Delay-Dependent Generalized H2 Control for Discrete- Time Fuzzy Systems with Infinite-Distributed Delays

In recent years, there has been significant interest in the study of stability analysis and con‐ troller synthesis for Takagi-Sugeno(T-S) fuzzy systems, which has been used to approximate certain complex nonlinear systems [1]. Hence it is important to study their stability analysis and controller synthesis. A rich body of literature has appeared on the stability analysis and synthesis problems for T-S fuzzy systems [2-6]. However, these results rely on the existence of a common quadratic Lyapunov function (CQLF) for all the local models. In fact, such a CQLF might not exist for many fuzzy systems, especially for highly nonlinear complex sys‐ tems. Therefore, stability analysis and controller synthesis based on CQLF tend to be more conservative. At the same time, a number of methods based on piecewise quadratic Lyapu‐ nov function (PQLF) for T-S fuzzy systems have been proposed in [7-14]. The basic idea of these methods is to design a controller for each local model and to construct a global piece‐ wise controller from closed-loop fuzzy control system is established with a PQLF. The au‐ thors in [7,13] considered the information of membership function, a novel piecewise continuous quadratic Lyapunov function method has been proposed for stability analysis of T-S fuzzy systems. It is shown that the PQLF is a much richer class of Lyapunov function candidates than CQLF, it is able to deal with a large class of fuzzy systems and obtained re‐ sults are less conservative.


Introduction
In recent years, there has been significant interest in the study of stability analysis and controller synthesis for Takagi-Sugeno(T-S) fuzzy systems, which has been used to approximate certain complex nonlinear systems [1].Hence it is important to study their stability analysis and controller synthesis.A rich body of literature has appeared on the stability analysis and synthesis problems for T-S fuzzy systems [2][3][4][5][6].However, these results rely on the existence of a common quadratic Lyapunov function (CQLF) for all the local models.In fact, such a CQLF might not exist for many fuzzy systems, especially for highly nonlinear complex systems.Therefore, stability analysis and controller synthesis based on CQLF tend to be more conservative.At the same time, a number of methods based on piecewise quadratic Lyapunov function (PQLF) for T-S fuzzy systems have been proposed in [7][8][9][10][11][12][13][14].The basic idea of these methods is to design a controller for each local model and to construct a global piecewise controller from closed-loop fuzzy control system is established with a PQLF.The authors in [7,13] considered the information of membership function, a novel piecewise continuous quadratic Lyapunov function method has been proposed for stability analysis of T-S fuzzy systems.It is shown that the PQLF is a much richer class of Lyapunov function candidates than CQLF, it is able to deal with a large class of fuzzy systems and obtained results are less conservative.
On the other hand, it is well known that time delay is a main source of instability and bad performance of the dynamic systems.Recently, a number of important analysis and synthesis results have been derived for T-S fuzzy delay systems [4-7, 11, 13].However, it should be pointed out that most of the time-delay results for T-S fuzzy systems are constant delay or time-varying delay [4-5, 7, 11, and 13].In fact, Distributed delay occurs very often in reality and it has been drawing increasing attention.However, almost all existing works on distributed delays have focused on continuous-time systems that are described in the form of either finite or infinite integral and delay-independent.It is well known that the discrete-time system is in a better position to model digitally transmitted signals in a dynamic way than its continuous-time analogue.Generalized H 2 control is an important branch of modern control theories, it is useful for handling stochastic aspects such as measurement noise and random disturbances [10].Therefore, it becomes desirable to study the generalized H 2 control problem for the discrete-time systems with distributed delays.The authors in [6] have derived the delay-independent robust H ∞ stability criteria for discrete-time T-S fuzzy systems with infinite-distributed delays.Recently, many robust fuzzy control strategies have been proposed a class of nonlinear discrete-time systems with time-varying delay and disturbance [15][16][17][18][19][20][21][22][23][24][25][26][27][28][29][30][31][32][33].These results rely on the existence CLKF for all local models, which lead to be conservative.It is observed, based on the PLKF, the delay-dependent generalized H 2 control problem for discrete-time T-S fuzzy systems with infinite-distributed delays has not been addressed yet and remains to be challenging.Motivated by the above concerns, this paper deals with the generalized H 2 control problem for a class of discrete time T-S fuzzy systems with infinite-distributed delays.Based on the proposed Delay-dependent PLKF(DDPLKF), the stabilization condition and controller design method are derived for discrete time T-S fuzzy systems with infinite-distributed delays.It is shown that the control laws can be obtained by solving a set of LMIs.
A simulation example is presented to illustrate the effectiveness of the proposed design procedures.

