Dissipative Hamiltonian Research Articles

Abstract Dissipative Hamiltonian Differential-Algebraic Equations Are Everywhere

Abstract Dissipative Hamiltonian Differential-Algebraic Equations Are Everywhere

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.

Differential–algebraic systems with dissipative Hamiltonian structure

Different representations of linear dissipative Hamiltonian and port-Hamiltonian differential–algebraic equations (DAE) systems are presented and compared. Using global geometric and algebraic points of view, translations between different representations are presented. Characterizations are also derived when a general DAE system can be transformed into one of these structured representations. Approaches for computing the structural information and the described transformations are derived that can be directly implemented as numerical methods. The results are demonstrated with a large number of examples.

On Non-Hermitian Positive (Semi)Definite Linear Algebraic Systems Arising from Dissipative Hamiltonian DAEs

We discuss different cases of dissipative Hamiltonian differential-algebraic equations and the linear algebraic systems that arise in their linearization or discretization. For each case we give examples from practical applications. An important feature of the linear algebraic systems is that the (non-Hermitian) system matrix has a positive definite or semidefinite Hermitian part. In the positive definite case we can solve the linear algebraic systems iteratively by Krylov subspace methods based on efficient three-term recurrences. We illustrate the performance of these iterative methods on several examples. The semidefinite case can be challenging and requires additional techniques to deal with the “singular part,” while the “positive definite part” can still be treated with the three-term recurrence methods.

Boundary-dependent dynamical instability of bosonic Green's function: Dissipative Bogoliubov–de Gennes Hamiltonian and its application to non-Hermitian skin effect

The energy spectrum of bosonic excitations from a condensate is given by the spectrum of a non-Hermitian Hamiltonian constructed from a bosonic Bogoliubov-de Gennes (BdG) Hamiltonian in general even though the system is essentially Hermitian. In other words, two types of non-Hermiticity can coexist: one from the bosonic BdG nature and the other from the open quantum nature. In this paper, we propose boundary-dependent dynamical instability. We first define the bosonic dissipative BdG Hamiltonian in terms of Green's function in Nambu space and discuss the correct particle-hole symmetry of the corresponding non-Hermitian Hamiltonian. We then construct a model of the boundary-dependent dynamical instability so that it satisfies the correct particle-hole symmetry. In this model, an anomalous term that breaks the particle number conservation represents the non-Hermiticity of the BdG nature, while a normal term is given by a dissipative Hatano-Nelson model. Thanks to the competition between the two types of non-Hermiticity, the imaginary part of the spectrum can be positive without the help of the amplification of the normal part and the particle-hole band touching that causes the Landau instability. This leads to the boundary-dependent dynamical instability under the non-Hermitian skin effect, -strong dependence of spectra on boundary conditions for non-Hermitian Hamiltonians-, of the Bogoliubov spectrum.

Dissipative Dynamics and Uncertainty Measures of a Charged Oscillator in the Presence of the Aharonov-Bohm Effect

We analyze the effects of dissipation in a charged oscillator in the presence of the Aharonov-Bohm effect by using time-dependent mass (m(t)) Hamiltonians. We consider two different models for the dissipative Hamiltonian and analyze the uncertainties (Δr and Δp) and the quantum mechanical expectation value of energy (⟨E⟩) in terms of time (t), damping parameters and flux parameter (v). For the Caldirola-Kanai model, we observe that the flux parameter v decreases the energy dissipation in a quantum dot for a certain range of t.

On the Energy Stable Approximation of Hamiltonian and Gradient Systems

Abstract A general framework for the numerical approximation of evolution problems is presented that allows to preserve an underlying dissipative Hamiltonian or gradient structure exactly. The approach relies on rewriting the evolution problem in a particular form that complies with the underlying geometric structure. The Galerkin approximation of a corresponding variational formulation in space then automatically preserves this structure which allows to deduce important properties for appropriate discretization schemes including projection based model order reduction. We further show that the underlying structure is preserved also under time discretization by a Petrov–Galerkin approach. The presented framework is rather general and allows the numerical approximation of a wide range of applications, including nonlinear partial differential equations and port-Hamiltonian systems. Some examples will be discussed for illustration of our theoretical results, and connections to other discretization approaches will be highlighted.

Approximation of stability radii for large-scale dissipative Hamiltonian systems

A linear time-invariant dissipative Hamiltonian (DH) system dot x = (J-R)Q x, with a skew-Hermitian J, a Hermitian positive semidefinite R, and a Hermitian positive definite Q, is always Lyapunov stable and under further weak conditions even asymptotically stable. By exploiting the characterizations from Mehl et al. (SIAM J. Matrix Anal. Appl. 37(4), 1625–1654, 2016), we focus on the estimation of two stability radii for large-scale DH systems, one with respect to non-Hermitian perturbations of R in the form R + BΔCH for given matrices B, C, and another with respect to Hermitian perturbations in the form R + BΔBH,Δ = ΔH. We propose subspace frameworks for both stability radii that converge at a superlinear rate in theory. The one for the non-Hermitian stability radius benefits from the DH structure-preserving model order reduction techniques, whereas for the Hermitian stability radius we derive subspaces yielding a Hermite interpolation property between the full and projected problems. With the proposed frameworks, we are able to estimate the two stability radii accurately and efficiently for large-scale systems which include a finite-element model of an industrial disk brake.

