Optimal Control in Real Time by a Linear System with Disturbance

A method of sequential synthesis of time optimal control by a linear system with unknown disturbance is considered. A system of linear algebraic equations is obtained that relays the increments of the phase coordinates to the increments of the initial conditions of the normalized adjoint system and that of the control completion time. The evaluations consist in solving repeatedly the system of linear algebraic equations and integrating the matrix differential equation on the displacement intervals of the control switching times and that of the final control time. A procedure of correcting the switching times and the completion time in moving the phase trajectory of the controlled object is examined. The simple and constructive conditions are obtained for: occurrence of discontinuous mode; moving the representative point along the switching manifolds; transformation of the optimal control structure in moving the phase trajectory of the system with uncontrollable disturbance. The computational algorithm is given. The sequence of controls is proved to converge locally, at a quadratic rate, and globally to time optimal control.