port-Hamiltonian Systems Research Articles

Structural Decomposition of the Passivity-Based Control System of Wind–Solar Power Generating and Hybrid Battery-Supercapacitor Energy Storage Complex

Wind–solar power generating and hybrid battery-supercapacitor energy storage complex is used for autonomous power supply of consumers in remote areas. This work uses passivity-based control (PBC) for this complex in accordance with the accepted energy management strategy (EMS). Structural and parametric synthesis of the overall PBC system was carried out, which was accompanied by a significant amount of research. In order to simplify this synthesis, a structural decomposition of the overall dynamic system of the object presented in the form of a port-Hamiltonian system, which was described by a system of differential equations of the seventh order, into three subsystems was applied. These subsystems are a wind turbine, a PV plant, and a hybrid battery-supercapacitor system. For each of the subsystems, it is quite simple to synthesize the control influence formers according to the interconnections and damping assignment (IDA) method of PBC, which locally performs the tasks set by the EMS. The results obtained by computer simulation of the overall and decomposed systems demonstrate the effectiveness of this approach in simplifying synthesis and debugging procedures of complex multi-physical systems.

Exergetic port-Hamiltonian systems for multibody dynamics

AbstractMultibody dynamics simulation plays an important role in various fields, including mechanical engineering, robotics, and biomechanics. Setting up computational models however becomes increasingly challenging as systems grow in size and complexity. Especially the consistent combination of models across different physical domains still requires significant effort. This motivates the study of formal languages that enable a compositional approach to modeling multiphysical systems with basic guarantees. The paper shows how multibody systems, or more precisely assemblies of rigid bodies connected by lower kinematic pairs, can be described as Exergetic Port-Hamiltonian Systems (EPHS). The EPHS modeling language features a straightforward graphical syntax for expressing the energy-based interconnection of hierarchically nested subsystems. This reduces cognitive load and facilitates clearer communication among experts, nonexperts, and computational tools. Hierarchical nesting of systems enables abstraction of lower-level details and promotes the reuse of models at different levels of complexity. At the lowest level, there are three basic kinds of systems, representing energy storage and reversible/irreversible energy exchange. The structured approach guarantees fundamental properties of macroscopic systems, such as conservation of energy and nonnegative entropy production. In combination with the compositional syntax, this makes building and modifying models simpler and less error-prone.

Adaptive exponential tracking control of port-Hamiltonian system via contraction and timed IDA-PBC method

Adaptive exponential tracking control of port-Hamiltonian system via contraction and timed IDA-PBC method

Operator splitting for coupled linear port-Hamiltonian systems

Operator splitting methods tailored to coupled linear port-Hamiltonian systems are developed. We present algorithms that are able to exploit scalar coupling, as well as multirate potential of these coupled systems. The obtained algorithms preserve the dissipative structure of the overall system and are convergent of second order. Numerical results for coupled mass–spring–damper chains illustrate the computational efficiency of the splitting methods compared to a straight-forward application of the implicit midpoint rule to the overall system.

Improved a posteriori error bounds for reduced port-Hamiltonian systems

Projection-based model order reduction of dynamical systems usually introduces an error between the high-fidelity model and its counterpart of lower dimension. This unknown error can be bounded by residual-based methods, which are typically known to be highly pessimistic in the sense of largely overestimating the true error. This work applies two improved error bounding techniques, namely (a) a hierarchical error bound and (b) an error bound based on an auxiliary linear problem, to the case of port-Hamiltonian systems. The approaches rely on a secondary approximation of (a) the dynamical system and (b) the error system. In this paper, these methods are adapted to port-Hamiltonian systems. The mathematical relationship between the two methods is discussed both theoretically and numerically. The effectiveness of the described methods is demonstrated using a challenging three-dimensional port-Hamiltonian model of a classical guitar with fluid–structure interaction.

Interconnection of irreversible port Hamiltonian systems

This paper shows how the interconnection of two controlled Irreversible port Hamiltonian Systems has to be state and co-state modulated in order to ensure the closed-loop Irreversible port Hamiltonian structure, satisfying the first and second laws of Thermodynamics. It proposes a precise parametrization of this modulation from the open-loop systems structures in order to guarantee the consistency of the closed loop energy and entropy balance equations. The results are illustrated by means of the examples of a heat-exchanger, a gas-piston system and a chemical reaction.

Stability analysis of a stochastic port-Hamiltonian car-following model

Port-Hamiltonian systems are pertinent representations of many nonlinear physical systems. In this study, we formulate and analyse a general class of stochastic car-following models with a systematic port-Hamiltonian structure. The model class is a generalisation of classical car-following approaches, including the optimal velocity model of Bando et al (1995 Phys. Rev. E 51 1035), the full velocity difference model of Jiang et al (2001 Phys. Rev. E 64 017101), and recent stochastic following models based on the Ornstein–Uhlenbeck process. In contrast to traditional models where the interaction is totally asymmetric (i.e. depending only on the speed and distance to the predecessor), the port-Hamiltonian car-following model also depends on the distance to the follower. We determine the exact stability condition of the finite system with N vehicles and periodic boundaries. The stable system is ergodic with a unique Gaussian invariant measure. Other properties of the model are studied using numerical simulation. It turns out that the Hamiltonian component improves the flow stability and reduces the total energy in the system. Furthermore, it prevents the problematic formation of stop-and-go waves with oscillatory dynamics, even in the presence of stochastic perturbations.

