871 Δ119862119863= +01119862

119863 Δ119903ini = minus1000m

The simulation results of the optimal guidance algorithmare shown in Figures 4 5 6 7 and 8 The altitude velocityflight path angle angle of attack and dynamic pressure arerespectively shown in Figures 4 to 8 As expected the finalstate is achieved under aerodynamic bias and initial stateerror Figure 8 shows that the state inequality constraint issatisfied The dynamic pressure is less than 140000 Pa fordifferent cases

Table 2 shows the terminal error and the fuel cost ofthe optimal guidance algorithm The terminal accuracy ishigh for different cases Moreover the fuel cost of theoptimal guidance algorithm is lower than that of the trackingguidance algorithm A greater initial state error will result inmore fuel savings In summarywith lower fuel cost the resultsof the proposed optimal guidance algorithm satisfies all theequality and inequality constraints The simulation resultsshow the great potential for the final flight experimental test

7 Conclusion

In this paper we present an optimal guidance algorithmfor air-breathing launch vehicle based on optimal trajectorycorrection The proposed guidance algorithm corrects thereference trajectory in every guidance cycle to satisfy the opti-mality condition of the optimal feedback control problem

0 14012010080604020

18000

21000

24000

27000

30000

33000

Alti

tude

(m)

Time (s)

Case 1Case 2Case 3

Case 4Case 5

Figure 4 The altitude of the simulation results

1200

1400

1600

1800

2000

2200Ve

loci

ty (m

s)

Time (s)0 14012010080604020

Case 1Case 2Case 3

Case 4Case 5

Figure 5 The velocity of the simulation results

By linearizing the optimality condition the linear TPBVPis obtained for optimal trajectory correction The solutionof the linear TPBVP is derived using the Simpson rule Thenew trajectory is generated by a linear search for the stepsize of the solution Smooth Hamiltonian approximation isimplemented to the inequality constraints The sufficiencycondition for the global convergence of the guidance algo-rithm is given in this paper

Finally simulations for 5 different cases of modeling andinitial state error are presented Compared with the trackingguidance method the simulation results prove the low fuelcost and high precision of the proposed optimal guidancemethod

10 Mathematical Problems in Engineering

Table 2 The simulation results for optimal guidance algorithm

Case Δ119903119891(m) ΔV

119891(ms) Δ120574

119891(deg) Fuel cost (kg) Fuel cost (tracking method) (kg)

1 minus2820 0136 020 457219 4574622 859 minus003 minus015 458314 4628393 minus1508 007 020 458474 4605664 10816 047 minus023 460292 4776005 minus3243 019 minus016 460133 468894

0

2

4

6

8

Time (s)

Flig

ht p

ath

angl

e (D

EG)

minus2

minus4

0 14012010080604020

Case 1Case 2Case 3

Case 4Case 5

Figure 6 The flight path angle of the simulation results

Ang

le o

f atta

ck (D

EG)

0

1

2

3

4

minus1

minus2

minus3

minus4

minus5

minus6

Time (s)0 14012010080604020

Case 1Case 2Case 3

Case 4Case 5

Figure 7 The angle of attack of the simulation results

Acknowledgments

This work was supported in part by the National NatureScience Foundation of China nos 61203081 and 61174079Doctoral Fund of Ministry of Education of China

20000

40000

60000

80000

100000

120000

140000

Dyn

amic

pre

ssur

e (Pa

)

Time (s)0 14012010080604020

Case 1Case 2Case 3

Case 4Case 5

Figure 8 The dynamic pressure of the simulation results

no 20120142120091 andPrecisionmanufacturing technologyand equipment for metal parts no 2012DFG70640

References

[1] H J Henry Jr and C H McLellan ldquoAir-breathing launch vehi-cle for earth-orbit shuttle New technology and developmentapproachrdquo Journal of Aircraft vol 8 no 5 pp 381ndash387 1971

[2] J E CorbanA J Calise andGA Flandro ldquoReal-time guidancealgorithm for aerospace plane optimal ascent to low earth orbitrdquoin Proceedings of the American Control Conference pp 2475ndash2481 Pittsburgh PA USA June 1989

[3] C H Yam D D Lorenzo and D Izzo ldquoLow-thrust trajectorydesign as a constrained global optimization problemrdquo Journalof Aerospace Engineering G vol 225 no 11 pp 1243ndash1251 2011

[4] B A Conway ldquoA survey of methods available for the numericaloptimization of continuous dynamic systemsrdquo Journal of Opti-mization Theory and Applications vol 152 no 2 pp 271ndash3062012

[5] C H Chen and O L Mangasarian ldquoSmoothing methodsfor convex inequalities and linear complementarity problemsrdquoMathematical Programming vol 71 no 1 pp 51ndash69 1995

[6] X Chen ldquoSmoothing methods for nonsmooth nonconvexminimizationrdquo Mathematical Programming vol 134 no 1 pp71ndash99 2012

[7] W Wang Y Shang and Y Zhang ldquoA filled function approachfor nonsmooth constrained global optimizationrdquoMathematical

Mathematical Problems in Engineering 11

Problems in Engineering vol 2010 Article ID 310391 9 pages2010

[8] L I Rozonoer ldquoLinear-quadratic optimization and some gen-eral hypotheses on optimal controlrdquoMathematical Problems inEngineering vol 5 no 4 pp 275ndash289 1999

[9] J T Betts ldquoSurvey of numerical methods for trajectory opti-mizationrdquo Journal of Guidance Control and Dynamics vol 21no 2 pp 193ndash207 1998

[10] G Q Huang Y P Lu and Y Nan ldquoA survey of numericalalgorithms for trajectory optimization of flight vehiclesrdquo ScienceChina-Technological Sciences vol 55 no 9 pp 2538ndash2560 2012

[11] Y Ulybyshev ldquoDiscrete pseudocontrol sets for optimal controlproblemsrdquo Journal of Guidance Control and Dynamics vol 33no 4 pp 1133ndash1142 2010

[12] M Vasile and F Zuiani ldquoMulti-agent collaborative search anagent-based memetic multi-objective optimization algorithmapplied to space trajectory designrdquo Journal of Aerospace Engi-neering G vol 225 no 11 pp 1211ndash1227 2011

[13] M Pontani and B A Conway ldquoParticle swarm optimizationapplied to space trajectoriesrdquo Journal of Guidance Control andDynamics vol 33 no 5 pp 1429ndash1441 2010

[14] Y P Ulybyshev ldquoOptimization of low-thrust orbit transfers withconstraintsrdquo Cosmic Research vol 50 no 5 pp 376ndash390 2012

[15] M Chen R Mei and B Jiang ldquoSliding mode control for a classof uncertainMIMOnonlinear systemswith application to near-space vehiclesrdquoMathematical Problems in Engineering vol 2013Article ID 180589 9 pages 2013

[16] H Zhang Y Shi M Q Xu and H T Cui ldquoObserver-basedtracking controller design for networked predictive controlsystems with uncertain markov delaysrdquo in Proceedings of theAmerican Control Conference pp 5682ndash5687 2012

[17] H Zhang and Y Shi ldquoRobust H-infinity sliding-mode controlfor markovian jump systems subject to intermittent observa-tions and partially known transition probabilitiesrdquo Systems ampControl Letters vol 62 no 12 pp 1114ndash1124 2013

[18] Z Shuai H Zhang J Wang J Li and M Ouyang ldquoCombinedAFS andDYC control of four-wheel-independent-drive electricvehicles over CAN network with time-varying delaysrdquo IEEETransactions on Vehicular Technology no 99 p 1 2013

[19] H Zhang Y Shi and A Saadat Mehr ldquoRobust static outputfeedback control and remote PID design for networked motorsystemsrdquo IEEETransactions on Industrial Electronics vol 58 no12 pp 5396ndash5405 2011

[20] H Zhang Y Shi and M X Liu ldquoH-infinity step trackingcontrol for networked discrete-time nonlinear systems withintegral and predictive actionsrdquo IEEE Transactions on IndustrialInformatics vol 9 no 1 pp 337ndash345 2013

[21] X J Wang and S P Wang ldquoAdaptive fuzzy robust controlfor a class of nonlinear systems via small gain theoremrdquoMathematical Problems in Engineering vol 2013 Article ID201432 11 pages 2013

[22] V Brinda S Dasgupta and M Lal ldquoTrajectory optimizationand guidance of an air breathing hypersonic vehiclerdquo in Pro-ceedings of the 14th AIAAAHI International Space Planes andHypersonics Systems Technologies Conference vol 2 pp 856ndash869 November 2006

[23] B Tian and Q Zong ldquoOptimal guidance for reentry vehiclesbased on indirect legendre pseudospectral methodrdquoActa Astro-nautica vol 68 no 7-8 pp 1176ndash1184 2011

[24] M H Graszliglin J Telaar and UM Schottle ldquoAscent and reentryguidance concept based on NLP-methodsrdquo Acta Astronauticavol 55 no 3ndash9 pp 461ndash471 2004

[25] P Lu and B Pan ldquoHighly constrained optimal launch ascentguidancerdquo Journal of Guidance Control and Dynamics vol 33no 2 pp 404ndash414 2010

[26] P Lu H Sun and B Tsai ldquoClosed-loop endo-atmosphericascent guidancerdquo in Proceedings of the AIAA Guidance Navi-gation and Control Conference Monterey Calif USA 2002

[27] J Oscar Murillo and P Lu ldquoFast ascent trajectory optimizationfor hypersonic air-breathing vehiclesrdquo in Proceedings of theAIAA Guidance Navigation and Control Conference pp 5ndash8Toronto Canada 2010

[28] M Marrdonny and M Mobed ldquoA guidance algorithm forlaunch to equatorial orbitrdquo Aircraft Engineering and AerospaceTechnology vol 81 no 2 pp 137ndash148 2009

[29] T Yamamoto and J Kawaguchi ldquoA new real-time guidancestrategy for aerodynamic ascent flightrdquo Acta Astronautica vol61 no 11-12 pp 965ndash977 2007

[30] C Filici andR S Sanchez Pena ldquoOnline guidance updates usingneural networksrdquo Acta Astronautica vol 66 no 3-4 pp 477ndash485 2010

[31] P T Boggs A J Kearsley and J W Tolle ldquoA global convergenceanalysis of an algorithm for large-scale nonlinear optimizationproblemsrdquo SIAM Journal on Optimization vol 9 no 4 pp 833ndash862 1999

[32] S Schlenkrich and A Walther ldquoGlobal convergence of quasi-Newton methods based on adjoint Broyden updatesrdquo AppliedNumerical Mathematics vol 59 no 5 pp 1120ndash1136 2009

[33] S Schlenkrich A Griewank and A Walther ldquoOn the localconvergence of adjoint Broyden methodsrdquo Mathematical Pro-gramming vol 121 no 2 pp 221ndash247 2010

[34] S Lian and L Zhang ldquoA simple smooth exact penalty functionfor smooth optimization problemrdquo Journal of Systems Science ampComplexity vol 25 no 3 pp 521ndash528 2012

[35] S Keshmiri R Colgren and M Mirmirani ldquoDevelopment ofan aerodynamic database for a generic hypersonic air vehiclerdquoin Proceedings of the AIAA Guidance Navigation and ControlConference pp 3978ndash3998 San Francisco Calif USA August2005

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

Discrete MathematicsJournal of

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Page 6: Research Article Optimal Ascent Guidance for Air-Breathing ...downloads.hindawi.com/journals/mpe/2013/313197.pdfResearch Article Optimal Ascent Guidance for Air-Breathing Launch Vehicle

6 Mathematical Problems in Engineering

where 120578 is the penalty parameter satisfied 120578 gt max |120582| Wedenote

119910 = [119909119896

119906119896

] 119889119896= [

Δ119909

Δ119906] (23)

where [Δ119909 Δ119906 120582]119879 is the solution of the linear TPBVP (21)

The minimizer of penalty function 119901119896(119909 119906) corresponds

to the solution of problem (11) Backtracking line search wasused for large-scale nonlinear optimization problems in [31]The step size 119897

119896is determined by backtracking line search that

satisfies

119897119896= max 119897 isin 119878 119901

119896+1(119910119896+ 119897119889119896)

le 119901119896+1

(119910119896) + 1198971205761120575119901119896+1

(119910119896 119889119896)

119878 = 119897 119897 = 1205762

119898 119898 = 0 1 2

(24)

where 1205761

isin (0 1) 1205762

isin (0 1) and 120575119901119896+1

(119910119896 119889119896) is the

variational of the functional 119901119896+1

(119910) caused by variational119889119896of 119910 on 119910

119896 Considering 119889

119896as the solution of (21)

120575119901119896+1

(119910119896 119889119896) le 0 is established The new state and guidance

command of the reference trajectory are generated as follows

[119909119896+1

119906119896+1

] = [119909119896

119906119896

] + 119897119896[Δ119909

Δ119906] (25)

4 The Global Convergence of the Algorithm

In [32 33] the convergence of the off-board algorithm wasanalyzed But the relationship between the global conver-gence and the guidance cycle 120591 has not been discussed for on-board optimization in the previous literatures In this sectionwe state and prove the global convergence of the on-boardalgorithm proposed in the preceding section The solutionsequence [119909

119896119906119896]119879 generated by the algorithm will converge

to the solution of the feedback optimal control problemUnlike off-board trajectory optimization the initial conditionof the feedback optimal control problem depends on thecurrent state of the vehicle in real time For the 119896th guidancecycle the optimal feedback control problem is described as(13)Thepenalty function119901

119896for this feedback optimal control

problem is given by (22) For the subsequent guidance cyclethe optimal feedback guidance problem is given by

min 119869 = 119882(119909 (119864119891)) + int

119864119891

119864119888119896+1

119892 (119909 119906) 119889119864

st = 119891 (119909 119906)

119909 (119864119888119896+1

) = 119909119888119896+1

119882 (119909 (119864119891)) = 0

(26)

where

119864119888119896+1

= 119864119888119896+ int

120591119896+Δ120591

120591119896

119889119864

119889120591119889120591 (27)

