Bounds for the solution of algebraic matrix equations arising in mathematical control theory

Abstract

The study of solution bounds of algebraic Lyapunov and Riccati equations are highly important in control problems, and have been an attractive re­ search topic over the past three decades. The solution bounds give solution estimates, and can also be applied to solve such problems involving these equations, hence a motivation for the research attraction. Besides, in control applications involving them, the exact solutions are often not required, but rather bounds of their solution, particularly when solving the equation is difficult.

Therefore, many papers have proposed solution bounds for these equations, mainly for a deterministic nominal system, when the exact values of the coef­ficient matrices of the equations are available. Additionally, some works have focused on solution bounds of these equations for perturbed systems, when only approximate values of the coefficient matrices are available, so they avail­ able are perturbed versions of their actual values; as a consequence of these perturbed coefficient matrices, the solution matrix also becomes perturbed, so it becomes of interest to estimate the disturbance range for the solution. Furthermore, fewer works have focused on solution bounds of coupled alge­braic Lyapunov and Riccati equations arising from stochastic systems, for both nominal and perturbed cases. In fact, it appears that there is no paper in the literature that studies solution bounds of perturbed coupled algebraic Riccati equations.

Finally, many existing bounds only exist under assumptions which are not always valid, many of which are not realistic in control problems involving each equation. Furthermore, some bounds do not appear to be as tight as others, some bounds require heavy and complicated calculations to deter­ mine, and some are not very concise. Therefore, this work seeks to obtain solution bounds for Lyapunov and Riccati equations, which are tighter, less restrictive, possibly simpler in calculation, and more concise than existing results. When possible, all derived results shall be compared with existing results to verify the advantage(s) of the new results.

Date of Award	2007
Original language	English
Supervisor	Peng Shi (Supervisor) & Ron Wiltshire (Supervisor)