Stability via closure relations with applications to dissipative and port-Hamiltonian systems

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Trajectory tracking PID passivity-based control of spacecraft formation flying around Sun-Earth L2 point in the port-Hamiltonian framework

Spacecraft formation flying for interferometric observations around the 2nd Lagrange point in the Sun-Earth system (SEL2) is currently focal point in deep space exploration research, which demands high precision in relative position control of spacecraft. This paper proposes the relative motion dynamics of satellite formations around the L2 point in the port-Hamiltonian framework, and establishes a high-precision nonlinear dynamics model. PID passivity-based control (PID-PBC) is widely used in engineering. However, existing methods of PID-PBC cannot address the trajectory tracking issues in port-Hamiltonian systems. Utilizing the contraction properties of the port-Hamiltonian system, this paper proposes the trajectory tracking PID-PBC (tPID-PBC) approach, effectively resolving trajectory tracking issues for formation dynamics around L2 point in port-Hamiltonian framework. The paper details explicit solutions of the Partial Differential Equations (PDE) for the tPID-PBC method and its controller structure, and references the trajectories of interferometric observation formations, to verify the method’s effectiveness through numerical simulation. The presented control approach is applicable across generic port-Hamiltonian systems, offering substantial theoretical value.

Port-Hamiltonian descriptor systems are relative generically controllable and stabilizable

AbstractThe present work is a successor of Ilchmann and Kirchhoff (Math Control Signals Syst 33:359–377, 2021) on generic controllability and of Ilchmann and Kirchhoff (Math Control Signals Syst 35:45–76, 2022) on relative generic controllability of linear differential-algebraic equations. We extend the result from general, unstructured differential-algebraic equations to differential-algebraic equations of port-Hamiltonian type. We derive results on relative genericity. These findings are the basis for characterizing relative generic controllability of port-Hamiltonian systems in terms of dimensions. A similar result is proved for relative generic stabilizability.

WITHDRAWN: Robust voltage regulation of DC-DC buck converter with ZIP load via an energy shaping control approach

ZIP loads (the parallel combination of constant impedance loads, constant current loads and constant power loads) exist widely in power system. In order to stabilize buck converter based DC distributed system with ZIP load, an adaptive energy shaping controller (AESC) is devised in this paper. Firstly, based on the assumption that the lumped disturbances are known, a full information controller is designed in the framework of the port Hamiltonian system via energy shaping technique. An obvious merit is that it is PI-type controller, which is easy to implement in practical application. Besides, using mathematical deductive method, an estimation of the domain of attraction is given to ensure the strict stability. Furthermore, to eliminate the influence of parameter perturbations on the system, a disturbance observer is proposed to reconstruct the lumped disturbances and then the estimated terms are introduced to above controller to form an adaptive control scheme. In addition, the strict stability analysis of the closed-loop system is given. Lastly, the simulation and experiment results are presented for assessing the designed controller.

Data-driven adjoint-based calibration of port-Hamiltonian systems in time domain

We present a gradient-based calibration algorithm to identify the system matrices of a linear port-Hamiltonian system from given input–output time data. Aiming for a direct structure-preserving approach, we employ techniques from optimal control with ordinary differential equations and define a constrained optimization problem. The input-to-state stability is discussed which is the key step towards the existence of optimal controls. Further, we derive the first-order optimality system taking into account the port-Hamiltonian structure. Indeed, the proposed method preserves the skew symmetry and positive (semi)-definiteness of the system matrices throughout the optimization iterations. Numerical results with perturbed and unperturbed synthetic data, as well as an example from the PHS benchmark collection [17] demonstrate the feasibility of the approach.

Neural network quaternion-based controller for port-Hamiltonian system

Abstract In this research article, a control approach for port-Hamiltonian PH systems based in a neural network (NN) quaternion-based control strategy is presented. First, the dynamics is converted by the implementation of a Poisson bracket in order to facilitate the mathematical model in order to obtain a feasible formulation for the controller design based on quaternion NNs. In this study, two controllers for this kind of of system are presented: the first one consists in the controller design for a PH system about its equilibrium points taking into consideration the position and momentum. This mean is achieved by dividing the quaternion neural controller into scalar and vectorial parts to facilitate the controller derivation by selecting a Lyapunov functional. The second control strategy consists in designing the trajectory tracking controller, in which a reference moment is considered in order to drive this variable to the final desired position according to a reference variable; again, a Lyapunov functional is implemented to obtain the desired control law. It is important to mention that both controllers take into advantage that the energy consideration and that the representation of many physical systems could be implemented in quaternions. Besides the angular velocity, trajectory tracking of a three-phase induction motor is presented as a third numerical experiment. Two numerical experiments are presented to validate the theoretical results evinced in this study. Finally, a discussion and conclusion section is provided.

