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
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