Duality on Orthogonality Conditions for Discrete-Time Optimal Control and Optimal Estimate

Abstract

In this paper, the discrete-time system is studied, and some orthogonality conditions are derived about response signals in optimal control. These conditions are just dual relations of well known orthogonality conditions on the linear least-square estimate. Assume that a control system has an inside driving signal (system noise) and an outside control input signal. The most basic orthogonality for the optimal system is stated as follows: The response for the impulse driving signal is orthogonal to the response for the posterior impulse control signal. This is the dual property of the following orthogonal projection theorem for the linear least-square estimate: The estimate error is orthogonal to prior observation. The above condition for the optimal control is simplified for a deterministic system with no deriving signal as: The initial state response is orthogonal to impulse responses. The duality is an extension of the well known duality written in terms of Riccati equations. Other orthogonality conditions are also clarified for the optimal control such as dual properties of orthogonality related to the innovation process.