A geometric approach to eigenstructure assignment for singular systems

Abstract

Moore's well-known result on the feedback assignment of eigenstructure in linear state-space systems [5] was generalized to descriptor systems in [9]. In this note we make some useful extensions to fill out the theory. To begin with, we do not assume controllability of the plant. It is shown that the maximum number of finite closed-loop eigenvalues is equal to dim (EL <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">*</sup> ), where L <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">*</sup> is the supremal <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">(A, E, R(B))-</tex> invariant subspace, in order to relate the possible closed-loop structure to the open-loop plant structure, we define a set of indexes for descriptor systems which are in many respects similar to the controllability indexes of state-space systems. In terms of these "controllability" indexes, we give a systematic approach to selecting the closed-loop eigenvectors. A major contribution of this note is the fact that it is geometric in nature, and hence avoids any decomposition of the system matrices into a special form (e.g., Weierstrass form [1]). The geometric setting introduced here should constitute a basis for further research in generalizing the well-known results in geometric system theory (disturbance decoupling, input/output decoupling, etc.) to singular systems.