Abstract
Generalization of linear system stability theory and LQ control theory are presented. It is shown that the partial stabilizability problem is equivalent to a Linear Matrix Inequality (LMI). Also, the set of all initial conditions for which the system is stabilizable by an open-loop control (the stabilizability subspace) is characterized in terms of a semi-definite programming (SDP). Next, we give a complete theory for an infinite-time horizon Linear Quadratic (LQ) problem with possibly indefinite weighting matrices for the state and control. Necessary and sufficient convex conditions are given for well-posedness as well as attainability of the proposed (LQ) problem. There is no prior assumption of complete stabilizability condition as well as no assumption on the quadratic cost. A generalized algebraic Riccati equation is introduced and it is shown that it provides all possible optimal controls. Moreover, we show that the solvability of the proposed indefinite LQ problem is equivalent to the solvability of a specific SDP problem.
Highlights
The present study brings a novel treatment of the stability and control of linear systems which are not necessary stabilizable
Theorem 6.4 Let k be the dimension of the stabilizability subspace Su and assume that the possibly indefinite Linear Quadratic (LQ) problem is attainable at k independent initial conditions v1, . . . , vk ∈ Su
In this study we have addressed one of the remaining problems in linear system theory: the problem of designing optimal stabilizing control laws for a possibly indefinite indefinite LQ problem involving possibly singular control
Summary
The present study brings a novel treatment of the stability and control of linear systems which are not necessary stabilizable. No convexity assumption is made on the proposed quadratic optimization problem and the control variable weight matrix may be singular. Since the innovative work [19] related to a definite linear quadratic problem with connection to the classical Riccati equation [13], much research has been devoted to the optimization of more general quadratic cost functionals. Such problems have a theoretical interest in their own right, and have natural applicability in several fields of system theory. In [14], the stabilizability condition is dropped but the definitness of the quadratic cost is assumed and the optimal control law is not necessarily stabilizing. I denotes the identity matrix, with size determined from the context
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