A New Approach to Stable Optimal Control of Complex Nonlinear Dynamical Systems

Abstract

This paper gives a simple approach to designing a controller that minimizes a userspecified control cost for a mechanical system while ensuring that the control is stable. For a usergiven Lyapunov function, the method ensures that its time rate of change is negative and equals a user specified negative definite function. Thus a closedform, optimal, nonlinear controller is obtained that minimizes a desired control cost at each instant of time and is guaranteed to be Lyapunov stable. The complete nonlinear dynamical system is handled with no approximations/linearizations, and no a priori structure is imposed on the nature of the controller. The methodology is developed here for systems modeled by secondorder, nonautonomous, nonlinear, differential equations. The approach relies on some recent fundamental results in analytical dynamics and uses ideas from the theory of constrained motion.