INERNAIONAL JOURNAL OF INFORMAION AND SYSEMS SCIENCES Volume, Number 3-4, Pages 339 346 c 005 Institute for Scientific Computing and Information SABILIY ANALYSIS FOR DISCREE -S FUZZY SYSEMS IAOGUANG YANG, IAODONG LIU, QINGLING ZHANG, AND PEIYONG LIU Abstract Quadratic stability problems for discrete -S fuzzy system have been studied in this paper New quadratic stability conditions for discrete - S fuzzy system, which are simpler and more relaxed than that in EKim and HLee 000, are proposed By using LMIs and MALAB, the controllers can be directly obtained he examples show that the design method proposed in this paper is more relaxed, efficient and optimal than the previous papers such as EKim and HLee 000 Key Words discrete -S fuzzy system, Schur complement, quadratic stability, linear matrix inequalitylmi) Introduction Since the inception of the fuzzy logic by Zadeh in 965, the fuzzy logic has found extensive applications in the areas of industrial systems and consumer products In recent years, it becomes quite popular to employ the so-called -S fuzzy model to represent or approximate a nonlinear system It is because the control technique based on -S fuzzy model allows the designers to analyze and design nonlinear systems by taking full advantage of the strength of modern linear control theory, what s more, different from the heuristics-based approach to fuzzy control system, it can provide a systematic design methodology for nonlinear fuzzy control systems his permits appliance of conventional linear system in analysis and design of the fuzzy control systems Because some researches consider little the interactions among the fuzzy subsystems, some results such as EKim and HLee 000, tend to give a more conservative condition In this paper, we propose a new quadratic stability condition which is simpler than that in and prove that if there exist matrices satisfying the conditions in, then there exist matrices satisfying the conditions in heorem of this paper and with an example, we illustrated that the new conditions in heorem are not equivalent to the conditions in Finally, based on the LMIs, the controller designing methods in the example, we show the effectiveness of the method in this paper he discrete -S fuzzy model and quadratic stability his paper considers the fuzzy dynamic model of -S described by the following fuzzy IF-HEN rules: IF ξ is M i and, and ξ p is M pi, HEN xt + ) = A t xt) + B t ut), zt) = C t xt), i =,,, r Received by the editors June, 004 and, in revised form, January, 005 000 Mathematics Subject Classification 93A30 his work is supported by National Nature Science Foundation of China under grant No 605740 and No 607404 and the Science Funds of Dalian Maritime University 339
340 G YANG, D LIU, Q L ZHANG, AND P Y LIU Where x R n is the state, z R q is the output, u R m is the input, A t R n n, B t R n m,ξ,, ξ p are premise variables We set ξ = ξ,, ξ p ) It is assumed that the premise variables do not depend on control variables and the disturbance hen the state equation and the output are defined as follows: ) xt + ) = λ t ξ)a t xt) + B t ut)), t= ) zt) = Here λ t ξt)) = λ t ξ)c t xt) t= β i ξt)) r j= β jξt)), β jξt)) = p M kjξt)) M kj ) is the membership function of fuzzy set M kj In the following we always assume that: 3) λ i ξt)) =, λ k ξt)) 0, k =,, r i= Definition For discrete -S fuzzy system ), when ut) 0, if there exists and a positive-definite matrix such that V xt)) < 0, V xt)) = x t)xt), then discrete -S fuzzy system ) is called quadratically stable 3 he stability condition he following lemma is similar to the paper 3 Lemma If there exist matrices F i, i =,,, r and a positive definite matrix such that R R r 4) < 0, R r R rr k= 5) R ii = G iig ii, i =,,, r, 6) R ij = G ij + G ji ) G ij + G ji ), i < j G ij = A i + B i F j hen for discrete -S fuzzy system ), the state feedback is ut) = λ i ξt))f i xt) Stabilizes the closed-loop system xt + ) = λ i ξt))λ j ξt))a i + B i F j )xt) j= i= i=
