port-Hamiltonian Formulation Research Articles

Port-hamiltonian formulation of an electrical circuit using different kinds of states

This work focuses on a dynamic electrical circuit whose dynamics are affine in the control input. Such dynamics are considered to be re-expressed in a canonical form, namely the port-Hamiltonian (pH) representation with dissipation, where the Hamiltonian is a quadratic function and has the unit of energy or power. On this basis, it allows revealing the transformation of energy (or power) inside the system, including the energy supply, storage and dissipation, thereby facilitating Lyapunov-based or energy-related control approaches for stabilization and optimization purposes. Two pH representations are proposed and compared; the first one is established with difficult-to-measure states while the second one is obtained with easier-to-measure states.

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 formulations of the incompressible Euler equations with a free surface

In this paper, we present port-Hamiltonian formulations of the incompressible Euler equations with a free surface governed by surface tension and gravity forces, modelling e.g. capillary and gravity waves and the evolution of droplets in air. Three sets of variables are considered, namely (v,Σ), (η,ϕ∂,Σ) and (ω,ϕ∂,Σ), with v the velocity, η the solenoidal velocity, ϕ∂ a potential, ω the vorticity, and Σ the free surface, resulting in the incompressible Euler equations in primitive variables and the vorticity equation. First, the Hamiltonian formulation for the incompressible Euler equations in a domain with a free surface combined with a fixed boundary surface with a homogeneous boundary condition will be derived in the proper Sobolev spaces of differential forms. Next, these results will be extended to port-Hamiltonian formulations allowing inhomogeneous boundary conditions and a non-zero energy flow through the boundaries. Our main results are the construction and proof of Dirac structures in suitable Sobolev spaces of differential forms for each variable set, which provides the core of any port-Hamiltonian formulation. Finally, it is proven that the state dependent Dirac structures are related to Poisson brackets that are linear, skew-symmetric and satisfy the Jacobi identity.

Discrete-Time Port-Hamiltonian Systems for Power and Energy Applications

The Port-Hamiltonian formalism has demonstrated to be useful in different applications, allowing to explore structural properties to obtain stable controls. A common approach is to design controls in a continuous domain although, in practice, they are implemented in discrete time. However, the study of discrete-time port-Hamiltonian systems is challenging since the system may lose structural properties such as passivity after discretization. This paper shows that passivity is conserved for the backward Euler discretization method when the Hamiltonian is convex, and its gradient is Lipschitz. This result is relevant for power systems applications such as the control of two-area systems, high-voltage direct-current transmission, and microgrids. Although our method is simple and preserves passivity, it does not preserve the symplectic structure of the hamiltonian.

Contraction Theory and Differential Passivity in the port-Hamiltonian formalism

In this work, we recall the concept of contractive dynamics and its natural extension to the notion of Differential dissipativity/passivity via the use of the so-called prolonged (or extended) system dynamics, obtained by lifting the system to the tangent bundle of the underlying manifold. The new concept of dissipative Differential Hamiltonian dynamics is proposed providing a weaker notion of contractive dynamical system. Furthermore, the Differential Hamiltonian notion is extended to the definition of Differentially passive port-Hamiltonian system. We describe explicit conditions to exploit the ‘natural’ Differential Hamiltonian function as differential storage function for the port-Hamiltonian system dynamics.

STOCHASTIC GALERKIN METHOD AND PORT-HAMILTONIAN FORM FOR LINEAR FIRST-ORDER ORDINARY DIFFERENTIAL EQUATIONS

We consider linear first-order systems of ordinary differential equations (ODEs) in port-Hamiltonian (pH) form. Physical parameters are remodeled as random variables to conduct an uncertainty quantification. A stochastic Galerkin projection yields a larger deterministic system of ODEs, which does not exhibit a pH form in general. We apply transformations of the original systems such that the stochastic Galerkin projection becomes structure-preserving. Furthermore, we investigate meaning and properties of the Hamiltonian function belonging to the stochastic Galerkin system. A large number of random variables implies a high-dimensional stochastic Galerkin system, which suggests itself to apply model order reduction (MOR) generating a low-dimensional system of ODEs. We discuss structure preservation in projection-based MOR, where the smaller systems of ODEs feature pH form again. Results of numerical computations are presented using two test examples.

