Filtering for Discrete Fuzzy Stochastic Time-Delay Systems with Sensor Saturation

Abstract

This paper addresses the ℋ_∞ filtering problem for discrete fuzzy stochastic systems with time-varying delay and sensor saturation. Random noise depending on state and external disturbance is also taken into account. A decomposition approach is employed to solve the characteristic of sensor saturation. The scaled small gain (SSG) theorem is extended to the stochastic systems, which is employed to handle with the time-varying delay by transforming the original system into the form of an interconnected system consisting of two subsystems. By the proposed Lyapunov-Krasovskii function, the scaled small gains of the subsystems are analyzed, respectively. Sufficient conditions for the stochastic stability of the filtering error system with a prescribed ℋ_∞ level are established such that the gains of the ℋ_∞ filter can be obtained explicitly. Finally, simulation results are presented to demonstrate the effectiveness of the proposed approach.

1. Introduction

In recent years, the control and filtering problems for Takagi-Sugeno (T-S) fuzzy systems, which have been employed to solve a great deal of issues in control and filtering fields, have drawn a great attention of many researchers [1–4]. Described by a set of If-Then rules, the T-S fuzzy system can approximate any given continuous function in a compact set of ℛ^𝑛 at any preciseness [5–7]. Consequently, it has become one of the most useful and popular research platforms in fuzzy logic control (FLC), since many nonlinear systems can be analyzed by using the properties of conventional linear systems via T-S fuzzy model. A great number of significant results have been reported in the literatures [8–11]. In particular, a great effort has been devoted to the filtering problems for T-S fuzzy systems. Among the existing filtering approaches, the Kalman filtering is popular and useful due to its easy implementing and good performance in many engineering problems [12]. However, the Kalman filtering has strict constraints that a precise model of the system is available and all error terms and measurements have Gaussian distributions, which are hard to reach in many practical dynamics. Meanwhile, ℋ_∞ filtering, which only requires that the noise sources to be arbitrary signals with bounded energy or bounded average power instead of Gaussian noises, has arguably become another useful and important filtering method for T-S fuzzy systems.

On the other hand, a large number of practical engineering problems encounter time-delay phenomenon, which is always a source of poor performance and even instability of the systems [13, 14]. Over the past few years, a great interest has been devoted to time-delay systems, and many significant results have been reported by the Lyapunov method [3, 15–17]. On the other hand, an input-output (IO) approach developed from the SSG theorem that has been introduced for constant delay [18, 19] and then extended to time-varying delay [20]. In contrast to the time-independent methods [15], less conservative results could be achieved by this approach, which analyzes the stability of the system by transforming the system into the form of two interconnected subsystems [20, 21], especially for small time delays [22]. It should be noted that a proper approximation to the time delay plays the key role in this method, and many approximation approaches are proposed to pursue smaller approximation error. Recently, a method for estimating the time-varying delay by the upper and lower delay bounds was introduced in [23], which has a much smaller resulting error than the other one-term ones [24–26]. This motivates this research to extend this approach to ℋ_∞ filtering problems for T-S fuzzy systems.

In practical engineering, sensor saturation is one of the most popular phenomena that limits the performance of sensors on various aspects significantly. When sensors cannot provide the enough amplitude signals due to the physical or safety constraints [18, 27], a complicated controller is needed for a good resulting performance. In recent years, researchers have paid a great attention to filtering problems with sensor saturation [28, 29]. However, it should be noted that in [28] the output of the system is only the sum of a linear term and a nonlinear one, which means that the output model is a special class of nonlinear sensor model, rather than a general form, and the filtering method in [28] cannot be utilized for the stochastic systems directly [30, 31].

Motivated by the above reasons, we investigate the ℋ_∞ filtering problem for discrete-time T-S fuzzy stochastic systems with time-varying delay and senor saturation. A decomposition approach is applied to handle with the characteristic of the sensor saturation, and random noise subject to state and external-disturbance is also considered. A new model transformation is employed to transform the original system into a form of two interconnected subsystems. Based on the SSG theorem developed to stochastic systems and the proposed Lyapunov-Krasovskii function, the SSG of each subsystem is analyzed to establish the sufficient conditions under which the filtering error system is stochastically stable with a prescribed ℋ_∞ performance level 𝛾. Then, the corresponding ℋ_∞ filter design technique is proposed. Finally, an illustrative example is presented to show the effectiveness of the proposed method.

The remainder of this paper is organized as follows. In Section 2, the problem of ℋ_∞ filtering is formulated for T-S fuzzy stochastic systems with time-varying delay and sensor saturation. Section 3 presents the results for ℋ_∞ performance analysis and ℋ_∞ filter design. A numerical example is provided to demonstrate the effectiveness of the designed filtering technique in Section 4, and we conclude the paper with Section 5.

Notation. Throughout this paper, ℝ^𝑛 represents the 𝑛-dimensional Euclidean space, ℝ^𝑛×𝑚 is the set of all 𝑛×𝑚 real matrices, and the superscripts “−1” and “𝑇”, respectively, stand for the matrix inverse and matrix transpose. Sym{𝐴} is the shortened notation for 𝐴+𝐴^𝑇, and the notation 𝑃>0 (resp., 𝑃≥0), for 𝑃∈ℝ^𝑛×𝑛, means that 𝑃 is real symmetric and positive definite (resp., semidefinite). The symmetric elements of the symmetric matrix are represented by an asterisk (∗), and the block diagonal matrices are denoted by diag{⋯}. 𝐆₁∘𝐆₂ means the series connection of mapping 𝐆₁ and 𝐆₂⋅𝐄{⋅} denotes the expectation operator with respect to probability measure, and for vector 𝑥(𝑘), ‖𝑥‖_𝐸2=𝐄{∑^∞_𝑛=0‖𝑥(𝑛)‖²}^1/2.

