Abstract
This paper addresses the problem of robust control design via the proportional-spatial derivative (P-sD) control approach for a class of nonlinear distributed parameter systems modeled by semilinear parabolic partial differential equations (PDEs). By using the Lyapunov direct method and the technique of integration by parts, a simple linear matrix inequality (LMI) based design method of the robust P-sD controller is developed such that the closed-loop PDE system is exponentially stable with a given decay rate and a prescribed performance of disturbance attenuation. Moreover, a suboptimal controller is proposed to minimize the attenuation level for a given decay rate. The proposed method is successfully employed to address the control problem of the FitzHugh-Nagumo (FHN) equation, and the achieved simulation results show its effectiveness.
1. Introduction
A significant research area that has received a lot of attention over the past few decades is the control design for distributed parameter systems (DPSs) modeled by parabolic partial differential equations (PDEs). These DPSs can be applied to describe many industrial processes, such as thermal diffusion processes, fluid, and heat exchangers [1–4]. The key characteristic of DPSs is space distribution, which causes their outputs, inputs, process states, and parameters to be spatially varying. On the other hand, external disturbances and nonlinear phenomena appear in most real systems. In this situation, the study of robust control design for nonlinear parabolic PDE systems is of theoretical and practical importance.
Significant research results have been reported in the past few decades for DPSs [1–3, 5–18]. The most interesting results within these research activities are those developed on the basis of PDE model [9–18]. For example, Krstic and Smyshlyaev have developed nonadaptive and adaptive kernel-based backstepping methods for linear boundary control PDE systems [9–11]. Fridman and Orlov [12] have presented exponential stabilization with performance in terms of linear matrix inequalities (LMIs) for uncertain semilinear parabolic and hyperbolic systems via a robust collocated static output feedback boundary controller. These results [9–12] are only applicable for boundary control PDE systems. Motivated by significant recent advances in actuation and sensing technology, particularly the advances of microelectromechanical systems, it is possible to manufacture large arrays of microsensors and actuators with integrated control circuitry (for control applications of such devices, see [13] and the references therein). Hence, the problems on distributed control theory and design for PDE systems have received a great deal of attention [1–3, 14–18]. For example, Orlov et al. have developed state feedback tracking control design [3] for an uncertain heat diffusion process and exponential stabilization [14] for an uncertain wave equation via distributed dynamic input extension. Wang, Wu, and Li have established sufficient conditions of distributed exponential stabilization via simple fuzzy proportional state feedback controllers for first-order hyperbolic PDE systems [15–17] and via a fuzzy proportional-spatial derivative (P-sD) for semi-linear parabolic PDE system [18]. Wu, Wang, and Li [19] have proposed a Lyapunov-based distributed fuzzy controller design with constraint for semi-linear first-order hyperbolic PDE systems. Notice that the results reported in [15–19] are presented in terms of spatial differential linear matrix inequalities (SDLMIs), which can be only approximately solved on the basis of standard finite difference method and the existing convex optimization techniques [20, 21]. Despite these efforts, to the best of the authors’ knowledge, there are still few results on the robust control design via the original PDE model of semi-linear parabolic PDE systems with external disturbances, which motivates this study.
In this study, we will deal with the problem of robust control design for a class of semi-linear parabolic PDE systems with external disturbances via P-sD control approach. Based on the Lyapunov direct method and integration by parts, a sufficient condition for the exponential stabilization with a given decay rate and a prescribed performance of disturbance attenuation is presented in terms of standard LMIs. Moreover, a suboptimal controller is proposed to minimize the attenuation level for a given decay rate. Finally, the simulation study on the robust control of a semi-linear parabolic PDE system represented by FitzHugh-Nagumo (FHN) equation is provided to show the effectiveness of the proposed method.
The remainder of this paper is organized as follows. The problem formulation and preliminaries are given in Section 2. The robust P-sD control design is provided in Section 3. Section 4 presents an example to illustrate the effectiveness of the proposed method. Finally, Section 5 offers some conclusions.