SABILIY ANALYSIS FOR DISCREE -S FUZZY SYSEMS 34 Proof :If a Lyapunov function candidate is chosen as V xt)) = x t)xt), then V xt)) = V xt + )) V xt)) = x t + )xt + ) x t)xt) = λ i ξt))λ j ξt))g ij xt) i= j= i= j= = x t) = λ i ξt))λ j ξt))g ij xt) x t)xt) i= j= λ i ξt))λ j ξt))g ijg ij ) xt) λ i ξt))x t) G iig ii xt) + i= λ i ξt))λ j ξt))x t) i<j Gij ) ) + G ji Gij + G ji xt) λ x R R r λ x λ r x R r R rr λ r x < 0 QED In Lemma, 5) and 6) are nonlinear matrix inequalities If for > 0, define F i = M i, i =,,, r, substituting into 5) and 6), they can be converted to linear matrix inequalities using Schur complement, hence there exists the following theorem heorem If there exist matrices M i, Z, Y ij, Z is a positive definite matrix, Y ii are symmetric matrices, Y ji = Yij, i j, i, j =,,, r, satisfy the following LMIs: 7) 8) 9) Y ii Z ZA i + Mi B i A i Z + B i M i Z < 0, Y ij Z ZA i + Mj B i + ZA j + M ) i B j A iz + B i M j + A j Z + B j M i ) Z Y Y r < 0, Y r Y rr then for discrete -S fuzzy system ),when the state feedback ut) = λ i ξt))f i xt), i= 0,
34 G YANG, D LIU, Q L ZHANG, AND P Y LIU stabilizes the closed-loop system xt + ) = λ i ξt))λ j ξt))a i + B i F j )xt) i= j= Here F i = M i, i =,,, r Proof: let = Z,F i = M i Z Pre- and postmultiply 7) and 8) with and pre- and postmultiply 9) with diag,, ), we have 0) A i + Fi Bi ) Ai + F i B i ) < Y ii, i j, i, j =,,, r, ) A i + Fj Bi + A j + Fi Bj ) ) A i + B i F j + A j + B j F i ) Y ij +Y ij, Y Y r < 0 Y r Y rr i j If a Lyapunov function candidate is chosen as V xt)) = x t)xt), then we have V xt)) = V xt + )) V xt)) = λ i ξt))x t)a i + Fi Bi )A i + B i F i ) xt) < = i= + i= j= λ i ξt))λ j ξt))x t) A i + Fj Bi + A j + Fi Bj λ i ξt))x t)y ii xt) i= + < 0 i= j= λ x λ r x ) A i + B i F j + A j + B j F i ) xt) λ i ξt))λ j ξt))x t) Y ij + Y ij ) xt) Y Y r Y r Y rr λ x λ r x QED states the following he equilibrium of discrete -S fuzzy system ) xt + ) = h i ξ) A i xt) + B i ut)) i= is quadratically stabilizable via the fuzzy control ut) = h i ξt)) F i xt)) i=
SABILIY ANALYSIS FOR DISCREE -S FUZZY SYSEMS 343 in the large if there exist symmetric matrices Q and H ij sand matrices N is such that Q > 0 3) 4) 5) Q H ii QA i Ni B i A i Q B i N i Q > 0i =,, r), Q H ij QA i Nj B i + QA j N ) i B j A iq B i N j + A j Q B j N i ) Q H H H r H H H r > 0 H r H r H rr i < j r) 0, Proposition If there exist matrices satisfying the conditions in then there exist matrices satisfying the conditions in heorem and they are not equivalent Proof: Since H ij s in are symmetric matrices, hence, H ij = H ij + Hij and if matrices Q,H ij s,n is satisfy, then Z = Q,Y ij = H ij, M i= Ni satisfies heorem in this paper In Example, the solution matrices of heorem in this paper are different from the solution matrices of herefore, they are not equivalent QED Since each H ij, i j in is symmetric matrix and each Y ij, i j in heorem is not necessary symmetric matrix, hence there are more variables in Y ij than H ij his implies that the conditions in heorem admit great more freedom or dimension) in guaranteeing the stability of the fuzzy control systems than Lemma Assume that the number of rules that fire for all t is less than or equal to s < s rhe equilibrium of the discrete fuzzy control system described by )is asymptotically stable in the large if there exists a common positive definite matrix and a common positive semidefinite matrix Q such that 6) G iig ii + s )Q < 0, 7) ) ) Gij + G ji Gij + G ji Q 0, i < j, G ij = A i + B i F j Using Schur complements, matrices inequalities in lemma are converted linear matrices inequalities, finding P > 0,Y > 0 and M i i =,, r) such that P s )Y P A 8) i + Mi B i > 0, A i P + B i M i P 9) P + Y ZA i + Mj B i + ZA j + M ) i B j A iz + B i M j + A j Z + B j M i ) P P =, M i = K i P, Y = P QP i < j 0,
