Research ArticleFinite-Time Antisaturation Tracking Control for HypersonicVehicle with Uncertain Dynamics

Yuwei Cui ,1,2 Yushan He,3 Aijun Li,1 Jun Li,3 and Shuo Song3

1School of Automation, Northwestern Polytechnical University, Xi’an Shaanxi 710072, China2AVIC Xi'an Flight Automatic Control Research Institute, Xi'an Shaanxi 710065, China3School of Automation, Harbin Institute of Technology, Harbin 150001, China

Correspondence should be addressed to Yuwei Cui; cuiyuwei0505@mail.nwpu.edu.cn

Received 20 August 2020; Revised 10 October 2020; Accepted 24 October 2020; Published 9 November 2020

Academic Editor: Wu Zhonghua

Copyright © 2020 Yuwei Cui et al. This 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.

This paper concentrates on the problem of finite-time altitude and velocity tracking control for hypersonic flight vehicles thatencounter unmodeled dynamics and input saturations. An adaptive neural finite-time backstepping control strategy isconstructed by designing modified virtual commands and compensation signals. The minimum learning parameter algorithmbased on a radial basis function is employed to approximate the unknown dynamics with low computational burden.Furthermore, an auxiliary system is established to cope with the nonlinearity caused by actuator saturation. It is concluded by aLyapunov-based analysis that the finite-time stability is guaranteed under the developed architecture. Finally, numeralsimulation is provided to demonstrate the effectiveness of the proposed controller.

1. Introduction

In the research of near space aircrafts, generic hypersonicvehicles (GHVs) have become a hotspot in countries all overthe world. It owes much to the significant advantages in flightspeed and concealment, which makes GHVs possess highapplication value in both military and civilian fields. How-ever, the unique integrated design and complex flight envi-ronment have brought tremendous challenges to thecontrol of GHVs. Facing such a dynamic system with strongcoupling, fast time-varying dynamics, high nonlinearity, anduncertainties, scholars have contributed strenuous attempts,including but not limited to sliding mode control [1–3],dynamic compensation control (DCC) [4–6], and adaptivecontrol [7–9].

The backstepping method is an effective strategy to copewith higher-order nonlinear systems, which is widely utilizedin trajectory tracking control for GHVs. However, repeatedderivation for virtual commands leads to a problem called“explosion of complexity (EOC).” The solution for this obsta-cle is a dynamic surface control (DSC), in which the low-passfilter is introduced to provide the differential signal of virtual

command. In [10], a compound control algorithm based ondynamic compensation was proposed so that the desired tra-jectory is tracked under strong coupling and high nonlinear-ity. After that, Zhao and Li designed a robust controller forGHVs with actuator failures, in which the trajectory trackingerrors are restricted with a prescribed range [11]. Unfortu-nately, the above results are not satisfactory enough in termsof convergence speed, which is especially important for aGHV in continual rapid motion. For this reason, an adaptiverobust control strategy was constructed for finite-time track-ing missions when external disturbances and model uncer-tainties were encountered [12]. What is more, Wu et al.established a fixed-time sliding mode backstepping frame-work, and the time-varying disturbances were estimated viaa modified observer [3].

Meanwhile, another ineluctable problem is that the non-linearity and uncertainty in the GHV dynamics must be paidfull attention. As a result, it is difficult to achieve the controlobjectives through traditional control methods deeply relyingon the accurate model information. To relax the conserva-tism, fuzzy logic control (FLC) [13–15] and neural network(NN) [1, 16, 17] have become excellent instruments for

HindawiInternational Journal of Aerospace EngineeringVolume 2020, Article ID 8852339, 13 pageshttps://doi.org/10.1155/2020/8852339

approximation of unknown dynamics. In [13], two fuzzylogic systems were developed to handle the aerodynamicuncertainties of the GHV model. Meanwhile, the constraintsof the attitude tracking errors are guaranteed by virtue of thenonlinear mapping method. On the other hand, aiming at theunknown measurement noises, Liu et al. exploited an adap-tive fuzzy approximator, which could provide a safeguardfor the performances of the proposed controller [14]. Differ-ent from the above methodologies, the advantage of theradial basis function neural network (RBFNN) is that it canapproximate nonlinear smooth functions with arbitrary pre-cision. Therefore, it is extensively applied in the tracking con-trol of GHV. In [16], adaptive neural networks wereemployed to tackle unmodeled dynamics, while the unavail-able external disturbances were estimated accurately via anovel disturbance observer. To proceed further, Ping et al.formulated a neuroadaptive sliding mode control strategyto eliminate the system uncertainties, where the nonlinearityarising from the actuator failures was alleviated significantly[1].

Besides the capability of antiuncertainty, another impor-tant indicator is the stabilization rate. To the best of ourknowledge, compared with an asymptotically stable control,the finite-time control performs better in terms of rapidityand robustness. On the basis of a finite time disturbanceobserver [18], constructs a trajectory tracking backsteppingalgorithm for a hypersonic flight vehicle with the presenceof time-varying environmental disturbance. Subsequently,the finite-time tracking control problem of flexible GHV isinvestigated in [19], while the unavailable system parametersare taken into consideration. To proceed further, the finite-time tracking control scheme is further extended to the caseof unknown actuator failures in [12], where the fixed-timeobserver is adopted to estimate fault information and otheruncertain parameters.

It is noteworthy that another caveat here is input satura-tion constraint. From a practical point of view, it is impossi-ble for actuators to output infinite control torque due to therestriction of mechanical structure. In [20], an adaptive slid-ing mode control algorithm was proposed for flexible GHVs,in which adaptive laws were designed to cope with the actu-ator saturations. Subsequently, Sun et al. constructed abackstepping-based dynamic surface control technique toachieve the altitude and velocity tracking control for GHVs[21]. Not only the phenomenon of EOC is removed by afirst-order filter, but also the errors induced by the filter areeliminated via designing compensation signals. Moreover, aspecific auxiliary system is developed to conquer input satu-ration. With further exploration, an adaptive antisaturationcontroller for GHVs is established in [22], which can ensurethat the system still has the capability of stable tracking evenin the presence of actuator faults.

Motivated by the above observations, this paperaddresses the finite-time trajectory tracking control forGHVs exposed to unmodeled dynamics and input satura-tions. On the basis of the longitudinal model of GHVs, thealtitude and velocity tracking control is achieved via theDCC-based algorithm design. The parameter uncertaintiesare approximated by RBFNN, and actuator saturations are

handled via an auxiliary system. The main contributions ofthe research are summarized as follows:

(1) Compared with [10, 11], the DCC applied in thispaper ensures the global finite-time convergence oftracking errors. In addition, the problem of EOC isrelaxed, and all the tracking errors and estimationerrors will converge to a tiny region containing theorigin in finite time

(2) Different from [16], the minimum learning parame-ter (MLP) algorithm-based RBFNN is employed toapproximate the unknown dynamics. In this way,only a single parameter is adaptively updated ratherthan the whole weight matrix; thus, the excessiveoccupation of the computational resource is avoidedeffectively

(3) In consideration of the antisaturation control in [23],the nonlinearity arising from actuator saturation iscompensated for effectively, while the auxiliary vari-able is restricted to a bounded range. In this paper,the hyperbolic tangent function-based modified aux-iliary relaxes the conservatism creatively

The remainder of this paper is structured as follows: Section2 shows the mathematical model and preliminaries; Section 3details the structure and design procedure of the controller;Section 4 theoretically authenticates the stability of the pro-posed controller; finally, a simulation example is provided inSection 5 and the conclusion is shown in Section 6.

