Determining the order of linear dynamic systems (Q1180950)

Consider the dynamic system \(\dot X=AX+Bu\), \(y=C^ T X\) where \(A\in R^{m\times m}\) is a Frobenius matrix, \(B\in R^ m\) and \(C\in R^ m\). Here \(X\in R^ m\) is the state vector and \(u\) and \(y\) are the input and output of the system. The author proves that the order of \(A\) satisfies the inequality \(c_ 0 m^{-1}\leq(\| A\|/c)^ m-1\) where \(c_ 0=(\alpha\| A\|-c)/c\), \(0<\alpha\leq 1\), \(c>0\). An iterative algorithm for the solution of the inequality is presented. The properties of the algorithm are also studied.