New sufficient conditions for robust H∞ fuzzy hyperbolic tangent control of uncertain nonlinear systems with time-varying delay

Fuzzy Sets and Systems 161 (2010) 1993 – 2011 www.elsevier.com/locate/fss

New sufficient conditions for robust H ∞ fuzzy hyperbolic tangent control of uncertain nonlinear systems with time-varying delay Huaguang Zhanga,d,∗ , Xinrui Liua , Qingxian Gonga, b , Bing Chenc a School of Information Science and Engineering, Northeastern University, Shenyang, Liaoning 110004, PR China b Changchun Institute of Technology, Changchun, Jilin 130012, PR China c Institute of Complexity Science, Qingdao University, Qingdao 266071, PR China d Key Laboratory of Integrated Automation of Process Industry (Northeastern University) of National Education Ministry,

Shenyang, Liaoning 110004, China Received 4 November 2007; received in revised form 9 June 2009; accepted 29 October 2009 Available online 26 November 2009

Abstract This paper considers the delay-dependent robust H∞ control problem based on fuzzy hyperbolic model (FHM) for a class of uncertain nonlinear systems with time-varying delay. Firstly, FHM modeling method is presented for uncertain nonlinear systems with time-varying delay. Secondly, the new delay-dependent sufficient condition for the existence of a state feedback fuzzy hyperbolic tangent controller is proposed in terms of linear matrix inequalities (LMIs) by constructing a novel Lyapunov–Krasovskii functional. The robust FHM-based control law can guarantee that the closed-loop system is robustly asymptotically stable with a prescribed H∞ index. As more information of the FHM and the size of time-delay are taken into account, the proposed conditions are new and less conservative. Finally, three illustrative examples are given to show the effectiveness of using FHM-based controllers. © 2009 Published by Elsevier B.V. Keywords: Fuzzy hyperbolic model; Robust H∞ control; Asymptotically stable; Time-varying delay; Parameter uncertainties

1. Introduction In the past decades, fuzzy logic control has become an effective approach to control complex and ill-defined systems for which the application of conventional techniques is infeasible. Takagi–Sugeno (T–S) fuzzy models as the most popular fuzzy model have attracted many researchers. Fruitful significant results have been achieved (see [3,13] and literatures therein). In general, T–S fuzzy models are used to approximate the nonlinear systems, then the fuzzy controllers are designed. Linear matrix inequalities (LMIs)-based stability conditions were developed using the Lyapunov second method for T–S fuzzy models-based control systems [8,11]. On the other hand, time-delays are frequently encountered in various engineering, communication, and biological systems, which are often the cause for instability and poor performance of systems. Moreover uncertainties are not avoidable in control systems due to modeling errors, measurement errors, and so on. The two factors usually influence the practical control performances. Therefore, stability analysis and designing of robust controllers for uncertain nonlinear systems with time-delay are practically ∗ Corresponding author. Tel.: +86 2483687762; fax: +86 2483671498.

E-mail address: [email protected] (H. Zhang). 0165-0114/$ - see front matter © 2009 Published by Elsevier B.V. doi:10.1016/j.fss.2009.10.025

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H. Zhang et al. / Fuzzy Sets and Systems 161 (2010) 1993 – 2011

important and have attracted considerable attention over the past years, see [1,3,5–7,12,13,16–18,22] and the references therein. Generally speaking, stability analysis and fuzzy controller design methods are provided in the sense of delay-independent stability [5,12,16], and in the sense of delay-dependent stability [1,3,4,6,7,13,17,18,22]. The delay-dependent stabilization is concerned with the size of the time delay and usually provides an upper bound of the time-delay, while the delay-independent approach provides stability conditions irrespective of the size of the delay, which may lead to comparatively conservative stability analysis results, particularly when the delay is small. A new continuous-time fuzzy model, called FHM, has been proposed in [26,28,29]. As same as T–S fuzzy models, the FHM is the universal approximator [27]. Therefore, the FHM can be used to establish the model for the nonlinear system with arbitrarily good approximation. The FHM can be seen as a neural network model [29], the parameters of which can be trained using Back-Propagation (BP) algorithm. To the best of the authors’ knowledge, there are two approaches to construct fuzzy models (rules) for nonlinear systems [19]: (Case 1) Derivation from given nonlinear system equations. In this case, nonlinear dynamic models are mathematical descriptions of the nonlinear systems, i.e. model-based systems. (Case 2) Identification (fuzzy modeling) using input–output data. The main procedure consists of two parts: structure identification and parameter identification. The identification approach to fuzzy modeling is suitable for complex systems that are unable or too difficult to be represented by analytical or physical models, i.e. model-free systems. In case 2, both T–S models and FHM are not the exact representation of the nonlinear systems, but the arbitrarily good approximation. However, the FHM-based controller has its own distinguishing characteristics. Firstly, neither structure identification nor completeness design of premise variables space is required when the FHM is used to approximate the nonlinear systems, therefore the computational effort of modeling the FHM is lower than modeling the T–S models. The FHM-based control may be the best choice for fast time-variable industrial processes with disturbances. Secondly, less computational effort is required when FHM is used since only one LMI needs to be solved. However, the number of LMIs that need to be solved when the T–S model is used increases significantly as the number of fuzzy rules increases, especially when it comes to complex nonlinear systems which require a lot of fuzzy rules to accurately approximate. Moreover, the increasing number of LMIs may result in infeasible LMI solutions. Last but not least, the FHM-based controller we designed is naturally fuzzy nonlinear saturated controller, which is suitable for applying to practical systems. Actually, most practical systems are model-free systems, rather than model-based systems. Therefore, the advantages of FHM-based controller are more obvious. The FHM can be used as the effective alternative to substitute T–S models for some cases, but not for all cases. In addition, considerable attention has been paid to the H∞ control problem in recent years, and different approaches were derived in [2,6,18,24]. Some literatures focused on eliminating disturbance about H∞ control problem. It is refereed as disturbance attenuation, see [9,10] for example. In [15,23], H∞ control problem was considered for FHM systems without time-delay and parameter uncertainties. In [14], the guaranteed cost control problem was concerned for a class of time-delay FHM systems, and the delay-independent sufficient condition for the existence of controller was obtained. However, [14] did not discuss how to model FHM for time-delay systems. Moreover, the sufficient condition of controller was independent of the time-delay, so the result might be conservative. In [25], delay-dependent robust H∞ control based on FHM with multiple constant delays was investigated. In [21], robust H∞ condition based on time-delay FHM was given, the computational effort of which was heavier since the number of variables need to be solved was more. To the best of the authors’ knowledge, the problem of H∞ FHM-based control for uncertain nonlinear systems with time-varying delay has not been fully investigated, which motivates the present study. In this paper, our main objective is to investigate H∞ FHM-based control problem for a class of uncertain nonlinear systems with time-varying delay. Firstly, the FHM modeling method for uncertain nonlinear systems with time-varying delay is presented briefly. Secondly, the new delay-dependent sufficient condition for the existence of a robust state feedback controller is proposed in terms of LMIs by constructing a novel Lyapunov–Krasovskii functional. The controller can guarantee the closed-loop system is robustly asymptotically stable with a prescribed H∞ index. The proposed LMI-based condition can be efficiently solved with global convergence guarantee using convex optimization techniques such as the interior point algorithm. Finally, three illustrative examples are given to show the effectiveness of using FHM-based controllers. As more information of the FHM and the size of time-delay are taken into account compared with our earlier work [14,23], presented results are less conservative. The rest of this paper is organized as follows. In Section 2, the FHM modeling method for uncertain nonlinear systems with time-varying delay are given, and robust H∞ control problem is formulated. Main results are presented in Section 3. In Section 4, simulation study is shown. The conclusions are drawn in Section 5.

