Robust 𝑙₂−𝑙_∞ Filtering for Takagi-Sugeno Fuzzy Systems with Norm-Bounded Uncertainties

Abstract

We study the filter design problem for Takagi-Sugeno fuzzy systems which are subject to norm-bounded uncertainties in each subsystem. As we know that the Takagi-Sugeno fuzzy linear systems can be used to represent smooth nonlinear systems, the studied plants can also be uncertain complex systems. We suppose to design a filter with the order of the original system which is also dependent on the normalized fuzzy-weighting function; that is, the filter is also a Takagi-Sugeno fuzzy filter. With the augmentation technique, an uncertain filtering error system can be obtained and the system matrices in the filtering error system are reorganized into two categories (without uncertainties and with uncertainties). For the filtering error system, we have two objectives. (1) The first one is that the filtering error system should be robust stable; that is, the filtering error system is stable though there are uncertainties in the original system. (2) The second one is that the robust energy-to-peak performance should be guaranteed. With the well-known Finsler’s lemma, we provide the conditions for the robust energy-to-peak performance of the filtering error system in which three slack matrices are introduced. Finally, a numerical example is used to show the effectiveness of the proposed design methodology.

1. Introduction

It is well known that the Takagi-Sugeno (T-S) linear systems can be used to approximate smooth complex systems. Therefore, recently, fuzzy systems based on Takagi-Sugeno (T-S) [1–8] model have attracted a lot of attention [9–16]. In [17], the parameterized linear matrix inequality techniques were used in fuzzy control system design. In [18], the stability analysis and synthesis of fuzzy singularly perturbed systems were investigated. In [19], by assuming that there are some uncertainties in the controller, the robust nonfragile 𝐻_∞ filtering of fuzzy systems with linear fractional parametric uncertainties was studied. In [20], the fuzzy guaranteed cost control for nonlinear systems with time-varying delay was exploited. In [21], the variable structure system approach was used in the stabilization of nonlinear systems. The tracking control problem was studied in [22]. The authors in [23] studied the robust quadratic-optimal control for TS-fuzzy-model-based dynamic systems by considering both elemental parametric uncertainties and norm-bounded approximation error simultaneously. In the filtering aspect for T-S fuzzy systems or nonlinear systems, we can see that robust filter design approaches were proposed in [24] and filtering problem for discrete-time T-S fuzzy systems with time-varying delays was studied in [25].

The filtering problem is quite important since this technique can be used to reduce the cost and improve the measuring performance. Therefore, it is unsurprising that a lot of effort has been made during the past few decades. Among all the filtering work, the Kalman filtering [26] has been utilized for many industrial and astronautics projects since it was proposed in the 1960s. To design the Kalman filter, it requires the precise linear system model and the covariance of the measurement noise at each sampling time. If the system is a nonlinear one, the traditional Kalman cannot be directly applied. Moreover in many practical applications, the above requirements cannot be satisfied at the same time and the on-line computation cannot be guaranteed [27, 28].

If the external disturbance is energy-bounded, we have three additional strategies: energy-to-energy filtering, energy-to-peak filtering, and peak-to-peak filtering in the new filtering aspects [29–31]. In the energy-to-peak filtering [32–37], the infinity norm of the final filtering error aimes to be minimized when the energy of the external noise is bounded. When the peak value of the final filtering error is required to be within a certain range or required to be minimized, the energy-to-peak filtering is one of the best strategies.

In this paper, we study the robust filtering problem for T-S fuzzy discrete-time systems. We proposed to design a full-order T-S fuzzy filter such that the filtering error can satisfy the energy-to-peak performance. By using the augmentation method, we derive a filtering error system. For the filtering error system, we set two objectives. (1) The first one is that the filtering error system should be robust stable. (2) The second one is that the robust energy-to-peak performance should be guaranteed. To achieve the objective, with two well-known lemmas, we offer the sufficient conditions which can guarantee the above two objectives. With the proposed new conditions in which there are several slack matrices to reduce the conservativeness, we provide the design method of the filter parameters of the T-S fuzzy filter. Finally, an illustrative example is used to show the efficacy of the proposed design methodology. The main contributions of this paper can be organized as follows. (1) We introduce a lot of slack matrices in the robust energy-to-peak performance for T-S fuzzy systems. These matrices can be used to reduce the conservativeness of the results in robust control and filtering. (2) Compared with the work in the existing paper, the norm-bounded uncertainties are considered in the T-S fuzzy systems, which have be great potential to applied in filtering problem for complex systems.

