Rational representations and controller synthesis of behaviors

This paper considers linear dynamical systems restricted to square integrable trajectories. Following the behavioral formalism, a number of relevant classes of linear and shift-invariant L 2 systems are defined. It is shown that rational functions, analytic in specific half-spaces of the complex plane, prove most useful for representing such systems. For various classes of L 2 systems, this paper provides a complete characterization of system equivalence in terms of rational kernel representations of L 2 systems. In addition, a complete solution is given for the problem when selected (non-manifest) variables of an L 2 system can be completely eliminated from their behavior. This elimination theorem has considerable independent interest in general modeling problems. It is shown that the elimination result is key in the solution of the problem for realizing an arbitrary L 2 system as the interconnection of a given L 2 system and a to-be-synthesized L 2 system. In the context of control, this problem amounts to characterizing the existence and parameterization of all controllers that, after interconnection with a given plant, constitute a desired controlled system.