Control of port-Hamiltonian differential-algebraic systems and applications

Abstract

We discuss the modelling framework of port-Hamiltonian descriptor systems and their use in numerical simulation and control. The structure is ideal for automated network-based modelling 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 analysed, and 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 modelling 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.

A geometric framework for discrete time port‐Hamiltonian systems

“…It is well known that the loss‐less interconnection of continuous‐time pH systems using Dirac subspaces preserves the pH system structure [11, 12]. In this section, we present an analogous interconnection result for discrete‐time pH systems.…”

“…As a special case, we also discuss norm preserving interconnections. It is shown in [11, 12] that interconnections using Dirac structures not only preserve the passivity of the interconnected system but also the pH system structure. With these results in mind, it is interesting to analyze whether this property holds for the scattering pH formulation of discrete‐time systems using contractive interconnections.…”

Rotational shallow water equations with viscous damping and boundary control: structure-preserving spatial discretization

“…Now let us investigate the linear dissipation model induced by the Navier-Stokes equation, averaged on a slice of fluid: the full model will be recalled first in Section 2.2.1, and recast as a port-Hamiltonian system (an explicit distributed pHs, giving rise to a pH-ODE once spatially discretized) under the form J − R, helping prove dissipativity in an easy way. Then in Section 2.2.2, enlightening a factorization of R as GSG * gives rise to the definition of physically meaningful dissipation ports and a port-Hamiltonian system with an extended structure matrix J e (an implicit distributed pHs, leading to a pH-DAE once spatially discretized, see [38] for a review), allowing a more straightforward computation of the boundary terms in the energy balance: here the tangential component of the velocity does play a role as an additional boundary control port.…”

“…Another source of difficulty would be to use the formulation involving extra dissipation ports, some of them being symmetric tensors, see Section 2.2.2; thus, it has been preferred to work with the J − R formulation, which is of much smaller dimension. The wide class of linear pH-DAEs is already challenging and constitutes an active research topic: in particular, the index of such DAEs can be at most 2 [48,Corollary 51]; the interested reader may refer to [38,49,50] and the many references therein.…”

Distributed Adaptive Formation Control for Uncertain Point Mass Agents With Mixed Dimensional Space

“…The incorporation of energy in the pH framework suits well to the formation control problem as it can be formulated as a design problem of virtual mechanical spring coupling where the minimum energy (associated with the equilibrium point) corresponds to the desired formation shape. In the presence of algebraic constraints, which may arise from physical/interconnection constraints, the pH framework leads to pH differential algebraic equations (pHDAE) [16], [17].…”