Mathematical Problems in Engineering

Volume 2013 (2013), Article ID 874085, 18 pages

http://dx.doi.org/10.1155/2013/874085

## Control of Pairwise Distributable Large-Scale TS Fuzzy Systems

Department of Cybernetics and Artificial Intelligence, Faculty of Electrical Engineering and Informatics, Technical University of Košice, Letná 9, 042 00 Košice, Slovakia

Received 12 July 2013; Revised 1 October 2013; Accepted 3 October 2013

Academic Editor: Hongli Dong

Copyright © 2013 Anna Filasová and Dušan Krokavec. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

The paper presents new conditions suitable in design of the stabilizing state controller for a class of continuous-time nonlinear systems, which are representable by pairwise distributable Takagi-Sugeno models. Taking into account the affine properties of the TS model structure and applying the pairwise subsystems fuzzy control scheme relating to the parallel distributed output compensators, the extended bounded real lemma form and the sufficient design conditions for pairwise decentralized control are outlined in terms of linear matrix inequalities. The proposed procedure decouples the Lyapunov matrix and the system parameter matrices in the LMIs and, using free tuning parameter, provides the way to obtain global stability of such large-scale TS systems and optimizes subsystems interaction norm bounds.

#### 1. Introduction

A number of problems that arise in state control can be reduced to a handful of standard convex and quasiconvex problems that involve matrix inequalities (LMI). It is known that the optimal solution can be computed by using the interior point methods [1], which converge in polynomial time with respect to the problem size. Thus, efficient interior point algorithms have recently been developed and further development of algorithms for these standard problems is an area of active research. For this approach, the stability conditions may be expressed in terms of LMIs, which have a notable practical interest due to the existence of numerical solvers. Some progress review in this field can be found, for example, in [2, 3] and the references therein.

Based on the concept of quadratic stability, the control design problems, in respect of the norm of the closed-loop system transfer matrix, are transferred into a standard LMI optimization task, which includes bounded real lemma (BRL) formulation. The first version of BRL presented simple condition under which a transfer function was contractive on the imaginary axis of the complex variable plain. Using this formulation, it was possible to determine the norm of a transfer function, and the BRL has become a significant element to show and prove that the existence of feedback controllers, resulting in a closed-loop transfer matrix having the norm less than a given upper bound, is equivalent to the existence of a solution of certain LMIs. Motivated by the underlying ideas, the technique for BRL representation was extended to state feedback control design and stayed preferable for systems with time-varying parameters [4–6]. When used in robust analysis of linear systems with polytopic uncertainties, as the number of polytops increases, the solution turned out to be very conservative. To reduce conservatism inherent in such use of quadratic methods, the equivalent LMI representations of BRL for continuous-time as well as discrete-time uncertain systems were introduced [7–10]. Moreover, exploiting the sector nonlinearity approach to obtain Takagi-Sugeno (TS) models from the nonlinear system equations [11], suitable BRLs as well as enhanced BRL representation guaranteeing quadratic performances for the closed-loop nonlinear systems are exploited in fuzzy control [12–14].

In last years, modern control methods have found their way into design of interconnected systems and led to a wide variety of new concepts and results. In particular, paradigms of LMIs and norm have appeared to be very attractive due to their promise of handling systems with relative high dimensions, and design of partly decentralized schemes substantially minimized the information exchange between subsystems of a large scale system. With respect to the existing structure of interconnections in a large-scale system, it is generally impossible to stabilize all subsystems and the whole system simultaneously by using decentralized controllers, since the stability of interconnected systems is not only dependent on the stability degree of subsystems but is also closely dependent on the interconnections [15–17]. Analogous principles were applied in decentralized fuzzy control [18].

Considering the decomposition-based control strategy [19] and including into design step the effects of subsystem pairs interconnections [20], a pairwise decentralized control problem was proposed for linear large-scale systems in [21] and for linear large-scale systems with polytopic uncertainties in [22], respectively. Introducing the results as a pairwise partially decentralized control, in [21] there were formulated design conditions for a linear control, while the design conditions given in [22] reflect the robust linear control design task, both solved in the frames of LMI representations. Since the feedback control for TS models is the so-called parallel distributed compensation (PDC) control, the above results have to be significantly reformulated considering the pairwise distributable large-scale TS fuzzy systems.