344 G YANG, D LIU, Q L ZHANG, AND P Y LIU Obtained the local feedback gains K i, a common positive definite matrix P and a common positive semidefinite matrix Q:K i = M i P, P =, Q = Y By lemma, for α <, since the condition V xt)) α )V xt))is equivalent to G iip G ii α P + s )Q < 0 ) ) Gij+Gji Gij+Gji P α P Q 0, i < j G ij = A i + B i F j herefore, attenuation design problem for response speed in discrete fuzzy control system can be indicated: satisfy 0) min βi =,,, r) P > 0, Y 0 βp s )Y P A i + Mi B i A i P + B i M i P > 0 ) βp + Y ZA i + Mj B i + ZA j + M ) i B j A 0 iz + B i M j + A j Z + B j M i ) P i < j β = α, 0 < β < Lemma 34 If there exists positive definite matrix and a series of symmetry matrices ij such that ) G iig ii + ij < 0 ) ) Gij + G ji Gij + G ji 3) + ij 0 r r 4) = > 0 r r rr then there is the same conclusion in lemma Lemma 4 If there exists a positive definite matrix and a symmetry matrix Q in lemma, then there must be a positive definite matrix P and a symmetry matrix such that ) 4), and there is the same conclusion as in lemma Proof: Assume r = s If there exists positive definite matrix and a symmetry matrix Q satisfying 6) and 7), then can find small Q ε such that Q ε > 0, Q ε 0, G iig ii + r )Q + Q ε < 0, ) ) Gij + G ji Gij + G ji Q 0, i < j, G ij = A i + B i F j
SABILIY ANALYSIS FOR DISCREE -S FUZZY SYSEMS 345 Since 6) is a inequality, if choice ij as ii = r )Q + Q ε, ij = Q, then r r = r r rr r )Q + Q ɛ Q Q Q r )Q + Q ɛ Q = Q Q r )Q + Q ɛ Following proof > 0 Let z = z,, zr, since z r )Q + Q ɛ Q Q z z Q r )Q + Q ɛ Q z = zr Q Q r )Q + Q ɛ z z z r = z i z j ) Qz i z j ) + zi Q ε z i > 0 i<j i= herefore, lemma 4 holds QED 4 A Simulation Example Example : Consider the following fuzzy system: IF x is M, HEN xt + ) = A xt) + B ut) IF x is M, HEN xt + ) = A xt) + B ut) Where A = 05 0 05, A = 0, B = By heorem, we have the following solution matrices: Z = 9686 767 767 4690, B =, M = 47694 09665, M = 3075 8400 5965 070, Y = 070 3788 505764 3078 8049 477 Y =, Y 3078 385649 = 477 0555 F = 0789 04895, F = 0707 09665, F = 05634, F = 076 Best value of t in the LMI program of MALAB:-69043,,
346 G YANG, D LIU, Q L ZHANG, AND P Y LIU By, we have the following solution matrices: 80097 468 Z =, M 468 0470 = 36074 4765, M = 303 40364 33869 5637, Y =, 5637 57857 39399 840 73565 40307 Y =, Y 840 5735 =, 40307 360 F = 36686 435, F = 3047 689, F = 39393, F = 4079 Best value of t in the LMI program of MALAB: -0775569 By Proposition and Example, we know that the solutions of heorem in this paper are more optimal than he feedback gains F, F obtained by heorem in Example are much less than those obtained by and the best value t is also much less that obtain by In, E Kim and H Lee proved that its h is more relaxed than the conditions derived in the previous papers herefore by Proposition and Example, we know that the controllers obtained by heorem are more relaxed, efficient and optimal than the designs derived earlier 5 Conclusions In this paper, quadratic stability conditions for discrete -S fuzzy system have been studied First, sufficient conditions for discrete -S fuzzy system are presented in terms of a set of nonlinear matrix inequalities, which guarantee the quadratic stability of the closed-loop fuzzy system And then, the nonlinear matrix inequalities are converted into LMIs using the Schur complement Finally, simulation results on discrete -S fuzzy system illustrate the effectiveness of the proposed design method References Kim, E and Lee, H, New Approaches to Relaxed Quadratic Stability Condition of Fuzzy Control Systems, IEEE rans Fuzzy Syst, 8: 53-533, 000 Wang, L, Fuzzy System and Fuzzy Control, singhua University Publishing Company, 003 3 Liu, D and Zhang, QL, Approaches to Quadratic Stability Conditions And H Control Designs for -S Fuzzy Systems, IEEE rans Fuzzy Syst, 6): 830-839, 003 4 ong, SC, Wang,, Wang, YP and ang, J, Design And Stability Analysis of Fuzzy Control Systems, Since Press, 004 School of Science, Northeastern University, Shenyang 0004, China; Department of Mathematics, Dalian Maritime University, Dalian 606, China E-mail: yxg978@sohucom