EXTENSION OF S-PROCEDURE IN THE ANALYSIS OF MULTIVARIABLE CONTROL SYSTEMS

Abstract

Let q(x) and pi(x),i = 1, …, m be quadratic forms of real variables x ∈ Rn. In many problems of investigation of stability and estimation of the attraction domain and attainability sets, one faces the following question: under what conditions, do inequalities pi(x) ≥ 0,i = 1, …, m, x ≠ 0 imply the inequality q(x) > 0? The commonly used S-procedure method (Yakubovich, 1977) consists in checking of whether there exist values τi ≥ 0 such that the quadratic form is positive definite. It is well known that, if m ≥ 2, the S-procedure gives us only sufficient conditions for positive definiteness of the quadratic form q(x) under the constraints pi(x) ≥ 0, i = 1, …, m. These conditions are necessary only for m = 1. This property is called “lossness” of the S-procedure for multiple constraints. The use of only sufficient conditions leads to additional conservatism of stability criteria and attraction domains estimation. Necessary and sufficient conditions are obtained in (Rapoport, 1989) and (Rapoport, 1996) for a special case of quadratic constraints represented as products of two linear forms. This paper further extends those results. The special case of m = 2, where additional conditions were imposed on the quadratic forms p1(x) and p2(x) to make conditions (1) necessary and sufficient, has been addressed in (Polyak, 1998). In this paper, the losslessness of the S-procedure for m = 2 is proved under less restrictive additional conditions. A case of one general-form quadratic constraint and m – 1 constraints presented as products of two linear forms is also considered.