Mathematical Problems in Engineering

Volume 2015 (2015), Article ID 380903, 12 pages

http://dx.doi.org/10.1155/2015/380903

## Robust Exponential Synchronization for a Class of Master-Slave Distributed Parameter Systems with Spatially Variable Coefficients and Nonlinear Perturbation

^{1}School of Informatics, Linyi University, Linyi 276005, China^{2}Provincial Key Laboratory for Network Based Intelligent Computing, Jinan 250022, China^{3}School of Science, Linyi University, Linyi 276005, China^{4}Department of Electrical and Computer Engineering, University of Rhode Island, Kingston, RI 02881, USA^{5}Science and Technology on Underwater Acoustic Antagonizing Laboratory, Systems Engineering Research Institute of CSSC, Beijing 100036, China^{6}School of Automobile Engineering, Linyi University, Linyi 276005, China^{7}School of Electronic and Information Engineering, Xi’an Jiaotong University, Xi’an 710049, China

Received 16 October 2014; Revised 12 April 2015; Accepted 15 April 2015

Academic Editor: Anna Vila

Copyright © 2015 Chengdong Yang et al. 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

This paper addresses the exponential synchronization problem of a class of master-slave distributed parameter systems (DPSs) with spatially variable coefficients and spatiotemporally variable nonlinear perturbation, modeled by a couple of semilinear parabolic partial differential equations (PDEs). With a locally Lipschitz constraint, the perturbation is a continuous function of time, space, and system state. Firstly, a sufficient condition for the robust exponential synchronization of the unforced semilinear master-slave PDE systems is investigated for all admissible nonlinear perturbations. Secondly, a robust distributed proportional-spatial derivative (P-sD) state feedback controller is desired such that the closed-loop master-slave PDE systems achieve exponential synchronization. Using Lyapunov’s direct method and the technique of integration by parts, the main results of this paper are presented in terms of spatial differential linear matrix inequalities (SDLMIs). Finally, two numerical examples are provided to show the effectiveness of the proposed methods applied to the robust exponential synchronization problem of master-slave PDE systems with nonlinear perturbation.

#### 1. Introduction

The study on master-slave systems has become more important for theoretical and practical points in many fields, including communication, mechanical systems, robotics, chemical reactions, and biological systems [1–9]. Ever since the discovery of Christian Huygens in 1665 on the synchronization of two pendulum clocks [10], synchronization has received considerable attention for a long time as a typical collective behavior and a basic motion in nature with potential applications in many different areas including secure communication, chaos generators design, chemical reactions, biological systems, and information science [11–26]. The theory of synchronization for master-slave systems, which aims to control the slave system so that the output of the slave system follows the output of the master system [27], is a recent research area extensively investigated nowadays in many industrial and technical processes, such as unmanned aerial vehicle (UAV) team, vehicular platoons, rendezvous of space shuttles, and many other practical control systems (see, e.g., [28, 29] and the references therein).

The existing works most considered dynamical behavior of master-slave systems described by ordinary differential equations (ODEs) or delay differential equations (DDEs), and a variety of synchronization criteria have been presented [14–16]. In practice, however, the outputs, inputs, and process states with relevant parameters usually vary both temporally and spatially in nature, and thereby their behavior depending on time and spatial position could be described by distributed parameter systems (DPSs) modeled by partial differential equations (PDEs). Unfortunately, few works have investigated the synchronization of master-slave PDE systems.

As a result of the infinite-dimensional nature of master-slave PDE systems, the existing finite-dimensional control theory and techniques for the master-slave ODE systems are difficult to be directly employed for the control design of master-slave PDE systems. In this situation, it is important to study the synchronization problem of master-slave PDE systems. Therefore, some researchers have paid attention to the study of synchronization of master-slave PDE systems [17–29], where “design-then-reduce” approach was employed to take the full advantage of the original PDE model for the controller design [30–34]. References [35–37] researched synchronization of neural networks with reaction-diffusion terms. Yuan et al. proposed synchronization of the coupled distributed parameter system with time delay via P-sD control [38]. Yang et al. proposed exponential synchronization for complex spatiotemporal networks with space-varying coefficients via P-sD control [39]. Wang et al. studied exponential synchronization for networked linear PDE systems via boundary control [40]. However, to the best of our knowledge, few results are available on the exponential synchronization of master-slave PDE systems with nonlinear perturbation and spatially variable coefficients, which motives the present investigation.

In this paper, we will deal with the problem of robust exponential synchronization for master-slave parabolic PDE systems with spatially variable coefficients and spatiotemporally variable nonlinear perturbation. Initially, a master-slave parabolic PDE model is discussed, and then a synchronization error dynamic of PDE systems is developed in spatial domain. Then, on the basis of Lyapunov’s direct method and the technique of integration by parts, a spatial differential linear matrix inequality (SDLMI) based condition for the robust exponential synchronization of the unforced semilinear master-slave PDE systems is studied. Once the unforced semilinear master-slave PDE systems cannot achieve robust exponential synchronization, distributed proportional-spatial derivative (P-sD) control design is developed to achieve that the closed-loop slave system which is exponentially synchronized with the master system with a given decay rate for all admissible nonlinear perturbations in terms of SDLMI. Furthermore, the SDLMI optimization problem can be approximately solved by the finite difference method and LMI optimization techniques [41, 42]. Finally, the simulation study on the exponentially synchronous control of a master-slave PDE systems with nonlinear perturbation is given to show the effectiveness of the proposed design method.

