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Distributed piecewise filtering design for largescale networked nonlinear systems
EURASIP Journal on Advances in Signal Processing volume 2016, Article number: 51 (2016)
Abstract
This paper investigates the problem of distributed piecewise filtering for discretetime largescale nonlinear systems. The considered largescale system is composed of a number of nonlinear subsystems and exchanges its information through communication network. Each nonlinear subsystem is described by a TakagiSugeno (TS) model, and datapacket dropouts happen intermittently in communication network, and its stochastic variables are assumed to satisfy the Bernoulli randombinary 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.
Introduction
In practical application, some complex systems, such as transportation systems, power systems, communication networks, and industrial processes, are referred to as largescale systems [1, 2]. Due to strong interconnection and high dimensionality, largescale 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 largescale 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 largescale systems [3]. The decentralized control is firstly to partition the overall control problem of a largescale 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 largescale 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, TakagiSugeno (TS) model has been proved to be a powerful solution to represent any smooth nonlinear functions at any preciseness [8, 9]. The TS model employs a group of IFTHEN 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. TS 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 TS 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 pointtopoint connections due to their great advantages, such as simple maintenance and installation, and low cost [20–24]. Unfortunately, the networkinduced 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, TS fuzzy control has been developed to investigate largescale nonlinear systems [30–35]. To mention a few, some results on analysis and synthesis methods for decentralized control of largescale systems have been presented in [30–32]. In [33, 34], the decentralized filtering problem was studied for the discretetime largescale system with timevarying delay. To the best knowledge of the authors, few results on the distributed filtering design have been given for largescale networked TS 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 discretetime largescale nonlinear systems. The largescale system is composed of several nonlinear subsystems and exchanges its information through communication network. Each nonlinear subsystem is described by a TS model, and datapacket dropouts occur intermittently in communication network, and its stochastic variables satisfy the Bernoulli randombinary 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 ndimensional 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 squaresummable infinite vector sequences over \([0,\infty).\) diag{ ···} is a blockdiagonal 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.
Model description and problem formulation
This paper considers a class of discretetime largescale systems, which consist of N nonlinear subsystems, the ith nonlinear subsystem is described by the TS model as below.
Plant rule : IF ζ _{ i1}(t) is and ζ _{ i2}(t) is and ⋯ and ζ _{ ig }(t) is , THEN
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
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 TS fuzzy subsystem as below:
where
In this paper, we will address the distributed filtering design problem of the discretetime largescale 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 lth 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 TS fuzzy system in (3) can be rewritten as
where
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:
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 \(\mathcal {S}_{ij}.\)
Given the largescale fuzzy system (5) with the premise variable space partition, we propose a distributed piecewise filter of the following form:
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 fullorder filter, and 1≤n _{ fi }<n _{ xi } for the reducedorder 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 dataloss condition based on a stochastic approach, thus, it yields [38]:
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
In addition, let us define
It is easy to see from (11) that
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 largescale fuzzy system in (5) and distributed piecewise filter in (8), the filtering error system is given by
where
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
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) The filtering error system in (13) is stochastically stable in the sense of Definition 1;

■■■
2) Under zeroinitial 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.
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 fullorder and reducedorder filters by solving a number of LMIs.
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:
where
Proof.
Choose the following piecewise Lyapunov functional (PLF):
where , are positive definite symmetric Lyapunov matrices.
Define Δ V _{ i }(t)=V _{ i }(t+1)−V _{ i }(t), one has
It follows from the error system in (13) that
where .
Note that
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
Similarly, by introducing \(0<M_{ikj}<M_{ik0}\in \Re ^{n_{wi}\times n_{wi}},\) it yields
Given the following index
It follows from (19)–(21) and (23)–(25) that
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
and
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 zeroinitial 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 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:
where
Furthermore, a distributed piecewise filter in the form of (8) is given by
Proof.
For matrix inequality linearization purpose, the multipliers \(\mathcal {G}_{i},i\in \mathcal {N}\) are firstly specified by
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
by Γ, it yields
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
It is easy to see that the matrix variable is absorbed by the filter gain variables when introducing
with .
It is noted that the first blockrow 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,
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
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.
Simulation examples
In the following, let us consider two examples to illustrate the result proposed in this paper.
Example 1.
Consider a discretetime largescale TS model with three fuzzy rules as below:
Plant rule : IF x _{ i1}(t) is , THEN
where
for the first subsystem, and
for the second subsystem.
Figure 2 shows the membership functions. According to the premise variable space partition, there are three subspaces for each subsystem:
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 fullorder 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 fullorder filter and \(\gamma _{\min }=6.5599\) for the reducedorder filter are obtained, and the corresponding filter gains are
for fullorder filter and
for the reducedorder case.
Example 2.
Consider a modified doubleinverted pendulum, which is connected by a spring [34]. The motional equations of the interconnected pendulum are given by
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/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 TS 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 closedloop TS fuzzy subsystems are stable. Then, by discretizing the TS fuzzy systems with sampling period T=0.01 s, the discretetime interconnected TS fuzzy system can be obtained as below:
Plant rule : IF \(\left \vert x_{i1}(t)\right \vert \) is , THEN
where
for the first subsystem and
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:
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 fullorder 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
and
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 datapacket dropouts are shown in Figs. 6 and 7, respectively. Then, it can also be shown in Fig. 8 that the performance is satisfactory under zeroinitial conditions thus verifying the effectiveness of the distributed filtering design.
Conclusions
This paper has investigated the distributed filtering design for discretetime largescale TS 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.
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
where \(\bar {d}=d_{2}d_{1}+1.\)
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Acknowledgements
The authors wish to thank the EditorinChief, 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).
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Shao, Z., Chen, T. Distributed piecewise filtering design for largescale networked nonlinear systems. EURASIP J. Adv. Signal Process. 2016, 51 (2016). https://doi.org/10.1186/s1363401603502
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Keywords
 Largescale fuzzy systems
 Distributed filter
 Piecewise Lyapunov function