Open Access

Distributed piecewise filtering design for large-scale networked nonlinear systems

EURASIP Journal on Advances in Signal Processing20162016:51

https://doi.org/10.1186/s13634-016-0350-2

Received: 4 January 2016

Accepted: 12 April 2016

Published: 27 April 2016

Abstract

This paper investigates the problem of distributed piecewise filtering for discrete-time large-scale nonlinear systems. The considered large-scale system is composed of a number of nonlinear subsystems and exchanges its information through communication network. Each nonlinear subsystem is described by a Takagi-Sugeno (T-S) model, and data-packet dropouts happen intermittently in communication network, and its stochastic variables are assumed to satisfy the Bernoulli random-binary distribution. Our objective is to design a distributed piecewise filter such that the filtering error system is stochastically stable with an performance. Based on a piecewise Lyapunov function and some convexifying techniques, less conservative results are developed for the distributed piecewise filtering design of the considered system in the form of linear matrix inequalities (LMIs). The effectiveness of the proposed method is validated by two examples.

Keywords

Large-scale fuzzy systemsDistributed filterPiecewise Lyapunov function

1 Introduction

In practical application, some complex systems, such as transportation systems, power systems, communication networks, and industrial processes, are referred to as large-scale systems [1, 2]. Due to strong interconnection and high dimensionality, large-scale systems lead to severe difficulties for their analysis and control synthesis. To date, three main control approaches, centralized, decentralized, and distributed control, have been proposed for large-scale systems with interconnection. Since the centralized control suffers from the excessive information processing and heavy computational burdens, there has been recently an increasing interest in the use of decentralized control for large-scale systems [3]. The decentralized control is firstly to partition the overall control problem of a large-scale system into several independent or almost independent subproblems. Then, instead of a single controller, a set of independent controllers can be designed to achieve the overall control of large-scale system [4]. However, the decentralized control strategy appears weaker stability margins and performance, especially when the interconnections among subsystems are strong [5]. In the distributed control, the supplemental feedbacks with the interconnected information are provided for the local controllers to enhance the requirements of stability and performance. As a result, the distributed control avoids those shortages appearing in both centralized and decentralized controls [6, 7].

On the other hand, an important issue is to consider the control problems of nonlinear systems because most control plants are nonlinear. Recently, Takagi-Sugeno (T-S) model has been proved to be a powerful solution to represent any smooth nonlinear functions at any preciseness [8, 9]. The T-S model employs a group of IF-THEN fuzzy rules to describe the global behavior of the nonlinear system in which a number of linear models are connected smoothly by fuzzy membership functions. T-S fuzzy approach combining the merits of both fuzzy logic theory and linear system theory is successfully implemented in embedded microprocessors and is widely applied in a variety of engineering fields [1013]. During the past few years, a great number of results on function approximation, systematic stability analysis, controller and filtering design for T-S fuzzy systems have been reported in the open literature [1419].

With the rapid development of digital technology, in the feedback loops, communication networks are often used instead of point-to-point connections due to their great advantages, such as simple maintenance and installation, and low cost [2024]. Unfortunately, the network-induced imperfections, such as quantization errors, packet dropouts, and time delays, can degrade significantly the performance of control systems and may even lead to instability [2529]. Recently, based on fuzzy/piecewise Lyapunov functions, some results on stability analysis and controller synthesis of fuzzy systems have been presented. It has been demonstrated that the inherently conservatism in common Lyapunov function can be relaxed by using piecewise/fuzzy Lyapunov functions. More recently, T-S fuzzy control has been developed to investigate large-scale nonlinear systems [3035]. To mention a few, some results on analysis and synthesis methods for decentralized control of large-scale systems have been presented in [3032]. In [33, 34], the decentralized filtering problem was studied for the discrete-time large-scale system with time-varying delay. To the best knowledge of the authors, few results on the distributed filtering design have been given for large-scale networked T-S fuzzy systems by using piecewise Lyapunov function, which motivates us for the research presented in this paper.

This paper will deal with the distributed filtering problem for discrete-time large-scale nonlinear systems. The large-scale system is composed of several nonlinear subsystems and exchanges its information through communication network. Each nonlinear subsystem is described by a T-S model, and data-packet dropouts occur intermittently in communication network, and its stochastic variables satisfy the Bernoulli random-binary distribution. Based on a piecewise Lyapunov functional (PLF) and some convexifying techniques, the distributed filtering design result will be proposed. It will be shown that the filtering error system is stochastically stable with an performance, and the filtering gains can be given by the form of LMIs. Two simulation examples will be presented to demonstrate the advantage of the proposed methods.

Notations.

\(\Re ^{n\times m}\) is the n-dimensional Euclidean space and \(\Re ^{n\times m}\) denotes the set of n×m matrices. P>0 (≥0) means that matrix P is positive definite (positive semidefinite). Sym{A} denotes \(A+A^{T}.\mathbf {I}_{n}\) and 0 m×n are the n×n identity matrix and m×n zero matrix, respectively. The subscripts n and m×n are omitted when the size is irrelevant or can be determined from the context. For matrices \(A\in \Re ^{n\times n},A^{-1}\) and A T denote the inverse and transpose of the matrix A, respectively. \(l_{2}[0,\infty)\) is the space of square-summable infinite vector sequences over \([0,\infty).\) diag{ ···} is a block-diagonal matrix. The notation \(\left \Vert \cdot \right \Vert \) denotes the Euclidean vector norm, and \(\left \Vert \cdot \right \Vert _{2}\) is the usual \(l_{2}[0,\infty)\) norm. \(\mathbb {E}\left \{ \cdot \right \} \) denotes the mathematical expectation. The notation denotes the symmetric terms.

2 Model description and problem formulation

This paper considers a class of discrete-time large-scale systems, which consist of N nonlinear subsystems, the ith nonlinear subsystem is described by the T-S model as below.

Plant rule : IF ζ i1(t) is and ζ i2(t) is and and ζ ig (t) is , THEN
(1)

where \(i\in \mathcal {N}:=\{1,2,\ldots,N\}\); denotes the lth fuzzy inference rule; r i is the number of fuzzy inference rules; are fuzzy sets; \(x_{i}(t)\in \Re ^{n_{xi}},y_{i}(t)\in \Re ^{n_{yi}}\), and \(z_{i}(t)\in \Re ^{n_{zi}}\) are the system state, the measured output, and the estimated signal, respectively; \(w_{i}(t)\in \Re ^{n_{wi}}\) is the disturbance input, which belongs to \(l_{2}[0,\infty);\zeta _{i}(t):=[\zeta _{i1}(t),\zeta _{i2}(t),\ldots,\zeta _{ig}(t)]\) are some measurable variables of the ith subsystem; (A il ,B il ,C il ,D il ,L il ) denotes the lth local model for the ith subsystem; \(\bar {A}_{ikl}\) is the interconnection matrix between the ith and kth subsystems.

Let us define \(\mu _{il}\left [ \zeta _{i}(t)\right ]\) as the normalized membership function of the inferred fuzzy set ; it yields
$$ {\fontsize{9pt}{6pt}{}{\begin{aligned} \mu_{il}\left[ \zeta_{i}(t)\right] :=\frac{{\prod}_{\phi=1}^{g} \mu_{\textit{il}\phi}\left[ \zeta_{\textit{i}\phi}(t)\right] }{{\sum}_{\varsigma =1}^{r_{i}}{\prod}_{\phi=1}^{g}\mu_{\textit{i}\varsigma\phi}\left[ \zeta_{\textit{i}\phi}(t)\right] }\ \geq0,\sum\limits_{l=1}^{r_{i}}\mu_{il}\left[ \zeta_{i}(t)\right] =1 \end{aligned}}} $$
(2)

where \(\mu _{\textit {il}\phi }\left [ \zeta _{\textit {i}\phi }(t)\right ] \) denotes the grade of membership of ζ i ϕ (t) in . For convenience, in the sequel, the argument of \(\mu _{il}\left [ \zeta _{i}(t)\right ] \) will be dropped for the situations without ambiguity, i.e., we denote \(\mu _{il}:=\mu _{il}\left [ \zeta _{i}(t)\right ].\)

