Author name used in this publication: F. H. F. LeungAuthor name used in this publication: P. K. S. TamVersion of RecordPublishe

Lam, HK

Leung, FHF

Tam, PKS

Author name used in this publication: F. H. F. LeungAuthor name used in this publication: P. K. S. TamVersion of RecordPublishedVoR allowe

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Lyapunov Function Based Design of Robust Fuzzy Controllers for Uncertain Nonlinear Systems: Distinct Lyapunov Functions' F. H. F. Leung, The Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong H. K. Lam, The Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong P. K. S.  Tam, The Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong Abstract This paper presents the stability and robustness analyses of an uncertain fuzzy control system which is formed by an uncertain fuzzy plant model and a fuzzy controller. The fuzzy plant model with parameter uncertainties describes exactly the behavior of an uncertain nonlinear plant. Three design approaches are introduced to close the feedback loop. Based on the Lyapunov's stabili9 theory, new stability criteria and robust areas are to be derived without resorting to a common Lyapunov function. An application example on stabilizing an uncertain nonlinear mass-spring-damper system will be given to illustrate the merit. 1. Introduction As fuzzy control was found capable of tackling uncertain nonlinear systems, it has become a hot topic of research. The design is usually by heuristic methods. Although they are simple and easy to understand, the stability and performance are not guaranteed. To prove the stability, researchers had derived different conditions based on sliding mode control technique [3] and adaptive technique [4]. One significant work [l] proposed the use of a fuzzy model. If this fuzzy model can describe exactly the system dynamics, a stability condition can be derived by finding a common Lyapunov's function [2, 51. However, the condition is valid to systems without parameter uncertainties. In this paper, we analyze the stability and robustness of an uncertain fuzzy control system. The fuzzy model is modified to one with parameter uncertainties such that it can exactly describe the behavior of an uncertain nonlinear system [6]. Based on this modified fuzzy model, stability conditions and robust area are derived by applying Lyapunov's stability theory. Unlike [l], the Lyapunov's functions used no longer need to be common. 2. Fuzzy plant model and fuzzy controller ' The authors are with the Department of Electronic Engineering, The Hong Kong Polytechnic University. This work was supported by the Research Grant of The Hong Kong Polytechnic University with the project account code 0350 525 A3 420. An uncertain multivariable fuzzy control system can be regarded as consisting of a fuzzy plant model and a fuzzy controller closing the feedback loop. 2.1. Fuzzy plant model with Uncertainties Let p be the number of fuzzy rules describing the uncertain nonlinear plant. The i-th rule is of the following format, Rule i: IF x1 is MI' and . . . and x,, is M,,' where ML is a fuzzy term of rule i corresponding to the stake xk,  k = 1, ..., n, i = 1, ..., p;  AAi E%"Xn and are the uncertainties of A'ER"'" and mi , - p m  respectively; XE%"'' is the system state B; E 3 nxm vector and UE%" is the input vector. The inferred system states are given by THEN X = ( A ' + A A ' ) X + ( B '   AB')^ (1) i ( t )  = E wi ( X ) ( ( A ;  + AA; )x( t )  + (B' + mi ) u( t ) )  (2) i=I  2.2. Fuzzy controller A fuzzy controller with c fuzzy rules is to be designed for the plant. Thej-th rule of the fuzzy controller is of the following format: Rule j: IF x1 is N) and . . . and x,, is N,' where Nd is a fuzzy term of rule j corresponding to the state xp, /.? = 1, ..., n, j = 1, ..., c; GJ E % ~ ~ "  is the feedback gain of rulej, r E Rmxi s the input vector. The inferred output of the fuzzy controller is given by THEN u = G i x + r  (5 )  U = i m J ( G j x + r )  j=  I 0-7803-4863-Xl98 $10.0001998 iEEE 511 Authorized licensed use limited to: Hong Kong Polytechnic University. Downloaded on July 7, 2009 at 04:37 from IEEE Xplore.  