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Proceedings of the 5IhWorld Congress on Intelligent Controland Automation, J u n e 15-19,2004, Hangzhou. P.R.ChinaResearch on the Stability of Overload-Controlfor Aerodynamic Missiles

Wenjin Gu, H ongc ha o Z ha o a nd Y un a n HuFaculty 301Naval A eronautical Engineering InstituteYantai, Shandong Province, China

gwj@public.ytptt.sd.cn Abstract - The overload-control approach was developed andapplied to aerodynamic missiles in order to solve the problem ofnonminimum phase of the acceleration control. It controlled theoverload and the angle acceleration, hut it didn t control theattitude variables in order to acquire high maneuverability.Sliding mode control SMC) was adopted in this approach todesign a control law of the fin deflection. The c ontro l law realizedan integrative control of the overload and the angle acceleration.Stability analysis for the overload-control system was providedconsidering the uncertainties of aerodynamic coefficients. The

analysis shows that the attitude v ariables ofaerodynam ic missilesare asymptotically stable. The validity of the overload-control isdemonstrated by numerical simulation.Index Terms - Nonminimum phase. Accelerat ion controI .0verIoad-confro.lSl id ing mode coniroL

I. INTRODUCTIONRecently, acceleration control for aerodynamic missileshas been greatly appreciated. Unfortunately, the dynamiccharacteristics from the control tin deflection to missileacceleration are of nonminimum phase for aerodynamic tail-

controlled configuration. This arises from the fact that an upelevator produces a downward force until the angle-of-attackincreases to result in a net upward force [I]. The plantinversion control and output-redefinition technique wereapplied to stabilize the zero dynamics, thus the nonminimumphase characteristic was removed in [2]. Lee and Ha [3]applied the partial linearization and singular perturbationtechnique to transform the missile model with an output ofacceleration into a simplified system, which not only was ofminimum phase, but also realized the input-outputlinearization. A function approximation technique wasintroduced and a feedback-linearizing controller was designedin [4], by which the missile model was transformed into aparametric affine weak-minimum-phase model.Though above techniques are different, their basic idea isto design the control laws according to missile accelerationand attitude variables of the system (i.e. attitude angle andangle velocity), consequently the control laws is complex anddifficult to be applied in the engineering. In addition,introducing the attitude variables to the acceleration controlweakens the missile s maneuverability and loses the meaningof the acceleration control. Can we find a control method thathandles the nonminimum phase characteristic withoutcontrolling the attitude variables? The ans wer is positive.

The authors have proposed a control approach ooverload-control i n [ 5 ] and [6], which only controls theoverload and the angle acceleration of aerodynamic missilesHere the overload and the angle acceleration are treated aoutputs of the missile model. In a missile s angle moving, theangle acceleration is a variable corresponding to theacceleration, which can he measured by a sensor of angleacceleration. The overload represents the nondimensionaacceleration. M oreover, the missile s maneuve rability isusually evaluated by the overload; therefore the overload-control is a control approach with high maneuverability.This paper is organized as follows. In Section 11, themissile dynamics will be presented for aerodynamic tail-controlled configuration. In Section 111 the sliding-modecontrol (SMC) law and the stability analysis will be provided.Section IV will illustrate a simulation example. Conclusionwill follow in Section V.

11. MISSILE YNAMICSWe only take the pitch channel of aerodynamic missilesas an example to research. For adopting the overload-controlthe missile acceleration is replaced by the overload. Accordingto the research method and definition of notations of Chineseresearchers [7], the nonlinear differential equations for thepitch dynamics is given by

1, (Psin +Y - m g c o s 8 )

P s i n + Ym gIn =

where the meanings of all notations are described in [7]Equation (1) can be expressed in the form of [3] and [4],

