Structural Controllability of a Linear Time-Invariant System with Auxiliary Variables

Abstract

Structural controllability is studied for a system described in the following form with auxiliary variable w: x=A₁x+B₁u+C₁w and w=A₂x+B₂u+C₂w, where the nonzero entries of the coeffiicient matrices A_i, B_i, C_i (_i=1 2) are taken for independent parameters. This setting reflects the physical consideration that the entries of the coefficient matrices A and B of the reduced form x=Ax+Bu, which is obtained by eliminating w above, are usually subject to algebraic constraints and do not represent independent elemental physical parameters. A necessary and sufficient condition, which is a natural extension of the presently known results, for the structural controllability is given in terms of graph-theoretic conditions. Specifically, in terms of a directed graph G with nodes corresponding to variables x_i, u_i and w_i and with arcs to the nonzero entries of matrices A_i, B_i and C_i (i=1, 2), the controllability of the nonzero modes is shown to be equivalent to the condition that each node x_i is accessible from some node u_j by a directed path on G.