MATRIX-METHOD FOR EIGENVALUE ASSIGNMENT - THE SINGLE-INPUT CASE

The eigenvalue assignment/pole placement procedure has found applicati on in a wide variety of control problems. The associated literature is rather extensive with a number of techniques discussed to that end. I n this paper a method for assigning eigenvalues to a Linear Time Invar iant (LTI) single input system is proposed. The algorithm determines a matrix, which has eigenvalues at the desired locations. It is obtaine d from the knowledge of the open-loop system and the desired eigenvalu es. Solution of the matrix equation, involving unknown controller gain s, open-loop system matrices and desired eigenvalues, results in the s tate feedback controller. The proposed algorithm requires the closed-l oop eigenvalues to be different from those of the open-loop case. This apparent constraint is easily overcome by a negligible shift in the v alues. Two examples are considered to verify the proposed algorithm. T he first one pertains to the in-plane libration of a Tethered Satellit e System (TSS) while the second is concerned with control of the short period dynamics of a flexible airplane. Finally, the method is extend ed to determine the Controllability Grammian, corresponding to the spe cified closed-loop eigenvalues, without computing the controller gains .