The polynomial approach to the LQ non-Gaussian regulator problem through output injection

In this paper, an improved approach for the solution of the regulator problem for linear discrete-time dynamical systems with non-Gaussian disturbances and quadratic cost functional is proposed. It is known that a sub-optimal recursive control can be derived from the classical LQG solution by substituting the linear filtering part with a quadratic, or in general polynomial, filter. However, we show that when the system is not asymptotically stable the polynomial control does not improve over the classical LQG solution, due to the lack of the internal stability of the polynomial filter. In order to enlarge the class of systems that can be controlled, we propose a new method based on a suitable rewriting of the system by means of an output injection term. We show that this allows to overcome the problem and to design a polynomial optimal controller also for non asymptotically stable systems. Numerical results show the effectiveness of the method.