2. Problem Formulation

In this paper, we consider a class of uncertain discrete-time T-S fuzzy systems as follows.

Plant Rule  𝑖. If 𝜂₁(𝑘) is 𝒩_𝑖1 and 𝜂₂(𝑘) is 𝒩_𝑖2… and 𝜂_𝑠(𝑘) is 𝒩_𝑖𝑠, then 𝑥_𝑘+1=𝐴_𝑖+Δ𝐴_𝑖𝑥_𝑘+𝐵_𝑖+Δ𝐵_𝑖𝜔_𝑘,𝑦_𝑘=𝐶_𝑖𝑥_𝑘+𝐷_𝑖𝜔_𝑘,𝑧_𝑘=𝐺_𝑖𝑥_𝑘, (1) where 𝒩_𝑖𝑗 for 𝑖=1,…𝑠₁,  𝑗=1,…,𝑠 are the fuzzy sets; 𝑠 denotes the number of the premise variables; 𝑠₁ denotes the number of fuzzy rules; 𝜂_𝑘=[𝜂_1𝑘,𝜂_2𝑘,…,𝜂_𝑠𝑘] denotes the premise variable vector; 𝑥_𝑘 is the system state; 𝜔_𝑘 is the external input which are bounded. Δ𝐴_𝑖 and Δ𝐵_𝑖 denote the norm-bounded uncertainties satisfying Δ𝐴_𝑖,Δ𝐵_𝑖=𝑄_𝑖𝐻(𝑘)𝑋_𝑖,𝑌_𝑖, (2) where  𝑄_𝑖,  𝑋_𝑖,  𝑌_𝑖  are given matrices and the time-varying 𝐻(𝑘) is bounded as 𝐻^𝑇(𝑘)𝐻(𝑘). In addition,  𝐴_𝑖,  𝐵_𝑖,  𝐶_𝑖,  𝐷_𝑖, and 𝐺_𝑖 are matrices with appropriate dimensions and fixed values.

In order to deal with the fuzzy rules, similar with the existing work on the T-S fuzzy systems, we can choose the normalized fuzzy-membership functions as follows: 𝜇_𝑖𝜂(𝑘)=∏^𝑠_𝑗=1𝜃_𝑖𝑗𝜂_𝑗(𝑘)∑^𝑠1_𝑖=1∏^𝑠_𝑗=1𝜃_𝑖𝑗𝜂_𝑗(𝑘), (3) where 𝜃_𝑖𝑗(𝜂_𝑗(𝑘)) is the grade of the membership of 𝜂_𝑗(𝑘) in the 𝑖th fuzzy rule. After the definition, we can see that the normalized fuzzy-membership functions have the following properties: 0≤𝜇_𝑖𝜂(𝑘)≤1,_𝑠1𝑖=1𝜇_𝑖𝜂(𝑘)=1. (4)

The principle of the filter design problem in this paper can be shown in Figure 1. The uncertain T-S fuzzy system is driven by the external input 𝜔_𝑘. The output of the system is measured and denoted by 𝑦_𝑘 which is used to drive the filter. 𝑧_𝑘 is the output of the filter and 𝑧_𝑘 is the estimation of 𝑧_𝑘.

Figure 1 

Filtering problem for uncertain T-S fuzzy systems.

In order to utilize the on-line information of the T-S fuzzy system, we propose to design a full-order T-S fuzzy filter which has the following structure: Filter rule  𝑖: if 𝜃₁(𝑘) is 𝒩_𝑖1, and 𝜃₂(𝑘) is 𝒩_𝑖2,… and 𝜃_𝑠(𝑘) is 𝒩_𝑖𝑠, then 𝑥_𝑘+1=𝐴_𝑓𝑖𝑥_𝑘+𝐵_𝑓𝑖𝑦_𝑘,𝑧_𝑘=𝐺_𝑓𝑖𝑥_𝑘,𝑖=1,…,𝑠₁, (5) where 𝑥_𝑘 denotes the state in the filter;  𝐴_𝑓𝑖,  𝐵_𝑓𝑖, and  𝐺_𝑓𝑖 are filter parameters to be determined in the filter design.

