Stability Analysis for the Discrete-Time T-S Fuzzy System with Stochastic Disturbance and State Delay

In this paper, we study the stability for discrete-time Takagi and Sugeno (T-S) fuzzy systems with perturbation disturbance and time delay. Stochastic delay-dependent stability criteria are derived for stochastic T-S fuzzy systems with time-invariant and time-varying delays, respectively. For the time-varying delay case, a novel fuzzy Lyapunov–Krasovskii functional (LKF) without requiring all the involved symmetric matrices to be positive definite is constructed to reduce the conservatism. These stability conditions are then represented in terms of finite linear matrix inequalities (LMIs), which can be solved efficiently by using standard LMIS optimisation techniques. Two numerical examples are given to illustrate the feasibility of the proposed method.


Introduction
In 1985, Takagi and Sugeno first proposed the method to use fuzzy systems to approximate nonlinear systems [1]. Since then, T-S fuzzy systems have attracted great attention of a wide range of scholars because they can provide effective measures for the control of nonlinear systems. Rich results are presented in [2][3][4][5]; H ∞ control designs are studied in [5][6][7][8]; fault detection has been investigated in [9,10] and sliding mode control based on fuzzy model can be found in [11,12].
The traditional T-S fuzzy dynamic mode is described by a family of fuzzy IF-THEN rules that represent local linear input-output relations of a linear system. While the good local linearity may be broken by the stochastic noise. Stochastic noise is an ideal signal to simulate irregular internal and external interference. Around the problems of stochastic systems, lots of efficient approaches have been proposed by scholars. Asynchronous output feedback control based on the stochastic T-S fuzzy model is presented in [13]. Su et al. [14] studies H ∞ model reduction of T-S fuzzy stochastic systems. For more results about the stochastic T-S fuzzy system, one may refer to [15,16] and the reference therein.
As is well-known, time delay may cause the instability of the systems. Stability results can be classified into two types: delay-independent stability and delay-dependent stability. Most researchers concentrate on studying delay-dependent stability because of its less conservative property. Various approaches have proposed for presenting the stability conditions of the discrete systems with time delay, see in [17][18][19][20][21] and the references therein. Although it is feasible to use a full Lyapunov matrix to analyse the stability of the discrete time-delay systems, the computational complexity caused by this method is high. It makes the use of Lyapunov-Krasovskii functional has become popular, which provides an effective way in obtaining delay-dependent stability results for the discrete time-delay systems. In the case of time-varying delays, delay-dependent stability conditions of discrete-time T-S fuzzy systems with stochastic disturbance are given in [22], which is derived by the construction of a fuzzy Lyapunov-Krasovskii functional. It is worth noting that they need requiring all the involved symmetric matrices in a chosen fuzzy Lyapunov-Krasovskii functional to be positive definite. Such a requirement can lead to the conservatism in the stability criteria.
Motivation by the aforementioned analysis, we intend to investigate the stochastic stability for discrete-time Takagi and Sugeno (T-S) fuzzy systems with stochastic perturbation and time delay. Stochastic delay-dependent stability criteria are derived for stochastic T-S fuzzy systems with time-invariant and time-varying delays, respectively. In this paper, we mainly analyse the stability of the discrete T-S fuzzy systems with stochastic disturbance and time-varying delays. Different fuzzy Lyapunov-Krasovskii functions are constructed for constant time delay and time-varying delay, respectively.
For the time-varying delay case, a novel fuzzy Lyapunov-Krasovskii functional (LKF) without requiring all the involved symmetric matrices to be positive definite is constructed to reduce the conservatism. These stability conditions are then represented in terms of finite linear matrix inequalities (LMIs), which can be solved efficiently by using standard LMIS optimisation techniques. Finally, two numerical examples are given to illustrate the feasibility of the proposed method.
The remaining parts of this paper are organised as follows. In the second part, the formation and preparation of discrete-time T-S fuzzy systems with stochastic disturbance and time-varying delays are introduced. The delay-dependent stability analysis is given in the section there. In the fourth part, we provide some simulation results to verify the effectiveness of the method. The last section draws the conclusion of this paper.
Notation: The notations that are used throughout this paper are fairly standard. The superscript 'T' stands for matrix transposition; R n denotes the n-dimensional Euclidean space; the notation P > 0(≥ 0) means that P is real symmetric and positive definite (semidefinite); and R m×n is the set of all real matrices of dimension m × n; and in symmetric block matrices or long matrix expressions, we use an asterisk ( * ) to represent a term that is induced by symmetry; diag{· · · } stands for a block-diagonal matrix; Matrices, if their dimensions are not explicitly stated, are assumed to be compatible for algebraic operations.