Notation:
The superscript "T" stands for matrix transposition, R n denotes the n-dimensional Euclidean space, R n×m is the set of all n×m real matrices, I is an identity matrix, the notation P>0(P≥0) means that P is symmetric and positive(nonnegative) definite, diag{…} stands for a block diagonal matrix.Z -denotes the set of negative integers.For symmetric block matrices, the notation * is used as an ellipsis for the terms that are induced by symmetry.In addition, matrices, if not explicitly stated, are assumed to have compatible dimensions.

Problem Formulation
The following discrete-time T-S fuzzy dynamic systems with infinite-distributed delays [6] can be used to represent a class of complex nonlinear time-delay systems with both local analytic linear models and fuzzy inference rules: : Remark 1.The delay term ∑ d =1 +∞ μ d x(t − d ) in the fuzzy system (1), is the so-called infinitely distributed delay in the discrete-time setting.The description of the discrete-time-distributed delays has been firstly proposed in the [6], and we aim to study the generalized H 2 control problem for discrete-time fuzzy systems with such kind of distributed delays in this paper, which is different from one in [6].
Remark 2. In this paper, similar to the convergence restriction on the delay kernels of infinite-distributed delays for continuous-time systems, the constants μ d (d =1,2, …)are assumed to satisfy the convergence condition (2), which can guarantee the convergence of the terms of infinite delays as well as the DDPLKF defined later.
By using a standard fuzzy inference method, that is using a center-average defuzzifiers product fuzzy inference, and singleton fuzzifier, the dynamic fuzzy model (1) can be expressed by the following global model: where h j (s(t)) = ω j (s(t)) , ω j (s(t)) = ∏ i=1 g F ji (s(t)), with F ji (s(t)) being the grade of membership of s i (t)inF ij ,ω j (s(t)) ≥ 0 has the following basic property: and therefore In order to facilitate the design of less conservative H 2 controller, we partition the premise variable space Ω ⊆ R s into m polyhedral regions Ω i by the boundaries [7] where v is the set of the face indexes of the polyhedral hull with satisfying Based on the boundaries (6), m independent polyhedral regionsΩ l , l ∈ L = {1,2 ⋯ m} can be obtained satisfying , , , where L denotes the set of polyhedral region indexes.
In each region Ω l , we define the set Considering ( 5) and ( 8), in each region Ω l , we have and then, the fuzzy infinite-distributed delays system (1) can be expressed as follows: Advances in Discrete Time Systems Remark 3.According to the definition of ( 8), the polyhedral regions can be divided into two folds: operating and interpolation regions.For an operating region, the set M(l) contains only one element, and then, the system dynamic is governed by the s-th local model of the fuzzy system.For an interpolation region, the system dynamic is governed by a convex combination of several local models.
In this paper, we consider the generalized H 2 controller design problem for the fuzzy system (1) or equivalently (10), give the following assumptions.
Assumption 1.When the state of the system transits from the region Ω l to Ω j at the time t, the dynamics of the system is governed by the dynamics of the region model of Ω l at that time t.
For future use, we define a setΘthat represents all possible transitions from one region to itself or another regions, that is , } Here l = j, when the system stays in the same region Ω l , and l ≠ j, when the system transits from the region Ω l to another one Ω j .
Considering the fuzzy system (10), choose the following non-fragile piecewise state feedback controller here ΔK l are unknown real matrix functions representing time varying parametric uncertainties, which are assumed to be of the form ( ) , ( ) ( ) , ( ) where E l , H l are known constant matrices, and U l (t) ∈ R l 1 ×l 2 are unknown real time varying matrix satisfyingΔU l T (t)ΔU l ≤ I .
Then, the closed-loop T-S system is governed by Before formulation the problem to be investigated, we first introduce the following concept for the system (14).
Definition 1. [10] Let a constant γ > 0 be given.The closed-loop fuzzy system ( 14) is said to be stable with generalized H 2 performance if both of the following conditions are satisfied: • The disturbance-free fuzzy system is globally asymptotically stable.
• Subject to assumption of zero initial conditions, the controlled output satisfies for all non-zero v ∈ I 2 .Now, we introduce the following lemmas that will be used in the development of our main result.
Lemma 1. [6] Let M ∈ R n×n be a positive semi-definite matrix, x i (t) ∈ R n and constant a i > 0(i = 1, 2, ⋯ ), if the series concerned is convergent, then we have Lemma 2. [14] For the real matrices P 1 , P 2 , P 3 , P 4 , A, A d , B, X j ( j = 1, ⋯ ,5) and ) with compatible dimensions, the inequalities show in ( 17) and ( 18) at the following are equivalent, where U is an extra slack nonsingular matrix.
Advances in Discrete Time Systems