H. Zhang et al. / Fuzzy Sets and Systems 161 (2010) 1993 – 2011

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Notation: Throughout this paper, A T stands for transpose of matrix A, I is an identity matrix with appropriate dimension, Rn denotes the n dimensional Euclidean vector space, and the notation P > 0, for P ∈ Rn×n means that P is symmetric and positive definite matrix. The notation “∗” in the symmetric matrix stands for the transposed element. If not explicitly stated, all matrices are assumed to have compatible dimensions. 2. System description and preliminaries In this section, we give the method to model FHM for the nonlinear time-varying delay systems, and the robust H∞ control problem is formulated. 2.1. FHM for nonlinear time-varying delay systems The FHM modeling method for nonlinear systems without delay was given in [26,29]. Similarly, we discuss how to model FHM for uncertain nonlinear systems with time-varying delay. The following definition is addressed. Definition 1. Consider the original nonlinear systems with n input variables x(t) ¯ = [x¯ 1T (t), . . . , x¯ nT (t)]T and n output T T T ˙¯ = [x˙¯ (t), . . . , x˙¯ n (t)] , the generalized input variables are defined as x1 (t) = x¯1 (t) − d11 , . . . , xw1 (t) = variables x(t) 1 x¯1 (t) − d1w1 , . . . , xw1 +1 (t) = x¯2 (t) ¯n (t) − dn1 , . . . , xm¯ (t) = x¯n (t) − dnwn , x(t) = ¯ n +1 (t) = x n− d21 , . . . , xm−w w is the number of generalized input variables, wi (i = 1, . . . , n) repre[x1T (t), . . . , xmT¯ (t)]T , where m¯ = i=1 i sents the number of transformations associated with each xi and di j (i = 1, . . . , n, j = 1, . . . , wi ) are constants that define the transformations. The following group of fuzzy rule base is called hyperbolic type fuzzy rule base (HFRB), if it satisfies the following three conditions: (1) For each output variable x˙¯ l , l = 1, 2, . . . , n, the corresponding group of fuzzy rules have the following form: R j : IF x1 (t) is Fx1 and x2 (t) is Fx2 , . . . , and xm¯ (t) is Fxm¯ and x1 (t − h(t)) is Fd x1 and x2 (t − h(t)) is Fd x2 , . . . , and xm¯ (t − h(t)) is Fd xm¯ , THEN x˙¯ l = clFx + · · · + clFx + clFd x + · · · + clFd x + clFx + · · · + clFx + clFd x + · · · + clFd x , m¯

1

j = 1, . . . , 2

2m¯

m¯

1

m¯

1

m¯

1

.

(1)

where Fxi and Fd xi are fuzzy sets of xi , and xi (t − h(t)), which include Pxi , Pd xi (positive) and N xi , Nd xi (negative) subsets, respectively. h(t) is the time-varying delay with 0 < h(t)≤h¯ < ∞. (2) The constant terms clFx and clFd x in the THEN-part correspond to Fxi and Fd xi in the IF-part, respectively. That i

i

is, if the linguistic value of Fxi , Fd xi term in the IF-part are Pxi , Pd xi , respectively, corresponding clFx , clFd x must i

i

appear as clPx , clPd x in the THEN-part, respectively; if the linguistic value of Fxi , Fd xi term in the IF-part are N xi , i

i

Nd xi , respectively, corresponding clFx , clFd x must appear as clNx , clNd x in the THEN-part, respectively; if there are i

i

i

i

no Fxi , Fd xi in the IF-part, clF xi , clFd x do not appear in the THEN-part. The conditions of the uncertain terms clFx i

and clFd x are the same as that of clFx and clFd x , respectively. i

i

i

i

(3) There are two fuzzy rules in each rule base. Thus, there are a total of 22m¯ input variable combinations of all the possible Pxi , Pd xi , N xi and Nd xi in the IF-part. We need n HFRBs to describe the systems with n output variables. Lemma 1. Given n HFRBs, if the membership functions of Pxi , N xi , Pd xi and Nd xi are defined as  Px (xi (t)) = e−1/2(xi (t)−ki ) ,  Nx (xi (t)) = e−1/2(xi (t)+ki ) , 2

i

2

i

 Pd x (xi (t − h(t))) = e−1/2(xi (t−h(t))−ki ) ,  Nd x (xi (t − h(t))) = e−1/2(xi (t−h(t))+ki ) , 2

i

2

i

(2)

where i = 1, 2, . . . , m, ¯ ki are positive constants. Then, the system can be derived as the following form: x(t) ˙ = V + V + ( A + A) tanh(K x(t)) + (Ad + Ad ) tanh(K x(t − h(t))),

(3)

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where V = [v1 , . . . , v1 , . . . , vn , . . . , vn ]T , vi =       w1

wn

i i i m¯ ci  Px j + c N x j + c Pd x j + c Nd x j

V = [v1 , . . . , v1 , . . . , vn , . . . , vn ] , vi =       T

w1

⎡

wn

q11 . . . q1m¯

⎢ ⎢ .. .. ⎢ . . ⎢ ⎢ q11 . . . ⎢ ⎢ . .. A=⎢ . ⎢ .. ⎢ ⎢ qn1 . . . ⎢ ⎢ . .. ⎢ . . ⎣ . qn1 . . . ⎡ qd11 ⎢ . ⎢ . ⎢ . ⎢ ⎢ qd11 ⎢ ⎢ . Ad = ⎢ ⎢ .. ⎢ ⎢ qdn1 ⎢ ⎢ . ⎢ . ⎣ .

⎤

⎥ .. ⎥ . ⎥ ⎥ q1m¯ ⎥ ⎥ .. ⎥ ⎥ , . ⎥ ⎥ qn m¯ ⎥ ⎥ .. ⎥ ⎥ . ⎦ qn m¯ m× ¯ m¯

⎤ . . . qd1m¯ .. .. ⎥ ⎥ . . ⎥ ⎥ . . . qd1m¯ ⎥ ⎥ .. .. ⎥ ⎥ . . ⎥ ⎥ . . . qdn m¯ ⎥ ⎥ .. .. ⎥ ⎥ . . ⎦

qdn1 . . . qdn m¯ qdi j =

ciPd x j

− ciNd x j 2

,

2

j=1

i i i m¯ ci  Px j + c N x j + c Pd x j + c Nd x j

2

j=1

⎤ ⎡ qd11 . . . qd1m¯ q11 ⎥ ⎢ . ⎢ . . . ⎢ . ⎢ . .. .. ⎥ ⎥ ⎢ . ⎢ . ⎥ ⎢ ⎢ ⎢ qd11 . . . qd1m¯ ⎥ ⎢ q11 ⎥ ⎢ ⎢ ⎥ ⎢ . ⎢ . .. .. ⎥ ⎢ . . Ad = ⎢ , A = . . ⎥ ⎢ . ⎢ . ⎥ ⎢ ⎢ ⎢ qdn1 . . . qdn m¯ ⎥ ⎢ qn1 ⎥ ⎢ ⎢ ⎥ ⎢ . ⎢ . .. .. ⎥ ⎢ . ⎢ . . . ⎦ ⎣ . ⎣ . qdn1 . . . qdn m¯ m× qn1 ¯ m¯ ⎡

, qi j =

ciPx − ciNx j

2

j

, qi j =

, i = 1, . . . , n,

⎤ . . . q1m¯ .. .. ⎥ ⎥ . . ⎥ ⎥ . . . q1m¯ ⎥ ⎥ .. .. ⎥ ⎥ , . . ⎥ ⎥ . . . qn m¯ ⎥ ⎥ .. .. ⎥ ⎥ . . ⎦ . . . qn m¯ m× ¯ m¯

ciPx − ciNx j

2

j

,

m× ¯ m¯

, qdi j = (ciPd x − ciNd x )/2, i = 1, . . . , n, j = 1, . . . , m¯ j

j

K = diag[k1 , . . . , km¯ ], and tanh(K x) is defined by tanh(K x) = [tanh T (k1 x1 ), tanh T (k2 x2 ), . . . , tanh T (km¯ xm¯ )]T . Proof. The proof can be obtained from [26,29].  Eq. (3) is called the time-varying delay FHM. From Definition 1, if vi = 0, vi = 0, i = 1, . . . , n, a homogeneous time-varying delay FHM can be obtained: x(t) ˙ = (A + A) tanh(K x(t)) + ( Ad + Ad ) tanh(K x(t − h(t))).

