Port Hamiltonian Research Articles

Asymptotic stability of port-hamiltonian systems with constant inputs

In this paper, the asymptotic stability of Port-Hamiltonian (PH) systems with constant inputs is studied. Constant inputs are useful for stabilizing systems at their nonzero equilibria and can be realized by step signals. To achieve this goal, two methods based on integral action and comparison principle are presented in this paper. These methods change the convex Hamiltonian function and the restricted damping matrix of the previous results into a Hamiltonian function with a local minimum and a positive semidefinite matrix, respectively. Due to common conditions of Hamiltonian function and damping matrix, the proposed method asymptotically stabilizes more classes of PH systems with constant inputs than the existing methods. Finally, the validity and advantages of the presented methods are shown in an example.

A Lyapunov Approach to Robust Regulation of Distributed Port–Hamiltonian Systems

This article studies robust output tracking and disturbance rejection for boundary-controlled infinite-dimensional Port–Hamiltonian systems including second-order models such as the Euler–Bernoulli beam equation. The control design is achieved using the internal model principle and the stability analysis using a Lyapunov approach. Contrary to existing works on the same topic, no assumption is made on the external well-posedness of the considered class of PDEs. The results are applied to robust tracking of a piezo actuated tube used in atomic force imaging.

Reduced Order LQG Control Design for Infinite Dimensional Port Hamiltonian Systems

This article proposes a method that combines linear quadratic Gaussian (LQG) control design and structure preserving model reduction for the reduced order control of infinite dimensional port Hamiltonian systems (IDPHS).For that purpose the weighting operators used in LQG control design are chosen such that the resulting dynamic controller is passive and the closed-loop system equivalent to control by interconnection. The method of Petrov-Galerkin is then used to approximate the balanced realization of the IDPHS by a finite dimensional port Hamiltonian system and to provide the associated reduced order LQG controller. The main advantages of the proposed method are that, first, both control and reduction are driven by closed-loop performances and that, second, due to the passivity properties of the controller the closed-loop stability is guaranteed when the finite dimensional controller is applied to the infinite dimensional system.

Infinite dimensional model of a double flexible-link manipulator: The Port-Hamiltonian approach

This paper proposes a modular and control oriented model of a double flexible-link manipulator that stems from the modelling of a spatial flexible robot. The model consists of the power preserving interconnection between two infinite dimensional systems describing the beam’s motion and deformation with a finite dimensional nonlinear system describing the dynamics of the actuated rotating joints. To derive the model, Timoshenko’s assumptions are made for the flexible beams. Using Hamilton’s principle, the dynamic equations of the system are derived and then written in the Port-Hamiltonian (PH) framework through a proper choice of the state variables. These so called energy variables allow to write the total energy as a quadratic form with respect to a state dependent energy matrix. The resulting model is shown to be a passive system, a convenient property for control design purposes.

Stabilization of Port Hamiltonian Chaotic Systems with Hidden Attractors by Adaptive Terminal Sliding Mode Control.

In this study, the design of an adaptive terminal sliding mode controller for the stabilization of port Hamiltonian chaotic systems with hidden attractors is proposed. This study begins with the design methodology of a chaotic oscillator with a hidden attractor implementing the topological framework for its respective design. With this technique it is possible to design a 2-D chaotic oscillator, which is then converted into port-Hamiltonia to track and analyze these models for the stabilization of the hidden chaotic attractors created by this analysis. Adaptive terminal sliding mode controllers (ATSMC) are built when a Hamiltonian system has a chaotic behavior and a hidden attractor is detected. A Lyapunov approach is used to formulate the adaptive device controller by creating a control law and the adaptive law, which are used online to make the system states stable while at the same time suppressing its chaotic behavior. The empirical tests obtaining the discussion and conclusions of this thesis should verify the theoretical findings.

Port Hamiltonian systems with moving interface: a phase field approach

In this paper, we give a formulation of distributed parameter systems with a moving diffuse interface using the Port Hamiltonian formalism. For this purpose, we suggest to use the phase field modeling approach. In the first part we recall the phase field models, in particular the Cahn–Hilliard and Allen–Cahn equations, and show that they may be expressed in terms of a dissipative Hamiltonian system. In the second part we show how this Hamiltonian model may be extended to a Boundary Port Hamiltonian System and illustrate the construction on the example of crystallization.

