The Limiting Form of Linear Regulators and the Output Zeroing Problem

Abstract

The problem to be considered in this paper is to find general duality between a control problem and a state estimation in linear time invariant systems.
A dual form of the filter equation, derived previously by present authors for a linear stochastic system with completely unknown disturbances, is introduced. Then it is shown that the above dual equation yields the solution of an output zeroing problem.
On the other hand, a limiting form can be derived from a standard linear regulator problem by letting the weighting matrix associated with the state vector tend to infinity. Futhermore, it is poited out that the limiting form also resolves itself into the solution to the output zeroing problem. From this observation, it can be shown that the above limiting form of standard linear regulator coincides with the dual form of the filter equation with completely unknown diturbances. Coincidentally, the solution of the output zeroing problem which minimizes the input cost function is explicitly obtained.