Generic observability for port-Hamiltonian descriptor systems

The present work is a successor of Ilchmann and Kirchhoff (Math Control Signals Syst 33:359–377, 2021. https://doi.org/10.1007/s00498-021-00287-x), Ilchmann and Kirchhoff (Math Control Signals Syst 35:45–76, 2023. https://doi.org/10.1007/s00498-021-00287-x) on (relative) generic controllability of unstructured linear differential-algebraic systems and of Ilchmann et al. (Port-Hamiltonian descriptor systems are generically controllable and stabilizable. Submitted to Mathematics of Control, Signals and Systems, 2023. https://arxiv.org/abs/2302.05156) on (relative) generic controllability of port-Hamiltonian descriptor systems. We extend their results to (relative) genericity of observability. For unstructured differential-algebraic systems, criteria for (relative) generic observability are derived from Ilchmann and Kirchhoff (Math Control Signals Syst 35:45–76, 2023. https://doi.org/10.1007/s00498-021-00287-x) using duality. This is not possible for port-Hamiltonian systems. Hence, we tweak the results of Ilchmann et al. (Port-Hamiltonian descriptor systems are generically controllable and stabilizable. Submitted to Mathematics of Control, Signals and Systems, 2023. https://arxiv.org/abs/2302.05156) and derive similar criteria as for the unstructured case. Additionally, we consider certain rank constraints on the system matrices.

Port-Hamiltonian discontinuous Galerkin finite element methods

Abstract A port-Hamiltonian (pH) system formulation is a geometrical notion used to formulate conservation laws for various physical systems. The distributed parameter port-Hamiltonian formulation models infinite dimensional Hamiltonian dynamical systems that have a nonzero energy flow through the boundaries. In this paper, we propose a novel framework for discontinuous Galerkin (DG) discretizations of pH-systems. Linking DG methods with pH-systems gives rise to compatible structure preserving semidiscrete finite element discretizations along with flexibility in terms of geometry and function spaces of the variables involved. Moreover, the port-Hamiltonian formulation makes boundary ports explicit, which makes the choice of structure and power preserving numerical fluxes easier. We state the Discontinuous Finite Element Stokes–Dirac structure with a power preserving coupling between elements, which provides the mathematical framework for a large class of pH discontinuous Galerkin discretizations. We also provide an a priori error analysis for the port-Hamiltonian discontinuous Galerkin Finite Element Method (pH-DGFEM). The port-Hamiltonian discontinuous Galerkin finite element method is demonstrated for the scalar wave equation showing optimal rates of convergence.

Port-Hamiltonian system: structure recognition and applications

In this paper, we continue to consider the problem of recovering the port-Hamiltonian structure for an arbitrary system of differential equations. We complement our previous study on this topic by explaining the choice of machine learning algorithms and discussing some details of their application. We also consider the possibility provided by this approach for a potentially new definition of canonical forms and classification of systems of differential equations.

Stochastic stabilization based on kinetic-potential energy shaping for stochastic mechanical port-Hamiltonian systems

This paper extends a kinetic-potential energy shaping method to stochastic mechanical port-Hamiltonian systems. The kinetic-potential energy shaping brings a new class of Lyapunov function candidates involving a cross term of the position and the momentum without solving partial differential equations for deterministic port-Hamiltonian systems. However, the conventional kinetic-potential energy shaping does not necessarily work for stochastic port-Hamiltonian systems due to energy increase by stochastic noise. Therefore, we first provide a modification properly compensated by using stochastic generalized canonical transformations. Then, two stochastic stability results are presented. The first result shows a necessary condition for stochastic asymptotic stability for an equilibrium state. The other one shows a necessary condition for stochastic bounded stability for a target state, which is not necessarily an equilibrium point.

Manifold turnpikes of nonlinear port-Hamiltonian descriptor systems under minimal energy supply

Turnpike phenomena of nonlinear port-Hamiltonian descriptor systems under minimal energy supply are studied. Under assumptions on the smoothness of the system nonlinearities, it is shown that the optimal control problem is dissipative with respect to a manifold. Then, under controllability assumptions, it is shown that the optimal control problem exhibits a manifold turnpike property.

The collective dynamics of a stochastic Port-Hamiltonian self-driven agent model in one dimension

This paper studies the collective motion of self-driven agents in a one-dimensional space with periodic boundaries, using a stochastic Port-Hamiltonian system (PHS) with symmetric nearest-neighbor interactions and additive Brownian noise as an external input. In the case of a quadratic potential the PHS is an Ornstein-Uhlenbeck process for which we explicitly determine the distribution for any time t ≥ 0 and in the limit t → ∞. In particular, we characterize the collective motion by showing that the agents’ positions tend to build exactly one cluster. This is confirmed in simulations that show rapid and coordinated motion among agents, driven by noise, despite the absence of a preferred direction of motion in the model. Remarkably, the theoretical properties observed in the Ornstein-Uhlenbeck process also emerge in simulations of the nonlinear model incorporating a general interaction potential.

Exponential Decay Rate of Linear Port-Hamiltonian Systems: A Multiplier Approach

Exponential Decay Rate of Linear Port-Hamiltonian Systems: A Multiplier Approach