port-Hamiltonian Representation 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.

A systematic methodology for port-Hamiltonian modeling of multidimensional flexible linear mechanical systems

This article introduces a novel systematic methodology for modeling a class of multidimensional linear mechanical systems that directly allows to obtain their infinite-dimensional port-Hamiltonian representation. While the approach is tailored to systems governed by specific kinematic assumptions, it encompasses a wide range of models found in current literature, including ℓ-dimensional elasticity models (where ℓ = 1, 2, 3), vibrating strings, torsion in circular bars, classical beam and plate models, among others. The methodology involves formulating the displacement field using primary generalized coordinates via a linear algebraic relation. The non-zero components of the strain tensor are then calculated and expressed using secondary generalized coordinates, enabling the characterization of the skew-adjoint differential operator associated with the port-Hamiltonian representation. By applying Hamilton's principle and employing a specially developed integration by parts formula for the considered class of differential operators, the port-Hamiltonian model is directly obtained, along with the definition of boundary inputs and outputs. To illustrate the methodology, the plate modeling process based on Reddy's third-order shear deformation theory is presented as an example. To the best of our knowledge, this is the first time that a port-Hamiltonian representation of this system is presented in the literature.

Port-Hamiltonian fluid–structure interaction modelling and structure-preserving model order reduction of a classical guitar

ABSTRACT A fluid–structure interaction model in a port-Hamiltonian representation is derived for a classical guitar. After discretization, we combine the laws of continuum mechanics for solids and fluids within a unified port-Hamiltonian (pH) modelling approach by adapting the equations through an appropriate coordinate transformation on the second-order level. The high-dimensionality of the resulting system is reduced by model order reduction. The article focuses on pH-systems in different state transformations, a variety of basis generation techniques as well as structure-preserving model order reduction approaches that are independent from the projection basis. As main contribution, a thorough comparison of these method combinations is conducted. In contrast to typical frequency-based simulations in acoustics, transient time simulations of the system are presented. The approach is embedded into a straightforward workflow of sophisticated commercial software modelling and flexible in-house software for multi-physics coupling and model order reduction.

Sampled-Data Steering of Unicycles via PBC

In this letter, on the basis of a recently proposed discrete-time port-Hamiltonian representation of sampled-data dynamics, we propose a new time-varying digital feedback for steering mobile robots. The quality of the proposed passivity-based control is validated and compared through simulations with the existing literature and the continuous-time implementation using the unicycle as a case study.

Distributed Control and Optimization of DC Microgrids: A Port-Hamiltonian Approach

This article proposes a distributed secondary control scheme that drives a dc microgrid to an equilibrium point where the generators share optimal currents, and their voltages have a weighted average of nominal value. The scheme does not rely on the electric system topology nor its specifications; it guarantees plug-and-play design and functionality of the generators. First, the incremental model of the microgrid system with constant impedance, current, and power devices is shown to admit a port-Hamiltonian (pH) representation, and its passive output is determined. The economic dispatch problem is then solved by the Lagrange multipliers method; the Karush-Kuhn-Tucker conditions and weighted-average formation of voltages are then formulated as the control objectives. We propose a control scheme that is based on the Control by Interconnection design philosophy, where the consensus-based controller is viewed as a virtual pH system to be interconnected with the physical one. We prove the regional asymptotic stability of the closed-loop system using Lyapunov and LaSalle theorems. Equilibrium analysis is also conducted based on the concepts of graph theory and economic dispatch. Finally, the effectiveness of the presented scheme for different case studies is validated with a test microgrid system, simulated in both MATLAB/Simulink and OPAL-RT environments.

Port Hamiltonian formulation of the solidification process for a pure substance: A phase field approach*

In this paper we suggest a Port Hamiltonian model of the solidification process of water, using the phase field approach. Firstly, the Port Hamiltonian formulation of the dynamics of the phase field variable, governed by the Allen-Cahn equation, is recalled. It is based on adding to the phase field variable, its gradient, and extending the system with its dynamics. Secondly, the model is completed by the energy balance equation for the heat conduction and the complete Port Hamiltonian model is derived. Thirdly an Algebro-differential Port Hamiltonian representation is suggested, where the Port Hamiltonian system is defined on a Lagrangian submanifold, allowing to use directly the variables defining the thermodynamical data.

