Controller synthesis for L2 behaviors using rational kernel representations

This paper considers the controller synthesis problem for the class of linear time-invariant L2 behaviors. We introduce classes of LTI L2 systems whose behavior can be represented as the kernel of a rational operator. Given a plant and a controlled system in this class, an algorithm is developed that produces a rational kernel representation of a controller that, when interconnected with the plant, realizes the controlled system. This result generalizes similar synthesis algorithms in the behavioral framework for infinitely smooth behaviors that allow representations as kernels of polynomial differential operators.