Output Feedback Fuzzy H Control of Nonlinear Systems with Time-Varying Delayed State

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1 8 Transactions on Control Automation and Systems Engineering Vol. No. December 000 Output Feedback Fuzzy H Control of Nonlinear Systems with Time-Varying Delayed State Kap Rai Lee Abstract: This paper presents an output feedback fuzzy H control problem for a class of nonlinear systems with time-varying delayed state. The Takagi-Sugeno fuzzy model is employed to represent a nonlinear systems with time-varying delayed state. Using a single quadratic Lyapunov function the globally exponential stability and disturance attenuation of the closed-loop fuzzy control system are discussed. Suf cient conditions for the existence of fuzzy H controllers are given in terms of matrix inequalities. Constructive algorithm for design of fuzzy H controller is also developed. A simulation example is given to illustrate the performance of the proposed design method. Keywords: Nonlinear systems time delay systems output feedback fuzzy H control linear matrix inequality I. Introduction There have been signi cant research efforts on the stability analysis and systematic design for fuzzy control systems [1]-[]. These methods are conceptually simple and straightforward. The nonlinear system is represented by a Takagi- Sugeno(T-S)-type fuzzy model. And then the control design is carried out on the basis of the fuzzy model via the so-called parallel distributed compensation scheme. Since uncertainties are frequently a source of instability Tanaka et al. [5] [] presented stability analysis for a class of uncertain nonlinear systems and method for designing robust fuzzy controllers to stabilize the uncertain nonlinear systems. However all the above design method has to predetermine the state feedback gains before checking the stability condition of the closed-loop system. In real control problems all of the states are not available thus it is necessary to design output feedback controller. Ma et al. [8] presented the analysis and design of the fuzzy controller and fuzzy observer on the basis of T-S fuzzy model using separation property. Tanaka et al. [9] also presented systematic design method of the fuzzy regulator and fuzzy observer on the basis of T-S fuzzy model. The H control approach is concerned with the design of controller which stabilizes a system while satisfying an H - norm bound constraint on disturbance attenuation [10]-[1]. Over past a few years H control of T-S fuzzy model are paid a lot of attention in fuzzy control [1]-[15]. Han and Feng [1] and Hong and Langari [1] presented H controller design for fuzzy dynamic systems with state feedback. Chen et al. [15] presented the design method of an observer-based fuzzy H controller via an LMI approach. Since time delay is frequently a source of instability and encountered in various engineering systems the H control problem for delayed systems has received considerable attention over the last few decades [1]-[18]. However for a fuzzy control system there are few publica- Manuscript received: July Accepted: Dec Kap Rai Lee: Dept. of Computer Applied Control Doowon Technical College * This work was supported by