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

This paper presents a design framework of discrete-time regulators for linear port-Hamiltonian boundary control systems. The contribution is twofold. At first, a discrete-time approximation of the plant dynamics originally described by a linear PDE with boundary actuation is introduced. The discretisation is performed in time only. Thus, the 'distributed nature' of the state is maintained. Such a system inherits the passivity of the original one and is well-posed, i.e. the 'next' state always exists. In the second part, instead, the control design problem is tackled. Initially, the characterisation of discrete-time, linear controllers in the port-Hamiltonian form that render the closed-loop dynamics asymptotically stable is presented. Then, the control by energy-shaping and damping injection paradigm is extended to deal with this novel class of distributed-parameter systems. A numerical example illustrates the effectiveness of the proposed framework.