Jacquelien M. A. Scherpen

Marko Seslija, Jacquelien M. A. Scherpen, Arjan van der Schaft

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.

Krishna Chaitanya Kosaraju, Michele Cucuzzella, Jacquelien M. A. Scherpen et al.

This paper deals with a class of Resistive-Inductive-Capacitive (RLC) circuits and switched RLC (s-RLC) circuits modeled in Brayton Moser framework. For this class of systems, new passivity properties using a Krasovskii's type Lyapunov function as storage function are presented. Consequently, the supply-rate is a function of the system states, inputs and their first time-derivatives. Moreover, after showing the integrability property of the port-variables, two simple control methodologies called output shaping and input shaping are proposed for regulating the voltage in RLC and s-RLC circuits. Global asymptotic convergence to the desired operating point is theoretically proved for both proposed control methodologies. Moreover, robustness with respect to load uncertainty is ensured by the input shaping methodology. The applicability of the proposed methodologies is illustrated by designing voltage controllers for DC-DC converters and DC networks.

Marko Seslija, Arjan van der Schaft, Jacquelien M. A. Scherpen

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.

Michele Cucuzzella, Riccardo Lazzari, Yu Kawano et al.

This work deals with the design of a robust and decentralized passivity-based control scheme for regulating the voltage of a DC microgrid through boost converters. A Krasovskii-type storage function is proposed and a (local) passivity property for DC microgrids comprising unknown 'ZIP' (constant impedance 'Z', constant current 'I' and constant power 'P') loads is established. More precisely, the input port-variable of the corresponding passive map is equal to the first-time derivative of the control input. Then, the integrated input port-variable is used to shape the closed loop storage function such that it has a minimum at the desired equilibrium point. Convergence to the desired equilibrium is theoretically analyzed and the proposed control scheme is validated through experiments on a real DC microgrid.

Yu Kawano, Jacquelien M. A. Scherpen

In this paper, we present an empirical balanced truncation method for nonlinear systems with linear time-invariant input vector field components. First, we define differential reachability and observability Gramians. They are matrix valued functions of the state trajectory (i.e. the initial state and input trajectory) of the original nonlinear system, and it is difficult to find them as functions of the initial state and input. The main result of this paper is to show that for a fixed state trajectory, it is possible to compute the values of these Gramians by using impulse and initial state responses of the variational system. Therefore, balanced truncation is doable along the fixed state trajectory without solving nonlinear partial differential equations, differently from conventional nonlinear balancing methods. We further develop an approximation method, which only requires trajectories of the original nonlinear systems. Our methods are demonstrated by an RL network along a trajectory.

Krishna Chaitanya Kosaraju, Yu Kawano, Jacquelien M. A. Scherpen

In this paper we introduce a new notion of passivity which we call Krasovskii's passivity and provide a sufficient condition for a system to be Krasovskii's passive. Based on this condition, we investigate classes of port-Hamiltonian and gradient systems which are Krasovskii's passive. Moreover, we provide a new interconnection based control technique based on Krasovskii's passivity. Our proposed control technique can be used even in the case when it is not clear how to construct the standard passivity based controller, which is demonstrated by examples of a Boost converter and a parallel RLC circuit.

H. Jardon-Kojakhmetov, Jacquelien M. A. Scherpen, D. del Puerto-Flores

In this document, we deal with the stabilization problem of slow-fast systems (or singularly perturbed Ordinary Differential Equations) at a non-hyperbolic point. The class of systems studied here have the following properties: 1) they have one fast variable and an arbitrary number of slow variables, 2) they have a non-hyperbolic singularity of the fold type at the origin. The presence of the aforementioned singularity complicates the analysis and the controller design of such systems. In particular, the classical theory of singular perturbations cannot be used. We show a novel design process based on geometric desingularization, which allows the stabilization of a fold point of singularly perturbed control systems. Our results are exemplified on an electric circuit.

Kees Loeff, Matin Jafarian, Jacquelien M. A. Scherpen

This document presents the data for a single-phase distribution bus (based on IEEE 37 bus) together with the model and data for a PV inverter and active and reactive power loads.

Jochem van der Veen, Pablo Borja, Jacquelien M. A. Scherpen

In this work, we propose a passivity-based control approach that addresses the trajectory tracking problem for a class of mechanical systems that comprises a broad range of robotic arms. The resulting controllers can be naturally saturated and do not require velocity measurements. Moreover, the proposed methodology does not require the implementation of observers, and the structure of the closed-loop system permits the identification of a Lyapunov function, which eases the convergence analysis. To corroborate the effectiveness of the methodology, we perform experiments with the Philips Experimental Robot Arm.

Xiaodong Cheng, Yu Kawano, Jacquelien M. A. Scherpen

This paper proposes a general framework for structure-preserving model reduction of a secondorder network system based on graph clustering. In this approach, vertex dynamics are captured by the transfer functions from inputs to individual states, and the dissimilarities of vertices are quantified by the H2-norms of the transfer function discrepancies. A greedy hierarchical clustering algorithm is proposed to place those vertices with similar dynamics into same clusters. Then, the reduced-order model is generated by the Petrov-Galerkin method, where the projection is formed by the characteristic matrix of the resulting network clustering. It is shown that the simplified system preserves an interconnection structure, i.e., it can be again interpreted as a second-order system evolving over a reduced graph. Furthermore, this paper generalizes the definition of network controllability Gramian to second-order network systems. Based on it, we develop an efficient method to compute H2-norms and derive the approximation error between the full-order and reduced-order models. Finally, the approach is illustrated by the example of a small-world network.

Nima Monshizadeh, Claudio De Persis, Arjan J. van der Schaft et al.

We consider a network-preserved model of power networks with proper algebraic constraints resulting from constant power loads. Both for the linear and the nonlinear differential algebraic model of the network, we derive explicit reduced models which are fully expressed in terms of ordinary differential equations. For deriving these reduced models, we introduce the "projected incidence" matrix which yields a novel decomposition of the reduced Laplacian matrix. With the help of this new matrix, we provide a complementary approach to Kron reduction which is able to cope with constant power loads and nonlinear power flow equations.