Nonlinear Model Predictive Control UAV

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Nonlinear Model Predictive Control Techniquefor Unmanned Air Vehicles

Nathan Slegers

University of Alabama at Huntsville, Huntsville, Alabama 35899Jason Kyle

Oregon State University, Corvallis, Oregon 97331-4501

and

Mark Costello

Georgia Institute of Technology, Atlanta, Georgia 30332

DOI: 10.2514/1.21531

A nonlinear model predictive control strategy is developed and subsequently specialized to autonomous aircraft

that can be adequately modeled with a rigid 6-degrees-of-freedom representation. Whereas the general air vehicle

dynamic equations are nonlinear and nonaffine in control, a closed-form solution for the optimal control input is

enabled by expandingboththe outputand control in a truncatedTaylorseries.The closed-form solution forcontrolis

relatively simple to calculate and well suited to the real time embedded computing environment. An interesting

feature of this control law is that the number of Taylor series expansion terms can be used to indirectly penalize

control action. Also, ill conditioning in the optimal control gain equation limits practical selection of the number ofTaylor series expansion terms. These claims are substantiated through simulation by application of the method to a

parafoil and payload aircraft as well as a glider.

Nomenclature

b = spanc = chordd = control flap widthFA = aerodynamic force coefficient vectorFc = aerodynamic control force coefficient matrixI = inertia matrix of system with respect to its mass

centerI

B, J

B, K

B= body frame unit vectors

II, JI, KI = inertial frame unit vectorsL, M, N = aerodynamic moment components in the body

reference frameMA = aerodynamic moment coefficient vectorMc = aerodynamic control moment coefficient matrixMi = relative degree for the ith outputp, q, r = components of angular velocity of the system in

body reference frameQ = diagonal tracking error weighting matrixqi = tracking error weight on the ith diagonal ofQRi = number of Taylor expansion terms in the ith

output and desired trajectory approximationSi = number of Taylor expansion terms in the ith

control approximation

SR = reference areau = control vectorub, vb, wb = components of velocity vector of the system mass

center in the body frame

VA = total aerodynamic velocity of the parafoil andpayload system

X, Y, Z = aerodynamic force components in the bodyreference frame

xI, yI, zI = components of position vector of the system masscenter in an inertial frame

, , = Euler roll, pitch, and yaw angles of system

I. Introduction

U nmanned air vehicles (UAVs) are providing improved capa-bility in performing a diverse set of military missions such asreconnaissance, targeting, and civilian missions such as borderpatroland environmental sensing. More aggressive use of UAVs promisesto yield new capability such as adaptive battlefield communicationnetworks, autonomous equipment delivery, flexible autonomoustraffic monitoring, and massively parallel autonomous search andrescue. A key element in these systems in the autonomous flight con-trol system. Many different control strategies have been developedspecifically for highly nonlinear dynamic systems such as UAVsincluding feedback linearization, sliding-mode control, and fuzzylogic [13].

Model predictive control laws use a dynamic model of the plant to

project the state into the future and subsequently use the estimatedfuture state and the desired future state to determine control action.This control technique has been successfully applied to manydifferent dynamic systems.

In standard linear model predictive control, the plant is modeledas a discrete linear system [4]. For the linear case when a finitehorizon quadratic cost function is minimized, the optimal controlsequence can be solved in closed form. When the plant is nonlinear,the optimal control sequence is, in general, not possible to obtain inclosed form. However, in some special cases, closed-form optimalcontrol sequences are possible. To solve these types of nonlinearoptimal control problems requires difficult and laborious numericaloptimization algorithms. An approach for solving the optimalproblem was given by Chen and Allgwer [5]. However, significantnumerical computations are still required for all but the simplestcases and are not practical for real time implementation. Anotherapproach is based on solving the state-dependent Ricatti equation[6,7]. The state-dependent Ricatti equation technique solves the

Received 2 December 2005;revision received 14 February 2006;acceptedfor publication 17 March 2006. Copyright 2006 by the American Instituteof Aeronautics andAstronautics, Inc. Allrights reserved.Copies of this papermaybe made forpersonal or internal use, on condition that the copier pay the$10.00 per-copy fee to the Copyright Clearance Center, Inc., 222 RosewoodDrive, Danvers, MA 01923; include the code $10.00 in correspondence withthe CCC.

