Input-output invariants for linear multivariable systems

The problem of parameterization of the input-output relation of constant finite-dimensional linear multivariable systems is considered. As a first result it is shown that a precisely defined set of entries of the Markov parameters of a system constitutes a complete set of independent invariants of the system. Specializing this result a new complete set of invariants is derived in which the input and output Kronecker indices and a canonical permutation constitute the structural invariants, whereas the set of numerical parameters in the set of invariants directly defines the parameters in a related new canonical form. The number of numerical parameters involved may be strictly less than the number of parameters in existing canonical forms. The results have been obtained by formulating a realization problem in terms of Rosenbrock's concept of a system matrix. Prototype algorithms for obtaining the proposed invariants from a state-space description or from a sequence of Markov parameters are presented.