2. System Description and Preliminaries

2.1. Longitudinal Model of GHV. According to the longitudi-nal model of GHVs in [24], the velocity and attitude dynamicequations are given as

_V = T cos α −Dm

−μ sin γ

r2ð1Þ

_γ = L + T sin α

mV−

μ −V2r� �

cos γVr2

, ð2Þ

_h =V sin γ, ð3Þ_α =Q − _γ, ð4Þ

_Q =Myy

Iyy, ð5Þ

which include five rigid-body states: the velocityV , the flight-path angle γ, the altitude h, the angle of attack α, and thepitch rate Q. Meanwhile, four forces or moments are intro-duced as the thrust T , the drag D, the lift moment L, andthe pitching moment Myy. Other parameters m, μ, and Iyydenote the mass, the gravitational constant, and the momentof inertia, respectively. The relevant aerodynamic coefficientis shown as

L = �QSCL, ð6Þ

2 International Journal of Aerospace Engineering

D = �QSCD, ð7ÞT = �QSCT , ð8Þ

Myy = �QS�c CM αð Þ + CM δð Þ + CM Qð Þ½ �, ð9Þr = h + re, ð10Þ

CL = 0:6203α, ð11ÞCD = 0:6450α2 + 0:0043378α + 0:003772, ð12Þ

CT =0:022576β, β < 1,0:0224 + 0:00336β, β > 1,

(ð13Þ

CM αð Þ = −0:035α2 + 0:036117α + 5:3261 × 10−6, ð14Þ

CM qð Þ = �cq −0:6796α2 + 0:3015α − 0:2289� �

2V , ð15Þ

CM δeð Þ = ce δe − αð Þ, ð16Þce = 0:0292, ð17Þ

where the dynamic pressure �Q satisfies �Q = ð1/2ÞρV2 with ρdenoting the density of air; S and re denote the reference areaand the radius of the Earth, respectively; and�c and ce are bothconstants.

It is noteworthy that most of the existing studies aboutGHV control lack relevant discussions on actuator satura-tion. In practice, there is no doubt that the actuator can onlyoutput a limited control torque due to the inherent mechan-ical structure. This paper will provide a solution for trackingcontrol for GHVs in the presence of input saturation con-straints and external disturbances. To this end, δe is definedas

δe = sat u1ð Þ + d1, ð18Þ

where d1 represents the lumped external disturbance. Thesaturation nonlinearity satðuÞ is expressed as

sat uð Þ =sign uð Þum, u ≥ um,u, u < um,

(ð19Þ

where um denotes the upper bound of satðuÞ and u is thedesired control input.

2.2. NN-Based Radial Basis Function. In this paper, theunmodeled dynamics of GHVs are approximated viaRBFNN. The basic principle of RBFNN is expounded as thefollowing lemma.

Lemma 1 (see [25]). By defining a basis function HðxÞ, anarbitrary continuous smooth function f ðxÞ can be written inthe following form:

f xð Þ =W∗TH xð Þ + ο, 0 < οj j ≤O, ð20Þ

where x = ½x1, x2; ;⋯, xm� represents the input vector and

W∗ = ½w1,w2,⋯,wp�T is the ideal weight matrix; ο and Odenote the approximation error and its upper bound, respec-tively; and hðxÞ = ½h1ðxÞ, h2ðxÞ,⋯, hmðxÞ�T is selected as theGaussian basis function, whose definition is shown as

hi xð Þ = exp −x − cik k222b2i

� �, i = 1,⋯, p, ð21Þ

where ci ∈ Rm and b ∈ Rp denote the center vector and thewidth of Gaussian basis function, respectively.

2.3. Other Preliminaries

Lemma 2 (see [26]). For x ∈ R and μ > 0, the hyperbolic tan-gent function possesses the following property:

0 < xj j − x tanh μxð Þ ≤ Kμ, ð22Þ

with K = 0:2785.

Assumption 3. The external disturbances in the GHV systemis unknown but bounded, i.e., 0 < jdij ≤Di, i = 1, 2, in whichDi are positive constants.

Remark 4. In contrast to the traditional aircraft, GHVmainlyworks in the near-space area, which possesses the character-istics of complicated environment and dramatically changingaerodynamics. The actual external disturbance torque is usu-ally difficult to observe accurately. Therefore, it is necessaryto enhance the controllers’ robustness to disturbances anduncertainties.

3. Finite-Time Controller Design

In this section, a finite-time tracking controller is developedin virtue of the MLP-based adaptive backstepping strategy.To complete the trajectory tracking control, two subsystemsare formulated according to the longitudinal model of GHVin Section 2. For each subsystem, the novel control com-mands and adaptive laws are designed to ensure the globalfinite-time stability of a closed-loop system. One caveat hereis that the altitude tracking dynamic is a three-order system,in which the effect of EOC is too severe to be neglected.Therefore, the compensation signals are constructed on thebasis of a command-filtered control. Meanwhile, a RBFNN-based MLP algorithm is employed to approximate unknowndynamics, such that the burden of computation is relaxedsignificantly. Input saturations are addressed by resorting toan auxiliary system. All the approximation errors are elimi-nated along with the unknown external disturbances throughthe design of the adaptive laws.

3.1. Altitude Control. As the reference trajectory of altitude isdenoted by hd , the corresponding flight path angle γd can be

3International Journal of Aerospace Engineering

written as

γd = arcsin−kh~h − khh tanh ~h

� �+ _hd

V

0@

1A, ð23Þ

where kh and khh are positive constants and γd ∈ ð−90°, 90°Þ.Considering the GHV model in Section 2, there are three

states in the altitude tracking control system, including γ, ϑp,and q. Hence, a vector is introduced as

x = x1, x2, x3½ �T = γ, ϑp, q� T , ð24Þ

where ϑp = α + γ. Subsequently, according to Equations (2),(3), (4) and (5), the altitude tracking subsystem of GHVcan be expressed as

_x1 = f1 x1, Vð Þ + g1 Vð Þx2,_x2 = x3,_x3 = f3 x1, x2, x3, Vð Þ + g3 Vð Þδe = f3 x1, x2, x3, Vð Þ + g3 Vð Þ sat u1ð Þ + d1½ �,

8>><>>:

ð25Þ

with

g1 Vð Þ = 0:31015ρVSm

,

f1 x1, Vð Þ = T sin α

mV−

μ‐V2r� �

cos γVr2

,

f3 x1, x2, x3, Vð Þ = 0:5ρV2S CM αð Þ + CM qð Þ − ceαð ÞIyy

,

g3 Vð Þ = �qS�cceIyy

:

ð26Þ

Assumption 5. The uncertain term giðVÞ possesses an upperbound �gi, which satisfies 0 < jgiðVÞj ≤ �gi, i = 1, 3.

