arXiv subjects

Arjan van der Schaft

Publications and source records attributed to Arjan van der Schaft.

At least 19 recordsLinked to original sources

arXiv2026

Suboptimal control by primal-dual gradient dynamics

This note generalizes the port-Hamiltonian formulation of the continuous time primal-dual gradient algorithm for static constrained convex optimization to the convex optimal control problem.The resulting dynamics is shown to be a port-Hamiltonian system of partial differential equations, involving ordinary physical time as well 'algorithmic' time. Convergence to the optimal control solution is indicated, and it is argued that sub-optimal control strategies could be derived starting from the partial differential equation formulation.

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arXiv2026

On Port-Hamiltonian Formulation of Hysteretic Energy Storage Elements: The Backlash Case

This paper presents a port-Hamiltonian formulation of hysteretic energy storage elements. First, we revisit the passivity property of backlash-driven storage elements by presenting a family of storage functions associated to the dissipativity property of such elements. We explicitly derive the corresponding available storage and required supply functions `a la Willems [1], and show the interlacing property of the aforementioned family of storage functions sandwiched between the available storage and required supply functions. Second, using the proposed family of storage functions, we present a port-Hamiltonian formulation of hysteretic inductors as prototypical storage elements in port-Hamiltonian systems. In particular, we show how a Hamiltonian function can be chosen from the family of storage functions and how the hysteretic elements can be expressed as port-Hamiltonian system with feedthrough term, where the feedthrough term represents energy dissipation. Correspondingly, we illustrate its applicability in describing an RLC circuit (in parallel and in series) containing a hysteretic inductor element.

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arXiv2026

Hopfield neural networks as port-Hamiltonian and gradient systems

The structure of continuous Hopfield networks is revisited from a system-theoretic point of view. After adopting a novel electrical network interpretation involving nonlinear capacitors, it is shown that Hopfield networks admit a port-Hamiltonian formulation provided an extra passivity condition is satisfied. Subsequently it is shown that any Hopfield network can be represented as a gradient system, with Riemannian metric given by the inverse of the Hessian matrix of the total energy stored in the nonlinear capacitors. On the other hand, the well-known 'energy' function employed by Hopfield turns out to be the dissipation potential of the gradient system, and this potential is shown to satisfy a dissipation inequality that can be used for analysis and interconnection.

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arXiv2025

Singular networks and ultrasensitive terminal behaviors

Negative conductance elements are key to shape the input-output behavior at the terminals of a network through localized positive feedback amplification. The balance of positive and negative differential conductances creates singularities at which rich, intrinsically nonlinear, and ultrasensitive terminal behaviors emerge. Motivated by neuromorphic engineering applications, in this note we extend a recently introduced nonlinear network graphical modeling framework to include negative conductance elements. We use this extended framework to define the class of singular networks and to characterize their ultra-sensitive input/output behaviors at given terminals. Our results are grounded in the Lyapunov-Schmidt reduction method, which is shown to fully characterize the singularities and bifurcations of the input-output behavior at the network terminals, including when the underlying input-output relation is not explicitly computable through other reduction methods.

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arXiv2025

Symmetry in linear physical systems

Physical systems with symmetry arise abundantly in applications, and are endowed with interesting mathematical structures. The present paper focusses on linear reciprocal and input-output Hamiltonian systems. Their characterization is studied from an input-output as well as from a state point of view. Geometrically, it turns out that they both define Lagrangian subspaces with corresponding generating functionals. Furthermore, the relations with time reversibility are analyzed. The system classes under consideration are expected to admit scalable control laws, and to be important building blocks in design.

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arXiv2025

Maximum Power Transfer for Nonlinear State Space Systems

The classical Maximum Power Transfer theorem of linear electrical network theory is generalized to the setting of a nonlinear state space system connected to a source. This yields a state space version of the input-output operator results of Wyatt (1988). Key tool in the analysis is the formulation of a Hamiltonian input-output system, which is closely related to Pontryagin's Maximum principle. The adjoint variational system incorporated in this system defines an optimal load. The structure of such an optimal load is investigated for classes of physical systems.

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arXiv2024

Port-Hamiltonian nonlinear systems

Control theory often takes the mathematical model of the to-be-control-led system for granted. In contrast, port-Hamiltonian systems theory bridges the gap between modelling and control for physical systems. It provides a unified framework for the modelling of complex multiphysics systems. At the same time it offers powerful tools for analysis and control by identifying the underlying physical structure, as reflected in, e.g., energy balance and other conserved quantities. This leads to control schemes that \emph{exploit} the physical structure, instead of compensating for it. As a result, the derived control laws tend to be simple, physically interpretable, and robust with respect to physical parameter variations. In this paper, after introducing port-Hamiltonian systems, the focus is on 'control by interconnection' for set-point stabilization of nonlinear physical systems. Most of this theory is well-established, but novel developments using 'energy ports' instead of 'power ports' are also included.

