Fuzzy Neural Network-Based Sliding Mode

Control for Missile's Overload Control SystemHongchao Zhao

Department of Scientific ResearchNaval AeronauticalEngineering InstituteYantai, China, 264001

E-mail: navyzhc@l63.com

Hongyun Yu

School of Mathematics & InformationYantai Normal UniversityYantai, China, 264001

E-mail: xiaofish_er@l63.com

Wenjin GuDepartment of Control Engineering

Naval AeronauticalEngineering InstituteYantai, China, 264001

E-mail: kzgwj@sina.com

Abstract-In order to remove the nonminimum phasecharacteristics of missile overload output, theoutput-redefinition technique is applied to the missile'soverload control system. Because the overload control systemdoesn't control missile's attitude angle and angle rate, they aretreated as uncertainty of the system. In order to improve therobustness of the overload control system, a fuzzy neuralnetwork-based sliding mode control method is used to designit. The sliding mode control method is used to control themodified output with the fuzzy neural network as anapproximator for the unknown uncertainty bound. Asimulation example is Mustrated to show the validity of thedesigned overload control system.

I. INTRODUCTION

Recently, acceleration control of aerodynamic missileshas been greatly appreciated. Unfortunately, the dynamiccharacteristics from the control fm deflection to missileacceleration are of nonminimum phase for tail-controlledconfigurations [1]. The nonminimum phase plant hasunfavorable properties in view of control system design.The plant inversion control with output-redefinition methodwas applied to remove the nonminimum phasecharacteristics in [2] and [3]. Lee and Ha [4] applied thepartial linearization and singular perturbation-like techniqueto transform the missile model with an output ofacceleration into minimum phase system. Gu et al [5]proposed an overload control method for aerodynamicmissiles. As a novel acceleration control method, it onlycontrols missile normal overload and angle accelerationwithout controlling the attitude angle and angle rate. It alsoremoves the nonminimum phase characteristics. However,the authors applied the pseudo-inverse to dispose of theattitude variables and omitted the actuator dynamics, sotheir deducing was not exact, but approximate. Theuncertainties of the missile dynamics were considered andtheir bounds were assumed to be known in [5], but in factthe bounds were difficult to know in advance for practicalapplications.

In this work, the output-redefinition approach is appliedto removes the nonminimum phase characteristics of the

missile overload output. Because the overload controlsystem doesn't control the missile's attitude variables, theyare treated as the uncertainty. In order to improve therobustness of the overload control system, a fuzzy neuralnetwork-based sliding mode control approach [6] [7] isutilized to design it. The performance of the designedoverload control system will be verified through numericalsimulation.

II. MISSILE DYNAMICS

For the derivation of missile dynamics, the followingassumptions are made:Assumption 1: The missile body has pitch and yaw

symmetry.Assumption 2: The missile is roll-position stabilized.Form above assumptions, the characteristics of pitch and

yaw dynamics of the missile are almost the same, thestructure of pitch control system can be converted to that ofyaw control system with ease. Hence, only the pitchdynamics is considered. For small angle-of-attack andsideslip angle, the pitch dynamics of aerodynamic missileswith actuator dynamics is described as follows [5]:

=az - a34ae- a35 6

ICz = a24 + a22COz +a254z6z = -w5z + wuc

V Vny =-a34a+-a356z

g g

(1)

where a is the angle-of-attack, Co_, is the pitch rate, tzis the fin deflection angle, w is the actuator bandwidth,uc is the control input, V is the flight speed, g is the

gravity acceleration, n, is the normal overload, (O, is

the pitch angle acceleration, a22, a24, a25, a34 and

a35 are aerodynamic coefficients. In order to help the state

0-7803-9422-4/05/$20.00 Q2005 IEEE1786

variables represent the performances of a missile directly,we take a coordinate transformation on the system (1) as[aC wZ 6]T <- [a o, ]T Note that the relativedegree of the system is 1, so there is no need to transform itinto the normal form, i.e. ny is theoutputand [a CO]constitutes the zero dynamics.

hy = Q1 +JP ny +bu, (2)Q= Q + P2 ny

where 17=[ ] = 34T 1

Q1= [Va34 W5/g Va34g], b =Va35 w6/g,

[ga25%a35)j Q2 a24 - a25a34/a35 a22]It is easy to see that the eigenvalues of Q2 coincide

with the open transmission zeros. If the missile is atail-controlled configuration, it has unstable zero dynamics.In next section, we will stabilize the zero dynamics by theoutput-redefinition method.