2. Problem Formulation

We consider the following nonlinear fuzzy dynamic described by a Takagi-Sugeno (T-S) stochastic model:

𝑅^𝑖: If 𝜁₁(𝑘)  is  𝐹^𝑖₁,𝜁₂(𝑘)𝑖𝑠𝐹^𝑖₂  and  …, and  𝜁_𝑠(𝑘)  is  𝐹^𝑖_𝑠, Then 𝑥(𝑘+1)=𝐴_𝑖𝑥(𝑘)+𝐴_𝑑𝑖𝑥𝑘−𝜏_𝑘+𝐵_𝑖𝑣(𝑘)+𝐸_𝑖𝑥(𝑘)+𝐺_𝑖𝑣(𝑘𝑤(𝑘),𝑦(𝑘)𝐶)=𝜙_𝑖𝑥(𝑘)+𝐷_𝑖𝑣(𝑘),𝑧(𝑘)=𝐿_𝑖𝑥(𝑘), (1) where 𝑅^𝑖, 𝑖∈{1,2,…,𝑛}, denotes the 𝑖th fuzzy inference rule, 𝑛 is the number of the inference rules, 𝐹^𝑖_𝑗, 𝑗∈{1,2,…,𝑠}, are fuzzy sets, and 𝜁(𝑘)=[𝜁₁(𝑘),𝜁₂(𝑘),…,𝜁_𝑠(𝑘)] are some measurable premise variables of the system. 𝑥(𝑘)∈ℝ^𝑛 is the state vector, 𝑦(𝑘)∈ℝ^𝑚 is the measured output, and 𝑧(𝑘)∈ℝ^𝑝 is the signal to be estimated. 𝐴_𝑖∈ℝ^𝑛×𝑛, 𝐴_𝑑𝑖∈ℝ^𝑛×𝑛, 𝐵_𝑖∈ℝ^𝑛×𝑞, 𝐶_𝑖∈ℝ^𝑚×𝑛, 𝐷_𝑖∈ℝ^𝑚×𝑞, 𝐿_𝑖∈ℝ^𝑝×𝑛, 𝐸_𝑖∈ℝ^𝑛×𝑛, and 𝐺_𝑖∈ℝ^𝑛×𝑞 are known constant matrices. 𝜏_𝑘 is the time-varying delay satisfying 𝑑₁≤𝜏_𝑘≤𝑑₂, (2) where 𝑑₁>0 and 𝑑₂>0 denote the lower and upper bounds of the delays, respectively. We define the delay interval as 𝑑₁₂=𝑑₂−𝑑₁.

In the system (1), 𝑤(𝑘) is a random process with standard one dimension on a probability space (Ω,ℱ,𝒫), where Ω denotes the sample space, and 𝒫 is the probability measure on ℱ, which is the 𝜎-algebra of subsets of the sample space. The sequence of 𝑤(𝑘) is generated by (𝑤(𝑘))_𝑘∈𝐍, where 𝐍 is the set of natural numbers, such that 𝐄{𝑤(𝑘)}=0, 𝐄{𝑤(𝑘)²}=1, and 𝐄{𝑤(𝑖)𝑤(𝑗)}=0 for 𝑖≠𝑗.

The exogenous disturbance 𝑣(𝑘)∈ℝ^𝑞 is assumed to belong to ℒ_𝐸2([0,∞);ℝ^𝑞), which denotes the space of 𝑘-dimensional nonanticipatory square-integrable process 𝑓(⋅)=(𝑓(𝑘))_𝑘∈𝐍 on 𝐍 with respect to (ℱ_𝑘∈𝐍), such that 𝑓²_𝐸2=𝐄^∞_𝑘=0𝑓(𝑘)²=^∞_𝑘=0𝐄𝑓(𝑘)²<∞. (3)

The defuzzified output of the T-S fuzzy system (1) is inferred as Σ∶𝑥(𝑘+1)=^𝑟_𝑖=1ℎ_𝑖𝜃_𝑘𝐴_𝑖𝑥(𝑘)+𝐴_𝑑𝑖𝑥𝑘−𝜏_𝑘+𝐵_𝑖𝑣(𝑘)+𝐸_𝑖𝑥(𝑘)+𝐺_𝑖𝑣(𝑘𝑤(𝑘,𝑦(𝑘)=))_𝑟𝑖=1ℎ_𝑖𝜃_𝑘𝜙𝐶_𝑖𝑥(𝑘)+𝐷_𝑖𝑣(𝑘,𝑧(𝑘)=_𝑟𝑖=1)ℎ_𝑖𝜃_𝑘𝐿_𝑖𝑥(𝑘), (4) where ℎ_𝑖𝜃_𝑘=𝜇_𝑖𝜃_𝑘∑^𝑟_𝑗=1𝜇_𝑗𝜃_𝑘,𝜇_𝑖𝜃_𝑘=^𝑠_𝑖=1𝐹^𝑖_𝑗𝜃_𝑗𝑘. (5) Then, we have 𝜇_𝑖(𝜃_𝑘)≥0, (𝑖=1,2,…,𝑟) and ∑^𝑟_𝑖=1𝜇_𝑖(𝜃_𝑘)>0, which implies that ℎ_𝑖(𝜃_𝑘)≥0(𝑖=1,2,…,𝑟) and ∑^𝑟_𝑖=1ℎ_𝑖(𝜃_𝑘)=1.