Finite-time robust control for transformerless H-bridge cascaded STATCOM with star configuration

This article presents a finite-time robust control (FTRC) of a transformerless STATCOM based on a cascaded multilevel H-bridge converter (CMHC) with star configuration. The FTRC is first proposed for the current loop control of a CMHC-based transformerless STATCOM by using the finite time robust control theory. Taking the parameters, perturbations and external disturbances into account and using coordinate transformation method, the nonlinear dynamic model of the CMHC-based transformerless STATCOM is transformed into a standard nonlinear port-controlled dissipative Hamiltonian (PCDH) structure. Based on the PCDH structure, an FTRC is designed for the CMHC-based transformerless STATCOM to improve the transient stability and oscillation damping of power system. Finally, the simulation results demonstrate that the FTRC has better dynamic performance and strong robustness in comparison with the passivity-based control of the CMHC-based transformerless STATCOM.

PCDH-based nonlinear H ∞ control of VSC-HVDC with wind power integrated

Based on the port-controlled dissipative Hamiltonian (PCDH) system, a novel nonlinear (NN) control is designed for the wind farm side (WFS) voltage source converter (VSC) of VSC-based high-voltage direct current (VSC-HVDC) transmission system connected large wind farm. Firstly, the uncertain nonlinear dynamical model of WFS VSC in the VSC-HVDC is established by taking external disturbances and parameter variations into account. Then, the dissipative Hamiltonian structure of the WFS VSC system is constructed by means of variable transformation. Thirdly, on the basis of the obtained dissipative Hamiltonian structure, a NN control is put forward. Finally, the simulation results show that the proposed NN control gains the advantage over the interconnection and damping assignment passivity-based (IDA-PB) control.

An improved nonlinear robust control design for grid-side converter of VSC-HVDC connected to wind power generation system

This article proposes an improved nonlinear (IN) robust control strategy for the grid-side voltage-source converter (GSVSC) of a VSC-based high-voltage direct current (VSC-HVDC) transmission system connected to a large wind farm by the using Hamiltonian function method. With the help of variable transformation, the nonlinear model with parameter uncertainties and external disturbances of the GSVSC is changed into the port-controlled dissipative Hamiltonian (PCDH) system. Based on the PCDH system, an IN robust control law is established. In order to demonstrate the effectiveness and robustness of the IN robust control, the backstepping power control (BPC), which is developed by using the backstepping design procedure in the sense of Lyapunov stability theorem for the GSVSC to satisfy the control objectives of a stable HVDC bus and the grid connection with a unity power factor, is selected as a comparison object. Generally speaking, the system used by the backstepping method must have special strict feedback of the lower triangular structure, but the wind farm connected to the power grid with the VSC-HVDC is a multivariable structure and highly coupled nonlinear system. So, the BPC cannot effectively maintain and utilize the nonlinear characteristics of the VSC-HVDC system, while this nonlinear physical structure characteristics is very useful for the design of a nonlinear controller. The greatest advantage of the Hamilton function method is that it can effectively keep and utilize the nonlinear characteristics of the system. The simulation results indicate that the proposed IN control designed by using the Hamilton function method gains the advantage over the BPC under a variety of operating conditions.

Global continuous robust finite‐time control design for unified power flow controller

SummaryThis article proposes a global continuous robust finite‐time control (GCRFTC) for a unified power flow controller (UPFC) to improve the transient stability and oscillation damping of power system. First, by using an appropriate coordinate transformation, the nonlinear system of the UPFC is transformed into a nonlinear port‐controlled dissipative Hamiltonian (PCDH) system. Second, based on the PCDH system, by utilizing the Hamiltonian structural properties and the “energy shaping plus damping injection” technique, a proper form of a Hamiltonian function is obtained for the PCDH system. Third, using the Hamiltonian function method, finite‐time stability criterion and robust control technique, the GCRFTC is designed and theoretically proved, and the chattering phenomena and the high‐order frequencies of the power system are avoided effectively. Lastly, a six‐bus and two power plants power system with a UPFC is used to test the effectiveness and robustness of the GCRFTC. Simulation results show that the speed, overshoot, and settling time of the response and the transient conditions of the GCRFTC are further improved in comparison with that of the robust finite‐time power flow control.

