dynamic surface backstepping control design for one hypersonic vehicle

7/30/2019 Dynamic Surface Backstepping Control Design for One Hypersonic Vehicle

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Proceedings of the 2009 IEEEInternational Conference on Mechatronics and AutomationAugust9 - 12, Changchun, China

Dynamic Surface Backstepping Control Design for OneHypersonic VehicleChen lie

Department ofControl EngineeringNaval Aeronautical andAstronautical University

Yantai,Shandong Province, [email protected] Jinhua

Department ofControl EngineeringNaval Aeronautical andAstronautical University

Yantai,Shandong Province, [email protected]

Pan ChangpengDepartment ofCommandNaval Aeronautical and Astronautical University

Yantai,Shandong Province, Chinanavypcp@ tom.com

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Abstract - Based on multi layer neural networks, feedbacklinearization technology, and backs tepp ing design method , anovel robust adaptive control design method is proposed for onehypersonic vehicle(HSV) uncertain MIMO nonaffine blockcontrol system. Multilayer neural networks are used to identifythe nonlinear uncerta inties of the system. A continuous robustterm is adopt to minify the inf luence of the mul ti laye r neuralnetworks construction error, the dynamic surface controlstrategy to eliminate "the explosion of terms" by introducing aseries of f irst order f il ters to obtain the different ia tion of thevirtual control inputs. Finally, nonlinear six-degree-of-freedom(6-DOF) numer ica l simulation results for a HSV model arepresented to demonstrate the effectiveness of the proposedmethod.

Index Terms -hypersonic vehicle ,.neural networks ,.dynamicsurface ,.backstepping

of a 6-DOF HSV model is preformed to verify theeffectiveness of the proposed algorithm and the conclusionsare given.II. NONLINEAR HSV MODEL WITH UNCERTAINTIES

Winged-cone '! ' is one of the main investigate objectswith its hypersonic mach number and open aerodynamicparameters, but most research is based on its longitudinalchannel model. In this paper we finished Winged-cone'sthree-axes, highly nonlinear model with general set ofuncertainties. The nonlinear dynamic equations are given asfollows(the full HSV model is given in appendix, andparameters are obtained from literature [1]):Xl T, (xl'x2 )x2 == h (X2)+ gI (X2)X3+ WI (x2)u

We assume that .h ' gI , hI ' J. ' f3 and g2 areunknown and h ==hO +4h, gI == gIO + 'h == h o + ~ ,g2 ==g20 + ~ g 2 , h I = hlO + ~ h I

where flO , gIO , wlO , f20 , g20 are nominal systemparameters, the others are uncertainties. The meaning of thesymbols used in this paper can be founded in appendix andliterature[1].The task of the controller is to track the commandssignals when aerodynamic model uncertainties exist.Assumption 1: Ignoring the effect of the fin deflection onthe aerodynamic force and viewing it as a part of theuncertainties, namely WI (Xl) == 0 .

where, (3)

q3-r X2 = [a6a 6r ]T

x -[v-I. INTRODUCTION

Hypersoninc vehicles Control technology has attracted alot of attention in recent years because of highly nonlineardynamics and high Mach velocity[1,2,3]. Wang[4] adoptedadaptive sliding controller to analyze the longitudinaldynamics of a generic hypersonic air vehicle. Shtessel'F'designed inner and outer sliding mode controller loop toinvestigate the reentry of hypersonic vehiles, Austin[6] usefuzzy logic control to compensate the unknown nonlinearities,genetic algorithm to optimize the scaling parameters of fuzzycontroller. a kind of neural adaptive controller for hypersonicvehicle is proposed by Xu, Mirmirani and Ioannou [7] A fewrecent contributions have also attempted design directly onnonlinear models [8-10]. Most research results were restricted tolinearly parametrized and state feedback linearizable.Different from above work, The paper investigate aMIMO nonaffine nonlinear hypersonic vehicle model withgeneral set of uncertainties. Dynamic surface backsteppingcontroller is designed based on Radial-Basis-Function (RBF)neural networks approach. Stability analysis is also shown bythe Lyapunov stability theory. Finally, a numerical simulation

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III. CONTROLLERDESIGNA. RBF Neural networkIn the control ler design RBF NN is used to approximatethe unknown function rjJ(o) . In a compact set, RBF NN can beexpressed as

(-) =WS T (- ) (4)Where WE RPis the tuning weight matrix and rjJ(-) is

a vector composed of Gauss functions. Given a smoothfunction rjJ(-) , where n is a compac t subset of R m , and5 > 0 , there exist a Gauss function vector S :R" H RPand an optimal weight matrix W* such thatI(x) - W' ST(x)1 :-:::; l5, Vx En.Define

(10)X 2d O is the certain part of x;, Multilayer NN:

*T (*T )SI ~ XI + 1 can be used to approach the uncertainpart of X; ,where XI is the input of above NN, I1/ 0 .

Then we choose desired virtual control value as follow:

0< giO < Igi ( ~ , X i + I ) : = 8f ( ~ , X i + 1 )/8xi+1~ g i u s;> 0 tomake

h(X)-W*ST(X) I1h(x) (5) Where II > 0 is designed parameter, 'i is the inductedrobust item and will be embodied as follow.

