arXiv subjects

Arjan van der Schaft

Publications and source records attributed to Arjan van der Schaft.

At least 73 records · Page 4Linked to original sources

arXiv2016

Nonlinear Analysis of an Improved Swing Equation

In this paper, we investigate the properties of an improved swing equation model for synchronous generators. This model is derived by omitting the main simplifying assumption of the conventional swing equation, and requires a novel analysis for the stability and frequency regulation. We consider two scenarios. First we study the case that a synchronous generator is connected to a constant load. Second, we inspect the case of the single machine connected to an infinite bus. Simulations verify the results.

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arXiv2016

Optimal power dispatch in networks of high-dimensional models of synchronous machines

This paper investigates the problem of optimal frequency regulation of multi-machine power networks where each synchronous machine is described by a sixth order model. By analyzing the physical energy stored in the network and the generators, a port-Hamiltonian representation of the multi-machine system is obtained. Moreover, it is shown that the open-loop system is passive with respect to its steady states which implies that passive controllers can be used to control the multi-machine network. As a special case, a distributed consensus based controller is designed that regulates the frequency and minimizes a global quadratic generation cost in the presence of a constant unknown demand. In addition, the proposed controller allows freedom in choosing any desired connected undirected weighted communication graph.

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arXiv2016

Interconnections of input-output Hamiltonian systems with dissipation

Recently, negative imaginary and counter-clockwise systems have attracted attention as an interesting class of systems, which is well-motivated by applications. In this paper first the formulation and extension of negative imaginary and counter-clockwise systems as (nonlinear) input-output Hamiltonian systems with dissipation is summarized. Next it is shown how by considering the time-derivative of the outputs a port-Hamiltonian system is obtained, and how this leads to the consideration of alternate passive outputs for port-Hamiltonian systems. Furthermore, a converse result to positive feedback interconnection of input-output Hamiltonian systems with dissipation is obtained, stating that the positive feedback interconnection of two linear systems is an input-output Hamiltonian system with dissipation if and only if the systems themselves are input-output Hamiltonian systems with dissipation. This implies that the Poisson and resistive structure matrices can be redefined in such a way that the interaction between the two systems only takes place via the coupling term in the Hamiltonian of the interconnected system. Subsequently, it is shown how network interconnection of (possibly nonlinear) input-output Hamiltonian systems with dissipation results in another input-output Hamiltonian system with dissipation, and how this leads to a stability analysis of the interconnected system in terms of the Hamiltonians and output mappings of the systems associated to the vertices, as well as the topology of the network.

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arXiv2016

Stability Analysis of Networked Systems Containing Damped and Undamped Nodes

This paper answers the question if a qualitatively heterogeneous passive networked system containing damped and undamped nodes shows consensus in the output of the nodes in the long run. While a standard Lyapunov analysis shows that the damped nodes will always converge to a steady-state value, the convergence of the undamped nodes is much more delicate and depends on the parameter values of the network as well as on the topology of the graph. A complete stability analysis is presented based on an eigenvector analysis involving the mass values and the topology of both the original graph and the reduced graph obtained by a Kron reduction that eliminates the damped nodes.

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arXiv2015

A Port-Hamiltonian Approach to Optimal Frequency Regulation in Power Grids

This paper studies the problem of frequency regulation in power grids, while maximizing the social welfare. Two price-based controllers are proposed; the first one an internal-model-based controller and the second one based on a continuous gradient method for optimization. Both controllers can be implemented in a fully distributed fashion, with freedom in choosing a controller communication network. As a result, two real-time dynamic pricing models described by port- Hamiltonian systems are obtained. By coupling with the port- Hamiltonian description of the physical network we obtain a closed-loop port-Hamiltonian system, whose properties are ex- ploited to prove asymptotic stability of the set of optimal points. Numerical results show the performance of both controllers in a simple case study.

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arXiv2015

A geometric approach to differential Hamiltonian systems and differential Riccati equations

Motivated by research on contraction analysis and incremental stability/stabilizability the study of 'differential properties' has attracted increasing attention lately. Previously lifts of functions and vector fields to the tangent bundle of the state space manifold have been employed for a geometric approach to differential passivity and dissipativity. In the same vein, the present paper aims at a geometric underpinning and elucidation of recent work on 'control contraction metrics' and 'generalized differential Riccati equations'.

