Geometric Integration Methods that Preserve Lyapunov Functions

Geometric integration methods that unconditionally contract volume

This is a call for contributions to a special issue ofJournal of Physics A: Mathematical and General entitled `Geometric Numerical Integration of Differential Equations'. This issue should be a repository for high quality original work. We are interested in having the topic interpreted broadly, that is, to include contributions dealing with symplectic or multisymplectic integration; volume-preserving integration; symmetry-preserving integration; integrators that preserve first integrals, Lyapunov functions, or dissipation; exponential integrators; integrators for highly oscillatory systems; Lie-group integrators, etc. Papers on geometric integration of both ODEs and PDEs will be considered, as well as application to molecular-scale integration, celestial mechanics, particle accelerators, fluid flows, population models, epidemiological models and/or any other areas of science. We believe that this issue is timely, and hope that it will stimulate further development of this new and exciting field.

We consider the geometric numerical integration of Hamiltonian systems subject to both equality and ``hard” inequality constraints. As in the standard geometric integration setting, we target long-term structure preservation. Additionally, however, we also consider invariant preservation over persistent, simultaneous, and/or frequent boundary interactions. Appropriately formulating geometric methods for these cases has long remained challenging due the inherent nonsmoothness and one-sided conditions that they impose. To resolve these issues we thus focus both on symplectic-momentum preserving behavior and the preservation of additional structures, unique to the inequality constrained setting. Toward these goals we introduce, for the first time, a fully nonsmooth, discrete Hamilton's principle and obtain an associated framework for composing geometric numerical integration methods for inequality-equality--constrained systems. Applying this framework, we formulate a new family of geometric numerical integra...

Stability analysis of discrete T-S fuzzy system with the common Lyapunov function approach and the fuzzy Lyapunov function approach is computationally attractive but conservative. Based on the definition of a discrete piecewise fuzzy Lyapunov function at each overlapping rules group, this paper introduces a new sufficient stability condition that only needs satisfying the stability condition of the fuzzy Lyapunov function approach in each maximal overlapped-rules group (MORG). A simulation example shows the condition is less conservative and easier to check than earlier approaches.

In this paper, Lyapunov functions are generated to determine the regions of asymptotic stability of power systems under transient disturbances. With suitable assumptions, the swing equations of the synchronous machines connected to a power system are second- or third-order nonlinear autonomous differential equations. The Lyapunov functions V, employed to determine the domains of asymptotic stability of these nonlinear differential equations, are simple quadratic forms, whose coefficients are chosen so that the Routh-Hurwitz criteria are satisfied for the corresponding linear differential equations. For a synchronous machine swinging against an infinite bus, three typical Lyapunov functions are generated, taking transient saliency and positive-sequence damping into account. The application of Aizerman's method to the same problem leads to a fourth Lyapunov function. The domains of stability given by these Lyapunov functions are compared with the actual stability region obtained by numerical integration. Consideration of one time constant of the prime-mover governor leads to a third-order differential equation. A Lyapunov function is generated, and the stability surface obtained is verified by computing the swing curves numerically for various initial conditions. Finally, it is explained how some of these techniques could be extended for generating Lyapunov functions for multimachine systems.

SummaryThis paper presents a general framework for analyzing stability of linear impulsive stochastic systems (LISSs). Some simple mean square stability criteria for the three types of LISSs are firstly derived by analyzing an equivalent system. By exploring the hybrid characteristics of impulsive systems, the novel quasi‐periodic composite polynomial Lyapunov function and the time‐varying discretized Lyapunov function are developed, which leads to unified dwell‐time–based criteria for mean square stability and almost sure stability of LISSs without imposing the stability condition on continuous‐ and discrete‐time dynamics. Next, based on the established stability criteria, the synthesis problem of state‐feedback controller is solved. The computational complexity and the comparison with existing results on the deterministic systems are discussed. Finally, numerical examples are provided to illustrate the usefulness of the proposed results.

The dynamics of many systems are described by ordinary differential equations (ODE). Solving ODE's with standard methods (i.e. numerical integration) needs a high amount of computing time but only a small amount of storage memory. For some applications, e.g. short time weather forecast or real time robot control, long computation times are prohibitive. Is there a method which uses less computing time (but has drawbacks in other aspects, e.g. memory), so that the computation of ODE's gets faster? We will try to discuss this question for the method of a neural network which was trained on ODE dynamics and compare both methods using the same approximation error. In many cases, as for physics engines used in computer games, the shape of the approximation curve is important and not the exact values of the approximation. Therefore, we introduce as error measure the subjective error based on the Total Least Square Error (TLSE) which gives more consistent results than the standard error. Finally, we derive a method to evaluate where neural nets are advantageous over numerical ODE integration and where this is not the case.

