LQ control of descriptor systems by cancelling structure at infinity

ISSN： 1366-5820

Source： International Journal of Control, Vol.79, Iss.3, 2006-03, pp. : 224-238

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Abstract

This paper presents results concerning LQ control of arbitrary descriptor systems and a numerical algorithm for achieving it. The algorithm consists of two steps. The first step is, using the Silverman algorithm (orthogonal version), transformation of the problem of LQ control of descriptor system, into a problem of LQ control of a strictly proper system (with singular or non-singular new matrix S 22 ). If new S 22 is singular, we perform the second step, i.e. solving the LQ singular control problem using again the Silverman algorithm. Regarding the existing results on the LQ singular control problem, we have new results: a counter example that this problem may not have a stable solution, most general solvability conditions (with or without stability), and a new numerical algorithm based on using orthogonal matrices. When stabilizability conditions are satisfied, the algorithm places simultaneously stable poles in the closed-loop system. We show that the traditional Riccati equation associated to this problem has not a solution (although the problem has), and introduce a generalization of Riccati equation, together with a method for its solving, as well as a generalization of spectral factorization of Popov function. Examples that cannot be solved by the existing methods are presented.