Assistant Professor, Department of Mechanical and AerospaceEngineering. Member AIAA.

Research Assistant, Department of Mechanical Engineering. MemberAIAA.Sikorsky Associate Professor, School of Aerospace Engineering.

Associate Fellow AIAA.

JOURNAL OF GUIDANCE, CONTROL, AND DYNAMICSVol. 29, No. 5, SeptemberOctober 2006

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nonlinear problem by satisfying thealgebraic Ricatti equation at eachtime step. However, this methodalso requires a significantamount ofonline iterative computations and is not feasible for many real timeapplications. Nonlinear model-based controllers with increasedefficiency have been considered where predicted control moves arecentered about an optimal control law. The predicted optimal controllaw is then continuously modified at future times to ensure anappropriate control [810]. This process allows the majority ofcomputations to be calculated offline. Improvements on standard

receding horizon controllers for nonlinear systems have beenestablished by Magni et al. [11]. They demonstrated that using twodistinct horizons, a prediction horizon and a shorter control horizon,can increase the domain of attraction for short control horizons. Asuboptimal nonlinear model predictive control (NMPC) techniquealso has been proposed by Xin and Balakrishnan [12] that rewritesthe original equation in a linearlike form and solves the HamiltonJacobiBellman equation by power series expansion; however, it islimited to control affine systems. Suboptimal NMPC techniques fornonaffine systems have been proposed by Patwardhan et al. [13] aswell as Patwardhan and Madhavan [14]. An algorithm proposed byRajendra et al. [15] works for nonaffine systems and requires lesscomputation than other methods, butan iterative solution is required.An ideal NMPC algorithm displays the closed-form solutionproperties of linear MPC while being applicable to a wide range of

nonlinear systems. Chen [16] developed a NMPC technique that canbe applied to a general nonlinear plant by using Taylor expansion ofthe plant output and control. It is assumed that control weighting isnot in the performance index and that control order is the differencebetween the Taylor series expansion and the relative degree of thesystem. This technique was developed for a general nonaffine singleinput single output system.

The work reported here presents a nonlinear model predictivecontrol law inthe same spiritas Chens algorithm except it iscast inaMIMO setting. After establishing the general algorithm it isspecialized to air vehicles that are adequately described dynamicallywith a rigid 6-degrees-of-freedom representation. It is shown that apenalty for control can be implemented through the selection of thenumber of Taylor series expansion terms even with control beingabsent in the cost function. The methodology is successfullyexercised on two systems, namely, a parafoil and payload aircraftalong with a glider. Performance characteristics of the controlstrategy are presented through dynamic simulation results of both airvehicles.

II. Nonlinear Model Predictive Control Algorithm

Consider a generalnonlinear system ofthe formas given inEq. (1)with N inputs and P outputs. Notice that the assumed system is notaffine in control.

_x fx; u y hx (1)

For a nonlinear system, it is well known that after a specificnumber of derivatives of the output, control inputs appear. Therelative degree Mi is the number of time derivatives of the ith outputrequired for control to appear. When the control appears, it generallyappears in a nonlinear manner. If successive time derivatives areobtained after therelative degree,time derivatives of control will alsoappear. Time derivatives of an output can be expressed as in Eq. (2)when the number of time derivatives of the output is less than Mi,where ij is a function of only the state vector when j is less than the

relative degree.

djyidtj

ij (2)

Note that iMi is a function of both the state and the control. Ingeneral, when j is greater than the relative degree, the output can beexpressed as in Eq. (3).

djyidtj

ij i1dMij _u1

dtMi j iN

dMij _uN

dtMij(3)

Assume the input and output are sufficiently differentiable withrespect to time so that the input equations can be approximated by aTaylor series of orderRi, where Ri Mi. The output equations areapproximated by Taylor series of order S Ri Mi, where each Riis selected so thatS is the same for all output equations.