Remark 6. As pointed above, there exists high uncertainty inthe GHV model, due to not only the strong coupling andnonlinearity caused by its own structure but also the fast timevariability caused by the high-speed flight. Accordingly, it isdifficult to obtain an accurate model for GHVs. In this paper,a RBFNN-based control strategy is proposed to facilitate con-troller design. Concretely speaking, the functions f i and giwith i = 1, 3 possess unknown structures, which will beapproximated by NN. And the resulting approximation errorwill be eliminated by resorting to the design of adaptive laws.

With the introduction of reference trajectory of states, thetracking errors are defined as

ε1 = x1 − x1d ,ε2 = x2 − x2d ,

ε3 = x3 − x3d − ξ1 tanh χ1ð Þ,ð27Þ

where

_χ1 =cosh2χ1

ξ1−ζ1 tanh χ1ð Þ + sat u1ð Þ − u1½ �, χ1 0ð Þ = 0,

ð28Þ

with the designed parameters satisfying ξ1, ζ1 > 0, xid beingthe designed trajectory.

Remark 7. In [23], the auxiliary system and the transformederror are defined as _χ = −ζχ + satðuÞ − u and ε = x − xd − ξχ, respectively. However, this solution is effective only if theauxiliary variable χ is bounded, while the dynamics (27)and (28) employs the hyperbolic tangent function to con-struct a bounded compensation term ζ tanh ðχÞ, so that theconservatism of above assumption is relaxed.

Taking the time derivative of equations and substitutingEquation (25) yields

_ε1 = f1 + g1x2 − _x1d ,_ε2 = x3 − _x2d ,

_ε3 = f3 + g3 sat u1ð Þ + d1½ � − _x3d + ζ1 tanh χ1ð Þ − sat u1ð Þ + u1:

ð29Þ

For the purpose of relaxing the issue of EOC, a first-orderlow-pass filter is introduced as

c2 _x2d + x2d = x2c, x2d 0ð Þ = 0,c3 _x3d + x3d = x3c, x3d 0ð Þ = 0,

ð30Þ

where x2c, x3c are both output signals and the filter parame-ters satisfy c2, c3 > 0. Resulting error variables are defined as

y2 = x2d − x2c, y3 = x3d − x3c: ð31Þ

Step 1. Taking the unavailable dynamic f1 into account andusing Lemma 2 yield

f1 =W∗T1 H1 xð Þ + ο1, 0 < ο1j j ≤O1,

  W∗T1 H1

≤ W∗T1

H1k k ≤w1h1,ð32Þ

where h1 = kH1k and w1 ≥ kW∗T1 k.

4 International Journal of Aerospace Engineering

The compensation signal is designed as

_φ1 = −ϑ1φ1 − ϑ2 tanh φ1ð Þ − g1ε1y2j j + 0:5y22φ1

+ y2, ð33Þ

where ϑ1, ϑ2 are both positive constants.Subsequently, the adaptive laws and the virtual command

are designed as

_w1 = α1 ε1j jh1 − r1w1ð Þ, ð34Þ

_R1 = β1 ε1j j − r2R1� �

, ð35Þ

x2c =1g1

_x1d − k1ε1 − k2 tanh ε1ð Þ − w1h1 tanh ε1ð Þ½

− R1 tanh ε1ð Þ + l1φ1,

ð36Þ

in which R1 =O1; w1 and R1 denote estimations ofw1 and R1,respectively; and parameters α1, β1, r1, r2, k1, k2, l1 are alldesigned as positive constants.

Remark 8. Different from the conventional algorithm ofRBFNN, MLP algorithms select the upper bound of theweight matrix norm as the online updating parameter.Hence, only a single parameter is estimated rather than thewhole weight matrix, so that the excessive occupation of thecomputational resource is relaxed significantly.

To show the stability of ε1, the Lyapunov function (LF) ischosen as

F1 =12 ε

21 +

12φ

21: ð37Þ

Differentiating F1 and using Equations (33) and (40)yield

_F1 = ε1 _ε1 + φ1 _φ1 = ε1 f1 + g1 ε2 + x2dð Þ − _x1d½ � + φ1

� −ϑ1φ1 − ϑ2 tanh φ1ð Þ − g1ε1y2j j + 0:5y22φ1

+ y2

� �= ε1 f1 + g1ε1ε2 + g1ε1y2 + ε1

� −k1ε1 − k2 tanh ε1ð Þ − w1h1 tanh ε1ð Þ − R1 tanh ε1ð Þ + l1φ1� − ϑ1φ

21 − ϑ2φ1 tanh φ1ð Þ − g1ε1y2j j − 1

2 y22 + φ1y2 ≤ ε1 f1

+ g1ε1ε2 − k1ε21 − k2ε1 tanh ε1ð Þ − w1h1ε1 tanh ε1ð Þ

− R1ε1 tanh ε1ð Þ + ε21 +14 l

21φ

21 − ϑ1φ

21 − ϑ2φ1 tanh φ1ð Þ + 1

2φ21:

ð38Þ

According to Lemma 2, one has

−k2ε1 tanh ε1ð Þ≤−k2 ε1j j + k2ϖ,−w1h1ε1 tanh ε1ð Þ≤−w1h1 ε1j j + w1h1ϖ,

−R1ε1 tanh ε1ð Þ≤−R1 ε1j j + R1ϖ,−ϑ2φ1 tanh φ1ð Þ≤−ϑ2 φ1j j + ϑ2ϖ:

ð39Þ

Substituting Equation (39) into Equation (38) leads to

_F1 ≤ ε1 W∗T1 H1 xð Þ + ο1

� �+ g1ε1ε2 − k1 − 1ð Þε21 − k2 ε1j j + k2ϖ

− w1h1 ε1j j + w1h1ϖ − R1 ε1j j + R1ϖ − ϑ1 −14 l

21 − 1

� �φ21

− ϑ2 φ1j j + ϑ2ϖ ≤ ε1j jw1h1 + ε1j jR1 + g1ε1ε2 − k1 − 1ð Þε21

− k2 ε1j j + k2ϖ − w1h1 ε1j j + h21w21

2 + ϖ2

2 − R1 ε1j j + R212

+ ϖ2

2 − ϑ1 −14 l

21 −

12

� �φ21 − ϑ2 φ1j j + ϑ2ϖ = ε1j j~w1h1

+ ε1j j~R1 − k1 − 1 − 12 �g1

� �ε21 +

12 �g1ε

22 − k2 ε1j j −

� ϑ1 −14 l

21 −

12

� �φ21 − ϑ2 φ1j j + h21w

21

2 + R212 + ϖ2

+ k2ϖ + ϑ2ϖ:

ð40Þ

Step 2. For the purpose of the stabilization of ε2, the virtualcommand x3d is given as

x3c = −k3ε2 − k4ε2 tanh ε2ð Þ + _x2d + l2φ2, ð41Þ

where k3, k4, and l2 are all positive parameters.The filter compensation signal is expressed as