In this paper, the sensor saturation is described by the saturation function 𝜙(⋅)∶ℝ^𝑚→ℝ^𝑚 satisfying 𝜙∈[𝐾₁,𝐾₂], for some given diagonal matrices 𝐾₁≥0, 𝐾₂≥0 with 𝐾₂>𝐾₁, such that 𝜙(𝜁)−𝐾₁𝜁^𝑇𝜙(𝜁)−𝐾₂𝜁≤0,∀𝜁∈ℝ^𝑚. (6) Furthermore, the nonlinear function 𝜙(𝜁) can be decomposed into a linear and a nonlinear part 𝜙(𝜁)=𝜙_𝑠(𝜁)+𝐾₁𝜁, (7) where the nonlinearity 𝜙_𝑠(𝜁) belongs to Φ_𝑠, where Φ_𝑠=𝜙_𝑠∶𝜙^𝑇_𝑠(𝜁)𝜙_𝑠(𝜁)−𝐾𝜁≤0,𝐾≜𝐾₂−𝐾₁. (8)

For the fuzzy system (1), we construct the following filter with full order to estimate the system states 𝑥(𝑘) from measured output 𝑦(𝑘): 𝑥()=ℱ∶𝑘+1∑^𝑟_𝑖=1ℎ_𝑖𝜃_𝑘𝐴_Fi𝑥(𝑘)+𝐵_Fi𝑦(𝑘,𝑧(𝑘)=_𝑟)_𝑖∑₌₁ℎ_𝑖𝜃_𝑘𝐿_𝑖𝑥(𝑘), (9) where 𝑥(𝑘)∈ℝ^𝑛 is the filter state vector, 𝑧(𝑘)∈ℝ^𝑝 is the estimate of 𝑧(𝑘), and 𝐴_𝐹∈ℝ^𝑛×𝑛 and 𝐵_𝐹∈ℝ^𝑛×𝑚 are the filter gains to be determined.

Define the following error variables: 𝑒_𝑥(𝑘)=𝑥(𝑘)−𝑥(𝑘),𝑒(𝑘)=𝑥^𝑇(𝑘)𝑒^𝑇_𝑥(𝑘)^𝑇,𝑒_𝑧(𝑘)=𝑧(𝑘)−𝑧(𝑘). (10) Combining the system (1) and the filter (9) yields the filtering error system as follows: 𝑒()=ℰ∶𝑘+1∑^𝑟𝑟_𝑖=1∑_𝑗=1𝐴_𝑖𝑗𝑒(𝑘)+𝐴_𝑑𝑖𝐼_𝐾𝑒𝑘−𝜏_𝑘+𝐵_𝑖𝑗𝑣(𝑘)𝐸_𝑖𝐼_𝐾𝑒(𝑘)+𝐺_𝑖𝑣(𝑘𝑤(𝑘)+)𝐵_Fi𝜙𝐶_𝑗𝑒(𝑘),𝑒_𝑧(𝑘)=∑^𝑟_𝑖=1ℎ_𝑖𝜃_𝑘𝐿_𝑖𝑒(𝑘), (11) where 𝐴_𝑖𝑗=𝐴_𝑖0𝐴_𝑖−𝐴_Fi−𝐵_Fi𝐾₁𝐶_𝑗𝐴_Fi,𝐴_𝑑𝑖=𝐴_𝑑𝑖𝐴_𝑑𝑖,𝐵_𝑖𝑗=𝐵_𝑖𝐵_𝑖−𝐵_Fi𝐷_𝑗,𝐸_𝑖=𝐸_𝑖𝐸_𝑖,𝐺_𝑖=𝐺_𝑖𝐺_𝑖,𝐵_Fi=0−𝐵_Fi,𝐶_𝑗=𝐶_𝑗0,𝐿_𝑖=0𝐿_𝑖,𝐼_𝐾=𝐼_𝑛0. (12)

Furthermore, the following definitions will be used in this paper.

Definition 1 (see [32]). For the system in (1) with 𝜔(𝑘)=0, the system is stochastically stable if for any 𝜓(𝑘)∈ℝ^𝑛, 𝐄^∞_𝑘=0‖𝑥(𝑘)‖²≤𝜃𝐄‖𝑥(0)‖². (13)

Definition 2 (see [33]). A mapping 𝐆∶𝑢(𝑘)→𝑦(𝑘) is said to be input-output stable in the mean square if there exists 𝛾≥0, such that 𝑦(𝑘)_𝐸2=‖𝐆(𝑢)‖_𝐸2≤𝛾‖𝑢(𝑘)‖_𝐸2. (14)
Then, the main objective of this paper is as follows.
ℋ_∞  Filtering Problem. Given a scalar 𝛾>0, the gains 𝐴_Fi and 𝐵_Fi of the filter (9) are designed such that for any time-delay 𝜏(𝑘) satisfying (2), the filter error system (11) is stochastically stable under 𝑣(𝑘)=0, and for any given integer 𝛾, the following ℋ_∞ performance index ‖𝑧(𝑘)−𝑧(𝑘)‖_𝐸2≤𝛾‖𝑣(𝑘)‖_𝐸2 (15) holds for any function 𝜙(⋅)∈[𝐾₁,𝐾₂].

