On Rational State Space Realizations

Abstract

. It is investigated when a polynomial inputoutput differential equation can be realized in rational, explicit state space form, i.e. so that all components of the right hand side are rational functions of the states. In the case where there are no inputs the problem is showed to be equivalent to a famous problem in algebraic geometry, which is solved only in various special cases. For systems with inputs the problem is more complicated, as is the discrete time case. An interpretation of the Luroth problem in terms of observability is made. Keywords: Realization theory, polynomial systems, algebraic observability, algebraic geometry, rational varieties 1 Introduction The problem of state space realization of nonlinear systems is a classical one, extensively discussed in the literature. Depending on what kinds of realizations we allow, the problem becomes more or less difficult. The most conservative generalization of the Kalmanian state space theory for linear systems to nonlinear sys..