_φ2 = −ϑ3φ2 − ϑ4 tanh φ2ð Þ − ε2y3j j + 0:5y23φ2

+ y3, ð42Þ

where ϑ3, ϑ4 are both positive parameters.The second LF is selected, and its derivatives are pre-

sented as

F2 =12 ε

22 +

12φ

22, ð43Þ

_F2 = ε2 _ε2 + φ2 _φ2 = ε2 ε3 + x3d − _x2dð Þ + φ2

� −ϑ3φ2 − ϑ4 tanh φ2ð Þ − 12φ2

y23 + y3

� �= ε2ε3 + ε2y3 − k3ε

22 − k4ε2 tanh ε2ð Þ − g1ε1ε2 + ε2l2φ2

− ϑ3φ22 − ϑ4φ2 tanh φ2ð Þ − ε2y3j j − 1

2 y23 + φ2y3 ≤ ε2ε3

− k3ε22 − k4ε2 tanh ε2ð Þ − g1ε1ε2 + ε22 +

14 l

22φ

22 − ϑ3φ

22

− ϑ4φ2 tanh φ2ð Þ + 12φ

22:

ð44ÞIt can be derived from Lemma 2 that

−k4ε1 tanh ε2ð Þ≤−k4 ε2j j + k4ϖ,−ϑ4φ2 tanh φ2ð Þ≤−ϑ4 φ2j j + ϑ4ϖ:

ð45Þ

5International Journal of Aerospace Engineering

Substituting the result into Equation (44), it obtains

_F2 ≤ ε2ε3 − k3 − 1 − 12 �g1

� �ε22 +

12 �g1ε

21 − k4 ε2j j + k4ϖ −

� ϑ3 −14 l

22 −

12

� �φ22 − ϑ4 φ2j j + ϑ4ϖ:

ð46Þ

Step 3. In this part, the uncertain dynamic f3 is approximatedvia RBFNN, which is shown as

f3 =W∗T3 H3 xð Þ + ο3, 0 < ο3j j ≤O3, ð47Þ

W∗T3 H3

≤ W∗T3

H3k k ≤w3h3, ð48Þwhere h3 = kH3k and w3 ≥ kW∗T

3 k.Owing to the effect of external disturbances, it is defined

according to Assumptions 3 and 5 that R3 = �g3½um +D1� +O3 + ζ1 − um. Afterward, the adaptive laws and the controlinput are given as

_w3 = α3 ε3j jh3 − r3w3ð Þ, ð49Þ

_R3 = β3 ε3j j − r4R3� �

, ð50Þ

u1 =1g3

−k5ε3 −k6ε3

ε3j j + ρ1−

w3h3ε3ε3j j + ρ2

−R3ε3ε3j j + ρ3

+ _x3d

� �,

ð51Þ

ρ1 =b1

1 + k6, ρ2 =

b21 + w3h3

, ρ3 =b3

1 + R3, ð52Þ

where parameters α3, β3, r3, r4, k5, k6, ρ1, ρ2, ρ3 are all posi-tive constants.

Remark 9. Considering the controller (51), several ingredi-ents should to be emphasized as follows: (i) The term −k5ε3and the term −ðk6ε3/ðjε3j + ρ1ÞÞ guarantee the finite timeconvergence of ε3. Meanwhile, the term −ðw3h3ε3/ðjε3j + ρ2ÞÞ with the construction (49) is utilized to estimate the weightparameter of NN, and the term −ðR3ε3/ðjε3j + ρ3ÞÞ is forunknown lumped disturbances. (ii) The introduction of var-iables εi ði = 1, 2, 3Þ not only leads to the attenuation of chat-tering phenomenon but also improves the controllability ofsteady-state errors. This point will be proved in subsequentderivations.

Consequently, the third LF and its derivative are pre-sented as

F3 =12 ε

23, ð53Þ

_F3 = ε3 _ε3 = ε3½ f3 + g3½satðu1Þ + d1� − _x3d + ζ1 tanh ðχ1Þ− satðu1Þ + u1� = ε3 f3 − k5ε

23 − k6ε

23/jε3j + ρ1 − w3h3ε

23/jε3j +

ρ2 − R3ε23/jε3j + ρ3 − ε2ε3 + ε3½g3½satðu1Þ + d1� + ζ1 tanh ðχ1

Þ − satðu1Þ�:

Taking Equations (47) and (48) into account, it leads to

_F3 = ε3 W∗T3 H3 xð Þ + ο3

� �− k5ε

23 −

k6ε23

ε3j j + ρ1−

w3h3ε23

ε3j j + ρ2

−R3ε

23

ε3j j + ρ3− ε2ε3 + ε3 g3 sat u1ð Þ + d1½ � + ζ1 tanh χ1ð Þ½

− sat u1ð Þ� ≤ ε3j jw3h3 − k5ε23 −

k6ε23

ε3j j + ρ1−

w3h3ε23

ε3j j + ρ2

−R3ε

23

ε3j j + ρ3− ε2ε3 + ε3 g3 sat u1ð Þ + d1½ � +O3½

+ ζ1 tanh χ1ð Þ − sat u1ð Þ� ≤ ε3j jw3h3 − k5ε23 −

k6ε23

ε3j j + ρ1

−w3h3ε

23

ε3j j + ρ2−

R3ε23

ε3j j + ρ3− ε2ε3 + ε3j jR3:

ð54Þ

Some of the items in Equation (54) are scaled as

−k6ε

23

ε3j j + ρ1= −k6 ε3j j + k6ρ1 ×

ε3j jε3j j + ρ1

= −k6 ε3j j + k6b11 + k6

× ε3j jε3j j + ρ1

≤ −k6 ε3j j + b1,

−w3h3ε

23

ε3j j + ρ2= −w3h3 ε3j j + w3h3ρ2 ×

ε3j jε3j j + ρ2

= −w3h3 ε3j j + w3h3b21 + w3h3

× ε3j jε3j j + ρ2

≤ −w3h3 ε3j j + b2,

−R3ε

23

ε3j j + ρ3= −R3 ε3j j + R3ρ3 ×

ε3j jε3j j + ρ3

= −R3 ε3j j + R3b31 + R3

× ε3j jε3j j + ρ3

≤ −R3 ε3j j + b3:

ð55Þ

With the substitution of these results, Equation (54)becomes

_F3 ≤ ε3j jw3h3 − k5ε23 − k6 ε3j j + b1 − w3h3 ε3j j + b2 − R1 ε3j j

+ b3 − ε2ε3 + ε3j jR3 ≤ ε3j j~w3h3 + ε3j j~R3 − k5 −12

� �ε23

+ 12 ε

22 − k6 ε3j j + b1 + b2 + b3:

ð56Þ

3.2. Velocity Control. As shown in Section 2, the closed-loopsystem for velocity tracking control is expressed as

_V = f v + sat u2ð Þ + d2, ð57Þ

6 International Journal of Aerospace Engineering

with

f v =T cos θ − γð Þ −D½ �

m− g sin γ − u2: ð58Þ

According to Lemma 2, the unavailable state f v can bewritten as

f v =W∗Tv Hv xð Þ + οv , 0 < οv ≤Ov , ð59Þ

W∗Tv Hv

≤ W∗Tv

Hvk k ≤wvhv, ð60Þ

where hv = kHvk and wv ≥ kW∗Tv k.

