Control of state and input covariances for dynamic systems

The set of all state and input covariances that a linear closed-loop system may possess is characterized. For any choice of state covariance within this set, all dynamic controllers which assign those covariances are parameterized. The number of free parameters in these controllers is related to the excess of the controller order over the sum of the number of controls and measurements. If the desired state covariance is not assignable, it is also proved that for the state feedback case the controllers presented are the closest solutions to the corresponding Lyapunov equation in the norm sense