Doowon Technical College Grant. tions on H control design for delayed systems. In this paper we design an output feedback fuzzy H controller for fuzzy dynamic system with time-varying delayed state. The controller design is carried out on the basis of the T-S fuzzy model and the resulting controller is tuned on-line based on fuzzy operations. A suf cient condition is derived such that the closed-loop system is globally exponentially stable and L gain of the input-output map is bounded. Based on the derivation constructive algorithm of LMI-based fuzzy H controller are presented. II. Problem formulation The continuous fuzzy dynamic model proposed by Takagi and Sugeno is described by fuzzy IF-THEN rules which represented local linear input-output relations of nonlinear system. Consider a nonlinear system with time varying delay that can be described by the following T-S fuzzy model with time-varying delay: Plant Rule i: IF z 1(t) is M i1 and and z g(t) is M ig THEN ẋ(t) = A i x(t) + A di x(t d(t)) + B 1i w 1 (t) + B i u(t) (1) e 1(t) = C ix(t) y(t) = C yi x(t) + w (t) x(t) = 0 t 0 e (t) = u(t) i = 1 r M ij is the fuzzy set x(t) R n is the state vector u(t) R k 1 is the control input vector [w1 T (t) w T (t)] T R p is the square-integrable disturbance input vector y(t) R k is the measured output [e T 1 (t) e T (t)] T R q is the controlled output vector r is the number of IF-THEN rules z 1 z g are some measurable system variables i.e. the premise variables and all matrices are constant matrices with appropriate dimensions d(t) are the time-varying delay with following assumptions: 0 d(t) < d(t) β < 1. () Let µ i (z(t)) be the normalized membership function of the inferred fuzzy set h i(z(t)) i.e. µ i(z(t)) = h i(z(t))/ h i(z(t)) ()

2 9 Transactions on Control Automation and Systems Engineering Vol. No. December 000 h i(z(t)) = gy j=1 M ij(z j(t)) z(t) = [z 1 (t) z (t) z g (t)] T. () M ij (z j (t)) is the grade of membership of z j (t) in M ij. It is assumed that h i(z(t)) 0 i = 1 r h i (z(t)) > 0 (5) for all t. Then we can obtain the following conditions: µ i(z(t)) 0 i = 1 r µ i (z(t)) = 1 () for all t. Let P R r be the set of membership function satisfying (). Given a pair of (y(t) u(t)) by using a center average defuzzifer product inference and singleton fuzzi er the dynamic fuzzy model (1) can be expressed by the following global model ẋ(t) = A(µ)x(t) + A d (µ)x(t d(t)) + B 1 (µ)w 1 (t) e 1 (t) = C(µ)x(t) e (t) = u(t) +B (µ)u(t) () y(t) = C y(µ)x(t) + w (t) µ = [µ 1 µ µ r ] P A(µ) = B 1 (µ) = C(µ) = E x (µ) = µ i(z(t))a i A d (µ) = µ i (z(t))b 1i B (µ) = µ i(z(t))c i C y(µ) = µ i(z(t))a di µ i (z(t))b i µ i(z(t))c yi µ i (z(t))e xi. (8) As a fuzzy H controller of the fuzzy system (1) we consider the following output feedback controller: ˆx(t) = A k (µ)ˆx(t) + B k (µ)y(t); ˆx(0) = 0 (9) u(t) = C k (µ)ˆx(t) matrix functions A k (µ) B k (µ) and C k (µ) are to be determined and tuned on-line based on fuzzy operation i.e. A k (µ) B k (µ) and C k (µ) are function of membership functions. From () and (9) we obtain the following closed-loop system ζ = e(t) = ζ(t) = 0 t 0 Â(µ)ζ(t) + Â1(µ)x(t d(t)) + ˆB(µ)w(t) Ĉ(µ)ζ(t) (10) (t) = [x T (t) ˆx T (t)] T w(t) = [w T 1 (t) w T (t)] T and e(t) = [e