Port-Hamiltonian FE models for filaments

In this article, we present the port-Hamiltonian representation, the structure preserving discretization and the resulting finite-dimensional state space model of one-dimensional filaments based on a mixed finite element formulation. Due to the fact that the equations of motion of a filamentous body are based on the theory of geometrically nonlinear mechanical systems, the port-Hamiltonian formulation is expressed by means of its co-energy (effort) variables. The resulting port-Hamiltonian state space model features a quadratic Hamiltonian and the nonlinearity is reflected in the state dependence of its interconnection matrix. Numerical experiments generated with FEniCS illustrate the properties of the resulting finite element models.

Port-Hamiltonian Mathematical Model of a Fluid Ring Attitude System

In this article, we propose a mathematical model using the port-Hamiltonian formalism for a satellite’s three-axis attitude system comprising fluid rings. Fluid rings are an alternative to reaction wheels used for the same purpose, since, for the same mass, they can exert a greater torque than a reaction wheel as the fluid can circulate the periphery of the satellite. The port-Hamiltonian representation lays the foundation for a posterior controller that is feasible, stable, and robust based on the interconnection of the system to energy shaping and/or damping injection components, and by adding energy routing controllers. The torques exerted by the fluid rings are modeled using linear regression analysis on the experimental data got from a prototype of a fluid ring. Since the dynamics of turbulent flows is complex, the torques obtained by the prototype lead to a simpler first approach, leaving its uncertainties to a controller. Thus, the attitude system model could be tested in a future prototype before considering a spatial environment.

A unified port-Hamiltonian approach for modelling and stabilizing control of engineering systems

This work deals with systems whose dynamics are affine in the control input. Such dynamics are considered to be significantly differentially expressed in a canonical form, namely the quadratic (pseudo) port-Hamiltonian representation, in order to explore further some structural properties usable for the tracking-error passivity-based control design. Different kinds of linear and nonlinear engineering systems including an open isothermal homogeneous system and a continuous biochemical fermenter are used to illustrate the approach.

Stochastic Bounded Stability in the Three-Body Problem with Probabilistic Uncertainty Using Port-Hamiltonian Representation

Stochastic Bounded Stability in the Three-Body Problem with Probabilistic Uncertainty Using Port-Hamiltonian Representation

Limit Cycles in Locally Hamiltonian Systems with Dissipation

Abstract We consider systems that possess a local but non-global port-Hamiltonian representation. We show that non-globality is a consequence of the phase-space topology and that local port-Hamiltonian systems may transgress the dissipation inequality satisfied by their global counterparts. Far from being a defect, non-globality extends the modeling power of port-Hamiltonian systems by providing a framework to model cyclic physical processes for energy conversion and extraction.

Nonlinear state feedback control design for port-Hamiltonian systems with unstructured component

Abstract This paper considers the problem of state feedback controller design to stabilize generalized Hamiltonian systems with an unstructured component. This class of models enable one to exploit the structure of port-Hamiltonian systems for feedback control design while relaxing the constraint of deriving an exact structured port-Hamiltonian representation. For a given stabilizable nonlinear system, and with some assumptions on the unstructured part of the dynamics, a stabilizing control law is designed and asymptotic stability of a desired equilibrium of the system is demonstrated. A numerical illustration of the proposed approach is presented to demonstrate the design method.

Explicit and Implicit IDA-PBC Design and Implementation for a Portal Crane

The interconnection and damping assignment passivity-based control (IDA-PBC) is well-known for regulating the behavior of nonlinear systems. In underactuated mechanical systems (UMSs), its application requires the satisfaction of matching conditions, which in many cases demands to solve partial differential equations (PDEs). Only recently, the IDA-PBC method has been extended to UMSs in implicit representation, where the system dynamics are described by a set of differential-algebraic equations. In some system classes, this implicit model allows to circumvent the PDE obstacle and to construct an output-feedback law. This paper discusses the design and real-system implementation of the total energy shaping IDA-PBC with an optimal local performance for a portal crane system in implicit port-Hamiltonian representation. The implicit controller is compared with the simplified (explicit) IDA-PBC, introduced by Xue and Zhiyong (2017), which also shapes the total energy and avoids PDEs.

