Arjan van der Schaft

On Representations and Integrability of Mathematical Structures in Energy-Conserving Physical Systems

In the present paper we elaborate on the underlying Hamiltonian structure of interconnected energy-conserving physical systems. It is shown that a power-conserving interconnection of port-controlled generalized Hamiltonian systems leads to an implicit generalized Hamiltonian system, and a power-conserving partial interconnection to an implicit port-controlled Hamiltonian system. The crucial concept is the notion of a (generalized) Dirac structure, defined on the space of energy-variables or on the product of the space of energy-variables and the space of flow-variables in the port-controlled case. Three natural representations of generalized Dirac structures are treated. Necessary and sufficient conditions for closedness (or integrability) of Dirac structures in all three representations are obtained. The theory is applied to implicit port-controlled generalized Hamiltonian systems, and it is shown that the closedness condition for the Dirac structure leads to strong conditions on the input vector fields.

Port-Hamiltonian Systems Theory: An Introductory Overview

An up-to-date survey of the theory of port-Hamiltonian systems is given, emphasizing novel developments and relationships with other formalisms. Port-Hamiltonian systems theory yields a systematic framework for network modeling of multi-physics systems. Examples from different areas show the range of applicability. While the emphasis is on modeling and analysis, the last part provides a brief introduction to control of port-Hamiltonian systems.

Hamiltonian discretization of boundary control systems

A fundamental problem in the simulation and control of complex physical systems containing distributed-parameter components concerns finite-dimensional approximation. Numerical methods for partial differential equations (PDEs) usually assume the boundary conditions to be given, while more often than not the interaction of the distributed-parameter components with the other components takes place precisely via the boundary. On the other hand, finite-dimensional approximation methods for infinite-dimensional input-output systems (e.g., in semi-group format) are not easily relatable to numerical techniques for solving PDEs, and are mainly confined to linear PDEs. In this paper we take a new view on this problem by proposing a method for spatial discretization of boundary control systems based on a particular type of mixed finite elements, resulting in a finite-dimensional input-output system. The approach is based on formulating the distributed-parameter component as an infinite-dimensional port-Hamiltonian system, and exploiting the geometric structure of this representation for the choice of appropriate mixed finite elements. The spatially discretized system is again a port-Hamiltonian system, which can be treated as an approximating lumped-parameter physical system of the same type. In the current paper this program is carried out for the case of an ideal transmission line described by the telegraphers equations, and for the two-dimensional wave equation.

Symmetries and conservation laws for Hamiltonian Systems with inputs and outputs: A generalization of Noether's theorem

The definition of symmetries and conservation laws for autonomous (i.e. without external (cases) Hamiltonian systems are generalized to Hamiltonian systems with inputs and outputs. It is shown that a symmetry implies the existence of a conservation law and vice versa; thereby generalizing Noethers theorem for autonomous Hamiltonian systems.

Speed Observation and Position Feedback Stabilization of Partially Linearizable Mechanical Systems

The problems of speed observation and position feedback stabilization of mechanical systems are addressed in this paper. Our interest is centered on systems that can be rendered linear in the velocities via a (partial) change of coordinates. It is shown that the class is fully characterized by the solvability of a set of partial differential equations (PDEs) and strictly contains the class studied in the existing literature on linearization for speed observation or control. A reduced order globally exponentially stable observer, constructed using the immersion and invariance methodology, is proposed. The design requires the solution of another set of PDEs, which are shown to be solvable in several practical examples. It is also proven that the full order observer with dynamic scaling recently proposed by Karagiannis and Astolfi obviates the need to solve the latter PDEs. Finally, it is shown that the observer can be used in conjunction with an asymptotically stabilizing full state--feedback interconnection and damping assignment passivity--based controller preserving asymptotic stability.

Passive output feedback and port interconnection

Abstract In this paper, the design of an intrinsically passive controller having a Hamiltonian structure plus dissipation will be presented. This controller will be intrinsically passive since his coupling to the plant will be through a power port. It will be shown that this does not in general imply the necessity of measuring the port variables for implementation purposes.

On the Mathematical Structure of Balanced Chemical Reaction Networks Governed by Mass Action Kinetics

Motivated by recent progress on the interplay between graph theory, dynamics, and systems theory, we revisit the analysis of chemical reaction networks described by mass action kinetics. For reaction networks possessing a thermodynamic equilibrium we derive a compact formulation exhibiting at the same time the structure of the complex graph and the stoichiometry of the network, and which admits a direct thermodynamical interpretation. This formulation allows us to easily characterize the set of positive equilibria and their stability properties. Furthermore, we develop a framework for interconnection of chemical reaction networks, and we discuss how the formulation leads to a new approach for model reduction.

A passivity-based approach to reset control systems stability

The stability of reset control systems has been mainly studied for the feedback interconnection of reset compensators with linear time-invariant systems. This work gives a stability analysis of reset compensators in feedback interconnection with passive nonlinear systems. The results are based on the passivity approach to L2-stability for feedback systems with exogenous inputs, and the fact that a reset compensator will be passive if its base compensator is passive. Several examples of full and partial reset compensations are analyzed, and a detailed case study of an in-line pH control system is given.

Structure preserving model reduction of port-Hamiltonian systems by moment matching at infinity

It is shown that by use of the Kalman-decomposition an uncontrollable and/or unobservable port-Hamiltonian system is reduced to a controllable/observable system that inherits a port-Hamiltonian structure. Energy and co-energy variable representations for port-Hamiltonian systems are discussed and the reduction procedures are used for both representations. These exact reduction procedures motivate two approximate reduction procedures structure preserving for a general port-Hamiltonian system in scattering representation, Effort- and Flow-constraint reduction methods.

Conservation Laws and Lumped System Dynamics

Physical systems modeling, aimed at network modeling of complex multi-physics systems, has especially flourished in the fifties and sixties of the 20-th century, see e.g. [11, 12] and references provided therein. With the reinforcement of the ’systems’ legacy in Systems & Control, the growing recognition that ’control’ is not confined to developing algorithms for processing the measurements of the system into control signals (but instead is concerned with the design of the total controlled system), and facing the complexity of modern technological and natural systems, systematic methods for physical systems modeling of large-scale lumpedand distributed-parameter systems capturing their basic physical characteristics are needed more than ever.