Trajectory Tracking Control Of Port-Controlled Hamiltonian Systems With Nonholonomic Constraints

The conventional canonical transformation in classical mechanics was to transform a set of Hamilton's canonical equations into another one i.e. It preserves the Hamiltonian structure of dynamical systems in classical mechanics. This method belongs to the theory of geometric control for nonlinear systems. A.J.van der Schaft [6] showed that the unity feedback will render the input and output signals of the generalized Hamiltonian system converging to 0 via the passivity property. Moreover, if system is zero-state detectable, the unity feedback will stabilize the system. Generally, however, most mechanical systems do not satisfy it. We can use generalized canonical to transform the original Hamiltonian system into another Hamiltonian system which preserve the inner property. K.Fujimoto[20] introduced the canonical transformation for PCH systems, which preserves the Hamiltonian structure of the original systems .Hamiltonian systems with dynamic constraints are divided into two kinds .They are holononomic and noholonomic constraints systems. So far, a relatively systemic theory has been found for holonomic systems. Nonholonomic constraints exist in mobile robots, aircrafts and artificial satellites. Because of the difficulty of control, there are much challenges on the research and realization to the nonholonomic systems. Especially the stabilization and trajectory tracking of the systems are studied deeply.In this paper, we give some researches on generalized canonical transformations which preserve inner property and stabilization and trajectory of nonholonomic Hamiltonian systems. The main contents:1. Port-controlled Hamiltonian systems with dissipation is used to be the object, on which we had construct generalized canonical transformations transform the original into what we need.2. Port-controlled Hamiltonian systems with affine constraints is used to be the object, on which we first use generalized canonical transformations to transform the system on the dimensional,then use the passivity and zero-state detectable conditions to stabilize the system ,the unity feedback will stabilize the system .Next we use generalized canonical transformations to transform the system into an error system,then trajectory tracking becomes stabilize the error system .3. A ball on a rotating table is used to be the object, on which we use generalized canonical transformations and use the passivity and zero-state detectable conditions to the stabilize the system and trajectory tracking of the system .