By using a standard fuzzy inference, we obtain the ith global T-S fuzzy subsystem as below:
$$ {\fontsize{9pt}{6pt}{} {\begin{aligned} \left\{ \begin{array} [c]{l} x_{i}\left(t+1\right) =A_{i}(\mu_{i})x_{i}\left(t\right) +\sum \limits_{\substack{k=1\\k\neq i}}^{N}\bar{A}_{ik}(\mu_{i})x_{k}(t)+B_{i} (\mu_{i})w_{i}\left(t\right) \\ y_{i}(t)=C_{i}(\mu_{i})x_{i}(t)+D_{i}(\mu_{i})w_{i}\left(t\right) \\ z_{i}(t)=L_{i}(\mu_{i})x_{i}(t),i\in\mathcal{N} \end{array} \right. \end{aligned}}} $$
(3)
where
$$ {\fontsize{8.7pt}{6pt}{} {\begin{aligned} \left\{ \begin{array} [c]{ccc} A_{i}(\mu_{i}):=\sum\limits_{l=1}^{r_{i}}\mu_{il}A_{il}, & \bar{A}_{ik} (\mu_{i}):=\sum\limits_{l=1}^{r_{i}}\mu_{il}A_{ikl}, & B_{i}(\mu_{i}):=\sum\limits_{l=1}^{r_{i}}\mu_{il}B_{il}\\ C_{i}(\mu_{i}):=\sum\limits_{l=1}^{r_{i}}\mu_{il}C_{il}, & D_{i}(\mu_{i}):=\sum\limits_{l=1}^{r_{i}}\mu_{il}D_{il} & L_{i}(\mu_{i}):=\sum \limits_{l=1}^{r_{i}}\mu_{il}L_{il}. \end{array} \right. \end{aligned}}} $$
(4)

In this paper, we will address the distributed filtering design problem of the discrete-time large-scale fuzzy system in (3) based on a piecewise Lyapunov functional (PLF). For each nonlinear subsystem \(i\in \mathcal {N}\), we follow the idea proposed in [36, 37], where the premise variable space is partitioned into two different kinds of regions: fuzzy regions and crisp regions. The region with \(0<\mu _{il}\left [\zeta _{i}(t)\right ] <1\) is defined as the fuzzy region, where the system dynamics are governed by a convex combination of several local models dropped into that region. In addition, the crisp region is the region where \(\mu _{il}\left [ \zeta _{i}(t)\right ] =1\) for some rules l, and the rest of membership functions equal to zero. The system dynamics in crisp region are governed by the l-th local model within that region.

Let be the premise variable space partition for the ith subsystem, and be the set of region indices. Based on the partition policy, the global T-S fuzzy system in (3) can be rewritten as
(5)
where
$$ {\fontsize{7.8pt}{6pt}{} {\begin{aligned} \left\{ \begin{array}{ccc} \mathcal{A}_{ij}:=\sum\limits_{m\in\mathcal{I}_{i}\left(j\right) }\mu_{im}A_{im}, & \mathcal{B}_{ij}:=\sum\limits_{m\in\mathcal{I}_{i}\left(j\right) }\mu_{im}B_{im} & \mathcal{\bar{A}}_{ikj}:=\sum\limits_{m\in \mathcal{I}_{i}\left(j\right) }\mu_{im}A_{ikm}\\ \mathcal{C}_{ij}:=\sum\limits_{m\in\mathcal{I}_{i}\left(j\right) }\mu_{im}C_{im}, & \mathcal{D}_{ij}:=\sum\limits_{m\in\mathcal{I}_{i}\left(j\right) }\mu_{im}D_{im} & \mathcal{L}_{ij}:=\sum\limits_{m\in\mathcal{I} _{i}\left(j\right) }\mu_{im}L_{im} \end{array} \right. \end{aligned}}} $$
(6)

with \(0<\mu _{im}\left [ \zeta _{i}(t)\right ] <1,\sum _{m\in \mathcal {I}_{i}\left (j\right) }\mu _{im}\left [ \zeta _{i}(t)\right ] =1.\) For each local region \(\mathcal {S}_{ij},\mathcal {I}_{i}\left (j\right) \) contains the matrix indices used in that region. For a crisp region, \(\mathcal {I}_{i}\left (j\right) \) contains a single index.

For convenience, we denote a new set Ω i , which represents all possible region transitions for the ith subsystem:
(7)

where js when ζ i (t) transits from the region \(\mathcal {S}_{ij}\) to \(\mathcal {S}_{is},\) and j=s when ζ i (t) stays in the same region \(\mathcal {S}_{ij}.\)

Given the large-scale fuzzy system (5) with the premise variable space partition, we propose a distributed piecewise filter of the following form:
(8)

where \(x_{Fi}(t)\in \Re ^{n_{fi}}\) is the filter state, \(z_{Fi}(t)\in \Re ^{n_{zi}}\) is an estimation of z i (t),y Fi (t) is the measured output applied to filter, and (A Fij ,B Fij ,B Fikj , C Fij ) are filter gains to be designed, where n fi =n xi for the full-order filter, and 1≤n fi <n xi for the reduced-order one.

Here, we assume that the data loss happens in the communication links between the filter and physical plain, the measured output y i (t) is no longer equivalent to y Fi (t). The ith filtering subsystem with unreliable communication network is shown in Fig. 1, where the ith filter takes all measured outputs via unreliable communication links. This paper models the data-loss condition based on a stochastic approach, thus, it yields [38]:
$$ y_{Fi}(t)=\alpha_{i}\left(t\right) y_{i}(t),i\in\mathcal{N} $$
(9)
Fig. 1

The ith subsystem with unreliable communication network

where \(\alpha _{i}\left (t\right) \) is the independent Bernoulli processes, which represent the unreliable condition of the links from the sensor to the filter. Specifically, \(\alpha _{i}\left (t\right) \equiv 0\) when the link fails, i.e., data are lost, and \(\alpha _{i}\left (t\right) \equiv 1\) represents successful transmission, and \(\alpha _{i}\left (t\right) \) is supposed to be given by
$$ {\fontsize{9.1pt}{6pt}{} {\begin{aligned} \begin{array}{ll} \text{Prob}\left\{ \alpha_{i}\left(t\right) =1\right\} =\mathbb{E} \left\{ \alpha_{i}\left(t\right) \right\} =\bar{\alpha}_{i}, &\text{Prob}\left\{ \alpha_{i}\left(t\right) =0\right\} =1-\bar{\alpha}_{i}. \end{array} \end{aligned}}} $$
(10)
In addition, let us define
$$ \tilde{\alpha}_{i}\left(t\right) =\alpha_{i}\left(t\right) -\bar{\alpha }_{i}. $$
(11)
It is easy to see from (11) that
$$ \mathbb{E}\left\{ \tilde{\alpha}_{i}\left(t\right) \right\} =0,\mathbb{E}\left\{ \tilde{\alpha}_{i}\left(t\right) \tilde{\alpha} _{i}\left(t\right) \right\} =\bar{\alpha}_{i}\left(1-\bar{\alpha} _{i}\right). $$
(12)
By defining \(\bar {x}_{i}(t)=\left [{x_{i}^{T}}(t) \; x_{Fi}^{T}(t)\right ]^{T},\bar {z}_{i}(t)=z_{i}(t)-z_{Fi}(t),\) and based on the large-scale fuzzy system in (5) and distributed piecewise filter in (8), the filtering error system is given by
(13)
where
(14)

Definition 1.

Let \(\tilde {x}\left (t\right) =\left [\bar {x}_{1}^{T}\left (t\right) \; \bar {x}_{2}^{T}\left (t\right) \; \cdots \;\bar {x}_{N}^{T}\left (t\right)\right ]^{T}\) and \(\tilde {w}\left (t\right) =\left [{w_{1}^{T}}\left (t\right) \; {w_{2}^{T}}\left (t\right) \; \cdots \; w_{N}^{T}\left (t\right)\right ]^{T}.\) Then, the filtering error system in (13) is stochastically stable in the mean square if there exists matrix W>0 such that
$$ \mathbb{E}\left\{ \left. \sum\limits_{t=0}^{\infty}\left\vert \tilde {x}\left(t\right) \right\vert^{2}\right\vert \tilde{x}\left(0\right) \right\} <\tilde{x}^{T}\left(0\right) W\tilde{x}\left(0\right) $$
(15)

for any initial condition \(\tilde {x}\left (0\right) \) when \(\tilde {w}\left (t\right) \equiv 0.\)

Now, we formulate the distributed piecewise filtering problem as below.