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arXiv2015

Complex and detailed balancing of chemical reaction networks revisited

The characterization of the notions of complex and detailed balancing for mass action kinetics chemical reaction networks is revisited from the perspective of algebraic graph theory, in particular Kirchhoff's Matrix Tree theorem for directed weighted graphs. This yields an elucidation of previously obtained results, in particular with respect to the Wegscheider conditions, and a new necessary and sufficient condition for complex balancing, which can be verified constructively.

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arXiv2015

A network dynamics approach to chemical reaction networks

A crisp survey is given of chemical reaction networks from the perspective of general nonlinear network dynamics, in particular of consensus dynamics. It is shown how by starting from the complex-balanced assumption the reaction dynamics governed by mass action kinetics can be rewritten into a form which allows for a very simple derivation of a number of key results in chemical reaction network theory, and which directly relates to the thermodynamics of the system. Central in this formulation is the definition of a balanced Laplacian matrix on the graph of chemical complexes together with a resulting fundamental inequality. This directly leads to the characterization of the set of equilibria and their stability. Both the form of the dynamics and the deduced dynamical behavior are very similar to consensus dynamics. The assumption of complex-balancedness is revisited from the point of view of Kirchhoff's Matrix Tree theorem, providing a new perspective. Finally, using the classical idea of extending the graph of chemical complexes by an extra 'zero' complex, a complete steady-state stability analysis of mass action kinetics reaction networks with constant inflows and mass action outflows is given.

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arXiv2014

Stability of dynamical distribution networks with arbitrary flow constraints and unknown in/outflows

A basic model of a dynamical distribution network is considered, modeled as a directed graph with storage variables corresponding to every vertex and flow inputs corresponding to every edge, subject to unknown but constant inflows and outflows. We analyze the dynamics of the system in closed-loop with a distributed proportional-integral controller structure, where the flow inputs are constrained to take value in closed intervals. Results from our previous work are extended to general flow constraint intervals, and conditions for asymptotic load balancing are derived that rely on the structure of the graph and its flow constraints.

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arXiv2014

Structure-preserving model reduction of physical network systems by clustering

In this paper, we establish a method for model order reduction of a certain class of physical network systems. The proposed method is based on clustering of the vertices of the underlying graph, and yields a reduced order model within the same class. To capture the physical properties of the network, we allow for weights associated to both the edges as well as the vertices of the graph. We extend the notion of almost equitable partitions to this class of graphs. Consequently, an explicit model reduction error expression in the sense of H2-norm is provided for clustering arising from almost equitable partitions. Finally the method is extended to second-order systems.

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arXiv2013

On differential passivity of physical systems

Differential passivity is a property that allows to check with a pointwise criterion that a system is incrementally passive, a property that is relevant to study interconnected systems in the context of regulation, synchronization, and estimation. The paper investigates how restrictive is the property, focusing on a class of open gradient systems encountered in the coenergy modeling framework of physical systems, in particular the Brayton-Moser formalism for nonlinear electrical circuits.

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arXiv2013

Explicit Simplicial Discretization of Distributed-Parameter Port-Hamiltonian Systems

Simplicial Dirac structures as finite analogues of the canonical Stokes-Dirac structure, capturing the topological laws of the system, are defined on simplicial manifolds in terms of primal and dual cochains related by the coboundary operators. These finite-dimensional Dirac structures offer a framework for the formulation of standard input-output finite-dimensional port-Hamiltonian systems that emulate the behavior of distributed-parameter port-Hamiltonian systems. This paper elaborates on the matrix representations of simplicial Dirac structures and the resulting port-Hamiltonian systems on simplicial manifolds. Employing these representations, we consider the existence of structural invariants and demonstrate how they pertain to the energy shaping of port-Hamiltonian systems on simplicial manifolds.

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arXiv2013

Reaction-Diffusion Systems as Complex Networks

The spatially distributed reaction networks are indispensable for the understanding of many important phenomena concerning the development of organisms, coordinated cell behavior, and pattern formation. The purpose of this brief discussion paper is to point out some open problems in the theory of PDE and compartmental ODE models of balanced reaction-diffusion networks.