Generalized Maxwell viscoelasticity for geometrically exact strings: Nonlinear port-Hamiltonian formulation and structure-preserving discretization

This contribution proposes a nonlinear and dissipative infinite-dimensional port-Hamiltonian (PH) model for the dynamics of geometrically exact strings. The mechanical model provides a description of large deformations including finite elastic and inelastic strains in a generalized Maxwell model. It is shown that the overall system results from a power-preserving interconnection of PH subsystems. By using a structure-preserving mixed finite element approach, a finite-dimensional PH model is derived. Eventually, midpoint discrete derivatives are employed to deduce an energy-consistent time-stepping method, which inherits discrete-time dissipativity for the irreversible system. An example simulation illustrates the numerical properties of the present approach.

Port-Hamiltonian Macroscopic Modelling based on the Homogenisation Method: case of an acoustic pipe with a porous wall

This paper addresses linear propagation in an acoustic pipe with a porous wall, a common scenario in wooden wind instruments. First, a scale separation technique is proposed for dissipative propagation within the wall: the material is modelled as a periodic assembly of identical microscopic cells, forming a network of channels filled with air. It is shown that the resulting PDE admits a port-Hamiltonian formulation, of which the state, flow, effort, Hamiltonian, and Differential connection operator are structured using powers of the scale parameter. The resulting macroscopic description, derived from the governing equations at the two lowest orders, manifests as a constrained port-Hamiltonian system involving a Lagrange multiplier. As an example, using an academic cell geometry, we determine the effective wavenumber and dissipation coefficient of a straight tube with a porous wall.

Stochastic Galerkin method and port-Hamiltonian form for linear dynamical systems of second order

We investigate linear dynamical systems of second order. Uncertainty quantification is applied, where physical parameters are substituted by random variables. A stochastic Galerkin method yields a linear dynamical system of second order with high dimensionality. A structure-preserving model order reduction (MOR) produces a small linear dynamical system of second order again. We arrange an associated port-Hamiltonian (pH) formulation of first order for the second-order systems. Each pH system implies a Hamiltonian function describing an internal energy. We examine the properties of the Hamiltonian function for the stochastic Galerkin systems. We show numerical results using a test example, where both the stochastic Galerkin method and structure-preserving MOR are applied.

Design of a Port-Hamiltonian Control for an Alt-Azimuth Liquid–Mirror Telescope

In this work, we design a control strategy to be applied in a port-Hamilton representation of a liquid-mirror telescope for an alt-azimuth configuration. Starting from a dynamical model for an alt-azimuth liquid-mirror telescope based on Lagrange mechanics, a transformation to the port-Hamilton form is made. Such a dynamical model is obtained by computing the kinetic and potential energy of the telescope and substituting them in the Euler–Lagrange equation of motion. Then, for the transformation to the port-Hamiltonian form, we obtain the relation between the Hamiltonian and the Lagrangian. The resulting open-loop model based on the Hamiltonian function is controlled using an extension of the interconnection and damping-assignment passivity-based control aiming for a robust and accurate steady behavior in the closed loop while tracking a star’s position. For comparison purposes, two different control strategies are applied to the Lagrangian model, inverse-dynamics control and sliding mode super-twisting control. Since the light is collected by the principal mirror of the telescope while tracking a star, we make a description of the liquid mirror’s behavior. The tracking star’s position is described as a function of the observer’s position and the star’s coordinates as well as the date of observation. The simulations’ results show that the port-Hamilton control has a good transitory and steady response as well as great accuracy competing with that of inverse-dynamics control but with greater robustness and no chattering drawback.

