Minimum Effort Control Using Variational Methods of Analytical Mechanics A New Approach For Optimal Control

Abstract

Modern optimal control theory involves adjoining the already known equations of motion of a dynamic system to the objective function using dynamic costates; this is done in order to constrain the optimal control solutions to satisfy the equations of motion. The use of costates increases the number of variables and hence increases the complexity of the problem. On the other hand, variational methods of analytical mechanics finds the equations of motion by minimizing an action functional of the dynamic system, realizing control forces as external input to the system. In this paper a new disruptive approach for computing the optimal control is presented. This approach adopts the variational methods of analytical mechanics to derive equations for the control, in addition to the equations of motion. This is achieved by recognizing the control actuator as part of the dynamic system. In addition to the kinetic energy and potential energy, the action functional in this new approach includes additional energy terms that represent the control energy of the system. Two different methods are presented to write the modified action functional. The proposed approach is a significant departure from the modern optimal control theory, and it eliminates the need for costates when solving for the control. In this paper, a case study is presented to demonstrate the new approach.