A Joined Geometric Structure for Hamiltonian and Gradient Control Systems

Abstract

A unified framework is presented to describe (possibly dissipative) Hamiltonian control systems and gradient control systems, as well as their interconnections. An example of the Hamiltonian systems considered are mechanical systems with nonholonomic kinematic constraints and damping. As an example of a gradient system the Brayton-Moser model for electrical RLC circuits is given. The framework is based on the new geometric notion of a Dirac structure with respect to a non-canonical bilinear form. Dissipativity can be directly related to the structural properties of the non-canonical bilinear form. Applications of the framework to control purposes are briefly discussed. In particular, energy shaping passivity based control for Hamiltonian systems translates into the recently proposed power shaping control for RLC circuits