Robust H∞ control for nonlinear uncertain stochastic T–S fuzzy systems with time delays

Applied Mathematics Letters 24 (2011) 1986–1994

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Robust H∞ control for nonlinear uncertain stochastic T–S fuzzy systems with time delays T. Senthilkumar, P. Balasubramaniam ∗ Department of Mathematics, Gandhigram Rural University, Gandhigram - 624 302, Tamilnadu, India

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Article history: Received 4 August 2010 Received in revised form 17 February 2011 Accepted 16 May 2011 Keywords: Robust H∞ control Stochastic systems Nonlinear disturbance Takagi–Sugeno (T–S) fuzzy systems Linear matrix inequalities (LMIs)

abstract This work deals with the problem of robust H∞ control for nonlinear uncertain stochastic Takagi–Sugeno (T–S) fuzzy systems with time delays. The system under consideration involves parameter uncertainties, Itô-type stochastic disturbances and nonlinear disturbances. On the basis of a Lyapunov–Krasovskii functional (LKF), the delay-dependent robust H∞ control scheme is presented in terms of linear matrix inequalities (LMIs). By solving these LMIs, a desired controller can be obtained. Finally, a numerical example is given to illustrate the effectiveness of the proposed method. © 2011 Elsevier Ltd. All rights reserved.

1. Introduction In recent years, the well-known Takagi–Sugeno (T–S) fuzzy model [1] has been recognized as a popular and powerful tool in approximating and describing complex nonlinear systems. Many issues related to the stability analysis and control synthesis of T–S fuzzy systems have been reported over the past two decades; see, for example, [2–9] and references therein. The T–S fuzzy model approach has been extended to deal with nonlinear systems with time delays. Various methodologies have been proposed for investigation of T–S fuzzy systems with time delay in wide research topics, for example stability and stabilization (see [10–12]) and H∞ control problems (see [5,9,13,14]). Recently H∞ control for an uncertain T–S fuzzy system with time-varying delays and nonlinear perturbations has been considered in [15]. In most of the relevant literature some stability and controllability conditions have been derived by means of linear matrix inequalities (LMIs) using Matlab LMI toolbox, which can be conveniently exploited. The phenomena of time delay are often encountered in various practical systems, such as aircraft stabilization, chemical engineering systems, the inferred grinding model, manual control, nuclear reactors, the population dynamic model, rolling mills, ship stabilization, and systems with lossless transmission lines. In practical systems, the analysis of a mathematical model is usually an important job for a control engineer, for controlling a system. However, the mathematical model always contains some uncertain elements. Therefore, the problems of stability analysis and controller design for fuzzy dynamic time delay uncertain systems are practically important and have attracted considerable attention over the past years; see for example [10,13,16–18] and references therein. On the other hand, stochastic systems have been found successful in many branches of science and engineering applications; hence the control and filtering problems for stochastic systems, particularly the Itô-type stochastic systems, have stirred up a lot of interest [19–22]. Recently, there has been growing attention paid to the study of stochastic T–S fuzzy systems. It is known that a class of nonlinear stochastic systems can be approximated by the T–S fuzzy model. Thus, we can deal with the stability analysis and controller design problems of nonlinear stochastic systems via a fuzzy logic approach.

∗

Corresponding author. Tel.: +91 451 2452371; fax: +91 451 2453071. E-mail addresses: [email protected] (T. Senthilkumar), [email protected] (P. Balasubramaniam).

0893-9659/$ – see front matter © 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.aml.2011.05.023