Δ120591 is the guidance cycle and 119909119888119896+1

is the new state of vehiclefrom navigation system at the time 120591

119896+ Δ120591 The penalty

function for optimal problem (24) is given by

119901119896+1

(119909 119906) = 119869 (119909 119906) + 12057810038171003817100381710038171003817119882(119909119896(119864119891))10038171003817100381710038171003817

+ 120578int

119864119891

119864119888119896+1

1003817100381710038171003817119891 (119909 119906) minus 1003817100381710038171003817 119889119864

+ 1205781003817100381710038171003817119909 (119864119888119896+1) minus 119909119888

119896+1

1003817100381710038171003817

(28)

We define the entire penalty function as follows

119901119896(119909 119906) = 119901

119896(119909 119906)

+

119896minus1

sum

119894=0

int

119864119888119894+1

119864119888119894

[1205781003817100381710038171003817119891 (119909119894 119906119894) minus 119894

1003817100381710038171003817 + 119892 (119909119894 119906119894)] 119889119864

(29)

The entire penalty function 119901119896(119909 119906) includes the penalty

function on the whole integration interval [1198640 119864119891]

Theorem 1 If the minimizer of 119901119896(119909119896 119906119896) is exited 119892(119909 119906) ge

0 and Δ120591 is selected and satisfies

120578100381710038171003817100381710038171003817119909119888119896+1

minus 119909119888119896minus int119864119888119896+1

119864119888119896119896119889119864

100381710038171003817100381710038171003817

minus120575119901119896+1

(119910119896 119889119896)

lt 1198971198961205763 (30)

where 1205763isin (0 120576

1) The solution sequence [119909

119896119906119896]119879 generated

from (24) will converge to the solution of the feedback optimalcontrol problem

Proof From (22) (28) (29) and

119909119896(119864119888119896+1

) = 119909119896(119864119888119896) + int

119864119888119896+1

119864119888119896

119896119889119864 (31)

we can obtain that

119901119896+1

(119909119896 119906119896) minus 119901119896(119909119896 119906119896)

= 120578

100381710038171003817100381710038171003817100381710038171003817

119909119896(119864119888119896) minus 119909119888

119896

minus(119909119888119896+1

minus 119909119888119896minus int

119864119888119896+1

119864119888119896

119896119889119864)

100381710038171003817100381710038171003817100381710038171003817

minus 1205781003817100381710038171003817119909119896 (119864119888119896) minus 119909119888

119896

1003817100381710038171003817

(32)

Mathematical Problems in Engineering 7

From (32) we obtain

limΔ120591rarr0

[119901119896+1

(119909119896 119906119896) minus 119901119896(119909119896 119906119896)] = 0

lim119891(119909119896119906119896)minus 119909119896rarr0

[119901119896+1

(119909119896 119906119896) minus 119901119896(119909119896 119906119896)] = 0

(33)

119901119896+1

(119909119896 119906119896) minus 119901119896(119909119896 119906119896)

= 120578

100381710038171003817100381710038171003817100381710038171003817

119909119896(119864119888119896) minus 119909119888

119896+1+ int

119864119888119896+1

119864119888119896

119896119889119864

100381710038171003817100381710038171003817100381710038171003817

minus1003817100381710038171003817119909119896 (119864119888119896) minus 119909119888

119896

1003817100381710038171003817

le 1205781003817100381710038171003817119909119896 (119864119888119896) minus 119909119888

119896

1003817100381710038171003817

+

100381710038171003817100381710038171003817100381710038171003817

119909119888119896+1

minus 119909119888119896minus int

119864119888119896+1

119864119888119896

119896119889119864

100381710038171003817100381710038171003817100381710038171003817

minus1003817100381710038171003817119909119896 (119864119888119896) minus 119909119888

119896

1003817100381710038171003817

= 120578

100381710038171003817100381710038171003817100381710038171003817

119909119888119896+1

minus 119909119888119896minus int

119864119888119896+1

119864119888119896

119896119889119864

100381710038171003817100381710038171003817100381710038171003817

(34)

From conditions (24) and (30) it is obtained that

119901119896+1

(119909119896+1

119906119896+1

)

le 119901119896+1

(119909119896 119906119896) + 1198971198961205761120575119901119896+1

(119889119896)

= 119901119896(119909119896 119906119896) + 1198971198961205761120575119901119896+1

(119889119896)

+ 120578

100381710038171003817100381710038171003817100381710038171003817

119909119896(119864119888119896) minus 119909119888

119896

minus (119909119888119896+1

minus 119909119888119896minus int

119864119888119896+1

119864119888119896

119896119889119864)

100381710038171003817100381710038171003817100381710038171003817

minus1003817100381710038171003817119909119896 (119864119888119896) minus 119909119888

119896

1003817100381710038171003817

le 119901119896(119909119896 119906119896) + 120578

100381710038171003817100381710038171003817100381710038171003817

119909119888119896+1

minus 119909119888119896minus int

119864119888119896+1

119864119888119896

119896119889119864

100381710038171003817100381710038171003817100381710038171003817

+ 1198971198961205761120575119901119896+1

(119910119896 119889119896)

le 119901119896(119909119896 119906119896) + 120578

100381710038171003817100381710038171003817100381710038171003817

119909119888119896+1

minus 119909119888119896minus int

119864119888119896+1

119864119888119896

119896119889119864

100381710038171003817100381710038171003817100381710038171003817

+ 1198971198961205763120575119901119896+1

(119910119896 119889119896)

lt 119901119896(119909119896 119906119896)

(35)

Thus the sequence 119901119896(119909119896 119906119896) 119896 = 1 2 3 is monotoni-

cally decreasing Considering the existence of a minimizer119901119896(119909119896 119906119896) converges Thus the following equation is estab-

lished

lim119896rarrinfin

(1198971198961205763120575119901119896+1

(119910119896 119889119896) minus 1198971198961205761120575119901119896+1

(119910119896 119889119896)) = 0 (36)

From (36) we derive

lim119896rarrinfin

(119897119896120575119901119896+1

(119910119896 119889119896)) = 0 (37)

If 119889119896

= 0 [119909119896

119906119896]119879 is not the minimizer of penalty function

then 120575119901119896+1

(119910119896 119889119896) = 0 Considering that 119897

119896is bounded away

from zero we derive

lim119896rarrinfin

119889119896= 0

lim119896rarrinfin

120575119901119896+1

(119910119896 119889119896) = 0

lim119896rarrinfin

[119896minus 119891 (119909

119896 119906119896)] = 0

lim119896rarrinfin

(119896+ 119867119896

119909) = 0

lim119896rarrinfin

119867119896

119906= 0

(38)

Thus the solution sequence [119909119896

119906119896]119879 generated by the

algorithmwill converge to the solution of the optimal controlproblem

5 Smooth Approximation forInequality Constraints

The nondifferentiable TPBVP from optimal control problem(11) with the inequality constraints brings difficulties for theproposed algorithm which can only solve a smooth problemSmooth approximations for nondifferentiable optimizationproblems have been studied in [5 6 34] In this sectionwe introduce the smooth approximation method for theinequality constraints If the inequality constraint function isabout the state and the guidance command

119862 (119909 119906) le 0 (39)

the Hamiltonian for the optimal control problem (11) withinequality constraints (39) is given by

119867 = 119892 (119909 119906) + 120582119879119891 (119909 119906) + 120583max 119862 (119909 119906) 0 (40)

However the Hamiltonian (40) is a nondifferentiable func-tion which brings difficulty for the proposed algorithmFor the smooth Hamiltonian approximation a differentiablefunction is substituted for max119862(119909 119906) 0 We denote thenondifferentiable function as follows

max 0 119910 = int

119910

minusinfin

119908 (119911) 119889119911 (41)

where 119908(119911) is the step function

119908 (119911) = 1 if 119911 ge 0

0 if 119911 lt 0(42)

Considering 119904(119911 119886) as the approximation function of 119908(119911)the differentiable approximation to max0 119910 is obtained as

119891119901(119910 119886) asymp int

119910

minusinfin

119904 (119911 119886) 119889119911 = 119910 +1

119886In (1 + 119890

minus119886119910) (43)

8 Mathematical Problems in Engineering

where 119904(119911 119886) is Sigmoid function

119904 (119911 119886) =1

1 + 119890minus119886119911 119886 gt 0 (44)

The Properties of 119891119901(119910 119886) 119886 gt 0 are as follows [5]

(1) 119891119901(119910 119886) is 119899-times continuously differentiable for any

positive integer 119899 with 120597119891119901(119910 119886)120597119910 = 1(1 + 119890

minus119886119910)

and 1205972119891119901(119910 119886)120597119910

2= 119886119890minus119886119910

(1 + 119890minus119886119910

)2

(2) 119891119901(119910 119886) is strictly convex and strictly increasing onR

(3) 119891119901(119910 119886) gt max0 119910

(4) lim119886rarrinfin

119891119901(119910 119886) = max0 119910

(5) 119891119901(119910 119886) gt 119891

119901(119910 1198861) for 119886 lt 119886

1 119910 isin R

Substituting (43) into (40) the following differentiableHamiltonian approximation is obtained

= 119892 (119909 119906) + 120582119879119891 (119909 119906) + 120583 [119862 +

1

119886In (1 + 119890

minus119886119862)] (45)

If the inequality constraint function is a 119902th-order statevariable inequality constraint given by

119862 (119909) le 0 (46)

If 119901 times derivatives of 119862(119909) are required before 119906 appearsexplicitly in the result the Hamiltonian with inequalityconstraints (46) is given by

119867 = 119892 (119909 119906) + 120582119879119891 (119909 119906) + 120583119862

(119902) (47)

where

120583 = 0 if 119862 lt 0 (48)

Considering the differentiable approximation to 120583 as follows

120583 = 120583119908 (119862) asymp 1205831

1 + 119890minus119886119862(49)

the following differentiable Hamiltonian approximation isobtained

= 119892 (119909 119906) + 120582119879119891 (119909 119906) + 120583

1

1 + 119890minus119886119862119862(119902) (50)

With the differentiable Hamiltonian approximation theinequality constraints are considered in the proposed guid-ance algorithm

6 Simulation Results

61 Numerical Calculation Based on Linear TPBVPDiscretiza-tion In this section the linear TPVBP is transformed intolinear equations using the Simpson rule which correspondsto the three-point Newton-Cotes quadrature rule The Simp-son rule for differential equation 119889119883119889119864 = 119910(119864) is given by

int

119887

119886

119910 (119864) 119889119864 =119887 minus 119886

6[119910 (119886) + 4119910(

119886 + 119887

2) + 119910 (119887)] (51)

From (51) we obtain

119883(119887) minus 119883 (119886) =119887 minus 119886

6[119910 (119886) + 4119910(

119886 + 119887

2) + 119910 (119887)] (52)

We disperse the normalized energy 119864 into119873 equal segmentsas [1198640

1198641

sdot sdot sdot 119864119873] with 119864

0= 119864119888119896and 119864

119873= 119864119891 The

differential equation 119889119883119889119864 = 119910(119883 119864) can be transformedinto the following equation

119883(119864119894) minus 119883 (119864

119894minus1)

=119864119894minus 119864119894minus1

6[119910 (119883

119894 119864119894)

+ 4119910 (119883119894+ 119883119894minus1

2119864119894+ 119864119894minus1

2)

+ 119910 (119883119894minus1

119864119894minus1

)] 119894 = 1 2 3 119873

(53)

Based on (53) the linear TPBVP (21) can be transformed intolinear equations described by

119860119883 = 119861 (54)

The linear equations can be solved by Gaussian eliminationFinally we can obtain the solution Δ119909(119864) Δ119906(119864) and

120582(119864) of the linear TPBVP (21) from the solution of the linearequations through interpolating

62 GHV Model The proposed method is applied to theGenericHypersonicVehicle (GHV) to verify the effectivenessby flight simulation In order to develop a complete setof aerodynamic coefficients experimental longitudinal andlateral-directional aerodynamics were obtained for the GHVby using six Langley wind tunnels [35] The aerodynamicdatabase of the GHV is shown in [35] The GHV model withscramjet propulsion system is used for the simulation where119878ref = 33473m2 and the full-scale weight119898ini = 110000 kg

According to the characterization of scramjet propulsionsystem described in [2 27] the development of the propul-sion system begins with the following equation

119879 = 05120601119868sp (120601119872119886) 120588V1198920119862119879 (120572119872119886) (55)

where 120601 is the throttle commandwhich varies from 0 to 1Thecoefficient 119862

119879depends on 120572 and119872

119886as follows

119862119879= minus00012119872

1198861205722+ 0008120572 + 14 minus 01119872

119886 (56)

The impulse 119868sp is given by the following equation

119868sp = minus811198722

119886+ 1004119872

119886minus 52 + 100120601 (57)

63 Simulation Results for Modeling and Initial State ErrorConsidering the high cost of the flight experimental testresearchers do flight numerical simulation hardware-in-the-loop simulation and final flight experimental test to verifythe guidance algorithm step by step In this section flight

Mathematical Problems in Engineering 9

Table 1 Initial and required final states of the vehicle

119903ini(m)

Vini(ms)

120574ini(deg)

119903119903119891

(m)V119903119891

(ms)120574119903119891

(deg)20000 1000 0 35000 2000 0

numerical simulation results for the GHV with modelingerror and initial state error are given to show the effectivenessof the proposed guidance algorithm For the modeling errorcaused by the wind tunnel experimental test the aerody-namic coefficients bias is considered in the flight simulationThe simulations end once the final normalized energy 119864

119891

is reached The optimal guidance algorithm updates thetrajectory every guidance cycle which is 03 s The initial andrequired final states of the air-breathing vehicle are shownin Table 1 The number of the discrete nodes 119873 = 10 Thedynamic pressure inequality constraint is given by

119876 =1

2120588V2 le 140000Pa (58)

To assess the capability of the proposed algorithm to dealwith disturbance the aerodynamic coefficients bias and theinitial state error in the simulations are given by followingcases

