Discussion on: 'An Algorithm for Solving a Perturbed Algebraic Riccati Equation'

摘要

For a summary of the state of the art of generalized and perturbed matrix Riccati equations we refer the reader to Sections 6.7-9 of [1] (see also [4]); for convenience of the reader we mention below some of the key facts from the theory of generalized Riccati equations: 1. The solutions of (1) and (2) depend monotonically on T', moreover, the solutions of (2) depend monotonically on X_f - this results from the general comparison theorem ([1, Theorem 6.7.33]). 2. Necessary and sufficient conditions for the existence of solutions of perturbed matrix Riccati equations can be formulated via generalized algebraic Riccati inequalities. For example, Eq. (4) has at least one positive semidefinite solution if and only if (see [11, Theorem 3.1]) there exist matrices X_1,X_2 ∈H~n such that X_1 ≤ X_2 and R(X_2) ≤ 0 ≤ R(X_1); this solution is not necessarily stabilizing. Notice that on account of (3) the solvability of ± R(X) ≤ 0 is (in case of a linear function Ⅱ) equivalent to the solvability of a linear matrix inequality (LMI).