In the filter design, we wish the estimate 𝑧_𝑘 can track the exact value 𝑧_𝑘 which is the most interesting signal. To fulfill this wish, we can define and study the filtering error as 𝑒_𝑘∶=𝑧_𝑘−𝑧_𝑘. With the definition of the filtering error, we can have the following filtering error system as 𝜉_𝑘+1=𝐴𝜇_𝑘𝜉_𝑘+𝐵𝜇_𝑘𝜔_𝑘,𝑒_𝑘=𝐺𝜇_𝑘𝜉_𝑘. (6) Here, we have the following notations: 𝜉_𝑘=𝑥_𝑘𝑥_𝑘,𝐴𝜇_𝑘=𝐴𝜇_𝑘+Δ𝐴𝜇_𝑘0𝐵_𝑓𝜇_𝑘𝐶𝜇_𝑘𝐴_𝑓𝜇_𝑘,𝐵𝜇_𝑘=𝐵𝜇_𝑘+Δ𝐵𝜇_𝑘𝐵_𝑓𝜇_𝑘𝐷𝜇_𝑘,𝐺𝜇_𝑘=𝐺𝜇_𝑘−𝐺_𝑓𝜇_𝑘. (7) For the simplicity, we use the following notations to represent these parameters: 𝐴𝜇(𝑘)=^𝑠1_𝑖=1𝜇_𝑖𝜂(𝑘)𝐴_𝑖,Δ𝐴𝜇(𝑘)=^𝑠1_𝑖=1𝜇_𝑖𝜂(𝑘)Δ𝐴_𝑖,𝐵𝜇(𝑘)=^𝑠1_𝑖=1𝜇_𝑖𝜂(𝑘)𝐵_𝑖,Δ𝐵𝜇(𝑘)=^𝑠1_𝑖=1𝜇_𝑖𝜂(𝑘)Δ𝐵_𝑖,𝐶𝜇(𝑘)=^𝑠1_𝑖=1𝜇_𝑖𝜂(𝑘)𝐶_𝑖,𝐷𝜇(𝑘)=^𝑠1_𝑖=1𝜇_𝑖𝜂(𝑘)𝐷_𝑖,𝐺𝜇(𝑘)=^𝑠1_𝑖=1𝜇_𝑖𝜂(𝑘)𝐺_𝑖,𝐴_𝑓𝜇(𝑘)=^𝑠1_𝑖=1𝜇_𝑖𝜂(𝑘)𝐴_𝑓𝑖,𝐵_𝑓𝜇(𝑘)=^𝑠1_𝑖=1𝜇_𝑖𝜂(𝑘)𝐵_𝑓𝑖,𝐺_𝑓𝜇(𝑘)=^𝑠1_𝑖=1𝜇_𝑖𝜂(𝑘)𝐺_𝑓𝑖. (8)(9)

As the norm-bounded uncertainties are involved in the filtering error system, we reorganize the system as 𝜉_𝑘+1=𝐴𝜇_𝑘+Δ𝐴𝜇_𝑘𝜉_𝑘+𝐵𝜇_𝑘+Δ𝐵𝜇_𝑘𝜔_𝑘,𝑒_𝑘=𝐺𝜇_𝑘𝜉_𝑘, (10) where 𝐴𝜇_𝑘=𝐴𝜇_𝑘0𝐵_𝑓𝜇_𝑘𝐶𝜇_𝑘𝐴_𝑓𝜇_𝑘,Δ𝐴𝜇_𝑘=Δ𝐴𝜇_𝑘0=𝑄𝜇00^𝑘𝐻(𝑘)𝑋𝜇_𝑘,𝑄𝜇_𝑘=𝑄𝜇_𝑘0,𝑋𝜇_𝑘=𝑋𝜇_𝑘0,𝐵𝜇_𝑘=𝐵𝜇_𝑘𝐵_𝑓𝜇_𝑘𝐷𝜇_𝑘,𝐵𝜇_𝑘=Δ𝐵𝜇_𝑘0=𝑄𝜇_𝑘𝐻(𝑘)𝑌𝜇_𝑘. (11)

In the filtering work, the main objective is to obtain a smallest filtering error. The question becomes how to evaluate the external input 𝜔_𝑘 and the filtering error 𝑒_𝑘? Inspired by the filtering work in [35], we use the energy-to-peak gain 𝛾 which has the following meaning: if the initial conditions for the system and the filter are all zeros, the energy-to-peak gain from the external disturbance 𝜔_𝑘 to the defined filtering error 𝑒_𝑘 should be smaller than the preset energy-to-peak index 𝛾; that is, ‖𝑒‖_∞<𝛾‖𝜔‖₂. (12)

To make the design objective more clear, the objectives of the paper are summarized in the following two points. (1)The uncertain filtering error system in (6) is robustly stable for zero disturbance input 𝜔_𝑘. (2)For zero-initial conditions for the original system and the filter, the energy-to-peak gain from the external input to the filtering error should be no greater than a prescribed value 𝛾; that is, ‖𝑒‖_∞<𝛾‖𝜔‖₂. (13)

In order to have a better result, we also include the following lemmas.

