port-Hamiltonian Systems Research Articles

Extended Balancing of Continuous LTI Systems: A Structure-Preserving Approach

In this article, we treat extended balancing for continuous-time linear time-invariant systems. We take a dissipativity perspective, thus, resulting in a characterization in terms of linear matrix inequalities. This perspective is useful for determining a priori error bounds. In addition, we address the problem of structure-preserving model reduction of the subclass of port-Hamiltonian systems. We establish sufficient conditions to ensure that the reduced-order model preserves a port-Hamiltonian structure. Moreover, we show that the use of extended Gramians can be exploited to get a small error bound and, possibly, to preserve a physical interpretation for the reduced-order model. We illustrate the results with a large-scale mechanical system example. Furthermore, we show how to interpret a reduced-order model of an electrical circuit again as a lower dimensional electrical circuit.

A Case Study of Port-Hamiltonian Systems With a Moving Interface

A Case Study of Port-Hamiltonian Systems With a Moving Interface

On the Generating Functions of Irreversible port-Hamiltonian Systems⋆

We study the geometric structure of the drift dynamics of Irreversible port-Hamiltonian systems. This drift dynamics is defined with respect to a product of Poisson brackets, reflecting the interconnection structure and the constitutive relations of the irreversible phenomena occuring in the system. We characterise this product of Poisson brackets using a covariant 4-tensor and an associated function. We derive various conditions for which this 4-tensor and the associated function may be reduced to a product of almost Poisson brackets.

An ISS property of mechanical port-Hamiltonian systems with KPES for output feedback control

This paper proves a stability property of mechanical port-Hamiltonian systems with kinetic potential energy shaping (KPES) with respect to external disturbances using the input-to-state stability (ISS) characterization. Such analysis is quite useful when constructing an output feedback controller consisting of a KPES state feedback controller and a state observer, since the stability of the resulting output feedback closed-loop system can be guaranteed by such stability by regarding the estimation error of the observer as the external disturbance to the KPES state feedback system. If this state feedback system is ISS with respect to the state estimation error and if this error is asymptotically stable, then the origin of the closed-loop system is asymptotically stable. Therefore, it is possible to obtain an output feedback controller consisting of a KPES state feedback controller with any asymptotic state observer. Numerical simulations verify the ISS property of the KPES state feedback system.

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.

Adaptive Energy-Based Control for Buck Converter With a Class of Nonlinear Loads

In this brief, the problem of voltage regulation of buck converter with a class of nonlinear loads is addressed by using an adaptive energy-based controller. It is noted that the existing results mainly cope with converters feeding one or several nonlinear loads. Other types of loads do exist in power systems. Therefore, based on the passivity theory and a coordinate transformation, this brief proposes a new energy-based controller to stabilize buck converter with a class of nonlinear loads. It is ensured that the closed-loop system is a port-Hamiltonian system with exponential convergence. Then, the estimated terms generated by a designed disturbance observer are forwarded to energy-based controller to reduce the influence of the time-varying disturbances on the system. The experiment test is conducted by means of the setup of buck converter with a constant power load. The simulation and experimental results verify the effectiveness of the designed controller.

Mixed-dimensional geometric coupling of port-Hamiltonian systems

We propose a new interconnection relation for infinite-dimensional port-Hamiltonian systems that enables the coupling of ports with different spatial dimensions by integrating over the surplus dimensions. To show the practical relevance, we apply this interconnection to a model system of an actively cooled gas turbine blade. We also show that this interconnection relation behaves well with respect to a discretization in finite element space, ensuring its usability for practical applications.

An Overview on Irreversible Port-Hamiltonian Systems.

A comprehensive overview of the irreversible port-Hamiltonian system’s formulation for finite and infinite dimensional systems defined on 1D spatial domains is provided in a unified manner. The irreversible port-Hamiltonian system formulation shows the extension of classical port-Hamiltonian system formulations to cope with irreversible thermodynamic systems for finite and infinite dimensional systems. This is achieved by including, in an explicit manner, the coupling between irreversible mechanical and thermal phenomena with the thermal domain as an energy-preserving and entropy-increasing operator. Similarly to Hamiltonian systems, this operator is skew-symmetric, guaranteeing energy conservation. To distinguish from Hamiltonian systems, the operator depends on co-state variables and is, hence, a nonlinear-function in the gradient of the total energy. This is what allows encoding the second law as a structural property of irreversible port-Hamiltonian systems. The formalism encompasses coupled thermo-mechanical systems and purely reversible or conservative systems as a particular case. This appears clearly when splitting the state space such that the entropy coordinate is separated from other state variables. Several examples have been used to illustrate the formalism, both for finite and infinite dimensional systems, and a discussion on ongoing and future studies is provided.