(4)

Remark 1. It has been proved that the FHM can uniformly approximate any nonlinear function over a compact set to any degree of accuracy, i.e. the FHM is a universal approximator (see [27, Section 4.3.3] for details). Using the linear transformation of x¯i , the enough number of fuzzy rules is chosen until the FHM approximates the nonlinear system at an arbitrary accuracy. The transformation number wi of the state x¯i is the parameter chosen by designers. It can be seen that the bigger the wi , the smaller the modeling error. Meanwhile, it means that more fuzzy rules and higher dimensional system matrices A, Ad are used, therefore the controller design is more difficult. However, an analytical way has not been found to obtain an optimum wi for the FHM modeling and the controller design. If the modeling error is satisfied, the parameters can be chosen as wi = 1, di1 = 0, i.e. x = [x¯i , . . . , x¯n ]T in (4).

H. Zhang et al. / Fuzzy Sets and Systems 161 (2010) 1993 – 2011

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There is no essential difference between the model of (3) and (4), so the problem discussed in this paper are based on the time-varying delay FHM described in (4). Remark 2. No matter whether original nonlinear systems involve the parameter uncertainties or not, x and x˙ of original nonlinear systems as input and output data are used, then the parameters A, Ad , K of (4) can be obtained by BP algorithm. That is to say, in the time-varying delay FHM (4), x(t) ˙ = A tanh(K x(t)) + Ad tanh(K x(t − h(t))) can be used to model original uncertain nonlinear systems with time-varying delay. And the parameter uncertainties A, Ad of the time-varying delay FHM (4) are used to represent the modeling errors between original nonlinear systems and the FHM. The robust fuzzy controller is designed to compensate the approximation error. 2.2. Robust H∞ controller design To compensate the approximation error, the following uncertain time-varying delay FHM concerning with the controller is used: x(t) ˙ = (A + A(t)) tanh(K x(t)) + ( Ad + Ad (t)) tanh(K x(t − h(t))) + (B + B(t))u(t) + Bw w(t),

(5)

z(t) = C tanh(K x(t)) + Du(t) + Dw w(t), ¯ 0], x(t) = (t), ∀t ∈ [−h,

(6)

where x(t) ∈ Rm¯ , u(t) ∈ Rm , z(t) ∈ R p denote the state vector, input vector and controlled output vector, respectively; ¯ m¯ , Ad ∈ w(t) ∈ Rq denotes the exogenous disturbance, which can be unknown but belongs to L 2 [0, ∞); A ∈ Rm× m× ¯ m¯ m×m ¯ m×q ¯ p×m¯ p×m p×q ,B ∈ R , Bw ∈ R ,C ∈ R , D ∈ R , Dw ∈ R are known real constant matrices; h(t) is R ˙ the bounded time-varying delay and is assumed to satisfy 0 < h(t)≤h¯ < ∞ and |h(t)| ≤ , where h¯ and  are known constant scalars. The condition (t) is given by initial vector function, which is continuous for −h¯ ≤ t ≤ 0; ¯ m¯ ¯ m¯ ¯ A(t) ∈ Rm× , Ad (t) ∈ Rm× and B(t) ∈ Rm×m are time-varying parameter uncertainty matrices and satisfy the condition [A(t) Ad (t) B(t)] = M F(t)[N Nd Nb ],

(7)

where M, N , Nd and Nb are known real constant matrices of appropriate dimension, and F(t) is an unknown matrix function satisfying F T (t)F(t) ≤ I . For convenience, let A := A(t), Ad := Ad (t), B := B(t). The aim of this paper is to construct a memoryless feedback control law for system (5) and (6) in FHM form as u(t) = G tanh(K x(t)),

(8)

where G ∈ Rm×m¯ is the controller gain matrix to be determined, such that the resulting closed-loop system x(t) ˙ = ( A + A + (B + B)G) tanh(K x(t)) + ( Ad + Ad ) tanh(K x(t − h(t))) + Bw w(t), z(t) = (C + DG) tanh(K x(t)) + Dw w(t).

(9) (10)

satisfies the following requirements: (1) When w(t) ≡ 0, the closed-loop system is asymptotically stable. (2) Under the zero-initial condition, for any nonzero w(t) ∈ L 2 [0, ∞), the controlled output satisfies z(t)2 < ∞ w(t)2 for any given disturbance attenuation level  > 0, where z(t)2 = ( 0 z T (t)z(t) dt)1/2 . 3. Main results To state our main results, the following lemmas are useful in the proofs of our results. Lemma 2 (Wang et al. [20]). Given appropriate dimension matrices M, E and F satisfying F T F ≤ I , for any real scalar  > 0, the following result holds: M F E + E T F T M T ≤ M M T + −1 E T E.

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H. Zhang et al. / Fuzzy Sets and Systems 161 (2010) 1993 – 2011

Lemma 3 (Chen et al. [3]). For any constant positive definite matrix W ∈ Rm×m , a scalar  > 0, a function  : [0, ] → R+ , and the vector function : [ − (), ] → Rm such that the integrations in the following are well defined, then  T    ()



−()



T (s)W (s) ds ≥

−()

(s) ds



W

−()

(s) ds .

Lemma 4. For any vector ∈ Rn , T = [ 1 · · · n ], and diagonal positive definite matrix X, the following result holds: ˙ ˙ T ( )X tanh( ) ≤ ˙ T X ˙ . tanh Proof. ˙ T ( )X tanh( ) ˙ tanh =



˙ 1 cosh2 ( 1 )

≤ ˙ T X ˙ .



 ···

˙ n cosh2 ( n )



 X

˙ 1 cosh2 ( 1 )



 ···

˙ n cosh2 ( n )

T



The robust stabilization problem of (5) will be analyzed when w(t) = 0. Theorem 1. Given scalars  and h¯ > 0, the system (9) (when w(t) = 0) is robustly asymptotically stable for any delay ˙ h(t) satisfying 0 < h(t) ≤ h¯ and |h(t)| ≤ , if there exist a diagonal matrix X > 0, matrices S > 0 ( = 1, 2, 3), L and the scalars 1 > 0, 2 > 0 such that the following LMI holds: ⎤ ⎡ S3 (K AX + K B L)T X N T + L T NbT X N T + L T NbT 11 K Ad X + h¯ −1 X ⎥ ⎢ 22 −S3 (K Ad X )T X NdT X NdT ⎥ ⎢ ∗ ⎥ ⎢ ¯ −1 S2 + S3 ⎥ ⎢ ∗ ∗ − h 0 0 0 ⎥ < 0, (11) ⎢ ⎥ ⎢ ∗ 0 0 ∗ ∗ 33 ⎥ ⎢ ⎥ ⎢ 0 ∗ ∗ ∗ −1 I ⎦ ⎣ ∗ ∗ ∗ ∗ ∗ ∗ −2 I where 11 = K AX + X A T K + K B L + L T B T K + S1 + h¯ S2 + 1 K M M T K − h¯ −1 X, 22 = −(1 − )S1 + S3 − h¯ −1 X, 33 = −h¯ −1 X + 2 K M M T K . Moreover, the controller can be chosen as (8) with G = L X −1 . Proof. When w(t) = 0, the closed-loop system (9) can be described as x(t) ˙ = ( A + A + (B + B)G) tanh(K x(t)) + ( Ad + Ad ) tanh(K x(t − h(t))).