Lemma 1 (see [38]). Given 𝑥∈ℝ^𝑛, Θ=Θ^𝑇∈ℝ^𝑛×𝑛, and ℬ∈ℝ^𝑚×𝑛, if  rank(ℬ)<𝑛, the following conditions are equivalent: (i)𝑥^TΘ𝑥<0,  ∀ℬ𝑥=0,  𝑥≠0, (ii)∃𝒳∈ℝ^𝑛×𝑚 such that Θ+𝒳ℬ+ℬ^T𝒳^T<0.

Lemma 2. Suppose that Θ=Θ^𝑇, 𝒬 and 𝒳 are real matrices with compatible dimensions, and 𝐻(𝑘) is a time-varying matrix satisfying 𝐻^𝑇(𝑘)𝐻(𝑘)<𝐼. The following condition Θ+𝒬𝐻(𝑘)𝒳+𝒬^T𝐻^T(𝑘)𝒳^T<0 (14) holds if and only if there exists a positive scaler 𝜀 such that Θ𝒬𝜀𝒳^T∗−𝜀𝐼0∗∗−𝜀𝐼<0 (15) is satisfied. Here, we use an asterisk (∗) as an ellipsis for the terms that are introduced by symmetry.

Remark 3. Though the T-S fuzzy system is used to approximate the nonlinear systems, we still introduce the norm-bounded uncertainties in the system models. The main reason is due to the approximate error and the uncertain values for each linear model. This uncertain T-S fuzzy model is more generalized and can be used to model more complex systems. The norm-bounded uncertainties have been widely used in the linear systems to cover more applications; see [39, 40] and the references therein.

3. Main Results

3.1. Robust Stability and Robust Energy-to-Peak Performance Analysis

We firstly propose a new theorem which can guarantee the robust stability and the robust energy-to-peak performance of the filtering error system.

Theorem 4. Consider an uncertain T-S fuzzy system in (1) and suppose that the filter parameters  𝐴_𝑓(𝜇_𝑘),  𝐵_𝑓(𝜇_𝑘),  𝐺_𝑓(𝜇_𝑘), and a positive 𝛾 are given. The filtering error system in (6) is robustly stable with a prescribed robust energy-to-peak performance index 𝛾, if there exist matrices 𝑃(𝜇_𝑘)=𝑃^𝑇(𝜇_𝑘)>0, 𝑃(𝜇_𝑘+1)=𝑃^𝑇(𝜇_𝑘+1)>0, 𝐿(𝜇_𝑘), 𝑀(𝜇_𝑘), 𝑁(𝜇_𝑘), and 𝜀 such that 𝒥₁=^{𝑃𝜇𝑘+1+sym𝐿𝜇𝑘𝜇−𝐿𝑘𝐴𝜇𝑘+𝑀𝑇𝜇𝑘𝜇−𝐿𝑘𝐵𝜇𝑘+𝑁𝑇𝜇𝑘𝐿𝜇𝑘𝑄𝜇𝑘0𝜇∗−𝑃𝑘−sym𝑀𝜇𝑘𝐴𝜇𝑘𝜇−𝑀𝑘𝐵𝜇𝑘−𝑁𝜇𝑘𝐴𝜇𝑘𝑇𝑀𝜇𝑘𝑄𝜇𝑘𝑋−𝜀𝑇𝜇𝑘∗∗−𝐼−sym𝑁𝜇}_𝑘^𝐵𝜇_𝑘^𝑁𝜇_𝑘^𝑄𝜇_𝑘^{−𝜀𝑌𝑇𝜇}_{𝑘∗∗∗−𝜀𝐼0∗∗∗∗−𝜀𝐼}<0, (16)   −𝑃𝜇_𝑘𝐺^𝑇𝜇_𝑘∗−𝛾²𝐼. (17)