T 1 (t) e T (t)] T Â(µ) = A(µ) B (µ)c k (µ) B k (µ)c y (µ) A k (µ) Â 1 (µ) = A T d (µ) 0 T ˆB(µ) = Ĉ(µ) = B1 (µ) 0 0 B k (µ) C(µ) 0 0 C k (µ) (11). We de ne a variable including all parameters of controller K(µ) = 0 C k (µ) B k (µ) A k (µ) and we introduce the abbreviations: A(µ) 0 A o (µ) = B (µ) 0 B 00 (µ) = 0 I 0 I D yo(µ) = D 0 (µ) = I 0 B 1o (µ) = C 00 (µ) = C 1o(µ) = (1) B1(µ) 0 Cy (µ) 0 0 I C(µ) 0 (1) then the closed-loop matrices of (10) can be written as Â(µ) = A o (µ) + B 00 (µ)k(µ)c 00 (µ) ˆB(µ) = B 1o (µ) + B 00 (µ)k(µ)d yo (µ) Ĉ(µ) = C 1o(µ) + D 0(µ)K(µ)C 00(µ). (1) Note that (1) involves only plant data and all matrices of (1) are af ne form of the controller data K(µ). For a given γ we de ne L gain γ-performance of the system (10) as the quantity Z T 0 Z T e(t) dt γ [ w(t) dt] (15) 0 for all T > 0 and all w L [0 T ] denotes the Euclidean norm. This paper addresses designing an output feedback fuzzy H controller (9) for the system () such that the closed-loop system is globally exponentially stable and achieves L gain γ-performance (Globally ES-γ). III. Stability and L norm analysis Lemma 1 : Consider the unforced system of (10). If there exist matrices P > 0 and S 11 > 0 and positive scalars λ and α satisfying the following inequalities " Â(µ) T P + PÂ(µ) + Ŝ + Ŝ1 PÂ1(µ) Â T 1 (µ)p Ŝ11 for all µ P β is de ned in () and S11 0 αi 0 Ŝ = Ŝ 1 = Ŝ 11 = (1 β)s 11 # 0 (1) (1)

3 ICASE : The Institute of Control Automation and Systems Engineers KOREA Vol. No. December then the equilibrium of the unforced system of (10) is globally exponentially stable. Proof: De ne a Lyapunov functional V (ζ t) = ζ T (t)p ζ(t) + Z t t d(t) x(τ) T S 11x(τ)dτ (18) P > 0 and S 11 > 0. Then there exist scalars δ 1 > 0 and δ > 0 such that δ 1 ζ V (ζ t) δ ζ. If there exists scalar α > 0 such that V (ζ t) α x then the unforced system of (10) is globally exponentially stable [19]. From assumption () V (ζ t) ζ T (t)pζ(t) + ζ T (t)p ζ(t) + x(t) T S 11x(t) x(t d(t)) T Ŝ 11 x(t d(t)) := V a (ζ t). (19) By using the Schur complement[0] the condition V a(ζ t) α x for all µ is equivalent to (1). Lemma : Consider the system (10) and let γ > 0 be a given scalar. If there exist matrices P > 0 and S 11 > 0 and positive scalar λ and α satisfying the following inequalities for all µ P Â(µ) T P + PÂ(µ) + Ŝ + Ŝ1 PÂ1(µ) P ˆB(µ) Ĉ T (µ) Â T 1 (µ)p Ŝ11 ˆB T (µ)p 0 γ I 0 Ĉ(µ) I 0 (0) Ŝ Ŝ1 and Ŝ11 are given by (1) then the corresponding closed-loop system (10) is globally exponentially stable and achieves L gain γ-performance for all w(t). Proof: The matrices P > 0 and S 11 > 0 and positive scalars λ and α satisfying (0) also satisfy the inequalities (1). Using a Lyapunov functionals (18) (19) and following condition J a(t) := V a(ζ t) + e T (t)e(t) γ w T (t)w(t) 0 (1) inequality (0) is obtained. Since V (x(t )) > 0 the condition (1) implies R T 0 e(t) dt γ [ R T 0 w(t) dt]. IV. Fuzzy model-based robust H controller design By applying the result of lemma. we present a suf cient condition of the existence of fuzzy H controllers for the T-S fuzzy model () and explain how to construct H controllers. De nition 1: Given the system () and γ > 0. The Globally ES-γ problem is solvable if there exist a nite dimensional controller (9) and matrices P > 0 and S 11 > 0 and positive scalars λ and α satisfying the inequality (0) for all µ P. Using the notations of (1) and (1) de ne the left-hand side of inequality (0) as G(µ) := Φ(µ) + ΣΠ(µ)K(µ)Θ T (µ) +Θ(µ)K T (µ)π T (µ)σ T () 5 and Φ(µ) = Σ = Diag(P I I I) Π(µ) = [B T 00(µ) D T 0(µ)] T Θ(µ) = [C 00 (µ) 0 D yo (µ) 0] T () Φ 1 1(µ) PÂ1(µ) PB 1o (µ) C T 10(µ) Â T 1 (µ)p Ŝ11 B T 1o(µ)P 0 γ I 0 C 1o(µ) I 5 Φ 11 (µ) = A T o (µ)p + PA o (µ) + Ŝ + Ŝ1. () Now de ne matrices Π (µ) and Θ (µ) such that for all µ Π T (µ)π(µ) = 0 Θ T (µ)θ(µ) = 0 and [Π(µ) Π (µ)] [Θ(µ) Θ (µ)] are full column rank. Then Π T (µ) = Θ T (µ) = I 0 B (µ) I 0 I 0 I 0 I 0 I 0 C T y (µ) I 0 I 0 0 I 0 I 5 5.(5) Since both Π (µ) and Θ (µ) are full column rank for all µ P ; it is clear that if G(µ) < 0 then Θ T (µ)g(µ)θ (µ) < 0 and Π T (µ)σ 1 G(µ)Σ 1 Π (µ) < 0 which is equivalent to for all µ P. Π T (µ)σ 1 Φ(µ)Σ 1 Π (µ) < 0 () Θ T (µ)φ(µ)θ (µ) < 0 () Theorem 1: Given the system () and γ > 0 Globally ESγ Problem is solvable if and only if there exist matrices X( R n ) > 0 and Y( R n ) > 0 satisfying following inequalities for all µ P ; Λ(µ) A d (µ) B 1 (µ) YC T (µ) Y A T d (µ) Ŝ11 0 B T 1 (µ) 0 γ I C(µ)Y I 0 Y 0 (S 11 + αi) 1 < 0 (8) Γ(µ) XA d (µ) XB 1(µ) C T (µ) A T d (µ)x Ŝ11 B T 1 (µ)x 0 γ I 0 C(µ) I X I I Y 5 5 < 0 (9) > 0 (0)

4 51 Transactions on Control Automation and Systems Engineering Vol. No. December 000 for some martix S 11 > 0 and positive scalars λ and α Λ(µ) = YA T (µ) + A(µ)Y B (µ)b T (µ) (1) Γ(µ) = A(µ) T X + XA(µ) + S 11 + αi γ C T y (µ)c y (µ). Furthermore one n-dimensional strictly proper controller that solves the feedback problem is de ned as: A k (µ) := A(µ) γ Z 1 C T y (µ)c y (µ) B (µ)b T (µ)y 1 +γ B 1(µ)B 1(µ) T Y 1 + A d (µ)ŝ 1 11 A T d (µ)y 1 Z 1 H(µ) () B k (µ) := γ Z 1 C T y (µ) C k (µ) := B T (µ)y 1 Z := X Y 1 and H(µ) = [Y 1 A(µ) + A T (µ)y 1 Y 1 B (µ)b T (µ)y 1 +A d (µ)ŝ 1 11 A T d (µ))y 1 + C T (µ)c(µ) + S 11 + αi] +Y 1 (γ B 1 (µ)b T 1 (µ) + λ 1 HH T. () Proof: ( ) Let P R (n+m) (n+m) be the positive de nite matrix that satis es the Lemma.. Similarly to the proof procedure of [?] we de ne Q := P 1 and partition P as X X Y Y P = X T Q = X Y T () Y X Y R n n and X Y R m m. By the matrix inversion lemma it follows that X Y 1 0 which is LMI in equation (0). From () and () carrying out the algebraic manipulations it can readily be seen by taking Schur complements of the resulting expressions in () and () that the conditions (8) and (9) are satis ed. ( ) We verify that the controller given in () satis es the Globally ES-γ problem using X (X Y 1 ) P := (X Y 1 ) X Y 1 > 0. (5) We rede ne the left-hand side of inequality (0) as Υ(µ) := ÂT (µ)p + PÂ(µ) + Ŝ + Ŝ1 + ĈT (µ)ĉ(µ) + γ P ˆB(µ) ˆB T (µ)p + PÂ1(µ)Ŝ 1 11 ÂT 1 (µ)p () the closed-loop matrices  Â1 ˆB and Ĉ are de ned in (11). Partition Υ into n n blocks Υ 11 Υ 1 and Υ. De ne a transformation I 0 T := () I I and transformed state-space data Ã(µ) := T 1 Â(µ)T B(µ) := T 1 ˆB(µ) C(µ) := Ĉ(µ)T à 1 (µ) := T 1  1(µ) (8) P(µ) := T T P(µ)T S := T T ŜT S1 := T T Ŝ 1T. Then T T Υ(µ)T < 0 can be written as à T (µ) P + PÃ(µ) + S + S 1 + C T (µ) C(µ) +γ P B(µ) BT (µ) P + PÃ1(µ)Ŝ 1 11 ÃT 1 (µ) P < 0. (9) Note Υ(µ) < 0 if and only if T T