0-7803-8273-01041820.00 02004 IEEE 5450

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where the meanings of all notations are also described in [ 7 ] .From [X we can get the following proposition.Proposition 2.1: Suppose the linear approximation isasymptotically stable, i.e. either the pair A , B ) is controllableor - in case the pair A . B ) is not controllable - theuncontrollable modes correspond to eigenvalues with negativereal part. Then, any linear feedback that asymptoticallystabilizes the linear approximation is also able toasymptotically stabilize the original nonlinear system, at leastlocally. If the pair A , B ) is not controllable and there existuncontrollable modes associated with positive real part, thenthe original nonlinear system canno t he stabilized at all.Proofi The proof had already been provided in [XIIt is based on this Proposition that [ 2 ] directly introducesthe linear missile dynamics to research. We can also researchthe linear approximation of ( I ) and look for a control law thatcan asymptotically stabilize (I). According to [7], the linearapproximation of (1) is

& =o, a,,a -a,$:&. = azIa+ a 2 2 0 z+ a2&, 3)[ i , ,= :a j6a+ ;a35s .

where the aerody namic coefficients are listed in Table 1.System 3) may he rewritten as(4)= A X BUY =cx Du '

where X = [a ;' is the state vector of 3), Y = n,, z ]'is the output vector, control input is U = S,, the coefficients

controllable. Considering the uncertainties o f the missilemodel, System 4) can be changed to he5 )x = A M)X + ( B AB)uY = C X D u

where A4 and B a r e the uncertainties matrices. which aredefined as, respectively, M =

111. SMC ANDSTABILITY ANALYSISFor system 9,we apply SMC to design the control law.Assumption 3.1: The actuator dynamics is fast enough toAssumpiion 3.2: The uncertainties satisfy the bounded

We make two assumptions as follows.be neglected.condition, that is

(6)a x { / 4 , I , IA O I A 4 } 5max{l&, I, I l } 0 , nvd and &, are the desired valuesrespectively. The derivative of a X) is (let dn , , / d r=O,d r j , I d = 0 )

where

They only have relation to the aerodynamic coefficients. Wecan get D , > 0 for < 0 in Table 1. Without loss ofgenerality. we choose 0 >0 (let c , < -c2 a,, g/(vaj,) ).Substituting the states equations of 5) into (8) gives

10)U = f a , u . )+A f (a , m;)+(D,a>,- D,a,, + D,Aa,, + D N A a 2 , ) u 'where

f ( a w,) = D,a,, -D,a,,)a + (0, + D,a2, w,, (1 a )A f a , o,)= D,Aa, + D , A a a , ) a + D , A a a , 2 w ; . ( I l b )

We construct a Lyapunov function asV ( X )=lu(X)12/2 (12)

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whose derivative is

where, the term im s the estimation of equivalent control,which do esn t contain the uncertainties, and all uncertaintiesare captured by the SMC term us In I 6), we take k >0 ,

=m, D,+D, )la +m,D, loz + % ( D e +D, IL, I.Substituting (14), 1 5 ) and (16) into (13), along with (6) ,gives

V = o & < - k o Z 0 , 17)Consequently, system ( 5 ) is asymptotically stable, that is,

the state variables a and U are asymptotically stable.From 15), w e h o w that i is a function of the state

variables. The overload-control approach d oesn t co ntrol thestate variables, so i r q as to he transformed to a function ofthe outputs n and 6: . Synthesizing the second equation andthe third equation of 3) with removing 6, gives

- - . .

.=K ,,(D ,a, -D,a,)+Kz,(D +D,a,,) . (21c)Dua,5 - D,%Obviously, according to (14), (16) and Z I ) , the controlaws only have relation to n y , r j , and the aerodynamic

coefficients. So to speak, the overload-control can ensure thathe attitude variables a and mz are asymptotically stablethough it only controls the overload n and the anglacceleration bZo f a missile.

Synthesizing above discussion, we can get the followingtheorem.heorem 3.1: For the linear approximation system 9,apply SMC to control the outputs n and hZ With th

uncertainties satisfying Assumption 3.2, we define thswitching function (7) and design the control laws (14), (16and (21). Then, the state variables of the system ( 5 ) arasymptotically stable.Proofi This is a direct consequence of (12) and (17)according to the Lyapunov stability theorem.Synthesizing Proposition 2.1 and Theorem 3.1, we cadraw a conclusion: the control laws (14), (16) and (21) caalso asymptotically stabilize the original nonlinear system ( lthus the nonminimum phase characteristic in the acceleratiocontrol is removed.

