LIAPUNOV AND COVARIANCE CONTROLLERS

The early work of Liapunov produced some of the most powerful tools for stability analysis that remain to this day. To capture the class of all stabilizing controllers one would be well served by posing the problem in terms of the existence of a Liapunov function, since a Liapunov function is known to exist for stable systems. For linear systems, this paper derives the set of all quadratic Liapunov functions for output feedback control problems, and in this way parametrizes the set of all stabilizing controllers of fixed order. This is a unifying framework from which all other controllers can be produced by special choices of the free parameters in these controllers (we will show how to choose the free parameters to produce all covariance controllers and all H(infinity) controllers of fixed order). These results also apply to robustness analysis, and provide a closed form expression for the set of all stabilizing real structured perturbations. Due to the assignment of a matrix property to the system (e.g. covariance matrix), this approach lends itself naturally to mixed problems with multiple objectives.