State feedback output regulation for a class of hyperbolic PDE systems

This work focuses on the development of a state feedback regulator for a class of first order hyperbolic PDE systems with spatially varying coefficients and single point observation. The plant is assumed to be exponentially stabilizable and driven by a linear finite dimensional exosystem which is neutral stable and generates the reference signal and disturbance for the plant. The regulator design problem is to control the fixed plant such that the plant output tracks the reference signal and/or rejects the disturbance. Under the standard assumption of stabilizability, this work shows that the solvability of a constrained Sylvester equation is sufficient to guarantee the solvability of the regulator problem. Moreover, the Riccati and Lyapunov equations are utilized to provide a choice of stabilizing feedback gain which guarantees the closed-loop stability. An adequate numerical example of constant tracking for the first order hyperbolic PDE system with spatially varying coefficients is explored within the proposed regulator design.

I. Introduction

The dynamics of many relevant chemical, petrochemical, pharmaceutical processes can be described by transport-reactions models that take the mathematical form of partial differential equations (PDEs) (see [18], [14], [7], [22]). The majority of control related research efforts dealing with PDEs has been made based on control methods for distributed parameter systems (DPS) which inevitable include the infinite-dimensional nature of these systems (see [8], [5], [4], [9]). The complexity of infinite dimensional system setting in particular needs to be addressed in the framework of essential control problems such as stabilization of the system and/or servo-problem.