Stability radii for real linear Hamiltonian systems with perturbed dissipation

We study linear dissipative Hamiltonian (DH) systems with real constant coefficients that arise in energy based modeling of dynamical systems. We analyze when such a system is on the boundary of the region of asymptotic stability, i.e., when it has purely imaginary eigenvalues, or how much the dissipation term has to be perturbed to be on this boundary. For unstructured systems the explicit construction of the real distance to instability (real stability radius) has been a challenging problem. We analyze this real distance under different structured perturbations to the dissipation term that preserve the DH structure and we derive explicit formulas for this distance in terms of low rank perturbations. We also show (via numerical examples) that under real structured perturbations to the dissipation the asymptotical stability of a DH system is much more robust than for unstructured perturbations.

Stability Radii for Linear Hamiltonian Systems with Dissipation Under Structure-Preserving Perturbations

Dissipative Hamiltonian (DH) systems are an important concept in energy based modeling of dynamical systems. One of the major advantages of the DH formulation is that system properties are encoded in an algebraic way. For instance, the algebraic structure of DH systems guarantees that the system is automatically stable. In this paper the question is discussed when a linear constant coefficient DH system is on the boundary of the region of asymptotic stability, i.e., when it has purely imaginary eigenvalues, or how much it has to be perturbed to be on this boundary. For unstructured systems this distance to instability (stability radius) is well understood. In this paper, explicit formulas for this distance under structure-preserving perturbations are determined. It is also shown (via numerical examples) that under structure-preserving perturbations the asymptotical stability of a DH system is much more robust than under general perturbations, since the distance to instability can be much larger when structure-preserving perturbations are considered.

Dissipative Hamiltonian realisation and robust H<SUB align="right">∞ control of induction motor considering iron losses for electric vehicles

The dissipative Hamiltonian realisation and robust H∞ control of induction motor considering iron losses for electric vehicle are investigated in this paper. First, the dissipative Hamiltonian of the electric vehicle drive system is obtained based on the system’s mathematical model in a synchronously rotating frame. Then, a robust co-ordinated tracking controller is designed based on the dissipative Hamiltonian form. One part of the controller is designed by using the method of interconnection and damping assignment to ensure the system’s stability, and another part is designed by using the Hamiltonian system’s robust H∞ technique to attenuate external disturbances. The simulation results show that the controller proposed in the paper works very well in robust tracking of induction motor.

Studying Complexity for a Modified Dissipative Hamiltonian System: From Lyapunov Stability at the Origin to a Limit Cycle and Chaos

Hamiltonian formalism is commonly adopted to model a wide range of physical systems. In this paper, a particular case of a simple dissipative Hamiltonian system with modified damping matrix is analysed in terms of stability properties. It is shown that, dependent from the selected system parameters, we can obtain a system response that varies from a stable equilibrium at the origin, to a stable limit cycle and finally to chaotic behaviour.

Random Attractors for the Dissipative Hamiltonian Amplitude Equation Governing Modulated Wave Instabilities with Additive Noise

In this paper, we study the random dynamical system (RDS) generated by the dissipative Hamiltonian amplitude equation with additive noise defined on the periodic boundaries. We investigate the existence of a compact random attractor for the RDS associated with the equation through introducing two functions and one process in E0=H1×L2. The compactness of the RDS is established by the decomposition of solution semigroup.

Stochastic Behavior of Dissipative Hamiltonian Systems with Limit Cycles

AbstractIn this paper noise analysis of non-linear oscillator circuits are discussed. For this purpose the physical based concept of dissipative Hamiltonian systems should be applied to a certain class of systems with limit cycle behavior.

Adiabatic approximation for quantum dissipative systems: Formulation, topology, and superadiabatic tracking

A generalized adiabatic approximation is formulated for a two-state dissipative Hamiltonian which is valid beyond weak dissipation regimes. The history of the adiabatic passage is described by superadiabatic bases as in the nondissipative regime. The topology of the eigenvalue surfaces shows that the population transfer requires, in general, a strong coupling with respect to the dissipation rate. We present, furthermore, an extension of the Davis-Dykhne-Pechukas formula to the dissipative regime using the formalism of Stokes lines. Processes of population transfer by an external frequency-chirped pulse-shaped field are given as examples.

Macroscopic Superpositions and Entanglement of Mesoscopic Squeezed Vacuum States in Dissipative Cavity QED System

A scheme has been proposed for generating the macroscopic superpositions and the entanglement between the mesoscopic squeezed vacuum states by considering the two-photon interaction of N two-level atoms in a dissipative cavity with high quality factor assisted by a strong driving field. A number of multiparty entangled states between the atoms and the squeezed vacuum states, among the atoms, and among the squeezed vacuum states, can be prepared by virtue of a specific choice of the cavity detuning and the detuning of applied coherent field under the dissipation condition. The corresponding analytical expressions of the influence can be given. Moreover, we can also give a series of macroscopic entangled states between the usual coherent states and the squeezed vacuum states using the combination of the dissipative one-photon interaction Hamiltonian with the dissipative two-photon interaction Hamiltonian. We also discuss the experimental feasibility. Our scheme can be realized in the current techniques on the cavity QED.