Estimability and regulability of linear systems

A linear state-space system is said to be estimable if in estimating its state from its output the posterior error covariance matrix is strictly smaller than the prior covariance matrix. It is said to be regulatable if the quadratic cost of the state feedback control is strictly smaller than the cost when no feedback is used. Estimability and regulability are shown to be dual properties, equivalent to the nonreducibility of the Kalman filter and of the optimal linear quadratic regulator, respectively