port-Hamiltonian Systems Research Articles

Passivity-based sliding mode control for mechanical port-Hamiltonian systems

Passivity-based sliding mode control for mechanical port-Hamiltonian systems

Jet space extensions of infinite-dimensional Hamiltonian systems

We analyze infinite-dimensional Hamiltonian systems corresponding to partial Differential equations on one-dimensional spatial domains formulated with formally skew-adjoint Hamiltonian operators and non-quadratic Hamiltonian density. In various applications, the Hamiltonian density can depend on spatial derivatives of the state such that these systems can not straightforwardly be formulated as boundary port-Hamiltonian system using a Stokes-Dirac structure. In this work, we show that any Hamiltonian system of the above class can be reformulated as a Hamiltonian system on the jet space, in which the Hamiltonian density only depends on the extended state variable itself and not on its derivatives. Consequently, well-known geometric formulations with Stokes-Dirac structures are applicable. Additionally, we provide a similar result for dissipative systems. We illustrate the developed theory by means of the the Boussinesq equation, the dynamics of an elastic rod and the Allen-Cahn equation.

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.

Remarks on the geometric structure of port-Hamiltonian systems

We study the geometric structure of port-Hamiltonian systems. Starting with the intuitive understanding that port-Hamiltonian systems are “in between” certain closed Hamiltonian systems, the geometric structure of port-Hamiltonian systems must be “in between” the geometric structures of the latter systems. These are Courant algebroids; and hence the geometric structures should be related by Courant algebroid morphisms. Using this idea, we propose a definition of an intrinsic geometric structure and show that it is unique, if it exists.

A behavioural approach to port-controlled systems

We give insight in the structure of port-Hamiltonian systems as control systems in between two closed Hamiltonian systems. Using the language of category theory, we identify systems with their behavioural representation and view a port-control structure with desired structural properties on a given closed system as an extension of this system which itself may be embedded in a “larger” closed system. The latter system describes the nature of the ports (e.g. Hamiltonian, metriplectic etc.). This point of view allows us to describe meaningful port-control structures for a large family of systems, which is illustrated with Hamiltonian and metriplectic systems.

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.

Passivity-based second-order sliding mode control via the homogeneous Lyapunov approach for mechanical port-Hamiltonian systems

In this work, a new second-order sliding mode controller for mechanical port-Hamiltonian systems is proposed. The authors’ former paper proposed a passivity-based sliding mode controller based on the kinetic-potential energy shaping (KPES). This controller is able to achieve only first-order sliding mode control since the KPES allows us to embed a subsystem, whose dimension is the same as that of the input, into the closed-loop system. The present paper extends the KPES to incorporate a higher-order subsystem in the closed-loop system, which enables us to obtain the subsystem that can realize second-order sliding mode control. The proposed controller is a unification of a passivity-based controller and a second-order sliding mode controller which does not cause undesirable chattering phenomena. It ensures finite-time convergence of the subsystem and asymptotic stability of the entire closed-loop system by utilizing two Lyapunov functions. A numerical example demonstrates the effectiveness of the proposed method.

Port-Hamiltonian modeling of large-scale curling HASEL actuators

This paper presents a modeling methodology to enhance the dynamic performance of the mechanical component of finite-dimensional curling HASEL (Hydraulically Amplified Self-Healing Electrostatic) actuators within the port-Hamiltonian systems framework. The proposed approach entails replacing the sheet dynamics that limit deformation in a low-scale model with those derived from a large-scale discretized beam model. By making a few additional assumptions compared to the original low-scale HASEL model, the resulting interconnected system is established by aligning the states of the mechanical component in the low-scale model with those of the large-scale beam model in a straightforward manner. To validate the effectiveness of the methodology, simulated examples are provided along with a comparison to experimental results.

Stability of port-Hamiltonian systems with mixed time delays subject to input saturation

In this paper, we investigate the stability of port-Hamiltonian systems with mixed time-varying delays as well as input saturation. Three types of time delays, including state delay, input delay, and output delay, are all assumed to be bounded. By introducing the output feedback control law and utilizing serval Lyapunov–Krasovskii functionals, we present three delay-dependent stability criteria in terms of the linear matrix inequality. Meanwhile, we use Wirtinger’s inequality, constraint conditions, and Lyapunov–Krasovskii functionals of triple and quadruple integral form to obtain less conservative results. Some numerical examples demonstrate and support our results.

On discrete-time dissipative port-Hamiltonian (descriptor) systems

Port-Hamiltonian (pH) systems have been studied extensively for linear continuous-time dynamical systems. This manuscript presents a discrete-time pH descriptor formulation for linear, completely causal, scattering passive dynamical systems based on the system coefficients. The relation of this formulation to positive and bounded real systems and the characterization via positive semidefinite solutions of Kalman–Yakubovich–Popov inequalities is also studied.