Modeling of Multi Terminal VSC-HVDC links using port-Hamiltonian Systems

In recent years, different strategies have been used to model HVDC systems. A usual strategy is to look for the models simplification (or linearization) to do a small-signal analysis. However, VSC-HVDC multiterminal links have intrinsically non-linear characteristics that need to be considered. Port-Hamiltonian (pH) systems provide a modeling methodology that allows incorporating the elements of the HVDC system in a modular way while explicitly providing the energy function, internal interconnection, and dissipation, which is helpful for analysis and control. This paper presents the modeling of a multiterminal HVDC network commonly used in the Electrical Power Systems literature from the pH perspective. It aims to use the pH structure to incorporate elements into the system while preserving its properties. Numerical simulations are carried out to show a comparison between the common model and the pH representation and it is shown that the latter reproduces the same dynamics of the former without loss of information but gaining in simplicity and structure.

Observer design for a class of nonlinear Hamiltonian systems based on energy function structure

In this paper the observer design problem for a class of nonlinear port-Hamiltonian system is addressed. The class of nonlinear Hamiltonian system is based on the structure of energy function where it is possible to pull the state apart at measured and unmeasured states which generates two interconnected Hamiltonian systems. Exploiting the properties of the class of nonlinear Hamiltonian systems a reduced order observation scheme is proposed. Finally the observer design proposed in this paper is evaluated in a magnetic levitator in order to prove its the convergence properties, also it is comparing the observer convergence characteristics and the observer performance among the observation schemes that are reported in the literature.

Stochastic Port-Hamiltonian Systems

In the present work we formally extend the theory of port-Hamiltonian systems to include random perturbations. In particular, suitably choosing the space of flow and effort variables we will show how several elements coming from possibly different physical domains can be interconnected in order to describe a dynamic system perturbed by general continuous semimartingale. Relevant enough, the noise does not enter into the system solely as an external random perturbation, since each port is itself intrinsically stochastic. Coherently to the classical deterministic setting, we will show how such an approach extends existing literature of stochastic Hamiltonian systems on pseudo-Poisson and pre-symplectic manifolds. Moreover, we will prove that a power-preserving interconnection of stochastic port-Hamiltonian systems is a stochastic port-Hamiltonian system as well.

Dual field structure-preserving discretization of port-Hamiltonian systems using finite element exterior calculus

In this paper we propose a novel approach to discretize linear port-Hamiltonian systems while preserving the underlying structure. We present a finite element exterior calculus formulation that is able to mimetically represent conservation laws and cope with mixed open boundary conditions using a single computational mesh. The possibility of including open boundary conditions allows for modular composition of complex multi-physical systems whereas the exterior calculus formulation provides a coordinate-free treatment. Our approach relies on a dual-field representation of the physical system that is redundant at the continuous level but eliminates the need of mimicking the Hodge star operator at the discrete level. By considering the Stokes-Dirac structure representing the system together with its adjoint, which embeds the metric information directly in the codifferential, the need for an explicit discrete Hodge star is avoided altogether. By imposing the boundary conditions in a strong manner, the power balance characterizing the Stokes-Dirac structure is then retrieved at the discrete level via symplectic Runge-Kutta integrators based on Gauss-Legendre collocation points. Numerical experiments validate the convergence of the method and the conservation properties in terms of energy balance both for the wave and Maxwell equations in a three dimensional domain. For the latter example, the magnetic and electric fields preserve their divergence free nature at the discrete level.

A combined Control by Interconnection—Model Predictive Control design for constrained Port-Hamiltonian systems

This paper proposes a Control by Interconnection design, for a class of constrained Port-Hamiltonian systems, which is based on an associated Model Predictive Control optimization problem. This associated optimization problem allows to consider both state and input constraints simultaneously. Based on the first order Karush–Kuhn–Tucker optimality condition, the primal–dual gradient method is then used to build a passive feedback controller, derived from the MPC-induced optimization problem. The resulting passive controller is coupled with the original Port-Hamiltonian system through a power-preserving interconnection, in order to guarantee both the closed-loop stability and constraints satisfaction, but not the optimality anymore. Comments on parameters tuning for the proposed control design, together with validations of the approach through simulations first on a linear LC circuit, then on a nonlinear Permanent Magnet Synchronous Motor and comparisons with a classical MPC design, are provided to discuss the effectiveness of the approach. • A controller combining CbI and MPC is proposed to tackle PHs with constraints. • The MPC principle is used to deal with both state and input constraints. • The CbI takes advantage of the PHs passivity for convergence and stability analysis. • The primal–dual gradient method significantly reduces the computational effort. • The explicit form of the controller facilitates implementation and maintenance.

A Unified Passivity-Based Framework for Control of Modular Islanded AC Microgrids

Voltage and frequency control in an islanded ac microgrid (ImG) amounts to stabilizing an <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">a priori</i> unknown ImG equilibrium induced by loads and changes in topology. This article puts forth a unified control framework, which, while guaranteeing such stability, allows for modular ImGs interconnecting multiple subsystems, that is, dynamic <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$RLC$ </tex-math></inline-formula> lines, nonlinear constant impedance, current, power (ZIP) and exponential (EXP) loads, and inverter-based distributed generation units (DGUs) controlled with different types of primary controllers. The underlying idea of the framework is based on the equilibrium-independent passivity (EIP) of the ImG subsystems, which enables stability certificates of ImG equilibria without explicit knowledge of these equilibria. In order to render DGUs EIP, we propose a decentralized controller synthesis algorithm based on port-Hamiltonian systems (PHSs). We also show that EIP, being the key to stability, provides a general framework, which can embrace other solutions available in the literature. Furthermore, we provide a novel argument based on LaSalle’s theorem for proving asymptotic voltage and frequency stability. Finally, we analyze the impact of actuator saturation on the stability results by exploiting the inherent EIP properties of the PHS DGU model. Theoretical findings are backed up by realistic simulations based on the medium-voltage CIGRE benchmark network.

