STABILIZATION OF DYNAMICAL SYSTEMS BY DIRECT MINIMIZATION OF KINETIC ENERGY

Abstract

It was established long ago that the stability of continuous time dynamical systems can be studied by using positive non-increasing energy functions, which came to be known as Lyapunov functions. While the process of establishing these functions for linear systems has been easy, the case has been different for nonlinear systems where trial and error methods are still used. This paper revives the application of the system kinetic energy as a Lyapunov function for stability analysis of that particular system. The kinetic energy was the foundation that gave rise to the notion of the generalized energy functions, which became prominent in linear optimal control with names such as cost functions or objective functions. Although generalized energy functions have been very successful with linear systems, no equivalent generalized energy function has ever been developed for nonlinear systems. This can be attributed to the fact that the generalized energy function expressions for linear systems have the same structure as the kinetic energy for all linear systems, while the corresponding expression for nonlinear systems depend on the system itself; therefore, it is difficult to develop one generalized energy function structure for all nonlinear systems. The paper recalls the use of kinetic energy in studying the stability of continuous time dynamical systems, and it develops the optimality equation for stabilization of such systems. The paper shows further that most existing stabilizing controllers for linear systems can be defined in terms of this optimality equation. The paper closes by providing some examples of using kinetic energy in analyzing and stabilizing some known nonlinear systems.