Restrictions apply.In this section, the system stability and the robustness ('I are analyzed. Theorem 1 to 4 summarize the analysis results for the three design approaches respectively. (8) Theorem 1 and 2 are directly extended from El], whereas Theorem 3 and 4 describe the less conservative conditions. '4 i m I ( x )  = 1 ,  mj(x> E io, 11 for all j PN{ ( X l )  O PN; (x, )o** -PUN; (%I) c (PN; (x1) O PN; (x, >O"."!$q: (x ,  1) "(x)= /=1 p . ( x ~ )  is the grade of membership. N: 3. Stability and robustness analysis The stability and robustness of the uncertain fuzzy control system are to be analyzed in this section. Three cases of controller design approaches will be investigated. 3.1. General Design Approach (GDA) General design approach allows differences in the rule antecedents between the fuzzy plant model and the fuzzy controller. This approach gives designers the largest freedom on controller design. From (1) to (8), the closed- loop fuzzy system is given by, (9) (10) x = f w'mj((Hg + AHi)x + (B' + AB' )r) ;=I j=1 H" '' = A'+B'Gj,AH'i = A A ' + m ' G i  3.2. Parallel Design Approach (PDA) Parallel design approach uses the same rule antecedents of the plant model in the fuzzy controller. Hence, some of the terms in (9) can be grouped together. This makes the stability criterion to be satisfied more easily. The closed- loop fuzzy system is given by, x =  $w'w'(H" + A H H " ) x + * $ w ' w q J U  +AJJ")x+$w'(B' +AB')r (11) i=l ;< j , = I  _ _  HU + ~ j i  AHu + AHji (12) (13) J" = , d J u  = 2 2 H U  = A' + B ' G ~ ,  AHV = AA; + D ' G j  Theorem 1. given by (9) without uncertainty, i.e. AH'' = 0, is stable the following inequality holds: H" P + PH" < -&I for all i andj where P E  32"'" is a symmetric positive definite matrix, E is a positive value and I E 9Inxn is an identity matrix, 1\11 denotes the lz vector norm or induced matrix norm. Under PDA, the fizzy control system as given by (12) without uncertainty, i.e. AH" = 0 and AJ" = 0 ,  is stable if the following inequalities hold Under GDA, the fuzzy control system as T H " ~ P  + PHI' < --E1 for all i J ~ ~ P  + PJ" < -E][ for all i<j Under SDA, the fizzy control system of (15) without uncertainty, i.e. AH' = 0 ,  is stable if the following inequality holds: H ' ~ P  +P H ~  < -&I for all j Definition 1. The robust area of afizzy control system is defined as the area in the parameter space inside which uncertainties are allowed to exist without aflecting the system stability. Theorem 2. Under GDA, with the uncertain fizzy control system given by (9) the robust area is governed by, lbsTp + p ~ s  [I The uncertain fuzzy control system is stable if the I < - ( H ~ ~ P  + p ~ " )  - d for all i and j .  