Circulation algorithm for partial eigenstructure assignment via state feedback

This paper treats the partial eigenstructure assignment problem for multi-variable linear systems by state feedback. Both the open-loop and the closed-loop eigenvalues are allowed to have arbitrary geometric and algebraic multiplicities. Based on a previously proposed result on partial eigenstructure assignment, a new complete parametric solution is derived, which gives a complete general parameterization of the feedback gain and also provides the general parametric expression for all the admissible eigenvector matrices. The solution involves, instead of a matrix inverse of full-dimension, the inverse of a matrix with a much smaller dimension. By using this new solution repeatedly, a circulation algorithm is proposed, which is simple and dramatically reduces the computational load in the case of large scale systems. Furthermore, two important special cases are especially examined, and simple and direct solutions are proposed, which no longer needs computation of matrix inverses, and suits best in the proposed circulation algorithm. The approach provides all the design degrees of freedom which can be further utilized to achieve additional closed-loop system performance. An illustrative example is thoroughly examined, which demonstrates the procedure and the advantages of the proposed parametric approach.