yit yit dyidt

. . . Ri

Ri!dRi yidtRi

(4)

uit uit duidt

. . . S

S!

dSu

dtS(5)

Rather than solving the control input function uit over thecontrol horizon, the control is parameterized with its S derivatives.Because uit is not required to appear in the state equations in alinear manner, it can be considered to be a state variable. Thisconverts determination of a continuous function for control intodetermination of a finite and relatively small number of discreteparameters to determine control. Subsequently this converts theresulting optimal control problem into a discrete parameter

optimization problem. The parameters to be determined for theoptimal control problem then become

Ui duidt

d2uidt2

dSui

dtS

h iT

(6)

U U1 U2 UN

T (7)

Assuming that parameters in Eq. (6) defining the control sequenceare known, the control is expressed in Eq. (8).

uit

Zduidt

d2uidt2

. . . S1

S 1!

dSuidtS

d (8)

In practical applicationsthe control is only needed nearthe currenttime ( 0). In this case, only the first derivative is required todetermine the optimal control input.

uit2 duidt

t2 t1 uit1 (9)

To develop compact expressions for the control input, the Taylorseries expansion for the output in Eq. (4) are written in as

yit TiYi (10)

Ti 11!

2

2!

Ri

Ri!

h i(11)

Yi yitdyidt

dRi yi

dtRi

h iT

(12)

Consider the entire outputvector, approximated by a Taylor seriesup to orderRi (possibly different orders for each output).

yt

T1 0 0

0 T2. .

. ...

..

. . .. . .

.0

0 0 Tp

266664

377775

8>>>>>:

Y1Y2

..

.

YP

9>>>=>>>;

TY (13)

The matrix Thas dimensions ofP V, where V PNi1 Ri. Thedesired trajectory is approximated in the same manner.

yDt TYD (14)

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The tracking error can now be formed and expressed in compactnotation

et YDt Yt T YD Y (15)

In model predictive control, a quadraticcost function is minimizedover afinite horizon.

J 12

ZT2

T1

et TQet d

1

2

ZT2

T1

YD YT TTQ T YD Y d (16)

Tracking error cost is weighted with the possibly time-dependentpositive definite matrix Q. Selecting the components ofQ to be timevarying allows initial or final tracking performance to be more

heavily weighted. Because only T, and possibly Q, depend on , thecost function can be rewritten in Eq. (17).

J1

2 YD Y

T YD Y (17)

The matrix i is given in Eq. (18) for a constant diagonal Qmatrix.

i qi

Zt2

t1

TTi Ti d

qi

t2t110!0!

t22

t21

20!1!. . .

tRi 1

2t

Ri 1

1

Ri10!Ri!

t22

t21

21!0!

t32

t31

31!1!

tRi 2

2t

Ri 2

1

Ri21!Ri!

t

Ri 1

2t

Ri 1

1

Ri1Ri!0!

tR22

tR21

Ri2Ri !1!

t2Ri 1

2t

2Ri 1

1

2Ri1Ri!Ri!

266664

377775 (18)

1 0

2

. ..

0 P

26664

37775 (19)

Note that the matrix can be computed in closed form andstored.

For the parameters U to be an optimal solution, the cost function inEqs. (1719) must satisfy the gradient condition

@J

@ U YD Y

T@ Y

@U 0 (20)

The optimal control input is determined by considering Eq. (20).Partition Y, YD, and @ Y=@ Uinto upper and lower sections as shownin Eqs. (2123).

Yi

YUiYLi

8>>>>>>>>>>>>>>>>>>>>>>>>>>>:

i0x

..

.

iMi1x

iMi x; u1N

iMi1x; u1N i1x _u1 iNx _uN

...

iRi x; u1N; . . . ; dS1u1N=dt

S1 i1xdSu1dtS

iNxdSuN

dtS

9>>>>>>>>>>>>>>=

>>>>>>>>>>>>>>;

(21)

@Yi@Uj

8>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>:

0 0 0

0 0 ...

..

. ... . .

.