3. Filtering Performance Analysis

In this section, we focus on the filter performance analysis based on the developed SSG theorem. By the developed SSG theorem to the stochastic systems and a new model transformation to the system ℰ in (11), the sufficient conditions are proposed, under which the system ℰ is stochastically stable with a prescribed ℋ_∞ performance index.

3.1. SSG Theorem for Stochastic Systems

Lemma 3 (stochastic small gain theorem [34]). Consider the following interconnected system consisting of two subsystems in Figure 1: 𝒮₁∶𝜀(𝑡)=𝐆𝜖(𝑡),𝒮₂∶𝜖(𝑡)=Δ𝜀(𝑡). (16) Then, the closed loop systems are input-output stable in mean square for all the subsystems 𝐆 and Δ satisfying 𝐆(𝑢₁)_𝐸2≤𝑘₁𝑢_1𝐸2,𝑘₁Δ𝑢>0,_2𝐸2≤𝑘₂𝑢_2𝐸2,𝑘₂>0 (17) if 𝑘₁𝑘₂<1.

Figure 1 

The interconnection subsystems.

Proof. Considering (16) yields 𝑦_𝐸2≤𝑘₁𝑢_1𝐸2=𝑘₁𝑢−Δ(𝑦)_𝐸2≤𝑘₁‖𝑢‖_𝐸2+𝑘₂𝑦_𝐸2. (18) Obviously, 𝑦_𝐸2≤𝑘₁1−𝑘₁𝑘₂‖𝑢‖_𝐸2, (19) which implies that the closed loop is input-output stable in mean square.

Corollary 4 (stochastic SSG theorem [34]). Consider the system described in (16). The system is input-output stable in mean square if ‖𝑇_𝜖∘𝐆∘𝑇_𝜀‖<1 holds for some nonsingular matrices 𝑇_𝑦 and 𝑇_𝛿 with ‖𝑇_𝜀∘Δ∘𝑇_𝜖‖≤1.

The proof of Corollary 4 can be readily obtained by following the similar lines of Lemma 3.

3.2. A New Model Transformation

To transform the systems (11) into the form of (16), we express 𝑥(𝑘−𝜏_𝑘) as 𝑥𝑘−𝜏_𝑘=12𝑥𝑘−𝑑₁+12𝑥𝑘−𝑑₂+𝑑₁₂2𝜖(𝑘), (20) where (1/2)[𝑥(𝑘−𝑑₁)+𝑥(𝑘−𝑑₂)] is the approximation of 𝑥(𝑘−𝜏_𝑘), and (𝑑₁₂/2)𝜖(𝑘) denotes the approximation error. Then, the system ℰ can be rewritten as follows: 𝑒()=ℰ∶𝑘+1∑^𝑟𝑟_𝑖=1∑_𝑗=1𝐴_𝑖𝑗𝑒(𝑘)+12𝐴_𝑑𝑖×𝑥𝑘−𝑑₁+𝑥𝑘−𝑑₂+𝑑₁₂𝜖(𝑘+)𝐵_𝑖𝑗𝑣(𝑘)+𝐸_𝑖𝐼_𝐾𝑒(𝑘)+𝐺_𝑖𝑣(𝑘(𝑘))×𝑤,+𝐵_Fi𝜙𝐶_𝑗𝑒(𝑘),𝑒_𝑧(𝑘)=∑^𝑟_𝑖=1ℎ_𝑖𝜃_𝑘𝐿_𝑖𝑒(𝑘). (21) Define 𝜀(𝑘)=𝐄{𝑥(𝑘+1)}−𝑥(𝑘), and we have 𝜖(𝑘)=2𝑑₁₂𝑥𝑘−𝜏_𝑘−12𝑥𝑘−𝑑₁−12𝑥𝑘−𝑑₂=1𝑑₁₂^{𝑘−𝜏𝑘−1}_{𝑖=𝑘−𝑑2}𝜀(𝑖)−^{𝑘−𝑑1−1}_{𝑖=𝑘−𝜏𝑘}𝜀(𝑖)=1𝑑₁₂^{𝑘−𝑑1−1}_{𝑖=𝑘−𝑑2}𝛽(𝑖)𝜀(𝑖), (22) where 𝛽(𝑖)≜1,when𝑖⩽𝑘−𝜏_𝑘−1,−1,when𝑖>𝑘−𝜏_𝑘−1. (23) Then, obviously the system (21) can be rewritten in the form as follows: ℰ₁∶𝜎(𝑘)=^𝑟𝑟_𝑖=1𝑗=1𝐆𝜉(𝑘),ℰ₂∶𝜖(𝑘)=Δ𝜀(𝑘), (24) where𝜉(𝑘)=𝑒^𝑇(𝑘)𝑥^𝑇𝑘−𝑑₁𝑥^𝑇𝑘−𝑑₂𝜖(𝑘)𝑣(𝑘)𝜙_𝑠(𝐶_𝑗𝑥(𝑘))^𝑇,𝜎(𝑘)=𝑒^𝑇(𝑘+1)𝜀^𝑇(𝑘)𝑒^𝑇_𝑧(𝑘)^𝑇,𝐆=𝐆^𝑇₁𝐆^𝑇𝐆^𝑇₃^𝑇,𝐆₁=𝐴_𝑖𝑗12𝐴_𝑑𝑖12𝐴_𝑑𝑖𝑑₁₂2𝐴_𝑑𝑖𝐵_𝑖𝑗𝐵_Fi,𝐆₂=𝐴_𝑖1−𝐼02𝐴_𝑑𝑖12𝐴_𝑑𝑖𝑑₁₂2𝐴_𝑑𝑖𝐵_𝑖0,𝐆₃=𝐿_𝑖0_{𝑝×(3𝑛+𝑚+𝑞)}. (25)