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arXiv2024

Kron reduction of nonlinear networks

Kron reduction is concerned with the elimination of interior nodes of physical network systems such as linear resistor electrical circuits. In this paper it is shown how this can be extended to networks with nonlinear static relations between the variables associated to the edges of the underlying directed graph.

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arXiv2024

Port-Hamiltonian systems with energy and power ports

We extend the port-Hamiltonian framework defined with respect to a Lagrangian submanifold and a Dirac structure by augmenting the Lagrangian submanifold with the space of external variables. The new pair of conjugated variables is called energy port. We show that in the most general case, the extension describes constrained Hamiltonian systems whose Hamiltonian function depends on inputs.

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arXiv2024

Segmented Model-Based Hydrogen Delivery Control for PEM Fuel Cells: a Port-Hamiltonian Approach

This paper proposes an extended interconnection and damping assignment passivity-based control technique (IDA-PBC) to control the pressure dynamics in the fuel delivery subsystem (FDS) of proton exchange membrane fuel cells. The fuel cell stack is a distributed parameter model which can be modeled by partial differential equations PDEs). In this paper, the segmentation concept is used to approximate the PDEs model by ordinary differential equations (ODEs) model. Therefore, each segments are having multiple ODEs to obtain the lump-sum model of the segments. Subsequently, a generalized multi-input multi-output lumped parameters model is developed in port-Hamiltonian framework based on mass balance to minimize the modeling error. The modeling errors arises due to the difference between spatially distributed pressures in FDS segments, and also due to the difference between the actual stack pressure and the measured output pressure of the anode. The segments interconnection feasibilities are ensured by maintaining passivity of each segment. With consideration of re-circulation and bleeding of the anode in the modeling, an extended energy-shaping and output tracking IDA-PBC based state-feedback controller is proposed to control the spatially distributed pressure dynamics in the anode. Furthermore, a sliding mode observer of high order is designed to estimate the unmeasurable pressures in FDS with known disturbances. Performance recovery of output feedback control is accomplished with explicit stability analysis. The effectiveness of the proposed IDA-PBC approach is validated by the simulation results.

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arXiv2023

Reciprocity of nonlinear systems

One of the key contributions of the 1972 seminal paper by Willems was the analysis of symmetry (also called reciprocity) of input-state-output systems, both from an external (input-output) and internal (state) point of view. The developed theory also included the combination of reciprocity with passivity, and the consideration of relaxation systems, which are passive reciprocal systems without any oscillatory behavior. The paper was motivated from a fundamental system-theoretic point of view (how is external structure reflected into internal structure), as well as by a wide range of application areas, including electrical network synthesis, thermodynamics, and viscoelastic materials. On the other hand, the obtained results are for linear systems, and the extension to the nonlinear case, even for subclasses of nonlinear systems, is far from trivial. The present paper aims at taking some steps into this direction.

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arXiv2023

Comparison of Non-deterministic Linear Systems by $(γ,δ)$-Similarity

We introduce $(γ,δ)$-similarity, a notion of system comparison that measures to what extent two stable linear dynamical systems behave similarly in an input-output sense. This behavioral similarity is characterized by measuring the sensitivity of the difference between the two output trajectories in terms of the external inputs to the two potentially non-deterministic systems. As such, $(γ,δ)$-similarity is a notion that characterizes \emph{approximation} of input-output behavior, whereas existing notions of simulation target equivalence. Next, as this approximation is specified in terms of the $L_2$ signal norm, $(γ,δ)$-similarity allows for integration with existing methods for analysis and synthesis of control systems, in particular, robust control techniques. We characterize the notion of $(γ,δ)$-similarity as a linear matrix inequality feasibility problem and derive its interpretation in terms of transfer matrices. Our study on the compositional properties of $(γ,δ)$-similarity shows that the notion is preserved through series and feedback interconnections. This highlights its potential application in compositional reasoning, namely abstraction and modular synthesis of large-scale interconnected dynamical systems. We further illustrate our results in an electrical network example.