Proof. By using Lemma 2, the condition in (16) holds if and only if the following condition is satisfied: 𝑃𝜇_𝑘+1+sym𝐿𝜇_𝑘−𝐿𝜇_𝑘𝐴𝜇_𝑘+𝑀^𝑇𝜇_𝑘−𝐿𝜇_𝑘𝐵𝜇_𝑘+𝑁^𝑇𝜇_𝑘∗−𝑃𝜇_𝑘−sym𝑀𝜇_𝑘𝐴𝜇_𝑘−𝑀𝜇_𝑘𝐵𝜇_𝑘−𝑁𝜇_𝑘𝐴𝜇_𝑘^𝑇∗∗−𝐼−sym𝑁𝜇_𝑘𝐵𝜇_𝑘<0. (18)  Choose a Lyapunov function candidate for the filtering error system in (6) as follows: 𝑉𝜉_𝑘,𝜇_𝑘,𝑘=𝜉^𝑇_𝑘𝑃𝜇_𝑘𝜉_𝑘, (19) where 𝑃(𝜇_𝑘)=𝑃^𝑇(𝜇_𝑘)>0 is a positive-definite matrix which is also dependent on the normalized fuzzy-weighting functions.
It can be seen that the matrix inequality (18) is rewritten as follows: Θ+𝒳ℬ+ℬ^𝑇𝒳^𝑇<0, (20) where 𝑃𝜇Θ=^𝑘+1𝜇00∗−𝑃_𝑘0,𝐿𝜇∗∗−𝐼𝒳=𝑀𝜇^𝑘_𝑘𝑁𝜇_𝑘,ℬ^𝑇𝜇_𝑘=𝐼−𝐴^𝑇𝜇_𝑘−𝐵^𝑇𝜇_𝑘, (21) and  𝐿(𝜇_𝑘),  𝑀(𝜇_𝑘),  and  𝑁(𝜇_𝑘) are matrices with appropriate dimensions.
Recalling the filtering error system (6), we can see that ℬ𝑥_𝑘 is equal to zero for all 𝑥_𝑘 which is defined as 𝑥^𝑇_𝑘∶=[𝜉^𝑇_𝑘+1𝜉^𝑇_𝑘𝜔^𝑇_𝑘]. According to Lemma 1, the condition (18) has an equivalent one: 𝑥^𝑇_𝑘Θ𝑥_𝑘<0,  ∀ℬ𝑥_𝑘=0,  𝑥_𝑘≠0.
Substituting the representative of Θ into 𝑥^𝑇_𝑘Θ𝑥_𝑘, we have the following inequality: 𝜉^𝑇_𝑘+1𝑃𝜇_𝑘+1𝜉_𝑘+1−𝜉^𝑇_𝑘𝑃𝜇_𝑘𝜉_𝑘−𝜔^𝑇_𝑘𝜔_𝑘<0. (22) When the external disturbance is zero, the inequality (22) becomes 𝜉^𝑇_𝑘+1𝑃𝜇_𝑘+1𝜉_𝑘+1−𝜉^𝑇_𝑘𝑃𝜇_𝑘𝜉_𝑘<0, (23) which implies that Δ𝑉=𝜉^𝑇_𝑘+1𝑃𝜇_𝑘+1𝜉_𝑘+1−𝜉^𝑇_𝑘𝑃𝜇_𝑘𝜉_𝑘. (24) According to the Lyapunov theory, the uncertain filtering error system is robustly stable.
In the sequel, we are going to establish the 𝑙₂−𝑙_∞ performance attenuation level 𝛾. Suppose that the nonzero external disturbance is no-zero. It is infered from (23) that the difference of the Lyapunov function satisfies Δ𝑉<𝜔^𝑇_𝑘𝜔_𝑘. (25) Under the assumption of the zero initial values for the filter and the original system, the Lyapunov function satisfies 𝑉𝜉_𝑘,𝜇_𝑘,𝑘=^𝑘−1_𝑖=0Δ𝑉<^𝑘−1_𝑖=0𝜔^𝑇_𝑖𝜔_𝑖. (26) In terms of the Schur complement, the inequality (17) implies that 𝐺^𝑇𝜇_𝑘𝐺𝜇_𝑘<𝛾²𝑃𝜇_𝑘. (27) Recalling the filtering error system (6), (19), and (26), we can conclude that 𝑒^𝑇_𝑘𝑒_𝑘=𝜉_𝑘^𝑇𝐺^𝑇𝜇_𝑘𝐺𝜇_𝑘𝜉_𝑘<𝛾²𝜉_𝑘^𝑇𝑃𝜇_𝑘𝜉_𝑘=𝛾²𝑉𝜉_𝑘,𝜇_𝑘,𝑘+𝜔^𝑇_𝑘𝜔_𝑘<𝛾^2∞_𝑖=0𝜔^𝑇_𝑖𝜔_𝑖. (28) Taking the supremum over time 𝑘>0 to the above inequality can result in the following condition ‖𝑒‖_∞<𝛾‖𝜔‖₂ for all nonzero 𝜔_𝑘∈𝑙₂[0,∞). Thus, the robust energy-to-peak performance can be guaranteed. This completes the proof.