(1) Δ119862119871= minus01119862

119871 Δ119862119863= +01119862

119863

(2) Δ119862119871= minus01119862

119871 Δ119862119863= +01119862

119863 Δ119903ini = +500m

(3) Δ119862119871= minus01119862

119871 Δ119862119863= +01119862

119863 Δ119903ini = minus500m

(4) Δ119862119871= minus01119862

119871 Δ119862119863= +01119862

119863 Δ119903ini = +1000m

(5) Δ119862119871= minus01119862

119871 Δ119862119863= +01119862

119863 Δ119903ini = minus1000m

The simulation results of the optimal guidance algorithmare shown in Figures 4 5 6 7 and 8 The altitude velocityflight path angle angle of attack and dynamic pressure arerespectively shown in Figures 4 to 8 As expected the finalstate is achieved under aerodynamic bias and initial stateerror Figure 8 shows that the state inequality constraint issatisfied The dynamic pressure is less than 140000 Pa fordifferent cases

Table 2 shows the terminal error and the fuel cost ofthe optimal guidance algorithm The terminal accuracy ishigh for different cases Moreover the fuel cost of theoptimal guidance algorithm is lower than that of the trackingguidance algorithm A greater initial state error will result inmore fuel savings In summarywith lower fuel cost the resultsof the proposed optimal guidance algorithm satisfies all theequality and inequality constraints The simulation resultsshow the great potential for the final flight experimental test

7 Conclusion

In this paper we present an optimal guidance algorithmfor air-breathing launch vehicle based on optimal trajectorycorrection The proposed guidance algorithm corrects thereference trajectory in every guidance cycle to satisfy the opti-mality condition of the optimal feedback control problem

0 14012010080604020

18000

21000

24000

27000

30000

33000

Alti

tude

(m)

Time (s)

Case 1Case 2Case 3

Case 4Case 5

Figure 4 The altitude of the simulation results

1200

1400

1600

1800

2000

2200Ve

loci

ty (m

s)

Time (s)0 14012010080604020

Case 1Case 2Case 3

Case 4Case 5

Figure 5 The velocity of the simulation results

By linearizing the optimality condition the linear TPBVPis obtained for optimal trajectory correction The solutionof the linear TPBVP is derived using the Simpson rule Thenew trajectory is generated by a linear search for the stepsize of the solution Smooth Hamiltonian approximation isimplemented to the inequality constraints The sufficiencycondition for the global convergence of the guidance algo-rithm is given in this paper

Finally simulations for 5 different cases of modeling andinitial state error are presented Compared with the trackingguidance method the simulation results prove the low fuelcost and high precision of the proposed optimal guidancemethod

10 Mathematical Problems in Engineering

Table 2 The simulation results for optimal guidance algorithm

Case Δ119903119891(m) ΔV

119891(ms) Δ120574

119891(deg) Fuel cost (kg) Fuel cost (tracking method) (kg)

1 minus2820 0136 020 457219 4574622 859 minus003 minus015 458314 4628393 minus1508 007 020 458474 4605664 10816 047 minus023 460292 4776005 minus3243 019 minus016 460133 468894

0

2

4

6

8

Time (s)

Flig

ht p

ath

angl

e (D

EG)

minus2

minus4

0 14012010080604020

Case 1Case 2Case 3

Case 4Case 5

Figure 6 The flight path angle of the simulation results

Ang

le o

f atta

ck (D

EG)

0

1

2

3

4

minus1

minus2

minus3

minus4

minus5

minus6

Time (s)0 14012010080604020

Case 1Case 2Case 3

Case 4Case 5

Figure 7 The angle of attack of the simulation results

Acknowledgments

This work was supported in part by the National NatureScience Foundation of China nos 61203081 and 61174079Doctoral Fund of Ministry of Education of China

20000

40000

60000

80000

100000

120000

140000

Dyn

amic

pre

ssur

e (Pa

)

Time (s)0 14012010080604020

Case 1Case 2Case 3

Case 4Case 5

Figure 8 The dynamic pressure of the simulation results

no 20120142120091 andPrecisionmanufacturing technologyand equipment for metal parts no 2012DFG70640

References

[1] H J Henry Jr and C H McLellan ldquoAir-breathing launch vehi-cle for earth-orbit shuttle New technology and developmentapproachrdquo Journal of Aircraft vol 8 no 5 pp 381ndash387 1971

[2] J E CorbanA J Calise andGA Flandro ldquoReal-time guidancealgorithm for aerospace plane optimal ascent to low earth orbitrdquoin Proceedings of the American Control Conference pp 2475ndash2481 Pittsburgh PA USA June 1989

[3] C H Yam D D Lorenzo and D Izzo ldquoLow-thrust trajectorydesign as a constrained global optimization problemrdquo Journalof Aerospace Engineering G vol 225 no 11 pp 1243ndash1251 2011

[4] B A Conway ldquoA survey of methods available for the numericaloptimization of continuous dynamic systemsrdquo Journal of Opti-mization Theory and Applications vol 152 no 2 pp 271ndash3062012

[5] C H Chen and O L Mangasarian ldquoSmoothing methodsfor convex inequalities and linear complementarity problemsrdquoMathematical Programming vol 71 no 1 pp 51ndash69 1995

[6] X Chen ldquoSmoothing methods for nonsmooth nonconvexminimizationrdquo Mathematical Programming vol 134 no 1 pp71ndash99 2012

[7] W Wang Y Shang and Y Zhang ldquoA filled function approachfor nonsmooth constrained global optimizationrdquoMathematical

Mathematical Problems in Engineering 11

Problems in Engineering vol 2010 Article ID 310391 9 pages2010

[8] L I Rozonoer ldquoLinear-quadratic optimization and some gen-eral hypotheses on optimal controlrdquoMathematical Problems inEngineering vol 5 no 4 pp 275ndash289 1999

[9] J T Betts ldquoSurvey of numerical methods for trajectory opti-mizationrdquo Journal of Guidance Control and Dynamics vol 21no 2 pp 193ndash207 1998

[10] G Q Huang Y P Lu and Y Nan ldquoA survey of numericalalgorithms for trajectory optimization of flight vehiclesrdquo ScienceChina-Technological Sciences vol 55 no 9 pp 2538ndash2560 2012

[11] Y Ulybyshev ldquoDiscrete pseudocontrol sets for optimal controlproblemsrdquo Journal of Guidance Control and Dynamics vol 33no 4 pp 1133ndash1142 2010

[12] M Vasile and F Zuiani ldquoMulti-agent collaborative search anagent-based memetic multi-objective optimization algorithmapplied to space trajectory designrdquo Journal of Aerospace Engi-neering G vol 225 no 11 pp 1211ndash1227 2011

[13] M Pontani and B A Conway ldquoParticle swarm optimizationapplied to space trajectoriesrdquo Journal of Guidance Control andDynamics vol 33 no 5 pp 1429ndash1441 2010

[14] Y P Ulybyshev ldquoOptimization of low-thrust orbit transfers withconstraintsrdquo Cosmic Research vol 50 no 5 pp 376ndash390 2012

[15] M Chen R Mei and B Jiang ldquoSliding mode control for a classof uncertainMIMOnonlinear systemswith application to near-space vehiclesrdquoMathematical Problems in Engineering vol 2013Article ID 180589 9 pages 2013

[16] H Zhang Y Shi M Q Xu and H T Cui ldquoObserver-basedtracking controller design for networked predictive controlsystems with uncertain markov delaysrdquo in Proceedings of theAmerican Control Conference pp 5682ndash5687 2012

[17] H Zhang and Y Shi ldquoRobust H-infinity sliding-mode controlfor markovian jump systems subject to intermittent observa-tions and partially known transition probabilitiesrdquo Systems ampControl Letters vol 62 no 12 pp 1114ndash1124 2013

[18] Z Shuai H Zhang J Wang J Li and M Ouyang ldquoCombinedAFS andDYC control of four-wheel-independent-drive electricvehicles over CAN network with time-varying delaysrdquo IEEETransactions on Vehicular Technology no 99 p 1 2013

[19] H Zhang Y Shi and A Saadat Mehr ldquoRobust static outputfeedback control and remote PID design for networked motorsystemsrdquo IEEETransactions on Industrial Electronics vol 58 no12 pp 5396ndash5405 2011

[20] H Zhang Y Shi and M X Liu ldquoH-infinity step trackingcontrol for networked discrete-time nonlinear systems withintegral and predictive actionsrdquo IEEE Transactions on IndustrialInformatics vol 9 no 1 pp 337ndash345 2013

[21] X J Wang and S P Wang ldquoAdaptive fuzzy robust controlfor a class of nonlinear systems via small gain theoremrdquoMathematical Problems in Engineering vol 2013 Article ID201432 11 pages 2013

[22] V Brinda S Dasgupta and M Lal ldquoTrajectory optimizationand guidance of an air breathing hypersonic vehiclerdquo in Pro-ceedings of the 14th AIAAAHI International Space Planes andHypersonics Systems Technologies Conference vol 2 pp 856ndash869 November 2006

[23] B Tian and Q Zong ldquoOptimal guidance for reentry vehiclesbased on indirect legendre pseudospectral methodrdquoActa Astro-nautica vol 68 no 7-8 pp 1176ndash1184 2011

[24] M H Graszliglin J Telaar and UM Schottle ldquoAscent and reentryguidance concept based on NLP-methodsrdquo Acta Astronauticavol 55 no 3ndash9 pp 461ndash471 2004

[25] P Lu and B Pan ldquoHighly constrained optimal launch ascentguidancerdquo Journal of Guidance Control and Dynamics vol 33no 2 pp 404ndash414 2010

[26] P Lu H Sun and B Tsai ldquoClosed-loop endo-atmosphericascent guidancerdquo in Proceedings of the AIAA Guidance Navi-gation and Control Conference Monterey Calif USA 2002

[27] J Oscar Murillo and P Lu ldquoFast ascent trajectory optimizationfor hypersonic air-breathing vehiclesrdquo in Proceedings of theAIAA Guidance Navigation and Control Conference pp 5ndash8Toronto Canada 2010

[28] M Marrdonny and M Mobed ldquoA guidance algorithm forlaunch to equatorial orbitrdquo Aircraft Engineering and AerospaceTechnology vol 81 no 2 pp 137ndash148 2009

[29] T Yamamoto and J Kawaguchi ldquoA new real-time guidancestrategy for aerodynamic ascent flightrdquo Acta Astronautica vol61 no 11-12 pp 965ndash977 2007

[30] C Filici andR S Sanchez Pena ldquoOnline guidance updates usingneural networksrdquo Acta Astronautica vol 66 no 3-4 pp 477ndash485 2010

[31] P T Boggs A J Kearsley and J W Tolle ldquoA global convergenceanalysis of an algorithm for large-scale nonlinear optimizationproblemsrdquo SIAM Journal on Optimization vol 9 no 4 pp 833ndash862 1999

[32] S Schlenkrich and A Walther ldquoGlobal convergence of quasi-Newton methods based on adjoint Broyden updatesrdquo AppliedNumerical Mathematics vol 59 no 5 pp 1120ndash1136 2009

[33] S Schlenkrich A Griewank and A Walther ldquoOn the localconvergence of adjoint Broyden methodsrdquo Mathematical Pro-gramming vol 121 no 2 pp 221ndash247 2010

[34] S Lian and L Zhang ldquoA simple smooth exact penalty functionfor smooth optimization problemrdquo Journal of Systems Science ampComplexity vol 25 no 3 pp 521ndash528 2012

[35] S Keshmiri R Colgren and M Mirmirani ldquoDevelopment ofan aerodynamic database for a generic hypersonic air vehiclerdquoin Proceedings of the AIAA Guidance Navigation and ControlConference pp 3978ndash3998 San Francisco Calif USA August2005

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Page 7: Research Article Optimal Ascent Guidance for Air-Breathing ...downloads.hindawi.com/journals/mpe/2013/313197.pdfResearch Article Optimal Ascent Guidance for Air-Breathing Launch Vehicle

Mathematical Problems in Engineering 7

From (32) we obtain

limΔ120591rarr0

[119901119896+1

(119909119896 119906119896) minus 119901119896(119909119896 119906119896)] = 0

lim119891(119909119896119906119896)minus 119909119896rarr0

[119901119896+1

(119909119896 119906119896) minus 119901119896(119909119896 119906119896)] = 0

(33)

119901119896+1

(119909119896 119906119896) minus 119901119896(119909119896 119906119896)

= 120578

100381710038171003817100381710038171003817100381710038171003817

119909119896(119864119888119896) minus 119909119888

119896+1+ int

119864119888119896+1

119864119888119896

119896119889119864

100381710038171003817100381710038171003817100381710038171003817

minus1003817100381710038171003817119909119896 (119864119888119896) minus 119909119888

119896

1003817100381710038171003817

le 1205781003817100381710038171003817119909119896 (119864119888119896) minus 119909119888

119896

1003817100381710038171003817

+

100381710038171003817100381710038171003817100381710038171003817

119909119888119896+1

minus 119909119888119896minus int

119864119888119896+1

119864119888119896

119896119889119864

100381710038171003817100381710038171003817100381710038171003817

minus1003817100381710038171003817119909119896 (119864119888119896) minus 119909119888

119896

1003817100381710038171003817

= 120578

100381710038171003817100381710038171003817100381710038171003817

119909119888119896+1

minus 119909119888119896minus int

119864119888119896+1

119864119888119896

119896119889119864

100381710038171003817100381710038171003817100381710038171003817

(34)

From conditions (24) and (30) it is obtained that

119901119896+1

(119909119896+1

119906119896+1

)

le 119901119896+1

(119909119896 119906119896) + 1198971198961205761120575119901119896+1

(119889119896)

= 119901119896(119909119896 119906119896) + 1198971198961205761120575119901119896+1

(119889119896)

+ 120578

100381710038171003817100381710038171003817100381710038171003817

119909119896(119864119888119896) minus 119909119888

119896

minus (119909119888119896+1

minus 119909119888119896minus int

119864119888119896+1

119864119888119896

119896119889119864)

100381710038171003817100381710038171003817100381710038171003817

minus1003817100381710038171003817119909119896 (119864119888119896) minus 119909119888

119896

1003817100381710038171003817

le 119901119896(119909119896 119906119896) + 120578

100381710038171003817100381710038171003817100381710038171003817

119909119888119896+1

minus 119909119888119896minus int

119864119888119896+1

119864119888119896

119896119889119864

100381710038171003817100381710038171003817100381710038171003817

+ 1198971198961205761120575119901119896+1

(119910119896 119889119896)

le 119901119896(119909119896 119906119896) + 120578

100381710038171003817100381710038171003817100381710038171003817

119909119888119896+1

minus 119909119888119896minus int

119864119888119896+1

119864119888119896

119896119889119864

100381710038171003817100381710038171003817100381710038171003817

+ 1198971198961205763120575119901119896+1

(119910119896 119889119896)

lt 119901119896(119909119896 119906119896)

