Control of port-Hamiltonian differential-algebraic systems and applications

Abstract

The modeling framework of port-Hamiltonian descriptor systems and their use in numerical simulation and control are discussed. The structure is ideal for automated network-based modeling since it is invariant under power-conserving interconnection, congruence transformations, and Galerkin projection. Moreover, stability and passivity properties are easily shown. Condensed forms under orthogonal transformations present easy analysis tools for existence, uniqueness, regularity, and numerical methods to check these properties.After recalling the concepts for general linear and nonlinear descriptor systems, we demonstrate that many difficulties that arise in general descriptor systems can be easily overcome within the port-Hamiltonian framework. The properties of port-Hamiltonian descriptor systems are analyzed, time-discretization, and numerical linear algebra techniques are discussed. Structure-preserving regularization procedures for descriptor systems are presented to make them suitable for simulation and control. Model reduction techniques that preserve the structure and stabilization and optimal control techniques are discussed.The properties of port-Hamiltonian descriptor systems and their use in modeling simulation and control methods are illustrated with several examples from different physical domains. The survey concludes with open problems and research topics that deserve further attention.

On discrete-time port-Hamiltonian (descriptor) systems

“…The same stability definition can be used for discrete-time descriptor systems, see also [17,43] for the continuous-time case. Definition 2.7.…”

“…It is well known that the interconnection of continuous-time pH systems using Dirac subspaces preserves the pH system structure [42,43]. In this section, we present an analogous type of interconnection for discrete-time systems.…”

The difference between port-Hamiltonian, passive and positive real descriptor systems

“…However, if Q is not invertible then this multiplication might enlarge the solution set of the descriptor system. The treatment of pH systems with singular Q is described in detail in [28, Section 6.3], see also [29].…”

“…For the numerical treatment it is beneficial to use only unitary or orthogonal transformations S and T to obtain the Kalman-like form given in Proposition 4.1. Such condensed forms which can be obtained using unitary or orthogonal transformations S and T were given in [11], see also [29,Section 7].…”

“…Port-Hamiltonian (pH) systems have been increasingly used in recent years as a unified structured framework for energy based modeling of systems, see e.g. [5,6,25,29,32,38,39]. This type of formulation has gained increased interest from engineers and mathematicians due to its modeling flexibility and robustness properties [5,27,28,30].…”