IV. N U M E R I C A LIMULATIONIn this section, an example of a supersonic aerodynamimissile is studied, whose pitch dynamics model is in the form

of 5). Two characteristic points are chosen from its flightrajectory of high altitude and low altitude respectively, whosaerodynamic data are different. The uncertainties matrices o5) AA and AB are assumed to he, respectively,

0.05sin(3t) 0 0.02 cos(2t)A A = [ OSsin(5f) O.lsin(2f) . (22According to Assumption 3.2, the boundaries are m, 0.5 anm2 O S respectively.

In the simulation, the control laws (14), (16) and (21) arapplied to design a controller. Step responses are shown iFig. 1 and Fig. 2.Fig. l(a) and Fig. 2(a) present the overloaresponse to command nyd= 4.0 in two characteristic pointsFig. I(b) and Fig. 2(b) present the angle acceleration responsto command chZd= 0 ( rads ) . The simulation results show thathe overload-control system is stable; and that it not only hagood tracking ability to the commands, hut also has stronrobustness against uncertainties.

V . CONCLUSIONSFor tail-controlled aerodynamic missiles, the acceleratiocontrol causes the problem of nonminimum phase. We hav

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proposed an overload-control, which only controls theoverload and the angle acceleration of aerodynamic missiles.The stability problem of state variables in the overload-controlsystem is researched. Because SMC is applied to syntheticallycontrol the o verload and the angle acceleration, the system hasstrong robustness. Finally, a simulation example is illustrated,and the results show the validity of the overload-control.However, as a new control approach, the overload-control isvery immature. The control laws are complex in the paper. Inthe further work, we will do research on the control laws, andwe will research the maneuverability of aerodynamic missilesunder the control of he overload-control.REFERENCES

[ I ] M. J. Tahk, M. M. Briggs, and P. K. A . Menon, Applications of plantinversion via state feedback to missile autopilot design, Proceedings ofthe 27 h CDC,Austin, Texas, pp. 730-735, December 1988,.[Z] I.-H. Ryu, C.-S. Park and M.-I . Tank, Plant inversion contml of tail-controlled missiles, AIAA-97-3766, pp. 1691-1696, 1997.[3] J.-I. Lee, and ILJ. Ha, Autopilot design for highly maneuvering STTmissiles via singular pemrhatim-like technique, IEEE Tronsociions on

Cont ro lSyslem Technology, vol. 1 o. 5 , pp. 527-541, 19Y9.[4] D. K. Chwa, and I. Y. Chai, New parametric affine modelling andcontrol for skid-to-am missiles, IEEE Transadionr on Conrrol SpsremsTechnology, vol. 9, no. 2, pp. 335-347,2001[5] Wenjin Gu, Shixing Wang, Yifei Zhang, and Jingmei Han, An overload-control realized by large-space maneuver, Journal o Nova/ Aeranoui icol

Engineering Academy, vol. IS. no. 4, pp. 4 0 1 4 0 4 , 2000 [RK ,, zk@F@*@, 15(4), 401-404 20003.

[6] Wenjin Gu; An overload-control of anti-ship missile, Journal o/Projecriles. Rockers, Mirsiles und Guidonce, ol. 22, no. I , pp. 14-18,14-18, 20021.M is i l z s , Beijing: Beijing Institute of Technology Press, 2000 [as ,

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2002 [En* , CmiT ffii3QMmll. z$W S~ . ~+R,2(1) ,[7] Xingfang Qian, Ruixiong Lin, and Yanan Zhao, Flirhl @namics o/

mi @>_X W, .b? fn+. , At : i W B I * + & i + i ,20001Press, 1995.[8] lsidori A Nonlinear conlrol sys/rm, 3* ed., London: Springer-Verlag

I1 2 3 4 5-1 Time 5)(8)

2 ,

-1.5: 2 3 4 5Time i s )ib)Fig. 1 Step responses for characteristic point ofhigh altifude

-1 1 2 3 4 5Time s i(4

. ,(b)Fig. 2 Step responses for charactenstic point of ow al t iade