Approximation Theory for Linear-Quadratic-Gaussian Control of Flexible Structures

Abstract

This paper presents approximation theory for the linear-quadratic-Gaussian optimal control problem for flexible structures whose distributed models have bounded input and output operators. The main purpose of the theory is to guide the design of finite-dimensional compensators that approximate closely the optimal compensator, which is infinite-dimensional. Design of the optimal compensator separates into an optimal linear-quadratic control problem and a dual optimal state estimation problem; the solution to each problem lies in the solution to an infinite-dimensional Riccati operator equation. The approximation scheme in the paper approximates the infinite-dimensional LQG problem with a sequence of finite-dimensional LQG problems defined for a sequence of finite-dimensional, usually finite-element or modal, approximations of the distributed model of the structure. Two Riccati matrix equations determine the solution to each approximating problem. The finite-dimensional equations for numerical approximation are developed, including formulas for converting matrix control and estimator gains to their functional representation to allow comparison of gains based on different orders of approximation. Convergence of the approximating control and estimator gains and of the corresponding finite-dimensional compensators is studied. Also, convergence and stability of the closed-loop systems produced with the finite-dimensional compensators are discussed. The convergence theory is based on the convergence of the solutions of the finite-dimensional Riccati equations to the solutions of the infinite-dimensional Riccati equations. A numerical example with a flexible beam, a rotating rigid body, and a lumped mass is given.