III. OVERLOAD CONTROL SYSTEM DESIGN

A. Stabilization ofZero DynamicsThe main idea of the output-redefinition technique is to

redefine the plant output such that the modified outputstabilizes the zero dynamics and that asymptotic tracking ofthe new output also ensure asymptotic tracking of theoriginal output. Because the overload control system onlycontrols ny and d)z, the modified output is defined as

; = r(Q1 + KQ2- PIK - KP2K)q+(PI + KP2)J + cbuc

4 = (Q2 - P2K)q+i-P24K

(6)

Since it can be shown that (Q2, P2) is a controllable

pair, we can assign arbitrary eigenvalues of Q2 - P2K inthe left half of the phase plane by choosing K, i.e.originally unstable zero dynamics is stabilized throughoutput-redefinition technique. K plays the role in onlystabilization of zero dynamics by pole placement. Thechoosing of K is discussed by Kim and Tahk [3] in detail.

For the further analysis, we introduce the following errorvariables:

_d =nnd 4=_d (7)

where the superscript "d" represents the desired value. From(3) and (5) we have

{d =knd + k2d = x(nyd+ K77d) (8)

The overload control system is designed to force-+0O, then according to (5) the error dynamics of ?1 is

== (Q2 - P2K)4 (9)

Since Q2-P2K is Hurwitz, q - 0, as t - oo .The integration of (5) and (8) yields

5 = K(ny + K4) (10)

t = klny + k2Cz

The last equation of (2) is

Z = a24 _ a2sa34L a35

(3)

227]+ ga25yi Va35

Substituting (4) into (3) gives

S = K(ny + KU)

where K = [k2V(a24a35 - a25a34)L k1Va35 + k2ga25

which implies== {/K-K7 (11)

Hence we conclude that hy - 0 as t -> oo. In the next(4) section the fuzzy neural network-based sliding mode control

method will be applied to design the overload controlsystem to force { ->0.

(5)

k2Va22a35k1Va35 + k2ga25

K = k + k ga25 Then the system (2) is transformedVa35

into (6) as follows.

B. Fuzzy Neural Network-Based Sliding Mode Control

We consider the 5 subsystem of (5). Because the

overload control system doesn't control a and (Os, i.e.

77, the 77-term is treated as the uncertainty. Then the {subsystem of (5) is transformed into a simple form asfollows,

=(, +KP2)F + Kbu, + d(Q) (12)

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where d(q) is the uncertainty,

d(q) = ATQ1 + KQ2 -PK - KP2K)q (13)The uncertainty is assumed to be bounded, i.e.d(q) 1. D, where D is an unknown function.A new switching surface without reaching phase is

designed as follows [8],

S(t)= (t) + p (r)dT -4(0) (14)

where p > 0, St(0) denotes the initial value at the initialtime t = 0 . It is obvious that at t = 0 the systemtrajectory is placed on the switching surface (14), namelyS(O) = 0. Thus the reaching time is eliminated and therobustness to the uncertainty is always ensured in the wholecontrol process. If the system trajectory stays on theswitching surface, namely S(t) = S(t) =0 , then theequivalent dynamics ofthe system (9) is governed by

{(O = -p{(O (15)

So the error Q(t) will converge to zero exponentially. Toguarantee that the system trajectory stays on the switchingsurface, we design a sliding mode controller with a fuzzyneural network approximator compensating the uncertainty.

Fuzzy neural network is generally a fuzzy inferencesystem constructed from the structure of neural network.Learning algorithms are used to adjust the weights of thefuzzy inference system [6]. Its output o(xf ) can beexpressed as [7]:

o(xf) = l(H;=) = 0 (x) (16

where A,A (xj) is the membership function of fuzzy

variable xj, h is the total number of the fuizzy rules, y7

is the point at which 1B'(5i) 1, Aj and B' denotefizzy sets of input and output variable, respectively,0 = [yl y2 ... yh ]T is a weight vector of the output layer,Xf E 9R" is input vector, and

(Xf) = [y1 V2 * * * JT is a fuzzy basis vector, where

VI' is defined as

i (x1) = H=lg,(Ij) (17)

The fuzzy neural network in the form of (16) can be usedas an approximator to approximate the uncertain nonlinearfunction to arbitrary accuracy as Lemma 1.

Lemma 1 [7]: For any given real continuous function %

on a compact set UC 91n and arbitrary small E > 0, thereexists an optimal parameter vector -0 , such that

SUp O*Tr (Xf )_(xf) j< EXfCU

(18)

Now we use the fuzzy neural network to approximate D.Its input vector is xf = [5 411T,and its output o(xf).

Form Lemma 1, for arbitrary small £ > 0, there exists anoptimal parameter vector 0*, O(Xf ) = 0*T#P(Xf ), suchthat

sup I 6*TV(xf) -DI< E (19)which yields

D <0*TV(xf)+e (20)

However, 0 can't be used directly, so we introduce Uas an estimate of 0 . The controller design and the stabilityanalysis are stated in the following theorem.

Theorem 1: For the uncertain system (12) with theswitching surface (14), the fuzzy neural network-basedsliding mode controller is designed as

= -(Qb)-'[(PI +KP2)4+p-_ed]- (rb)-1[UTVI(xf ) + 2]Sgn(S)

(21)

and the learning algorithm of the weight vector is chosen as

O=FISIW(xf) (22)

where r = diag(q71 72 ...*1h) is the learning rate matrix,then the error e(t) will converge to zero exponentially.