Furthermore, the SSG of the mapping Δ denoted by 𝛾(Δ_𝑇) has an upper bound.

Lemma 5. 𝛾(Δ_𝑇) has an upper bound 𝛾Δ_𝑇‖(𝑘=sup𝑇𝜖)‖_𝐸2‖𝑇𝜀(𝑘)‖_𝐸2≤1. (26)

Proof. According to the fact that |𝛽(𝑖)|=1, by using Jensen’s inequality, it can be derived under zero initial condition ‖𝑇𝜖(𝑘)‖²_𝐸2=𝐄^∞_𝑗=0𝜖^𝑇𝑗𝑗≤2𝑍𝜖𝑑²₁₂𝐄^∞_𝑖=0𝑑₁₂^{𝑖−𝑑1−1}_{𝑗=𝑖−𝑑2}𝛽²𝑗𝜖^𝑇𝑗𝜖𝑗=1𝑑₁₂𝐄^−𝑑1−1_{𝑗=−𝑑2}^∞_𝑖=0𝜖^𝑇𝜖≤1𝑖+𝑗𝑖+𝑗𝑑₁₂𝐄^−𝑑1−1_{𝑗=−𝑑2}^∞_𝑖=0𝜖^𝑇(𝑖)𝜖(𝑖)≤𝐄^∞_𝑗=0𝜀^𝑇𝑗𝑍𝜀𝑗=‖𝑇𝜀(𝑘)‖²_𝐸2, (27) where 𝑍=𝑇^𝑇𝑇, which implies 𝛾(Δ_𝑇)≤1. The proof is completed.

From Lemmas 3 and 5, the system ℰ in (11) is input-output stable in mean square if the SSG of the subsystem ℰ₁ satisfies the conditions in Lemma 3, which will be investigated in the following section.

3.3. ℋ_∞ Performance Analysis

In this section, the ℋ_∞ performance of the filtering error system is analyzed by the SSG of the subsystem ℰ₁ in (24) in order to establish the conditions under which the ℋ_∞ performance index is guaranteed for the filtering error system (11).

Theorem 6. Suppose that the system matrices 𝐴_𝑖, 𝐴_𝑑𝑖, 𝐵_𝑖, 𝐶_𝑖, 𝐷_𝑖,and𝐿_𝑖 represent a fixed system. Given the filter matrices 𝐴_Fi, 𝐵_Fi and integers 𝑑₂>𝑑₁≥0 and a constant 𝛾>0, the filtering error system (21) is input-output stable in mean square with an ℋ_∞ performance attention level 𝛾 for any time-varying delay satisfying (2) if there exist matrices 𝑃>0, 𝑄_𝑖>0, 𝑖=1,2, such that Ψ_𝑖𝑗≜Ξ₁Γ^𝑇₂𝑍Γ^𝑇₃Γ^𝑇₁𝑃∗diag−𝑍,−𝐼_𝑝,−𝑃<0, (28) where Ξ₁=Ξ₁₁000𝐼^𝑇_𝑘𝐸^𝑇_𝑖𝑃𝐺_𝑖𝐼^𝑇_𝑘𝐶^𝑇_𝑗𝐾^𝑇∗diag−𝑄₁,−𝑄₂,−𝑍,𝛼_𝑖,−2𝐼_𝑚,Γ₁=𝐴_𝑖𝑗12𝐴_𝑑𝑖12𝐴_𝑑𝑖12𝑑₁₂𝐴_𝑑𝑖𝐵_𝑖𝑗𝐵_Fi,Γ₂=𝐴_𝑖1−𝐼02𝐴_𝑑𝑖12𝐴_𝑑𝑖12𝑑₁₂𝐴_𝑑𝑖𝐵_𝑖0,Γ₃=𝐿_𝑖0_{𝑝×(3𝑛+𝑚+𝑞)},𝛼_𝑖=−𝛾²𝐼_𝑞+𝐺^𝑇_𝑖𝑃𝐺_𝑖,Ξ₁₁=−𝑃+𝐼^𝑇_𝐾𝑄₁+𝑄₂+𝐸^𝑇_𝑖𝑃𝐸_𝑖𝐼_𝐾. (29)

