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

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


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][2][3][4][5][6][7][8][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][12][13][14][15][16][17][18][19][20][21][22][23][24][25][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][15][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 masterslave 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][18][19][20][21][22][23][24][25][26][27][28][29], where "design-then-reduce" approach was employed to take the full advantage of the original PDE model for the controller design [30][31][32][33][34]. References [35][36][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 masterslave 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.

Problem Formulation
Consider the following synchronization scheme of semilinear master-slave PDE systems with spatially variable coefficients and spatiotemporally variables as follows.
We introduce the following definition of the exponential synchronization for the master-slave systems ( 2) and ( 3) in the sense of norm ‖ ⋅ ‖ 2 .
Definition 2. For a given constant  > 0, the master-slave systems ( 2)-( 3) achieve -exponential synchronization or exponential synchronization with a given decay rate , if there exists a constant  > 0 such that the following inequality holds for any initial condition e 0 (),  ∈ [ 1 ,  2 ]: It is easily seen from Definition 2 that the master-slave systems (2)-(3) achieve exponential synchronization with a given decay rate  if and only if the error system ( 6) is exponentially stable with a given decay rate .

Lemma 3. For any two square integrable vector functions a(𝑥), b(𝑥)
Proof.It is easily found that the inequality which implies Mathematical Problems in Engineering Integrating inequality (14) from  1 to  2 , we can obtain that which means that inequality (12) holds.The proof is complete.

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 P() > 0 is a real  ×  matrix function to be determined.The time derivative of () along the solution of system ( 10) is given by Substituting (19), (20) into (17), we obtain ( Theorem 4. Consider the unforced semilinear master-slave PDE systems ( 2) and ( 3) under Assumption 1.For a given scalar  > 0, the master-slave PDE systems ( 2) and ( 3) achieve exponential synchronization with a given decay rate , if there exist a matrix function P() > 0 and a scalar function () > 0 satisfying the following SDLMI in space: where Proof.Assume that SDLMI ( 23) is satisfied for matrix functions P() > 0 and () > 0. Using Schur complement, the following inequality is achieved if SDLMI ( 23) is satisfied for each  ∈ [ 1 ,  2 ]: Substituting inequality ( 25) into (21) yields Integration of ( 26) from 0 to  derives Since P() > 0 is a spatially continuous matrix function of  defined on [ 1 ,  2 ], it is easily observed that () given by ( 18) satisfies the following inequality: where Therefore, we have Thus, from (30) and Definition 2, the unforced semilinear master-slave PDE systems (2) and (3) achieve exponential synchronization with a given decay rate .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 masterslave 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].

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 U of the slave system S (3) as shown in Figure 1: where K(), L() ∈ R × ,  ∈ [ 1 ,  2 ], are continuous matrices to be determined.
Remark 5.The controller U (31) of the slave system S needs distributed sensing and actuation, whereas it has become feasible nowadays to produce integrated control circuitry with a large number of microsensors and actuators.It has been pointed out in [33,34,38,39] that the controller U (31) with spatial derivative term provides more spatial performance and that is why the controller U (31) is chosen in this paper.
Using the Schur complement, SDLMI (37) is equivalent to the inequality where pre-and postmultiplying (40) by diag{Q −1 (), Q −1 ()}, respectively, with application of Assumption 6, and considering the property , and the commutative law of matrix multiplication [47], we can obtain we can get the following inequality from (40) and (42): Substituting inequality ( 43) into (35) yields The rest of proof of Theorem 7 is similar to that of Theorem 4, and thus we can also get the conclusion that the semilinear master-slave PDE systems (2) and (3) achieve exponential synchronization with a given decay rate  with the suitable controller (31).Moreover, from (41), we have (39).The proof is complete.
Theorem 7 presents an SDLMI-based condition for the existence of a distributed controller (31) for the exponential synchronization of the semilinear master-slave PDE systems (2) and (3).Explicit expressions of a desired controller are proposed when the SDLMI (37) is feasible.Remark 8. Notice that an SDLMI control design has been more recently proposed in [33,34] for a class of PDE systems.
Different from the stable control design for PDE system in [33,34], this paper provides an SDLMI-based sufficient condition on the exponential synchronization for masterslave PDE systems.
Remark 9. Notice that [38][39][40] dealt with exponential synchronization of linear models, while this paper considers exponential synchronization of the model with nonlinear perturbation.
Remark 10.For a special case, when the master system M: y  (, ) ≡ 0, we have z(, ) = e(, ), and then the exponential synchronization of the master-slave systems is equal to exponential stability of the slave system S.In other words, Theorem 4 provides the sufficient condition of exponential stability of the slave system S with a given decay rate  when the master system y(, ) ≡ 0, while Theorem 7 also contains the way of the P-sD controller (31) design for exponential stability.

Numerical Simulation
In this section, in order to show the effectiveness of Theorems 4 and 7, we consider the following two examples.
It is clear from Figure 6 that the proposed distributed state feedback controller (31) can ensure that the semilinear master-slave PDE systems (2) and (3) with the parameters given in (48) under the initial condition given in (49) achieve exponential synchronization with a given decay rate .Moreover, the evolution profile of the distributed control u(, ) is shown in Figure 7.

Conclusions
In this paper, we have addressed the exponential synchronization problem of semilinear master-slave PDE systems with spatially variable coefficients.An SDLMI-based sufficient condition for the robust exponential synchronization of the unforced semilinear master-slave PDE systems has been presented.Moreover, when the master-slave PDE systems are not exponentially synchronized, it has been proved in detail that P-sD controllers with appropriate gain parameters can guide the closed-loop master-slave PDE systems to exponentially synchronize with a given decay rate  if a given SDLMI condition is fulfilled.Finally, numerical examples showed the effectiveness of the proposed methods.