Kullback–Leibler life time testing

A numerical method for delayed partial differential equations describing infectious diseases

A control method based on quadratic stabilization of discrete switched affine system is proposed to control the current of three-phase APF(Active Power Filter) in this paper. Quadratic stability of switching equilibrium point where the state of the discrete switched affine system has to be driven to is assessed using a discrete Lyapunov function, and sufficient conditions are given. According to the dynamic characteristics of three-phase APF, a discrete switched affine system is established. And it is easy to prove that the discrete switched affine system of three-phase APF meet the conditions which make the system globally stable. The feasibility of the method has been verified by the results of Matlab simulation.

An active noise control algorithm is introduced based on adaptive wavelet networks using rationale functions with second-order poles wavelets. A novel network structure is derived using a nonlinear static mapping cascaded with an infinite impulse response filter wherein only input at the current step is needed to generate the output of the next sample. Therefore, incorporating tap delayed line of input–output of the physical system can be eliminated, avoiding the use of multidimensional wavelet networks. Online dynamic back-propagation learning algorithms (based on gradient descent method) are applied to adjust the network parameters. The local convergence of the closed-loop system is proved using discrete Lyapunov function. Simulations are carried out to compare the performance of proposed methods with other nonlinear algorithms (e.g. FxBPNN, Volterra, FLNN, and RFFLNN). Experiments are then conducted to evaluate the developed algorithms. Both simulation and experimental results show the superior performance of proposed method in terms of fast convergence rate and noise attenuation while avoiding curse of dimensionality.

A computational approach of feasible regions for parameter-dependent Lyapunov functions for linear systems based on real-root-classification (RRC) is proposed in this paper. We refer to such Lyapunov functions as feasible if there exists a parameter set that guarantees the existence of a Lyapunov function. In this sense, the stability of parameter-dependent linear ordinary differential equations (ODEs) and differential algebraic equations (DAEs) is considered. To this end, the existence condition of Lyapunov functions is given in form of an algebraic condition by means of a quantifier elimination (QE) formulation. We apply RRC to solve the QE problem to obtain algebraic necessary and sufficient existence conditions. Several numerical examples demonstrate that this approach produces non-conservative stability regions in the parameter space.

In this paper, a method is proposed for improving the accuracy of the stability criterion by enlarging the estimation of the stability region determined by means of Lyapunov's direct method using series expansion, and by putting it closer to the actual stability region. The Lyapunov function obtained by energy integration is used as the first estimation. The method is applied to a single-machine infinite bus power system, and the stability domain in the δ-ω plane is shown together with the actual stability domain obtained by numerical integration. It is shown that the application of the proposed method results in a considerable improvement in the stability boundary estimation over that given by the original Lyapunov function. Zubov's method, which obtains an accurate stability boundary using the series expansion, is related to the method proposed in this paper. The last section is, therefore, devoted to comparing the two methods and examining the possibility of their application to multi-machine power systems.

This study establishes a compartment model for the categorized COVID-19 risk area. In this model, the compartments represent administrative regions at different transmission risk levels instead of individuals in traditional epidemic models. The county-level regions are partitioned into High-risk (H), Medium-risk (M), and Low-risk (L) areas dynamically according to the current number of confirmed cases. These risk areas are communicable by the movement of individuals. An LMH model is established with ordinary differential equations (ODEs). The basic reproduction number R0 is derived for the transmission of risk areas to determine whether the pandemic is controlled. The stability of this LHM model is investigated by a Lyapunov function and Poincare–Bendixson theorem. We prove that the disease-free equilibrium (R0 < 1) is globally asymptotically stable and the disease will die out. The endemic equilibrium (R0 > 1) is locally and globally asymptotically stable, and the disease will become endemic. The numerical simulation and data analysis support the previous theoretical proofs. For the first time, the compartment model is applied to investigate the dynamics of the transmission of the COVID-19 risk area. This work should be of great value in the development of precision region-specific containment strategies.

The method of Lyapunov functions is used to investigate the stability of systems with distributed and lumped parameters, described by linear partial and ordinary differential equations. The original high-order partial differential equations are represented by a first-order system of partial differential evolution equations and constraint equations; to do that, we introduce additional variables. Passing to the first-order system of partial differential equations and representing ordinary differential equations in the normal Cauchy form, we obtain a possibility to construct the Lyapunov function as a sum of integral and classical quadratic forms and develop general methods of the investigation of the stability of a broad class of systems with distributed and lumped parameters. For example, we consider the stability of the work of the wind-driven lift pump and take into account the elasticity of the shaft transmitting the torque from the wind engine to the pump.