- Zhenhua Shao
^{1}Email author and - Tianxiang Chen
^{1}

**2016**:51

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

© Shao and Chen. 2016

**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

## 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 [10–13]. 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 [14–19].

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 [20–24]. 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 [25–29]. 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 [30–35]. To mention a few, some results on analysis and synthesis methods for decentralized control of large-scale systems have been presented in [30–32]. 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 *i*th nonlinear subsystem is described by the T-S model as below.

where \(i\in \mathcal {N}:=\{1,2,\ldots,N\}\);
denotes the *l*th 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 *i*th subsystem; (*A*
_{
il
},*B*
_{
il
},*C*
_{
il
},*D*
_{
il
},*L*
_{
il
}) denotes the *l*th local model for the *i*th subsystem; \(\bar {A}_{ikl}\) is the interconnection matrix between the *i*th and *k*th subsystems.

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 \(\mu _{il}:=\mu _{il}\left [ \zeta _{i}(t)\right ].\)

*i*th global T-S fuzzy subsystem as below:

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.

*i*th 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

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.

*Ω*

_{ i }, which represents all possible region transitions for the

*i*th subsystem:

where *j*≠*s* 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 \(\mathcal {S}_{ij}.\)

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.

*y*

_{ i }(

*t*) is no longer equivalent to

*y*

_{ Fi }(

*t*). The

*i*th filtering subsystem with unreliable communication network is shown in Fig. 1, where the

*i*th 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]:

###
**Definition**
**1**.

*W*>0 such that

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.

*γ*>0, design a distributed piecewise filter in the form of (8) such that the following two conditions are satisfied simultaneously.

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

- ■■■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**.

*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:

###
*Proof*.

*Δ*

*V*

_{ i }(

*t*)=

*V*

_{ i }(

*t*+1)−

*V*

_{ i }(

*t*), one has

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

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).

*t*≥0, under zero-initial conditions it yields

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**.

*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:

###
*Proof*.

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

*Γ*, it yields

*G*

_{ i j(4)}=

*G*

_{ i j(2)}. Thus, we can directly specify the multipliers \(\mathcal {G}_{i}\) as

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.

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**

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 .

*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

###
**Example**
**2**.

*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/s^{2}. 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}=−18*x*
_{11}−16*x*
_{12} and *u*
_{2}=−20*x*
_{21}−14*x*
_{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**

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
.

*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

*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.

## 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*

_{2}≥

*d*

_{1}≥1. Then, it yields

## 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

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