The velocity tracking error ev is introduced as

εv =V −Vd − ζ2 tanh χ2ð Þ, ð61Þ

where

_χ2 =cosh2χ2

ξ2−ζ2 tanh χ2ð Þ + sat u2ð Þ − u2½ �, χ2 0ð Þ = 0,

ð62Þ

with the parameters satisfying ξ2, ζ2 > 0.Differentiating εv along Equation (61) and using Equa-

tion (57) yield

_εv = f v + d2 − _Vd + ζ2 tanh χ2ð Þ + u2: ð63Þ

It is defined that Rv =D2 +Ov + ζ1. Subsequently, theadaptive laws and the control input are given as

_wv = αv εvj jhv − r5wvð Þ, ð64Þ

_Rv = βv εvj j − r6Rv

� �, ð65Þ

u2 = −k7εv −k8εv

εvj j + ρ4−

wvhvεvεvj j + ρ5

−Rvεvεvj j + ρ6

− _Vd , ð66Þ

ρ4 =b4

1 + k8, ρ5 =

b51 + wvhv

, ρ6 =b6

1 + Rv

, ð67Þ

where the parameters αv, βv, r5, r6, k7, k8, b4, b5, b6 are alldesigned as positive constants.

To illustrate the convergence of the velocity trackingerror, the LF is selected as

Fv =12 ε

2v: ð68Þ

Differentiating Fv along (78) yields

_Fv = εv _εv = εv f v + d2 − _Vd + ζ2 tanh χ2ð Þ + u2�

= εv f v − k7ε2v −

k8ε2v

εvj j + ρ4−

wvhvε2v

εvj j + ρ5−

Rvε2v

εvj j + ρ6+ εv d2 + ζ2 tanh χ2ð Þ½ �:

ð69Þ

With the substitution of Equations (59) and (60), it is fur-ther derived as

_Fv = εv W∗Tv Hv xð Þ + οv

� �− k7ε

2v −

k8ε2v

εvj j + ρ4−

wvhvε2v

εvj j + ρ5

−Rvε

2v

εvj j + ρ6+ εv d2 + ζ2 tanh χ2ð Þ½ � ≤ εvj jwvhv − k7ε

2v

−k8ε

2v

εvj j + ρ4−

wvhvε2v

εvj j + ρ5−

Rvε2v

εvj j + ρ6+ εv d2 +Ov½

+ ζ2 tanh χ2ð Þ� ≤ εvj jwvhv − k7ε2v −

k8ε2v

εvj j + ρ4−

wvhvε2v

εvj j + ρ5

−Rvε

2v

εvj j + ρ6+ εvj jRv:

ð70Þ

There holds the inequations that

−k8ε

2v

εvj j + ρ4= −k8 εvj j + k8ρ4ð Þ εvj j

εvj j + ρ4= −k8 εvj j + k8b4

1 + k8

× εvj jεvj j + ρ4

≤ −k8 εvj j + b4,

−wvhvε

2v

εvj j + ρ5= −wvhv εvj j + wvhvρ5ð Þ εvj j

εvj j + ρ5= −wvhv εvj j

+ wvhvb51 + wvhv

× εvj jεvj j + ρ5

≤ −wvhv εvj j + b5,

−Rvε

2v

εvj j + ρ6= −Rv εvj j + Rvρ6

� � εvj jεvj j + ρ6

= −Rv εvj j + Rvb61 + Rv

× εvj jεvj j + ρ6

≤ −Rv εvj j + b6:

ð71Þ

Substituting the results into Equation (70), one obtains

_Fv ≤ εvj jwvhv − k7ε2v − k8 εvj j + b4 − wvhv εvj j + b5 − Rv εvj j

+ b6 + εvj jRv ≤ εvj j~wvhv + εvj j~Rv − k7ε2v − k8 εvj j + b4

+ b5 + b6:

ð72Þ

Table 1: Initial values of states.

Item Value Unit

V 15060 ft/s

h 110000 ft

γ 0 rad

α 1:6325π/180 rad

q 0 rad/s

7International Journal of Aerospace Engineering

4. Stability Analysis

In this section, the stability of the developed controller isdemonstrated via Lyapunov-based analysis. It is concluded

that the control system is capable of maintaining globalfinite-time convergence. Furthermore, the altitude and veloc-ity tracking errors of GHV will converge to a tiny region con-taining the origin in finite time.

Table 2: Parameters for the designed control algorithm.

Section Parameters

Low-pass filter ε1 = 10, ε2 = 10Compensation signals ϑ1 = 10, ϑ2 = 0:5

Auxiliary systemξ1 = 2, ζ1 = 0:01ξ2 = 2, ζ2 = 0:01

Controller

kh = 1:3, khh = 13, k1 = 0:8, k2 = 0:2, l1 = 0:01, ϑ1 = 10,ϑ2 = 0:5, ϑ3 = 10, ϑ4 = 0:5, α1 = 50, r1 = 0:04, β1 = 10,r2 = 2, k3 = 1:5, k4 = 0:2, k5 = 2, k6 = 0:5, α3 = 0:005,r3 = 10, β3 = 0:05, r4 = 0:5, b1 = b2 = b3 = 0:001,

ξ1 = 2, ζ1 = 0:01, ξ2 = 2, ζ2 = 0:01, αv = 0:001, r5 = 10,βv = 0:001, r6 = 10, k7 = 0:6, k8 = 0:1,

b4 = b5 = b6 = 0:001

20

0

–20

–40

–60

–80

–1000 10 20

t (s)30 40 50

he (

m)

Figure 1: Trajectory tracking error of altitude of GHV.

v e (m

/s)

20

10

0

–10

–20

–30

–40

–50

–600 10 20

t (s)30 40 50

Figure 2: Trajectory tracking error of velocity of GHV.

0

0.05

0.1

0.15

0.2

–0.05

–0.1

–0.15

–0.20 10 20

t (s)30 40 50

𝛿e (

rad)

Figure 3: The control input of altitude subsystem.

𝛽

1

0.8

0.6

0.4

0.2

00 10 20

t (s)30 40 50

Figure 4: The control input of velocity subsystem.

8 International Journal of Aerospace Engineering

Theorem 10. For the GHV dynamics (25) and (57) withAssumptions 3 and 5, the global finite-time convergence isguaranteed under controllers (36), (41), (51), and (66); theadaptive laws (34) and(35), (49) and (50), and (64) and

(65); the compensation signals (33) and (42); and the auxiliarysystems (28) and (62).