(35)

Thus the sequence 119901119896(119909119896 119906119896) 119896 = 1 2 3 is monotoni-

cally decreasing Considering the existence of a minimizer119901119896(119909119896 119906119896) converges Thus the following equation is estab-

lished

lim119896rarrinfin

(1198971198961205763120575119901119896+1

(119910119896 119889119896) minus 1198971198961205761120575119901119896+1

(119910119896 119889119896)) = 0 (36)

From (36) we derive

lim119896rarrinfin

(119897119896120575119901119896+1

(119910119896 119889119896)) = 0 (37)

If 119889119896

= 0 [119909119896

119906119896]119879 is not the minimizer of penalty function

then 120575119901119896+1

(119910119896 119889119896) = 0 Considering that 119897

119896is bounded away

from zero we derive

lim119896rarrinfin

119889119896= 0

lim119896rarrinfin

120575119901119896+1

(119910119896 119889119896) = 0

lim119896rarrinfin

[119896minus 119891 (119909

119896 119906119896)] = 0

lim119896rarrinfin

(119896+ 119867119896

119909) = 0

lim119896rarrinfin

119867119896

119906= 0

(38)

Thus the solution sequence [119909119896

119906119896]119879 generated by the

algorithmwill converge to the solution of the optimal controlproblem

5 Smooth Approximation forInequality Constraints

The nondifferentiable TPBVP from optimal control problem(11) with the inequality constraints brings difficulties for theproposed algorithm which can only solve a smooth problemSmooth approximations for nondifferentiable optimizationproblems have been studied in [5 6 34] In this sectionwe introduce the smooth approximation method for theinequality constraints If the inequality constraint function isabout the state and the guidance command

119862 (119909 119906) le 0 (39)

the Hamiltonian for the optimal control problem (11) withinequality constraints (39) is given by

119867 = 119892 (119909 119906) + 120582119879119891 (119909 119906) + 120583max 119862 (119909 119906) 0 (40)

However the Hamiltonian (40) is a nondifferentiable func-tion which brings difficulty for the proposed algorithmFor the smooth Hamiltonian approximation a differentiablefunction is substituted for max119862(119909 119906) 0 We denote thenondifferentiable function as follows

max 0 119910 = int

119910

minusinfin

119908 (119911) 119889119911 (41)

where 119908(119911) is the step function

119908 (119911) = 1 if 119911 ge 0

0 if 119911 lt 0(42)

Considering 119904(119911 119886) as the approximation function of 119908(119911)the differentiable approximation to max0 119910 is obtained as

119891119901(119910 119886) asymp int

119910

minusinfin

119904 (119911 119886) 119889119911 = 119910 +1

119886In (1 + 119890

minus119886119910) (43)

8 Mathematical Problems in Engineering

where 119904(119911 119886) is Sigmoid function

119904 (119911 119886) =1

1 + 119890minus119886119911 119886 gt 0 (44)

The Properties of 119891119901(119910 119886) 119886 gt 0 are as follows [5]

(1) 119891119901(119910 119886) is 119899-times continuously differentiable for any

positive integer 119899 with 120597119891119901(119910 119886)120597119910 = 1(1 + 119890

minus119886119910)

and 1205972119891119901(119910 119886)120597119910

2= 119886119890minus119886119910

(1 + 119890minus119886119910

)2

(2) 119891119901(119910 119886) is strictly convex and strictly increasing onR

(3) 119891119901(119910 119886) gt max0 119910

(4) lim119886rarrinfin

119891119901(119910 119886) = max0 119910

(5) 119891119901(119910 119886) gt 119891

119901(119910 1198861) for 119886 lt 119886

1 119910 isin R

Substituting (43) into (40) the following differentiableHamiltonian approximation is obtained

= 119892 (119909 119906) + 120582119879119891 (119909 119906) + 120583 [119862 +

1

119886In (1 + 119890

minus119886119862)] (45)

If the inequality constraint function is a 119902th-order statevariable inequality constraint given by

119862 (119909) le 0 (46)

If 119901 times derivatives of 119862(119909) are required before 119906 appearsexplicitly in the result the Hamiltonian with inequalityconstraints (46) is given by

119867 = 119892 (119909 119906) + 120582119879119891 (119909 119906) + 120583119862

(119902) (47)

where

120583 = 0 if 119862 lt 0 (48)

Considering the differentiable approximation to 120583 as follows

120583 = 120583119908 (119862) asymp 1205831

1 + 119890minus119886119862(49)

the following differentiable Hamiltonian approximation isobtained

= 119892 (119909 119906) + 120582119879119891 (119909 119906) + 120583

1

1 + 119890minus119886119862119862(119902) (50)

With the differentiable Hamiltonian approximation theinequality constraints are considered in the proposed guid-ance algorithm

6 Simulation Results

61 Numerical Calculation Based on Linear TPBVPDiscretiza-tion In this section the linear TPVBP is transformed intolinear equations using the Simpson rule which correspondsto the three-point Newton-Cotes quadrature rule The Simp-son rule for differential equation 119889119883119889119864 = 119910(119864) is given by

int

119887

119886

119910 (119864) 119889119864 =119887 minus 119886

6[119910 (119886) + 4119910(

119886 + 119887

2) + 119910 (119887)] (51)

From (51) we obtain

119883(119887) minus 119883 (119886) =119887 minus 119886

6[119910 (119886) + 4119910(

119886 + 119887

2) + 119910 (119887)] (52)

We disperse the normalized energy 119864 into119873 equal segmentsas [1198640

1198641

sdot sdot sdot 119864119873] with 119864

0= 119864119888119896and 119864

119873= 119864119891 The

differential equation 119889119883119889119864 = 119910(119883 119864) can be transformedinto the following equation

119883(119864119894) minus 119883 (119864

119894minus1)

=119864119894minus 119864119894minus1

6[119910 (119883

119894 119864119894)

+ 4119910 (119883119894+ 119883119894minus1

2119864119894+ 119864119894minus1

2)

+ 119910 (119883119894minus1

119864119894minus1

)] 119894 = 1 2 3 119873

(53)

Based on (53) the linear TPBVP (21) can be transformed intolinear equations described by

119860119883 = 119861 (54)

The linear equations can be solved by Gaussian eliminationFinally we can obtain the solution Δ119909(119864) Δ119906(119864) and

120582(119864) of the linear TPBVP (21) from the solution of the linearequations through interpolating

62 GHV Model The proposed method is applied to theGenericHypersonicVehicle (GHV) to verify the effectivenessby flight simulation In order to develop a complete setof aerodynamic coefficients experimental longitudinal andlateral-directional aerodynamics were obtained for the GHVby using six Langley wind tunnels [35] The aerodynamicdatabase of the GHV is shown in [35] The GHV model withscramjet propulsion system is used for the simulation where119878ref = 33473m2 and the full-scale weight119898ini = 110000 kg

According to the characterization of scramjet propulsionsystem described in [2 27] the development of the propul-sion system begins with the following equation

119879 = 05120601119868sp (120601119872119886) 120588V1198920119862119879 (120572119872119886) (55)

where 120601 is the throttle commandwhich varies from 0 to 1Thecoefficient 119862

119879depends on 120572 and119872

119886as follows

119862119879= minus00012119872

1198861205722+ 0008120572 + 14 minus 01119872

119886 (56)

The impulse 119868sp is given by the following equation

119868sp = minus811198722

119886+ 1004119872

119886minus 52 + 100120601 (57)

63 Simulation Results for Modeling and Initial State ErrorConsidering the high cost of the flight experimental testresearchers do flight numerical simulation hardware-in-the-loop simulation and final flight experimental test to verifythe guidance algorithm step by step In this section flight

Mathematical Problems in Engineering 9

Table 1 Initial and required final states of the vehicle

119903ini(m)

Vini(ms)

120574ini(deg)

119903119903119891

(m)V119903119891

(ms)120574119903119891

(deg)20000 1000 0 35000 2000 0

numerical simulation results for the GHV with modelingerror and initial state error are given to show the effectivenessof the proposed guidance algorithm For the modeling errorcaused by the wind tunnel experimental test the aerody-namic coefficients bias is considered in the flight simulationThe simulations end once the final normalized energy 119864

119891

is reached The optimal guidance algorithm updates thetrajectory every guidance cycle which is 03 s The initial andrequired final states of the air-breathing vehicle are shownin Table 1 The number of the discrete nodes 119873 = 10 Thedynamic pressure inequality constraint is given by

119876 =1

2120588V2 le 140000Pa (58)

To assess the capability of the proposed algorithm to dealwith disturbance the aerodynamic coefficients bias and theinitial state error in the simulations are given by followingcases

(1) Δ119862119871= minus01119862

119871 Δ119862119863= +01119862

119863

(2) Δ119862119871= minus01119862

119871 Δ119862119863= +01119862

119863 Δ119903ini = +500m

(3) Δ119862119871= minus01119862

119871 Δ119862119863= +01119862

119863 Δ119903ini = minus500m

(4) Δ119862119871= minus01119862

119871 Δ119862119863= +01119862

119863 Δ119903ini = +1000m

(5) Δ119862119871= minus01119862

119871 Δ119862119863= +01119862

119863 Δ119903ini = minus1000m

The simulation results of the optimal guidance algorithmare shown in Figures 4 5 6 7 and 8 The altitude velocityflight path angle angle of attack and dynamic pressure arerespectively shown in Figures 4 to 8 As expected the finalstate is achieved under aerodynamic bias and initial stateerror Figure 8 shows that the state inequality constraint issatisfied The dynamic pressure is less than 140000 Pa fordifferent cases

Table 2 shows the terminal error and the fuel cost ofthe optimal guidance algorithm The terminal accuracy ishigh for different cases Moreover the fuel cost of theoptimal guidance algorithm is lower than that of the trackingguidance algorithm A greater initial state error will result inmore fuel savings In summarywith lower fuel cost the resultsof the proposed optimal guidance algorithm satisfies all theequality and inequality constraints The simulation resultsshow the great potential for the final flight experimental test

7 Conclusion

In this paper we present an optimal guidance algorithmfor air-breathing launch vehicle based on optimal trajectorycorrection The proposed guidance algorithm corrects thereference trajectory in every guidance cycle to satisfy the opti-mality condition of the optimal feedback control problem

0 14012010080604020

18000

21000

24000

27000

30000

33000

Alti

tude

(m)

Time (s)

Case 1Case 2Case 3

Case 4Case 5

Figure 4 The altitude of the simulation results

1200

1400

1600

1800

2000

2200Ve

loci

ty (m

s)

Time (s)0 14012010080604020

Case 1Case 2Case 3

Case 4Case 5

Figure 5 The velocity of the simulation results

By linearizing the optimality condition the linear TPBVPis obtained for optimal trajectory correction The solutionof the linear TPBVP is derived using the Simpson rule Thenew trajectory is generated by a linear search for the stepsize of the solution Smooth Hamiltonian approximation isimplemented to the inequality constraints The sufficiencycondition for the global convergence of the guidance algo-rithm is given in this paper

Finally simulations for 5 different cases of modeling andinitial state error are presented Compared with the trackingguidance method the simulation results prove the low fuelcost and high precision of the proposed optimal guidancemethod

10 Mathematical Problems in Engineering

Table 2 The simulation results for optimal guidance algorithm

Case Δ119903119891(m) ΔV

119891(ms) Δ120574

119891(deg) Fuel cost (kg) Fuel cost (tracking method) (kg)

1 minus2820 0136 020 457219 4574622 859 minus003 minus015 458314 4628393 minus1508 007 020 458474 4605664 10816 047 minus023 460292 4776005 minus3243 019 minus016 460133 468894

0

2

4

6

8

Time (s)

Flig

ht p

ath

angl

e (D

EG)

minus2

minus4

0 14012010080604020

Case 1Case 2Case 3

Case 4Case 5

Figure 6 The flight path angle of the simulation results

Ang

le o

f atta

ck (D

EG)

0

1

2

3

4

minus1

minus2

minus3

minus4

minus5

minus6

Time (s)0 14012010080604020

Case 1Case 2Case 3

Case 4Case 5

Figure 7 The angle of attack of the simulation results

Acknowledgments

This work was supported in part by the National NatureScience Foundation of China nos 61203081 and 61174079Doctoral Fund of Ministry of Education of China

20000

40000

60000

80000

100000

120000

140000

Dyn

amic

pre

ssur

e (Pa

)

Time (s)0 14012010080604020

Case 1Case 2Case 3

Case 4Case 5

Figure 8 The dynamic pressure of the simulation results

no 20120142120091 andPrecisionmanufacturing technologyand equipment for metal parts no 2012DFG70640

References

[1] H J Henry Jr and C H McLellan ldquoAir-breathing launch vehi-cle for earth-orbit shuttle New technology and developmentapproachrdquo Journal of Aircraft vol 8 no 5 pp 381ndash387 1971

[2] J E CorbanA J Calise andGA Flandro ldquoReal-time guidancealgorithm for aerospace plane optimal ascent to low earth orbitrdquoin Proceedings of the American Control Conference pp 2475ndash2481 Pittsburgh PA USA June 1989

[3] C H Yam D D Lorenzo and D Izzo ldquoLow-thrust trajectorydesign as a constrained global optimization problemrdquo Journalof Aerospace Engineering G vol 225 no 11 pp 1243ndash1251 2011

[4] B A Conway ldquoA survey of methods available for the numericaloptimization of continuous dynamic systemsrdquo Journal of Opti-mization Theory and Applications vol 152 no 2 pp 271ndash3062012

[5] C H Chen and O L Mangasarian ldquoSmoothing methodsfor convex inequalities and linear complementarity problemsrdquoMathematical Programming vol 71 no 1 pp 51ndash69 1995

[6] X Chen ldquoSmoothing methods for nonsmooth nonconvexminimizationrdquo Mathematical Programming vol 134 no 1 pp71ndash99 2012