Given the fuzzy filtering system shown in Fig. 1, assume that the communication link parameter \(\bar {\alpha }_{i}\) is available. Given a prescribed scalar γ>0, design a distributed piecewise filter in the form of (8) such that the following two conditions are satisfied simultaneously.
  1. ■■■

    1) The filtering error system in (13) is stochastically stable in the sense of Definition 1;

     
  2. ■■■
    2) Under zero-initial conditions, the estimated error \(\tilde {z}\left (t\right) \) satisfies
    $$ \left\Vert \tilde{z}\right\Vert_{\mathbb{E}}\leq\gamma\left\Vert \tilde {w}\right\Vert_{2}, $$
    (16)

    where \(\left \Vert \tilde {z}\right \Vert _{\mathbb {E}}:=\mathbb {E}\left \{ \sqrt {\sum \limits _{t=0}^{\infty }\left \vert \tilde {z}\left (t\right) \right \vert ^{2}}\right \}.\)

     

If both the conditions are satisfied, then the filtering error system in (13) is stochastically stable with an performance γ.

3 Main results

In this section, based on a piecewise Lyapunov functional (PLF), the performance analysis and design of the distributed piecewise filter will be developed for the considered system in (3). The filter gains will be given for both full-order and reduced-order filters by solving a number of LMIs.

3.1 3.1 Distributed filtering performance analysis

Here, we will present a distributed filtering performance analysis, and the result can be summarized in the following lemma.

Lemma 1.

Given the fuzzy system in (3) and distributed piecewise filter in (8), then the filtering error system in (13) is stochastically stable with an performance γ, if there exist matrices and matrix multiplier \(\mathcal {G}_{i}\in \Re ^{\left (2n_{xi}+2n_{fi} +n_{wi}\right) \times (n_{xi}+n_{fi})}\), \(i\in \mathcal {N}\), and matrices , such that for all , the following matrix inequalities hold:
$$ {\fontsize{9.8pt}{6pt}{} {\begin{aligned} \left[ \begin{array}{cccccc} \Theta_{ijs}+\text{Sym}\left\{ \Pi_{ij}\right\} & \Psi_{\left(1\right)} & f_{i}\Psi_{\left(2\right)} & \Psi_{\left(3\right)} & f_{i}\Psi_{\left(4\right)} & \Psi_{\left(5\right) }\\ \star & -\mathbb{H}_{ij} & 0 & 0 & 0 & 0\\ \star & \star & -\mathbb{H}_{ij} & 0 & 0 & 0\\ \star & \star & \star & -\mathbb{M}_{ij} & 0 & 0\\ \star & \star & \star & \star & -\mathbb{M}_{ij} & 0\\ \star & \star & \star & \star & \star & -\mathbf{I} \end{array} \right] <0, \end{aligned}}} $$
(17)
where
(18)

Proof.

Choose the following piecewise Lyapunov functional (PLF):
(19)

where , are positive definite symmetric Lyapunov matrices.

Define Δ V i (t)=V i (t+1)−V i (t), one has
$$ \begin{aligned} \Delta V_{i}(t)&=\bar{x}_{i}^{T}\left(t+1\right) P_{is}\bar{x}_{i}\left(t+1\right)\\ &\quad-\bar{x}_{i}^{T}\left(t\right) P_{ij}\bar{x}_{i}\left(t\right),(j,s)\in\Omega_{i}. \end{aligned} $$
(20)
It follows from the error system in (13) that
$$ \begin{aligned} \left[ \begin{array}{ccc} -\mathbf{I} & \mathbb{A}_{ij} & \mathbb{B}_{ij} \end{array} \right] \chi_{i}\left(t\right) &+\sum\limits_{\substack{k=1 \\k\neq i} }^{N}\mathbb{\bar{A}}_{ikj}x_{k}(t)\\&{+}\sum\limits_{\substack{k=1 \\k\neq i }}^{N}\mathbb{\bar{B}}_{ikj}w_{k}(t)=0, \end{aligned} $$
(21)

where .

Note that
$$ 2\bar{x}^{T}\bar{y}\leq\bar{x}^{T}M^{-1}\bar{x}+\bar{y}^{T}M\bar{y}, $$
(22)

where \(\bar {x},\bar {y}\in \Re ^{n}\) and matrix M=M T >0.

Define the matrix multipliers \(\mathcal {G}_{i}\in \Re ^{\left (2n_{xi} +2n_{fi}+n_{wi}\right)}\times (n_{xi}+n_{fi}),~i\in \mathcal {N}\), and the matrices \(0<H_{ikj}<H_{ik0}\in \Re ^{n_{xi} \times n_{xi} },\) and it follows from (21), (22) and Lemma A1 in the “Appendix” section that
$$ {\fontsize{7.99pt}{6pt}{} {\begin{aligned} 2\sum\limits_{i=1}^{N}{\chi_{i}^{T}} & \left(t\right) \mathcal{G}_{i} \sum\limits_{\substack{k=1 \\k\neq i}}^{N}\mathbb{\bar{A}}_{ikj} x_{k}(t)\\ & \leq\sum\limits_{i=1}^{N}\sum\limits_{\substack{k=1 \\k\neq i}}^{N}\chi_{i}^{T}\left(t\right) \mathcal{G}_{i}\mathbb{\bar{A}}_{ikj}H_{ikj} ^{-1}\left(\star\right) \chi_{i}\left(t\right) +\sum\limits_{i=1} ^{N}\sum\limits_{\substack{k=1 \\k\neq i}}^{N}{x_{k}^{T}}(t)H_{ikj} x_{k}(t)\\ & \leq\sum\limits_{i=1}^{N}\sum\limits_{\substack{k=1 \\k\neq i}}^{N}\chi_{i}^{T}\left(t\right) \mathcal{G}_{i}\mathbb{\bar{A}}_{ikj}H_{ikj} ^{-1}\left(\star\right) \chi_{i}\left(t\right) +\sum\limits_{i=1} ^{N}\sum\limits_{\substack{k=1 \\k\neq i}}^{N}{x_{k}^{T}}(t)H_{ik0} x_{k}(t)\\ & =\sum\limits_{i=1}^{N}\sum\limits_{\substack{k=1 \\k\neq i}}^{N}\chi_{i}^{T}\left(t\right) \mathcal{G}_{i}\mathbb{\bar{A}}_{ikj}H_{ikj} ^{-1}\left(\star\right) \chi_{i}\left(t\right) +\sum\limits_{i=1} ^{N}\sum\limits_{\substack{k=1 \\k\neq i}}^{N}{x_{i}^{T}}(t)H_{ki0}x_{i}(t). \end{aligned}}} $$
(23)
Similarly, by introducing \(0<M_{ikj}<M_{ik0}\in \Re ^{n_{wi}\times n_{wi}},\) it yields
$$ {\fontsize{7.71pt}{6pt}{} {\begin{aligned} 2\sum\limits_{i=1}^{N}{\chi_{i}^{T}} & \left(t\right) \mathcal{G}_{i} \sum\limits_{\substack{k=1 \\k\neq i}}^{N}\mathbb{\bar{B}}_{ikj} w_{k}(t)\\ & \leq\sum\limits_{i=1}^{N}\sum\limits_{\substack{k=1 \\k\neq i}}^{N}\chi_{i}^{T}\left(t\right) \mathcal{G}_{i}\mathbb{\bar{B}}_{ikj}M_{ikj} ^{-1}\left(\star\right) \chi_{i}\left(t\right) +\sum\limits_{i=1} ^{N}\sum\limits_{\substack{k=1 \\k\neq i}}^{N}{w_{k}^{T}}\left(t\right) M_{ikj}w_{k}\left(t\right) \\ & \leq\sum\limits_{i=1}^{N}\sum\limits_{\substack{k=1 \\k\neq i}}^{N}\chi_{i}^{T}\left(t\right) \mathcal{G}_{i}\mathbb{\bar{B}}_{ikj}M_{ikj} ^{-1}\left(\star\right) \chi_{i}\left(t\right) +\sum\limits_{i=1} ^{N}\sum\limits_{\substack{k=1 \\k\neq i}}^{N}{w_{k}^{T}}\left(t\right) M_{ik0}w_{k}\left(t\right) \\ & =\sum\limits_{i=1}^{N}\sum\limits_{\substack{k=1 \\k\neq i}}^{N}\chi_{i}^{T}\left(t\right) \mathcal{G}_{i}\mathbb{\bar{B}}_{ikj}M_{ikj} ^{-1}\left(\star\right) \chi_{i}\left(t\right) +\sum\limits_{i=1} ^{N}\sum\limits_{\substack{k=1 \\k\neq i}}^{N}{w_{i}^{T}}\left(t\right) M_{ki0}w_{i}\left(t\right). \end{aligned}}} $$
(24)
Given the following index
$$\begin{array}{*{20}l} J(t) & =\mathbb{E}\left\{ \sum\limits_{i=1}^{N}J_{i}(t)\right\} \\ & =\mathbb{E}\left\{ \sum\limits_{i=1}^{N}\sum\limits_{t=0}^{\infty}\left[ \bar{z}_{i}^{T}(t)\bar{z}_{i}(t)-\gamma^{2}{w_{i}^{T}}\left(t\right) w_{i}\left(t\right) \right] \right\}. \end{array} $$
(25)
It follows from (19)–(21) and (23)–(25) that
$$ {\fontsize{8pt}{6pt}{} {\begin{aligned} J(t) & \leq\mathbb{E}\left\{ J(t)+V(\infty)-V(0)\right\}\\ & =\mathbb{E}\left\{ \sum\limits_{i=1}^{N}\sum\limits_{t=0}^{\infty}\left[ \Delta V_{i}(t)+\bar{z}_{i}^{T}(t)\bar{z}_{i}(t)-\gamma^{2}{w_{i}^{T}}\left(t\right) w_{i}\left(t\right) \right] \right\}\\ & \leq\mathbb{E}\left\{ \sum\limits_{i=1}^{N}\sum\limits_{t=0}^{\infty} {\chi_{i}^{T}}\left(t\right) \left[ \Theta_{ijs}+\text{Sym}\left\{ \Pi_{ij}\right\} +\sum\limits_{\substack{k=1 \\k\neq i}}^{N}\mathcal{G} _{i}\mathbb{\bar{A}}_{ikj}H_{ikj}^{-1}\left(\star\right)\right.\right.\\ &\qquad\qquad\qquad\qquad\qquad\quad+\left.\left.\sum \limits_{\substack{k=1 \\k\neq i}}^{N}\mathcal{G}_{i}\mathbb{\bar{B}} _{ikj}M_{ikj}^{-1}\left(\star\right) \right] \chi_{i}\left(t\right) \right\} \\ & =\sum\limits_{i=1}^{N}\sum\limits_{t=0}^{\infty}{\chi_{i}^{T}}\left(t\right) \left\{ \Theta_{ijs}+\text{Sym}\left\{ \Pi_{ij}\right\} +\sum\limits_{\substack{k=1 \\k\neq i}}^{N}\Psi_{ikj\left(1\right) } H_{ikj}^{-1}\Psi_{ikj\left(1\right) }^{T}\right.\\&\qquad+\left.{f_{i}^{2}}\sum \limits_{\substack{k=1 \\k\neq i}}^{N}\Psi_{ikj\left(2\right) }H_{ikj} ^{-1}\Psi_{ikj\left(2\right) }^{T}\right. \left. +\sum \limits_{\substack{k=1 \\k\neq i}}^{N}\Psi_{ikj\left(3\right) }M_{ikj} ^{-1}\Psi_{ikj\left(3\right) }^{T}\right.\\&\qquad+\left.{f_{i}^{2}}\sum\limits_{\substack{k=1 \\k\neq i}}^{N}\Psi_{ikj\left(4\right) }M_{ikj}^{-1}\Psi_{ikj\left(4\right) }^{T}+\Psi_{(5)}\Psi_{(5)}^{T}\right\} \chi_{i}\left(t\right), \end{aligned}}} $$
(26)