0 0 0

i1 0 0 0@i

Mi 2

@u1j

i1 0 0

... . .. 0@i

Ri

@u1j

@iRi

@u2j

i1

9>>>>>>>>>>>>>>>>>=>>>>>>>>>>>>>>>>>;

0

i1

(22)

@ Y

@ U

0 0 0

11 12

1N

0 0 0

21 22

2N

..

.

0 0 0

P1 P2 PN

2666666666666664

3777777777777775

(23)

In a similar manner partition to be conformal with the vectors Y

and YD.

111 112

0

121 122

. ..

P11 P12

0

P21 P22

26666666664

37777777775

(24)

The format of@ Y=@ Uand allows for their multiplication to bewritten in compact form.

@ Y

@ U

112 0 0

. . .

122 0 0

0 2120 222

..

. . ..

0 0 P12. . .

0 0 P22

266666666666664

377777777777775

11 12

1N

21 22

2N

..

. ... . .

. ...

P1 P2

PN

26664

37775

RdY

(25)

With the selection ofRi such that the number of Taylor expansionterms in the control approximation for each control is equal and the

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system is square, i.e., the number of inputs and outputs are equal,dY is a square S P matrix and R has dimensions ofV P S P. The equivalent optimal condition is thenexpressed as follows.

YD YTRdY 0 (26)

Where dY1 exists, we have the following condition.

YD YTR 0 (27)

Because of the block structure R can be expanded as in Eq. (28).The resulting P conditions for optimality are shown in Eq. (29).

YUDiYLDi

YUiYLi

T

i12i22

0 (28)

YLi i22

1i12TYUDi Y

Ui Y

LDi (29)

If we are interested only in the current control input, then _u is theonly derivativerequired which is thefirst component ofYLi . Denotingthe first row of i22

1i12T as Ki and evaluating the first

component ofYLi the optimal solution for the first time derivative ofthe control u is given in Eqs. (3037).

B

11 11

..

. . .. ..

.

P1 PN

264

375 (30)

A 1M1 1 NMP1

T (31)

YL1D YLD11 Y

LDP1

T (32)

YUD

YUD1

..

.

YUDP

264

375 (33)

YU

YU1

..

.

YUP

264

375 (34)

UC _u1t _uNt

T (35)

K

K1 0 0

0 K2. .

. ...

..

. . .. . .

.0

0 0 KP

266664

377775 (36)

Using this notation, the MIMO control law is

UC B1KTY

UD Y

U YL1D A (37)

In the case of a SISO system, Eq. (37) reduces to the followingscalar result for the optimal control derivative (13).

_u 1

M1

KYUD Y

U dM1yD

dtM1 M1

(38)

III. Application to a Rigid Air Vehicle Model

Many air vehicles can be modeled as a rigid body possessing sixdegrees of freedom (DOF) including three inertial positioncomponents of the system mass center as well as the three Eulerorientation angles. Kinematic equations of motion for the generalsystem are provided in Eqs. (39) and (40).

8;

1

2SRVACD0 CD2

2

CDaa

8>>>:

ua

va

wa

9>>=>>;

(53)

8>>:Clb

Clp b2p

2VA

Cm0c Cmc Cmqc

2q

2VACnrb2r

2VA

9

>>=>>;

8>:

Clabd

0Cnab

d

9>=>;a

2

664

3

775(54)

Because of the symmetry of the parafoil system the inertia matrixtakes the form

I

IXX 0 IXZ0 IYY 0

IXZ 0 IZZ

24

35 (55)

I1

IXXI 0 IXZI

0 IYYI 0IXZI 0 IZZI

24 35(56)

For a SISO system the control derivatives are given in Eq. (38).The parafoil system has relative degree of 2, requiring threederivatives of the output equation to find 3 and 3. Taking the three

derivatives of the output and writing _L, _M,and _Nin compact form, y:::

can be written in the desired form of Eq. (3) with 3 and 3 given inEqs. (60) and (61).

