High Order Approximations for LQR Control and LQG Estimation of Convection Diffusion Systems

Abstract

In this paper we discuss higher order methods for approximating linear control systems governed by convection diffusion equations. In particular, we employ a hp - finite element method to construct low order approximations of the Riccati equations that arise in linear quadratic control and optimal state estimation. The method is based on using high order piecewise polynomial approximations that achieves accuracy by combining mesh refinement (h → 0) with high order polynomials (p → +∞). By employing high order methods one can obtain accuracy with fewer degrees of freedom than can be achieved with mesh refinement alone. These methods are well known in the simulation community and can be used to achieve high order simulations, but hp – methods have not been investigated as potential methods for approximating control systems governed by partial differential equations. We employ the hp – finite element method to develop approximations of the infinite dimensional Riccati operators that define optimal LQR and LQG controllers. Convergence and rates of convergence are presented. A numerical example is provided to illustrate the convergence rates for the hp – method and we close with a discussion of how such methods might be used to enhance the development of reduced order models.