Dissipativity-Based Controller Design for Time-Delayed TS Fuzzy Switched Distributed Parameter Systems

This paper studies fuzzy controller design problem for a class of nonlinear switched distributed parameter systems (DPSs) subject to time-varying delay. Initially, the original nonlinear DPSs are accurately described by Takagi-Sugeno fuzzy model in a local region. On the basis of parallel distributed compensation technique, mode-dependent fuzzy proportional and fuzzy proportional-spatialderivative controllers are constructed, respectively. Subsequently, using single Lyapunov-Krasovskii functional and some matrix inequality methods, sufficient conditions that guarantee the stability and dissipativity of the closed-loop systems are presented in the form of linear matrix inequalities, which allow the control gain matrices to be easily obtained. Finally, numerical examples are provided to demonstrate the validity of the designed controllers.


Introduction
Different from lumped parameter systems, the dynamic behaviors of many industrial processes depend on not only time but also the space, for which the mathematical models are described by various partial differential equations (PDEs) such as parabolic, hyperbolic, elliptic, etc. [1].In the past decades, considerable research has been conducted for distributed parameter systems (DPSs), especially hyperbolic and parabolic PDE systems [2][3][4][5][6][7][8][9][10].To mention a few, the authors in [2] investigated the state estimation problem of linear parabolic PDE systems via using backstepping technique, [3] considered the robust controller design for hyperbolic DPSs, adaptive control approach of parabolic PDE systems was proposed in [4][5][6], and [10] investigated boundary control problem of parabolic PDE systems with parameter variations.However, most of the mentioned works are reported for linear PDE systems, and the control of nonlinear DPSs is still an open issue.
Since Takagi-Sugeno (T-S) fuzzy system was proposed, a huge body studies for nonlinear ordinary differential equation (ODE) systems have been carried out [11][12][13][14][15][16][17][18][19][20].Inspired by the processing method of nonlinear ODE systems, based on T-S fuzzy model, a few results have been obtained for nonlinear PDE systems recently.For instances, Wu et al. [21][22][23] realized exponential stabilization for a class of nonlinear parabolic PDE systems via designing a fuzzy controller and illustrated the application on nonisothermal plug-flow reactor, Zhu et al. [24] developed fuzzy control method subject to a class of nonlinear coupled ODE-PDE systems with input constraint.
It should be pointed out that switched PDE systems have not been paid enough attention.Switched system (SS) is an important class of hybrid systems and plays a key role in industrial process.SSs consist of numbers of subsystems, in which switching rule is utilized to indicate the active subsystem at each instant of time.For switched systems, switching signals are classified into arbitrary switching signal, timedependent signal, and state-dependent switching signal.As the fundamental problem, the stabilization of switched PDEs has been investigated in [25,26] and semiconductor power chip is used to illustrate the practical application, whereas, to the best of our knowledge, the control synthesis problem for switched PDE systems has not been fully investigated, which motivates us to do this work.
On another research front, proportional-spatialderivative (P-sD) controller has been considered in many DPSs [27][28][29] because of its advantages compared with proportional (P) controller.Thus, in this paper, we investigate the problem of control synthesis for switched nonlinear parabolic PDE systems with time delay, including both fuzzy P and P-sD control approaches.Based on single Lyapunov-Krasovskii functional and Wirtinger's inequality, sufficient conditions for stabilization and dissipative performance are obtained by using linear matrix inequalities.Finally, simulation studies are presented to show the effectiveness of the proposed strategies.
We organize the framework as follows: The model description is given in Section 2, controllers designs are presented in Section 3, and Section 4 provides simulations to illustrate the validity of the proposed control design approaches.Finally, conclusions are offered in Section 5.
Notations.The notation that have been used in this paper are fairly standard.R and R  denote the set of all real numbers and  dimensional Euclidean space.* stands for symmetric completion, for example,  + * =  +  T .

Preliminary System and T-S Fuzzy Model
In this note, a class of switched nonlinear distributed parameter systems with time delay is investigated, which can be described by the following parabolic partial differential equations (PDEs): (, ) = Θ  (, ) +  1 ( (, )) +   ( (,  −  ())) +   ( (, ))  (, ) subject to the Dirichlet boundary condition and the initial condition where (, ) ∈ R  denotes the state vector;  ∈ [ In this study, it is assumed that the switching signal  is arbitrary subjects to ]  () = 1, 0 ( ∈ ), which means that the -th subsystem is active or not, respectively.Thus the fuzzy switched PDE system (7) can be inferred as follows: The following definition and lemma will be used to obtain the main results.8) is said to be dissipative if the following inequality holds for a nonnegative real function (⋅) with (0) = 0, called the storage function,

