A Direct Discrete-time IDA-PBC Design Method for a Class of Underactuated Hamiltonian Systems

In this paper, a direct discrete time design method in the sense of passivity-based control (PBC) is investigated. This method, which is known as interconnection and damping (IDA), deals with the stabilization of under-actuated mechanical systems and, it is based on the modification of both the potential and kinetic energies. In order to give a direct discrete time IDA-PBC design method, the discrete time counterpart of matching conditions is derived using an appropriate discrete gradient. The discretetime matching conditions are obtained as a set of linear partial differential equations which can be solved off-line parametrically and a set of linear equations. The unknown parameters of linear partial differential equations and the linear equations have to be solved at each sampling time, to calculate control rule. Moreover, a design procedure is given to solve these matching conditions for a class of Hamiltonian systems. To illustrate the effectiveness and the appropriateness of the proposed method, the example of pendulum on a cart is considered.