Port-metriplectic neural networks: thermodynamics-informed machine learning of complex physical systems

We develop inductive biases for the machine learning of complex physical systems based on the port-Hamiltonian formalism. To satisfy by construction the principles of thermodynamics in the learned physics (conservation of energy, non-negative entropy production), we modify accordingly the port-Hamiltonian formalism so as to achieve a port-metriplectic one. We show that the constructed networks are able to learn the physics of complex systems by parts, thus alleviating the burden associated to the experimental characterization and posterior learning process of this kind of systems. Predictions can be done, however, at the scale of the complete system. Examples are shown on the performance of the proposed technique.

Port-Hamiltonian Formulations of Some Elastodynamics Theories of Isotropic and Linearly Elastic Shells: Naghdi–Reissner’s Moderately Thick Shells

The port-Hamiltonian system approach is intended to be an innovative and unifying way of modeling multiphysics systems, by expressing all of them as systems of conservation laws. Indeed, the increasing developments in recent years allow finding better control and coupling strategies. This work aimed to apply such an approach to Naghdi–Reissner’s five-kinematic-field shell model in linear elasticity, while including often-neglected higher-order intrinsic geometric coupling effects, therefore preparing the theoretical background required for the coupling (or interconnection) with an acoustic fluid model and the different types of interactions that can arise among them. The model derived thusly can be used for controller design in a wide variety of applications such as inflatable space structures, launcher tank vibration damping, payload vibration protection using smart materials, and many other related applications.

Pseudo-Hamiltonian neural networks with state-dependent external forces

Hybrid machine learning based on Hamiltonian formulations has recently been successfully demonstrated for simple mechanical systems, both energy conserving and not energy conserving. We introduce a pseudo-Hamiltonian formulation that is a generalization of the Hamiltonian formulation via the port-Hamiltonian formulation, and show that pseudo-Hamiltonian neural network models can be used to learn external forces acting on a system. We argue that this property is particularly useful when the external forces are state dependent, in which case it is the pseudo-Hamiltonian structure that facilitates the separation of internal and external forces. Numerical results are provided for a forced and damped mass–spring system and a tank system of higher complexity, and a symmetric fourth-order integration scheme is introduced for improved training on sparse and noisy data.

Energy-Based Modeling and Control of a Piezotube Actuated Optical Fiber

This article presents an energy-based modeling and control design method for a piezotube actuated optical fiber. A nonlinear infinite dimensional port Hamiltonian (pH) formulation of the 3-D flexible optical fiber is derived from the Cosserat rod dynamical equations. Then, the proposed infinite dimensional model is discretized using a pH structure and passivity preserving discretization method for the simulation and control design. This model is then validated against experimental data obtained using a built-in experimental setup equipped with a MEMS Analyzer. A complete pH formulation of the piezotube actuated optical fiber is proposed, combining the Cosserat rod model and actuator dynamics. This model is used for the end-point path control design using an interconnection and damping assignment passivity based control (IDA-PBC) method. Both the proposed pH model of the overall system and the controller are validated in simulation and against experimental results.

An economic cybernetic model for electricity market operation coupled with physical system dynamics

With the emergency of competitive markets as central operational mechanisms, the real-time market clearing results are used to determine the optimal power dispatch. It inevitably strengthens the coupling between the market update process and the physical response of the generators and networks. It necessitates the development of stability analysis for such coupled systems. Based on the primal–dual gradient method, an economic cybernetic model is established to simultaneously characterize the market operation and electromechanical power system dynamics. Inspired by modern control theory, the proposed dynamic model constructs a state space in which the actions of market participants are control signals, and the system states, such as nodal prices, bus voltages, and frequency, are treated as state variables. To promote the expandability, we also formulate the coupled dynamics in port-Hamiltonian form and provide detailed proof and sufficient conditions for its asymptotic stability by using properties of incremental passive systems. The whole transaction process implemented in a distributed manner aims at maximizing social welfare and frequency regulation when the economic-physical system reaches its equilibrium point.