Robust m a  3.3. Simplified Design Approach (SDA) Simplified design approach that requires the sub-system matrix B, and the fuzzy controller has the same number of in each rule of the fuzzy plant model has a common input rules with the same antecedents as the fuzzy plant model. The closed-loop fuzzy system is given by, as its maximum value, satisjies the following condition: [m"TP+pAH" <  AH^^^+^^^" <  AH^'^^+^^^" i andj  given by (12), the robust area is governed by, for all area ill I, I Under PDA, with the uncertain fizzy control system P ]=I x =  Cw'((H' +AH')x+(B'+AB')r) (14) I < -(H"=P +PHI' )- &I for all i Ib'l'P+PJ"[( I<-(JuTP+PJU)-~Iforall  i<j H i = A 1 + B G ' ,  AHJ =AAJ+ABG'  Because the input matrices are common, we have, B = B ' ,  AB=AB-' or, ]=I Cw'B' = B , /=1 C w'AB' = AB Robust m d  (16) The uncertain fuzzy control system is stable if the (17) uncertainty Ik"TP+PAH'lII  and I/41"TP+PAJ"ll, P P 3.4. Stability and Robustness Analysis 578 Authorized licensed use limited to: Hong Kong Polytechnic University. Downloaded on July 7, 2009 at 04:37 from IEEE Xplore.  Restrictions apply.their maximum values respectively, satisfy the following conditions: miiTp + pAHii < HiiTp + pAHii < AHiiTp + pmii for all i Ix. I, I I-- ijTp + PAJS~ < (IU"Tp + P A J ~ ~  < JblBTp + P A J ~ ~  for all i<j mu Rabauama Under SDA, with the uncertain fuzzy control system given by (15), the robust area is govemed by l P J T p  +p m j [  I < - ( H ~ ~ P  + HI) - d for  all j Robust m a  The uncertain jufuur control system is stable if the as its maximum value, satisfies the following condition: p + PAHJl < p T P  + PAHJI < PJ'P + PAHJ Theorem 3. Under GDA, the fuur control system as given by (9) without uncertainty, i.e. AHg = 0 ,  is stable if the following inequalities hold: for all j Inax I,, Iu I ?I P +P"EN<--d forall i, k = I ,  ...* p; j ,  1 = 1, ..., c. Under PDA, the fuzzy control system as given by (12) without uncertainty, i.e. AHii = 0 and AJg =0, is stable if the following inequalities hold: \ H k t T  P ct +P&H"<-d fori ,  k = I ,  ..., p;  j = 1, ..., c, Under SDA, the fuzzy control system of (15) without uncertainty, i.e. AHi =0, is stable if the following inequalities hold.: H kT P k +P'H*<--d +(I--)mut[L(HiTP' +PkH'). O]<-d p t* for all j ,  k = I ,  ..,, p. Theorem 4. Under GDA, with the uncertainfiuzy control system given by (91, the robust area is govemed by, Tpkl+ptlHjl I P X C  Robuam for all i, k = 1, ..., p;  j ,  1 = 1, ..., c, and ij # kl. The uncertain fuzzy control system is stable if the &TPkI +p"L\Hij with uncertainty II p c T p '  + P ~ A H ~  as its maximum value, satisfies the following condition: < + pkl"j-1 Robust area for all i and j .  given by (12). the robust area is govemed by, Under PDA, with the uncertain fuzzy control system Hbt'Pbt+PBHbt 1 <- - ( I -7 )L(JeTPbt  +P"JS)I-dforaIlk; i <  j bijTPbt +Pet J Y I -  +L(JSTPct  +Py J N  ) 20 for all i < j; k P 2  P The uncertain fuur control system is stable if the uncertainty I* + p'mul and b,'., + pM@ 1 and p + p f i ~ g i  as its maximum value respectively, sat i f i  the following conditions: lllpx mu .Tpu +pim~i<p + p u a ~ l  <p + p u m ~ l b ,  forg i ,  I I r J - T  P + P A J + ~ < ~  +P'AJJ"] I <p +P~AJ"IU_ IC. fwaicj; t Under SDA, with the uncertain fuay control system given by (15). the robust area is governed by H"P'+P'Hk 1 -(I--)&fH"Pk +P*H')-d IT k P P P + + H * ~ - - + ~ ( W ~ P ~  + P ~ W ) > O  for all k + i. The uncertain fuzzy control system is stable if the uncertainty 1 p . I i T p '  +PkAH'I, with I ~ A H ~ ~ P ~  + P ~ A H '  as its maximum value, satisjies the following condition: I, IPk +P"I <II..'TP' +P" 11111 for all k # i. 579 Authorized licensed use limited to: Hong Kong Polytechnic University. Downloaded on July 7, 2009 at 04:37 from IEEE Xplore.  Restrictions apply.Proof: In the following, we will prove the theorems above under GDA only. The proofs for PDA and SDA are similar to those of GDA and are omitted. Consider the Lyapunov's functions, Vk'=-x P x fo rk ,= l ,  ..., p a n d l = I ,  ..., c, k # i  and 1 T kl 2 l r j .  (18) Differentiate (1 8), we obtain, (19) L e t V k ' = V , P k l = P a n d  ~ ' i ~ p + p ~ ' i  + &jTp+pmBn I<-&I for all i andj. (21) From (20) and (21), I1 milx From (22), there are two cases to be investigated: r = 0 and r # 0 . For the former case, (22) becomes, (23) E E v 5 ---XTX = -- 2 2 Ilxll' 5 From (19), we have 1 1 v = y X T h  2 -&,,,, 2 (P)1$112 3 V 2 &,,n ( P ) l $ l l T  (24) where Ln(.) denotes the minimum eigenvalue of a matrix. From (23) and (24), where to is an arbitrary initial time. (25) implies llx(t)/l+ 0 as t 00 . For r # 0 , from (22) and (24), m a x j ~ ( 5 '  +m')rs- -- (I-e z,&P)(~-'*)) (27) From (27), 11x(?)ll is bounded if r is bounded. Hence, we can prove the condition of (21) is a condition for system stability. By assuming that the system has no parameter uncertainty, the stability condition for GDA in Theorem 1 is proved. The robust area under GDA in Theorem 2 can be proved by replacing AHQTP+PAH1' // with This ends the proofs of the -- !Jx(r) ls llx(t. z & p )  + I & ll max . Theorem 1 and Theorem 2. Robust area I1mTp + PAHQ (1 5 80 Authorized licensed use limited to: Hong Kong Polytechnic University. Downloaded on July 7, 2009 at 04:37 from IEEE Xplore.  Restrictions apply.It should be noted that if , the above condition is reduced to (21). Let &TPki + p k ~ H i j  + b i j T T P t l  + p k ~ m i j  I-_ for all i, j ,  k, I ,  0 # kl. Condition of (28) will lead to the conditions for GDA in Theorem 3 and Theorem 4. Under (28), E 2 i= I v k' 5 - - -XTX + 5 W i X T P  (B' + ")r (29) Compare (21) and (28), we find that the condition of (28) is less conservative than (21). (29) is similar to (22). Hence, the stability of the uncertain system can be proved for the two cases of r mentioned early. Recalling that the inequality (29) holds under the assumption that the product of the grades of membership wkm' is the largest, the system is stable only in a certain range of x that satisfies the assumption. This range of x will change as the largest product value of the grades of membership changes during operation. Hence, the stability is guaranteed for a local system corresponding to a particular range of x. To prove the overall system stability, (25) and (27), we can see that the norm Ib(r)ll is always exponentially decaying. The switching from one local system to another only results in a change of time constant only. Therefore, the system stability is guaranteed globally. This ends the proofs of Theorem 3 and 4. For the Theorem 3 and 4, the total number of Lyapunov's functions involved is pxc. 4. Application Example An application example on stabilizing an uncertain nonlinear mass-spring-damper system is given [5]. The behavior of this system can be described by where M is the mass and U is the force, f ix)  describes the spring nonlinearity and uncertainty, g(x,.i) describes the damper nonlinearity and uncertainty, and #(i) describes the input nonlinearity and uncertainty. Let g ( x , x )  = D(c,x + cz.i + c3(t)x) M i  + g ( x , i )  + f ( x )  = Q ( i ) u  (30) f ( x )  = K(c,x+':sx3 +c, ( t )x )  (31) #(i) =1.4387+c7i2 + c 8 ( i )  The operating range of the states is assumed to be within - 1.5 and 1.5. The parameters are chosen as follows: M = 1.0, D =  K=1.0, c1 =o, c2 = 1 ,  so that cy +c," ) sin( lot) cy +c: c3 ( t )  = -+(c,L -22 so that c6" +c," c6" + c I  )CO@) c6(t)=- + (c," -22 c7 = -0.13, L U  C6(t)E[C6 rC6 1 %  ) cos(5t)) cos(%) so that c8" +c: c8" +c: +(C: -- CfI ( t )  = (- c 8 ( i ) E [ c 8  ,c8 1,  c: =-0.1, ck =-1.26, c t  =-0.1, cy = 0.1, c t  = 1.26, cf = 0.1. It can be seen that the parameter uncertainties c3, c6 and c8 are modeled as functions of time in order to show the robustness of the designed controller. The nonlinear system then becomes P = - i - O . O l x - O . l ~ ~  - ~ ~ ( t ) X - - ~ ~ ( t ) ~ + ( 1 . 