An algorithm for the multi-input complex eigenvalue assignment problem

摘要

In this paper, a simple, efficient and accurate method for complex eigenvalue assignment in the multi-input linear system is presented. The method considers both cases: all eigenvalues need to assign are complex (with their complex conjugates). The other case is the arbitrary eigenvalues need to assign are mixed (some are real and the others are complex which are closed under complex conjugations). The algorithm proposed in this method is a modification to the one proposed by Arnold and Datta [IEEE Trans. Automat. Contr. 35 (10) (1986) 1149] which considers the case where all the eigenvalues need to assign are only real. Also, it is a kind of extension to the algorithms suggested by Ramadan [in press] and Ramadan and El-Shazly [in press] where arbitrary preassigned complex and mixed eigenvalues are assigned for single-input case respectively.The method first transforms the problem to block-Hessenberg form, of needed form, using orthogonal reduction, then a simple linear recursion is carried out to find the feedback matrix so that the closed-loop matrix has the prescribed eigenvalues. The main advantages of the algorithm, besides being simple and has recursive nature it avoids complex arithmetic and easy to program on computer. Numerical examples are given to illustrate the reduction process, the algorithm performance and the accuracy of the method.