Proof. The LF is constructed as

F4 = F1 + F2 + F3 + Fv +12α1

~w21 +

12α3

~w23 +

12αv

~w2v +

12β1

~R21

+ 12β3

~R23 +

12βv

~R2v:

ð73Þ

Taking the time derivation and substituting above results,it has:

_F4 = _F1 + _F2 + _F3 + _Fv −1α1

~w1 _w1 −1α3

~w3 _w3 −1αv

~wv_wv

−1β1

~R1_R1 −

1β3

~R3_R3 −

1βv

~Rv_Rv ≤ − k1 − 1 − �g1ð Þε21

− k3 − 1 − �g1ð Þε22 − k5 − 1ð Þε23 − k7ε2v − ϑ1 −

14 l

21 −

12

� �φ21

− ϑ3 −14 l

22 −

12

� �φ22 − k2 ε1j j − k4 ε2j j − k6 ε3j j − k8 εvj j

− ϑ2 φ1j j − ϑ4 φ2j j + h21w21

2 + R212 + r1w1 ~w1 + r3w3 ~w3

+ r5wv ~wv + r2R1~R1 + r4R3~R3 + r6Rv~Rv + ϖ2 + k2ϖ + k4ϖ

+ ϑ2ϖ + ϑ4ϖ + b1 + b2 + b3 + b4 + b5 + b6≤ − k1 − 1 − �g1ð Þε21 − k3 − 1 − �g1ð Þε22 − k5 − 1ð Þε23 − k7ε

2v

− ϑ1 −14 l

21 −

12

� �φ21 − ϑ3 −

14 l

22 −

12

� �φ22 − ~w2

1 − ~w23 − ~w2

v

− ~R21 − ~R

23 − ~R

2v + ~w2

1 +h21w

21

2 + r1w1 ~w1 + ~w23 + r3w3 ~w3

+ ~w2v + r5wv ~wv + ~R

21 +

R212 + r2R1~R1 + ~R

23 + r4R3~R3 + ~R

2v

+ r6Rv~Rv + ϖ2 + k2ϖ + k4ϖ + ϑ2ϖ + ϑ4ϖ + b1 + b2 + b3 + b4

+ b5 + b6:

ð74Þ

5

4

3

2

1

00 10 20

t (s)30 40 50

Figure 5: The estimation of R1.

2

1.5

1

0.5

0

–0.5

–1

–1.50 10 20

t (s)30 40 50

Figure 6: The estimation of R3.

0.14

0.12

0.1

0.08

0.06

0.04

0.02

00 10 20

t (s)30 40 50

Figure 7: The estimation of Rv .

0.6

0.5

0.4

0.3

0.2

0.1

00 10 20

t (s)30 40 50

Figure 8: The estimation of w1.

9International Journal of Aerospace Engineering

Take some of the items as quadratic functions to obtaintheir maximum:

~w21 +

h21w21

2 + r1w1 ~w1 = w1 −w∧1ð Þ2 + h21w21

2 + r1w1 w1 − w1ð Þ

= − r1 − 1 − h212

!w2

1 + r1 − 2ð Þw1w1

+w21 ≤

r21 − 2h21� �

w21

4r1 − 4 − 2h21,

~w23 + r3w3 ~w3 = w3 −w∧3ð Þ2 + r3w3 w3 − w3ð Þ

= − r3 − 1ð Þw23 + r3 − 2ð Þw3w3 +w2

3 ≤r23w

23

4 r3 − 1ð Þ ,

~w2v + r5wv ~wv ≤

r25w2v

4 r5 − 1ð Þ ,~R21 +

R212 + r2R1~R1 ≤

r22 − 2� �

R21

4r2 − 6 ,

~R23 + r4R3~R3 ≤

r24R23

4 r4 − 1ð Þ ,~R2v + r6Rv

~Rv ≤r26R

2v

4 r6 − 1ð Þ : ð75Þ

With the substitution of these results, one has

_F4 ≤ − k1 − 1 − �g1ð Þε21 − k3 − 1 − �g1ð Þε22 − k5 − 1ð Þε23 − k7ε2v

− ϑ1 −14 l

21 −

12

� �φ21 − ϑ3 −

14 l

22 −

12

� �φ22 − ~w2

1 − ~w23

− ~w2v − ~R

21 − ~R

23 − ~R

2v +

r21 − 2h21� �

w21

4r1 − 4 − 2h21+ r23w

23

4 r3 − 1ð Þ

+ r25w2v

4 r5 − 1ð Þ + r22 − 2� �

R21

4r2 − 6 + r24R23

4 r4 − 1ð Þ + r26R2v

4 r6 − 1ð Þ + ϖ2

+ k2ϖ + k4ϖ + ϑ2ϖ + ϑ4ϖ + b1 + b2 + b3 + b4 + b5 + b6≤ −κ1F4 + Δ1,

ð76Þ

where

κ1 = min 2 k1 − 1 − �g1ð Þ, 2 k3 − 1 − �g1ð Þ, 2 k5 − 1ð Þ, 2k7, 2ϑ1 −12 l

21 − 1

� �,

� 2ϑ3 −12 l

22 − 1

� �, 2α1, 2α3, 2αv , 2β1, 2β3, 2βv

�,

ð77Þ

Δ1 =r21 − 2h21� �

w21

4r1 − 4 − 2h21+ r23w

23

4 r3 − 1ð Þ + r25w2v

4 r5 − 1ð Þ + r22 − 2� �

R21

4r2 − 6

+ r24R23

4 r4 − 1ð Þ + r26R2v

4 r6 − 1ð Þ + ϖ2 + k2ϖ + k4ϖ + ϑ2ϖ + ϑ4ϖ

+ b1 + b2 + b3 + b4 + b5 + b6:

ð78Þ

The result indicates that the designed controller isasymptotically stable. Therefore, as long as the initial condi-tions are bounded, the estimation errors must have uniformultimate boundedness. Thus, there exist constants �wi, i = 1,3, v, satisfying �wi > wi and �wi >wi. �Ri, i = 1, 3, v, follow thesame principle, i.e.,

�wi − wi >wi − wi = ~wi, i = 1, 3, v,�Ri − Ri > Ri − Ri = ~Ri, i = 1, 3, v:

ð79Þ

Rebuild the LF as

F5 = F1 + F2 + F3 + Fv +1α1

�w1 −w∧1ð Þ2 + 1α3

�w3 −w∧3ð Þ2

+ 1αv

�wv −w∧vð Þ2 + 1β1

�R1 − R∧1� �2 + 1

β3�R3 − R∧3� �2

+ 1βv

�Rv − R∧v

� �2:

ð80Þ

4

3.5

3

2.5

2

1.5

1

0.5

00 10 20

t (s)30 40 50

Figure 9: The estimation of w3.

2.5

2

1.5

1

0.5

00 10 20

t (s)30 40 50

Figure 10: The estimation of wv .

