New Results on Pole-shifting for Parametrized Families of Systems*

This paper establishes new results regarding control problems for parametrized families of pairs ( 'systems') {(A(x), B(x)),xeX}, where A(x) is an n x n and B(x) is an n x m real matrix for each x (with n,m fixed integers), and the parameter x belongs to a manifold X. To be found is a new parametrized family {K(x),x~X} such that a given design criterion is satisfied by the closed-loop matrix A(x) +B(x)K(x) for all x, and the K(x) depend in a suitably 'nice' form on the parameter. The design criterion we shall be concerned with is that of pole assignment, and consists of obtaining arbitrary characteristic polynomials for the closed loop matrix. (Pole assignment problems are a central issue in the more 'classical' case studied in control theory, that of single systems; see e.g. [11].) 'Nice' will mean continuous, smooth (~**), or (real-)analytic. See [20] for an introduction to the topic of control of families of systems, including several motivating examples as well as a survey of other problems, different from pole-assignment, that are also of interest.