Fuzzy Controllers Design
In this section, two different fuzzy controllers will be proposed for the fuzzy switched distributed parameter system (8).Firstly, we consider designing a proportional control law in Section 3.1.
. .Fuzzy Proportional Controller Design.Using parallel distributed compensation (PDC) strategy, the following fuzzy controller is considered: Then, from ( 8) and ( 12), the following system is got: Theorem 3.For the fuzzy switched system ( ), if there exist symmetric positive matrices ,  and scalars   (,  = 1, 2, . . ., ,  = 1, 2, . . ., ) such that and where then for any switching signals, the controller ( ) can guarantee that the fuzzy switched system ( ) is asymptotically stable with dissipativity performance subject to given matrices ,  1 < 0, and Proof.According to Green's formula, Lemma 2, and Dirichlet boundary condition, we have For convenience, define and Choose a Lyapunov functional with the following form:

Complexity
Differentiating () along the solution of the system (13), we get Substituting   =    T   into ( 21), one has Inequality ( 22) can be also rewritten as follows: V () ≤ ∫ where To obtain the stability analysis results, we set (, ) ≡ 0, then the following inequality is got: Similar to the treatment in (22), inequality (25) implies that Thus, according to Lyapunov stability theory, inequalities ( 14) and (15) guarantee that the fuzzy switched system (13) is stable for any switching signal.
On the other hand, invoking ( 15) and ( 23), the following inequality is obtained: which can be rewritten as Under zero initial condition, integrating (28) on period [0, ], then it can be derived to It is obvious that system (13) is also dissipative.This completes the proof.
Note that the matrix inequalities in Theorem 3 is not linear; we give the following approach to solve it.Using Schur complements, we have where Pre-and postmultiply both sides of inequality (30) by the matrix diag{, , , }, where  =  −1 .Define Q = ; then we can obtain where The matrix  and   can be obtained by solving inequality (32); substituting  and   into we can easily obtain the controller gains.
Remark .It should be emphasized that the controller designed in Theorem 3 requires distributed sensors and actuators.With the development of advanced technique, it is possible to solve this problem, because there are more and more microsensors and microactuators manufactured in control circuitry.However, we are also looking for new method to avoid the drawback of the controller.
Remark .It is worth pointed out that the state feedback control method (12) cannot meet the engineering needs in many practical applications, especially for nonlinear spatially distributed processes.Therefore, we will consider the following fuzzy proportional-spatial-derivative (P-sD) controller.
. .Fuzzy P-sD Controller Design.In contrast to the proportional state feedback controller proposed in Section 3.1, the information of state spatial-derivative is added to fuzzy P-sD controller design, which will bring better dynamic performances.
Firstly, using PDC theory, the following fuzzy P-sD controller is constructed Combining ( 8) and ( 35 Theorem 6.Given matrices ,  1 < 0, and  2 =  T 2 , then for arbitrary switching signals, the fuzzy switched system ( ) is asymptotically stable and dissipative via using controller ( ), if there exist matrices  > 0,  > 0,   , and   such that the following inequality holds: where Proof.Construct the following Lyapunov functional candidate: By differentiating () along the solution of system (36), we can get Substituting (36) into inequality (40), one has According to Green's formula and Dirichlet boundary condition, it follows that In light of ( 41) and (42), one can obtain Define (, ) = [  T (, )  T  (, )  T (,  − ())  T (, )]; inequality (43) can be also written as follows: Set (, ) ≡ 0 to obtain the stability analysis results as follows: Based on inequality (37), the fuzzy switched system (36) is stable for any switching signal.The following proof process is similar to Theorem 3; thus it is omitted.
Setting   =    −1 and   =    −1 , following the similar line in Theorem 3, one has where  =  −1 and setting Q = , one can obtain where The matrices   ,   , , and Q can be obtained by solving inequalities (49), then substituting ,   , and   into we can easily get the controller gains.

Simulation Examples
Here, we consider the following fuzzy switched parabolic PDE systems with time delay: (, ) = A (, ) + subject to the Dirichlet boundary conditions and the initial conditions  1 (, 0) = 0.5 sin (2) , The membership functions are chosen as Finally, assume Now, we give the open-loop trajectories of ‖ 1 (, )‖ 2 , ‖ 2 (, )‖ 2 ,  1 (, ),  2 (, ), and the switching signal of system (52) in Figures 1-4.It is obvious that the system is unstable, which motives us to design controllers in Examples 7 and 8.
Example .In this part, we choose the fuzzy controller as By solving the inequality in Theorem Figures 5-7 present the closed-loop trajectories of ‖ 1 (, )‖ 2 , ‖ 2 (, )‖ 2 ,  1 (, ), and  2 (, ) of system (52) with the controller (59); the switching signal is shown in Figure 4.It is obvious that the system is stable.

Conclusions
In this paper, we have developed two fuzzy control laws for a class of T-S fuzzy switched distributed parameter systems (DPSs) with time delay.Using single Lyapunov-Krasovskii functional, sufficient conditions that ensure the DPSs stable and dissipative were established based on linear matrix inequalities.Furthermore, simulation studies have been shown to support the proposed techniques.It should be stressed that this is only a limited work for fuzzy switched DPSs.In future work, we will further investigate fuzzy control problem for Markovian jump DPSs and multi-Lyapunov functionals will be introduced to reduce the conservatism.