Proof. Choose the following Lyapunov-Krasovskii functional: 𝑉(𝑘)=𝑉₁(𝑘)+𝑉₂(𝑘) (30) with 𝑉₁(𝑘)=𝑒^𝑇(𝑘)(𝑘),𝑉𝑃𝑒₂(𝑘)=^𝑘−1_{𝑖=𝑘−𝑑1}𝑥^𝑇(𝑖)𝑄₁𝑥(𝑖)+^𝑘−1_{𝑖=𝑘−𝑑2}𝑥^𝑇(𝑖)𝑄₂𝑥(𝑖), (31) where 𝑃>0 and 𝑄_𝑖>0, 𝑖=1,2 are the matrices to be determined.
Define Δ𝑉(𝑘)=𝐄{𝑉(𝑘+1)}−𝑉(𝑘), then we have Δ𝑉₁(𝑘)=𝐄𝑒^𝑇(𝑘+1)𝑃𝑒(𝑘+1)−𝑒^𝑇(𝑘)(𝑘)≤𝑃𝑒_{𝑟𝑟𝑖=1𝑗=1}𝜉^𝑇(𝑘)Γ^𝑇₁𝑃Γ₁𝜉(𝑘)−𝑒^𝑇(𝑘)𝑃𝑒(𝑘)−2𝜙_𝑠𝐶_𝑗𝑥(𝑘)𝜙_𝑠𝐶_𝑗𝑥(𝑘)−𝐾𝐶_𝑗𝑥(𝑘),Δ𝑉₂(𝑘)=𝑥^𝑇(𝑘)𝑄₁𝑥(𝑘)−𝑥^𝑇𝑘−𝑑₁𝑄₁𝑥𝑘−𝑑₁+𝑥^𝑇(𝑘)𝑄₂𝑥(𝑘)−𝑥^𝑇𝑘−𝑑₂𝑄₂𝑥𝑘−𝑑₂. (32)
Then, define 𝐽_𝑠=𝐄^∞_𝑘=0𝜀^𝑇(𝑘)𝑍𝜀(𝑘)−𝜖^𝑇(𝑘)𝑍𝜖(𝑘)+𝑒^𝑇_𝑧(𝑘)𝑒_𝑧(𝑘)−𝛾²𝑣^𝑇(𝑘)𝑣(𝑘), (33) which together with (30) yields 𝐽_𝑠≤𝐽_𝑠+𝑉(∞)−𝑉(0)=𝐄^∞_𝑘=0Δ𝑉(𝑘)+𝜀^𝑇(𝑘)𝑍𝜀(𝑘)−𝜖^𝑇(𝑘)𝑍𝜖(𝑘)+𝑒^𝑇_𝑧(𝑘)𝑒_𝑧(𝑘)−𝛾²𝑣^𝑇(𝑘)𝑣(𝑘)=𝐄^∞𝑟_𝑘=0^𝑟_𝑖=1𝑗=1𝜉^𝑇(𝑘)Ξ₁+Γ^𝑇₁𝑃Γ₁+Γ^𝑇₂𝑍Γ₂+Γ^𝑇₃Γ₃𝜉(𝑘). (34) By the Schur complement, it can be verified that 𝐽_𝑠<0 is guaranteed by Ψ_𝑖𝑗<0, which implies that if 𝑍=𝑇^𝑇𝑇, we have 𝛾(𝐆_𝑇)<1, where 𝐆 is defined in (24). According to Lemmas 3 and 5, it is shown that the system ℰ in (11) is input-output stable in mean square.
Furthermore, 𝐽_𝑠<0 implies that ‖𝑇𝜀(𝑘)‖²_𝐸2+𝑒_𝑧(𝑘)²_𝐸2<‖𝑇𝜖(𝑘)‖²_𝐸2+𝛾²‖𝑣(𝑘)‖²_𝐸2, (35) which together with (30) yields ‖𝑒_𝑧(𝑘)‖²_𝐸2<𝛾²‖𝑣(𝑘)‖²_𝐸2. The proof is completed.

Theorem 6 shows the conditions under which the system ℰ in (11) is input-output stable in mean square with a prescribed ℋ_∞ performance index. Moreover, the conditions for stochastic stability of the system ℰ can be reached by the Lyapunov-Krasovskii functional method, which will be discussed in the following theorem.

Theorem 7. For the forward system ℰ in (24), if Theorem 6 holds, then, the system ℰ is stochastically stable for 𝑣(𝑘)=0.

Proof. For the LKF defined in (30), we have Δ𝑉_𝑐(𝑘)=Δ𝑉(𝑘)+𝜀^𝑇(𝑘)𝑍𝜀(𝑘)−𝜖^𝑇(𝑘)𝑍𝜖(𝑘)=𝜉^𝑇₁(𝑘)Ξ^∼₁+Γ^∼𝑇₁𝑃Γ^∼₁+Γ^∼𝑇₂𝑍Γ^∼₂𝜉₁(𝑘), (36) with 𝜉₁(𝑘)=𝑒^𝑇(𝑘)𝑥^𝑇(𝑘−𝑑₁)𝑥^𝑇(𝑘−𝑑₂)𝜖(𝑘)𝜙_𝑠(𝐶_𝑗𝑥(𝑘))^𝑇,_∼Ξ₁=Ξ₁₁000𝐼^𝑇_𝑘𝐶^𝑇_𝑗𝐾^𝑇∗diag−𝑄₁,−𝑄₂,−𝑍,−2𝐼_𝑚∼,Γ₁=𝐴_𝑖𝑗12𝐴_𝑑𝑖12𝐴_𝑑𝑖𝑑₁₂2𝐴_𝑑𝑖𝐵_Fi,_∼Γ₂=𝐴_𝑖1−𝐼02𝐴_𝑑𝑖12𝐴_𝑑𝑖𝑑₁₂2𝐴_𝑑𝑖0, (37) where Ξ₁₁ is defined in (28).

