Nonfragile Robust H ∞ Filter Design for a Class of Fuzzy Stochastic Systems with Stochastic Input-to-State Stability

The nonfragileH ∞ filtering problem for a kind of Takagi-Sugeno (T-S) fuzzy stochastic system which has a time-varying delay and parameter uncertainties has been studied in this paper. Sufficient conditions for stochastic input-to-state stability (SISS) of the fuzzy stochastic systems are obtained. Attention is focused on the design of a nonfragileH ∞ filter such that the filtering error system can tolerate some level of the gain variations in the filter and theH ∞ performance level also could be satisfied. By using the SISS result, the approach to design the nonfragile filter is proposed in terms of linear matrix inequalities. Finally, an illustrative example is given to demonstrate the effectiveness of the proposed method.


Introduction
As the performance of a control system is affected by parameter perturbations, exogenous disturbances, measurement errors, and other uncertainties, the research of the robust control problem has had a vital status in the studies of control theory.Considering Lyapunov stability theory which is not suitable for analyzing and processing the state responses of the system with perturbations, some new methods have been developed, such as input-to-state stability (ISS).Since Sontag presented the qualitative aspect of ISS for the system response to input with bounded magnitude in 1989 [1], ISS has become an essential concept in modern controller and filter design for the nonlinear system.The ISS problem has been extensively investigated by many authors [2][3][4][5][6] until now.At the same time, ISS has been studied for stochastic systems.Stochastic input-to-state stability (SISS) of Lure distributed parameter control system has been investigated in [7], and sufficient conditions for SISS in Hilbert space have been presented in terms of linear operator inequalities.In [8], SISS and the  ∞ filtering problem have been considered, and the filter has been designed in LMIs.A mean-square exponential ISS problem for stochastic delay neural networks has been investigated in [9].
On the other hand, fuzzy model could turn the nonlinear models into a linear representation by partitioning the original dynamic differential equations into linear ones [10].T-S fuzzy model [11] has been considered as an efficient technique to linearize the nonlinear systems.This model has been first put forward in the truck trailer system [12].And another typical application is in the stirred tank reactor system which has been addressed in [13].Until now, there have been a lot of results of T-S fuzzy system reported in literature.The stability and control problem have been investigated in [14][15][16][17][18][19] and the references therein.
Meanwhile, it is well known that state estimation can estimate the unavailable state variables or their linear combination for a given system [20,21], and it has been found in many practical applications over decades.As a branch of state estimation theory,  ∞ filter can process the estimation problem without exact statistical data for the external noise.This problem for the T-S fuzzy system has been addressed in [22][23][24][25][26][27]; and the robust filters for stochastic systems are designed in [28,29].During the filter design, gain perturbations are usually unavoidable.According to [30], those gain perturbations could destabilize the filtering error system even if they are very small, which makes the filter fragile.Hence, it is reasonable to design a filter that could tolerate some level of the gain variations, which is called nonfragile filter.The nonfragile filter has received considerable attention over the past two decades; refer to [31][32][33][34] and the references therein.From what is mentioned above, it is worth noting that T-S fuzzy model can be used to divide the nonlinear stochastic systems into several subsystems.The solution to fuzzy stochastic differential equations with local martingales has been presented in [35].The work in [36] has considered the robust fault detection problem for T-S fuzzy stochastic systems.And the stabilization for the fuzzy stochastic systems with delays has been investigated in [37][38][39].The control problem has been considered in [40][41][42][43][44][45].
Motivated by the above discussion, this paper will focus on the filter design for the fuzzy stochastic system, where few results have been found.The nonfragile fuzzy delaydependent  ∞ filter design for a T-S time-delay fuzzy stochastic system with norm-bounded parameter uncertainties is studied in this paper.The Lyapunov-Krasovskii functional technique is used and the sufficient conditions obtained are expressed in terms of linear matrix inequality (LMI) approach.This paper is organized as follows.Section 2 presents the problem formulation and preliminaries.Section 3 gives main results for the nonfragile filter design.In Section 4, a numerical example is shown to illustrate the effectiveness of the proposed methods.Section 5 concludes the paper.
Notation.The notation used in this paper is fairly standard.The superscript "" stands for matrix transposition.Throughout this paper, for real symmetric matrices  and , the notation  ≥  (resp.,  > ) means that the matrix  −  is positive semidefinite (resp., positive definite).R  denotes the -dimensional Euclidean space and R × denotes the set of all  ×  real matrices. stands for an identity matrix of appropriate dimension, while   ∈ R  denotes a vector of ones.The notation * is used as an ellipsis for terms that are induced by symmetry.diag(⋅ ⋅ ⋅ ) stands for a blockdiagonal matrix.| ⋅ | denotes the Euclidean norm for vectors and ‖ ⋅ ‖ denotes the spectral norm for matrices.L 2 [0, ∞) represents the space of square-integrable vector functions over [0, ∞).E(⋅) stands for the mathematical expectation operator.Matrix dimensions, if not explicitly stated, are assumed to be compatible for algebraic operations.