[7] W Wang Y Shang and Y Zhang ldquoA filled function approachfor nonsmooth constrained global optimizationrdquoMathematical

Mathematical Problems in Engineering 11

Problems in Engineering vol 2010 Article ID 310391 9 pages2010

[8] L I Rozonoer ldquoLinear-quadratic optimization and some gen-eral hypotheses on optimal controlrdquoMathematical Problems inEngineering vol 5 no 4 pp 275ndash289 1999

[9] J T Betts ldquoSurvey of numerical methods for trajectory opti-mizationrdquo Journal of Guidance Control and Dynamics vol 21no 2 pp 193ndash207 1998

[10] G Q Huang Y P Lu and Y Nan ldquoA survey of numericalalgorithms for trajectory optimization of flight vehiclesrdquo ScienceChina-Technological Sciences vol 55 no 9 pp 2538ndash2560 2012

[11] Y Ulybyshev ldquoDiscrete pseudocontrol sets for optimal controlproblemsrdquo Journal of Guidance Control and Dynamics vol 33no 4 pp 1133ndash1142 2010

[12] M Vasile and F Zuiani ldquoMulti-agent collaborative search anagent-based memetic multi-objective optimization algorithmapplied to space trajectory designrdquo Journal of Aerospace Engi-neering G vol 225 no 11 pp 1211ndash1227 2011

[13] M Pontani and B A Conway ldquoParticle swarm optimizationapplied to space trajectoriesrdquo Journal of Guidance Control andDynamics vol 33 no 5 pp 1429ndash1441 2010

[14] Y P Ulybyshev ldquoOptimization of low-thrust orbit transfers withconstraintsrdquo Cosmic Research vol 50 no 5 pp 376ndash390 2012

[15] M Chen R Mei and B Jiang ldquoSliding mode control for a classof uncertainMIMOnonlinear systemswith application to near-space vehiclesrdquoMathematical Problems in Engineering vol 2013Article ID 180589 9 pages 2013

[16] H Zhang Y Shi M Q Xu and H T Cui ldquoObserver-basedtracking controller design for networked predictive controlsystems with uncertain markov delaysrdquo in Proceedings of theAmerican Control Conference pp 5682ndash5687 2012

[17] H Zhang and Y Shi ldquoRobust H-infinity sliding-mode controlfor markovian jump systems subject to intermittent observa-tions and partially known transition probabilitiesrdquo Systems ampControl Letters vol 62 no 12 pp 1114ndash1124 2013

[18] Z Shuai H Zhang J Wang J Li and M Ouyang ldquoCombinedAFS andDYC control of four-wheel-independent-drive electricvehicles over CAN network with time-varying delaysrdquo IEEETransactions on Vehicular Technology no 99 p 1 2013

[19] H Zhang Y Shi and A Saadat Mehr ldquoRobust static outputfeedback control and remote PID design for networked motorsystemsrdquo IEEETransactions on Industrial Electronics vol 58 no12 pp 5396ndash5405 2011

[20] H Zhang Y Shi and M X Liu ldquoH-infinity step trackingcontrol for networked discrete-time nonlinear systems withintegral and predictive actionsrdquo IEEE Transactions on IndustrialInformatics vol 9 no 1 pp 337ndash345 2013

[21] X J Wang and S P Wang ldquoAdaptive fuzzy robust controlfor a class of nonlinear systems via small gain theoremrdquoMathematical Problems in Engineering vol 2013 Article ID201432 11 pages 2013

[22] V Brinda S Dasgupta and M Lal ldquoTrajectory optimizationand guidance of an air breathing hypersonic vehiclerdquo in Pro-ceedings of the 14th AIAAAHI International Space Planes andHypersonics Systems Technologies Conference vol 2 pp 856ndash869 November 2006

[23] B Tian and Q Zong ldquoOptimal guidance for reentry vehiclesbased on indirect legendre pseudospectral methodrdquoActa Astro-nautica vol 68 no 7-8 pp 1176ndash1184 2011

[24] M H Graszliglin J Telaar and UM Schottle ldquoAscent and reentryguidance concept based on NLP-methodsrdquo Acta Astronauticavol 55 no 3ndash9 pp 461ndash471 2004

[25] P Lu and B Pan ldquoHighly constrained optimal launch ascentguidancerdquo Journal of Guidance Control and Dynamics vol 33no 2 pp 404ndash414 2010

[26] P Lu H Sun and B Tsai ldquoClosed-loop endo-atmosphericascent guidancerdquo in Proceedings of the AIAA Guidance Navi-gation and Control Conference Monterey Calif USA 2002

[27] J Oscar Murillo and P Lu ldquoFast ascent trajectory optimizationfor hypersonic air-breathing vehiclesrdquo in Proceedings of theAIAA Guidance Navigation and Control Conference pp 5ndash8Toronto Canada 2010

[28] M Marrdonny and M Mobed ldquoA guidance algorithm forlaunch to equatorial orbitrdquo Aircraft Engineering and AerospaceTechnology vol 81 no 2 pp 137ndash148 2009

[29] T Yamamoto and J Kawaguchi ldquoA new real-time guidancestrategy for aerodynamic ascent flightrdquo Acta Astronautica vol61 no 11-12 pp 965ndash977 2007

[30] C Filici andR S Sanchez Pena ldquoOnline guidance updates usingneural networksrdquo Acta Astronautica vol 66 no 3-4 pp 477ndash485 2010

[31] P T Boggs A J Kearsley and J W Tolle ldquoA global convergenceanalysis of an algorithm for large-scale nonlinear optimizationproblemsrdquo SIAM Journal on Optimization vol 9 no 4 pp 833ndash862 1999

[32] S Schlenkrich and A Walther ldquoGlobal convergence of quasi-Newton methods based on adjoint Broyden updatesrdquo AppliedNumerical Mathematics vol 59 no 5 pp 1120ndash1136 2009

[33] S Schlenkrich A Griewank and A Walther ldquoOn the localconvergence of adjoint Broyden methodsrdquo Mathematical Pro-gramming vol 121 no 2 pp 221ndash247 2010

[34] S Lian and L Zhang ldquoA simple smooth exact penalty functionfor smooth optimization problemrdquo Journal of Systems Science ampComplexity vol 25 no 3 pp 521ndash528 2012

[35] S Keshmiri R Colgren and M Mirmirani ldquoDevelopment ofan aerodynamic database for a generic hypersonic air vehiclerdquoin Proceedings of the AIAA Guidance Navigation and ControlConference pp 3978ndash3998 San Francisco Calif USA August2005

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Page 8: Research Article Optimal Ascent Guidance for Air-Breathing ...downloads.hindawi.com/journals/mpe/2013/313197.pdfResearch Article Optimal Ascent Guidance for Air-Breathing Launch Vehicle

8 Mathematical Problems in Engineering

where 119904(119911 119886) is Sigmoid function

119904 (119911 119886) =1

1 + 119890minus119886119911 119886 gt 0 (44)

The Properties of 119891119901(119910 119886) 119886 gt 0 are as follows [5]

(1) 119891119901(119910 119886) is 119899-times continuously differentiable for any

positive integer 119899 with 120597119891119901(119910 119886)120597119910 = 1(1 + 119890

minus119886119910)

and 1205972119891119901(119910 119886)120597119910

2= 119886119890minus119886119910

(1 + 119890minus119886119910

)2

(2) 119891119901(119910 119886) is strictly convex and strictly increasing onR

(3) 119891119901(119910 119886) gt max0 119910

(4) lim119886rarrinfin

119891119901(119910 119886) = max0 119910

(5) 119891119901(119910 119886) gt 119891

119901(119910 1198861) for 119886 lt 119886

1 119910 isin R

Substituting (43) into (40) the following differentiableHamiltonian approximation is obtained

= 119892 (119909 119906) + 120582119879119891 (119909 119906) + 120583 [119862 +

1

119886In (1 + 119890

minus119886119862)] (45)

If the inequality constraint function is a 119902th-order statevariable inequality constraint given by

119862 (119909) le 0 (46)

If 119901 times derivatives of 119862(119909) are required before 119906 appearsexplicitly in the result the Hamiltonian with inequalityconstraints (46) is given by

119867 = 119892 (119909 119906) + 120582119879119891 (119909 119906) + 120583119862

(119902) (47)

where

120583 = 0 if 119862 lt 0 (48)

Considering the differentiable approximation to 120583 as follows

120583 = 120583119908 (119862) asymp 1205831

1 + 119890minus119886119862(49)

the following differentiable Hamiltonian approximation isobtained

= 119892 (119909 119906) + 120582119879119891 (119909 119906) + 120583

1

1 + 119890minus119886119862119862(119902) (50)

With the differentiable Hamiltonian approximation theinequality constraints are considered in the proposed guid-ance algorithm

6 Simulation Results

61 Numerical Calculation Based on Linear TPBVPDiscretiza-tion In this section the linear TPVBP is transformed intolinear equations using the Simpson rule which correspondsto the three-point Newton-Cotes quadrature rule The Simp-son rule for differential equation 119889119883119889119864 = 119910(119864) is given by

int

119887

119886

119910 (119864) 119889119864 =119887 minus 119886

6[119910 (119886) + 4119910(

119886 + 119887

2) + 119910 (119887)] (51)

From (51) we obtain

119883(119887) minus 119883 (119886) =119887 minus 119886

6[119910 (119886) + 4119910(

119886 + 119887

2) + 119910 (119887)] (52)

We disperse the normalized energy 119864 into119873 equal segmentsas [1198640

1198641

sdot sdot sdot 119864119873] with 119864

0= 119864119888119896and 119864

119873= 119864119891 The

differential equation 119889119883119889119864 = 119910(119883 119864) can be transformedinto the following equation

119883(119864119894) minus 119883 (119864

119894minus1)

=119864119894minus 119864119894minus1

6[119910 (119883

119894 119864119894)

+ 4119910 (119883119894+ 119883119894minus1

2119864119894+ 119864119894minus1

2)

+ 119910 (119883119894minus1

119864119894minus1

)] 119894 = 1 2 3 119873

(53)

Based on (53) the linear TPBVP (21) can be transformed intolinear equations described by

119860119883 = 119861 (54)

The linear equations can be solved by Gaussian eliminationFinally we can obtain the solution Δ119909(119864) Δ119906(119864) and

120582(119864) of the linear TPBVP (21) from the solution of the linearequations through interpolating

62 GHV Model The proposed method is applied to theGenericHypersonicVehicle (GHV) to verify the effectivenessby flight simulation In order to develop a complete setof aerodynamic coefficients experimental longitudinal andlateral-directional aerodynamics were obtained for the GHVby using six Langley wind tunnels [35] The aerodynamicdatabase of the GHV is shown in [35] The GHV model withscramjet propulsion system is used for the simulation where119878ref = 33473m2 and the full-scale weight119898ini = 110000 kg

According to the characterization of scramjet propulsionsystem described in [2 27] the development of the propul-sion system begins with the following equation

119879 = 05120601119868sp (120601119872119886) 120588V1198920119862119879 (120572119872119886) (55)

where 120601 is the throttle commandwhich varies from 0 to 1Thecoefficient 119862

119879depends on 120572 and119872

119886as follows

119862119879= minus00012119872

1198861205722+ 0008120572 + 14 minus 01119872

119886 (56)

The impulse 119868sp is given by the following equation

119868sp = minus811198722

119886+ 1004119872

119886minus 52 + 100120601 (57)

63 Simulation Results for Modeling and Initial State ErrorConsidering the high cost of the flight experimental testresearchers do flight numerical simulation hardware-in-the-loop simulation and final flight experimental test to verifythe guidance algorithm step by step In this section flight

Mathematical Problems in Engineering 9

Table 1 Initial and required final states of the vehicle

119903ini(m)

Vini(ms)

120574ini(deg)

119903119903119891

(m)V119903119891

(ms)120574119903119891

(deg)20000 1000 0 35000 2000 0

numerical simulation results for the GHV with modelingerror and initial state error are given to show the effectivenessof the proposed guidance algorithm For the modeling errorcaused by the wind tunnel experimental test the aerody-namic coefficients bias is considered in the flight simulationThe simulations end once the final normalized energy 119864

119891

is reached The optimal guidance algorithm updates thetrajectory every guidance cycle which is 03 s The initial andrequired final states of the air-breathing vehicle are shownin Table 1 The number of the discrete nodes 119873 = 10 Thedynamic pressure inequality constraint is given by

119876 =1

2120588V2 le 140000Pa (58)

To assess the capability of the proposed algorithm to dealwith disturbance the aerodynamic coefficients bias and theinitial state error in the simulations are given by followingcases

(1) Δ119862119871= minus01119862

119871 Δ119862119863= +01119862

119863

(2) Δ119862119871= minus01119862

119871 Δ119862119863= +01119862

119863 Δ119903ini = +500m

(3) Δ119862119871= minus01119862

119871 Δ119862119863= +01119862

119863 Δ119903ini = minus500m

(4) Δ119862119871= minus01119862

119871 Δ119862119863= +01119862

119863 Δ119903ini = +1000m

(5) Δ119862119871= minus01119862

119871 Δ119862119863= +01119862

119863 Δ119903ini = minus1000m

The simulation results of the optimal guidance algorithmare shown in Figures 4 5 6 7 and 8 The altitude velocityflight path angle angle of attack and dynamic pressure arerespectively shown in Figures 4 to 8 As expected the finalstate is achieved under aerodynamic bias and initial stateerror Figure 8 shows that the state inequality constraint issatisfied The dynamic pressure is less than 140000 Pa fordifferent cases

Table 2 shows the terminal error and the fuel cost ofthe optimal guidance algorithm The terminal accuracy ishigh for different cases Moreover the fuel cost of theoptimal guidance algorithm is lower than that of the trackingguidance algorithm A greater initial state error will result inmore fuel savings In summarywith lower fuel cost the resultsof the proposed optimal guidance algorithm satisfies all theequality and inequality constraints The simulation resultsshow the great potential for the final flight experimental test

7 Conclusion

In this paper we present an optimal guidance algorithmfor air-breathing launch vehicle based on optimal trajectorycorrection The proposed guidance algorithm corrects thereference trajectory in every guidance cycle to satisfy the opti-mality condition of the optimal feedback control problem