T. Senthilkumar, P. Balasubramaniam / Applied Mathematics Letters 24 (2011) 1986–1994

1987

For when time delays appear in a stochastic fuzzy system, a delay-independent stability condition has been given in [23]. The stabilization and H∞ control problems for stochastic T–S fuzzy systems have been investigated in [24,25] on the basis of the LMI approach developed in state feedback fuzzy controller design. A sliding mode fuzzy controller has been designed in [26] to stabilize a stochastic T–S fuzzy system with unknown nonlinearities and constant time delays. A guaranteed cost control problem for a class of uncertain stochastic fuzzy systems with multiple time delays has been discussed in [27]. To the best of the authors’ knowledge, robust H∞ control for uncertain stochastic T–S fuzzy systems with time delay and nonlinear disturbances has not yet been fully investigated and this will be the goal of this work. Motivated by the above discussion, in this work, we aim to investigate the problem of robust H∞ control for nonlinear uncertain stochastic T–S fuzzy systems with time delays. On the basis of a Lyapunov–Krasovskii functional (LKF), the delaydependent robust H∞ control scheme is presented in terms of LMIs. Finally, a numerical example is given to illustrate the effectiveness of the proposed method. Notation: Throughout this work, Rn and Rn×m denote, respectively, the n-dimensional Euclidean space and the set of all n × m real matrices. The superscript T denotes transpose and the notation X ≥ Y (respectively, X > Y ) where X and Y are symmetric matrices, means that X − Y is positive semi-definite (respectively, positive definite), and I denotes the identity matrix with appropriate dimensions. L2 [0, ∞) is the space of square integrable vectors. Let τ > 0 and C ([−τ , 0]; Rn ) denote the family of continuous functions ϕ from [−τ , 0] to Rn with the norm ‖ϕ‖ = sup−τ ≤θ≤0 |ϕ(θ )|, where | · | stands for the Euclidean vector norm and ‖ · ‖ represents the L2 [0, ∞) norm. Moreover, let (Ω , F , {Ft }t ≥0 , P ) be a complete probability space with a filtration {Ft }t ≥0 satisfying the usual conditions (i.e., the filtration contains all P -null sets and is right continuous). Denote by L2F0 ([−τ , 0]; Rn ) the family of all F0 measurable C ([−τ , 0]; Rn )-valued random variables ξ = {ξ (θ ) : −τ ≤ θ ≤ 0} such that sup−τ ≤θ≤0 E |ξ (θ )|p < ∞, where E {·} stands for the mathematical expectation operator with respect to the given probability measure P . The asterisk ‘‘∗’’ denotes a matrix that can be inferred by symmetry. Matrices, if not explicitly stated, are assumed to have compatible dimensions. 2. Problem formulation Consider the following description for nonlinear uncertain stochastic T–S fuzzy systems with time delays: Plant rule i: if θ1 (t ) is ηi1 and · · · · · · and θp (t ) is ηip , then

 (Σ ) : dx(t ) = (Ai + ∆Ai (t ))x(t ) + (Adi + ∆Adi (t ))x(t − τ ) + (B1i + ∆B1i (t ))u(t ) + Afi fi (x(t ), x(t − τ ), t )    + Bvi v(t ) dt + (Di + ∆Di (t ))x(t ) + (Ddi + ∆Ddi (t ))x(t − τ ) + (B2i + ∆B2i (t ))u(t ) dw(t ),

(1)

z (t ) = Ci x(t ) + Cdi x(t − τ ) + B3i u(t ),

(2)

x(t ) = φ(t ),

(3)

t ∈ [−τ , 0],

i = 1, 2, . . . , r

where θ1 (t ), . . . , θp (t ) are the premise variables and each ηij (j = 1, 2, . . . , p) is a fuzzy set. r is the number of if–then rules. x(t ) ∈ Rn is the state, u(t ) ∈ Rm is the control input, and v(t ) ∈ Rp is the disturbance input which belongs to L2 [0, ∞). z (t ) ∈ Rq is the controlled output, and ω(t ) is a one-dimensional Brownian motion defined on the probability space (Ω , F , {Ft }t ≥0 , P ), satisfying E {dw(t )} = 0, E {dw(t )2 } = dt. In the system (Σ ), Ai , Adi , B1i , Afi , Bv i , Di , Ddi , B2i , Ci , Cdi and B3i are known real constant matrices with appropriate dimensions. ∆Ai (t ), ∆Adi (t ), ∆B1i (t ), ∆Di (t ), ∆Ddi (t ) and ∆B2i (t ) are time-varying parameter uncertainties which are of the following form:



   ∆Ai (t ) ∆Adi (t ) ∆B1i (t ) ∆Di (t ) ∆Ddi (t ) ∆B2i (t ) = Ei Fi (t ) H1i H2i H3i H4i H5i H6i

(4)

where Ei , H1i , H2i , H3i , H4i , H5i and H6i are known real constant matrices with appropriate dimensions and Fi (t ) is an unknown real matrix function with a Lebesgue measurable function satisfying FiT (t )Fi (t ) ≤ I

∀ t.

