A homotopy method for eigenvalue assignment using decentralized state feedback

This paper addresses the following problem. Given an interconnected systemMcomposed ofNsubsystems of the formAᵢ + BᵢKᵢ,i = 1,..., N , (Aᵢ, Bᵢ), a controllable pair, and where the off diagonal blocks ofMlie in the image of the appropriate Bi, then is it possible to arbitrarily assign the characteristic polynomial ofMby a suitable selection of the characteristic polynomials ofAᵢ + BᵢKᵢ? Moreover, is it possible to compute the appropriate characteristic polynomials of theAᵢ + BᵢKᵢ(or equivalently construct the Ki) needed to do so? The first question is answered by constructing a mappingF: Rⁿ → Rⁿwhich maps a prescribed set ofnof the feedback gains (elements ofKᵢ, i=1,...,N) to thencoefficients of the characteristic polynomial ofM. The question then becomes, given ap ∈ Rⁿ, doesF(x) = phave a solution? The answer is found by constructing a homotopyH: Rⁿx[O.1] → RⁿwhereH(x,1)= F(x)andH(x,0)is some "trivial" function. Degree theory is then applied to guarantee that there exists anx(t)such thatH(x(t), t) = pfor alltin [0,1]. The parameterized Sard's theorem is then utilized to prove that (with probability 1)x(t)is a "smooth" curve, and hence can be followed numerically fromx(0)tox(1)by the solution of a differential equation (Davidenko's method).