(12)

A Lyapunov–Krasovskii functional candidate for (12) is chosen as follows:  t m¯  pi ln(cosh(ki xi (t))) + tanh T (K x(s))Q 1 tanh(K x(s)) ds V (t) = 2 i=1  0

+

−h¯

 +

t t+

tanh T (K x( ))Q 2 tanh(K x( )) d d T

t

tanh(K x(s)) ds 

+

t−h(t)



t−h(t) 0  t −h¯

t+



t

Q3

tanh(K x(s)) ds t−h(t)

˙ ˙ T (K x( ))P tanh(K x( )) d d, tanh

(13)

H. Zhang et al. / Fuzzy Sets and Systems 161 (2010) 1993 – 2011

1999

where scalars pi > 0 (i = 1, . . . , m), ¯ matrix Q  > 0 ( = 1, 2, 3), P is diagonal positive definite matrix, P = diag[ p1 , p2 , . . . , pm¯ ] and ki (i = 1, . . . , m) ¯ are defined in (2). Taking the derivative of V (t) with respect to t along the trajectory of (12) yields V˙ (t)|(12) = 2 tanh T (K x(t))K P[(A + A + (B + B)G) tanh(K x(t)) +( Ad + Ad ) tanh(K x(t − h(t)))] + tanh T (K x(t))Q 1 tanh(K x(t)) ˙ −(1 − h(t)) tanh T (K x(t − h(t)))Q 1 tanh(K x(t − h(t))) + h¯ tanh T (K x(t))Q 2 tanh(K x(t))  t tanh T (K x(s))Q 2 tanh(K x(s)) ds − t−h¯

˙ +2(tanh(K x(t)) − (1 − h(t)) tanh(K x(t − h(t))))T Q 3 ˙ T (K x(t))P tanh(K ˙ +h¯ tanh x(t)) −





t

tanh(K x(s)) ds t−h(t)

t t−h¯

˙ T (K x(s))P tanh(K ˙ tanh x(s)) ds,

(14)

From Lemma 3, we have  t − tanh T (K x(s))Q 2 tanh(K x(s)) ds t−h¯

≤−



t

tanh T (K x(s))Q 2 tanh(K x(s)) ds

t−h(t)

≤ −h¯ −1



T

t

tanh(K x(s)) ds



t−h(t)

and

 −

t t−h¯

t

Q2

tanh(K x(s)) ds.

(15)

t−h(t)

˙ ˙ T (K x(s))P tanh(K x(s)) ds tanh

≤ −h¯ −1 (tanh(K x(t)) − tanh(K x(t − h(t))))T P(tanh(K x(t)) − tanh(K x(t − h(t)))).

(16)

From Lemma 4, we have ˙ T (K x(t))P tanh(K ˙ h¯ tanh x(t))≤h¯ x(t) ˙ T K P K x(t). ˙ Moreover ˙ tanh T (K x(t − h(t)))Q 3 2h(t)



(17)

t

tanh(K x(s)) ds t−h(t)

˙ ≤ |h(t)| tanh T (K x(t − h(t)))Q 3 tanh(K x(t − h(t)))  t T  t ˙ +|h(t)| tanh(K x(s)) ds Q3 tanh(K x(s)) ds t−h(t)

t−h(t)

≤  tanh (K x(t − h(t)))Q 3 tanh(K x(t − h(t))) T  t  t tanh(K x(s))ds Q3 tanh(K x(s)) ds. + T

t−h(t)

t−h(t)

Substituting (15)–(18) into (14), we can obtain V˙ (t)|(12) ≤ 2 tanh T (K x(t))K P[(A + A + (B + B)G) tanh(K x(t)) +( Ad + Ad ) tanh(K x(t − h(t)))] + tanh T (K x(t))(Q 1 + h¯ Q 2 ) tanh(K x(t)) + tanh T (K x(t − h(t)))(−(1 − )Q 1 + Q 3 ) tanh(K x(t − h(t)))

(18)

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H. Zhang et al. / Fuzzy Sets and Systems 161 (2010) 1993 – 2011

  +



t

+2 tanh T (K x(t))Q 3

tanh(K x(s)) ds − 2 tanh T (K x(t − h(t)))Q 3

t−h(t)

T

t

tanh(K x(s)) ds

(−h¯ −1 Q 2 + Q 3 )

t−h(t)



t

tanh(K x(s)) ds t−h(t)

t

tanh(K x(s)) ds + h¯ x(t) ˙ T K P K x(t) ˙

t−h(t)

−h¯ −1 (tanh(K x(t)) − tanh(K x(t − h(t))))T P(tanh(K x(t)) − tanh(K x(t − h(t)))) = T (t) (t),

(19)

By Schur complement,  < 0 is equivalent to 



< 0. Where  t T  T T T , tanh(K x(s)) ds  (t) = tanh (K x(t)) tanh (K x(t − h(t))) ⎡

t−h(t)

⎤

11 12 13 14 ⎢ ∗  ⎥ 22 23 24 ⎥ ⎢ =⎢ ⎥, ∗ 33 0 ⎦ ⎣ ∗ ∗ ∗ ∗ 44 11 = K P(A + A + (B + B)G) + ( A + A + (B + B)G)T P K + Q 1 + h¯ Q 2 − h¯ −1 P, 12 = K P(Ad + Ad ) + h¯ −1 P, 13 = Q 3 , 14 = (A + A + (B + B)G)T K , 22 = −(1 − )Q 1 + Q 3 − h¯ −1 P, 23 = −Q 3 , 24 = ( Ad + Ad )T K , 33 = −h¯ −1 Q 2 + Q 3 , 44 = −h¯ −1 P −1 .  can be rewritten as  =  1 + 2 , where

(20)

⎡

⎡ ¯ ⎤ ¯ 12 13  ¯ 14 ¯ 11   11 K PAd  ⎢ ∗  ⎢ ∗ ⎥ ¯   0 22 23 24 ⎥ ⎢ ⎢ 1 = ⎢ ⎥ , 2 = ⎢ ⎣ ∗ ⎣ ∗ ∗ ∗ 33 0 ⎦ ∗ ∗ ∗ 44 ∗ ∗

0 (A + BG)T K 0 0

AdT K 0

∗

0

⎤ ⎥ ⎥ ⎥, ⎦

¯ 11 = K P(A + BG) + ( A + BG) P K + Q 1 + h¯ Q 2 − h P,  ¯ 12 = K P Ad + h¯ −1 P,  ¯ 14 = (A + BG)T K ,  ¯ 24 = AdT K ,  ¯  11 = K P(A + BG) + (A + BG)T P K .  ¯ −1

T

From (7) and Lemma 2, we have 2 = [(K P M)T 0 0 0]T F(t)[N + Nb G Nd 0 0] + [N + Nb G Nd 0 0]T F T (t)[(K P M)T 0 0 0] +[0 0 0 (K M)T ]T F(t)[N + Nb G Nd 0 0] + [N + Nb G Nd 0 0]T F T (t)[0 0 0 (K M)T ] T ≤ 1 [(K P M)T 0 0 0]T [(K P M)T 0 0 0] + −1 1 [N + Nb G Nd 0 0] [N + Nb G Nd 0 0] T +2 [0 0 0 (K M)T ]T [0 0 0 (K M)T ] + −1 2 [N + Nb G Nd 0 0] [N + Nb G Nd 0 0].

Based on (20) and (21), we have ⎡ ¯ ¯ 12 13 11 + 1 KPMMT P K  ⎢ ∗ 22 23 ⎢ ≤⎢ ⎣ ∗ ∗ 33

¯ 14  ¯ 24 

∗

44 + 2 K M M T K

∗

∗

0

(21)

⎤ ⎥ ⎥ ⎥ ⎦

−1 T T +−1 1 [N + Nb G Nd 0] [N + Nb G Nd 0] + 2 [N + Nb G Nd 0] [N + Nb G Nd 0] ¯ = .

(22)

H. Zhang et al. / Fuzzy Sets and Systems 161 (2010) 1993 – 2011

2001

The sufficient condition of V˙ (t)|(12) < 0 is that ¯ < 0. 

