Direct Optimal Tracking of Linear Systems using Assumed-Time-Modes Method

Direct optimal contol policies are developed for tracking control of spatially discrete linear systems. An assumed time-modes method in which the dependent variables of the dynamics problem are expanded in, terms of admissible basis functions in time is extended to include a similar representation of the control inputs. This approach together with direct use of Hamilton´s law of varying action allows explicit integration in time leading to the algebraic equations of motion. Expansion coefficients for the dependent variables and those for input forces constitute the states and controls respectively. Prescribed trajectories for tracking can be supplied analytically or as point-data and they are curve fitted for direct use. A typical performance measure of linear quadratic regulator theory is considered for the linear optimal tracking problem and it is trasformed into an algebraic objective function by using the assumed time-modes expansions. Resulting algebraic linear optimal trcking ing problem yields closed form solutions for the optimal control gains directly. The method is illustrated with simulation of a well-known two degrees of freedom problem.