where \(f_{i}=\sqrt {\bar {\alpha }_{i}\left (1-\bar {\alpha }_{i}\right) },\left \{ \Theta _{ijs},\Pi _{ij},\Psi _{ikj\left (1\right) },\Psi _{ikj\left (2\right) },\Psi _{ikj\left (3\right) },\right.\\\left.\Psi _{ikj\left (4\right) }\right \} \) are given by (18).

By using Schur complement lemma on (17), it is easy to see from (26) that the inequality (17) implies \(\mathbb {E}\left \{ \Delta V(t)\right \} <0\) when \(w_{i}\left (t\right) \equiv 0\). Thus, it yields
(27)
and
(28)
According to Definition 1, it is easy to see that the filtering error system is stochastically stable in the mean square. Then, considering \(\mathbb {E} \left \{ V\left (t\right) \right \} >0\) for all t≥0, under zero-initial conditions it yields
$$ \mathbb{E}\left\{ \sum\limits_{i=1}^{N}\sum\limits_{t=0}^{\infty}\bar{z} _{i}^{T}(t)\bar{z}_{i}(t)\right\} -\sum\limits_{i=1}^{N}\sum\limits_{t=0} ^{\infty}\gamma^{2}{w_{i}^{T}}\left(t\right) w_{i}\left(t\right) <0, $$
(29)

which means \(\left \Vert \tilde {z}\right \Vert _{\mathbb {E}}<\gamma \left \Vert \tilde {w}\right \Vert _{2}\), thus completing this proof.

3.2 3.2 Distributed filtering design

In this subsection, we will consider the distributed filtering design for the system in (5). Based on Lemma 1, and by specifying the multiplier \(\mathcal {G}_{i}\), the nonlinear matrix inequalities are formulated into the linear ones, the corresponding result is summarized as below.

Theorem 1.