(23)

By Schur complement, (23) is equivalent to ⎡ ¯ ¯ 14 ¯ 12 13  (N + Nb G)T (N + Nb G)T 11 + 1 KPMMT P K  ⎢ ¯ 24  ∗ 22 23 NdT NdT ⎢ ⎢ ⎢ ∗ ∗ 33 0 0 0 ⎢ ⎢ T ∗ ∗ ∗ 44 + 2 K M M K 0 0 ⎢ ⎢ ⎣ ∗ ∗ ∗ ∗ −1 I 0 ∗ ∗ ∗ ∗ ∗ −2 I

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ < 0. ⎥ ⎥ ⎥ ⎦

(24)

Pre- and post-multiplying diag[P −1 , P −1 , P −1 , I, I, I ] to (24) and letting X = P −1 , S1 = P −1 Q 1 P −1 , S2 = −1 −1 Q P −1 and L = G P −1 , (11) can be obtained. It implies that the system (12) is asymptotically 2 P , S3 = P 3 stable with time-delay and parameter uncertainties. It completes the proof.  P −1 Q

Remark 3. Theorem 1 presents a sufficient condition for robust stabilization for a class of uncertain time-delay FHM systems. The first item of Lyapuvov–Krasovskii functional n (13) in the proof is different from that in the existing results ([14,23] for example), which was often chosen as 2 i=1 ( pi /ki ) ln(cosh(ki xi (t))). Thus, the corresponding result depends on K, which is the parameter of the FHM. Obviously, it may reduce conservativeness because of taking more information of the systems into account, and it can guarantee the designed controller to obtain better performance. The controller is called a K-dependent controller. Now we give the main theorem of the robust H∞ control problem. Theorem 2. Given scalars  > 0,  and h¯ > 0, the system (9) and (10) is robustly stabilizable with disturbance ¯ |h(t)| ˙ attenuation level  for any delay h(t) satisfying 0 < h(t) ≤ h, ≤ , if there exist a diagonal matrix X > 0, matrices S > 0 ( = 1, 2, 3), L and the scalars 1 > 0, 2 > 0 such that the following LMI holds: ⎡ ⎤ ¯4 ¯1 ¯ S3 (K AX + K B L)T K Ad X + h¯ −1 X ⎢ ¯2⎥ ⎢ ∗ −(1 − )S1 + S3 − h¯ −1 X ⎥ −S3 0 (K Ad X )T ⎢ ⎥ ⎢∗ ∗ −h¯ −1 S2 + S3 0 0 0 ⎥ ⎢ ⎥ ⎢ ⎥ < 0 (25) 2 I + DT D T ⎢∗ ⎥ ∗ ∗ − (K B ) 0 w w w ⎢ ⎥ ⎢∗ ⎥ −1 T ¯ ∗ ∗ ∗ − h X +  K M M K 0 2 ⎣ ⎦ ¯3 ∗ ∗ ∗ ∗ ∗ where ¯ = K AX + X A T K + K B L + L T B T K + S1 + h¯ S2 − h¯ −1 X + 1 K M M T K , ¯ 4 = K Bw + (C X + DL)T Dw , ¯ 1 = [(N X + Nb L)T (N X + Nb L)T (C X + DL)T ], ¯ 3 = diag[−1 I, −2 I, −I ]. ¯ 2 = [X N T X N T 0], d d And the controller can be chosen as (8) with G = L X −1 . Proof. To establish the H∞ performance for the corresponding closed-loop system (9) and (10), we introduce 

T

J (T ) = 0

(z T (t)z(t) − 2 w T (t)w(t)) dt

(26)

2002

H. Zhang et al. / Fuzzy Sets and Systems 161 (2010) 1993 – 2011

for any nonzero w(t) ∈ L 2 [0, ∞), where T > 0. Under zero initial condition, (26) can be expressed as  T (z T (t)z(t) − 2 w T (t)w(t) + V˙ (t)|(9) ) dt − V (T ) J (T ) = 0



T

≤

(z T (t)z(t) − 2 w T (t)w(t) + V˙ (t)|(9) ) dt,

(27)

0

where V (t) is defined in (13). Similar proof to that of Theorem 1 is used, and we have V˙ (t)|(9) ≤ 2 tanh T (K x(t))K P[(A + A + (B + B)G) tanh(K x(t)) +(Ad + Ad ) tanh(K x(t − h(t)))] + tanh T (K x(t))(Q 1 + h¯ Q 2 ) tanh(K x(t)) + tanh T (K x(t − h(t)))(−(1 − )Q 1 + Q 3 ) tanh(K x(t − h(t)))  t  tanh(K x(s)) ds − 2 tanh T (K x(t − h(t)))Q 3 +2 tanh T (K x(t))Q 3  +

t−h(t)

T

t

tanh(K x(s))ds

(−h¯ −1 Q 2 + Q 3 )

t−h(t)



t

tanh(K x(s)) ds t−h(t)

t

tanh(K x(s)) ds t−h(t)

+2 tanh T (K x(t))K P Bw w(t) + h¯ x(t) ˙ T K P K x(t) ˙ −1 T ¯ −h (tanh(K x(t)) − tanh(K x(t − h(t)))) P(tanh(K x(t)) − tanh(K x(t − h(t)))). Substituting (10) and (28) into (27), it follows  T ˜  (t) dt, J (T ) ≤ T (t)

(28)

(29)

0 

˜ < 0 is equivalent to  ˜ < 0. Where By Schur complement,    t T  T T T  (t) = tanh (K x(t)) tanh (K x(t − h(t))) , tanh(K x(s))ds ⎡

t−h(t)

11 12 13 14 ⎢ ⎢ ∗ 22 23 0 ⎢ T T T ˜ ∗ 33 0  (t) = [ (t) w (t)],  = ⎢ ⎢ ∗ ⎢ ∗ ∗ ∗ 44 ⎣ ∗ ∗ ∗ ∗

⎤ 15 ⎥ 25 ⎥ ⎥ 0 ⎥ ⎥, ⎥ 45 ⎦ 55

11 = K P(A + A + (B + B)G) + (A + A + (B + B)G)T P K + Q 1 + h¯ Q 2 −h¯ −1 P + (C + DG)T (C + DG), 12 = K P( Ad + Ad ) + h¯ −1 P, 13 = Q 3 , 14 = K P Bw + (C + DG)T Dw , 15 = (A + A + (B + B)G)T K , 22 = −(1 − )Q 1 + Q 3 − h¯ −1 P, 23 = −Q 3 , 25 = ( Ad + Ad )T K , 33 = −h¯ −1 Q 2 + Q 3 , T 45 = BwT K , 44 = −2 I + Dw Dw , 55 = −h¯ −1 P −1 .

H. Zhang et al. / Fuzzy Sets and Systems 161 (2010) 1993 – 2011

ˆ < 0 then  ˜ < 0, and In the same way, we have if  ⎡ ¯ 12 13 14  ¯ 15 (N + Nb G)T (N + Nb G)T ¯ 11   ⎢ ¯ 25 NdT NdT ⎢ ∗ 22 23 0  ⎢ ⎢ ∗ ∗ 33 0 0 0 0 ⎢ ⎢ ˆ =⎢ ∗ ∗ ∗ 44 45 0 0 ⎢ ⎢ ∗ ¯ ∗ ∗ ∗ 55 0 0 ⎢ ⎢ 0 ∗ ∗ ∗ ∗ −1 I ⎣ ∗ ∗ ∗ ∗ ∗ ∗ ∗ −2 I

2003

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥, ⎥ ⎥ ⎥ ⎥ ⎦

(30)

where ¯ 11 = K P(A + BG) + ( A + BG)T P K + Q 1 + h¯ Q 2 − h¯ −1 P + (C + DG)T (C + DG) + 1 KPMMT P K ,  ¯ 12 = K P Ad + h¯ −1 P,  ¯ 15 = ( A + BG)T K ,  ¯ 25 = AdT K ,  ¯ 55 = −h¯ −1 P −1 + 2 KMMT K .  The sufficient condition of J < 0 is that ˆ < 0. 