(5)

Assumption 1. There exist known real constant matrices G1i ∈ Rn×n and G2i ∈ Rn×n (i = 1, 2, . . . , r ) such that the unknown nonlinear vector function fi (·, ·, ·) ∈ Rn satisfies the following bounded condition:

|fi (x(t ), x(t − τ ), t )| ≤ |G1i x(t )| + |G2i x(t − τ )|.

(6)

The defuzzified output of the T–S fuzzy system (Σ ) is represented as follows:

ˆ ) : dx(t ) = (Σ

r −

hi (θ (t ))





Ai (t )x(t ) + Adi (t )x(t − τ ) + B1i (t )u(t ) + Afi fi (x(t ), x(t − τ ), t ) + Bv i v(t ) dt

i =1

   + Di (t )x(t ) + Ddi (t )x(t − τ ) + B2i (t )u(t ) dw(t ) ,

(7)

1988

T. Senthilkumar, P. Balasubramaniam / Applied Mathematics Letters 24 (2011) 1986–1994

z (t ) =

r −





hi (θ (t )) Ci x(t ) + Cdi x(t − τ ) + B3i u(t ) ,

(8)

i=1

x(t ) = φ(t ),

t ∈ [−τ , 0]

(9)

where Ai (t ) = Ai + ∆Ai (t ), Adi (t ) = Adi + ∆Adi (t ), B1i (t ) = B1i + ∆B1i (t ), Di (t ) = Di + ∆Di (t ), Ddi (t ) = Ddi + ∆Ddi (t ) and p ν (θ(t )) B2i (t ) = B2i + ∆B2i (t ), with hi (θ (t )) = ∑r i ν (θ(t )) , νi (θ (t )) = Πj=1 ηij (θj (t )), and ηij (θj (t )) is the grade of membership i=1 i

value of θj (t ) in ηij . In this work, we assume that νi (θ (t )) ≥ 0 for i = 1, 2, . . . , r and i=1 νi (θ (t )) > 0 for all t. ∑r Therefore, hi (θ (t )) ≥ 0 (for i = 1, 2, . . . , r ) and i=1 hi (θ (t )) = 1 for all t. In the sequel, for simplicity we use hi to represent hi (θ (t )). On the basis of the parallel distributed compensation schemes, a fuzzy model of a state feedback controller for the system ˆ ) is formulated as follows: (Σ

∑r

Control rule i: if θ1 (t ) is ηi1 and · · · · · · and θp (t ) is ηip , then u(t ) = Ki x(t ),

i = 1, 2, . . . , r .

(10)

The overall state feedback fuzzy control law is represented by u(t ) =

r −

hi Ki x(t )

(11)

i=1

where the Ki (i = 1, 2, . . . , r ) are the local control gains. Under control law (11), the overall closed-loop system is obtained as follows:

˜ ) : dx(t ) = (Σ

r − r −

hi hj

  (Ai (t ) + B1i (t )Kj )x(t ) + Adi (t )x(t − τ ) + Afi fi (x(t ), x(t − τ ), t ) + Bvi v(t ) dt

i=1 j=1    + (Di (t ) + B2i (t )Kj )x(t ) + Ddi (t )x(t − τ ) dw(t ) ,

z (t ) =

r − r −





(12)

hi hj (Ci + B3i Kj )x(t ) + Cdi x(t − τ ) ,

(13)

t ∈ [−τ , 0].

(14)

i=1 j=1

x(t ) = φ(t ),

The following definitions and lemmas are essential for the proof in the following sections. Definition 1 ([19]). The nominal system (7) and (9) with u(t ) = 0 and v(t ) = 0 is said to be mean square stable if for any ε > 0, there exists a δ(ε) > 0 such that E {|x(t )|2 } < ε, t > 0, when sup E {|φ(s)|2 } < δ(ε). −τ ≤s≤0

If, in addition lim E {|x(t )|2 } = 0

t −→∞

for any initial conditions, then the nominal system (7) and (9) with u(t ) = 0, v(t ) = 0 is said to be mean square asymptotically stable. The uncertain stochastic system (7) and (9) is said to be robustly stochastically stable if the system associated with (7) and (9) with u(t ) = 0 and v(t ) = 0 is mean square asymptotically stable for all admissible uncertainties satisfying (4)–(5).