Given the fuzzy system in (3) and a distributed filter in (8), the filtering error system in (13) is stochastically stable with an performance γ, if there exist matrices , and matrices \(\bar {A}_{Fij}\in \Re ^{n_{fi}\times n_{fi}}\), \(\left \{ \bar {B}_{Fij},\bar {B}_{Fij}\right \} \in \Re ^{n_{fi}\times n_{yi}}\), , and matrices , such that for all the following LMIs hold:
$$ {\fontsize{8.9pt}{6pt}{} {\begin{aligned} \left[ \begin{array}{cccccc} \Theta_{ijs}+\textit{\text{Sym}}\left\{ \bar{\Pi}_{im}\right\} & \bar{\Psi }_{\left(1\right)} & f_{i}\bar{\Psi}_{\left(2\right)} & \bar{\Psi }_{\left(3\right)} & f_{i}\bar{\Psi}_{\left(4\right)} & \Psi_{\left(5\right) }\\ \star & -\mathbb{H}_{ij} & 0 & 0 & 0 & 0\\ \star & \star & -\mathbb{H}_{ij} & 0 & 0 & 0\\ \star & \star & \star & -\mathbb{M}_{ij} & 0 & 0\\ \star & \star & \star & \star & -\mathbb{M}_{ij} & 0\\ \star & \star & \star & \star & \star & -\mathbf{I} \end{array} \right] <0, \end{aligned}}} $$
(30)
where
$$ \left\{ \begin{array}{l} \Theta_{i1}=\textit{diag}\left\{ P_{is} \; \sum\limits_{\substack{k=1\\k\neq i}}^{N}\bar{E}H_{ki0}\bar{E} ^{T}-P_{ij} \; \sum\limits_{\substack{k=1\\k\neq i}}^{N}M_{ki0}-\gamma^{2}\mathbf{I} \right\},\\ \bar{\Pi}_{im}=\left[ \begin{array}{ccccc} -G_{ij(1)} & -{KG}_{ij(2)} & \Theta_{i2}^{\left(14\right)} & K\bar{A}_{Fij} & G_{ij(1)}B_{im}+\bar{\alpha}_{i}K\bar{B}_{Fij}D_{im}\\ -G_{ij(3)} & -G_{ij(2)} & \Theta_{i2}^{\left(24\right)} & \bar{A}_{Fij} & G_{ij(3)}B_{im}+\bar{\alpha}_{i}\bar{B}_{Fij}D_{im}\\ -G_{ij(5)} & 0 & G_{ij(5)}A_{im} & 0 & G_{ij(5)}B_{im} \end{array} \right],\\ \Theta_{i2}^{\left(14\right) }=G_{ij(1)}A_{im}+\bar{\alpha}_{i}K\bar {B}_{Fij}C_{im},\Theta_{i2}^{\left(24\right) }=G_{ij(3)}A_{im}+\bar{\alpha }_{i}\bar{B}_{Fij}C_{im},\\ \bar{\Psi}_{\left(1\right) }=\underset{N-1}{\underbrace{\left[ \bar{\Psi }_{i1j\left(1\right) }\cdots\bar{\Psi}_{ikj\left(1\right),k\neq i}\cdots\bar{\Psi}_{iNj\left(1\right) }\right] }},\bar{\Psi}_{ikj\left(1\right) }=\left[ \begin{array}{c} G_{ij(1)}\bar{A}_{ikm}+\bar{\alpha}_{i}K\bar{B}_{Fikj}C_{km}\\ G_{ij(3)}\bar{A}_{ikm}+\bar{\alpha}_{i}\bar{B}_{Fikj}C_{km}\\ G_{ij(5)}\bar{A}_{ikm} \end{array} \right],\\ \bar{\Psi}_{\left(2\right) }=\underset{N-1}{\underbrace{\left[ \bar{\Psi }_{i1j\left(2\right) }\cdots\bar{\Psi}_{ikj\left(2\right),k\neq i}\cdots\bar{\Psi}_{iNj\left(2\right) }\right] }},\bar{\Psi}_{ikj\left(2\right) }=\left[ \begin{array}{c} K\bar{B}_{Fikj}C_{km}\\ \bar{B}_{Fikj}C_{km}\\ 0 \end{array} \right],\\ \bar{\Psi}_{\left(3\right) }=\underset{N-1}{\underbrace{\left[ \bar{\Psi }_{i1j\left(3\right) }\cdots\bar{\Psi}_{ikj\left(3\right),k\neq i}\cdots\bar{\Psi}_{iNj\left(3\right) }\right] }},\bar{\Psi}_{ikj\left(3\right) }=\left[ \begin{array}{c} \bar{\alpha}_{i}K\bar{B}_{Fikj}D_{km}\\ \bar{\alpha}_{i}\bar{B}_{Fikj}D_{km}\\ 0 \end{array} \right],\\ \bar{\Psi}_{\left(4\right) }=\underset{N-1}{\underbrace{\left[ \bar{\Psi }_{i1j\left(4\right) }\cdots\bar{\Psi}_{ikj\left(4\right),k\neq i}\cdots\bar{\Psi}_{iNj\left(4\right) }\right] }},\bar{\Psi}_{ikj\left(4\right) }=\left[ \begin{array}{c} K\bar{B}_{Fikj}D_{km}\\ \bar{B}_{Fikj}D_{km}\\ 0 \end{array} \right],\\ \Psi_{(5)}=\left[ \begin{array}{ccc} 0 & \left[ \begin{array} [c]{cc} L_{im} & -\bar{C}_{Fij} \end{array} \right] & 0 \end{array} \right]^{T},f_{i}=\sqrt{\bar{\alpha}_{i}\left(1-\bar{\alpha}_{i}\right) },\\ \bar{E}=\left[ \begin{array}{c} \mathbf{I}\\ 0 \end{array} \right],K=\left[ \begin{array}{cc} \mathbf{I}_{n_{fi}} & 0_{n_{fi}\times(n_{xi}-n_{fi})} \end{array} \right]^{T},\\ \mathbb{H}_{ij}=\textit{{\text{diag}}}\underset{N-1}{\underbrace{\left\{ H_{i1j}\cdots H_{ikj,k\neq i}\cdots H_{iNj}\right\} }},\mathbb{M} _{ij}=\textit{{\text{diag}}}\underset{N-1}{\underbrace{\left\{ M_{i1j}\cdots M_{ikj,k\neq i}\cdots M_{iNj}\right\} }}. \end{array} \right. $$
(31)
Furthermore, a distributed piecewise filter in the form of (8) is given by
(32)

Proof.

For matrix inequality linearization purpose, the multipliers \(\mathcal {G}_{i},i\in \mathcal {N}\) are firstly specified by
(33)

where \(K=\left [\mathbf {I}_{n_{fi}} \; 0_{n_{fi}\times (n_{xi}-n_{fi})}\right ]^{T},G_{ij(1)}\in \Re ^{n_{xi}\times n_{xi}},G_{ij(2)}\in \Re ^{n_{fi}\times n_{fi}},G_{ij(3)}\in \Re ^{n_{fi}\times n_{xi}},G_{ij(4)}\in \Re ^{n_{fi}\times n_{fi}}.\)

Then, similar to [39], defining \(\Gamma := \text {diag}\left \{\mathbf {I} _{n_{xi}},G_{ij(2)}G_{ij(4)}^{-1}\right \}\), and performing a congruence transformation to
(34)
by Γ, it yields
$$ {\fontsize{7.8pt}{6pt}{} {\begin{aligned} & \left[ \begin{array}{cc} \mathbf{I}_{n_{xi}} & 0\\ 0 & G_{ij(2)}G_{ij(4)}^{-1} \end{array} \right] \left[ \begin{array}{cc} G_{ij(1)}+G_{ij(1)}^{T} & {KG}_{ij(2)}+G_{ij(3)}^{T}\\ \star & G_{ij(4)}+G_{ij(4)}^{T} \end{array} \right] \left[ \begin{array}{cc} \mathbf{I}_{n_{xi}} & 0\\ 0 & G_{ij(4)}^{-T}G_{ij(2)}^{T} \end{array} \right]\\ & =\left[ \begin{array}{cc} G_{ij(1)}+G_{ij(1)}^{T} & {KG}_{ij(2)}G_{ij(4)}^{-T}G_{ij(2)}^{T}+G_{ij(3)} ^{T}G_{ij(4)}^{-T}G_{ij(2)}^{T}\\ \star & G_{ij(2)}G_{ij(4)}^{-T}G_{ij(2)}^{T}+G_{ij(2)}G_{ij(4)}^{-1} G_{ij(2)}^{T} \end{array} \right]\\ & :=\left[ \begin{array}{cc} G_{ij(1)}+G_{ij(1)}^{T} & K\bar{G}_{ij(2)}+\bar{G}_{ij(3)}^{T}\\ \star & \bar{G}_{ij(2)}+\bar{G}_{ij(2)}^{T} \end{array} \right],i\in\mathcal{N}. \end{aligned}}} $$
(35)
Without loss of generalit, we can specify G i j(4)=G i j(2). Thus, we can directly specify the multipliers \(\mathcal {G}_{i}\) as
(36)
It is easy to see that the matrix variable is absorbed by the filter gain variables when introducing
$$ \bar{A}_{Fij}=G_{ij(2)}A_{Fij},\bar{B}_{Fij}=G_{ij(2)}B_{Fij},\bar{B} _{Fikj}=G_{ij(2)}B_{Fikj} $$
(37)

with .

It is noted that the first block-row of \(\left \{ \mathbb {A}_{ij},\mathbb {\bar {A}}_{ikj},\mathbb {B}_{ij},\mathbb {\bar {B}}_{ikj}\right \} \) does not involve in the filter gain variables. Thus, the multipliers \(\mathcal {G} _{i},i\in \mathcal {N}\) is finally specified as below,
(38)

where \(G_{ij(5)}\in \Re ^{\left (n_{xi}+n_{fi}+n_{wi}\right) \times n_{xi}}.\)

Then, substituting (38) into (17), and by extracting the fuzzy basis functions, the inequality (30) can be obtained.

In addition, the inequality in (17) imply that
$$ P_{is}+\text{Sym}\left\{ \left[ \begin{array}{cc} -G_{ij(1)} & -{KG}_{ij(2)}\\ -G_{ij(3)} & -G_{ij(2)} \end{array} \right] \right\} <0 $$
(39)

with .

Due to the fact that P is >0, we have \(G_{ij(2)}+G_{ij(2)}^{T}>0,\) which means that the matrix variable G i j(2) are nonsingular. Thus, the filtering gains \(\left \{ A_{Fij},B_{Fij},B_{Fikj},C_{Fij}\right \} \) can be obtained by (32) and completing this proof.

4 Simulation examples

In the following, let us consider two examples to illustrate the result proposed in this paper.

Example 1.