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arXiv2023

Linear Boundary Port-Hamiltonian Systems with Implicitly Defined Energy

In this paper we extend the previously introduced class of boundary port-Hamiltonian systems to boundary control systems where the variational derivative of the Hamiltonian functional is replaced by a pair of reciprocal differential operators. In physical systems modelling, these differential operators naturally represent the constitutive relations associated with the implicitly defined energy of the system and obey Maxwell's reciprocity conditions. On top of the boundary variables associated with the Stokes-Dirac structure, this leads to additional boundary port variables and to the new notion of a Stokes-Lagrange subspace. This extended class of boundary port-Hamiltonian systems is illustrated by a number of examples in the modelling of elastic rods with local and non-local elasticity relations. Finally it shown how a Hamiltonian functional on an extended state space can be associated with the Stokes-Lagrange subspace, and how this leads to an energy balance equation involving the boundary variables of the Stokes-Dirac structure as well as of the Stokes-Lagrange subspace.

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arXiv2023

A port-Hamiltonian framework for displacement-based and rigid formation tracking

This paper proposes a passivity-based port-Hamiltonian (pH) framework for multi-agent displacement-based and rigid formation control and velocity tracking. The control law consists of two parts, where the internal feedback is to track the velocity and the external feedback is to achieve formation stabilization by steering variables of neighboring agents that prescribe the desired geometric shape. Regarding the external feedback, a general framework is proposed for all kinds of formations by means of the advantage that the pH model is energy-based and coordinate-free. To solve the issue that the incidence matrix is not of full column rank over cyclic graphs, the matrix property is used to prove the convergence to the target sets for the displacement-based formation, while for rigid formations, the algebraic conditions of infinitesimal rigidity are investigated to achieve asymptotic local stability. Furthermore, the rigid formation with heterogeneous constraints is further investigated under this framework and the asymptotic local stability is proved under a mild assumption. Simulations are performed to illustrate the effectiveness of the framework.

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arXiv2023

Angle-based formation stabilization and maneuvers in port-Hamiltonian form with bearing and velocity measurements

This paper proposes a port-Hamiltonian framework for angle-based formation stabilization and maneuvers using bearing and velocity measurements with an underlying triangulated Laman graph. The corresponding port-Hamiltonian controller is designed using virtual couplings on the errors of angle constraints in angle space and then the angle constraints and agent actuators are mapped by the constraint Jacobian, which can be applied to other formation constraints. In addition, due to the fact that the port-Hamiltonian model allows for complex and heterogeneous agent dynamics, our framework can be extended to networks with different agent dynamics and formation constraints. To avoid unavailable distance terms in the control law, an estimator is designed based on port-Hamiltonian theory and the property that energy is coordinate-free for different sensor modalities using bearing and velocity measurements, which permits our framework to inject damping for the formation maneuvers. Furthermore, several maneuvers are analyzed under both considerations of stabilization and transient performance. Simulations are performed to illustrate the effectiveness of the approach.

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arXiv2022

Linear port-Hamiltonian DAE systems revisited

Port-Hamiltonian systems theory provides a systematic methodology for the modeling, simulation and control of multi-physics systems. The incorporation of algebraic constraints has led to a multitude of definitions of port-Hamiltonian differential-algebraic equations (DAE) systems. This paper presents extensions of results in Gernandt, Haller & Reis (2021) and Mehrmann & Van der Schaft (2022) in the context of maximally monotone structures and shows that any such space can be written as composition of a Dirac and a resistive structure. Furthermore, appropriate coordinate representations are presented as well as explicit expressions for the associated transfer functions.

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arXiv2022

Port-Hamiltonian systems and monotonicity

The relationships between port-Hamiltonian systems modeling and the notion of monotonicity are explored. The earlier introduced notion of incrementally port-Hamiltonian systems is extended to maximal cyclically monotone relations, together with their generating functions. This gives rise to new classes of incrementally port-Hamiltonian systems, with examples stemming from physical systems modeling as well as from convex optimization. An in-depth treatment is given of the composition of maximal monotone and maximal cyclically monotone relations, where in the latter case the resulting maximal cyclically monotone relation is shown to be computable through the use of generating functions. Furthermore, connections are discussed with incremental versions of passivity, and it is shown how incrementally port-Hamiltonian systems with strictly convex Hamiltonians are (maximal) equilibrium independent passive. Finally, the results on compositionality of monotone relations are employed for a convex optimization approach to the computation of the equilibrium of interconnected incrementally port-Hamiltonian systems.

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arXiv2022

A Lagrange subspace approach to dissipation inequalities

The standard dissipation inequality for passivity is extended from storage functions to general Lagrange subspaces. This is shown to have some interesting consequences. A classical factorization result for passive systems is extended to this generalized case, making use of the newly defined concept of the Hamiltonian lift of a DAE system.

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