3.2. T-S Fuzzy Filter Design Approach

In the previous subsection, we assumed the parameters of the filter are given and obtained two conditions which can guarantee the robust stability and the robust energy-to-peak performance. In this subsection, we will propose the design method of the filter parameters based on the main results in the previous subsection.

Theorem 5. Consider a discrete-time robust T-S fuzzy system in (1) and give a positive scalar 𝛾. Then the filtering error system in (6) is robustly stable with a filter in the form of (5) if there exist matrices 𝑃_11,𝑖=𝑃^𝑇_11,𝑖>0, 𝑃_12,𝑖, 𝑃_22,𝑖=𝑃^𝑇_22,𝑖>0, 𝐿_11,𝑖, 𝐿₁₂, 𝐿_21,𝑖, 𝑀_11,𝑖, 𝑀_21,𝑖, 𝑁_1,𝑖, 𝐴𝐹_𝑖, 𝐵𝐹_𝑖, and 𝐺𝐹_𝑖,  ∀𝑖=1,…,𝑠₁,  𝑗=𝑖,…,𝑠₁,  𝑟=1,…,𝑠₁, such that the following conditions hold: −𝑃_11,𝑖−P_12,𝑖𝐺^𝑇_𝑖∗−𝑃_22,𝑖−𝐺𝐹^𝑇_𝑖∗∗−𝛾²𝐼<0,ℋ_{𝑖,𝑗,𝑟}+ℋ_{𝑗,𝑖,𝑟}<0, (29)(30) where ℋ_{𝑖,𝑗,𝑟}=Ω₁₁Ω₁₂Ω₁₃Ω₁₄Ω₁₅Ω₁₆0∗Ω₂₂Ω₂₃Ω₂₄Ω₂₅Ω₂₆0∗∗Ω₃₃Ω₃₄Ω₃₅Ω₃₆Ω₃₇∗∗∗Ω₄₄Ω₄₅Ω₄₆0∗∗∗∗Ω₅₅Ω₅₆Ω₅₇,Ω∗∗∗∗∗−𝜀𝐼0∗∗∗∗∗∗−𝜀𝐼₁₁=𝑃_11,𝑟+sym𝐿_11,𝑖,Ω₁₂=𝑃_12,𝑟+𝐿₁₂+𝐿^𝑇_21,𝑖,Ω₁₃=𝑀^𝑇_11,𝑖−𝐿_11,𝑖𝐴_𝑗−𝐵𝐹_𝑗𝐶_𝑖,Ω₁₄=𝑀^𝑇_21,𝑖−𝐴𝐹_𝑖,Ω₁₅=−𝐿_11,𝑖𝐵_𝑗−𝐵𝐹_𝑗𝐷_𝑖+𝑁^𝑇_1,𝑖,Ω₁₆=𝐿_11,𝑖𝑄_𝑗,Ω₂₂=𝑃_22,𝑟+sym𝐿₁₂,Ω₂₃=−𝐿_21,𝑖𝐴_𝑗−𝐵𝐹_𝑗𝐶_𝑖,Ω₂₄=−𝐴𝐹_𝑖,Ω₂₅=−𝐿_21,𝑖𝐵_𝑗−𝐵𝐹_𝑗𝐷_𝑖,Ω₂₆=𝐿_21,𝑖𝑄_𝑗,Ω₃₃=−𝑃_11,𝑖−sym𝑀_11,𝑖𝐴_𝑗,Ω₃₄=−𝑃_12,𝑖−𝐴^𝑇_𝑖𝑀^𝑇_21,𝑗,Ω₃₅=−𝑀_11,𝑖𝐵_𝑗−𝑁_1,𝑖𝐴_𝑗^𝑇,Ω₃₆=𝑀_11,𝑖𝑄_𝑗,Ω₃₇=−𝜀𝑋^𝑇_𝑖,Ω₄₄=−𝑃_22,𝑖,Ω₄₅=−𝑀_21,𝑖𝐵_𝑗,Ω₄₆=𝑀_21,𝑖𝑄_𝑗,Ω₅₅=−𝐼−sym𝑁_1,𝑖𝐵_𝑗,Ω₅₆=𝑁_1,𝑖𝑄_𝑗,Ω₅₇=−𝜀𝑌^𝑇_𝑖. (31) Moreover, the parameters for each subfilter can be determined by the following equations: 𝐴_𝑓𝑖=𝐿⁻¹₁₂𝐴𝐹_𝑖,𝐵_𝑓𝑖=𝐿⁻¹₁₂𝐵𝐹_𝑖,𝐺_𝑓𝑖=𝐺𝐹_𝑖. (32)

