Parametric eigenstructure assignment for linear systems via state-derivative feedback

The problem of eigenstructure assignment in linear systems via state-derivative feedback is considered. A new derivative feedback design framework named complementary system framework is proposed. By using this framework, notions of complementary controllability and complementary controllability indices of linear systems are introduced. A necessary and sufficient condition for the solvability of the eigenstructure assignment problem in complementary S-controllable (C-controllable) linear systems is then given by inequalities which involve the complementary controllability indices of a linear system, and a list of the degrees of invariant polynomials and two lists of non-negative integers to represent the finite non-zero, zero and infinite eigenvalue structure of the closed-loop system. Based on a simple parametric solution to a group of recursive equations, a complete parametric approach for solving the eigenstructure assignment problem is proposed. General parametric expressions for the closed-loop eigenvectors and the feedback gain matrix are established in terms of a group of parameter vectors. The approach generalises and improves the previous results in this area. Besides, the combined problem of simultaneously assigning dynamical order and finite eigenstructure in linear systems by state-derivative feedback is also considered. Application of the proposed approach to control of a three degrees of freedom mass-spring-dashpot system is discussed.