Eigenstructure assignment and response analysis in descriptor linear systems with state feedback control

A new complete parametric approach for eigenstructure assignment in controllable descriptor linear systems via static state feedback control is proposed. General complete parametric expressions in direct closed forms for the closed-loop eigenvectors associated with the finite closed-loop eigenvalues, and two simple complete parametric solutions for the feedback gain matrix K are established. The approach guarantees closed-loop regularity, arbitrary assignment of rank(E) finite closed-loop eigenvalues with arbitrary given algebraic and geometric multiplicities, and realizes elimination of all possible initial time impulsive responses. Moreover, it does not impose any condition on the closed-loop finite eigenvalues, suits both the continuous-time and discrete-time descriptor systems, and overcomes the drawbacks of some previous works. Based on this approach, general parametric expressions for the closed-loop left eigenvector matrix associated with the finite closed-loop eigenvalues are also established, and it is shown that the closed-loop system can be transformed equivalently into a canonical form that is clearly composed of a dynamical part and a non-dynamical part. With the help of this canonical form, a several analytical parametric expression of the closed-loop system is presented in terms of the closed-loop eigenvector matrices and the closed-loop Jordan matrix. The 'Method of Companion' is introduced, from the point of view of computation, to cope with inverses of all the matrices involved in this paper. Two examples illustrate the advantages of the proposed approaches.