(31)

−1 Pre- and post-multiplying diag[P −1 , P −1 , P −1 , I, I, I, I ] to (31), letting X = P −1 , S = P −1 Q −1  P ( = 1, 2, 3) −1 and L = G P , (25) can be obtained. It implies that J (T ) < 0, that is, for any T > 0 and any nonzero w(t) ∈ L 2 [0, ∞),  T  T z T (t)z(t) dt < 2 w T (t)w(t) dt, (32) 0

0

which yields  ∞  z T (t)z(t) dt < 2 0

∞

w T (t)w(t) dt.

0

In addition, it can be clearly seen that (11) holds from (25), that is, the closed-loop system is asymptotically stable when w(t) = 0. It completes the proof.  Theorem 2 presents the delay-dependent robust H∞ stability criterion for system (9) and (10). It is easy to get the following corollary for the FHM without delay from above proof. Corollary 1. Given scalars  > 0, the system (9) and (10) without delay is robustly stabilizable with disturbance attenuation level , if there exist a diagonal matrix X > 0, a matrix L and a scalar  > 0 such that the following LMI holds: ⎡ ⎤ 1 2 3 ⎢∗ 0 0 ⎥ 4 ⎢ ⎥ (33) ⎢ ⎥ < 0, ⎣ ∗ ∗ −I 0 ⎦ ∗

∗

∗

−I

where = K AX + X A T K + K B L + L T B T K + K M M T K , 1 = K Bw + (C X + DL)T Dw , T Dw . 2 = (N X + Nb L)T , 3 = (C X + DL)T , 4 = −2 I + Dw

And the controller gain of the FHM is G = L X −1 .

2004

H. Zhang et al. / Fuzzy Sets and Systems 161 (2010) 1993 – 2011

Robust H∞ FHM-based controller design procedure is summarized as follows: Step 1: If the mathematical models of the original nonlinear systems are known, input–output data can be obtained from the mathematical models. Otherwise, input–output data are obtained directly from the nonlinear systems. The distribution of input–output data pairs should be sufficient and proper. Based on input–output identification, the original nonlinear systems are modeled to the FHM. Generally speaking, the FHM is not the exact representation of the original nonlinear systems, but the arbitrarily good approximation. To compensate the approximation error, the uncertain timevarying delay FHM is used. Step 2: The robust H∞ FHM-based controller is designed. Step 3: Solving the LMI-based condition, the gain of controller can be obtained, which is applied to the original nonlinear systems. 4. Simulation In this section, three examples are provided to illustrate the effectiveness of the presented result in this paper. Example 1. Consider the following time-delay nonlinear systems, x˙¯ 1 (t) = (0.1 + Ao )x¯1 (t) − 0.0125x¯1 (t − h(t)) − 0.02 x¯2 (t) − 0.67x¯23 (t) −0.1x¯23 (t − h(t)) − 0.005x¯2 (t − h(t)), x˙¯ 2 (t) = x¯1 (t), z(t) = 0.02 x¯1 (t) + 0.4 x¯2 (t),

(34)

where Ao denote the parameter uncertainty of original nonlinear systems. Denote x1 (t) = x¯1 (t) − d11 , x2 (t) = x¯1 (t) − d12 , x3 (t) = x¯2 (t) − d21 , x4 (t) = x¯2 (t) − d22 , x(t) = [x1T (t), x2T (t), T x3 (t), x4T (t)]T , d11 = 0.7818, d12 = −0.7818, d21 = 1.8372, d22 = −1.8372. The fuzzy sets of every variable are positive and negative, and the HFRBs are given as follows: IF x1 (t) is Px1 , x2 (t) is Px2 , x3 (t) is Px3 , x4 (t) is Px4 , x1 (t − h(t)) is Pd x1 , x2 (t − h(t)) is Pd x2 , x3 (t − h(t)) is Pd x3 and x4 (t − h(t)) is Pd x4 , THEN x˙¯ 1 = C x11 + C x12 + C x3 + C x4 + Cd x1 + Cd x2 + Cd x3 + Cd x4 ; · · · IF x1 (t) is N x1 , x2 (t) is N x2 , x3 (t) is N x3 , x4 (t) is N x4 , x1 (t − h(t)) is Nd x1 , x2 (t − h(t)) is Nd x2 , x3 (t − h(t)) is Nd x3 and x4 (t − h(t)) is Nd x4 , THEN x˙¯ 1 = −C x11 − C x12 − C x3 − C x4 − Cd x1 − Cd x2 − Cd x3 − Cd x4 ; IF x1 (t) is Px1 , x2 (t) is Px2 , THEN x˙¯ 2 = C x21 + C x22 ; · · · IF x1 (t) is N x1 , x2 (t) is N x2 , THEN x˙¯ 2 = −C x21 − C x22 ; IF x1 (t) is Px1 , x2 (t) is Px2 , x3 (t) is Px3 , x4 (t) is Px4 , THEN z(t) = Dx1 + Dx2 + Dx3 + Dx4 ; · · · IF x1 (t) is N x1 , x2 (t) is N x2 , x3 (t) is N x3 , x4 (t) is N x4 , THEN z(t) = −Dx1 − Dx2 − Dx3 − Dx4 . Here, we choose membership functions of Pxi , N xi Pd x1 and Nd x1 (i = 1, 2, 3, 4) as (2). Then, we have the following models: x(t) ˙ = A tanh(K x(t)) + Ad tanh(K x(t − h(t))) + Bu(t) + Bw w(t), z(t) = C tanh(K x(t)), where

⎡

C x11 C x12 C x3 C x4

⎢ 1 ⎢ C x1 C x12 C x3 C x4 A=⎢ ⎢ C2 C2 0 0 ⎣ x1 x2 2 2 C x1 C x2 0 0

(35) ⎤ ⎥ ⎥ ⎥, ⎥ ⎦

⎡

Cd x1 Cd x2 Cd x3 Cd x4

⎢C ⎢ d x1 Cd x2 Cd x3 Cd x4 Ad = ⎢ ⎣ 0 0 0 0 0 0 0 0

⎤ ⎥ ⎥ ⎥, ⎦

K = diag[k1 k2 k3 k4 ], C = [Dx1 Dx2 Dx3 Dx4 ]. We choose that h(t) = 0.5 sin(t) + 10 and the initial condition (t) = [−1 1]T , ∀t ∈ [−10.5, 0]. Without regard to u 1 (t), u 2 (t) and w1 (t), xi (t), x˙i (t) of (34) as input and output data are used for t ∈ [0, 5], the parameters of the

H. Zhang et al. / Fuzzy Sets and Systems 161 (2010) 1993 – 2011

2005

x1

2

x1

FHM

x1

0 −2

0

0.5

1

1.5

2

2.5 3 time (sec)

3.5

4

4.5

5

4.5

5

2 x2

x1

FHM

0 −2

x1

0

0.5

1

1.5

2

2.5 3 time (sec)

3.5

4

error

0.05

e1

0

e2

−0.05 −0.1 0

0.5

1

1.5

2

2.5 3 time (sec)

3.5

4

4.5

5

Fig. 1. The states of the original system, the states of the FHM and the errors between them.

FHM (34) can be obtained using neural network BP algorithm [26] as follows: ⎡ ⎤ ⎡ ⎤ 0.5150 0.9665 −4.1503 −4.1859 −1.5596 −0.4810 −1.6719 −0.2552 ⎢ 0.5150 0.9665 −4.1503 −4.1859 ⎥ ⎢ −1.5596 −0.4810 −1.6719 −0.2552 ⎥ ⎢ ⎥ ⎢ ⎥ A=⎢ ⎥ , Ad = ⎢ ⎥, ⎣ 7.2838 7.6633 ⎦ ⎣ ⎦ 0 0 0 0 0 0 7.2838 7.6633 0 0 0 0 0 0 K = diag[0.0691 0.0657 1.4254 1.4511], C = [−0.6530 − 0.6530 − 0.9637 − 0.9637]. The states of the original systems, the states of the FHM and the errors between them are given in Fig. 1. Other parameters are given as follows:  T −1 −1 0 0 , Bw = [1 0 0 0]T . B= 0 0 1 1 Consider the modeling errors, we assume M = [2.6666 0 0 0]T , N = [1 0 0 0], Nd = [0 0 0 0], Nb = [0 0 0 0]. Choosing  = 0.2, and utilizing Theorem 2, the feedback controller can be obtained in form (8) with   7.0160 7.0145 −0.2119 −0.2017 G= . −6.2456 −6.4803 −4.9003 −4.9006 Setting F(t) = sin t, and the disturbance w(t) = cos(t)e−0.1t , then the controller is applied to the original systems (34). The simulation results are shown in Figs. 2 and 3, which demonstrate the effectiveness of the proposed design method. If the K-independent controller is used like [14,23], then the gain of controller   88.1439 93.6386 0.2653 0.1562 G= . 5.4949 14.1345 −57.3263 −57.7661 Compared with the K-dependent controller, the gain of K-independent controller is bigger, which can illustrate that the present approach can reduce conservativeness mentioned in Remark 3.