0 14012010080604020

18000

21000

24000

27000

30000

33000

Alti

tude

(m)

Time (s)

Case 1Case 2Case 3

Case 4Case 5

Figure 4 The altitude of the simulation results

1200

1400

1600

1800

2000

2200Ve

loci

ty (m

s)

Time (s)0 14012010080604020

Case 1Case 2Case 3

Case 4Case 5

Figure 5 The velocity of the simulation results

By linearizing the optimality condition the linear TPBVPis obtained for optimal trajectory correction The solutionof the linear TPBVP is derived using the Simpson rule Thenew trajectory is generated by a linear search for the stepsize of the solution Smooth Hamiltonian approximation isimplemented to the inequality constraints The sufficiencycondition for the global convergence of the guidance algo-rithm is given in this paper

Finally simulations for 5 different cases of modeling andinitial state error are presented Compared with the trackingguidance method the simulation results prove the low fuelcost and high precision of the proposed optimal guidancemethod

10 Mathematical Problems in Engineering

Table 2 The simulation results for optimal guidance algorithm

Case Δ119903119891(m) ΔV

119891(ms) Δ120574

119891(deg) Fuel cost (kg) Fuel cost (tracking method) (kg)

1 minus2820 0136 020 457219 4574622 859 minus003 minus015 458314 4628393 minus1508 007 020 458474 4605664 10816 047 minus023 460292 4776005 minus3243 019 minus016 460133 468894

0

2

4

6

8

Time (s)

Flig

ht p

ath

angl

e (D

EG)

minus2

minus4

0 14012010080604020

Case 1Case 2Case 3

Case 4Case 5

Figure 6 The flight path angle of the simulation results

Ang

le o

f atta

ck (D

EG)

0

1

2

3

4

minus1

minus2

minus3

minus4

minus5

minus6

Time (s)0 14012010080604020

Case 1Case 2Case 3

Case 4Case 5

Figure 7 The angle of attack of the simulation results

Acknowledgments

This work was supported in part by the National NatureScience Foundation of China nos 61203081 and 61174079Doctoral Fund of Ministry of Education of China

20000

40000

60000

80000

100000

120000

140000

Dyn

amic

pre

ssur

e (Pa

)

Time (s)0 14012010080604020

Case 1Case 2Case 3

Case 4Case 5

Figure 8 The dynamic pressure of the simulation results

no 20120142120091 andPrecisionmanufacturing technologyand equipment for metal parts no 2012DFG70640

References

[1] H J Henry Jr and C H McLellan ldquoAir-breathing launch vehi-cle for earth-orbit shuttle New technology and developmentapproachrdquo Journal of Aircraft vol 8 no 5 pp 381ndash387 1971

[2] J E CorbanA J Calise andGA Flandro ldquoReal-time guidancealgorithm for aerospace plane optimal ascent to low earth orbitrdquoin Proceedings of the American Control Conference pp 2475ndash2481 Pittsburgh PA USA June 1989

[3] C H Yam D D Lorenzo and D Izzo ldquoLow-thrust trajectorydesign as a constrained global optimization problemrdquo Journalof Aerospace Engineering G vol 225 no 11 pp 1243ndash1251 2011

[4] B A Conway ldquoA survey of methods available for the numericaloptimization of continuous dynamic systemsrdquo Journal of Opti-mization Theory and Applications vol 152 no 2 pp 271ndash3062012

[5] C H Chen and O L Mangasarian ldquoSmoothing methodsfor convex inequalities and linear complementarity problemsrdquoMathematical Programming vol 71 no 1 pp 51ndash69 1995

[6] X Chen ldquoSmoothing methods for nonsmooth nonconvexminimizationrdquo Mathematical Programming vol 134 no 1 pp71ndash99 2012

[7] W Wang Y Shang and Y Zhang ldquoA filled function approachfor nonsmooth constrained global optimizationrdquoMathematical

Mathematical Problems in Engineering 11

Problems in Engineering vol 2010 Article ID 310391 9 pages2010

[8] L I Rozonoer ldquoLinear-quadratic optimization and some gen-eral hypotheses on optimal controlrdquoMathematical Problems inEngineering vol 5 no 4 pp 275ndash289 1999

[9] J T Betts ldquoSurvey of numerical methods for trajectory opti-mizationrdquo Journal of Guidance Control and Dynamics vol 21no 2 pp 193ndash207 1998

[10] G Q Huang Y P Lu and Y Nan ldquoA survey of numericalalgorithms for trajectory optimization of flight vehiclesrdquo ScienceChina-Technological Sciences vol 55 no 9 pp 2538ndash2560 2012

[11] Y Ulybyshev ldquoDiscrete pseudocontrol sets for optimal controlproblemsrdquo Journal of Guidance Control and Dynamics vol 33no 4 pp 1133ndash1142 2010

[12] M Vasile and F Zuiani ldquoMulti-agent collaborative search anagent-based memetic multi-objective optimization algorithmapplied to space trajectory designrdquo Journal of Aerospace Engi-neering G vol 225 no 11 pp 1211ndash1227 2011

[13] M Pontani and B A Conway ldquoParticle swarm optimizationapplied to space trajectoriesrdquo Journal of Guidance Control andDynamics vol 33 no 5 pp 1429ndash1441 2010

[14] Y P Ulybyshev ldquoOptimization of low-thrust orbit transfers withconstraintsrdquo Cosmic Research vol 50 no 5 pp 376ndash390 2012

[15] M Chen R Mei and B Jiang ldquoSliding mode control for a classof uncertainMIMOnonlinear systemswith application to near-space vehiclesrdquoMathematical Problems in Engineering vol 2013Article ID 180589 9 pages 2013

[16] H Zhang Y Shi M Q Xu and H T Cui ldquoObserver-basedtracking controller design for networked predictive controlsystems with uncertain markov delaysrdquo in Proceedings of theAmerican Control Conference pp 5682ndash5687 2012

[17] H Zhang and Y Shi ldquoRobust H-infinity sliding-mode controlfor markovian jump systems subject to intermittent observa-tions and partially known transition probabilitiesrdquo Systems ampControl Letters vol 62 no 12 pp 1114ndash1124 2013

[18] Z Shuai H Zhang J Wang J Li and M Ouyang ldquoCombinedAFS andDYC control of four-wheel-independent-drive electricvehicles over CAN network with time-varying delaysrdquo IEEETransactions on Vehicular Technology no 99 p 1 2013

[19] H Zhang Y Shi and A Saadat Mehr ldquoRobust static outputfeedback control and remote PID design for networked motorsystemsrdquo IEEETransactions on Industrial Electronics vol 58 no12 pp 5396ndash5405 2011

[20] H Zhang Y Shi and M X Liu ldquoH-infinity step trackingcontrol for networked discrete-time nonlinear systems withintegral and predictive actionsrdquo IEEE Transactions on IndustrialInformatics vol 9 no 1 pp 337ndash345 2013

[21] X J Wang and S P Wang ldquoAdaptive fuzzy robust controlfor a class of nonlinear systems via small gain theoremrdquoMathematical Problems in Engineering vol 2013 Article ID201432 11 pages 2013

[22] V Brinda S Dasgupta and M Lal ldquoTrajectory optimizationand guidance of an air breathing hypersonic vehiclerdquo in Pro-ceedings of the 14th AIAAAHI International Space Planes andHypersonics Systems Technologies Conference vol 2 pp 856ndash869 November 2006

[23] B Tian and Q Zong ldquoOptimal guidance for reentry vehiclesbased on indirect legendre pseudospectral methodrdquoActa Astro-nautica vol 68 no 7-8 pp 1176ndash1184 2011

[24] M H Graszliglin J Telaar and UM Schottle ldquoAscent and reentryguidance concept based on NLP-methodsrdquo Acta Astronauticavol 55 no 3ndash9 pp 461ndash471 2004

[25] P Lu and B Pan ldquoHighly constrained optimal launch ascentguidancerdquo Journal of Guidance Control and Dynamics vol 33no 2 pp 404ndash414 2010

[26] P Lu H Sun and B Tsai ldquoClosed-loop endo-atmosphericascent guidancerdquo in Proceedings of the AIAA Guidance Navi-gation and Control Conference Monterey Calif USA 2002

[27] J Oscar Murillo and P Lu ldquoFast ascent trajectory optimizationfor hypersonic air-breathing vehiclesrdquo in Proceedings of theAIAA Guidance Navigation and Control Conference pp 5ndash8Toronto Canada 2010

[28] M Marrdonny and M Mobed ldquoA guidance algorithm forlaunch to equatorial orbitrdquo Aircraft Engineering and AerospaceTechnology vol 81 no 2 pp 137ndash148 2009

[29] T Yamamoto and J Kawaguchi ldquoA new real-time guidancestrategy for aerodynamic ascent flightrdquo Acta Astronautica vol61 no 11-12 pp 965ndash977 2007

[30] C Filici andR S Sanchez Pena ldquoOnline guidance updates usingneural networksrdquo Acta Astronautica vol 66 no 3-4 pp 477ndash485 2010

[31] P T Boggs A J Kearsley and J W Tolle ldquoA global convergenceanalysis of an algorithm for large-scale nonlinear optimizationproblemsrdquo SIAM Journal on Optimization vol 9 no 4 pp 833ndash862 1999

[32] S Schlenkrich and A Walther ldquoGlobal convergence of quasi-Newton methods based on adjoint Broyden updatesrdquo AppliedNumerical Mathematics vol 59 no 5 pp 1120ndash1136 2009

[33] S Schlenkrich A Griewank and A Walther ldquoOn the localconvergence of adjoint Broyden methodsrdquo Mathematical Pro-gramming vol 121 no 2 pp 221ndash247 2010

[34] S Lian and L Zhang ldquoA simple smooth exact penalty functionfor smooth optimization problemrdquo Journal of Systems Science ampComplexity vol 25 no 3 pp 521ndash528 2012

[35] S Keshmiri R Colgren and M Mirmirani ldquoDevelopment ofan aerodynamic database for a generic hypersonic air vehiclerdquoin Proceedings of the AIAA Guidance Navigation and ControlConference pp 3978ndash3998 San Francisco Calif USA August2005

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Page 9: Research Article Optimal Ascent Guidance for Air-Breathing ...downloads.hindawi.com/journals/mpe/2013/313197.pdfResearch Article Optimal Ascent Guidance for Air-Breathing Launch Vehicle

Mathematical Problems in Engineering 9

Table 1 Initial and required final states of the vehicle

119903ini(m)

Vini(ms)

120574ini(deg)

119903119903119891

(m)V119903119891

(ms)120574119903119891

(deg)20000 1000 0 35000 2000 0

numerical simulation results for the GHV with modelingerror and initial state error are given to show the effectivenessof the proposed guidance algorithm For the modeling errorcaused by the wind tunnel experimental test the aerody-namic coefficients bias is considered in the flight simulationThe simulations end once the final normalized energy 119864

119891

is reached The optimal guidance algorithm updates thetrajectory every guidance cycle which is 03 s The initial andrequired final states of the air-breathing vehicle are shownin Table 1 The number of the discrete nodes 119873 = 10 Thedynamic pressure inequality constraint is given by

119876 =1

2120588V2 le 140000Pa (58)

To assess the capability of the proposed algorithm to dealwith disturbance the aerodynamic coefficients bias and theinitial state error in the simulations are given by followingcases

(1) Δ119862119871= minus01119862

119871 Δ119862119863= +01119862

119863

(2) Δ119862119871= minus01119862

119871 Δ119862119863= +01119862

119863 Δ119903ini = +500m

(3) Δ119862119871= minus01119862

119871 Δ119862119863= +01119862

119863 Δ119903ini = minus500m

(4) Δ119862119871= minus01119862

119871 Δ119862119863= +01119862

119863 Δ119903ini = +1000m

(5) Δ119862119871= minus01119862

119871 Δ119862119863= +01119862

119863 Δ119903ini = minus1000m

The simulation results of the optimal guidance algorithmare shown in Figures 4 5 6 7 and 8 The altitude velocityflight path angle angle of attack and dynamic pressure arerespectively shown in Figures 4 to 8 As expected the finalstate is achieved under aerodynamic bias and initial stateerror Figure 8 shows that the state inequality constraint issatisfied The dynamic pressure is less than 140000 Pa fordifferent cases

Table 2 shows the terminal error and the fuel cost ofthe optimal guidance algorithm The terminal accuracy ishigh for different cases Moreover the fuel cost of theoptimal guidance algorithm is lower than that of the trackingguidance algorithm A greater initial state error will result inmore fuel savings In summarywith lower fuel cost the resultsof the proposed optimal guidance algorithm satisfies all theequality and inequality constraints The simulation resultsshow the great potential for the final flight experimental test

7 Conclusion

In this paper we present an optimal guidance algorithmfor air-breathing launch vehicle based on optimal trajectorycorrection The proposed guidance algorithm corrects thereference trajectory in every guidance cycle to satisfy the opti-mality condition of the optimal feedback control problem

0 14012010080604020

18000

21000

24000

27000

30000

33000

Alti

tude

(m)

Time (s)

Case 1Case 2Case 3

Case 4Case 5

Figure 4 The altitude of the simulation results

1200

1400

1600

1800

2000

2200Ve

loci

ty (m

s)

Time (s)0 14012010080604020

Case 1Case 2Case 3

Case 4Case 5

Figure 5 The velocity of the simulation results

By linearizing the optimality condition the linear TPBVPis obtained for optimal trajectory correction The solutionof the linear TPBVP is derived using the Simpson rule Thenew trajectory is generated by a linear search for the stepsize of the solution Smooth Hamiltonian approximation isimplemented to the inequality constraints The sufficiencycondition for the global convergence of the guidance algo-rithm is given in this paper

Finally simulations for 5 different cases of modeling andinitial state error are presented Compared with the trackingguidance method the simulation results prove the low fuelcost and high precision of the proposed optimal guidancemethod

10 Mathematical Problems in Engineering

Table 2 The simulation results for optimal guidance algorithm

Case Δ119903119891(m) ΔV

119891(ms) Δ120574

119891(deg) Fuel cost (kg) Fuel cost (tracking method) (kg)

