Quadratic Optimization for Fixed Order Linear Controllers via Covariance Control

There are only two main theories for the design of fixed order controllers: optimal LQG with fixed order and covariance control. Covariance control theory provides the class of all stabilizing controllers of fixed order. By parameterizing the set of all stabilizing fixed order controlers as covariance controllers, the free parameters in the covariance controller can be optimized. Hence, it is quite natural to ask how such controllers relate to the established literature on optimal fixed order controllers. This paper shows these relationships. Physical meaning (in terms of the controller covariance and correlation between plant and controller covariance) is given to the projection matrix of the optimal fixed order LQG controller. The optimal covariance controller guarantees stability. The connection is also shown between covariance control and the existing theories of optimal control of full order (LQG). Because the covariance controller allows feedforward control, the quadratic cost (for the case of full order controllers) is always equal or smaller than the standard LQG cost.