The remainder of this paper is organized as follows. The problem formulation and preliminaries are given in Section 2. Section 3 provides the exponential synchronization analysis of the unforced master-slave PDE systems. Robust P-sD controller is designed in Section 4. Section 5 presents two examples on master-slave PDE systems to illustrate the effectiveness of the proposed method. Finally, Section 6 offers some concluding remarks.

*Notations.* The following notations will be employed throughout this paper. , , and denote the set of all real numbers,* n*-dimensional Euclidean space, and the set of all matrices, respectively. and denote the Euclidean norm and inner product for vectors, respectively. Identity matrix, of appropriate dimensions, will be denoted by . For a symmetric matrix , (, , resp.) means that is positive definite (negative definite, negative semidefinite, resp.). The space-varying symmetric matrix function , , is positive definite (negative definite, negative semidefinite, resp.), if (, , resp.) for each . represents the eigenvalue of a matrix. The superscript “” is used for the transpose of a vector or a matrix. is a Sobolev space of absolutely continuous* n*-dimensional vector functions with square integrable derivatives of the order and with the norm . The symbol “” is used as an ellipsis in matrix expressions that are induced by symmetry; for example,

#### 2. Problem Formulation

Consider the following synchronization scheme of semilinear master-slave PDE systems with spatially variable coefficients and spatiotemporally variables as follows.

The master system is as follows:and the slave system is as follows:where , are the state vectors, the subscripts and stand for the partial derivatives with respect to , , respectively, and are the spatial position and time, respectively, and is the distributed control input. , , and , , are real known matrix functions. , , is a real known matrix function. is* n*-dimensional real vector-valued spatiotemporally variable nonlinear perturbation with for all and . Moreover, assume that satisfies the following Lipschitz condition as follows.

*Assumption 1. * is continuous with respect to its arguments and is locally Lipschitz continuous in on ; that is, there exists a scalar such that the following inequality holds for all , and : where . Therefore, we have thatDefine the synchronization error , and we have the following synchronization error system :where .

Choose as the state space and the trajectory segment as the state. Define the spatial differential operator in aswith its domainLet be control input space and the trajectory segment as the control input. Then the state-space description of the synchronization error system (6) can be rewritten as the following nonlinear abstract differential equation on the Hilbert space [43]:where , , , and .

Using (8), we get from (6) thatwhere .

We introduce the following definition of the exponential synchronization for the master-slave systems (2) and (3) in the sense of norm .

*Definition 2. *For a given constant , the master-slave systems (2)-(3) achieve * ρ-exponential synchronization* or

*exponential synchronization with a given decay rate*, if there exists a constant such that the following inequality holds for any initial condition , :

*ρ*It is easily seen from Definition 2 that the master-slave systems (2)-(3) achieve exponential synchronization with a given decay rate * ρ* if and only if the error system (6) is exponentially stable with a given decay rate

*.*

*ρ*For simplicity, when , the master-slave systems (2)-(3) are referred to as* unforced* master-slave systems.

Lemma 3. *For any two square integrable vector functions , , , the following inequality holds for any positive scalar function defined on interval :*

*Proof. *It is easily found that the inequality holds for any . Hence, which impliesIntegrating inequality (14) from to , we can obtain thatwhich means that inequality (12) holds. The proof is complete.

*3. Exponential Synchronization Analysis*

*This section aims to analyze exponential synchronization for the unforced semilinear master-slave PDE systems (2)-(3).*

*We consider the following Lyapunov functional for the unforced synchronization error system (10):where is a real matrix function to be determined. The time derivative of along the solution of system (10) is given byBy integrating by parts and taking into account the boundary condition in (10), we haveAccording to the definition of in (7) and considering (18), we haveBy Assumption 1 and Lemma 3, for any positive scalar function , , we have Substituting (19), (20) into (17), we obtainwhere and*

*Theorem 4. Consider the unforced semilinear master-slave PDE systems (2) and (3) under Assumption 1. For a given scalar , the master-slave PDE systems (2) and (3) achieve exponential synchronization with a given decay rate ρ, if there exist a matrix function and a scalar function satisfying the following SDLMI in space:where*

*Proof. *Assume that SDLMI (23) is satisfied for matrix functions and . Using Schur complement, the following inequality is achieved if SDLMI (23) is satisfied for each :Substituting inequality (25) into (21) yields Integration of (26) from to derivesSince is a spatially continuous matrix function of defined on , it is easily observed that given by (18) satisfies the following inequality:where and are two positive constants. Using (28), we can get the following relation:Therefore, we haveThus, from (30) and Definition 2, the unforced semilinear master-slave PDE systems (2) and (3) achieve exponential synchronization with a given decay rate * ρ*. The proof is complete.

*Theorem 4 presents an SDLMI-based condition for the robust exponential synchronization of the unforced semilinear master-slave PDE systems (2) and (3). Once the unforced semilinear master-slave PDE systems (2) and (3) cannot exponentially synchronize by themselves, it is desired to design a distributed P-sD controller in the semilinear master-slave PDE systems (2) and (3). Moreover, the SDLMI feasibility problem is approximately solved via the finite difference method and the existing LMI optimization techniques [41, 42].*

*4. P-sD Control Design*

*Once the semilinear master-slave PDE systems (2)-(3) are not exponentially synchronized by themselves, a state feedback control design is therein desired. The aim of this section is to propose a distributed P-sD state feedback control design method to achieve exponential synchronization for semilinear master-slave PDE systems (2)-(3).*

*This study considers the following distributed P-sD state feedback controller of the slave system (3) as shown in Figure 1:where , , , are continuous matrices to be determined.*