The Region of Poles for Discrete-Time LQ Regulators

Abstract

In the design of discrete-time optimal regulators the location of the optimal closed-loop poles is determined by the weighting matrices of the performance index. When the controlled object is given in such a way that certain mode is decomposed from the others, it is possible to specify the weighting matrices so that only the specified mode is altered. This paper aims at clarifying the relationship between the weighting matrices and the optimal poles. Assuming the procedure of moving one real or a complex conjugate pair of poles and then confining the discussion to the corresponding eigenspace, the method gives rise to a systematic characterization of optimality after pole-shifting. To this end, the basic property of the associated symplectic matrix, namely the fact that the optimal poles are given as the eigenvalues of the symplectic matrix, is effectively used. The result is applicable to determining the domain where the closed-loop pole can be optimal. This is an extension to the results of Solheim (1974) and Amin (1984).