Notations. The notations used throughout the paper are given as follows. , , and denote the set of all real numbers, -dimensional Euclidean space, and the set of all real matrices, respectively. Identity matrix, of appropriate dimension, will be denoted by . For a symmetric matrix , means that it is positive definite (semipositive definite, negative definite, and seminegative definite, resp.). is a Hilbert space of -dimensional square integrable vector functions , , with the inner product and norm: where and . The superscript “” is used for the transpose of a vector or a matrix. The symbol “” is used as an ellipsis in matrix expressions that are induced by symmetry; for example,
2. Preliminaries and Problem Formulation
Consider the following nonlinear DPSs modeled by semi-linear parabolic PDEs: subject to the homogeneous Neumann boundary conditions: and the initial condition: where is the state, the subscripts and stand for the partial derivatives with respect to , , respectively, and denote the position and time, respectively, and is the control input. is the controlled output. is the exogenous disturbance satisfying . is the nonlinear part in the system, which is a locally Lipschitz continuous function on and satisfies for all and . , , , , , , and are real-known matrices with appropriate dimensions.
This study considers the following P-sD state feedback controller: where and are control gain matrices to be determined. The controller structure is shown in Figure 1, in which the notation “” means a first-order spatial differentiator.
Remark 1. It must be stressed that the implementation of the controller (7) requires distributed sensing and actuation. Although this is normally recognized as a critical drawback, with recent advances in technological developments of microelectromechanical systems, it becomes feasible to manufacture large arrays of microsensors and actuators with integrated control circuitry, which can be used for the implementation of distributed feedback control loops in some practical applications (see [13] and the references therein). The signal can be obtained using the finite difference method. In addition, it has been pointed out in [18] that the controller (7) can provide more convenient spatial performance.
Substituting (7) into (3) and (4) leads to the following PDE: where and .
In order to attenuate the effect of , robust control will be employed in this paper to deal with the disturbance attenuation problem. Let us consider the following control performance for the closed-loop PDE system of the form (5), (6), and (8): where is a real matrix and is a prescribed level of disturbance attenuation. In general, it is desirable to make the attenuation level as small as possible to achieve the optimal disturbance attenuation performance.
For simplicity, when , the PDE system (3)–(6) is referred to as an unforced PDE system, while when , it is referred to as a disturbance-free PDE system. We introduce the following definitions.
Definition 2. Given a constant , the unforced disturbance-free PDE system of (5), (6), and (8) (i.e., and ) is said to be exponentially stable with a given decay rate , if there exists a constant such that the following inequality holds:
Definition 3. Given constants and , the unforced PDE system of (5), (6), and (8) is said to be exponentially stable with a given decay rate and -disturbance attenuation if the response satisfies (9) and the disturbance-free system is exponentially stable with a given decay rate .
Therefore, the objective of this study is to find a robust P-sD controller of the form (7) such that the resulting closed-loop system is exponentially stable and the performance is ensured for a prescribed disturbance attenuation level . To do this, the following assumption and lemma are useful for the development of the main results.
Assumption 4. There exists a scalar such that the following inequality holds for any : where .
Lemma 5. For any two square integrable vector functions , , , the following inequality holds for any positive scalar , :
Proof. It is easily verified that holds for any and any positive scalar . Therefore,
which implies
Integrating both sides of (14) from to , we can obtain that
which implies that the inequality (12) holds. The proof is complete.
3. Robust P-sD Control Design
The aim of this section is to develop a robust P-sD state feedback controller to not only exponentially stabilize the semi-linear PDE system (3)–(6) but also achieve the performance with a prescribed disturbance attenuation level .
Consider the following Lyapunov functional for the system (5), (6), and (8): where is a real gain matrix to be determined. The time derivative of along the solution of the system (5), (6), and (8) is given by Integrating by parts and taking into account (5) yield Applying Assumption 4, for any scalar , Substitution of (18) and (19) into (17) implies where and Combining (4) and (20) gives where and
From the above analysis, we have the following theorem.
Theorem 6. Consider the semi-linear PDE system (3)–(6) with the P-sD controller (7). For some given scalar and , the closed-loop PDE system is exponentially stable with a decay rate and the -disturbance attenuation, if there exist a matrix , matrices , , and a positive scalar satisfying the following LMI: where and . In this case, the gain matrices , can be constructed as
Proof. Set
By pre- and post-multiplying the matrix by the matrix , respectively, we get
Using the Schur complement two times, LMI (24) is equivalent to the inequality . Since and , we can get the inequality .
From the inequality and (22), we can drive
Since , we can obtain from (28) that
Integrating (29) from to yields
which implies
Since , we obtain (9) from (31).