Define (Ξ^∼₁+Γ^∼𝑇₁𝑃Γ^∼₁+Γ^∼𝑇₂𝑍Γ^∼₂) as Φ_𝑖𝑗. It can be shown that Φ_𝑖𝑗<0, if Theorem 6 holds. Then, we can always find a small scalar 𝜆>0, such that Φ_𝑖𝑗<−𝜆𝐼_𝑛000, (38) which implies that 𝐄𝑉_𝑐(𝑘+1)−𝑉_𝑐(𝑘)≤−𝜆𝑥^𝑇(𝑘)𝑥(𝑘)<0. (39)

For any positive integer 𝑅, by summing up the inequality on both sides from 𝑘=0,…,𝑅, we obtain 𝐄𝑉_𝑐(𝑅)−𝑉_𝑐(0)≤−𝜆^𝑅_𝑘=0𝑥^𝑇(𝑘)𝑥(𝑘)<0, (40) which implies that when 𝑅→∞, ^∞_𝑘=0𝐄𝑥^𝑇(𝑘)𝑥(𝑘≤1)𝜆𝐄𝑉_𝑐(0)−𝑉_𝑐(∞≤1)𝜆𝐄𝑉_𝑐(0)<𝜃𝐄‖𝑥(0)‖², (41) which indicates that the systems in (24) are stochastically stable according to Definition 1. The proof is completed.

Remark 8. It is well known that free-weighting matrices are introduced a lot to estimate the upper bound of the term (1/𝑑₁₂)∑^−𝑑1−1_{𝑖=−𝑑2}𝜖^𝑇(𝑘+𝑖)𝑍𝜖(𝑘+𝑖), while in this paper, Jensen’s inequality is employed instead of such weighting matrices, which reduce the NoV of the proposed conditions to 3𝑛²+3𝑛. It is remarkably smaller than the results in some existing literatures, such as 9𝑛²+3𝑛 [35], and 11𝑛²+2.5𝑛 [16], 13𝑛²+5𝑛 [36], and 31𝑛²+7𝑛 [17], which demonstrates the computational advantage of the proposed approach obviously [24].

4. ℋ_∞ Filter Design

This section is devoted to the ℋ_∞ filter design problem for the system (1) based on Theorem 6. Sufficient conditions for the existence of the ℋ_∞ filter are provided in the following theorem.

Theorem 9. Consider the discrete-time fuzzy stochastic system in (1), for a given constant 𝛾>0, if there exist matrices 0<𝑋∈ℝ^𝑛×𝑛, 0<𝑌∈ℝ^𝑛×𝑛, 𝐴_Fi∈ℝ^𝑛×𝑛, and 𝐵_Fi∈ℝ^𝑛×𝑚, such that the following LMI holds: Ψ^∼_𝑖𝑗≜^∼Ξ₁Γ^𝑇₂𝑍Γ^𝑇₃Γ^𝑇_𝐶𝑋Γ^𝑇_𝐶𝑌+Γ^𝑇_𝐹∗diag−𝑍,−𝐼_𝑝,−𝑋,−𝑌<0, (42) where Ξ^∼₁=^∼Ξ₁₁0_𝑛×4𝑛𝐸^𝑇_𝑖(𝑋+𝑌)𝐺_𝑖𝐶^𝑇_𝑗𝐾^𝑇∗diag𝑌,−𝑄₁,−𝑄₂,−𝑍,𝛼^∼_𝑖,−2𝐼_𝑚,Γ_𝐶=𝐴_𝑖012𝐴_𝑑𝑖12𝐴_𝑑𝑖12𝑑₁₂𝐴_𝑑𝑖𝐵_𝑖0,Γ_𝐹=−𝐵_Fi𝐾₁𝐶_𝑗−𝐴_Fi𝐴_Fi0_𝑛×3𝑛−𝐵_Fi𝐷_𝑗−𝐵_Fi,_∼𝛼_𝑖=−𝛾²𝐼_𝑞+𝐺^𝑇_𝑖(𝑋+𝑌)𝐺_𝑖,_∼Ξ₁₁=−𝑋+𝑄₁+𝑄₂+𝐸^𝑇_𝑖(𝑋+𝑌)𝐸_𝑖, (43) then, the system ℰ in (11) is stochastically stable with ℋ_∞ performance level 𝛾, and the filter gains are given by 𝐴_Fi=𝑌⁻¹𝐴_Fi,𝐵_Fi=𝑌⁻¹𝐵_Fi. (44)

