A homotopy method for eigenvalue assignment using decentralized state feedback

This paper addresses the following problem. Given an interconnected system composed of subsystems of the form , , a controllable pair, and where the off diagonal blocks of lie in the image of the appropriate Bi, then is it possible to arbitrarily assign the characteristic polynomial of by a suitable selection of the characteristic polynomials of ? Moreover, is it possible to compute the appropriate characteristic polynomials of the (or equivalently construct the Ki) needed to do so? The first question is answered by constructing a mapping which maps a prescribed set of of the feedback gains (elements of ) to the coefficients of the characteristic polynomial of . The question then becomes, given a , does have a solution? The answer is found by constructing a homotopy where and is some "trivial" function. Degree theory is then applied to guarantee that there exists an such that for all in [0,1]. The parameterized Sard\´s theorem is then utilized to prove that (with probability 1) is a "smooth" curve, and hence can be followed numerically from to by the solution of a differential equation (Davidenko\´s method).