Eigenstructure assignment in descriptor systems via state feedback - a new complete parametric approach

A new complete parametric approach for eigenstructure assignment in controllable linear descriptor systems, Ex = Ax + Bu via the static state feedback control u = Kx, is proposed. General complete parametric expressions in direct closed forms for the closed-loop eigenvectors associated with the finite closed-loop eigenvalues are presented based on a recently proposed complete parametric solution to the generalized Sylvester matrix equation AV + BW = EVF. These expressions are linear in a group of parameter vectors which represent the degree of the design freedom. A very simple complete parametric solution for the feedback gain matrix K is also established with the help of a proposed 'completing the square' technique. The approach guarantees arbitral assignment of rank(E) number finite closed-loop eigenvalues with arbitrary given algebraic and geometric multiplicities, and realizes elimination of all possible initial time impulsive responses. It also guarantees the closed-loop regularity when two simple constraints on the design parameters are added. The approach does not impose any condition on the closed-loop finite eigenvalues, and over comes the drawback of some previous work. Two examples illustrate the advantages of the proposed approach.