˜ ) with u(t ) = 0 is said to be robustly Definition 2 ([19]). Given a scalar γ > 0, the uncertain stochastic system (Σ stochastically stable with disturbance attenuation γ if it is robustly stochastically stable and under zero initial conditions, ‖z (t )‖E2 < γ ‖v(t )‖2 is satisfied for all nonzero v(t ) ∈ L2 [0, ∞) and all admissible uncertainties satisfying (4)–(5), where ∞

 ∫ ‖z (t )‖E2 = E

|z (t )|2 dt

 12

.

0

Lemma 1 (Schur Complement [28]). Given constant matrices M , P and Q with appropriate dimensions, where P T = P and Q T = Q > 0, then P + M T Q −1 M < 0 if and only if

[

P M

MT −Q

]

< 0 or

[ −Q MT

M P

]

< 0.

T. Senthilkumar, P. Balasubramaniam / Applied Mathematics Letters 24 (2011) 1986–1994

1989

Lemma 2 ([29]). For any vectors x, y ∈ Rn , matrices A, P , D, E and F that are real matrices of appropriate dimensions with P > 0, F T F ≤ I, and scalar ε > 0, the following inequalities hold: (a) 2xT DFEy ≤ ε −1 xT DDT x + ε yT E T Ey, (b) if P − εDDT > 0, then (A + DFE )T P −1 (A + DFE ) ≤ AT (P − ε DDT )−1 A + ε−1 E T E , (c) 2xT y ≤ xT P −1 x + yT Py. Lemma 3 ([18]). For any real matrices Xij for i, j = 1, 2, . . . , r , and Λ > 0 with appropriate dimensions, we have r − r − r − r −

hi hj hk hl XijT ΛXkl ≤

i=1 j=1 k=1 l=1

r − r −

hi hj XijT ΛXij

i=1 j=1

where hi (1 ≤ i ≤ r ) are defined as hi (θ (t )) ≥ 0,

∑r

i=1

hi (θ (t )) = 1.

3. The robust H∞ control problem In this section, a sufficient condition for the solvability of the robust H∞ control problem is proposed and an LMI approach for designing the desired state feedback fuzzy controllers is developed.

˜ ) is robustly stochastically stabilizable with Theorem 1. For given scalars γ > 0 and τ > 0, the closed-loop system (Σ disturbance attenuation γ if there exist matrices X > 0, Q˜ > 0, R˜ > 0, Yj and scalars εi > 0, ε1ij > 0, ε2ij > 0 (i, j = 1, 2, . . . , r ) such that the following LMI holds: √  ij   2XGT1i HijT X τX CijT ψ Adi X X Bv i (Di X + B2i Yj )T HijT √ 0 T T T T T ∗ 2XG2i XCdi  −2X + Q˜ −X 0 XDdi XH2i XH5i 0 0 0     1 ˜ ∗ ∗ (− 2X + R) 0 0 0 0 0 0 0 0 0    τ   ∗ ∗ −γ 2 I 0 0 0 0 0 0 0 0  ∗   T ∗ ∗ ∗ ε2ij Ei Ei − X 0 0 0 0 0 0 0  ∗ Ψ ij =  ∗ ∗ ∗ ∗ −ε1ij I 0 0 0 0 0 0  ∗  ∗ ∗ ∗ ∗ ∗ ∗ −ε2ij I 0 0 0 0 0    ∗ ∗ ∗ ∗ ∗ ∗ ∗ −εi I 0 0 0 0    ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ −Q˜ 0 0 0    ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ −τ R˜ 0 0    ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ −εi I 0 ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ −I < 0,

i, j = 1, 2, . . . , r

(15)

where

ψ ij = (Ai X + B1i Yj ) + (Ai X + B1i Yj )T + εi Afi ATfi + ε1ij Ei EiT ,  Hij = H1i X + H3i Yj , Hij = H4i X + H6i Yj , Cij = Ci X + B3i Yj . Moreover the state feedback gain can be constructed as Kj = Yj X −1 , j = 1, 2, . . . , r.