Inspired by the enhanced design conditions proposed in [13] in designing the optimal control of TS systems with quadratic optimality criterion, the paper is devoted to studying the partially decentralized control problems from the above given viewpoint for pairwise distributable large-scale TS fuzzy systems. Sufficient stability conditions are stated now as a set of LMIs to encompass the quadratic stability case in respect of the approach. Used structures in the presented forms enable us potentially to design systems with an embedded reconfigurable control structure property [21]. To the best of the authors’ knowledge, the paper presents a new formulation of the control principle of pairwise distributable large-scale TS fuzzy systems, newly defines the conditions of existence of solutions for the fuzzy control scheme relating to PDCs in such distributable structure, and offers the possibility of new ways to solve the problem of control law synthesis for TS fuzzy systems with a large number of membership functions.

The paper is organized as follows. In Section 2, the basis preliminaries, concerning the TS models and the norm problems, are presented with results on BRL and enhanced BRL for TS systems. Formulating the pairwise distributable large-scale TS fuzzy systems structure in Section 3 and continuing with this formalism in Section 4, the equivalent TS BRL design methods are outlined to possess the sufficient conditions for the pairwise decentralized control of given class of TS large-scale systems. Finally, the example is given in Section 5, to illustrate the feasibility and properties of the proposed method and some concluding remarks are stated in Section 6.

Throughout the paper, the following notations are used: , denote the transposes of the vector and matrix , respectively, , denotes a block diagonal matrix with blocks, , entails a row- and column-wise partitioned matrix, broken into blocks by partitioning its rows and columns into collection of row-groups and collection of column-groups, for a square matrix, means that is a symmetric negative definite matrix, the symbol indicates the th order unit matrix, denotes the set of real numbers, and refers to the set of real matrices.

#### 2. Preliminaries

##### 2.1. System Model

The class of TS systems, considered in the paper, is formed as follows: where , , and are vectors of the state, input, and measurable output variables, respectively, matrices , , and are real matrices, is the time variable, and is the weight for th fuzzy rule. Satisfying the following, by definition, the property is the vector of the premise variables, where , are the numbers of fuzzy rules and premise variables, respectively. It is supposed next that all premise variables are measurable and independent on (more details can be found, e.g., in [11, 13]).

##### 2.2. LMI Formulations for Performance

Let the TS systems model (1), (2) be considered (in this subsection only) in the next extended form where , , and is the disturbance input that belongs to and, using the same set of membership functions, the fuzzy state control law is defined as Evidently, the closed-loop system state variable dynamics is described by the next equation and also, owing to the symmetry in summations, by the symmetric equation where Thus, adding (8), (9) gives Rearranging the computation, (10) can be written as respectively.

Considering this, the next lemmas can be introduced.

Lemma 1 (bounded real lemma). *The closed-loop system (7), (5) is stable with the performance if there exist a positive definite matrix and a positive scalar such that
**
for all , and , , respectively, where , are identity matrices and is an upper bound of norm of the disturbance transfer matrix function.**Here and hereafter, denotes the symmetric item in a symmetric matrix.*

*Proof. *Defining Lyapunov function as
where , , , and evaluating the derivative of with respect to along a system trajectory, then it yields
Substituting (5) and (12) in (17), the next inequality is obtained:
and with the notation
it yields
where
Since (20) implies
for all and , , respectively, then using the Schur complement property, (22) can be written as
Defining the transform matrix as
premultiplying the left-hand side and the right-hand side of (23) by , gives
Thus, using the substitution then (25) and (26) imply (14) and (15), respectively. This concludes the proof.

*Remark 2. *Another form of bounded real lemma can be obtained using Lyapunov function of the form
(compare, e.g., [6, 12, 13, 18]), but the BRL forms implying from (27) result, in general, the higher norm upper bound then using the presented conditions (14), (15).

