Eigenstructure assignment in descriptor systems via proportional plus derivative state feedback

A complete parametric approach for eigenstructure assignment in the controllable linear descriptor system E(x) over dot = Ax + Bu via proportional plus derivative state feedback control u = Kx + L(x) over dot proposed. General complete parametric expressions in direct closed forms for the closed-loop eigenvectors are presented, which are linear in a group of parameter vectors {f(ij)(k)} which represent the degree of the design freedom. It is shown that the derivative feedback gain L may be taken to be an arbitrary real matrix satisfying a simple constraint, and the general complete expression for the proportional feedback gain K is composed of both the group of vectors {f(ij)(k)} and the derivative feedback gain L. By properly restricting the design parameters in the obtained general results, solutions to eigenstructure assignment in descriptor linear systems via constant-ratio proportional plus derivative state feedback and proportional plus partial derivative state feedback are obtained. The approach does not impose any condition on the closed-loop finite eigenvalues, assigns arbitrarily n finite closed-loop eigenvalues with arbitrary given algebraic and geometric multiplicities and guarantees the closed-loop regularity. An example shows the effect of the proposed approach.