˜ ): Proof. Choose the following LKF for system (Σ V (x(t ), t ) = xT (t )Px(t ) +

∫

t

xT (s)Qx(s)ds + t −τ

∫

0

−τ

∫

t

xT (s)Rx(s)dsdθ + t +θ

∫

t

x(s)ds

T ∫

t

t −τ



x(s)ds

P

(16)

t −τ

where P , Q and R are symmetric positive definite matrices with appropriate dimensions. It can be derived by using Itô’s differential formula [30] that dV (x(t ), t ) = LV (x(t ), t )dt + 2

r − r −

hi hj xT (t )P







Di (t ) + B2i (t )Kj x(t ) + Ddi (t )x(t − τ ) dw(t )

(17)

i =1 j =1

where

LV (x(t ), t ) = 2

r − r −

hi hj xT (t )P







Ai (t ) + B1i (t )Kj x(t ) + Adi (t )x(t − τ ) + Afi fi (x(t ), x(t − τ ), t ) + Bv i v(t )

i=1 j=1

+

r − r − r − r − i=1 j=1 k=1 l=1

hi hj hk hl





T

Di (t ) + B2i (t )Kj x(t ) + Ddi (t )x(t − τ )

P

1990

T. Senthilkumar, P. Balasubramaniam / Applied Mathematics Letters 24 (2011) 1986–1994

×







Dk (t ) + B2k (t )Kl x(t ) + Ddk (t )x(t − τ ) + xT (t )Qx(t ) − xT (t − τ )Qx(t − τ )

+ xT (t )τ Rx(t ) −

t

∫

t

 ∫





x(s)ds .

xT (s)Rx(s)ds + 2 x(t ) − x(t − τ ) P

(18)

t −τ

t −τ

For the positive scalars εi , ε1ij , from Lemmas 2 and 3, and using (6), we have 2

r −

hi xT (t )PAfi fi (x(t ), x(t − τ ), t ) ≤

i =1

r  −

2  ,



hi εi xT (t )PAfi ATfi Px(t ) + εi−1 |G1i x(t )| + |G2i x(t − τ )|

i=1

r  −





hi εi xT (t )PAfi ATfi Px(t ) + 2εi−1 xT (t )GT1i G1i x(t ) + xT (t − τ )GT2i G2i x(t − τ )

≤

(19)

i=1

and using (4), we have 2

r − r −





hi hj xT (t )P (∆Ai (t ) + ∆B1i (t )Kj )x(t ) + ∆Adi (t )x(t − τ )

i =1 j =1

=2

r − r −





hi hj xT (t )PEi Fi (t ) (H1i + H3i Kj )x(t ) + H2i x(t − τ ) ,

i=1 j=1 r − r −

≤





−1 hi hj ε1ij xT (t )PEi EiT Px(t ) + ε1ij (H1i + H3i Kj )x(t )

i=1 j=1

T   + H2i x(t − τ ) (H1i + H3i Kj )x(t ) + H2i x(t − τ ) .

(20)

Use Jensen’s inequality to obtain

∫

t

−

xT (s)Rx(s)ds ≤ −

t −τ

∫

1

τ

t

x(s)ds

T ∫

t



x(s)ds .

R

(21)

t −τ

t −τ

Again, from Lemmas 2 and 3, it is easy to show that for P −1 − ε2ij Ei EiT > 0 we can obtain r − r − r − r −



hi hj hk hl  Dij + Ei Fi (t )H ij

T   P  Dkl + Ek Fk (t )H kl

i=1 j=1 k=1 l=1

≤

r − r −





−1 T hi hj  DTij (P −1 − ε2ij Ei EiT )−1 H ij H ij Dij + ε2ij

(22)

i=1 j=1

where  Dij = [Di + B2i Kj Ddi ], H ij = [H4i + H6i Kj Substituting (19)–(22) into (18) results in

LV (x(t ), t ) ≤

r − r −

H5i ].

hi hj ξ T (t )Ψ˜ ij ξ (t )

(23)

i=1 j=1

where

Ψ˜ ij

 ij  ψ˜ PAdi P PBv i  ∗ −Q + 2εi−1 GT2i G2i −P 0    −1 T =  Γij Γij  + ε1ij 1 ∗ ∗ − R 0  τ ∗ ∗ ∗ 0  T  T  Dij H ij    − 1 − 1 T − 1 +  0  (P − ε2ij Ei Ei )  Dij 0 0 + ε2ij  0  H ij 0

0

0 ,



0



ξ T (t ) = xT (t ) xT (t − τ )

t

∫

T

x(s)ds

v T (t )



t −τ

with

 Γij = H1i + H3i Kj

H2i

0

0 ,



ψ˜ ij = P (Ai + B1i Kj ) + (Ai + B1i Kj )T P + εi PAfi ATfi P + Q + τ R + 2εi−1 GT1i G1i + ε1ij PEi EiT P .

