Connections between Constrained Motions of Mechanical Systems and Tracking Control of Nonlinear Systems
Firdaus E. Udwadia
Departments of Aerospace and Mechanical Engineering,
Civil Engineering, and Mathematics
University of Southern California, Los Angeles
This talk aims to expose the connections between the determination of the equations of motion of constrained systems and the problem of tracking control of nonlinear mechanical systems. The duality between the imposition of constraints on a mechanical system and trajectory requirements for tracking control is exposed through the use a simple example. It is shown that given a set of constraints, díAlembertís principle corresponds to the problem of finding the optimal tracking control of a mechanical system for a specific cost function that Nature seems to choose. Furthermore, the general equations for constrained motion of mechanical systems that do not obey díAlembertís principle yield, through this duality, the entire set of Lipschitz continuous controllers that permit exact tracking of the trajectory requirements. The way Nature seems to handle the tracking control problem of highly nonlinear systems suggests ways in which we can develop new control methods that do not make any approximations and/or linearizations related to the nonlinear systems, or their controllers. Examples of the tumbling control of a two-component spacecraft system in a non-uniform gravity field and the synchronization of chaotic gyroscopes are provided illustrating the application of these ideas.