A uniform algorithm for the transformation of multivariable systems into canonical forms (Q2277544)

Single-input time-invariant linear systems can be characterized by the matrix equation \((*)\quad \dot x=Ax+\lambda b\), where \(A\) denotes an \(n\times n\) matrix, \(x\) and \(b\) are \(n\times 1\), and \(\lambda\) is a scalar. In many cases, this equation has to be transformed by means of a similarity transformation into a canonical form. The purpose of this paper is to present a uniform treatment of a variety of known canonical forms. In the single-input case, the authors provide a method based on the factorization of the transformation matrix. This can be expressed as the product of the controllability matrix of the system (*) and of a Hessenberg matrix. Extensions of the transformation algorithm to multiple-input systems are also given.