Abstract:

The realization problem of continuous-time rational systems is to determine, for a response map from inputs to outputs, a rational system whose response map equals the considered response map. Such a system is then called a rational realization of the response map. The problem also includes the classification of all minimal realizations. The problem is inspired by the publications of R.E. Kalman on realization of linear systems, of E.D. Sontag on realization of discrete-time polynomial systems, and of Z. Bartosiewicz on continuous-time polynomial systems. It will be proven that there exists a finite-dimensional rational realization if and only if a subalgebra of the response map has a finite set of generators. A realization is minimal if and only if it is both algebraically controllable and algebraically observable. The latter system theoretic properties are characterized by algebraic conditions. The lecture includes algebraic preliminaries of rational systems. 

     Applications of realization theory include: 

(1) equivalent conditions for structural identifiability of rational systems, 

(2) the existence of polynomial or rational observers, and 

(3) a system identification algorithm for polynomial systems. The research is based on scientific cooperation with: Jana Nemcova and Mihaly Petreczky.