Control Design for a Class of Discrete-Time Port-Hamiltonian Systems

Control Design for a Class of Discrete-Time Port-Hamiltonian Systems

The difference between port-Hamiltonian, passive and positive real descriptor systems

The relation between passive and positive real systems has been extensively studied in the literature. In this paper, we study their connection to the more recently used notion of port-Hamiltonian descriptor systems. It is well-known that port-Hamiltonian systems are passive and that passive systems are positive real. Hence it is studied under which assumptions the converse implications hold. Furthermore, the relationship between passivity and KYP inequalities is investigated.

A flexible short recurrence Krylov subspace method for matrices arising in the time integration of port-Hamiltonian systems and ODEs/DAEs with a dissipative Hamiltonian

For several classes of mathematical models that yield linear systems, the splitting of the matrix into its Hermitian and skew Hermitian parts is naturally related to properties of the underlying model. This is particularly so for discretizations of dissipative Hamiltonian ODEs, DAEs and port-Hamiltonian systems where, in addition, the Hermitian part is positive definite or semi-definite. It is then possible to develop short recurrence optimal Krylov subspace methods in which the Hermitian part is used as a preconditioner. In this paper, we develop new, right preconditioned variants of this approach which, as their crucial new feature, allow the systems with the Hermitian part to be solved only approximately in each iteration while keeping the short recurrences. This new class of methods is particularly efficient as it allows, for example, to use few steps of a multigrid solver or a (preconditioned) CG method for the Hermitian part in each iteration. We illustrate this with several numerical experiments for large scale systems.

On Dirac structure of infinite-dimensional stochastic port-Hamiltonian systems

Stochastic infinite-dimensional port-Hamiltonian systems (SPHSs) with multiplicative Gaussian white noise are considered. In this article we extend the notion of Dirac structure for deterministic distributed parameter port-Hamiltonian systems to a stochastic ones by adding some additional stochastic ports. Using the Stratonovich formalism of the stochastic integral, the proposed extended interconnection of ports for SPHSs is proved to still form a Dirac structure. This constitutes our main contribution. We then deduce that the interconnection between (stochastic) Dirac structures is again a (stochastic) Dirac structure under some assumptions. These interconnection results are applied on a system composed of a stochastic vibrating string actuated at the boundary by a mass–spring system with external input and output. This work is motivated by the problem of boundary control of SPHSs and will serve as a foundation to the development of stabilizing methods.

Evaluation of a Passive-Based Controller for Power Monitoring in Networks of High Voltage Transmission Lines in Direct Current with Voltage Source Converters (VSC–HVDC)

In this paper a tracking Passivity--based Control (PBC) for a multiterminal Voltage Source Converter based High Voltage Direct Current Network (VSC--HVDC) is evaluated. First, the system (an actual multiterminal HVDC) described as a Port Hamiltonian (PH) System and the controller are presented and subsequently three scenarios are proposed to evaluate the closed--loop behavior, namely: (i) load changes; (ii) generation changes and, (iii) DC--bus changes. Numerical simulations are carried out and results shown the robustness of the closed--loop to cope with these common disturbances.

A Passivity-Based PI Control of Quasi-Resonant Buck Converter

In order to guarantee the stability of quasi-resonant converters, the zero-current switching quasi-resonant buck topology is supplied with a PI passivity-based controller in this study. An adequate and full-order model with constant steady-state solutions is generated, via the interconnection of port-Hamiltonian systems based on the principal structure of the converter. The passivity-based approach is employed to complete the control design, and numerical validation results are provided to ensure performance estimations.

Linear Matrix Inequality Design of Exponentially Stabilizing Observer-Based State Feedback Port-Hamiltonian Controllers

The design of an observer-based state feedback (OBSF) controller with guaranteed passivity properties for port-Hamiltonian systems (PHS) is addressed using linear matrix inequalities (LMIs). The observer gain is freely chosen and the LMIs conditions such that the state feedback is equivalent to control by interconnection with an input strictly passive (ISP) and/or an output strictly passive (OSP) and zero state detectable (ZSD) port-Hamiltonian controller are established. It is shown that the proposed controller exponentially stabilizes a class of infinite-dimensional PHS and asymptotically stabilizes a class of finite-dimensional non-linear PHS. A Timoshenko beam model and a microelectromechanical system are used to illustrate the proposed approach.

Weak Energy Shaping for Stochastic Controlled Port-Hamiltonian Systems

Weak Energy Shaping for Stochastic Controlled Port-Hamiltonian Systems

Operator splitting based dynamic iteration for linear differential-algebraic port-Hamiltonian systems

A dynamic iteration scheme for linear differential-algebraic port-Hamiltonian systems based on Lions–Mercier-type operator splitting methods is developed. The dynamic iteration is monotone in the sense that the error is decreasing and no stability conditions are required. The developed iteration scheme is even new for linear port-Hamiltonian systems governed by ODEs. The obtained algorithm is applied to a multibody system and an electrical network.

Cooperative Control of LQ-Feedback Linearization and Error Port-Hamiltonian System for PMSM with NDOB

Cooperative Control of LQ-Feedback Linearization and Error Port-Hamiltonian System for PMSM with NDOB