A homotopy-method for eigenvalue assignment using decentralized state feedback

This paper addresses the following problem: Given an arbitrary interconnected system M which is composed of N subsystems of the form Ai + BiKi, i=1,...,N, (Ai, Bi) a controllable pair, then is it possible to arbitrarily assign the characteristic polynomial of M by a suitable selection of the characteristic polynomials of Ai + BiKi? Moreover is it possible to compute the appropriate characteristic polynomials of the Ai + BiKi (or equivalently construct the Ki) needed to do so? The first question is answered by constructing a mapping F: Rn → Rn which maps a prescribed set of n of the feed-back gains (elements of Ki, i=1,...,N) to the n coefficients of the characteristic polynomial of M. The question then becomes, given a p → Rn, does F(x) = p have a solution? The answer is found by constructing a homotopy H: Rnx [0,1] → Rn where H(x, 1) = F(x) and H(x,0) is some "trivial" function. Degree theory is then applied to guarantee that there exists an x(t) such that H(x(t),t) = p for all t in [0,1]. The Parameterized Sard's Theorem is then utilized to prove that (with probability one) x(t) is a "smooth" curve and hence can be followed numerically from x(0) to x(1) by the solution of a differential equation (Davidenko's method).