Parametric eigenstructure assignment via state feedback: a simple numerically reliable approach

An extremely simple method for eigenstructure assignment in multivariable linear systems via state feedback controllers is presented. It is shown that as soon as a series of singular value decompositions are completed, the whole set of admissible right closed-loop eigenvectors as well as the whole set of state feedback gain matrices can be simply and neatly parameterized in terms of a group of parameter vectors. This group of parameter vectors provide all the design degrees of freedom and can be utilized to achieve additional desired specifications. The method involves mainly a series of singular value decompositions and the solution to a linear matrix equation and is numerically reliable. The proposed approach does not require the closed loop eigenvalues to be distinct or to be different from the open loop ones. Moreover, it allows repeated closed-loop eigenvalues, but produces only nondefective eigenstructure for robustness consideration. Based on the presented approach, two algorithms for robust pole assignment are proposed. Several computational examples demonstrate the numerical reliability of the proposed approach.