Proof. Define 𝑃 introduced in Theorem 6 as 𝑃=𝑋00𝑌, (45) where 0<𝑋∈ℝ^𝑛×𝑛 and 0<𝑌∈ℝ^𝑛×𝑛 are to be determined such that solvability of (28) is satisfied.
Then, we have 𝐴^𝑇_𝑖𝐴^𝑇_𝑖−𝐴^𝑇_Fi−𝐶^𝑇_𝑗𝐾^𝑇₁𝐵^𝑇_Fi0𝐴^𝑇_Fi=𝐴𝑋00𝑌_𝑇𝑖𝑋𝐴^𝑇_𝑖𝑌−𝐴^𝑇_Fi𝑌−𝐶^𝑇_𝑗𝐾^𝑇₁𝐵^𝑇_Fi𝑌0𝐴^𝑇_Fi𝑌,𝐵^𝑇_𝑖𝐵^𝑇_𝑖−𝐷^𝑇_𝑗𝐵^𝑇_Fi=𝐵𝑋00𝑌_𝑇𝑖𝑋𝐵^𝑇_𝑖𝑌−𝐷^𝑇_𝑗𝐵^𝑇_Fi𝑌. (46) Define 𝐴_Fi=𝑌𝐴_Fi, 𝐵_Fi=𝑌𝐵_Fi, which together with (28) yields (42). The proof is completed.

5. Numerical Example

In this section, an example is provided to demonstrate effectiveness of the filter design method proposed in the preceding sections.

Example 10. Consider the system (1) with the following parameters: 𝐴₁=,𝐴−0.28−0.01−0.02−0.2−0.25−0.290.03−0.04−0.23_𝑑1=,𝐸0.010.0200.0100.0100−0.01₁=−0.110.11.6600.46−0.470.47−0.11−0.43,𝐵₁=,𝐶−2.023.831.39₁=−0.110.05−0.10.690.350.48,𝐿₁=,𝐺0.10.10.10.120.10.19₁=−0.210.610.15^𝑇,𝐷₁=0.3−0.3^𝑇,𝐴₂=,𝐴−0.350.00.01−0.59−0.240.020.1−0.06−0.72_𝑑2=,𝐸0.010.0200.0100.01000.01₂=−0.12−0.110.380.110.64−0.18−0.31−0.63−0.6,𝐵₂=,𝐶2.04−3.25−0.93₂=−0.2−0.1−0.20.50.20.21,𝐿₂=,𝐺0.20.190.20.10.10.2₂=−0.432.110.51^𝑇,𝐷₂=0.10.47^𝑇, (47) and the saturation parameters are selected as 𝐾₁=0.6000.7,𝐾₂=0.8000.8, (48) the membership functions are ℎ₁(𝜃)=1−sin𝑥₁2,ℎ₁(𝜃)=1+sin𝑥₁2, (49) and the noises 𝑣(𝑘) are assumed to be 𝑣(𝑘)=1/(0.1+𝑘²), which satisfy the constraint in (3), and set 𝑑₁=1 and 𝑑₂=2.
To design the ℋ_∞ filter, by solving the LMI condition (42) under the prescribed disturbance level 𝛾=1.2, it can be obtained that 𝐴_𝐹1=,−0.0001−0.00080.00210.00010.0005−0.00130.00000.0002−0.0006𝐵_𝐹1=,−0.58771.44251.2489−2.11860.5082−0.5446𝐴_𝐹2=,−0.0002−0.00050.00120.00020.00000.00020.00000.0001−0.0002𝐵_𝐹2=.1.09492.0782−0.9802−3.52440.0898−1.1212 (50)
Assume the sensor saturation as 𝜙𝑦=𝐾₁+𝐾₂2𝐾𝑦+²−𝐾₁2sin𝑦, (51) which satisfies (8), and the initial conditions as 𝑥(0)=[1.50−1]^𝑇, 𝑥(−1)=𝑥(−2)=[10.1−0.6]^𝑇. By calculation, it can be obtained that the ‖𝑒_𝑧(𝑘)‖²_𝐸2=0.0095 and ‖𝑣(𝑘)‖²_𝐸2=3.0810 that yields 𝛾=0.0555 which is below the given 𝛾=1.2, which demonstrates effectiveness of the proposed filter design. In addition, the trajectories of state variable 𝑥(𝑘) and its estimate 𝑥(𝑘) are shown in Figures 2, 3, and 4, 𝑧(𝑘) and its estimate 𝑧(𝑘) are given in Figures 5 and 6, and 𝑒_𝑧(𝑘)=𝑧(𝑘)−𝑧(𝑘) is provided in Figure 7.

Figure 2 

𝑥1(𝑘) and 𝑥1(𝑘).

Figure 3 

𝑥2(𝑘) and 𝑥2(𝑘).

Figure 4 

𝑥3(𝑘) and 𝑥3(𝑘).

Figure 5 

𝑧1(𝑘) and 𝑧1(𝑘).

Figure 6 

𝑧2(𝑘) and 𝑧2(𝑘).

Figure 7 

𝑒𝑧1(𝑘) and 𝑒𝑧2(𝑘).

6. Conclusion

This paper addresses the ℋ_∞ filtering problem for T-S fuzzy systems subject to sensor saturation. The plant is considered with random noise depending on state and external-disturbance. The system is transformed into an input-output form consisted of two interconnected subsystems. Based on the SSG theorem developed to stochastic systems and the proposed Lyapunov-Krasovskii function, sufficient conditions under which the ℋ_∞ filter can be achieved with the prescribed ℋ_∞ performance index are established. Finally, a numerical example is presented to demonstrate the effectiveness of the proposed filter design scheme.