Mini-Workshop: Mathematics of Dissipation – Dynamics, Data and Control

Dissipation of energy — as well as its sibling the increase of entropy — are fundamental facts inherent to any physical system. The concept of dissipativity has been extended to a more general system theoretic setting via port-Hamiltonian systems and this framework is a driver of innovations in many of areas of science and technology. The particular strength of the approach lies in the modularity of modeling, the strong geometric, analytic and algebraic properties and the very good approximation properties.

Optimal Control of Port-Hamiltonian Descriptor Systems with Minimal Energy Supply

We consider the singular optimal control problem of minimizing the energy supply of linear dissipative port-Hamiltonian descriptor systems. We study the reachability properties of the system and prove that optimal states exhibit a turnpike behavior with respect to the conservative subspace. Further, we derive a input-state turnpike towards a subspace for optimal control of port-Hamiltonian ordinary differential equations with a feed-through term and a turnpike property for the corresponding adjoint states towards zero. In an appendix we characterize the class of dissipative Hamiltonian matrices and pencils.

A Port Hamiltonian approach to dynamical chemical process systems network modeling and analysis

This work focuses on the investigation of the applicability of port Hamiltonian systems (PHS) approach to model chemical process plant systems and the generalization of the PHS originally proposed for reversible thermodynamic systems to irreversible thermodynamic system characterized by coupled physicochemical phenomenon and significant irreversible entropy generation. To this end, the first part of this work presents a definition and analysis of the proposed generalized class of conservative pseudo port Hamiltonian system for general open multi-input multi-output (MIMO) irreversible thermodynamic system. In the second part of the work, the generalized PHS approach is applied to multiphase process systems with coupled irreversible entropy generation. The proposed modeling framework is extended to model chemical process systems network with recycle, which is prevalent in chemical process plants.

Path-Tracking Considering Yaw Stability With Passivity-Based Control for Autonomous Vehicles

In this paper, a new passivity-based control approach is designed to improve the robustness and stability of autonomous vehicles in performing path-tracking tasks. A port-Hamiltonian model is a special geometric structure and provides a systematic and insightful framework to describe many physical systems from an energy perspective. The passivity-based control establishes a new and simple structure to achieve the path-tracking task by the port-Hamiltonian structure. Firstly, the path-tracking error system is transformed into a port-Hamiltonian system with the perturbation. The energy-shaping method is utilized to ensure the asymptotic stability, and the stability proof of the closed-loop, path-tracking error system is given. The proposed real-time, passivity-based controller is directly dependent on the passive outputs, which is robust to parameter uncertainties caused by load changes. Moreover, the control performance of steering control will be degraded when the steering angle is saturated. The yaw-moment control is introduced to enhance the driving safety. In the wheel torque distribution, an optimization-based control allocation is designed to minimize the sum of tire loads and to make them more evenly distributed to each tire. Finally, simulations are implemented in MATLAB/Simulink and CarSim co-simulation to demonstrate the effectiveness of the designed strategy under different driving conditions.

Linear Port-Hamiltonian Systems Are Generically Controllable

The new concept of relative generic subsets is introduced. It is shown that the set of controllable linear finite-dimensional port-Hamiltonian systems is a relative generic subset of the set of all linear finite-dimensional port-Hamiltonian systems. This implies that a random, continuously distributed port-Hamiltonian system is almost surely controllable.

Memristor‐based disturbance rejection control for port‐Hamiltonian systems with locally fixed‐time convergence

The memristor is added to the port-Hamiltonian systems to improve the disturbance suppression performance in this paper. The concepts of locally fixed-time stability and locally fixed-time H ∞ $H_{\infty }$ control of port-Hamiltonian systems are presented. From this starting point, two novel memristor-based locally fixed-time controllers are designed to suppress the external disturbance of port-Hamiltonian systems via the interconnection and damping assignment passivity-based control technique. Comparing with the classical interconnection and damping assignment passivity-based control methodology without memristor, there are some advantages. First, the settling-time related to the memristor-based controllers in stabilizing the same port-Hamiltonian systems are shorter. Second, the memristor-based controllers can make the oscillation in the case of the strong periodic disturbance be better suppressed. Theoretical analysis shows that the memristor-based controller possesses good disturbance rejection performance owing to it can make the system states in a neighbourhood accelerate convergence to the desired equilibrium point. Two illustrative examples show that the theoretical results and the controllers designed in this paper work very well.