On the flat representation for a particular class of port-Hamiltonian systems

Abstract This paper pertains to the flat representation of a class of port-Hamiltonian (PH) systems and advocates the use of bicausality of Bond graphs for finding appropriate flat outputs. Systems which are differentially flat have several useful properties which can be exploited to generate, for example, optimal trajectories/profiles which ensure constraints satisfaction. For the special case of PH systems combining the power preserving property with the flatness properties leads to effective control strategies for multi-physical systems. Hence, the purpose of this paper is to explore the implications and features of a particular class of PH systems (which can be retrieved from a Bond Graph representation) in finding their flat output representation. We concentrate on the example of an electrical storage system of a DC microgrid to illustrate the proposed theory.

Linear Boundary Port Hamiltonian Systems defined on Lagrangian submanifolds

Recently Port Hamiltonian systems have been extended to encompass an implicit definition of the energy function of the system, by defining it in terms of a Lagrangian submanifold. In this paper, we extend the definition of Port Hamiltonian systems defined with respect to Lagrangian submanifold to a class of infinite-dimensional systems where the Lagrangian submanifold is defined by first-order differential operators. We show that this adds some port boundary variables and derive the energy balance equation. This construction is illustrated on the model of a flexible nanorod made of composite material.

Economic constrained optimization for power balancing in a DC microgrid: A multi-source elevator system application

This paper considers a discrete-time scheduling method for the power balancing of a continuous-time DC microgrid system. A high-order dynamics and a resistor network are used for modelling the electrical storage unit and the DC bus of the centralized microgrid system, respectively. A PH (Port-Hamiltonian) formulation on graphs is employed to explicitly describe the microgrid topology. This modelling approach allows us to derive a discrete-time model which preserves the power and energy balance of the physical system. Next, a constrained economic MPC (Model Predictive Control) using the proposed control model is formulated for efficiently managing the microgrid operation. The systematic combination of the network modelling method and optimization-based control allows us to generate the appropriate power profiles. Finally, the benefits of the proposed approach are validated through simulation and comparison results over a particular DC microgrid elevator system under different scenarios and using real numerical data.

Meshed DC microgrid hierarchical control: A differential flatness approach

In this paper, a meshed DC microgrid control architecture whose goal is to manage load balancing and efficient power distribution is introduced. A novel combination of port-Hamiltonian (PH) modeling with differential flatness and B-splines parametrization is introduced and shown to improve the microgrid's performance. A three layer supervision structure is considered: (i) B-spline parametrized flat output provide continuous profiles for load balancing and price reduction (high level); (ii) the profiles are tracked through a MPC implementation with stability guarantees (medium level); (iii) explicit switching laws applied to the DC/DC converters ensure appropriate power injection. Each level functions at a different time-scale (from slow to fast), and the control laws are chosen appropriately. The effectiveness of the proposed approach is evaluated by simulations over a DC microgrid composed by a collection of solar panels (PV), an energy storage system (ES), a utility grid (UG) and a consumers’ demand.

Model Order Reduction of Port-Hamiltonian Systems by Riemannian Modified Fletcher–Reeves Scheme

Based on the Riemannian modified Fletcher–Reeves conjugate gradient scheme, we propose a new model order reduction algorithm to reduce port-Hamiltonian (PH) systems by the two-sided projection. Making full use of the geometric notions and geometrical characteristics of the Stiefel manifold, the proposed algorithm is not only computationally efficient, but also globally convergent. Additionally, it can preserve the PH structure and the passivity of the original system. A numerical example is simulated to demonstrate the efficiency of our algorithm.

Discrete-time port-Hamiltonian systems: A definition based on symplectic integration

We introduce a new definition of discrete-time port-Hamiltonian (PH) systems, which results from structure-preserving discretization of explicit PH systems in time. We discretize the underlying continuous-time Dirac structure with the collocation method and add discrete-time dynamics by the use of symplectic numerical integration schemes. The conservation of a structural discrete-time energy balance – expressed in terms of the discrete-time Dirac structure – extends the notion of symplecticity of geometric integration schemes to open systems. We discuss the energy approximation errors in the context of the presented definition and show that their order for linear PH systems is consistent with the order of the numerical integration scheme. Implicit Gauss–Legendre methods and Lobatto IIIA/IIIB pairs for partitioned systems are examples for integration schemes that are covered by our definition. The statements on the numerical energy errors are illustrated by elementary numerical experiments.

Tracking-error control via the relaxing port-Hamiltonian formulation: Application to level control and batch polymerization reactor

In this paper, a tracking-error-based control design without feedback passivation stage is developed for the stabilization of physical and chemical nonlinear processes. To achieve control objectives, the system dynamics is first formulated under appropriate conditions as a relaxing (pseudo) port-Hamiltonian (PH) representation with some quadratic storage function, in which the positive semi-definite property of damping matrix may not be taken necessarily into account. Then, a reference trajectory expressed by a certain structure passing through a desired set-point (or containing an optimal profile) is suitably chosen. By adding a relevant damping injection, asymptotic and global convergence of error dynamics is guaranteed. Two case studies including the level control of a four-tank process of continuous time type and the optimal tracking of a batch polymerization reactor of discontinuous time type due to the nature of the process operation are used to illustrate the application and the effectiveness of the approach. Besides the control performance comparison with the conventional passivity-based control method and the robustness evaluation against disturbance and/or noise are included.