Problem Formulation and Preliminaries
Consider the time-delay T-S fuzzy stochastic system with time-varying parameter uncertainties in the following form: where  1 ,  2 ,  1 ,  2 , and  3 are known real constant matrices and the unknown time-varying matrix function satisfying   ()    () ≤  for all .Now, we consider a dynamical nonfragile fuzzy filter for system (Σ): in which the fuzzy rules have the same representations as in (1).Consider x() ∈ R  and ẑ() ∈ R  .  ,   ,   , and   are the filters needed to be determined.Δ  (), Δ  (), and Δ  () represent the unknown time-varying parameter uncertainties and are assumed to satisfy where  4 ,  4 ,  5 , and  6 are known real constant matrices and the unknown time-varying matrix function satisfying   ()    () ≤  for all .
Remark 1.There are two approaches to design the filter for fuzzy systems.One is dependent on the fuzzy rules when the fuzzy model is available while the other one is independent of the fuzzy rules.In this paper, we choose the first approach since the fuzzy model is known here and this approach is less conserve.So the nonfragile fuzzy rule-dependent filter is investigated in this paper as in (3).
The filtering error dynamic system can be written as where We intend to design a dynamical nonfragile fuzzy filter in the form of (3) in this paper, such that, for any scalar 0 ≤ ℎ 1 < ℎ 2 and a prescribed level of noise attenuation  > 0, the filtering error system ( Σ) could be stochastic input-to-state stability and the error system ( Σ) satisfies  ∞ performance.Now, we present the definitions and lemmas used in this paper, which help to complete the proof of the main results.
Remark 6.In the proof of the theorem, we adopt Newton-Leibnitz formula to reduce the conservatism.Moreover, the results obtained in this theorem can be further extended based on fuzzy or piecewise Lyapunov-Krasovskii function.Now, it is easy to see that Moreover, By the above formulas ( 15)-( 18), we can deduce that where  > 0 is a given positive scalar and From ( 10) and ( 20), we can deduce Together with ( 14), (22), and Lemma 4, the system ( Σ) is SISS.Now, we are in the position to proof that ( Σ) satisfies an  ∞ attenuation level.By Itô's formula, there is Now, we consider the  ∞ performance of the filtering error system ( Σ). Define () = E{∫  0 [z()  z() −  2 V()  V()]} and consider (23).It is obvious that Using the same method in (15), we can deduce the following formula: Then, applying the Schur complement formula to (10), there is for all  > 0. Therefore, for all () ̸ = 0, () < 0, which means that ( 9) is satisfied.This completes the proof.

Remark 7.
Since not all the delays begin at 0 moments, the delay we considered here contains both the upper bound and the lower bound, which is different from most of the existing works.Instead of the [0, ℎ) expression of the time delay, a more reliable sufficient condition is proposed in this paper.
Based on the above results, a sufficient condition for the solvability of robust  ∞ filtering problem for system ( Σ) is considered in the next theorem.Theorem 8. Consider the uncertain T-S fuzzy stochastic timevarying delay system ( Σ) and a constant scalar  > 0. The robust  ∞ filtering problem is solvable if there exist scalars  1 > 0,  2 > 0,  3 > 0,  4 > 0, and  5 > 0 and matrices , 1 ≤  ≤ }, and {  , 1 ≤  <  ≤ }, such that the following LMIs hold: where When the LMIs (27) are feasible, the nonfragile filter we desired here can be chosen as Proof.
By using the same methods in [31], it can be easily proven that the condition in Theorem 5 and the LMIs in (27) are equivalent.Then, we can conclude that the filtering error system ( is SISS with  ∞ performance level . Remark 9.The desired  ∞ filters can be constructed by solving the LMIs in (27), which can be implemented by using standard numerical algorithms, and no tuning of parameters will be involved.

Numerical Example
In this section, a numerical example is provided to show the effectiveness of the results obtained in the previous section.

Conclusion
This paper considers the nonfragile  ∞ filter design problem for the uncertain time-delay T-S fuzzy stochastic system.Sufficient conditions have been addressed to guarantee that the system ( Σ) is SISS.An LMI approach has been developed to design the fuzzy filter ensuring a prescribed  ∞ performance level of the filtering error system for all admissible uncertainties.Finally, a numerical example has been provided to show the effectiveness of the proposed filter design methods.