Consider a discrete-time large-scale T-S model with three fuzzy rules as below:

Plant rule : IF x i1(t) is , THEN
$$\left\{ \begin{array} [c]{l} x_{i}\left(t+1\right) =A_{il}x_{i}\left(t\right) +\sum \limits_{\substack{k=1\\k\neq i}}^{2}\bar{A}_{ikl}x_{k}(t)+B_{il}w_{i}\left(t\right) \\ y_{i}(t)=C_{il}x_{i}(t)+D_{il}w_{i}\left(t\right) \\ z_{i}(t)=L_{il}x_{i}(t),l=\left\{ 1,2,3\right\},i=\left\{ 1,2\right\} \end{array} \right. $$
where
$${\fontsize{7.97pt}{6pt}{} {\begin{aligned} & \left[ A_{11}\quad A_{12} \quad A_{13} \right] =\left[ \begin{array}{rr|rr|rr} 0.98 & 0.01 & 0.87 & 0.02 & 0.95 & 0.01\\ -0.18 & 0.74 & -0.12 & 0.83 & -0.16 & 0.86 \end{array} \right],\\ & \left[B_{11} \; B_{12} \; B_{13}\right] = \left[ \begin{array}{l|l|l} 0 & 0 & 0\\ 0.28 & 0.25 & 0.21 \end{array}\right],\\ & C_{11}=\left[1.03 \quad 0\right],C_{12}= \left[0.95 \quad 0\right],C_{13}=\left[0.89 \quad 0\right],\\ & D_{11}=0.74,D_{12}=0.69,D_{13}=0.58,L_{11}=0.85,L_{12}=0.97,L_{13}=1.04 \end{aligned}}} $$
for the first subsystem, and
$${\fontsize{7.99pt}{6pt}{} {\begin{aligned} & \left[ A_{21} \quad A_{22}\quad A_{23} \right] =\left[\begin{array}{rr|rr|rr} 0.75 & 0.02 & 0.83 & 0.01 & 0.92 & 0.02\\ -0.20 & 0.82 & -0.15 & 0.88 & -0.17 & 0.75 \end{array} \right],\\ & \left[ \begin{array}{rrr} B_{21} \quad B_{22} \quad B_{23} \end{array}\right] =\left[\begin{array}{l|l|l} 0 & 0 & 0\\ 0.21 & 0.28 & 0.23 \end{array} \right],\\ & C_{21}=\left[ \begin{array}{cc} 0.94 & 0 \end{array} \right],C_{22}=\left[ \begin{array}{cc} 0.97 & 0 \end{array} \right],C_{23}=\left[ \begin{array}{cc} 0.86 & 0 \end{array} \right],\\ & D_{21}=0.49,D_{22}=0.65,D_{23}=0.58,L_{21}=1.05,L_{22}=0.95,L_{23}=0.92 \end{aligned}}} $$
for the second subsystem.
Figure 2 shows the membership functions. According to the premise variable space partition, there are three subspaces for each subsystem:
$$\begin{array}{*{20}l} \mathcal{S}_{i1} & =\left\{ x_{i1}(t)|r_{i1}\leq|x_{i1}(t)|\leq r_{i2}\right\} \\ \mathcal{S}_{i2} & =\left\{ x_{i1}(t)|r_{i2}<|x_{i1}(t)|\leq r_{i3}\right\}\\ \mathcal{S}_{i3} & =\left\{ x_{i1}(t)|r_{i3}<|x_{i1}(t)|\leq r_{i4} \right\}. \end{array} $$
Fig. 2

Membership functions in example 1

It can be observed that \(\mathcal {S}_{i1}\) is a crisp region; both \(\mathcal {S}_{i2}\) and \(\mathcal {S}_{i3}\) are fuzzy regions. The region index set is .

Here, we consider the case of a full-order filter with a i =0.6. It is noted that the common Lyapunov function proposed in [40] is not applicable to the distributed filtering design for this case in this example. However, by applying Theorem 1, the feasible solutions of \(\gamma _{\min }=3.9555\) for the full-order filter and \(\gamma _{\min }=6.5599\) for the reduced-order filter are obtained, and the corresponding filter gains are
for full-order filter and
$${\fontsize{9.7pt}{6pt}{} {\begin{aligned} \left[ \begin{array}{rrr} A_{F11} & A_{F12} & A_{F13}\\ C_{F11} & C_{F12} & C_{F13}\\ A_{F21} & A_{F22} & A_{F23}\\ C_{F21} & C_{F22} & C_{F23}\\ B_{F11} & B_{F12} & B_{F13}\\ B_{F121} & B_{F122} & B_{F123}\\ B_{F21} & B_{F22} & B_{F23}\\ B_{F211} & B_{F212} & B_{F213} \end{array} \right] =\left[ \begin{array}{rrr} 0.6646 & 0.7019 & 0.6069\\ -1.1355 & -1.3025 & -7.8639\\ 0.3419 & 0.2325 & 0.6842\\ -2.8731 & -6.5894 & -3.2918\\ -0.4212 & -0.3742 & -0.5079\\ -0.0007 & -0.0009 & -0.0014\\ -0.4754 & -0.6358 & -0.5075\\ -0.0001 & -0.0003 & -0.0004 \end{array} \right] \end{aligned}}} $$
for the reduced-order case.

Example 2.

Consider a modified double-inverted pendulum, which is connected by a spring [34]. The motional equations of the interconnected pendulum are given by
$${{} {\begin{aligned} \left\{ \begin{array}{l} \dot{x}_{i1}=x_{i2}\\ \dot{x}_{i2}=\frac{1}{100J_{i}}u_{i}-\frac{kr^{2}}{4J_{i}}x_{i1}+\left[ \frac{m_{i}gr}{J_{i}}-\frac{kr^{2}}{4J_{i}}x_{i2}\right] \sin(x_{i1})\\\qquad\qquad\qquad\qquad\quad\quad+\frac{1}{J_{i}}x_{i2}+\sum\limits_{\substack{k=1\\k\neq i}}^{2}\frac {3kr^{2}}{J_{k}}x_{k1},i=\{1,2\} \end{array} \right. \end{aligned}}} $$
where x i1 and x i2 denote the angle from the vertical and the angular velocity, respectively.

In this simulation, the moments of inertia are J 1=2 kg and J 2=2.5 kg; the masses of two pendulums are chosen as m 1=2 kg and m 2=2.5 kg; the constant of the connecting torsional spring is k=8 N ·m/rad; the length of the pendulum is r=1 m; the gravity constant is g=9.8 m/s2. Here, the interconnected pendulum is linearized around the origin, \(x_{i1}=\left (\pm 80^{\circ},0\right) \) and \(x_{i1}=\left (\pm 88^{\circ},0\right) \); each pendulum is described by the T-S model with three fuzzy rules. Given u 1=−18x 11−16x 12 and u 2=−20x 21−14x 22, it can be known that these two closed-loop T-S fuzzy subsystems are stable. Then, by discretizing the T-S fuzzy systems with sampling period T=0.01 s, the discrete-time interconnected T-S fuzzy system can be obtained as below:

Plant rule : IF \(\left \vert x_{i1}(t)\right \vert \) is , THEN
$$\left\{ \begin{array}{l} x_{i}\left(t+1\right) =A_{il}x_{i}\left(t\right) +\sum \limits_{\substack{k=1\\k\neq i}}^{2}\bar{A}_{ik}x_{k}(t)+B_{il}w_{i}\left(t\right) \\ y_{i}(t)=C_{il}x_{i}(t)+D_{il}w_{i}\left(t\right) \\ z_{i}(t)=L_{il}x_{i}(t),l=\left\{ 1,2,3\right\},i=\left\{ 1,2\right\} \end{array} \right. $$
where
$${\fontsize{6.41pt}{6pt}{} {\begin{aligned} & \left[ A_{11}\quad A_{12} \quad A_{13}\right] = \left[ \begin{array}{rr|rr|rr} 1 & 0.0120 & 1 & 0.0120 & 1 & 0.0120\\ -1.3200 & -0.1540 & -1.1818 & -0.1658 & -1.3760 & -0.0352 \end{array}\right],\\ & \bar{A}_{12}=\left[ \begin{array}{rr} 0 & 0\\ 0.12 & 0 \end{array} \right],B_{1l}=\left[ \begin{array}{c} 0\\ 0.5 \end{array} \right],C_{1l}=\left[ \begin{array}{cc} 1 & 0 \end{array} \right],D_{1l}=1,L_{1l}=\left[ \begin{array}{cc} 0 & 1 \end{array} \right] \end{aligned}}} $$
for the first subsystem and
$${\fontsize{6.41pt}{6pt}{} {\begin{aligned} & \left[A_{21} \quad A_{22} \quad A_{23} \right] =\left[ \begin{array}{rr|rr|rr} 1 & 0.0120 & 1 & 0.0120 & 1 & 0.0120\\ -1.3760 & -0.0352 & -1.2378 & -0.0447 & -1.2485 & -0.0448 \end{array} \right],\\ & \bar{A}_{21}=\left[\begin{array}{cc} 0 & 0\\ 0.096 & 0 \end{array} \right],B_{2l}=\left[ \begin{array}{c} 0\\ 0.4 \end{array} \right],C_{2l}=\left[ \begin{array}{cc} 1 & 0 \end{array} \right],D_{2l}=1,L_{2l}=\left[ \begin{array}{cc} 0 & 1 \end{array} \right] \end{aligned}}} $$
for the second subsystem.
Figure 3 shows the membership functions. Based on the premise variable space partition, it can be known that there exist three subspaces for each subsystem:
$$\begin{array}{*{20}l} \mathcal{S}_{i1} & =\left\{ x_{i1}(t)|r_{i1}<\left\vert x_{i1} (t)\right\vert \leq r_{i2}\right\} \\ \mathcal{S}_{i2} & =\left\{ x_{i1}(t)|r_{i2}<\left\vert x_{i1} (t)\right\vert \leq r_{i3}\right\} \\ \mathcal{S}_{i3} & =\left\{ x_{i1}(t)|r_{i3}<\left\vert x_{i1} (t)\right\vert <r_{i4}\right\} \end{array} $$
Fig. 3

Membership functions in example 2

where r i1=0,r i2=8°,r i3=80°, and r i4=88°. It is shown that both \(\mathcal {S}_{i1}\) and \(\mathcal {S}_{i3}\) are crisp regions and \(\mathcal {S}_{i2}\) is a fuzzy region. The region index set is .

