Adaptive Neural Control Design for Nonlinear Distributed Parameter Systems With Persistent Bounded Disturbances

Abstract

In this paper, an adaptive neural network (NN) control with a guaranteed L(infinity)-gain performance is proposed for a class of parabolic partial differential equation (PDE) systems with unknown nonlinearities and persistent bounded disturbances. Initially, Galerkin method is applied to the PDE system to derive a low-order ordinary differential equation (ODE) system that accurately describes the dynamics of the dominant (slow) modes of the PDE system. Subsequently, based on the low-order slow model and the Lyapunov technique, an adaptive modal feedback controller is developed such that the closed-loop slow system is semiglobally input-to-state practically stable (ISpS) with an L(infinity)-gain performance. In the proposed control scheme, a radial basis function (RBF) NN is employed to approximate the unknown term in the derivative of the Lyapunov function due to the unknown system nonlinearities. The outcome of the adaptive L(infinity)-gain control problem is formulated as a linear matrix inequality (LMI) problem. Moreover, by using the existing LMI optimization technique, a suboptimal controller is obtained in the sense of minimizing an upper bound of the L(infinity)-gain, while control constraints are respected. Furthermore, it is shown that the proposed controller can ensure the semiglobal input-to-state practical stability and L(infinity)-gain performance of the closed-loop PDE system. Finally, by applying the developed design method to the temperature profile control of a catalytic rod, the achieved simulation results show the effectiveness of the proposed controller.