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arXiv2012

Hamiltonian Perspective on Compartmental Reaction-Diffusion Networks

Inspired by the recent developments in modeling and analysis of reaction networks, we provide a geometric formulation of the reversible reaction networks under the influence of diffusion. Using the graph knowledge of the underlying reaction network, the obtained reaction-diffusion system is a distributed-parameter port-Hamiltonian system on a compact spatial domain. Motivated by the need for computer based design, we offer a spatially consistent discretization of the PDE system and, in a systematic manner, recover a compartmental ODE model on a simplicial triangulation of the spatial domain. Exploring the properties of a balanced weighted Laplacian matrix of the reaction network and the Laplacian of the simplicial complex, we characterize the space of equilibrium points and provide a simple stability analysis on the state space modulo the space of equilibrium points. The paper rules out the possibility of the persistence of spatial patterns for the compartmental balanced reaction-diffusion networks.

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arXiv2012

Model-order reduction of biochemical reaction networks

In this paper we propose a model-order reduction method for chemical reaction networks governed by general enzyme kinetics, including the mass-action and Michaelis-Menten kinetics. The model-order reduction method is based on the Kron reduction of the weighted Laplacian matrix which describes the graph structure of complexes in the chemical reaction network. We apply our method to a yeast glycolysis model, where the simulation result shows that the transient behaviour of a number of key metabolites of the reduced-order model is in good agreement with those of the full-order model.

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arXiv2012

A Graph-Theoretical Approach for the Analysis and Model Reduction of Complex-Balanced Chemical Reaction Networks

In this paper we derive a compact mathematical formulation describing the dynamics of chemical reaction networks that are complex-balanced and are governed by mass action kinetics. The formulation is based on the graph of (substrate and product) complexes and the stoichiometric information of these complexes, and crucially uses a balanced weighted Laplacian matrix. It is shown that this formulation leads to elegant methods for characterizing the space of all equilibria for complex-balanced networks and for deriving stability properties of such networks. We propose a method for model reduction of complex-balanced networks, which is similar to the Kron reduction method for electrical networks and involves the computation of Schur complements of the balanced weighted Laplacian matrix.

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arXiv2012

Reduction of Stokes-Dirac structures and gauge symmetry in port-Hamiltonian systems

Stokes-Dirac structures are infinite-dimensional Dirac structures defined in terms of differential forms on a smooth manifold with boundary. These Dirac structures lay down a geometric framework for the formulation of Hamiltonian systems with a nonzero boundary energy flow. Simplicial triangulation of the underlaying manifold leads to the so-called simplicial Dirac structures, discrete analogues of Stokes-Dirac structures, and thus provides a natural framework for deriving finite-dimensional port-Hamiltonian systems that emulate their infinite-dimensional counterparts. The port-Hamiltonian systems defined with respect to Stokes-Dirac and simplicial Dirac structures exhibit gauge and a discrete gauge symmetry, respectively. In this paper, employing Poisson reduction we offer a unified technique for the symmetry reduction of a generalized canonical infinite-dimensional Dirac structure to the Poisson structure associated with Stokes-Dirac structures and of a fine-dimensional Dirac structure to simplicial Dirac structures. We demonstrate this Poisson scheme on a physical example of the vibrating string.

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arXiv2012

Discrete Exterior Geometry Approach to Structure-Preserving Discretization of Distributed-Parameter Port-Hamiltonian Systems

This paper addresses the issue of structure-preserving discretization of open distributed-parameter systems with Hamiltonian dynamics. Employing the formalism of discrete exterior calculus, we introduce a simplicial Dirac structure as a discrete analogue of the Stokes-Dirac structure and demonstrate that it provides a natural framework for deriving finite-dimensional port-Hamiltonian systems that emulate their infinite-dimensional counterparts. The spatial domain, in the continuous theory represented by a finite-dimensional smooth manifold with boundary, is replaced by a homological manifold-like simplicial complex and its augmented circumcentric dual. The smooth differential forms, in discrete setting, are mirrored by cochains on the primal and dual complexes, while the discrete exterior derivative is defined to be the coboundary operator. This approach of discrete differential geometry, rather than discretizing the partial differential equations, allows to first discretize the underlying Stokes-Dirac structure and then to impose the corresponding finite-dimensional port-Hamiltonian dynamics. In this manner, a number of important intrinsically topological and geometrical properties of the system are preserved.

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