Invariants and canonical forms for systems structural and parametric identification

Abstract

The basic definitions regarding invariant functions and canonical forms for an equivalence relation on a generic set are first recalled. With reference to observable state space models and to the equivalence relation induced by a change of basis it is then shown how the image of a complete set of independent invariants for the considered equivalence relation can be used to parametrize a subset of canonical forms in the given set. Then the set of polynomial input-output models of the type P( z) y( t)= Q( z) u( t) and the equivalence relation induced by the premultiplication of P and Q by a unimodular matrix are considered and canonical forms parametrized by a complete set of independent invariants introduced. Since the two sets of canonical forms share common sets of complete independent invariants, very simple algebraical links between state space and input-output canonical forms can be deduced. The previous results are used to design efficient algorithms solving the problem of the canonical structural and parametric realization and identification of generic input-output sequences generated by a linear, discrete, time-invariant multivariable system. The results obtained in the identification of a real process are then reported.