Matched Disturbance Rejection for a Class of Nonlinear Systems

In this paper, we present a method to robustify asymptotically stable nonlinear systems by adding an integral action that rejects unknown additive disturbances. The proposed approach uses a port-Hamiltonian (pH) representation of the open-loop dynamics, which, relying on the asymptotic stability property, is guaranteed to exist. The integral action controller preserves the pH structure, and, by adding a suitable cross term between the plant and the controller states to the closed-loop energy function, it avoids the unnatural coordinate transformation used in the past. The controller is shown to be robust against some common types of modeling uncertainty, including unknown friction dynamics in mechanical systems.

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.

Tracking-error-based control of a chemical reactor using decoupled dynamic variables

Abstract This work proposes an advanced control strategy for an open non-isothermal homogeneous reactor by integrating alternative model of the system dynamics based on variant and invariant states with the tracking-error-based control method in the framework of port-Hamiltonian representation. More precisely, a linear transformation is used to transform the original dynamics into a new model, where the enthalpy, reaction variant and reaction invariant are the state variables. Then, the transformed model is formulated into the port-Hamiltonian structure before the tracking-error-based approach is applied to design the controllers for the purpose of stabilization. The numerical simulations for the first order reaction modelled with the continuous stirred tank reactor having the multiplicity behavior subsequently show that the proposed feedback law is able to stabilize the system at the desirable equilibrium point.

Feedback passivation plus tracking-error-based multivariable control for a class of free-radical polymerisation reactors

ABSTRACTThis paper proposes a tracking-error-based multivariable control to stabilise a nonlinear system at the desired trajectory (including the open-loop unstable equilibrium manifold). The control approach is developed on the basis of feedback passivation and then applied to stabilise globally exponentially a class of free-radical polymerisation reactors. More precisely, under certain conditions the system dynamics can be rendered strictly input/output passive through the use of an appropriate input coordinate transformation. A canonical form related to the so-called port-Hamiltonian representation of passive system is consequently derived and provides physical interpretations such as dissipative/non-dissipative term and supply rate. A feedback law based on tracking-error is then designed for the global exponential stabilisation at a reference trajectory passing through the desired set-point. The theoretical developments are illustrated for polystyrene production in a continuous stirred tank reactor. Numerical simulations show that the system trajectory converges globally exponentially to the reference trajectory despite effects of disturbance.

Tracking error plus damping injection control of non-minimum phase processes

This work proposes a passivity-based approach to deal with the output-tracking-error problem for a large class of nonlinear chemical processes including non-minimum phase systems. More precisely, in that framework, the system dynamics is firstly written into the relaxing (pseudo) port-Hamiltonian representation which does not necessarily require the positive semi-definite property of the damping matrix. Then, a reference trajectory associated with a certain structure passing through a desired equilibrium point (i.e., the set-point) is chosen so that the error dynamics can be globally asymptotically stabilized at the origin thanks to the assignment of an appropriate damping injection. This method is subsequently illustrated for a benchmark of multiple reactions systems, namely Van de Vusse reaction system. The numerical simulations show the applications of the proposed approach.

Representation of heat exchanger networks using graph formalism

This contribution addressed the systematic representation of heat exchanger networks thanks to graph formalism. The energy representation of one heat exchanger is presented based on the incidence matrices of the graph related to heat transfer and the graph of the heat convection. Then the global heat exchanger network is obtained from the heat exchanger graph models as well as the incidence matrices of the interconnection obtained by adding additional vertices corresponding to the connector elements. The Port Hamiltonian representation of the heat network exchanger is given in the case of constant pressure. Then we show the incremental passivity of one compartment of the heat exchanger.

Passivity Analysis and Port-Hamiltonian Formulation of the Müller-Achenbach-Seelecke Model for Shape Memory Alloys: the Isothermal Case

In this work we investigate passivity of the Müller-Achenbach-Seelecke Model for shape memory alloy (SMA) wires. Such a model represents an effective control-oriented tool to describe the hysteretic dynamics of SMAs, but its highly complex structure makes it difficult to analyze fundamental properties such as energetic consistency, i.e., passivity. Focusing on the isothermal case, we propose a suitable storage function which permits to analyze passivity of the SMA model and to quantify energy dissipation due to hysteresis. In addition, a port-Hamiltonian representation of the resulting passive model is developed. This result permits to quantitatively assess energetic performance of SMAs when operating, e.g., as passive non-viscous dampers, and opens up the possibility of using powerful tools from port-Hamiltonian theory to analyze and design control systems for SMA actuators.