T. Senthilkumar, P. Balasubramaniam / Applied Mathematics Letters 24 (2011) 1986–1994

1991

Note that r − r − r − r −

z T (t )z (t ) =

hi hj hk hl ξ T (t )C˜ ijT C˜ kl ξ (t ) ≤

i=1 j=1 k=1 l=1

T



where C˜ ijT = Ci + B3i Kj Cdi 0 0

r − r −

hi hj ξ T (t )C˜ ijT C˜ ij ξ (t )

(24)

i=1 j=1

.

Now, we set J (t ) = E

∫ t 

 

z T (s)z (s) − γ 2 v T (s)v(s) ds

(25)

0

where t > 0. Since V (φ(t ), 0) = 0 under the zero initial condition, that is, φ(t ) = 0 for t ∈ [−τ , 0], then by Itô’s formula, it follows that J (t ) = E

∫ t 

 





z T (s)z (s) − γ 2 v T (s)v(s) + LV (x(s), s) ds − E V (x(t ), t ) ,

0

≤E

∫ t 

 

z T (s)z (s) − γ 2 v T (s)v(s) + LV (x(s), s) ds ,

0

≤E

t

∫

ξ T (s)Ψˆ ij ξ (s)ds



(26)

0

where Ψˆ ij = Ψ˜ ij + C˜ ijT C˜ ij + diag (0 0 0 − γ 2 I ). Pre-multiplying and post-multiplying Ψˆ ij by diag (X X X I ), letting P = X −1 , Yj = Kj X , using the inequalities (see [27])

  −2X +R−1 ≥ −XRX , −2X +Q −1 ≥ −XQX , from the Schur complement lemma and LMI (15), we obtain E LV (x(t ), t ) < 0, where R−1 = R˜ and Q −1 = Q˜ . This implies that J (t ) < 0 for t > 0; therefore, we have ‖z (t )‖E2 < γ ‖v(t )‖2 . Therefore ˜ ) is robustly stochastically stable by Definition 2 and [31], we can conclude that the closed-loop stochastic fuzzy system (Σ with disturbance attenuation γ .  Remark 1. It is shown in Theorem 1 that the H∞ control problem addressed is solvable if the set of LMIs (15) are feasible. ˜ ), taking In this work, the unknown nonlinear perturbations are considered as a special case in Theorem 1. In system (Σ v(t ) = 0, the LMI conditions for the robust stabilization problem can be obtained by using Theorem 1. Remark 2. In order to prove the mean square exponential stability, we modify the Lyapunov functional (16) as V¯ (x(t ), t ) = ˜ ) is eα t V (x(t ), t ). Then for LV¯ (x(t ), t ) = eα t (α V (x(t ), t ) + LV (x(t ), t )), it is not difficult to show that system (Σ exponentially stable in mean square. The detailed proof is omitted due to limitations of space. 4. A numerical example In this section, we provide an example to demonstrate the effectiveness of the proposed method. ˜ ) with parameters defined by Consider the uncertain stochastic T–S fuzzy system (Σ

