Discrete-Time Synergetic Optimal Control of Nonlinear Systems

Discrete-time synergetic optimal control strategies are proposed for a class of nonlinear discrete-time dynamic systems, in which a special performance index is used that results in closed-form solutions of the optimal problems. Reduced-dimension aggregated variables representing a combination of the actual controlled plant variables are used to define the performance index. The control law optimizing the performance index for the respective nonlinear dynamic system is derived from the associated first-order difference equation in terms of the aggregated variables. Some connections betweendiscrete-time synergetic control and the discrete-time linear quadratic regulator aswell as discrete-time variable-structure sliding-mode controls are established. A control design procedure leading to closedloop stability of a class of nonlinear systems with matched nonlinearities is presented. For these types of systems, the discrete-time synergetic optimal control strategy for tracking problems is developed by incorporating integral action.The closed-loop stability depends uponproper construction of the aggregated variables so that the closed-loop nonlinear system on themanifold specified by the aggregated variables is asymptotically stable. An algorithm for the construction of such a stabilizingmanifold is given. Finally, the results are illustratedwith an example that involves a discrete-time nonlinear helicopter model.