10 International Journal of Aerospace Engineering

Differentiating F5 along Equation (80) yields

_F5 = _F1 + _F2 + _F3 + _Fv −2α1

�w1 − w1ð Þ _w1 −2α3

�w3 − w3ð Þ _w3

−2αv

�wv − wvð Þ _wv −2β1

�R1 − R1� � _R1 −

2β3

�R3 − R3� � _R3

−2βv

�Rv − Rv

� � _Rv ≤ − k1 − 1 − �g1ð Þε21 − k3 − 1 − �g1ð Þε22

− k5 − 1ð Þε23 − k7ε2v − ϑ1 −

14 l

21 −

12

� �φ21

− ϑ3 −14 l

22 −

12

� �φ22 − k2 ε1j j − k4 ε2j j − k6 ε3j j − k8 εvj j

− ϑ2 φ1j j − ϑ4 φ2j j − ε1j jh1 �w1 − w1ð Þ − ε3j jh3 �w3 − w3ð Þ− εvj jhv �wv − wvð Þ − ε1j j �R1 − R1

� �− ε3j j �R3 − R3

� �− εvj j �Rv − Rv

� �+ 2r1 �w1 − w1ð Þw1 + 2r3 �w3 − w3ð Þw3

+ 2r5 �wv − wvð Þwv + 2r2 �R1 − R1� �

R1 + 2r4 �R3 − R3� �

R3

+ 2r6 �Rv − Rv

� �Rv +

h21w21

2 + R212 + ϖ2 + k2ϖ + k4ϖ

+ ϑ2ϖ + ϑ4ϖ + b1 + b2 + b3 + b4 + b5 + b6:

ð81Þ

In particular, the results hold as

2r1 �w1 − w1ð Þw1 +h21w

21

2 = − 2r1 −h212

!w2

1 + 2r1 �w1w1 ≤2r21 �w2

14r1 − h21

,

2r3 �w3 − w3ð Þw3 = −2r3w23 + 2r3 �w3w3 ≤

r3 �w23

2 ,

2r5 �wv − wvð Þwv ≤r5 �w

2v

2 , 2r2 �R1 − R1� �

R1 +R212 ≤

2r22�R21

4r2 − 1 ,

2r4 �R3 − R3� �

R3 ≤r4�R

23

2 , 2r6 �Rv − Rv

� �Rv ≤

r6�R2v

2 : ð82Þ

Substituting these results into Equation (81), it leads to

_F5 ≤ −k2 ε1j j − k4 ε2j j − k6 ε3j j − k8 εvj j − ϑ2 φ1j j − ϑ4 φ2j j− ε1j jh1 �w1 − w1ð Þ − ε3j jh3 �w3 − w3ð Þ − εvj jhv �wv − wvð Þ− ε1j j �R1 − R1

� �− ε3j j �R3 − R3

� �− εvj j �Rv − Rv

� �+ 2r21 �w2

14r1 − h21

+ r3 �w23

2 + r5 �w2v

2 + 2r22�R21

4r2 − 1 + r4�R23

2 + r6�R2v

2+ ϖ2 + k2ϖ + k4ϖ + ϑ2ϖ + ϑ4ϖ + b1 + b2 + b3 + b4 + b5+ b6 ≤ −κ2F5 + Δ2,

ð83Þ

where

As a result, it is indicated that all the tracking errors andestimation errors converge into a tiny region in finite time.The proof of Theorem 10 is completed.

Remark 11. Consequently, it is necessary to make a guidingexplanation for the selection of controller parameters froma theoretical perspective. Considering Equations (83), (84)and (85), the overall feedback gain depends on ki, which indi-cates that the larger ki leads to the faster stabilization rate,and accordingly, the required control torque will increase.On the other hand, the design parameters bi play a significantrole in steady-state error. Obviously, if the value of bi is small,the steady-state error of the closed-loop system will decrease,while the chattering phenomenon of the actuator willincrease. Therefore, the best choice is to compromise thetwo indicators and determine the values of these parameters,

so that the controller possesses a satisfactory overallperformance.

5. Numerical Simulations

In this section, a simulation example is provided to demon-strate the stability and effectiveness of the designed control-ler. According to the dynamics (1), (2), (3), (4), (5), (6), (7),(8), (9), (10), (11), (12), (13), (14), (15), (16) and (17) ofGHV, the initial values and desired trajectories are given tofacilitate the implementation of the tracking controller, whilevarious performances will be simultaneously recorded andexhibited. Necessarily, the initial values of system states areset as Table 1, and the designed parameters of the controlalgorithm are shown in Table 2.

κ2 = minffiffiffi2

pk2,

ffiffiffi2

pk4,

ffiffiffi2

pk6,

ffiffiffi2

pk8,

ffiffiffi2

pϑ2,

ffiffiffi2

pϑ4,

ffiffiffiffiffiα1

pε1j jh1,

ffiffiffiffiffiα3

pε3j jh3,

ffiffiffiffiffiαv

pεvj jhv,

ffiffiffiffiffiβ1

pε1j j,

ffiffiffiffiffiβ3

qε3j j,

ffiffiffiffiffiβv

qεvj j

�, ð84Þ

Δ2 =2r21 �w2

14r1 − h21

+ r3 �w23

2 + r5 �w2v

2 + 2r22�R21

4r2 − 1 + r4�R23

2 + r6�R2v

2 + ϖ2 + k2ϖ + k4ϖ + ϑ2ϖ + ϑ4ϖ + b1 + b2 + b3 + b4 + b5 + b6: ð85Þ

11International Journal of Aerospace Engineering

Subsequently, to test the trajectory tracking performanceand robustness against disturbance, the reference altitudeand velocity signals, the lumped external disturbances, andthe saturation constraint of actuator are defined as

Vd = 15110,hd = 110100,

d1 = 0:01 ∗ sin 0:1 ∗ tð Þ,d2 = 0:01 ∗ cos 0:1 ∗ tð Þ,

u1m = 0:2rad,u2m = 0:9:

ð86Þ

Consequently, the main results are given in Figures 1–10.To start with, curves of trajectory tracking errors are

given in Figures 1 and 2, which indicate that the altitudeand velocity tracking missions are completed within 10 s.Undoubtedly, the finite-time controller shows great superior-ity in contrast with ordinary asymptotic convergence. After-wards, the feedback input signals of the closed-loop systemare shown in Figures 3 and 4. Practically speaking, the alti-tude control torque and velocity control torque convergesinto a tiny region containing origin within 10 s. It is notewor-thy that the saturation nonlinearity of actuators is wellaccommodated during the control process. What is more,Figures 5–7 give the curves of estimated values R1, R3, andRv, which converge to a constant in finite time. The adaptiveparameters of RBFNNs are addressed in Figures 8–10. Obvi-ously, estimations of the norm of the weight matrix willalways remain bounded under the proposed method. To con-clude, performances of the proposed controller satisfy thedesign requirements.

6. Conclusion

This paper investigates the trajectory tracking control ofGHVs exposed to system uncertainties and input saturationconstraints. On the basis of the longitudinal model of GHVs,the command filter-based adaptive neural controller is con-structed to guarantee the finite-time stability of the closed-loop system. Specially, considering the defect of EOC associ-ated with a conservative backstepping design, the low-passfilters are introduced and compensation signals are designedto avoid the repeat derivation on virtual command. Mean-while, the MLP algorithm-based function approximation isutilized to cope with the inaccurate model information, inwhich the tedious analytical computation is simplified signif-icantly. The input nonlinearity caused by actuator saturationis addressed via a modified auxiliary system. Finally, the rig-orous mathematical analysis and simulation results are pro-vided to confirm the validity of the proposed architecture.

Data Availability

No data were used to support to this study.

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding publication of this article.

Acknowledgments

This work is cosupported by the Defense Industrial Technol-ogy Development Program (JCKY2016205C013) and Aero-nautical Science Foundation of China (2015ZC18005).