Fixed-time stabilization control for port-Hamiltonian systems

In this paper, the locally fixed-time and globally fixed-time stabilization problems for the port-Hamiltonian (PH) systems via the interconnection and damping assignment passivity-based control technique are discussed. The definitions of fixed-time stability region (or region of attraction) and fixed-time stability boundary are given in this paper. From this starting point, the sufficient condition of globally fixed-time attractivity of a prespecified locally fixed-time stability region is obtained. Combining the locally fixed-time stability and the globally fixed-time attractivity of a prespecified locally fixed-time stability region, the globally fixed-time stabilization problem for PH system is effectively solved. Furthermore, the globally fixed-time control scheme independent of locally fixed-time stability region has also been derived by constructing a novel Lyapunov function. A illustrative example shows that the results obtained in this paper work very well in fixed-time control design of PH systems.

Structure Preserving Truncation of Nonlinear Port Hamiltonian Systems

In this paper, we present a novel balancing method for nonlinear port Hamiltonian systems based on the Hamiltonian and the controllability function. This corresponding balanced truncation method results in a reduced-order model that is still in port Hamiltonian form in contrast to the traditional balanced truncation method based on the controllability and observability functions.

Port Hamiltonian Modeling of a Cable Driven Robot

In this paper we present a generic Port-Hamiltonian (PH) model that includes cable dynamics (in particular elasticity and couplings with the platform and all cables among each other) of a cable-driven parallel robot (CDPR), which is used as a motion simulator. Moreover we consider changes in the cable parameters, i.e. it’s elasticity, mass and length when the cables are wound/unwound from the winches. To the best of our knowledge nobody considered such a detailed and generic model of a CDPR in PH structure before. Since motion simulators are built to mimic systems with different physical properties, PH modeling can pave the way for physics-shaping controllers.

PI simultaneous stabilization and set-point output regulation of Port-Hamiltonian systems

This paper is concerned with the problems of simultaneous stabilization and set-point output regulation of N Port-Hamiltonian (PH) systems. First, incremental models of PH systems are constructed and the passivity of incremental systems is established with some reasonable assumptions. Then a proportional plus integral (PI) controller is proposed to simultaneously stabilize incremental systems, and the stability of the closed-loop system is proved by Lyapunov function approach. The simple case with two PH systems is studied emphatically, and the obtained results are also naturally extended to the general case of N PH systems. Finally, the validity and applicability of the proposed PI controller is verified through an illustrative example.

Power balancing in a DC microgrid elevator system through constrained optimization

This paper considers the problem of power balancing in a DC microgrid. A PH (port Hamiltonian) formalism is used to describe the system components and interconnections. Energy and power conservation are kept for the discretized model. An economic model predictive controller is used for scheduling the microgrid power management. The proposed approach is validated through simulation results on a particular DC microgrid elevator system.

Modeling, discretization and motion control of a flexible beam in the port-Hamiltonian framework

In this paper, we present an approach to solve the feedforward motion control problem for a flexible beam, modeled with linear Timoshenko beam theory. The originality lies in the fact that all design steps, from modeling, over discretization to feedforward control are executed within the port-Hamiltonian (PH) framework. To obtain a finite-dimensional PH model which is suitable for inversion-based feedforward control design, a geometric pseudo-spectral discretization is performed. The feedforward control is tested with a plant model implemented in standard FEM software. The results of this paper will be amended by feedback control to achieve highly dynamic motion control on a lab test rig which is currently under construction.

Towards Ocean Grazer’s Modular Power Take-Off System Modeling:a Port-Hamiltonian Approach

This paper presents a modular modeling framework for the Ocean Grazer’s Power Take-Off (PTO) system, which operates as an array of point-absorber type devices connected to a hydraulic system. The modeling is based on the port-Hamiltonian (PH) framework that enables energy-based analysis and control of the PTO system. Firstly, a modular model of a point-absorber hydraulic system, which represents the main building block of the PTO, is presented. The model consists of wave-mechanical and hydraulic subsystems that are interconnected with a transformer-type interconnection. Secondly, we show passivity of the point-absorber hydraulic element and the accumulation of potential energy, which is due to the novel pumping mechanism of the point-absorber. Finally, we illustrate these properties through simulation results.