He U P U A P U A
where He { * } stands for * + * T .

Main Results
Based on the proposed partition method, the following DDPLKF is proposed to develop the stability condition for the closed-loop system of ( 14).
The following theorem shows that the desired controller parameters and considered controller uncertain can be determined based on the results of Theorem 1.This can be easily proved along the lines of Theorem 1, and we, therefore, only keep necessary details in order to avoid unnecessary duplication.
Theorem 2. Consider the uncertain terms (12).Given a constantγ > 0, the closed-loop fuzzy system ( 14) with infinite-distributed delays is stable with generalized H 2 performanceγ, if there exists a set of positive definite matricesP l , Q, Z > 0, the nonsingular matrix F and matrices X li , Y li , M l , l ∈ L , i = 1,2,3,4satisfying the following LMIs: where Furthermore, the control law is given by Proof.In ( 20) and ( 21), replace K l ¯ withK l + Δ K l , and then by S-procedure, we can easily obtain the results of this theorem, and the details are thus omitted.Corollary 1.Consider the uncertain terms (12).Given a constantγ > 0, the closed-loop fuzzy system ( 14) with infinite-distributed delays is stable with generalized H 2 performanceγ, if there exists a set of positive definite matricesP l , Q, Z > 0, the nonsingular matrix F and matrices X li , Y li , M l , l ∈ L , i = 1,2,3,4satisfying the following LMIs: where

Numerical Examples
In this section, we will present two simulation examples to illustrate the controller design method developed in this paper.
Example 1.Consider the following modified Henon system with infinite distributed delays and external disturbance where the constant c ∈ 0,1 is the retarded coefficient.
Assume thats(t) ∈ − 1,1 .The nonlinear term s 2 (t) can be exactly represented as where theh 1 (s(t)), h 2 (s(t)) ∈ 0,1 , andh 1 (s(t)) + h 2 (s(t)) = 1.By solving the equations, the membership functions h 1 (s(t))and h 2 (s(t))are obtained as It can be seen from the aforementioned expressions that h 1 (s(t)) = 1 and h 2 (s(t)) = 0 when s(t) = − 1, and that h 1 (s(t)) = 0 and h 2 (s(t)) = 1 whens(t) = 1.Then the nonlinear system in (48) can be approximately represented by the following T-S fuzzy model: where The subspaces can be described by Simulation results with the above solutions for the H 2 controller designs are shown Fig. 1 and Fig.We expanded the state space from [-1,1] to [− 3,3], the membership functions are given as   Using the Theorem 2 and Corollary 1, respectively, the achievable minimum performance index for the H 2 controller can be obtained and is summarized in Table 1.

Conclusions
This paper presents delay-dependent analysis and synthesis method for discrete-time T-S fuzzy systems with infinite-distributed delays.Based on a novel DDPLKF, the proposed stability and stabilization results are less conservative than the existing results based on the CLKF and delay independent method.The non-fragile stated feedback controller law has been developed so that the closed-loop fuzzy system is generalized H 2 stable.It is also shown that the controller gains can be determined by solving a set of LMIs.A simulation example was presented to demonstrate the advantages of the proposed approach.

Remark 4 .
If the global state space replace the transitionsΘand allP l s in Theorem 2 become a commonP, Theorem 2 is regressed to Corollary 1, shown in the following.

Delay-Dependent Generalized H 2 Figure 1 .
Figure 1.The state evolution x 1 (t) of controlled system.

Figure 3 .
Figure 3. Membership functions and partition of subspaces.