Υ(µ)T < 0. Denote the left-hand side of (9) as Υ and partition it into blocks Υ 11 Υ 1 Υ R n n. Using the controller () it can be shown that H(µ) H(µ) Υ(µ) =. (0) H(µ) Υ 11 H(µ) Using the Schur complement Υ(µ) < 0 is equivalent to (8) and (9). Theorem.1 is existence condition of robust H control for the global fuzzy model. It is not easy to nd the matrices X and Y from the global fuzzy model. From theorem.1 we derive the following condition which can be expressed for each rule of fuzzy model. Theorem : Consider the system () with assumption (). Then the Globally ES-γ problem is solvable if there exist common matrices X > 0 Ỹ > 0 S 11 > 0 and positive scalars γ α λ satisfying the following matrix inequalities: In here Ψ ij = Ω ij = Ψ ii < 0 i = 1... r Ψ ij + Ψ ji < 0 i < j < r (1) Ω ii < 0 i = 1... r Ω ij + Ω ji < 0 i < j < r () X I 0. () I Ỹ Λ ij B 1i ỸC T i Ỹ Ỹ B T 1 γi 0 i C i Ỹ 0 λi Ỹ S Ỹ 0 α 1 I Γ ij I XAdi XB1i C T i I α 1 I 0 A T d X i 0 (1 β) S11 B T 1 X i γi 0 C i 0 λi 5 () 5 (5) Λ ij = ỸAT i + A iỹ λb i B T j + (1 β) 1 A di S 1 11 AT d j Γ ij = XA i + A T i X + S 11 γc T y i C yj. () Furthermore positive de nite matrices X Y S 11 and positive scalars α γ are given by X = λ X Y = λ 1 Ỹ S 11 = λ S 11 α = λ α γ = λ γ () and a suitable controller is expressed in (). Proof: The inequality (8) and (9) of Theorem.1 are equivalent to YA T (µ) + A(µ)Y B (µ)b T (µ) + (1 β) 1 A d (µ)s 1 11 A T d (µ) + γ B 1(µ)B T 1 (µ) + YC T (µ)c(µ)y + YS 11 Y + αyy < 0 (8) XA(µ) + A T (µ)x + S 11 + αi γ C T y (µ)c y(µ) + (1 β) 1 XA T d (µ)s 1 11 A T d (µ)x + γ XB 1(µ)B T 1 (µ)x + C T (µ)c(µ) < 0. (9)

5 ICASE : The Institute of Control Automation and Systems Engineers KOREA Vol. No. December First multiply λ to (8) and λ 1 to (9). Next let X = λ 1 X Ỹ = λy S11 = λ 1 S 11 α = λ 1 α and γ = λ 1 γ. Then using Schur complement and notation (8) inequalities (8) and (9) are equivalent j=1 j=1 µ i (z(t))µ j (z(t))ψ ij < 0 µ i (z(t))µ j (z(t))ω ij < 0. (50) Inequalities (50) can be also written as + + µ i (z(t))µ i (z(t))ψ ii i<j µ i(z(t))µ j(z(t))(ψ ij + Ψ ji) < 0 µ i(z(t))µ i(z(t))ω ii (51) i<j µ i (z(t))µ j (z(t))(ω ij + Ω ji ) < 0. From (51) we get (1) and (). Note that the inequality (1) is an LMI for Ỹ S 1 11 and γ α λ and inequality () is an LMI for X S 11 and γ α λ. However (1) and () are not LMIs in terms of variable S 11 simultaneously. Thus we present a constructive algorithm for nding solution of the above nonconvex problems. The procedure is summarized as follows: [Procedure] Step 1: Find the regions D s = { S 11 Ỹ > 0 γ > 0 α > 0 λ > 0 (1)} (5) Step : Find the regions D s = { S 11 X > 0 γ > 0 α > 0 λ > 0 ()} (5) Step : Obtain the intersection of D s and D s ˆD s = D s D s (5) Step : Compute X Ỹ and α γ λ suct that min S 11 ˆD s [trace( γ) + trace(λ)] (55) subject to (1) () Step 5: Compute positive de nite matrices X Y and positive scalars α γ from () Step. Construct nally controller from () with X Y α γ and λ. Remark 1: In step the existence of the set ˆD s does not imply that the matrix inequalities (1)-() are solvable but a necessary condition for the solvability of (1)-(). Remark : In step the minimization of (55) is not convex problem in term of S 11. However it is not dif cult to nd the