Proof: Choose the following Lyapunov function

V=S2 /2+ jT] /-2 (23)

where 0 = 0- *, and 0 = 0. The time derivative of (23)is

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V= SS+OdpThOrl

=S- +KJi + ibuC + d(U)

_- + p f]+ 0 I-,

Using (21) (22) with (20) in (24) gives

V=S[d(q) -(OI,(Xf ) +A) Sgn(S)] + Tr-l1 S | [eOTV(Xf ) + ]- | S [eTW(Xf ) + 2]+ a

=-(2-A) | S +jT[r 1-1 S (xf

(24)

0

(25)

=-(2-e) IS 1< 0

Note that E is arbitrary small, e <2 is certainlysatisfied. If and only if S =0, V=0, so the systemtrajectory out of the switching surface will converge to itand then stay on it, namely S(t) = S(t)= 0, and

according to (15) the error et(t) will converge to zero

exponentially. This completes the proof.

IV. NUMERICAL SIMULATION

To evaluate the performance of the designed overloadcontrol system, a supersonic anti-ship missile is researchedin the simulation. For the sake of secrecy, the values of theaerodynamic coefficients of pitch dynamics (1) can't beprovided. We use the fuzzy neural network-based slidingmode controller (17) with (10) and (18) to design theoverload control system and perform the simulation. Sincein steady state the pitch angle acceleration goes to zero,

its desired value is chosen as Ci = rad/s2. The desired

value of normal overload is chosen as ny = 1.0 . From (7)

d =k, . k1 is determined by the simulation as

kj = 0.45. Simulation results are shown in Fig. 1 and Fig.2. Fig. 1 shows the errors of modified output and originaloutput converge to zero. Fig. 2 shows the system trajectorydeparts from the switching surface after the initial timewhile it fast converges to the switching surface. The systemtrajectory stays on the switching surface in almost the wholecontrol process. The Simulation results show the desigedoverload control system has good stability and robustness tothe uncertainty.

a.S.g

0.20

-0.2

-0.4

-0.6-0.8

-1

0.1

0.05

0

-0.05

-0.1,

3 1 2 3 4Time (s)

Fig. 1. Errors of5 and ny

N -

n x n n Au 1 2Time (s)

Fig. 2. Response of switching surface

V. CONCLUSIONS

In this paper, the aerodynamic missile's dynamics withactuator dynamics is researched. A fuzzy neuralnetwork-based sliding mode control method is developedfor the design of the overload control system. The constraintof demanding prior knowledge on upper bound of thelumped uncertainty is relaxed form general sliding modecontrol through the using of a fuzzy neural networkapproximator in sliding mode control. A simulation exampleis illustrated, and the simulation results show that the fuzzyneural network-based sliding mode control method has bothgood tracking performance and strong approximation abilityto the lumped uncertainty.

ACKNOWLEDGMENT

The authors would like to thank Anli Shang, Jinyong Yuand YumanYuan for their valuable comments and assistanceduring this work.

REFERENCES

[1] M. J. Tahk, M. M. Briggs, P. K. A. Menon, "Applications of plantinversion via state feedback to missile autopilot design." Proceedings ofthe 27th CDC, Austin, Texas, 1988, pp. 730-735.

[2] J. H. Ryu, C. S. Park, M. J. Tank, "Plant inversion control oftail-controlled missiles." AIA-97-3 766, 1997.

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error of eerror of ny

v

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.r J 4

[3] S. G. Kim, M. J. Tank, "Output-redefinition based on robust zerodynamics." AIAA-98-4493, 1998.

[4] J. I. Lee, I. J. Ha, "Autopilot design for highly maneuvering STTmissiles via singular perturbation-like technique." IEEE Transactionson Control System Technology, vol. 7, no. 5, pp. 527-541, 1999.

[5] Wenjin Gu, Hongchao Zhao, Yunan Hu, "Research on the stability ofoverload-control for aerodynamic missiles." Proceeding of the 5thWCICA Conference, Hangzhou, 2004, pp. 5450-5453.

[6] W. Y Wang, C. Y. Cheng, Y. CG Leu, "An online GA-basedoutput-feedback direct adaptive fuzzy-neural controller for uncertainnonlinear systems." IEEE Transactions on Systems, Man, andCybernetics, vol. 34, no. 1, pp. 334-345, 2004.

[7] W. Y. Wang, M. L. Chan, C. C. James Hsu, et al, "H,, tracking-basedsliding mode control for uncertain nonlinear systems via an adaptivefuzzy-neural approach." IEEE Transactions on Systems, Man, andCybernetics, vol. 32, no. 4, pp. 483-492, 2002.

[8] K K Shyu and H J. Shieh, "A new switching surface sliding-mode speedcontrol for induction motor drive systems." IEEE Trans. on PowerElectronics, vol. 11, no. 4, pp. 660-667, 1996.

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