1 minus2820 0136 020 457219 4574622 859 minus003 minus015 458314 4628393 minus1508 007 020 458474 4605664 10816 047 minus023 460292 4776005 minus3243 019 minus016 460133 468894

0

2

4

6

8

Time (s)

Flig

ht p

ath

angl

e (D

EG)

minus2

minus4

0 14012010080604020

Case 1Case 2Case 3

Case 4Case 5

Figure 6 The flight path angle of the simulation results

Ang

le o

f atta

ck (D

EG)

0

1

2

3

4

minus1

minus2

minus3

minus4

minus5

minus6

Time (s)0 14012010080604020

Case 1Case 2Case 3

Case 4Case 5

Figure 7 The angle of attack of the simulation results

Acknowledgments

This work was supported in part by the National NatureScience Foundation of China nos 61203081 and 61174079Doctoral Fund of Ministry of Education of China

20000

40000

60000

80000

100000

120000

140000

Dyn

amic

pre

ssur

e (Pa

)

Time (s)0 14012010080604020

Case 1Case 2Case 3

Case 4Case 5

Figure 8 The dynamic pressure of the simulation results

no 20120142120091 andPrecisionmanufacturing technologyand equipment for metal parts no 2012DFG70640

References

[1] H J Henry Jr and C H McLellan ldquoAir-breathing launch vehi-cle for earth-orbit shuttle New technology and developmentapproachrdquo Journal of Aircraft vol 8 no 5 pp 381ndash387 1971

[2] J E CorbanA J Calise andGA Flandro ldquoReal-time guidancealgorithm for aerospace plane optimal ascent to low earth orbitrdquoin Proceedings of the American Control Conference pp 2475ndash2481 Pittsburgh PA USA June 1989

[3] C H Yam D D Lorenzo and D Izzo ldquoLow-thrust trajectorydesign as a constrained global optimization problemrdquo Journalof Aerospace Engineering G vol 225 no 11 pp 1243ndash1251 2011

[4] B A Conway ldquoA survey of methods available for the numericaloptimization of continuous dynamic systemsrdquo Journal of Opti-mization Theory and Applications vol 152 no 2 pp 271ndash3062012

[5] C H Chen and O L Mangasarian ldquoSmoothing methodsfor convex inequalities and linear complementarity problemsrdquoMathematical Programming vol 71 no 1 pp 51ndash69 1995

[6] X Chen ldquoSmoothing methods for nonsmooth nonconvexminimizationrdquo Mathematical Programming vol 134 no 1 pp71ndash99 2012

[7] W Wang Y Shang and Y Zhang ldquoA filled function approachfor nonsmooth constrained global optimizationrdquoMathematical

Mathematical Problems in Engineering 11

Problems in Engineering vol 2010 Article ID 310391 9 pages2010

[8] L I Rozonoer ldquoLinear-quadratic optimization and some gen-eral hypotheses on optimal controlrdquoMathematical Problems inEngineering vol 5 no 4 pp 275ndash289 1999

[9] J T Betts ldquoSurvey of numerical methods for trajectory opti-mizationrdquo Journal of Guidance Control and Dynamics vol 21no 2 pp 193ndash207 1998

[10] G Q Huang Y P Lu and Y Nan ldquoA survey of numericalalgorithms for trajectory optimization of flight vehiclesrdquo ScienceChina-Technological Sciences vol 55 no 9 pp 2538ndash2560 2012

[11] Y Ulybyshev ldquoDiscrete pseudocontrol sets for optimal controlproblemsrdquo Journal of Guidance Control and Dynamics vol 33no 4 pp 1133ndash1142 2010

[12] M Vasile and F Zuiani ldquoMulti-agent collaborative search anagent-based memetic multi-objective optimization algorithmapplied to space trajectory designrdquo Journal of Aerospace Engi-neering G vol 225 no 11 pp 1211ndash1227 2011

[13] M Pontani and B A Conway ldquoParticle swarm optimizationapplied to space trajectoriesrdquo Journal of Guidance Control andDynamics vol 33 no 5 pp 1429ndash1441 2010

[14] Y P Ulybyshev ldquoOptimization of low-thrust orbit transfers withconstraintsrdquo Cosmic Research vol 50 no 5 pp 376ndash390 2012

[15] M Chen R Mei and B Jiang ldquoSliding mode control for a classof uncertainMIMOnonlinear systemswith application to near-space vehiclesrdquoMathematical Problems in Engineering vol 2013Article ID 180589 9 pages 2013

[16] H Zhang Y Shi M Q Xu and H T Cui ldquoObserver-basedtracking controller design for networked predictive controlsystems with uncertain markov delaysrdquo in Proceedings of theAmerican Control Conference pp 5682ndash5687 2012

[17] H Zhang and Y Shi ldquoRobust H-infinity sliding-mode controlfor markovian jump systems subject to intermittent observa-tions and partially known transition probabilitiesrdquo Systems ampControl Letters vol 62 no 12 pp 1114ndash1124 2013

[18] Z Shuai H Zhang J Wang J Li and M Ouyang ldquoCombinedAFS andDYC control of four-wheel-independent-drive electricvehicles over CAN network with time-varying delaysrdquo IEEETransactions on Vehicular Technology no 99 p 1 2013

[19] H Zhang Y Shi and A Saadat Mehr ldquoRobust static outputfeedback control and remote PID design for networked motorsystemsrdquo IEEETransactions on Industrial Electronics vol 58 no12 pp 5396ndash5405 2011

[20] H Zhang Y Shi and M X Liu ldquoH-infinity step trackingcontrol for networked discrete-time nonlinear systems withintegral and predictive actionsrdquo IEEE Transactions on IndustrialInformatics vol 9 no 1 pp 337ndash345 2013

[21] X J Wang and S P Wang ldquoAdaptive fuzzy robust controlfor a class of nonlinear systems via small gain theoremrdquoMathematical Problems in Engineering vol 2013 Article ID201432 11 pages 2013

[22] V Brinda S Dasgupta and M Lal ldquoTrajectory optimizationand guidance of an air breathing hypersonic vehiclerdquo in Pro-ceedings of the 14th AIAAAHI International Space Planes andHypersonics Systems Technologies Conference vol 2 pp 856ndash869 November 2006

[23] B Tian and Q Zong ldquoOptimal guidance for reentry vehiclesbased on indirect legendre pseudospectral methodrdquoActa Astro-nautica vol 68 no 7-8 pp 1176ndash1184 2011

[24] M H Graszliglin J Telaar and UM Schottle ldquoAscent and reentryguidance concept based on NLP-methodsrdquo Acta Astronauticavol 55 no 3ndash9 pp 461ndash471 2004

[25] P Lu and B Pan ldquoHighly constrained optimal launch ascentguidancerdquo Journal of Guidance Control and Dynamics vol 33no 2 pp 404ndash414 2010

[26] P Lu H Sun and B Tsai ldquoClosed-loop endo-atmosphericascent guidancerdquo in Proceedings of the AIAA Guidance Navi-gation and Control Conference Monterey Calif USA 2002

[27] J Oscar Murillo and P Lu ldquoFast ascent trajectory optimizationfor hypersonic air-breathing vehiclesrdquo in Proceedings of theAIAA Guidance Navigation and Control Conference pp 5ndash8Toronto Canada 2010

[28] M Marrdonny and M Mobed ldquoA guidance algorithm forlaunch to equatorial orbitrdquo Aircraft Engineering and AerospaceTechnology vol 81 no 2 pp 137ndash148 2009

[29] T Yamamoto and J Kawaguchi ldquoA new real-time guidancestrategy for aerodynamic ascent flightrdquo Acta Astronautica vol61 no 11-12 pp 965ndash977 2007

[30] C Filici andR S Sanchez Pena ldquoOnline guidance updates usingneural networksrdquo Acta Astronautica vol 66 no 3-4 pp 477ndash485 2010

[31] P T Boggs A J Kearsley and J W Tolle ldquoA global convergenceanalysis of an algorithm for large-scale nonlinear optimizationproblemsrdquo SIAM Journal on Optimization vol 9 no 4 pp 833ndash862 1999

[32] S Schlenkrich and A Walther ldquoGlobal convergence of quasi-Newton methods based on adjoint Broyden updatesrdquo AppliedNumerical Mathematics vol 59 no 5 pp 1120ndash1136 2009

[33] S Schlenkrich A Griewank and A Walther ldquoOn the localconvergence of adjoint Broyden methodsrdquo Mathematical Pro-gramming vol 121 no 2 pp 221ndash247 2010

[34] S Lian and L Zhang ldquoA simple smooth exact penalty functionfor smooth optimization problemrdquo Journal of Systems Science ampComplexity vol 25 no 3 pp 521ndash528 2012

[35] S Keshmiri R Colgren and M Mirmirani ldquoDevelopment ofan aerodynamic database for a generic hypersonic air vehiclerdquoin Proceedings of the AIAA Guidance Navigation and ControlConference pp 3978ndash3998 San Francisco Calif USA August2005

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Page 10: Research Article Optimal Ascent Guidance for Air-Breathing ...downloads.hindawi.com/journals/mpe/2013/313197.pdfResearch Article Optimal Ascent Guidance for Air-Breathing Launch Vehicle

10 Mathematical Problems in Engineering

Table 2 The simulation results for optimal guidance algorithm

Case Δ119903119891(m) ΔV

119891(ms) Δ120574

119891(deg) Fuel cost (kg) Fuel cost (tracking method) (kg)

1 minus2820 0136 020 457219 4574622 859 minus003 minus015 458314 4628393 minus1508 007 020 458474 4605664 10816 047 minus023 460292 4776005 minus3243 019 minus016 460133 468894

0

2

4

6

8

Time (s)

Flig

ht p

ath

angl

e (D

EG)

minus2

minus4

0 14012010080604020

Case 1Case 2Case 3

Case 4Case 5

Figure 6 The flight path angle of the simulation results

Ang

le o

f atta

ck (D

EG)

0

1

2

3

4

minus1

minus2

minus3

minus4

minus5

minus6

Time (s)0 14012010080604020

Case 1Case 2Case 3

Case 4Case 5

Figure 7 The angle of attack of the simulation results

Acknowledgments

This work was supported in part by the National NatureScience Foundation of China nos 61203081 and 61174079Doctoral Fund of Ministry of Education of China

20000

40000

60000

80000

100000

120000

140000

Dyn

amic

pre

ssur

e (Pa

)

Time (s)0 14012010080604020

Case 1Case 2Case 3

Case 4Case 5

Figure 8 The dynamic pressure of the simulation results

no 20120142120091 andPrecisionmanufacturing technologyand equipment for metal parts no 2012DFG70640

References

[1] H J Henry Jr and C H McLellan ldquoAir-breathing launch vehi-cle for earth-orbit shuttle New technology and developmentapproachrdquo Journal of Aircraft vol 8 no 5 pp 381ndash387 1971

[2] J E CorbanA J Calise andGA Flandro ldquoReal-time guidancealgorithm for aerospace plane optimal ascent to low earth orbitrdquoin Proceedings of the American Control Conference pp 2475ndash2481 Pittsburgh PA USA June 1989

[3] C H Yam D D Lorenzo and D Izzo ldquoLow-thrust trajectorydesign as a constrained global optimization problemrdquo Journalof Aerospace Engineering G vol 225 no 11 pp 1243ndash1251 2011

[4] B A Conway ldquoA survey of methods available for the numericaloptimization of continuous dynamic systemsrdquo Journal of Opti-mization Theory and Applications vol 152 no 2 pp 271ndash3062012

[5] C H Chen and O L Mangasarian ldquoSmoothing methodsfor convex inequalities and linear complementarity problemsrdquoMathematical Programming vol 71 no 1 pp 51ndash69 1995

[6] X Chen ldquoSmoothing methods for nonsmooth nonconvexminimizationrdquo Mathematical Programming vol 134 no 1 pp71ndash99 2012

[7] W Wang Y Shang and Y Zhang ldquoA filled function approachfor nonsmooth constrained global optimizationrdquoMathematical

Mathematical Problems in Engineering 11

Problems in Engineering vol 2010 Article ID 310391 9 pages2010

[8] L I Rozonoer ldquoLinear-quadratic optimization and some gen-eral hypotheses on optimal controlrdquoMathematical Problems inEngineering vol 5 no 4 pp 275ndash289 1999

[9] J T Betts ldquoSurvey of numerical methods for trajectory opti-mizationrdquo Journal of Guidance Control and Dynamics vol 21no 2 pp 193ndash207 1998

[10] G Q Huang Y P Lu and Y Nan ldquoA survey of numericalalgorithms for trajectory optimization of flight vehiclesrdquo ScienceChina-Technological Sciences vol 55 no 9 pp 2538ndash2560 2012

[11] Y Ulybyshev ldquoDiscrete pseudocontrol sets for optimal controlproblemsrdquo Journal of Guidance Control and Dynamics vol 33no 4 pp 1133ndash1142 2010

[12] M Vasile and F Zuiani ldquoMulti-agent collaborative search anagent-based memetic multi-objective optimization algorithmapplied to space trajectory designrdquo Journal of Aerospace Engi-neering G vol 225 no 11 pp 1211ndash1227 2011

[13] M Pontani and B A Conway ldquoParticle swarm optimizationapplied to space trajectoriesrdquo Journal of Guidance Control andDynamics vol 33 no 5 pp 1429ndash1441 2010

[14] Y P Ulybyshev ldquoOptimization of low-thrust orbit transfers withconstraintsrdquo Cosmic Research vol 50 no 5 pp 376ndash390 2012

[15] M Chen R Mei and B Jiang ldquoSliding mode control for a classof uncertainMIMOnonlinear systemswith application to near-space vehiclesrdquoMathematical Problems in Engineering vol 2013Article ID 180589 9 pages 2013

[16] H Zhang Y Shi M Q Xu and H T Cui ldquoObserver-basedtracking controller design for networked predictive controlsystems with uncertain markov delaysrdquo in Proceedings of theAmerican Control Conference pp 5682ndash5687 2012

[17] H Zhang and Y Shi ldquoRobust H-infinity sliding-mode controlfor markovian jump systems subject to intermittent observa-tions and partially known transition probabilitiesrdquo Systems ampControl Letters vol 62 no 12 pp 1114ndash1124 2013

