POLE ASSIGNMENT FOR SECOND-ORDER SYSTEMS

Abstract

This paper contains some results for pole assignment problems for the second-order system Mẍ(t)+Dẋ(t)+Kx (t)=Bu (t). Specifically, Algorithm 0 constructs feedback matrices F1 and F2 such that the closed-loop quadratic pencil Pc(λ)=λ2M+λ (D+BF2)+(K+BF1) has a desired set of eigenvalues and the associated eigenvectors are well-conditioned. The method is a modification of the SVD-based method proposed by Juang and Maghami [1, 2] which is a second-order adaptation of the well-known robust eigenvalue assignment method by Kautsky et al. [3] for first-order systems. Robustness is achieved by minimising some not-so-well-known condition numbers of the eigenvalues of the closed-loop second-order pencil. We next consider the partial pole assignment problem. In 1997, Datta, Elhay and Ram proposed three biorthogonality relations for eigenvectors of symmetric definite quadratic pencils [4]. One of these relations was used to derive an explicit solution to the partial pole assignment problem by state feedback for the related single-input symmetric definite second-order control system. The solution shed new light on the stabilisation and control of large flexible space structures, for which only one small subset of the spectrum needs to be reassigned while retaining the complementary part of the spectrum. In this paper, the method has been generalised for multi-input and non-symmetric quadratic pencils. Finally, we discuss briefly the output feedback pole assignment problem.