minimum γ + λ because the computation can be executed within the searching region of S 11 in step. V. Design example We will design a fuzzy H controller for the following nonlinear system; ẋ 1 (t) = 0.115x 1 (t) 0.015x 1 (t d(t)) 0.0x (t) 0.x (t) 0.1x (t d(t)) 0.005x (t d(t)) + w(t) + u(t) ẋ (t) = x 1 (t) y(t) = x (t) e 1(t) = x (t) e (t) = u(t) time-varying delay is d(t) = + 0.5cos(0.9t). The purpose of control is to achieve closed loop stability and to attenuate the in uence of the exogenous input disturbance w(t) on the penalty variable [e T 1 (t) e T (t)] T. It is also assumed that x (t) is measurable and x 1 (t) [ ] x (t) [ ]. Using the same procedure as in [?] the nonlinear term can be represented as 0.x (t) = M 11 0 x (t) (1 M 11) 1.505x (t). 0.1x (t d(t)) = M 11 0 x (t d(t)) (1 M 11 ) 0.5 x (t d(t)). By solving the equation M 11 is obtained as follows: M 11 (x (t)) = 1 x (t).5 M 1(x (t)) := 1 M 11(x (t)) = x.5. M 11 and M 1 can be interpreted as membership functions of fuzzy set. By using these fuzzy sets the nonlinear system can be presented by the following uncertain T-S fuzzy model Plant Rule 1: IF x (t) is M 11 THEN ẋ(t) = A 1 x(t) + A d1 x(t d(t)) + B 11 w(t) + B 1 u(t) y(t) = C y1 x(t) e 1 (t) = C 1 x(t) e (t) = u(t) Plant Rule : IF x (t) is M 1 THEN ẋ(t) = A x(t) + A d x(t d(t)) + B 1 w(t) + B u(t) y(t) = C y x(t) e 1(t) = C x(t) e (t) = u(t) x(t) = [x 1 (t) x (t)] T A 1 = A d1 =

6 5 Transactions on Control Automation and Systems Engineering Vol. No. December A = A d = B 11 = B 1 = [1 0] T B 1 = B = [1 0] T C y1 = C y = [0 1] C 1 = C = [0 1]. Let S 11 = ki for simplicity. From step 1 step and step of procedure we can obtain the set ˆD s = { S 11 S 11 = ki 10 9 < k < }. The minimization of step is attained at S 11 = 0.19 and the minimum value of γ is.. The values of λ α and X Y S 11 are λ =.5 α = S 11 = X = Y = The simulation results for the nonlinear systems are shown in Fig. 1. For these simulations the initial value of the states are x 1 (0) = 1 and x (0) = 1.. And The disturbance signal w(t) is de ned by 0. sec t sec w(t) = 0 otherwise. The designed fuzzy H controller stabilizes the the nonlinear system and attains disturbance attenuation effect. VI. Conclusion In this paper we have developed output feedback fuzzy H controller design method for nonlinear systems with timevarying delayed state. We have obtained suf cient conditions for the existence of fuzzy H controller such that the closedloop fuzzy system is globally exponentially stable and achieves a prescribed level of disturbance attenuation. The derived suf cient conditions are not LMI in all variables. Thus we presented a constructive algorithm for nding solution of the nonconvex problems. The controller is directly obtained from the derived LMI condition. The resulting controller is nonlinear and tuned on-line based on fuzzy operation. Through an example the validity was demonstrated. [] C. L. Chen P. C. Chen and C. K. Chen Analysis and design of fuzzy control systems Fuzzy Sets Syst. vol. 5 pp [] H. O. Wang K. Tanaka and M. F. Grif n An approach to fuzzy control of nonlinear systems: Stability and design issues IEEE Trans. Fuzzy Syst. vol. no. 1 pp. 1- Feb [5] K. Tanaka and M. Sano A robust stabilization