W; is the estimate of ~ is the estimated value of.Defining estimation error as ~ =W; - ,~ = ~ - , then the dynamic function of error statevalue ZI ' then

B. Dynamic surface backstepping controller designStep 1: Define error surface Zi = ~ - - \ i , - \ t = { ~ f1J ~ ] T i s

the system anticipant track, ZI derivative is

2"1 = - ~ Z I +glA, (Z2 +Xz - -Sd )= - ~ Z I +glA, (Z2 -llzl)+glA, [ f i { ( S I - ~ W ~ ) + ~ T ~ ~ T ~ +d1 - &1+1j ]

(6) SupposeDefining VI =-X l d +klzl, kl >O.By usingassumption 2 and 8vI /8x2 =0 ,we can get the following

inequalitya[J;(Xl'X2)+VI]/aX2 >glo >0

From literature [11] we can get that there has an idealvirtual control function x = x; (XI' VI ) , so

Suppose

1 2 1 - T - I - 1 {-T - I -}~ =--ZI + - ~ r +-tr ~ r V I ~ (12)2g lA 2 21

r WI = > 0,r VI = > 0 are parameters to bedesigned, Taking the time derivative of ~ ,we have

~ =rWI [ - ( SI _ S ; ~ T XI )ZI - o - W I ~ ] lit >a (14)

~ = r VI [ _ X I ~ T S;ZI - o - V I ~ ]

According to function (6),(7),so that2"1 = -klzl + glA, (x 2 - x;) (9)

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Step 2: Define error state value as Z2 =x2 - X 2d ' X 2dcan be obtained by the following first-order filter by statevariablex2


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(16)

So

By the error surfaces Z2 =x2 - x2dSo we have22=flO (.\2) +glO (.\2).\3 -iy +(I:f;. (.\2)+4?, (.\2).\3)

_ ( ) ( ) . (15)- ho x2 + glO x2 x3 - x2d - L1where L1 = -(L1J; (x 2 ) + L1g, (X 2)X3) is the uncertain term.By employing the neural networks to approximate L1.

We choose desired virtual control value as follow:~ = - ~ f ~ -U;S2(C';X2)-rz]

Candidate Lyapunov function and adaptive laws and therobust term are selected similar as step I.Step 3: Define the error surfaces Z3 =x3 - X3d . In this

step, we consider the third equation of (8) to design the actualcontrol input for the hypersonic aircraft. Let X3d pass througha first-order filter, a new state variable X3d can be obtained

IV. N UMERICAL SIMULAnON RESULTSIn order to validate the effectiveness and validity of theproposed method, nonlinear 6-DOF simulation results for aHSV model are presented.The controller design parameters are chosen as follows:II=11,/2 =13,/3 =7.3 ' k =20 , (Tw, =(Tv; =0.01 '7];=1,i=1,2,3 ,and X l d =[5130 2 2]. Simulationresults are shown in the following figures.,: /" (

"\E '

00 1 2 3 ! S 6 1 8 9 10t(s)

(17) Fig.1 Tracking curve of rAfter differentiating the error surface Z3 ' we get

We choose the control inputU = - g ~ ~ (! 3 Z3 +ho - X3d ) + g ~ O Z 2

)+kz3 + ~ 83 V; X 3 + r3

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(19).QC60 1 2 3 ! 5 6 1 8 9 10t(s)Fig.2 Tracking curve of P

13 , k > 0 is design parameters. Choose the followingadaptive laws and the robust termW; = r W3 [( S3 - S ~ ~ T Z 3 )z; - ( T W 3 ~ ]~ =r V3 [ Z 3 ( S ~ ~ Z 3 r - ( T V 3 ~ ]

1 3 = ; ( 1 1 U { ~ 1 1 : 1 1 A ; 1 1 2 1 1 ~ r ? A ; 1 1 : +2)/7B (20)Suppose

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0\ '

00 1 2 3 4 5 6 1 8 9 10t(s)Fig.3 Tracking curve of a

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FigA Tracking curve of fJ 1 [qSCLa+Tsina-mgcosycos,u]mvcos fJ ._l_[qSCypfJ- Tcosa sin fJ+mg cos ysin,u]mv '

1;3/ 13 = _l_[qSCLa(tan y sin zz + tan13) - mgcosycosu tan 13mv '

omv

qSCy O,.mv

1 sin atgfJ1o cosao sinasecfJqSCLoamvcosfJqSCy,oa

mv

+qSCy,pf3tany cos ,ucosf3+ T(sina tan ysin,u + sina tan 13-cos a tan y cos,usin13)]

r- cosa tg fJ

g l ( XI ) = sin acosasecfJ

qSCL,omvcos fJqSCy,o

10 1 '1 3 4 5 6 7 8 9 10t(s)

o ~ ' - - - - - - - -

'()20 1 2 3 4 5 6 7 8 9 10t(s)Fig6 Tracking curve orr

o\;1 \ - - -- - - - --1

Fig.S Tracking curve ofq

.150 1 '1 3 4 5 6 7 8 9 10t(s)Fig.7 ~ tuning weight

From the simulation results, it can be shown that thecontrol system has good stability, performance androbustness, even if the large model uncertainties exist.V .CONCLUSIONS

This paper presented a nonlinear adaptive controller for aHSV with a general set of uncertainties. The nonlinearadaptive controller is designed using dynam ic surfacebackstepping control techniques and fully tuned RBF neuralnetworks. Lyapunov stability theory is used to prove thestability of the system and derive the tuning rules for updatingall parameters of the RBF neural networks. Finally, anonlinear 6-DOF numerical simulation of a HSV model isperformed to demonstrate the good performance of theproposed method.

