Modeling and Control Design of Port-Hamiltonian Systems in Discrete-Time

This paper aims to describe a synthesis procedure for discrete-time, energy-based regulators for continuous-time port-Hamiltonian systems. The methodology consists of three steps. The first deals with the definition of a discrete-time approximation of the plant, which is subsequently employed in the development of the control law. The discrete-time model is obtained from the continuous-time dynamics by replacing the gradient of the Hamiltonian function with a discrete gradient. In this way, passivity, with the energy as storage function, is preserved, although the resulting state equation is in implicit form. The second step concerns the control synthesis and extends the continuous-time energy-shaping plus damping injection design technique to the proposed class of discrete-time port-Hamiltonian systems. Finally, the last step addresses the interconnection between the digital controller and the continuous-time plant. The coupling is implemented via a zero-order hold and relies on the solution of an optimisation problem that determines the “best” and “minimal” correction to be applied to the nominal control action in order to achieve the same performance as that obtained when the regulator is connected in closed loop with the discrete-time model of the plant. This is the reference scenario used to develop and tune the control law. The complete procedure (time discretisation, control design, and coupling implementation) is illustrated through an example.