Journal of Mathematical Sciences, Vol. 89, No. 6, 1998 AN ALGORITHM FOR THE SINGLE-INPUT PARTIAL POLE ASSIGNMENT PROBLEM A. Yu. Yeremin, N. L. Zamarashkin, and S. A. Kharchenko UDC 519.614 An algorithm for translating unstable eigenvalues of single-input partial pole assignment problems based on rank-one transformations is suggested. Special attention is paid to the practical case where translations are constructed using inexact spectral information provided by the Arnoldi procedure. Estimates of the resulting perturbations of stable and translated poles are derived. These estimates depend on the accuracy of spectral information about the unstable poles. The algorithm proposed is illustrated with numerical examples. Bibliog- raphy: 11 titles. 1. INTRODUCTION Modern applied problems of automatic control are characterized by the complexity of the objects under consideration and by rigorous accuracy criteria. For this reason, in algebraic formulations of these problems specific features appear, which must be taken into account in constructing numerical algorithms. As is well known, under certain assumptions, the problem of optimal control is equivalent to the problem of pole assignment. The most general formulation of the latter is as follows. Given matrices A E C "xn and B E C nxm and a set of complex numbers {~i}in=l , find a block vector Y E C mxn such that the spectrum of the matrix A + BY of a closed loop coincides with the set {#i}P=I. Now, many algorithms for solving the problem of pole assignment are available. Naturally, these algo- rithms have been developed by mathematicians and, hence, are oriented to the general algebraic formulation. We believe that this is the reason why existing algorithms are not especially attractive for specialists in control theory. For instance, from the practical standpoint, every algorithm considered in the good survey [2] possesses at least one of the following drawbacks: 9 the algorithm requires a considerable body of information on the spectral properties of the operator; 9 the algorithm deteriorates the inherent stability of the initial problem; 9 the algorithm is strongly based on a matrix representation of the operator, i.e., it utilizes certain matrix decompositions. These features of algorithms are unacceptable for experts in automatic control theory for the following two reasons: 9 The size of applied problems is extremely large, which is related to the methods for approxi- mating physical objects that are usually described by partial differential equations. Therefore, since computer resources are limited, not every algebraic operation is feasible. For this reason, in what follows we assume that the only one allowable operation is matrix-vector multiplication. 9 Usually, one needs to transform a matrix of an open loop, which has only a few unstable poles and is close to a normal matrix, to a stable matrix that is also close to a normal one. Of course, in constructing such a transformation, we do not want to solve a more difficult problem of exact pole assignment. The assumption that the matrix of a closed loop is close to a normal one is essential because otherwise short-pulse overloads arise in the system. In addition, we note that the closeness of the original matrix of the system to a normal and stable one is determined by the high stability of the original technical design of the system under development (provided that someone has not intended to create an unstable configuration). Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 229, 1995, pp. 5-28. Original article submitted June 16, 1995. 1072-3374/98/8906-1591520.00 9 Plenum Publishing Corporation 1591