1 . fJW31 =-[qSCu (tan ysm,u + tan )mv "+qSCy,o tan r cos ,ucos fJ]W32 =_I_[qSCL.o (tanysin,u+tanfJ)mv a+qSCy,oatan r cos u cos fJ]W33 = _1_[qSCL o.(tan ysin,u + tan fJ)mv "+qSCy0 tan y cos u cos fJ]. ,.

APPENDIXThe winged-cone dynamics model

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q b S G , ~ qbSG,6a qbSG,4I p I p I pg2= g21 g22 g23

q S ( ~ , ~ +Xcgc:;,,6e ) q S ( ~ , 6 a +Xcgc:;,,6a) q S ( ~ , ~ +xcgc:;,,4)If" 1,. 1,.

REFERENCES[1] J. D.Shaughnessy, S. Z. Pinckney, J. D. Mcminn, C. I. Cruz and M. L.Kelley. Hypersonic Vehicle Simulation Model: Winged-ConeConfiguration. NASA TM-I02610,1990.[2] I. M. Gregory, R. S. Chowdhry, J. D. McMinn and 1. D. Shaughnessy.Hypersonic vehicle model and control law development using Hcoand J.!synthesis. NASA TM-4562, 1994.[3] H. Bushcek and A. J. Cal ise. Uncer ta in ty model ing and f ixed-o rdercon trol le r design for a hyperson ic vehicle model. AIAA Journal ofGuidance, Control, and Dynamics, Vol. 20. No.1, pp. 42-48,1997.5[4] Q. Wang and R. F. Stengel, Robust non linear control of a hyperson icaircraft. Journal of Guidance, Control and Dynamics, Vol. 23, No.4, pp.577-584,2000.[5] Y. Shtessel and J. McDuffie. Sliding mode control of the X-33 vehicle inlaunch and re-entry modes. in Proc. AIAA Guidance, Navigation, andControl Conference and Exhibit, 1998, pp. 1352-1362.[6] Austin, K.J. and Jacobs, P.A. (2001) 'Application of Genetic Algorithmsto Hypersonic Flight control', IFSAWorld Congress and 20th NAFIPSInternational Conference, Vancouver, BC, Canada, July, pp.2428 {2433.[7] Xu, H., Mirmirani, M., and Ioannou, P., "Adaptive Sliding Mode ControlDesign for a Hypersonic Flight Vehicle," Journal of Guidance, Control,and Dynamics, Vol. 27, No.5, 2004, pp. 829-38.[8] Tournes, C. , Landrum, D. B. , Shtessel, Y., and Hawk, C. W., "RamjetPowered Reusable Launch Vehicle Control by Sliding Modes,"Journal ofGuidance, Control, and Dynamics, Vol. 21, No.3, 1998,pp. 409-15.

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[9] Parker, J. T., Serrani, A., Yurkovich, S., Bolender, M. A., and Doman, D.B., "Control-oriented Modeling of an Airbreathing Hypersonic Vehicle,"Journal of Guidance, Control, and Dynamics, Vol. 30, No.3, 2007, pp.856-869. doi: 10.2514/ 1.27830[10]Fiorentini, L., Serrani, A., Bolender, M., and Doman, D., "NonlinearRobust/Adaptive Controller Design for an Airbreathing HypersonicVehicle Model," AIAA Paper 2007-6329,2007.[l1]Ge, S. S., and Wang, C. Adaptive NN control of uncertain nonlinear purefeedback systems. Automatica, 2002, 38 (4):671-682[12]Michael A. Bolender ,David B. Doman ,A Non-L inear Model for theLongitudinal Dynamics of a Hypersonic Air-breathing Vehicle,AIAAGuidance, Navigation, and Control Conference and Exhibit 15-18 August2005, San Francisco, California' AIAA 2005-6255[13]D. Swaroop, J. K. Hedrick, P. P. Yip and J. C. Gerdes, "Dynamic surfacecontrol for a class of nonlinear systems," IEEE Trans. Autom. Contr. Vol.45 no. 10, pp. 1893-1899,2000.[l4]P. P. Yip and 1. K. Hedrick, "Adaptive dynamic surface control:asimplified algorithm for adaptive backstepping control of nonlinearsystems," Int. J. Contr. Vol. 71, no. 5, pp. 959-979, 1998.

dynamic surface backstepping control design for one hypersonic vehicle

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