Lemma 3 (enhanced bounded real lemma). *The closed-loop system (7), (5) is stable with the performance if for given , , there exist positive definite matrices and a positive scalar such that**for all and , , respectively.*

*Proof. *Since (12) implies
then, with arbitrary regular symmetric square matrices , it yields Thus, adding (32), as well as its transposition to (17), and substituting (5) into (17) give
and using the notation
can be obtained the following:
whereSince (35) implies
for all and , , respectively, using the Schur complement property, (37) can be written as Since , are supposed to be regular and symmetric, the transform matrix can be defined as
and pre-multiplying the left-hand and the right-hand side of (38) by gives
Thus, using the notation
(40), (41) imply (29), (30). This concludes the proof.

This enhanced form of the bounded real lemma eliminates products of a Lyapunov matrix and the system matrix parameters , , , and in the LMI stability conditions. When used in the synthesis of controllers or observers for TS systems (that are, evidently, the systems with polytopic uncertainties), the enhanced form of BRL gives solutions that are less conservative than ones given by the standard form of BRL. In that sense, the enhanced form of BRL can be preferred in TS systems analysis and design [13] giving less conservative equivalency to the synthesis based on the parameter dependent Lyapunov functions principle [4].

#### 3. Pairwise Distributed Principle in Control Design

The main property of the pairwise distributable structure of TS large-scale systems is given by the next two lemmas.

Lemma 4. *Let the system (1), (2) is structured in subsystems in such way that, for , , ,
**
where , , , , , and , respectively, and , , and .**Then the th unforced subsystem pair in unforced system (1), (2), and (44) is stable if there exists a set of symmetric matrices
**
such that
**
where
*

*Proof. *Defining Lyapunov function as
where , , then the time derivative of along a solution of the system (1), (2) is
Considering of the next form
then the next separation is possible
Using (49) and writing (45) as
and considering that for an unforced regime , , then (55) implies (50). Subsequently, (1), (52), and (54) give
and the inequality (56) implies (48). This concludes the proof.

Lemma 5. *System (1), (2), (44) with the pairwise distributed control takes the form
**
where
*

*Proof. *Considering in (6) the same structure of , , as is defined for in (53), that is,
then the control law takes the form
where , , are non-zero gain matrices.

Exploiting the main diagonal block property (63), then
where for , ,
Defining for , and with respect to the notations (59) that
and combining (64) for and , is obtained the following:
respectively. Then, substituting (69) in (55) gives
where
Using (59) and denoting
then (71) implies (57).

Rewriting (72) in the form
and using (61), then (74) implies (60).

Finally, using (62), it can be written as
and (76) implies (58). This concludes the proof.

#### 4. Pairwise Control Law Parameter Design

The design conditions, formulated as the set of LMIs, imply from the next theorems.

Theorem 6. *Controlled subsystem pair (57), (58) in the system (1), (3) is stable with performances , if there exist a symmetric positive definite matrix , matrices , and positive scalars such that
**
for , , ,
**
for , , , , , **for , , , , , , , with , , , , and defined in (59), (61), and (62), respectively,
**
and where, for given , , , and , are not included into the structure of (78).**When the above conditions hold, the gain matrices are given by
*

*Proof. *Considering , with given in (60), as a generalized disturbance acting on the subsystem pair (57), (58), and introducing the notations
(57) takes the form
Analogously, (58) can be rewritten as
where
Since (56) also gives
defining the matrices
(18) giveswhere, for simplicity, the argument of membership functions was omitted.

Therefore, inserting appropriate into (14), (15), it is obtained the following: where
Thus, with the substitutions
(89)-(90) imply (78)-(79). This concludes the proof.

Theorem 7. *Controlled subsystem pair (57), (58) in the system (1), (3) is stable with performances , if for given , there exist symmetric positive definite matrices , matrices , and positive scalars such that
**
for , , , **for , , , , , **for , , , , , , , with , , , , and defined in (59), (61), and (62), respectively, and with
**
where, for given , , , and *