Rational representations and controller synthesis of L2L2 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 L2L2 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 L2L2 systems, this paper provides a complete characterization of system equivalence in terms of rational kernel representations of L2L2 systems. In addition, a complete solution is given for the problem when selected (non-manifest) variables of an L2L2 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 L2L2 system as the interconnection of a given L2L2 system and a to-be-synthesized L2L2 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.