Now, considering the case of full-order filter with a i =0.9, the common Lyapunov function proposed in [40] is not applicable to the distributed filtering design for this case in this example. However, by applying Theorem 1, the filtering performance \(\gamma _{\min }=3.3755\) is obtained, and the obtained filter gains are
$${\fontsize{6.7pt}{6pt}{} {\begin{aligned} & \left[ A_{F11} \quad A_{F12} \quad A_{F13} \right] =\left[ \begin{array}{rr|rr|rr} 0.9530 & 0.0064 & 0.9645 & 0.0061 & 0.9645 & 0.0060\\ -0.8230 & 0.8049 & -0.7860 & 0.7924 & -0.7696 & 0.7620 \end{array} \right],\\ & \left[ B_{F11} \quad B_{F12} \quad B_{F13} \right] =\left[ \begin{array}{l|l|l} -0.0462 & -0.0357 & -0.0356\\ -0.6725 & -0.7136 & -0.7026 \end{array}\right],\\ & \left[ B_{F121} \quad B_{F122} \quad B_{F123}\right] =\left[ \begin{array}{l|l|l} -0.0009 & -0.0009 & -0.0009\\ -0.1312 & -0.1346 & -0.1314 \end{array} \right],\\ & \left[ C_{F11} \quad C_{F12} \quad C_{F13} \right] =\left[ \begin{array}{rr|rr|rr} 0.0324 & -0.9570 & 0 & -1.0475 & -0.0006 & -1.0285 \end{array} \right], \end{aligned}}} $$
and
$${\fontsize{6.7pt}{6pt}{} {\begin{aligned} & \left[ A_{F21} \quad A_{F22} \quad A_{F23} \right] =\left[ \begin{array}{rr|rr|rr} 0.9626 & 0.0062 & 0.9634 & 0.0064 & 0.9634 & 0.0063\\ -0.7417 & 0.8157 & -0.6866 & 0.8275 & -0.6732 & 0.8048 \end{array} \right],\\ & \left[ B_{F21} \quad B_{F22} \quad B_{F23} \right] =\left[ \begin{array}{l|l|l} -0.0360 & -0.0361 & -0.0361\\ -0.5731 & -0.5792 & -0.5732 \end{array} \right],\\ & \left[ B_{F211} \quad B_{F212} \quad B_{F213} \right] =\left[ \begin{array}{l|l|l} -0.0009 & -0.0010 & -0.0009\\ -0.1094 & -0.1103 & -0.1085 \end{array} \right],\\ & \left[ C_{F21} \quad C_{F22} \quad C_{F23} \right] =\left[ \begin{array}{rr|rr|rr} 0.0132 & -1.0074 & 0 & -1.0111 & 0.0004 & -1.0007 \end{array} \right]. \end{aligned}}} $$
Given the initial conditions x 1(0)=[1.3,0] T ,x 2(0)=[1.1,0] T , and assume that the external disturbances satisfy \(w_{1}(t)=5\textit {e}^{-0.02t}\sin (t)\) and \(w_{2}(t)=5\textit {e}^{-0.02t}\cos (t),\) it is easy to see from Figs. 4 and 5 that the responses of he time responses of z i (t) and \({z}_{Fi}(t),i=\left \{ 1,2\right \} \) converge to zero. The data-packet dropouts are shown in Figs. 6 and 7, respectively. Then, it can also be shown in Fig. 8 that the performance is satisfactory under zero-initial conditions thus verifying the effectiveness of the distributed filtering design.
Fig. 4

Responses of z 1(t) and z F1(t)

Fig. 5

Responses of z 2(t) and z F2(t)

Fig. 6

Data-packet dropout for subsystem 1

Fig. 7

Data-packet dropout for subsystem 2

Fig. 8

Response of performance

5 Conclusions

This paper has investigated the distributed filtering design for discrete-time large-scale T-S fuzzy systems, which exchange their information through unreliable communication network. Based on a piecewise Lyapunov function and some convexifying techniques, less conservative results on the distributed piecewise filtering design were derived for the considered system in terms of LMIs. The effectiveness of the method proposed in this paper was validated by using two examples.

6 Appendix

Lemma A1.

[ 41 ] Given matrix \(0<W=W^{T}\in \Re ^{n\times n},\) two positive integers d 2 and d 1 satisfy d 2d 1≥1. Then, it yields
$${\fontsize{9pt}{6pt}{} {\begin{aligned} \left(\sum\limits_{n=d_{1}}^{d_{2}}x^{T}\left(n\right) \right)^{T}W\left(\sum\limits_{n=d_{1}}^{d_{2}}x\left(n\right) \right) \leq \bar{d}\sum\limits_{n=d_{1}}^{d_{2}}x^{T}\left(n\right) Wx\left(n\right) \end{aligned}}} $$
where \(\bar {d}=d_{2}-d_{1}+1.\)

Declarations

Acknowledgements

The authors wish to thank the Editor-in-Chief, Associate Editor, and anonymous reviewers for their helpful comments which have improved the paper.

The work was supported in part by the Natural Science Foundation of Fujian Province(2016J01267), and by the Fujian Provincial Major Scientific and Technological Projects (2014H6028), and by the Scientific Research Items of XMUT (XYK201401), and by the Advanced Research Project of XUMT (YKJ12010R).

Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors’ Affiliations

(1)
High-voltage Key Laboratory of Fujian Province, Xiamen University of Technology

