On minimality in the partial realization problem (Q581311)

Given a finite sequence \(M_ 1,...,M_ r\) of \(p\times m\) matrices, a dynamical system \(\Sigma =(A,B,C)\) is called a realization of \(M_ 1,...,M_ r\) if \(CA^{i-1}B=M_ i\) for \(i=1,...,r\). Minimal realizations, that is realizations of the smallest possible state space dimension, are important in many control problems. In this paper, the authors present an algorithm to reduce an arbitrary realization of \(M_ 1,...,M_ r\) to a minimal one. Furthermore a minimality criterion and a formula for the minimal state space dimension are obtained.