Complete eigenstructure assignment in high-order descript or linear systems via proportional plus derivative state feedback

A parametric approach for eigenstructure assignment in the controllable high-order descriptor linear system A(m)x((m)) +(...)+ A(1)x + A(0)x = Bu via proportional plus derivative state feedback control u = F(0)x + F(1)x +(...)+ F(m)x((m)) is proposed. General parametric expressions in the direct closed forms for the closed-loop eigenvectors are presented, which are linear in a group of parameter vectors {f(i)} which represent the degree of the design freedom. It is shown that the derivative feedback gain F-i, i = 0,1,(...),m-1 may be taken to be an arbitrary real matrix satisfying a simple constraint, and the general expression for the proportional and derivative feedback F-i, are composed of both the group of vectors {f(i)} and the derivative feedback gain F-m. The proposed approach does not impose any condition on the closed-loop finite eigenvalues, and assigns arbitrarily nm finite closed-loop eigenvalues. An example shows the effect of the proposed approach.