Stochastic Lyapunov Function Research Articles

Adaptive fuzzy control for permanent magnet synchronous motors considering input saturation in electric vehicle stochastic drive systems

In this paper, an adaptive fuzzy control method based on command filtering is proposed for permanent magnet synchronous motors (PMSMs) stochastic nonlinear systems (SNSs) with input saturation in electric vehicle. The unknown stochastic nonlinear functions are approached by the fuzzy logic system (FLS). Then, the command filtering error compensation technology is used to solve “explosion of complexity” problem and compensation signals are utilized to abate the influence of filtering errors. And stochastic Lyapunov functions are selected to analyze the stability of the system. In addition, the iron losses and input saturation for PMSMs in EV stochastic drive systems are considered, which makes the designed controller more practical. Finally, the effectiveness of the approach is intuitively illustrated by simulation results.

Dynamical behavior of a higher order stochastically perturbed SIRI epidemic model with relapse and media coverage.

This paper is intended to explore a higher order stochastically perturbed SIRI epidemic model with relapse and media coverage. Firstly, we derive sufficient criteria for the existence and uniqueness of an ergodic stationary distribution of positive solutions to the system by establishing a suitable stochastic Lyapunov function. Then we obtain adequate conditions for complete eradication and wiping out of the infectious disease. In a biological interpretation, the existence of a stationary distribution implies that the disease will prevail and persist in the long term. Finally, the theoretical results are illustrated by computer simulations, including two examples based on real-life disease.

Event-triggered fuzzy adaptive quantized control for nonlinear multi-agent systems in nonaffine pure-feedback form

In this paper, we address the problem of event-triggered fuzzy adaptive quantized control for stochastic nonlinear non-affine pure-feedback multi-agent systems. The fuzzy logic system is used to estimate the stochastic disturbance term and unknown nonlinear functions. A nonlinearity decomposition method of asymmetric hysteresis quantizer is proposed by applying sector bound property. Moreover, to reduce the communication burden, an adaptive event-triggered protocol with a varying threshold is constructed. Based on the backstepping technique and stochastic Lyapunov function method, a novel adaptive event-triggered fuzzy control protocol and adaptive laws are constructed. By using stochastic Lyapunov stability theory, it is demonstrated that all signals are bounded in the closed-loop systems in probability and all the outputs of followers converge to the neighborhood of the leader output. Simulation results illustrate the effectiveness of our proposed scheme.

Stochastic quasi-gradient methods: variance reduction via Jacobian sketching

We develop a new family of variance reduced stochastic gradient descent methods for minimizing the average of a very large number of smooth functions. Our method—JacSketch—is motivated by novel developments in randomized numerical linear algebra, and operates by maintaining a stochastic estimate of a Jacobian matrix composed of the gradients of individual functions. In each iteration, JacSketch efficiently updates the Jacobian matrix by first obtaining a random linear measurement of the true Jacobian through (cheap) sketching, and then projecting the previous estimate onto the solution space of a linear matrix equation whose solutions are consistent with the measurement. The Jacobian estimate is then used to compute a variance-reduced unbiased estimator of the gradient. Our strategy is analogous to the way quasi-Newton methods maintain an estimate of the Hessian, and hence our method can be seen as a stochastic quasi-gradient method. Our method can also be seen as stochastic gradient descent applied to a controlled stochastic optimization reformulation of the original problem, where the control comes from the Jacobian estimates. We prove that for smooth and strongly convex functions, JacSketch converges linearly with a meaningful rate dictated by a single convergence theorem which applies to general sketches. We also provide a refined convergence theorem which applies to a smaller class of sketches, featuring a novel proof technique based on a stochastic Lyapunov function. This enables us to obtain sharper complexity results for variants of JacSketch with importance sampling. By specializing our general approach to specific sketching strategies, JacSketch reduces to the celebrated stochastic average gradient (SAGA) method, and its several existing and many new minibatch, reduced memory, and importance sampling variants. Our rate for SAGA with importance sampling is the current best-known rate for this method, resolving a conjecture by Schmidt et al. (Proceedings of the eighteenth international conference on artificial intelligence and statistics, AISTATS 2015, San Diego, California, 2015). The rates we obtain for minibatch SAGA are also superior to existing rates and are sufficiently tight as to show a decrease in total complexity as the minibatch size increases. Moreover, we obtain the first minibatch SAGA method with importance sampling.

Dynamical Behavior of a Stochastic SIRC Model for Influenza A

In this paper, a stochastic SIRC epidemic model for Influenza A is proposed and investigated. First, we prove that the system exists a unique global positive solution. Second, the extinction of the disease is explored and the sufficient conditions for extinction of the disease are derived. And then the existence of a unique ergodic stationary distribution of the positive solutions for the system is discussed by constructing stochastic Lyapunov function. Furthermore, numerical simulations are employed to illustrate the theoretical results. Finally, we give some further discussions about the system.