4 3 8 7 - 0 . 1 3 i ~  +c,(~))u (32) which can exactly be represented by the following rules. Rule i: IF x is MI and is Mi THEN x = (A' +AA')x+(B' +")U, i = 1,2,3,4 (33) with the r-norm operation being chosen as the multiplication. The fuzzy rules of the fuzzy controller designed by GDA are defined as follows, Rule 1: F x i s  MI and f is Mi where the membership functions of M', , i= 1,2, 3,4, 01 = 2 2 L U  THEN u=Gix j =  1,2 ,3 ,4  (34) X 2  1, 2, are &I (X I  = &; ( x )  = l-- 2.25 ' A'=[ -2 0 1, .'=[ -2 "1, A3=[ - 2  "1, -2.01 1 -2.01 1 -2.235 1 B'=[ 1.4387 " 1. A4=[ -2.235 -  "1; 1 B2=[ 0.5613 1, B3=[ 1.4387 " 1, gains are designed as G"=[1.3971 -2.08521, GI2=[ 3.5810 -5.34471, GI3=[1.5535 -2.08521 and GI4=[3.9818 - 5.34471 so that H I '  = HZ2 = H33 =Ha = 58 1 Authorized licensed use limited to: Hong Kong Polytechnic University. Downloaded on July 7, 2009 at 04:37 from IEEE Xplore.  Restrictions apply.p = [ y ]  . Figure. 1 and 2 show the responses of x ( t )  and i ( t )  with (solid lines) and without (dotted lines) parameter uncertainties with initial states of x = [-1, - 11 . The analysis results are tabulated in Table I which shows that the system is stable according to Theorem 2. Moreover, from (25), the system performance can be predicted to lie inside the range: D -02 -0.4 -0.6 -0.8 IIX(t>li I fit?-$ where E is chosen as 0.01. . . - (35) 5. Conclusions The stability and robustness of uncertain fuzzy control systems have been analyzed. Three design approaches of fuzzy controller have been introduced and investigated. Stability conditions and robust areas for each design approaches have been derived based on Lyapunov’s stability theorem. The Lyapunov’s functions for different subsystems are allowed to be distinct, and as a result, less conservative stability conditions have been obtained. An application example on stabilizing an uncertain nonlinear mass-spring-damper system has been given. References [I] T. Takagi and M. Sugeno, “Fuzzy Identification of systems and its applications to modeling and control,” IEEE Trunr. Sys., Mun., Cybern., vol. smc-15 no. 1, pp. 116-132, Jan.,1985. [2] K. Tanaka and M. Sugeno, “Stability analysis and design of fuzzy control system,’’, Fuzzy Sets und Sysfms, vol. 45, pp. 135-156, 1992. [3] R. Palm, “Sliding mode fuzzy control,” in Proc. of IEEE Int. Conference on Fuzzy Systems, 1992, pp. 519-526. [4] L. X. Wang, “Stable adaptive fuzzy control of nonlinear systems,”, IEEE Trans. Fuzzy Sys., vol. 1, no. 2, pp. 146-155, May, 1993. [5] H. 0. Wang, K. Tanaka, and M. E Griffin, “An approach to fuzzy control of nonlinear systems: stability and the design issues,” IEEE Tt-uns. Fuzzy Sysr., vol. 4, no. 1, pp. 14-23, Feb., 1996. [6] H.K. Lam, F.H.F. Leung, and P.K.S. Tam, “Design of stable and robust fuzzy controllers for uncertain nonlinear systems: Three Cases,” Proceedings of IFAUAIRTC97, 1997, pp. 28-33. 2 4 6 8 lime (Sec.) D Figure 1. Response of x(t) of system with (solid line) and without (dotted line) parameter uncertainties. ‘0 2 4 6 a 10 Time (Sec.) Figure 2. Response of i ( t )  of system with (solid line) and without (dotted line) parameter uncertainties. Table 1. The stability and robustness analysis result. 582 Authorized licensed use limited to: Hong Kong Polytechnic University. Downloaded on July 7, 2009 at 04:37 from IEEE Xplore.  Restrictions apply.

Lyapunov function based design of robust fuzzy controllers for uncertain nonlinear systems : distinct Lyapunov functions

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