Nonlinear Feedback Control of Parabolic PDE Systems

Abstract

Transport-reaction processes with significant diffusive and dispersive mechanisms (e.g. packed-bed reactors, rapid thermal processing systems, chemical vapor deposition reactors, etc.) are typically characterized by strong nonlinearities and spatial variations, and are naturally modeled by nonlinear parabolic PDE systems. The main feature of parabolic PDEs is that the eigenspectrum of the spatial differential operator can be partitioned into a finite-dimensional slow one and an infinite-dimensional stable fast complement [18, 1, 29]. This implies that the dynamic behavior of such systems can be approximately described by finite-dimensional systems. Motivated by this, the standard approach to control parabolic PDEs involves the application of Galerkin’s method to the PDE system to derive ODE systems that describe the dynamics of the dominant (slow) modes of the PDE system, which are subsequently used as the basis for the synthesis of finite-dimensional controllers [20, 1, 26, 8]. However, there are two key controller implementation and closed-loop performance problems associated with this approach. First, the number of modes that should be retained to derive an ODE system that yields the desired degree of approximation may be very large, leading to high dimensionality of the resulting controllers [19]. Second, there is a lack of a systematic way to characterize the discrepancy between the solutions of the PDE system and the approximate ODE system in finite time, which is essential for characterizing the transient performance of the closed-loop PDE system.