Design and implementation of passivity-based controller for active suspension system using port-Hamiltonian observer

The objective of this study is to design and implement an observer for quarter-car active suspension system in Port-Hamiltonian form. A novel state observer is designed for active suspension system modelled in port-Hamiltonian form to estimate the states in presence of road disturbances. The observer is designed considering suspension deflection alone as the output, which is an easily measurable output. Performance of the proposed observer is evaluated experimentally with road disturbance input mimicking a sudden bump and a continuously varying road input, and proven to be effective in minimising the error dynamics in presence of bounded unmodelled disturbances. To prove the effectiveness of the state-estimator, an Interconnection and Damping Assignment Passivity Based Control (IDA-PBC) designed using the desired physical properties of the closed-loop system is implemented using the observer states. Experimental results of the controller implemented using the designed state observer show good improvement in the ride comfort, ride stability and suspension stroke of the active suspension system, which proves the effectiveness of the proposed port-Hamiltonian observer in terms of minimising the error dynamics.

Passivity, port-hamiltonian formulation and solution estimates for a coupled magneto-quasistatic system

In this paper, we study a quasilinear coupled magneto-quasistatic model from a systems theoretic perspective. First, by taking the injected voltages as input and the associated currents as output, we prove that the magneto-quasistatic system is passive. Moreover, by defining suitable Dirac and resistive structures, we show that it admits a representation as a port-Hamiltonian system. Thereafter, we consider dependence of the solution on initial and input data. We show that the current and the magnetic vector potential can be estimated by means of the initial magnetic vector potential and the voltage. We also analyse the free dynamics of the system and study the asymptotic behavior of the solutions for $ t\to\infty $.

Learning port-Hamiltonian Systems—Algorithms

In this article we study the possibilities of recovering the structure of port-Hamiltonian systems starting from “unlabelled” ordinary differential equations describing mechanical systems. The algorithm we suggest solves the problem in two phases. It starts by constructing the connectivity structure of the system using machine learning methods – producing thus a graph of interconnected subsystems. Then this graph is enhanced by recovering the Hamiltonian structure of each subsystem as well as the corresponding ports. This second phase relies heavily on results from symplectic and Poisson geometry that we briefly sketch. And the precise solutions can be constructed using methods of computer algebra and symbolic computations. The algorithm permits to extend the port-Hamiltonian formalism to generic ordinary differential equations, hence introducing eventually a new concept of normal forms of ODEs.

Learning port-Hamiltonian Systems—Algorithms

In this article we study the possibilities of recovering the structure of port-Hamiltonian systems starting from ``unlabelled'' ordinary differential equations describing mechanical systems. The algorithm we suggest solves the problem in two phases. It starts by constructing the connectivity structure of the system using machine learning methods -- producing thus a graph of interconnected subsystems. Then this graph is enhanced by recovering the Hamiltonian structure of each subsystem as well as the corresponding ports. This second phase relies heavily on results from symplectic and Poisson geometry that we briefly sketch. And the precise solutions can be constructed using methods of computer algebra and symbolic computations. The algorithm permits to extend the port-Hamiltonian formalism to generic ordinary differential equations, hence introducing eventually a new concept of normal forms of ODEs.

A Two-Dimensional port-Hamiltonian Model for Coupled Heat Transfer

In this paper, we construct a highly simplified mathematical model for studying the problem of conjugate heat transfer in gas turbine blades and their cooling ducts. Our simple model focuses on the relevant coupling structures and aims to reduce the unrelated complexity as much as possible. Then, we apply the port-Hamiltonian formalism to this model and its subsystems and investigate the interconnections. Finally, we apply a simple spatial discretization to the system to investigate the properties of the resulting finite-dimensional port-Hamiltonian system and to determine whether the order of coupling and discretization affect the resulting semi-discrete system.