2006

H. Zhang et al. / Fuzzy Sets and Systems 161 (2010) 1993 – 2011

States

0.5

x1 x2

0

−0.5 0

5

10

15

20 25 30 time (sec)

35

40

45

50

40

45

50

5

u

u1 u2 0

−5

0

5

10

15

20 25 30 time (sec)

35

Fig. 2. The trajectories of states and control inputs.

Output

1 z w

0.5 0 −0.5 −1 0

5

10

15

20 25 30 time (sec)

35

40

45

50

Output

6 ||z2|| r||w2||

4 2 0 0

5

10

15

20 25 30 time (sec)

35

40

45

50

Fig. 3. The trajectories of output and demonstration for z(t)2 < w(t)2 .

Furthermore, assume that x2 (t) ∈ [−1.5 1.5], by selecting the membership function as M11 (x2 (t)) = 1− x22 (t)/2.25, M12 (x2 (t)) = x22 (t)/2.25, (34) can be presented by the following T–S fuzzy models: IF x2 (t) is M11 , Then x(t) ˙ = ( A1 + A1 )x(t) + Ad1 x(t − h(t)) + B1 u(t) + Bw1 w(t), z(t) = C1 x(t); ˙ = (A2 + A2 )x(t) + Ad2 x(t − h(t)) + B2 u(t) + Bw2 w(t), z(t) = C2 x(t), IF x2 (t) is M12 , Then x(t) where       0.1 −0.02 −0.1125 −0.005 0.1 −1.527 A1 = , Ad1 = , A2 = , 1 0 0 0 1 0     −0.0125 −0.23 −1 0 Ad2 = , B1 = B2 = , 0 0 0 1   1 Bw1 = Bw2 = , A1 = H F(t)E x1 , A2 = H F(t)E x2 , H = [0.1 0]T , 0 E x1 = E x2 = [1 0],

F(t) = sin(t), C1 = C2 = [0.02 0.4].

H. Zhang et al. / Fuzzy Sets and Systems 161 (2010) 1993 – 2011

2007

1 States

0.5 0 xts1 xts2

−0.5 −1 −1.5 0

1

2

3

4

5 6 time (sec)

7

20

9

10

9

10

uts1 uts2

10 u

8

0 −10 −20 0

1

2

3

4

5

6

7

8

time (sec)

Fig. 4. The trajectories of states and control inputs based on T–S models.

Output

1

zts w

0.5 0 −0.5 −1 0

1

2

3

4

5

6

7

8

9

10

9

10

time (sec)

Output

30 20 ||z2||

10

r||w2||

0 0

1

2

3

4

5

6

7

8

time (sec)

Fig. 5. The trajectories of output and demonstration for z(t)2 < w(t)2 based on T–S models.

Choosing  = 0.5, we can obtain feasible solutions utilizing Theorem 2 in [24], and the response trajectories are shown in Figs. 4 and 5. As can be seen from the figures, the control inputs using T–S models are evidently bigger than that of using the FHM. Therefore, the control performances based on the FHM are better for the illustrative nonlinear systems. Choosing  = 1, we cannot obtain feasible solutions utilizing Theorem 2 in [2]. Example 2. Consider the following T–S fuzzy system: x(t) ˙ =

2 

i (Ai x(t) + Adi x(t − h(t)) + Bu(t)),

i=1

where



1 1 = 1 − 1 + exp(−3(2x2 − /2))



 ×

 1 , 2 = 1 −  1 . 1 + exp(−3(2x2 + /2))

(36)

2008

H. Zhang et al. / Fuzzy Sets and Systems 161 (2010) 1993 – 2011 8 6

x1 x2

States

4 2 0 −2 −4 −6 0

1

2

3

4

5 6 time (sec)

7

8

9

10

Fig. 6. The trajectories of states based on FHM in case 1 of Example 2.

(1) Case 1 [4]:   0 1 A1 = , 0.1 −2 B=

  0 1

 A2 =

 0 1 , 0 −0.5 − 0.015/

 Ad1 = Ad2 =

 0.1 0 , 0.1 −0.2

, h(t) = 26.

Here, the transformation of x is not used, i.e. wi = 1, di1 = 0, i = 1, 2, the FHM x˙ = (A + A) tanh(kx) + (Ad + Ad ) tanh(kx(t − h(t))) + Bu is used to model for (36). x and x˙ of (36) as input and output data are used for t ∈ [0, 50], using the similar modeling method, the parameters of the FHM can be obtained as follows:  A=

 −0.0150 0.9788 , 0.0641 −1.0466

 Ad =

 0.05885 0.0037 , 0.1977 −0.1762

  0 B= , 1

K = diag[2.2593, 1.8700].

The maximum identification errors are max |x1 − xˆ1 | = 0.2782(max |x1 | = 4.9759), max |x2 − xˆ2 | = 0.0382 (max |x2 | = 1), where xi denote the states of original system, xˆi denote the states of the FHM. To compensate the approximation error, we assume that A = H F(t)E A , Ad = H F(t)E d , H = [0.1 0]T , F(t) = sin(t), E A = E d = [1 0]. Then using Theorem 1 of this paper, we can obtain the controller gain G = [−36.6963 − 0.9774], which is applied to the original systems (36) and the simulation results are shown in Fig. 6. [4] presented new delay-dependent stabilization conditions of T–S fuzzy systems, and the condition is less conservative compared with the literatures therein. The upper delay bound in [4] is h max = 25.7865, which is bigger than the results of literatures therein. That is to say, using the FHM can allow the bigger upper delay bound h max than using T–S models in the illustrative example. (2) Case 2:         0 1 0 0 0 1 0 0 A1 = , Ad1 = , A2 = , Ad2 = , −1.4536 −0.7767 1.5086 0 −1 −77.67 1.5483 0   1 B= , h(t) = 2. 0

H. Zhang et al. / Fuzzy Sets and Systems 161 (2010) 1993 – 2011

2009

1 0.8 0.6

x1 x2

States

0.4 0.2 0 −0.2 −0.4 −0.6 −0.8 −1 0

1

2

3

4

5

6

7

8

9

10

time (sec)

Fig. 7. The trajectories of states based on FHM in case 2 of Example 2.