Proof. The proof of the theorem can be done by partitioning these matrices in Theorem 4 as 𝑃_𝑖=𝑃_11,𝑖𝑃_12,𝑖∗𝑃_22,𝑖,𝐿_𝑖=𝐿_11,𝑖𝐿₁₂𝐿_21,𝑖𝐿₁₂,𝑀_𝑖=𝑀_11,𝑖0𝑀_21,𝑖0,𝑁_𝑖=𝑁_1,𝑖0. (33) This completes the proof.

The minimal value for the robust energy-to-peak performance 𝛾 can be obtained in the following corollary.

Corollary 6. The minimum robust energy-to-peak performance index 𝛾 for the filtering error system in (6) can be found by solving the following minimization problem: min𝛾²,s.t.(29),(30)∀𝑖≤𝑗,𝑖,𝑟,𝑗=1,…,𝑠₁. (34)

4. Illustrative Example

In this section, we consider a discrete-time T-S fuzzy system as follows.

Plant Rule  1. If 𝑥_1𝑘 is 𝒩₁₁, then 𝑥_𝑘+1=𝑥0.600.15−0.250.05^𝑘+𝜔−0.20−0.05^𝑘,𝑦_𝑘=1.00.5𝑥_𝑘+0.01𝜔_𝑘,𝑧_𝑘=0.20.3𝑥_𝑘. (35)Plant Rule  2. If 𝑥_1𝑘 is 𝒩₂₁, then 𝑥_𝑘+1=𝑥0.40−0.20−0.200.20^𝑘+𝜔0.1−0.1^𝑘,𝑦_𝑘=−1.00.3𝑥_𝑘+0.03𝜔_𝑘,𝑧_𝑘=0.31.5𝑥_𝑘, (36)(37)(38) where 𝑥_1𝑘 is the first state of the system 𝒩₁₁=𝑥0.5+0.3sin_1𝑘𝑥_1𝑘,for𝑥_1𝑘≠0,1,for𝑥_1𝑘=0,𝒩₂₁=𝑥0.5−0.3sin_1𝑘𝑥_1𝑘,for𝑥_1𝑘≠0,0,for𝑥_1𝑘=0. (39)(40) For the previous discrete-time T-S fuzzy system, using Corollary 6, the obtained minimal 𝛾 is 0.1337 and the corresponding parameters of the filters are 𝐴_𝑓1=,𝐵−0.0810−1.34020.01830.2738_𝑓1=,𝐺−3.38040.5396_𝑓1=,𝐴−0.0550−0.3265_𝑓2=,𝐵−0.0165−0.18190.01200.1310_𝑓2=,𝐺0.4062−0.2234_𝑓2=−0.1315−1.1957. (41)

In order to verify the performance of the designed filter, it is assumed that the initial state of the system is [0.5,0.1]^𝑇. Figure 2 shows the filtering error. It can be seen that the filtering error converges to zero quickly.

Figure 2 

The filtering error in this example.

5. Conclusions

In this paper, we have investigated the robust energy-to-peak filtering problem for uncertain T-S fuzzy systems. The full-order filter is dependent on the normalized fuzzy-weighting functions. By using two novel lemmas, the conditions for the robust stability and the robust energy-to-peak performance of the filtering error systems were provided. The filter design method was proposed by using the partition method. The parameters of the T-S fuzzy filters can be obtained by solving LMIs.

Acknowledgment

This work is supported by Natural Science Foundation of Zhejiang Province (no. Y1100769).