Dynamic behaviors of a two-group stochastic SIRS epidemic model with standard incidence rates

This paper deals with a two-group stochastic SIRS epidemic model with standard incidence rates. On one hand, we establish sufficient conditions for the existence of a stationary distribution of the positive solutions to the model by constructing a suitable stochastic Lyapunov function. On the other hand, we show the sufficient conditions for the diseases to be extinct.

Stationary distribution of a stochastic chemostat model with Beddington–DeAngelis functional response

In this paper, we study a stochastic chemostat model with Beddington–DeAngelis functional response. By constructing a suitable stochastic Lyapunov function, we establish sufficient conditions for the existence of a unique ergodic stationary distribution to the model. The existence of a stationary distribution implies stochastic weak stability.

Dynamics of a stochastic multigroup S-DI-A model for the transmission of HIV

ABSTRACT This paper is concerned with a stochastic multigroup S-DI-A model for the transmission of HIV with general incidence rate. We obtain sufficient criteria for the existence and uniqueness of an ergodic stationary distribution of the positive solutions to the model by constructing a proper stochastic Lyapunov function. The stationary distribution implies that the diseases will be persistent in the long term. Some numerical simulations are introduced to illustrate our theoretical result.

Stochastic Characteristics and Optimal Control for a Stochastic Chemostat Model with Variable Yield

In this paper, a stochastic chemostat model with variable yield and Contois growth function is investigated. The yield coefficient depends on the limiting nutrient, and the environmental noises are given by independent standard Brownian motions. First, the existence and uniqueness of global positive solution are proved. Second, by using stochastic Lyapunov function, Itô’s formula, and some important inequalities, stochastic characteristics for the stochastic model are studied, including the extinction of micro-organism, the strong persistence in the mean of micro-organism and, the existence of a unique stationary distribution of the stochastic model. Third, the necessary condition of an optimal stochastic control for the stochastic model is established by Hamiltonian function. In addition, some numerical simulations are carried out to illustrate the theoretical results and the influence of the variable yield on the microorganism.

Dynamics of a stochastic tuberculosis transmission model with treatment at home

In this paper, we consider a stochastic tuberculosis model with two kinds of treatments, that is, treatment at home and treatment in hospital. Firstly, we obtain sufficient conditions for extinction and persistence in the mean of the diseases, then in the case of persistence, we establish sufficient conditions for the existence and uniqueness of an ergodic stationary distribution of the positive solutions to the model by constructing a suitable stochastic Lyapunov function. The theoretical results show that large white noise is beneficial for eliminating TB endemic while small white noise is not.

Stochastic dynamical analysis in a hybrid bioeconomic system with telephone noise and distributed delay

In this paper, a hybrid stochastic prey-predator system with commercial exploitation and distributed delay is established, where combined dynamic effects of telephone noise and Lévy jumps on stochastic population dynamics are taken into accounts. Stochastically ultimate boundedness of the solution is studied. Existence and uniqueness of globally positive solution to the proposed system are investigated. By constructing appropriate stochastic Lyapunov functions, existence of a stationary distribution to the positive solution is discussed. Furthermore, sufficient conditions for permanence in mean and extinction of corresponding population are derived. Numerical simulations are carried out to show consistency with theoretical analysis.

Robust finite‐time H∞ synchronization for uncertain discrete‐time systems with nonhomogeneous Markovian jump: Observer‐based case

SummaryIn this article, the problem of robust finite‐time H∞ synchronization control is investigated for a class of uncertain discrete‐time master‐slave systems with Markovian switching parameters in the observer‐based case. Parameter uncertainties are assumed to be norm‐bounded, and the polyhedral character is utilized to describe the transition probabilities of nonhomogeneous Markov chain. By using stochastic Lyapunov function method and finite‐time analysis techniques, novel sufficient conditions that include the master‐slave parameters are obtained for designing an observer‐based finite‐time H∞ synchronization control law in terms of linear matrix inequalities. The effectiveness of the proposed theoretical scheme is finally demonstrated by some simulations.