References

[1] Z. Ping, J. Ju, Y. Chaojun, and W. Chengyu, “Neuro-adaptivesliding mode fault-tolerant control of hypersonic vehicleunder actuator fault,” in 2019 Chinese Control And DecisionConference (CCDC), pp. 1558–1563, Nanchang, China, 2019.

[2] Y. Cheng, S. Yang, B. Xu, D. Zhang, and Y. Zhang, “Discretesliding mode adaptive control of GHVwith prediction model,”IEEE 28th International Symposium on Industrial Electronics(ISIE), 2019, pp. 2253–2258, Vancouver, BC, Canada, 2019.

[3] Y. Wu, F. Ma, X. Liu, Y. Hua, X. Liu, and G. Li, “Super twistingdisturbance observer-based fixed-time sliding mode backstep-ping control for air-breathing hypersonic vehicle,” IEEEAccess, vol. 8, pp. 17567–17583, 2020.

[4] X. Cheng, P. Wang, and G. Tang, “Adaptive back-steppingcontrol for flexible hypersonic vehicle using ESO,” in 20189th International Conference on Mechanical and AerospaceEngineering (ICMAE),, pp. 617–622, Budapest, Hungary, July2018.

[5] C. Li, J. Sun, and Y. Guo, “Adaptive back-stepping trackingcontrol of the hypersonic vehicle with input saturation,” in2019 IEEE 4th Advanced Information Technology, Electronicand Automation Control Conference (IAEAC), pp. 839–845,Chengdu, China, 2019.

[6] L. Dou, P. Su, Q. Zong, and Z. Ding, “Fuzzy disturbanceobserver-based dynamic surface control for air-breathinghypersonic vehicle with variable geometry inlets,” IET ControlTheory & Applications, vol. 12, no. 1, pp. 10–19, 2018.

[7] X. Bu, D. Wei, and G. He, “A robust constrained controlapproach for flexible air-breathing hypersonic vehicles,” Inter-national Journal of Robust and Nonlinear Control, vol. 30,no. 7, pp. 2752–2776, 2020.

[8] X. Bu, X. Yu, and H. Lei, “An adaptive critic design-basedfuzzy neural controller for hypersonic vehicles: predefinedbehavioral nonaffine control,” IEEE-ASME Transactions onMechatronics, vol. 24, no. 4, pp. 1871–1881, 2019.

[9] Z. Wu, J. Lu, Q. Zhou, and J. Shi, “Modified adaptive neuraldynamic surface control for morphing aircraft with inputand output constraints,” Nonlinear Dynamics, vol. 87, no. 4,pp. 2367–2383, 2017.

[10] H. Chi and F. Yu, “Compound control for hypersonic vehiclesbased on dynamic compensation,” Chinese automation con-gress (CAC), 2018, pp. 3523–3528, Xi’an, China, 2018.

[11] S. Zhao and X. Li, “Prescribed performance fault tolerant con-trol for hypersonic flight vehicles with actuator failures,” IEEEAccess, vol. 7, pp. 100187–100204, 2019.

[12] Y. Chen, Y.Wei, G. R. Duan, and L. G. Tan, “Robust finite timetracking control of flexible hypersonic vehicle with uncer-tainties,” in 2020 IEEE 4th Information Technology, Network-ing, Electronic and Automation Control Conference (ITNEC),pp. 233–239, Chongqing, China, June 2020.

12 International Journal of Aerospace Engineering

[13] X. Zhang, K. Chen, and W. Fu, “Fuzzy adaptive control forgeneric hypersonic vehicles with non-affine aerodynamiccharacteristics and output constraints,” in 2019 19th Interna-tional Conference on Control, Automation and Systems(ICCAS), pp. 71–75, Jeju, Republic of Korea, 2019.

[14] Y. Liu, J. Yi, and Z. Pu, “Desired compensation based adaptivefuzzy control for hypersonic vehicle with measurementnoises,” in 2018 IEEE International Conference on Fuzzy Sys-tems (FUZZ-IEEE),, pp. 1–7, Rio de Janeiro, Brazil, 2018.

[15] X. Li and G. Li, “Novel fuzzy approximation control schemefor flexible air-breathing hypersonic vehicles with non-affinedynamics and amplitude and rate constraints,” IEEE Access,vol. 7, pp. 73602–73616, 2019.

[16] Y. Ma and Y. Cai, “A fuzzy model predictive control basedupon adaptive neural network disturbance observer for a con-strained hypersonic vehicle,” IEEE Access, vol. 6, pp. 5927–5938, 2018.

[17] X. Bu, “Air-breathing hypersonic vehicles funnel control usingneural approximation of non-affine dynamics,” IEEE/ASMETransactions on Mechatronics, vol. 23, no. 5, pp. 2099–2108,2018.

[18] J. Wang, J. Wang, P. Yang, K. Liu, X. Yu, and J. Hu, “Adaptivesliding mode control for a flexible hypersonic vehicle withactuator saturation,” in 2016 35th Chinese Control Conference(CCC), pp. 647–651, Chengdu, China, July 2016.

[19] G. Jianguo, L. Yifei, and Z. Jun, “Finite time control for generichypersonic vehicle,” in 2018 Chinese Control And DecisionConference (CCDC), pp. 3727–3731, Shenyang, China, June2018.

[20] X. Yu, P. Li, and Y. Zhang, “The design of fixed-time observerand finite-time fault-tolerant control for hypersonic glidingvehicles,” IEEE Transactions on Industrial Electronics, vol. 65,no. 5, pp. 4135–4144, 2018.

[21] S. Jing-guang and S. Shen-min, “Tracking control of hyper-sonic vehicle with input saturation based on adaptive back-stepping method,” in 2017 36th Chinese Control Conference(CCC), pp. 622–628, Dalian, China, July 2017.

[22] H. Sun, S. Li, and C. Sun, “Adaptive fault-tolerant controllerdesign for airbreathing hypersonic vehicle with input satura-tion,” Journal of Systems Engineering and Electronics, vol. 24,no. 3, pp. 488–499, 2013.

[23] K. Esfandiari, F. Abdollahi, and H. A. Talebi, “Adaptive con-trol of uncertain nonaffine nonlinear systems with input satu-ration using neural networks,” IEEE Transactions on NeuralNetworks and Learning Systems, vol. 26, no. 10, pp. 2311–2322, 2015.

[24] J. T. Parker, A. Serrani, S. Yurkovich, M. A. Bolender, and D. B.Doman, “Control-oriented modeling of an air-breathinghypersonic vehicle,” Journal of Guidance, Control, andDynamics, vol. 30, no. 3, pp. 856–869, 2007.

[25] B. Xu, “Robust adaptive neural control of flexible hypersonicflight vehicle with dead-zone input nonlinearity,” NonlinearDynamics, vol. 80, no. 3, pp. 1509–1520, 2015.

[26] R. J. Wallsgrove and M. R. Akella, “Globally stabilizing satu-rated attitude control in the presence of bounded unknowndisturbances,” Journal of Guidance, Control, and Dynamics,vol. 28, no. 5, pp. 957–963, 2005.

13International Journal of Aerospace Engineering