Problem Formulation and Preliminaries
Consider a discrete nonlinear time-delay system that is represented by the following T-S fuzzy stochastic model with delay: Plant rule i: where i ∈ S {1, 2, . . . , r}, r is the number of IF-THEN rules, M ij is the fuzzy set, θ(k) = [θ 1 (k), θ 2 (k), . . . , θ g (k)] are the premise variables, x(k) ∈ R n is the state vector, ω(k) is a 1-D, zero mean Gaussian white noise sequence on a probability space ( , F, P) with E{ω(k)} = 0; and with M ij (θ j (k)) representing the grade of membership of θ j (k) in M ij . For simplicity, we will replace λ i (θ (k)) by λ i in some places. By definition, the fuzzy basis functions satisfy λ i ≥ 0(i ∈ S) and r i=1 λ i = 1. Then, the defuzzified output of the T-S fuzzy system (1) can be represented as Before proceeding, the following lemmas will be used to derive our main results.

Lemma 2.1 ([23]):
Assume that a ∈ R n a , b ∈ R n b , and N ∈ R n a ×n b . Then for any matrices X ∈ R n a ×n a , Y ∈ R n a ×n b , and Z ∈ R n b ×n b satisfying X Y Y T Z ≥ 0, the following inequality holds, i.e.

Lemma 2.3 ([4]): There exists a matrix X such that
if and only if there is q 12 ∈ R such that p 11 q 12 q 12 p 22 < 0, p 12 ≤ q 12 .
A sufficient condition for (7) and (8) is

Delay-Dependent Stability Analysis
In this section, we analyse the stability for the fuzzy time-delay system. Firstly, we analyse the case of constant time delay. The system of (3) can be transformed into the following compact form: where The following result on the bounding of cross products of vectors will be used in the proof of Theorem 3.1.
Denote η(l) x(l + 1) − x(l). Then for the fuzzy time-delay system (10), we have For convenience of notations, we make the following definitions: where are n × n, symmetric, and positive definite, Y i , G i , and F i , i ∈ S, are n × n. Then we have the following theorem.
such that the following inequalities hold: where Proof: From the facts we have Thus, it follows from (13) and (14) that ⎡ Premultiplying and postmultiplying (18) by T 1 and T T 1 , respectively, (19) by T 1 and T T 1 , respectively, (15) by T 2 and T T 2 , respectively, and considering (12), we have ⎡ where 11 By the Schur complement theorem, it follows that (20) and (21) are equivalent to Summing up (23) and (24), we have Substituting (11) into (10), we obtain where Then, we construct the fuzzy LKF where Along the trajectory of system (26) and taking expectation, we have where By using Lemma 2.1, we have with X(k), Y(k),and Z(l) satisfying (22). Considering (29) and (30), we have
The inequalities we got in Theorem 3.1 contain time-delay parameters. Those parameters are merely available online, therefore, it is impossible for us to check the feasibility of those inequalities. We need to transform those PLMIs [24] into strict LMIs, and then check their feasibility by computer software. Thus, we restrict ourselves to the case of Then, we have the following theorem. (10) is stochastically stable if for some scalar ε > 0, there exist matrices P i > 0, Q i > 0, X i > 0, Z i > 0, Y i , and G i , i ∈ S, satisfying the LMIs:

Theorem 3.2: The system in
Proof: Note that the matrices in inequality (13) of Theorem 3.1 can be unfolded as So we just need to satisfy and thus a sufficient condition for (46) is  (15) is satisfied. Therefore, it follows from Theorem 3.1 that the time-delay fuzzy system (10) is stochastically stable.
It is noted that we can reduce the number of LMIs by selecting a specific matrix. For example, if we take P i = P, Q i = Q, X i = X, Z i = Z, Y i = Y, and G i = G, i ∈ S, then the number would be greatly reduced. In this case, the fuzzy LKF (27) becomes a non-fuzzy one. Then, we can get a corollary as follows. (10) is stochastically stable if for some scalar ε > 0, there exist matrices P > 0, Q > 0, X > 0, Z > 0, Y, and G, satisfying the LMIs:

Corollary 3.3: The system in
Proof: The result in this corollary is a special case of Theorem 3.2; therefore, we omit the proof here.
Next, we will analyse the time-varying delay, we give the open-loop system of (3) in a compact form where

Theorem 3.4:
The system in (51) is stochastically stable if there exist matrices P = P T , Proof: Define a Lyapunov functional as where Under the condition of the theorem, we first show that there exists a scalar δ 1 > 0, such that For this purpose, we note that and Furthermore, Q 3 > 0 and h 1 ≤ h(k) ≤ h 2 together imply that Thus, we have Applying Lemma 2.2 and using the relations in (57), we obtain so, we have Then, it follows from (59)-(62) that This, together with (53) and (54), imply that there exists a scalar δ 1 > 0, such that (58) holds. Now, we show that there exists a scalar δ 2 > 0, such that We have By Lemma 2.2, one may have Next, we introduce several slack matrices to further reduce conservatism. According to the definition of x(i), for any matrices where Note that where Then it is derived from (65)-(72) that Rewriting On the other hand, by Lemma 2.3, there exists an X of appropriate dimensions such that (52) holds if and only if According to the Schur complement theorem, the system (75) is equivalent to Therefore, if the condition (52) is satisfied, so does the condition (76). By (74), there exists a scalar δ 2 > 0 such that V(k) ≤ −δ 2 x(k) 2 < 0 for x(k) = 0, which is concluded that the system in (51) is stochastically stable.

Illustrative Examples
In this section, two examples are employed to illustrate the method developed in Sections 3. Example 4.1 shows that this method is effective and the stability condition based on the new fuzzy LKF is less conservative than that based on non-fuzzy LKF. Example 4.2 shows that this method reduces the conservatism.

Example 4.1 ([22]):
Consider the discrete-time delay fuzzy system (10) with r = 2, τ = 1, ε = 50, However, we can prove that the LMIs of Corollary 3.3 are infeasible. This shows that with ε = 50, Theorem 3.2 guarantees the stability of the system while Corollary 3.3 cannot. As we expected, this method is effective and the proposed fuzzy LKF-based stability condition is less conservative than the non-fuzzy LKF-based one.

Conclusion
The stability of Discrete T-S fuzzy stochastic system with time delay is studied. The fuzzy stochastic turbulence considered in the new system has broadened the applications in the more complicated irregular internal and external interference cases. The symmetric matrices involved in the novel Lyapunov-Krasovskii functional get rid of the positive definiteness restrictions. Numerical experiments show that the stability condition, obtained by this new Lyapunov-Krasovskii functional, is less conservative.

Disclosure statement
No potential conflict of interest was reported by the author(s).

Notes on contributors
Honglin Luo, is a Math professor, the research interests include theory and algorithms of optimization, optimal control and fuzzy control.
Jiali Zheng, is a master degree candidate. Her research interest is in fuzzy control.