References

  1. D Siljak, Large-Scale Dynamic Systems: Stability and Structure (Elsevier/North-Holland, New York, 1978).MATHGoogle Scholar
  2. D Penkin, G Janssen, A Yarovoy, Source positioning in a large-scale tiny-sensor network of arbitrary topology. EURASIP J. Adv. Sig. Process. 57:, 1–11 (2014).Google Scholar
  3. Z Hu, C Feng, T Zhang, Q Niu, Y Chen, Nonlinear joint transmit-receive processing for coordinated multi-cell systems: centralized and decentralized. EURASIP J. Adv. Sig. Process. 10:, 1–14 (2015).Google Scholar
  4. L Bakule, Decentralized control: status and outlook. Annu. Rev. Control. 38(1), 71–80 (2014).View ArticleGoogle Scholar
  5. F Kazempour, J Ghaisari, Stability analysis of model-based networked distributed control systems. J. Process Control. 23(3), 444–452 (2013).View ArticleGoogle Scholar
  6. I Guerreiro, D Hui, C Cavalcante, A distributed approach to precoder selection using factor graphs for wireless communication networks. EURASIP J. Adv. Sig. Process. 83:, 1–18 (2013).Google Scholar
  7. H Maa, H Cakmak, F Bach, R Mikut, et al., Data processing of high-rate low-voltage distribution grid recordings for smart grid monitoring and analysis. EURASIP J. Adv. Sig. Process. 14:, 1–21 (2015).Google Scholar
  8. T Tanaka, HO Wang, Fuzzy Control Systems Design and Analysis: A Linear Matrix Inequality Approach (Wiley, New York, 2001).View ArticleGoogle Scholar
  9. J Qiu, H Tian, Q Lu, H Gao, Nonsynchronized robust filtering design for continuous-time T-S fuzzy affine dynamic systems based on piecewise Lyapunov functions. IEEE Trans. Cybern. 43(6), 1755–1766 (2013).View ArticleGoogle Scholar
  10. L Li, SX Ding, J Qiu, Y Yang, Y Zhang, Weighted fuzzy observer-based fault detection approach for discrete-time nonlinear systems via piecewise-fuzzy Lyapunov functions. IEEE Trans. Fuzzy Syst. doi:10.1109/TFUZZ.2016.2514371.
  11. L Li, SX Ding, J Qiu, Y Yang, Real-time fault detection approach for nonlinear systems and its asynchronous T-S fuzzy observer-based implementation. IEEE Trans. Cybern. doi:10.1109/TCYB.2015.2513438.
  12. M Wang, J Qiu, M Chadli, M Wang, A switched system approach to exponential stabilization of sampled-data T-S fuzzy systems with packet dropouts. IEEE Trans. Cybern. doi:10.1109/TCYB.2015.2498522.
  13. J Qiu, SX Ding, H Gao, S Yin, Fuzzy-model-based reliable static output feedback control of nonlinear hyperbolic PDE systems. IEEE Trans. Fuzzy Syst (2015). doi:10.1109/TFUZZ.2015.2457934.
  14. G Feng, A survey on analysis and design of model-based fuzzy control systems. IEEE Trans. Fuzzy Syst. 14(5), 676–697 (2006).View ArticleGoogle Scholar
  15. J Qiu, H Gao, S Ding, Recent advances on fuzzy-model-based nonlinear networked control systems: a survey. IEEE Trans. Ind. Electron (2015). doi:10.1109/TIE.2015.2504351.
  16. J Qiu, G Feng, H Gao, Nonsynchronized-state estimation of multichannel networked nonlinear systems with multiple packet dropouts via T-S fuzzy-affine dynamic models. IEEE Trans. Fuzzy Syst. 19(1), 75–90 (2011).View ArticleGoogle Scholar
  17. C Zhang, G Feng, J Qiu, W Zhang, T-S fuzzy-model-based piecewise output feedback controller design for networked nonlinear systems with medium access constraint. Fuzzy Sets Syst. 248(1), 86–105 (2014).MathSciNetView ArticleMATHGoogle Scholar
  18. M Chadli, T Guerra, LMI solution for robust static output feedback control of discrete Takagi-Sugeno fuzzy models. IEEE Trans. Fuzzy Syst. 20(6), 1160–1165 (2012).View ArticleGoogle Scholar
  19. T Wang, H Gao, J Qiu, A combined adaptive neural network and nonlinear model predictive control for multirate networked industrial process control. IEEE Trans. Neural Netw. Learn. Syst (2015). doi:10.1109/TNNLS.2015.2411671.
  20. Y Shi, B Yu, Output feedback stabilization of networked control systems with random delays modeled by Markov chains. IEEE Trans. Autom. Control. 54(7), 1668–1674 (2009).MathSciNetView ArticleGoogle Scholar
  21. Y Zhu, L Zhang, W Zheng, Distributed filtering for a class of discrete-time Markov jump Lure systems with redundant channels. IEEE Trans. Ind. Electron (2015). doi:10.1109/TIE.2015.2499169.
  22. J Qiu, G Feng, H Gao, Observer-based piecewise affine output feedback controller synthesis of continuous-time T-S fuzzy affine dynamic systems using quantized measurements. IEEE Trans. Fuzzy Syst. 20(6), 1046–1062 (2012).View ArticleGoogle Scholar
  23. Y Wei, J Qiu, HR Karimi, M Wang, A new design of filtering for continuous-time Markovian jump systems with time-varying delay and partially accessible mode information. Signal Process. 93(9), 2392–2407 (2013).View ArticleGoogle Scholar
  24. C Zhang, G Feng, J Qiu, Y Shen, Control synthesis for a class of linear network-based systems with communication constraints. IEEE Trans. Ind. Electron. 60(8), 3339–3348 (2013).View ArticleGoogle Scholar
  25. C Zhang, G Feng, H Gao, J Qiu, filtering for nonlinear discrete-time systems subject to quantization and packet dropouts. Fuzzy Sets Syst. 19(2), 353–365 (2011).View ArticleGoogle Scholar
  26. J Qiu, Y Wei, HR Karimi, New approach to delay-dependent control for continuous-time Markovian jump systems with time-varying delay and deficient transition descriptions. J. Frankl. Inst. 352(1), 189–215 (2015).MathSciNetView ArticleMATHGoogle Scholar
  27. Y Zhu, Q Zhang, Z Wei, L Zhang, Robust stability analysis of Markov jump standard genetic regulatory networks with mixed time delays and uncertainties. Neurocomputing. 110(13), 44–50 (2013).View ArticleGoogle Scholar
  28. Y Shi, T Chen, Optimal design of multichannel transmultiplexers with stopband energy and passband magnitude constraints. IEEE Trans. Circ. Syst. II Analog Digit. Signal Process. 50(9), 659–662 (2003).View ArticleGoogle Scholar
  29. Y Wei, J Qiu, HR Karimi, M Wang, New approach to delay-dependent filtering for discrete-time Markovian jump systems with time-varying delay and incomplete transition descriptions. Iet Control. Theory Appl. 7(5), 684–696 (2013).MathSciNetView ArticleGoogle Scholar
  30. F Hsiao, JH Wang, C Chen, Z Tsai, Robust stabilization of nonlinear multiple time-delay large-scale systems via decentralized fuzzy control. IEEE Trans. Fuzzy Syst. 13(1), 152–163 (2005).View ArticleGoogle Scholar
  31. C Hua, S Ding, Decentralized networked control system design using T-S fuzzy approach. IEEE Trans. Fuzzy Syst. 20(1), 9–21 (2012).MathSciNetView ArticleGoogle Scholar
  32. Z Zhong, J Yu, Y He, T Hayat, F Alsaadi, Fuzzy-model-based decentralized dynamic-output-feedback control for large-scale nonlinear systems with time-varying delays. Neurocomputing (2015). doi:10.1016/j.neucom.2015.08.062.
  33. Z Zhong, S Fu, T Hayat, F Alsaadi, G Sun, Decentralized piecewise fuzzy filtering design for discrete-time large-scale nonlinear systems with time-varying delay. J. Frankl. Inst. 352(9), 3782–3807 (2015).MathSciNetView ArticleGoogle Scholar
  34. H Zhang, H Zhong, C Dang, Delay-dependent decentralized filtering for discrete-time nonlinear interconnected systems with time-varying delay based on the T-S fuzzy model. IEEE Trans. Fuzzy Syst. 20(3), 431–443 (2012).View ArticleGoogle Scholar
  35. Z Zhong, S Fu, T Hayat, F Alsaadi, G Sun, New results on decentralized fuzzy filtering design for continuous-time large-scale nonlinear systems with time-varying delay. Circ. Syst. Signal Process. 35(2), 527–552 (2016).View ArticleGoogle Scholar
  36. M Johansson, A Rantzer, K-E Å rzén, Piecewise quadratic stability of fuzzy systems. IEEE Trans. Fuzzy Syst. 7(6), 713–722 (1999).View ArticleGoogle Scholar
  37. J Qiu, G Feng, H Gao, Static output feedback control of continuous-time T-S fuzzy affine systems via piecewise Lyapunov functions. IEEE Trans. Fuzzy Syst. 21(2), 245–261 (2013).View ArticleGoogle Scholar
  38. J Qiu, G Feng, H Gao, Fuzzy-model-based piecewise static-output-feedback controller design for networked nonlinear systems. IEEE Trans. Fuzzy Syst. 18(5), 919–934 (2010).View ArticleGoogle Scholar
  39. J Qiu, G Feng, J Yang, A new design of delay-dependent robust filtering for discrete-time T-S fuzzy systems with time-varying delay. IEEE Trans. Fuzzy Syst. 17(5), 1044–1058 (2009).View ArticleGoogle Scholar
  40. H Zhang, S Liu, D Liu, Fuzzy filter design for a class of nonlinear discrete-time systems with multiple time delays. IEEE Trans. Fuzzy Syst. 15(3), 453–469 (2007).View ArticleGoogle Scholar
  41. X Jiang, Q Han, X Yu, in Proceedings of the 2005 IEEE Conference on American Control, 4. Stability criteria for linear discrete-time systems with interval-like time-varying delay, (2005), pp. 2817–2822, doi:10.1109/ACC.2005.1470396.

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© Shao and Chen. 2016