مرکز منطقه ای اطلاع رساني علوم و فناوري - A homotopy method for eigenvalue assignment using decentralized state feedback

Abstract :

This paper addresses the following problem. Given an interconnected system $M$ composed of $N$ subsystems of the form $A_{i} + B_{i}K_{i}$ , $i = 1,..., N , (A_{i}, B_{i})$ , a controllable pair, and where the off diagonal blocks of $M$ lie in the image of the appropriate Bi, then is it possible to arbitrarily assign the characteristic polynomial of $M$ by a suitable selection of the characteristic polynomials of $A_{i} + B_{i}K_{i}$ ? Moreover, is it possible to compute the appropriate characteristic polynomials of the $A_{i} + B_{i}K_{i}$ (or equivalently construct the Ki) needed to do so? The first question is answered by constructing a mapping $F: R^{n} \\rightarrow R^{n}$ which maps a prescribed set of $n$ of the feedback gains (elements of $K_{i}, i=1,...,N$ ) to the $n$ coefficients of the characteristic polynomial of $M$ . The question then becomes, given a $p \\in R^{n}$ , does $F(x) = p$ have a solution? The answer is found by constructing a homotopy $H: R^{n}x[O.1] \\rightarrow R^{n}$ 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 1) $x(t)$ 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).

Keywords :

Distributed control, linear systems; Eigenstructure assignment, linear systems; State-feedback, linear systems; Aerospace engineering; Control systems; Eigenvalues and eigenfunctions; Electrical engineering; Helium; Interconnected systems; Laboratories; Missiles; Polynomials; State feedback;