Abstract

Most real world dynamical systems consist of subsystems from different physical domains, modelled by partial-differential equations, ordinary differential equations, and algebraic equations, combined with input and output connections. To deal with such complex system, in recent years the class of dissipative port-Hamiltonian (pH) descriptor systems has emerged as a very successful modeling methodology. The main reasons are that the network based interconnection of pH systems is again pH, Galerkin projection in PDE discretization and model reduction preserve the pH structure and the physical properties are encoded in the geometric properties of the flow as well as the algebraic properties of the equations. Furthermore, dissipative pH system form a very robust representation under structured perturbations and directly indicate Lyapunov functions for stability analysis.

Another advantage of energy based modeling via pH systems is that each separate model of a physical system can be a whole model catalog from which models can be chosen in an adaptive way within simulation and optimization methods.

We discuss the class of pH descriptor systems and illustrate how many classical real world mathematical models can be formulated in this class. We illustrate the results with some real world examples from gas transport and district heating systems and point out emerging mathematical challenges.