] [ ] [ ] [ ] −2 1.2 −8 1 1 0.3 1.5 1 A1 = , A2 = , Ad1 = , Ad2 = , 0.1 −3 0.5 −12 0.2 0.3 0.2 0.1 [ ] [ ] [ ] [ ] 0.1 0.3 2.5 0.1 0.1 2 0.01 0.02 B11 = , B12 = , Af 1 = , Af 2 = , 0.4 1 0.01 1 0.2 0.6 0.1 −1 [ ] [ ] [ ] [ ] 0.2 0 0.02 2 −0.2 2 −0.01 0.1 Bv 1 = , Bv 2 = , D1 = , D2 = , 0 0.1 −0.25 0.25 1.09 0.31 0 0.25 [ ] [ ] [ ] [ ] −0.001 0.18 0.2 0.1 0.01 0.3 0.6 0.2 Dd1 = , Dd2 = , B21 = , B22 = 0.9 0.1 0.3 1 0.2 −0.1 −0.1 0.2 [ ] [ ] [ ] [ ] 0.2 1 −0.1 0.2 0.2 −0.4 0.1 0.01 C1 = , C2 = , Cd1 = , Cd2 = , 0.2 0.1 −1 0.2 0.1 0.2 0.2 0.3 [ ] [ ] [ ] [ ] 0.5 −0.2 0.2 0 0.1 0 0.2 −0.1 B31 = , B32 = , E1 = E2 = , H11 = , 0.4 −0.5 0 0.1 0 0.1 0.3 0.2 [ ] [ ] [ ] [ ] 0.1 −0.2 0.4 0.2 0.2 −0.1 0.1 −0.3 H12 = , H21 = , H22 = , H31 = , 0.1 0.1 0.1 0.1 0.1 0.3 0.2 0.1 [

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Fig. 1. State response of the closed-loop system.

Fig. 2. Control input.

H32

0.2 = 0.01

H52

0.1 = 0

0.1 , −0.01

[ [

Fi (t ) =

]

0.2 , 0.2

H41

]

0.2 sin(t ) 0

[

0.2 = −0.2

[

0.3 = −1.5

[

H61

]

0 , 0.2 cos(t )

0.1 , 0.1

]

H42

]

1 , 0.1

0.2 = 0.1

[

0.01 = 0.1

[

H62

0.1 , −0.1

]

0.02 , 0.2

]

0.1 = 0.1

[

H51

0.2 , 0.3

0.1 G1i = G2i = 0

[

]

]

0 , 0.1

i = 1, 2.

The purpose is to design a memoryless state feedback controller such that, for all admissible uncertainties, the resulting closed-loop system is robustly stochastically stable with disturbance attenuation γ . In this example, we assume that the noise attenuation level γ = 0.5 and the maximum time delay τ = 0.1433. Using Matlab LMI toolbox to solve the LMI (15), we obtain the following solution:

[ X =

1.0437 −0.2958

ε1 = 0.1916,

] −0.2958 , 1.9266

Q˜ =

0.6885 −0.4687

[

] −0.4687 , 0.9623

R˜ =

[

1.2294 −0.7323

] −0.7323 , 1.3356

ε2 = 2.7412.

By Theorem 1, we can obtain the desired state feedback fuzzy controller as follows:

[ −0.9874 K1 = K2 = −5.4440

] −0.5666 . −5.1786

For convenience of simulation, define the membership functions h1 (x1 (t )) = sin2 (x1 (t )) and h2 (x1 (t )) = cos2 (x1 (t )); the initial function is φ(t ) = [−2, 3]T ; the time delay τ = 0.1433 and the disturbance input is assumed to be v1 (t ) =

T. Senthilkumar, P. Balasubramaniam / Applied Mathematics Letters 24 (2011) 1986–1994

1993

Fig. 3. Controlled output.

, v2 (t ) = 1+1t 2 , t ≥ 0. In this example, the nonlinear disturbance signals are taken as fi (x(t ), x(t − τ ), t ) = [0.1 sin x1 (t ) 0.1 sin x2 (t − τ )]T , i = 1, 2. Then, with the state feedback fuzzy controller as defined earlier, the simulation 1 0.2+t

result for the state response of the closed-loop system is given in Fig. 1. Further, the control input and the controlled output are given in Figs. 2 and 3 respectively. From these simulation results, it can be seen that the designed H∞ controller satisfies the specified requirements. 5. Conclusions In this work, the problem of robust H∞ control for nonlinear uncertain stochastic T–S fuzzy systems with time delays has been studied. On the basis of LKF, a delay-dependent robust H∞ control scheme is presented in terms of LMIs. It has been shown that a desired state feedback fuzzy controller can be constructed when the given LMIs are feasible. Finally, a numerical example is given to illustrate the effectiveness of the proposed method. Acknowledgments The authors are very grateful to referees for their valuable comments and suggestions for improving the manuscript. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19]

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