[18] Z Shuai H Zhang J Wang J Li and M Ouyang ldquoCombinedAFS andDYC control of four-wheel-independent-drive electricvehicles over CAN network with time-varying delaysrdquo IEEETransactions on Vehicular Technology no 99 p 1 2013

[19] H Zhang Y Shi and A Saadat Mehr ldquoRobust static outputfeedback control and remote PID design for networked motorsystemsrdquo IEEETransactions on Industrial Electronics vol 58 no12 pp 5396ndash5405 2011

[20] H Zhang Y Shi and M X Liu ldquoH-infinity step trackingcontrol for networked discrete-time nonlinear systems withintegral and predictive actionsrdquo IEEE Transactions on IndustrialInformatics vol 9 no 1 pp 337ndash345 2013

[21] X J Wang and S P Wang ldquoAdaptive fuzzy robust controlfor a class of nonlinear systems via small gain theoremrdquoMathematical Problems in Engineering vol 2013 Article ID201432 11 pages 2013

[22] V Brinda S Dasgupta and M Lal ldquoTrajectory optimizationand guidance of an air breathing hypersonic vehiclerdquo in Pro-ceedings of the 14th AIAAAHI International Space Planes andHypersonics Systems Technologies Conference vol 2 pp 856ndash869 November 2006

[23] B Tian and Q Zong ldquoOptimal guidance for reentry vehiclesbased on indirect legendre pseudospectral methodrdquoActa Astro-nautica vol 68 no 7-8 pp 1176ndash1184 2011

[24] M H Graszliglin J Telaar and UM Schottle ldquoAscent and reentryguidance concept based on NLP-methodsrdquo Acta Astronauticavol 55 no 3ndash9 pp 461ndash471 2004

[25] P Lu and B Pan ldquoHighly constrained optimal launch ascentguidancerdquo Journal of Guidance Control and Dynamics vol 33no 2 pp 404ndash414 2010

[26] P Lu H Sun and B Tsai ldquoClosed-loop endo-atmosphericascent guidancerdquo in Proceedings of the AIAA Guidance Navi-gation and Control Conference Monterey Calif USA 2002

[27] J Oscar Murillo and P Lu ldquoFast ascent trajectory optimizationfor hypersonic air-breathing vehiclesrdquo in Proceedings of theAIAA Guidance Navigation and Control Conference pp 5ndash8Toronto Canada 2010

[28] M Marrdonny and M Mobed ldquoA guidance algorithm forlaunch to equatorial orbitrdquo Aircraft Engineering and AerospaceTechnology vol 81 no 2 pp 137ndash148 2009

[29] T Yamamoto and J Kawaguchi ldquoA new real-time guidancestrategy for aerodynamic ascent flightrdquo Acta Astronautica vol61 no 11-12 pp 965ndash977 2007

[30] C Filici andR S Sanchez Pena ldquoOnline guidance updates usingneural networksrdquo Acta Astronautica vol 66 no 3-4 pp 477ndash485 2010

[31] P T Boggs A J Kearsley and J W Tolle ldquoA global convergenceanalysis of an algorithm for large-scale nonlinear optimizationproblemsrdquo SIAM Journal on Optimization vol 9 no 4 pp 833ndash862 1999

[32] S Schlenkrich and A Walther ldquoGlobal convergence of quasi-Newton methods based on adjoint Broyden updatesrdquo AppliedNumerical Mathematics vol 59 no 5 pp 1120ndash1136 2009

[33] S Schlenkrich A Griewank and A Walther ldquoOn the localconvergence of adjoint Broyden methodsrdquo Mathematical Pro-gramming vol 121 no 2 pp 221ndash247 2010

[34] S Lian and L Zhang ldquoA simple smooth exact penalty functionfor smooth optimization problemrdquo Journal of Systems Science ampComplexity vol 25 no 3 pp 521ndash528 2012

[35] S Keshmiri R Colgren and M Mirmirani ldquoDevelopment ofan aerodynamic database for a generic hypersonic air vehiclerdquoin Proceedings of the AIAA Guidance Navigation and ControlConference pp 3978ndash3998 San Francisco Calif USA August2005

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Page 11: Research Article Optimal Ascent Guidance for Air-Breathing ...downloads.hindawi.com/journals/mpe/2013/313197.pdfResearch Article Optimal Ascent Guidance for Air-Breathing Launch Vehicle

Mathematical Problems in Engineering 11

Problems in Engineering vol 2010 Article ID 310391 9 pages2010

[8] L I Rozonoer ldquoLinear-quadratic optimization and some gen-eral hypotheses on optimal controlrdquoMathematical Problems inEngineering vol 5 no 4 pp 275ndash289 1999

[9] J T Betts ldquoSurvey of numerical methods for trajectory opti-mizationrdquo Journal of Guidance Control and Dynamics vol 21no 2 pp 193ndash207 1998

[10] G Q Huang Y P Lu and Y Nan ldquoA survey of numericalalgorithms for trajectory optimization of flight vehiclesrdquo ScienceChina-Technological Sciences vol 55 no 9 pp 2538ndash2560 2012

[11] Y Ulybyshev ldquoDiscrete pseudocontrol sets for optimal controlproblemsrdquo Journal of Guidance Control and Dynamics vol 33no 4 pp 1133ndash1142 2010

[12] M Vasile and F Zuiani ldquoMulti-agent collaborative search anagent-based memetic multi-objective optimization algorithmapplied to space trajectory designrdquo Journal of Aerospace Engi-neering G vol 225 no 11 pp 1211ndash1227 2011

[13] M Pontani and B A Conway ldquoParticle swarm optimizationapplied to space trajectoriesrdquo Journal of Guidance Control andDynamics vol 33 no 5 pp 1429ndash1441 2010

[14] Y P Ulybyshev ldquoOptimization of low-thrust orbit transfers withconstraintsrdquo Cosmic Research vol 50 no 5 pp 376ndash390 2012

[15] M Chen R Mei and B Jiang ldquoSliding mode control for a classof uncertainMIMOnonlinear systemswith application to near-space vehiclesrdquoMathematical Problems in Engineering vol 2013Article ID 180589 9 pages 2013

[16] H Zhang Y Shi M Q Xu and H T Cui ldquoObserver-basedtracking controller design for networked predictive controlsystems with uncertain markov delaysrdquo in Proceedings of theAmerican Control Conference pp 5682ndash5687 2012

[17] H Zhang and Y Shi ldquoRobust H-infinity sliding-mode controlfor markovian jump systems subject to intermittent observa-tions and partially known transition probabilitiesrdquo Systems ampControl Letters vol 62 no 12 pp 1114ndash1124 2013

[18] Z Shuai H Zhang J Wang J Li and M Ouyang ldquoCombinedAFS andDYC control of four-wheel-independent-drive electricvehicles over CAN network with time-varying delaysrdquo IEEETransactions on Vehicular Technology no 99 p 1 2013

[19] H Zhang Y Shi and A Saadat Mehr ldquoRobust static outputfeedback control and remote PID design for networked motorsystemsrdquo IEEETransactions on Industrial Electronics vol 58 no12 pp 5396ndash5405 2011

[20] H Zhang Y Shi and M X Liu ldquoH-infinity step trackingcontrol for networked discrete-time nonlinear systems withintegral and predictive actionsrdquo IEEE Transactions on IndustrialInformatics vol 9 no 1 pp 337ndash345 2013

[21] X J Wang and S P Wang ldquoAdaptive fuzzy robust controlfor a class of nonlinear systems via small gain theoremrdquoMathematical Problems in Engineering vol 2013 Article ID201432 11 pages 2013

[22] V Brinda S Dasgupta and M Lal ldquoTrajectory optimizationand guidance of an air breathing hypersonic vehiclerdquo in Pro-ceedings of the 14th AIAAAHI International Space Planes andHypersonics Systems Technologies Conference vol 2 pp 856ndash869 November 2006

[23] B Tian and Q Zong ldquoOptimal guidance for reentry vehiclesbased on indirect legendre pseudospectral methodrdquoActa Astro-nautica vol 68 no 7-8 pp 1176ndash1184 2011

[24] M H Graszliglin J Telaar and UM Schottle ldquoAscent and reentryguidance concept based on NLP-methodsrdquo Acta Astronauticavol 55 no 3ndash9 pp 461ndash471 2004

[25] P Lu and B Pan ldquoHighly constrained optimal launch ascentguidancerdquo Journal of Guidance Control and Dynamics vol 33no 2 pp 404ndash414 2010

[26] P Lu H Sun and B Tsai ldquoClosed-loop endo-atmosphericascent guidancerdquo in Proceedings of the AIAA Guidance Navi-gation and Control Conference Monterey Calif USA 2002

[27] J Oscar Murillo and P Lu ldquoFast ascent trajectory optimizationfor hypersonic air-breathing vehiclesrdquo in Proceedings of theAIAA Guidance Navigation and Control Conference pp 5ndash8Toronto Canada 2010

[28] M Marrdonny and M Mobed ldquoA guidance algorithm forlaunch to equatorial orbitrdquo Aircraft Engineering and AerospaceTechnology vol 81 no 2 pp 137ndash148 2009

[29] T Yamamoto and J Kawaguchi ldquoA new real-time guidancestrategy for aerodynamic ascent flightrdquo Acta Astronautica vol61 no 11-12 pp 965ndash977 2007

[30] C Filici andR S Sanchez Pena ldquoOnline guidance updates usingneural networksrdquo Acta Astronautica vol 66 no 3-4 pp 477ndash485 2010

[31] P T Boggs A J Kearsley and J W Tolle ldquoA global convergenceanalysis of an algorithm for large-scale nonlinear optimizationproblemsrdquo SIAM Journal on Optimization vol 9 no 4 pp 833ndash862 1999

[32] S Schlenkrich and A Walther ldquoGlobal convergence of quasi-Newton methods based on adjoint Broyden updatesrdquo AppliedNumerical Mathematics vol 59 no 5 pp 1120ndash1136 2009

[33] S Schlenkrich A Griewank and A Walther ldquoOn the localconvergence of adjoint Broyden methodsrdquo Mathematical Pro-gramming vol 121 no 2 pp 221ndash247 2010

[34] S Lian and L Zhang ldquoA simple smooth exact penalty functionfor smooth optimization problemrdquo Journal of Systems Science ampComplexity vol 25 no 3 pp 521ndash528 2012

[35] S Keshmiri R Colgren and M Mirmirani ldquoDevelopment ofan aerodynamic database for a generic hypersonic air vehiclerdquoin Proceedings of the AIAA Guidance Navigation and ControlConference pp 3978ndash3998 San Francisco Calif USA August2005

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Page 12: Research Article Optimal Ascent Guidance for Air-Breathing ...downloads.hindawi.com/journals/mpe/2013/313197.pdfResearch Article Optimal Ascent Guidance for Air-Breathing Launch Vehicle

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


AFTERMARKET NSTALLATION ANUAL ASCENT ASCENT XL · 2018. 11. 26. · 052553-021r4 Printed in USA November, 2018 AFTERMARKET INSTALLATION MANUAL ASCENT & ASCENT XL SLIDE-OUT COVER RV

AFTERMARKET NSTALLATION ANUAL ASCENT ASCENT XL · 2018. 11. 26. · 052553-021r4 Printed in USA November, 2018 AFTERMARKET INSTALLATION MANUAL ASCENT & ASCENT XL SLIDE-OUT COVER RV

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Ascent Advertising

Ascent plan

Ascent plan

Notice - First-Ascent Inc.Cry History Visualization Few ☓ ☓ Difference Alert from usual ☓ ☓ ☓ Life Cycle Advice ☓ ☓ ☓ Night Cry Report ☓ ☓ ☓ Breathing Detection

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Ascent - res.cloudinary.com

Ascent - res.cloudinary.com

SITAG AScenT - Felix Bureautique SAsitaG asCent | D | SITAG AScenT üBeRZeUGT DURch FUnKTIoneLLe RAFFIneSSe. exakt angepasste tische, Beistelltische, korpusse, Besprechungs-tische

SITAG AScenT - Felix Bureautique SAsitaG asCent | D | SITAG AScenT üBeRZeUGT DURch FUnKTIoneLLe RAFFIneSSe. exakt angepasste tische, Beistelltische, korpusse, Besprechungs-tische

ASCENT courseware mapping reference - ASCENT- Center for Technical

ASCENT courseware mapping reference - ASCENT- Center for Technical

ASCENT IMPACT - Ascent Group

ASCENT IMPACT - Ascent Group

· PDF file Evolve Your Breathing Essential Techniques for Optimal Performance "Breathing exercises alone, if done RIGHT, will make many a

· PDF file Evolve Your Breathing Essential Techniques for Optimal Performance "Breathing exercises alone, if done RIGHT, will make many a

Building the Adolescent Voice Part A. Breathing, Breathing, Breathing!

Building the Adolescent Voice Part A. Breathing, Breathing, Breathing!

Ascent 456

Ascent 456

Optimal Trajectory Planning for the Apollo Moon Landing ... · Optimal Trajectory Planning for the Apollo Moon Landing: Descent, Ascent, and Aborts Aero-Astro 16.323 Optimal Control

Optimal Trajectory Planning for the Apollo Moon Landing ... · Optimal Trajectory Planning for the Apollo Moon Landing: Descent, Ascent, and Aborts Aero-Astro 16.323 Optimal Control

Ascent Manufacturing

Ascent Manufacturing

Design and Analysis of Optimal Ascent Trajectories for ...

Design and Analysis of Optimal Ascent Trajectories for ...

Emanation Ascent

Emanation Ascent

ascent WaKeBoaRd toWeR - pwbimageserver.com · installation guide 1 GUIDE#: BT-ascEnT-005 ascent WaKeBoaRd toWeR installation suPPoRt

ascent WaKeBoaRd toWeR - pwbimageserver.com · installation guide 1 GUIDE#: BT-ascEnT-005 ascent WaKeBoaRd toWeR installation suPPoRt

Ascent exports

Ascent exports

ASCENT 40 & ASCENT 100

ASCENT 40 & ASCENT 100

Introducing ascent

Introducing ascent

ascent - mediafilelibrary.myasealive.com

ascent - mediafilelibrary.myasealive.com