A generalized singular value approach to the localization of closed loop eigenvalues for multivariable discrete-time systems

A method for eigenvalue localization for discrete-time systems which does not require explicit placement of the closed-loop eigenvalues is presented. The method is based on a simultaneous diagonalization accomplished by means of the generalized singular value decomposition (GSVD). The GSVD is introduced and applied to the system matrix and modified control distribution matrix to show that a proper choice of the full state feedback gain matrix leads to a diagonalized closed-loop system matrix. Furthermore, there exists a particular choice of the feedback gain matrix which reduces the closed-loop singular values as compared with those of the open loop. This fact is used to suggest an iterative algorithm for localization of the closed-loop eigenvalues