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On the computation of a time-varying optimal gain based on exact discretization of a matrix differential Riccati equation

An exact discrete-time model of a matrix differential Riccati equation that has recently been proposed is applied to a finite-horizon optimal control problem. This discretization method can always be used for a usual backward-in-time computation of a time-varying feedback control gain, which can be stored then in a memory for later use. Unlike existing methods that involve errors, the proposed algorithm yields gain values that are exact to those of the continuous-time case at discrete-time instants for any discrete-time period. Furthermore, an approach is suggested for extending this method to an on-line computation of the optimal gain, by converting the boundary value condition into an initial value condition using the backward differential Riccati equation. It was found from simulation studies that the proposed on-line algorithm can yield the correct gain sequence under a certain condition, eliminating the need to prepare the gain sequence a priori. However, a further investigation is needed to clarify this condition.