Next, we will show the exponential stability with a given decay rate of the disturbance-free system of (5), (6), and (8). When , inequality (20) can be rewritten as
We can easily derive from . Hence, the inequality (32) can be further written as
Integration of (33) from to yields
Since , it is easily observed that given by (16) satisfies the following inequality:
where and are two positive scalars. Inequalities (34) and (35) imply
Thus, from Definition 2 and (36), the disturbance-free system of (5), (6), and (8) is exponentially stable with a given decay rate . From Definition 3, the closed-loop system (5), (6), and (8) is exponentially stable with a given decay rate and -disturbance attenuation. Moreover, from (26), we have (25). The proof is complete.
From Theorem 6, since the controller (7) has been shown to be an effective control which can attenuate the effect of uncertain external disturbances, it is appealing to eliminate the influence brought by external disturbances as possible, that is, making the attenuation level as small as possible. To achieve this goal, for a given decay rate , setting , we consider the following minimization optimization problem: subject to the following LMI
Remark 7. Notice that the control design proposed in this paper is different from the results reported in [18, 19]. The result in [18] only considers simple exponential stabilization for a class of semi-linear parabolic PDE systems. The main difference between the result in this study and [19] is that the system under consideration in the latter one is a class of semi-linear first-order hyperbolic PDE systems, whereas the system addressed in this study is a class of semi-linear parabolic PDE systems. On the other hand, different from the SDLMI-based control designs in [18, 19], the main result of this study is presented in terms of standard LMI, which can be directly verified via the existing convex optimization techniques [20, 21].
4. Simulation Study on the FHN Equation
To illustrate the effectiveness of the proposed methods, the control problem of the FHN equation is considered, which is a wavy behavior model extensively applied to excitable media in biology [22] and chemistry [23]. The FHN equation has the following closed-form description: subject to the boundary conditions and the initial conditions where and are the state variables and is the manipulated input. , , and denote the independent time, space variables, and the length of the spatial domain, respectively. and are the initial conditions.
To more intuitively illustrate the effectiveness of the proposed design method, for the above values, we first verify through simulation that the operating steady states and of the system (39)–(41) are unstable ones. The initial conditions in (41) are assumed to be and . The length of spatial domain is set to be 20; that is, . The disturbance input is chosen as . Figure 2 shows the open-loop profiles of the evolution of and starting from the initial conditions. It is easily observed from Figure 2 that the equilibria and of the system (39)–(41) are unstable ones and , , , .
(a)
(b)
Equations (39) can be rewritten as the form of PDE (3) with the following parameters: where and is a nonlinear term. The controlled output is chosen as . Hence, the parameter matrices in (4) are chosen as and .
From Figure 2, we can easily observe that , , , . Let . The parameter satisfying Assumption 4 is chosen as We first show the effectiveness of the proposed design method. Set . Solving the optimization problem (37), we can get the optimized level of attenuation as . Setting and solving LMI (24), the control gain matrices in (7) can be derived as follows: Applying the P-sD controller (7) with the control gain matrices given in (44) to the semi-linear PDE system (39)–(41), the closed-loop profiles of evolution of and are shown in Figure 3, which implies that the proposed P-sD controller (7) with the control gain matrices given in (44) can stabilize the semi-linear PDE system (39)–(41). Moreover, profile of evolution of is shown in Figure 4.
(a)
(b)
Define the function as Figure 5 shows the value of . From this figure, we can see that for all time , which implies that the control performance in (9) is ensured.
5. Conclusions
In this paper, we have addressed the problem of robust P-sD state-feedback controller design for a class of semi-linear parabolic PDE systems with external disturbances. Based on the Lyapunov technique, the robust P-sD state-feedback controller design is formulated as a standard LMI optimization problem. The proposed controller can not only exponentially stabilize the semi-linear PDE system but also satisfy the performance in (9). The influence caused by external disturbances is eliminated as possible by the minimization optimization problem. Finally, the developed design method is successfully applied to the control of the FHN equation, and the achieved simulation results illustrate its effectiveness. Compared to one node in the paper, it is interesting to study the collective control in a coupled network with multiple nodes described by nonlinear parabolic PDEs in the future work.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgments
This work was supported in part by the National Natural Science Foundation of China under Grant 61273012, in part by the College Science and Technology Plan Project of Shandong Province under Grants J12LN91 and J13LN84, in part by the Science and Technology Development Plan Project of Shandong Province under Grant 2012YD01052, and in part by the Applied Mathematics Enhancement Program of Linyi University. The authors gratefully acknowledge the insightful comments and suggestions of the Associate Editor and anonymous reviewers, which helped improve the quality of this study.