ASYMPTOTIC BEHAVIOUR OF THE STOCHASTIC MAKI–THOMPSON MODEL WITH A FORGETTING MECHANISM ON OPEN POPULATIONS

AbstractRumours have become part of our daily lives, and their spread has a negative impact on a variety of human affairs. Therefore, how to control the spread of rumours is an important topic. In this paper, we extend the classic Maki–Thompson model from a deterministic framework to a stochastic framework with a forgetting mechanism, because real-world person-to-person communications are inevitably affected by random factors. By constructing suitable stochastic Lyapunov functions, we show that the asymptotic behaviour of the stochastic rumour model is governed by the basic reproductive number. If this number is less than one, then the solution of the stochastic rumour model oscillates around the rumour-free equilibrium under extra mild conditions, indicating the extinction of the rumour with a probability of one. Otherwise, the solution always fluctuates around the endemic equilibrium under certain parametric restrictions, implying that the rumour will continually persist. In addition, we discuss a possible intervention strategy that stops the spread of rumours by strengthening the intensity of white noise, which is very different from the deterministic rumour model without white noise. Also, numerical simulations are conducted to support our analytical results.

Neuroadaptive finite-time output feedback control for PMSM stochastic nonlinear systems with iron losses via dynamic surface technique

In this paper, an observer-based adaptive neural network finite-time dynamic surface control method is proposed for the position tracking control of PMSM stochastic nonlinear systems with iron losses. First, the finite-time technology is used to realize the fast and effective tracking of the desired signal and make the system have better robust performance. Then, the adaptive neural network (NN) technology and state observer are applied to approximating the uncertain nonlinear functions and estimating the immeasurable states, respectively. And, the dynamic surface control (DSC) technology is used to resolve the “explosion of complexity” problem. In addition, the influence of iron losses and stochastic disturbances in the system is considered, and a quartic stochastic Lyapunov function is established to analyze the stability of the system. Finally, the simulation results show the effectiveness of the proposed method.

Threshold Analysis and Stationary Distribution of a Stochastic Model with Relapse and Temporary Immunity

In this paper, a stochastic model with relapse and temporary immunity is formulated. The main purpose of this model is to investigate the stochastic properties. For two incidence rate terms, we apply the ideas of a symmetric method to obtain the results. First, by constructing suitable stochastic Lyapunov functions, we establish sufficient conditions for the extinction and persistence of this system. Then, we investigate the existence of a stationary distribution for this model by employing the theory of an integral Markov semigroup. Finally, the numerical examples are presented to illustrate the analytical findings.

The effect of environmental noise on threshold dynamics for a stochastic viral infection model with two modes of transmission and immune impairment

Abstract A stochastic viral infection model with two modes of transmission and immune impairment is proposed in this paper, which contains four variables; uninfected cells, infected cells, virus particles and cytotoxic T lymphocytes. We first investigate the exponential stability of the model, and further give sufficient conditions for the extinction and persistence of the disease. By constructing a suitable stochastic Lyapunov function, we then prove the existence of a unique ergodic stationary distribution of the model. More importantly, a stochastic threshold R 0 W is presented, which plays the similar role as R 0 in determining the persistence and extinction of the infected cells. As an application of the method proposed in this paper, the existence of an ergodic stationary distribution and a stochastic positive periodic solution of the model under the influence of colored noise and seasonal fluctuations are studied respectively. Finally, we carry out some numerical simulations, showing that environmental noise may suppress the spread of disease.

Dynamics of a multigroup SIQS epidemic model under regime switching

In this paper, we study the dynamical behavior of a multigroup SIQS epidemic model, which is formulated as a piecewise deterministic Markov process. Sufficient conditions for extinction and persistence in the mean of the diseases are obtained. In addition, in the case of diseases persistence, we establish sufficient conditions for the existence of positive recurrence of the solutions to the model by using the stochastic Lyapunov function with regime switching.

On constrained Volterra cubic stochastic operators

We consider constrained Volterra cubic stochastic operators and construct several Lyapunov functions for the constrained Volterra cubic stochastic operators. We prove that such kind operators do not have periodic trajectories. Finally, we show that the set of all constrained Volterra cubic stochastic operators is a convex compact set and find the extreme points of this set.

A stochastic SIRS epidemic model with logistic growth and general nonlinear incidence rate

In this paper, we consider a stochastic SIRS epidemic model with logistic growth and general nonlinear incidence rate. We establish sufficient conditions for the existence and uniqueness of an ergodic stationary distribution of the positive solutions to the stochastic model by constructing a suitable stochastic Lyapunov function, which provides us a good description of persistence. We find that the hypothetical conditions on the nonlinear function are relative weak and valid for many forms of incidence rate.

Stationary distribution of stochastic NP ecological model under regime switching

Marine ecosystem is inevitably disturbed by random factors. This article discusses a nutrient–phytoplankton (NP) model with TPP which is disturbed by both white and telegraph noises. Firstly, we get sufficient conditions for the existence of ergodic stationary distribution by constructing a suitable stochastic Lyapunov function with regime switching. And then we establish sufficient conditions for extinction of the phytoplankton. Finally, the results of this paper are verified by numerical simulation.