A Direct Optimal Tracking Methodology for Nonlinear Dynamic Systems

A completely algebraic direct optimal tracking methodology is developed and illustrated for the control of nonlinear, time-varying, spatially discrete mechanical systems. System dynamics is obtained from the work-energy equation given by Hamilton´s Law, using an assumed time-modes approach. Expansion coefficients of admissible time-modes for the dependent variables of the dynamic system and those for input forces constitute the states and controls respectively. This representation permits explicit integration in time producing a set of algebraic state equations, which replaces the conventional first order differential state equations. The tracks (design response) can be specified analytically or supplied as point data in time and they are curve fitted. The usual integral form of a quadratic regulator performance measure employed in the optimal tracking problem is also converted to an equivalent algebraic one. It is shown that the proposed methodology results in a completely algebraic optimality problem from which a closed form solution for nonlinear feedback control law is obtained directly. Simulation of a two degrees of freedom nonconservative system is included.