problem of fuzzy control systems and its application to backing up control of a truck-trailer IEEE Trans. Fuzzy Syst. vol. no. pp May 199. [] K. Tanaka T. Ikeda and H. O. Wang Robust stabilization of a class of uncertain nonlinear systems via fuzzy control: Quadratic stabilizability H control theory and linear matrix inequalities IEEE Trans. Fuzzy Syst. vol. no. 1 pp. 1-1 Feb [] B. S. Chen C. H. Lee and Y. C. Chang H tracking design of uncertain nonlinear SISO systems: Adaptive fuzzy approach IEEE Trans. Fuzzy Syst. vol. no. 1 pp. - Feb [8] X. J. Ma Z. Q. Sun and Y. Y. He Analysis and design of fuzzy controller and fuzzy observer IEEE Trans. Fuzzy Syst. vol. no. 1 pp [9] K. Tanaka T. Ikeda and H. Wang Fuzzy regulator and fuzzy observers: Relaxed stability conditions and LMIbased designs IEEE Trans. Fuzzy Syst. vol. no. pp [10] B. A. Francis A Course in H Control Theory Lecture Notes in Control Information Science vol. 8 Springer- Verlag Berlin 198. [11] M. Green and D. J. N. Limebeer Linear Robust Control Prentice-Hall Publishing Co. Englewood Cliff NJ [1] S. K. Nguang Global robust H control of a class of nonlinear systems Int. J. Robust and Nonlinear Contr. vol. pp [1] Z. Han and G. Feng State feedback H controller design of fuzzy dynamic systems using LMI techniques in Proc. FUZZ-IEEE pp [1] S. K. Hong and R. Langari Synthesis of an LMI-based fuzzy control system with guaranteed optimal H performance in Proc. FUZZ-IEEE pp [15] B. S. Chen C. S. Tseng and H. J. Uang Robustness design of nonlinear dynamic systems via fuzzy linear control IEEE Trans. Fuzzy Systems vol. no. 5 pp [1] M. S. Mahmoud and N. F. Al-Muthairi Design of robust controllers for time delay systems IEEE Trans. Automat. Contr. vol. 9 no. 1 pp May 199. References [1] K. Tanaka and M. Sugeno Stability analysis and design of fuzzy control systems Fuzzy Sets Syst. vol. 5 pp [] K. Tanaka and M. Sano Fuzzy stability criterion of a class of nonlinear systems Inform. Sci. vol. 1 pp Fig. 1. The simulation results of nonlinear systems.

7 ICASE : The Institute of Control Automation and Systems Engineers KOREA Vol. No. December Kap Rai Lee He was born in Korea on November 19. He received the B. S. M. S. and Ph. D. degrees in electronic engineering from Kyungpook National University Taegu Korea in and 1999 respectively. He was with the Agency for Defence Development from 1990 to He is currently an Assistant Professor in the Department of Computer Applied Control Doowon Technical College. His current research are robust control H / H control fuzzy control time-delay systems nonlinear systems eldbus network. [1] J. H. Kim E. T. Jeung and H. B. Park Robust control for parameter uncertain delay systems in state and control input Automatica vol. no. 9 pp [18] E. T. Jeung J. H. Kim and H. B. Park H output feedback controller design for linear systems with timevarying delayed state IEEE Trans. Automat. Contr. vol. no. pp [19] M. Vidysagar Nonlinear Systems Analysis Prentice-Hall Publishing Co. Englewood Cliff NJ 199. [0] S. Boyd L. E. Ghaoui E. Feron and V. Balakrishnan Linear Matrix Inequalities in System and Control Theory SIAM 199. [1] P. Gahinet A. Nemirovski A. J. Laub and M. Chilali LMI Control Toolbox The Math Works Inc