An invariant parameter in linear estimation and control

This paper considers the implementation difficulty of the joint observer-controller design, the forward and backward Kalman Bucy (KB) filter, and the (estimator-controller) LQG solution, with respect to the choice of the realization. By establishing an invariant parameter, it is shown that the implementation difficulty for the joint design in each of the three problems is invariant. This invariant parameter is the inner product of the gain vectors for the observer (forward KB filter) and the controller (backward KB filter). Next the balancedness in balanced realizations is interpreted from the invariant parameter viewpoint. It is shown that in general a balanced realization does not lead to a balanced observer-controller design. Also the concept of continuous-time balanced stochastic realizations is developed.