Table 1 The comparisons about the stability region, number of variables and time-delay (Example 3). Paper

Stability region for a ∈ [5, 10], b ∈ [5, 10], step = 0.2

Number of variables

Time-delay h(t)

[25] [21] Ours (Theorem 1)

– Part region (The detailed region is omitted here.) All region

5 10 5

h(t) is constant ¯ h(t) ˙ ≤ 0 < h(t)≤h, ¯ |h(t)| ˙ 0 < h(t)≤h, ≤

We cannot obtain the feasible solutions using Theorem 1 of [4]. Using the similar modeling method, the parameters of the FHM are     0.0304 0.0052 −0.0080 −0.0284 A= , Ad = , K = diag[0.2127, 0.4084]. −6.0063 −156.0680 8.2081 0.6301 The maximum identification errors are max |x1 − xˆ1 | = 0.0581 (max |x1 | = 1.3166), max |x2 − xˆ2 | = 0.0099 (max |x2 | = 1), where xi denote the states of original system, xˆi denote the states of the FHM. To compensate the approximation error, A and Ad are chosen as case 1. Using Theorem 1 of this paper, we can obtain the controller gain G = [−284.2323 12.0589], which is applied to (36) and the simulation results are shown in Fig. 7. Example 3. Consider the following FHM: x(t) ˙ = ( A + A) tanh(K x(t)) + ( Ad + Ad ) tanh(K x(t − h(t))) + Bu(t),

(37)

where  A=

 b 0.3643 , 1.2616 0

 Ad =

 −0.8583 −11.5762 , 0 0

 K =

 a 0 , 0 1

 B=

 0.4 0 , 0.1 −0.3

h(t) = 0.5 sin t + 2. A and Ad are chosen as case 1 of Example 2. In Table 1, the comparisons are given. It can be seen that the proposed stability conditions in Theorem 1 offer a larger stability region than [21,25], and the number of variables need to be solved is less than [21]. Therefore proposed conditions are new and less conservative compared with [21,25].

2010

H. Zhang et al. / Fuzzy Sets and Systems 161 (2010) 1993 – 2011

5. Conclusions In this paper, the delay-dependent robust H∞ FHM-based control problem has been considered for a class of uncertain nonlinear systems with time-varying delay. The FHM modeling method has been given for uncertain nonlinear systems with time-varying delay, then the new delay-dependent sufficient condition for the existence of a state feedback fuzzy hyperbolic tangent controller has been proposed in terms of linear matrix inequalities (LMIs) by constructing a novel Lyapunov–Krasovskii functional. The FHM-based control law can guarantee that the closed-loop system is robustly asymptotically stable with a prescribed H∞ performance index. Three illustrative examples have been given to show the effectiveness of using FHM-based controllers. Acknowledgements This work was supported by the National Natural Science Foundation of China (50977008, 60521003, 60774048), the Program for Cheung Kong Scholars, the National High Technology Research and Development Program of China (2009AA04Z127), National Basic Research Program of China (2009CB320601). References [1] Y.Y. Cao, Y.X. Sun, C.W. Cheng, Delay-dependent robust stabilization of uncertain systems with multiple state delays, IEEE Transactions on Automatic Control 43 (11) (1998) 1608–1612. [2] Y.Y. Cao, Y.X. Sun, J. Lam, Delayed-independent robust H∞ control for uncertain systems with time-varying delays, IEE Proceedings—Control Theory & Applications 145 (3) (1998) 338–343. [3] B. Chen, X.P. Liu, C. Lin, K.F. Liu, Robust H∞ control of Takagi–Sugeno fuzzy systems with state and input time delays, Fuzzy Sets and Systems 160 (4) (2009) 403–422. [4] B. Chen, X.P. Liu, S.C. Tong, New delay-dependent stabilization conditions of T–S fuzzy systems with constant delay, Fuzzy Sets and Systems 158 (2007) 2209–2224. [5] S.H. Esfahani, I.R. Petersen, An LMI approach to output-feedback guaranteed cost control for uncertain time-delay systems, International Journal of Robust and Nonlinear Control 10 (2) (2000) 157–174. [6] E. Fridman, U. Shaked, A descriptor system approach to H∞ control of linear time-delay systems, IEEE Transactions on Automatic Control 47 (2) (2002) 253–270. [7] E. Fridman, U. Shaked, An improved stabilization method for linear time-delay systems, IEEE Transactions on Automatic Control 47 (11) (2002) 1931–1937. [8] X.P. Guan, C.L. Chen, Delay-dependent guaranteed cost control for T–S fuzzy systems with time delays, IEEE Transactions on Fuzzy Systems 12 (2) (2004) 236–249. [9] J.B. He, Q.G. Wang, T.H. Lee, H∞ disturbance attenuation for state delayed systems, Systems & Control Letters 33 (2) (1998) 105–114. [10] H. Ito, Z.P. Jiang, Robust disturbance attenuation of nonlinear systems using output feedback and state-dependent scaling, Automatica 40 (9) (2004) 1621–1628. [11] X. Jang, Q.L. Han, Robust H∞ control for uncertain T–S fuzzy systems with interval time-varying delay, IEEE Transactions on Fuzzy Systems 15 (2) (2007) 321–331. [12] V. Kapila, W.M. Haddad, Memoryless H∞ controller for discrete time systems with time delay, Automatica 34 (9) (1998) 1141–1144. [13] C. Lin, Q.G. Wang, T.H. Lee, Delay-dependent LMI conditions for stability and stabilization of T–S fuzzy systems with bounded time-delay, Fuzzy Sets and Systems 157 (9) (2006) 1229–1247. [14] S.X. Lun, H.G. Zhang, Delayed-independent fuzzy hyperbolic guaranteed cost control design for a class of nonlinear continuous-time systems with uncertainties, Acta Automation Sinica 31 (5) (2005) 720–726. [15] M. Margaliot, G. Langholz, A new approach to fuzzy modeling and control of discrete-time systems, IEEE Transactions on Fuzzy Systems 11 (4) (2003) 486–494. [16] J.H. Park, Robust stabilization for dynamic systems with multiple time-varying delays and nonlinear uncertainties, Journal of Optimization Theory and Applications 108 (1) (2001) 155–174. [17] C.E.D. Souza, X. Li, Delay-dependent robust H∞ control of uncertain linear state-delayed systems, Automatica 35 (7) (1999) 1313–1321. [18] V. Suplin, E. Fridman, U. Shaked, H∞ control of linear uncertain time-delay systems-a projection approach, IEEE Transactions on Automatic Control 51 (4) (2006) 680–685. [19] K. Tanaka, H.O. Wang, Fuzzy Control Systems Design and Analysis: A Linear Matrix Inequality Approach, Wiley, New York, 2001. [20] Y. Wang, L.H. Xie, C.E.D. Souza, Robust control of a class of uncertain nonlinear systems, Systems & Control Letters 19 (2) (1992) 139–149. [21] Y.Z. Wang, H.G. Zhang, X.R. Liu, Robust H∞ control based on fuzzy hyperbolic model with time-delay, Progress in Natural Science 18 (2008) 1429–1435. [22] S.Y. Xu, J. Lam, Y. Zou, New results on delay-dependent robust H∞ control for systems with time-varying delays, Automatica 42 (2) (2006) 343–348. [23] J. Yang, D.R. Liu, J. Feng, et al., Controller design for a class of nonlinear systems based on fuzzy hyperbolic model, in: Proc. of the 6th World Congr. on Intelligent Control and Automation, 2006, pp. 873–877.

H. Zhang et al. / Fuzzy Sets and Systems 161 (2010) 1993 – 2011

2011

[24] Y.P. Yang, Y.G. Kao, C.C. Gao, Fuzzy Robust H∞ control for uncertain nonlinear systems with multiple delays, in: Proc. of IEEE Internat. Conf. on Control and Automation, 2007, pp. 1101–1105. [25] H.G. Zhang, Q.X. Gong, Y.C. Wang, Delay-dependent robust H∞ control for uncertain fuzzy hyperbolic systems with multiple delays, Progress in Natural Science 18 (2008) 97–104. [26] H.G. Zhang, X.Q. He, Fuzzy Adaptive Control Theory and its Application, Beijing Aeronautics and Astronomics University Press, Beijing, 2002 pp. 127–132 (in Chinese). [27] H.G. Zhang, D.R. Liu, Fuzzy Modeling and Fuzzy Control, Birkhauser, Boston, Basel, Berlin, 2006. [28] H.G. Zhang, Y.B. Quan, Modeling and control based on fuzzy hyperbolic model, Acta Automatica Sinica 26 (6) (2000) 729–735 (in Chinese). [29] H.G. Zhang, Y.B. Quan, Modeling, identification and control of a class of nonlinear systems, IEEE Transactions on Fuzzy Systems 49 (2) (2001) 349–354.