8>=>>;

(58)

8>:

_L_M_N

9>=>;

SRb

2d

8>>>>>:

b2

CL

bCLpp

2VA bCLrr

2VA

c

CM0 CM cCMqq

VA

b2

CN

bCNp p

2VA bCNr r

2VA

9>>>>=>>>>;

(65)

MC

dCLa

2 0 00 cCMf cCMebCNa

20 0

264 375 (66)

Fig. 8 Desired heading angle geometry for path tracking.

Fig. 9 Tracking a straight path and 27 deg turn.

Fig. 10 Yaw and desired yaw angle for path tracking.

Fig. 11 Yaw and desired yaw angle for path tracking.

Fig. 12 Glider schematic.



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The matrix Mc contains the systems control moment coefficientsand may in general be singular. In the glider considered, the elevatorand flap controls are redundant with respect to control moments,therefore the coefficient matrix is singular. In this case thepseudoinverse may be used in the optimal control solution tofind thebestfit solution.

The preceding system of equations describing the rigid glider arenumerically integrated using a fourth-order RungeKutta algorithmto generate trajectories of the system. One example scenario isexamined using the physical parameters provided in Table 3, and theaerodynamic properties provided in Table 4. In the followingresults,the control derivative is updated every 0.01 s.

Figures 1316 show simulation results for the glider tracking aconstant headingangle of 60 deg, roll angle of 0.34 times the headingangle error, and a pitch angle of 22:5 sint 6 deg. The values

1 2 1

are used in the error weighting matrix, the control is approximatedwith an eighth-order Taylor series expansion, and the predictionhorizon is 1.06.0 s.

As in the parafoil case, the prediction horizon and Taylor seriesexpansion orders are used to balance control magnitude and trackingerror. Heading angle and pitch angle converge to their desired valueswithin 7 s. To allow a more aggressive time varying pitch trajectory

to be easily tracked, a constant weighing factor of 2 is applied to thepitch angle error compared to unity for both the heading and rollangle errors.

VI. Conclusions

A nonlinear model predictive control strategy for tracking adesired orientation trajectory was developed for a general rigid airvehicle. The control strategy was simulated for an autonomous

parafoil and payload system and an autonomous glider. Theperformance of the controller was evaluated by varying predictionhorizons and number of Taylor expansion terms used for theapproximation of the output equation and the control sequence. Itwas observed that a penalty for control can be implemented throughthe selection of the number of Taylor series expansion terms evenwith control being absent in the cost function. The selection of thenumber of terms in the Taylor series expansion was limited only bythe conditioning of the matrix 22. Also, the limitation on the size ofS can be circumvented because a nearly equivalent controller to thatfor a large S and large prediction horizon can be found for a small Swith the appropriate prediction horizon. This observation leads totwo possible methods for selecting S and the prediction horizon. Thefirst method: select a desired prediction horizon specific to the plant

and then choose S to achieve an appropriate penalty on the controlsequence. The second method: ifS cannot be chosen large enough inthe first method, then a nearly equivalent controller can be designedby selecting S as large as reasonable so that the matrix 22 is well

Table 3 Glider physical properties

Variable Value Units

0.00238 slug ft3

Weight 1.99 lbf S 4.35 ft2

b 5.50 ft c 0.557 ft

IXX 0.0548 slug ft2

IYY 0.0288 slug ft2

IZZ 0.0813 slug ft2

IXZ 5:78e 005 slug ft2

IXY 5:05e 006 slug ft2

IYZ 1:93e 006 slug ft2

Table 4 Glider airfoil parameters

Main wing airfoil RG-15

Main wing flaps Trailing edge individual controlT-tail airfoil NACA 0009T-tail flaps Trailing edge elevator only

Fig. 14 Glider pitch angle time history.

Fig. 13 Glider heading angle time history. Fig. 15 Glider bank angle time history.

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conditioned and then choosing an appropriate prediction horizon soas to achieve a suitable control penalty. Another more complex

example is reported for an autonomous glider system in whichelevator andflap controls were redundant. The redundancy made thecontrol matrix MC singular. It was successfully demonstrated that apseudoinverse can be used.

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Fig. 16 Glider control deflection time history.


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