Conditions For Stochastic Stability Research Articles

Sliding Mode Control for Fuzzy Networked Semi-Markov Switching Models Under Cyber Attacks

Sliding mode control (SMC) is investigated for nonlinear networked stochastic switching systems subject to semi-Markov parameters, in which the model is formulated based on the Takagi-Sugeno (T-S) fuzzy strategy. The occurrences of actuator failures and deception attacks are considered within the framework of networked semi-Markov switching systems. A common sliding surface is designed to prevent instability caused by repeated jumps. Based on the generally uncertain transition rate, sufficient conditions for stochastic stability of the corresponding system are presented. Moreover, in the consideration of actuator failures and deception attacks, the feasibility of the SMC law drives the states toward the specified sliding region. The presented single link robot arm model demonstrates the effectiveness of the theoretical findings.

Robust stability and robust H2 performance of discrete‐time Markovian jump linear systems with uncertain transition probabilities

AbstractIn this article, the robust stochastic stability problem and robust performance problem of discrete‐time Markovian jump linear systems with uncertain transition probabilities are investigated. Based on the Kronecker product and analysis, perturbation bounds are obtained for the robust stochastic stability of the investigated system by transforming the problem into checking whether a class of uncertain matrices is nonsingular. Furthermore, a sufficient and necessary condition for robust stochastic stability is derived. Similarly, by converting the problem to the judgment of the nonsingularity of uncertain matrices, the exact upper and lower bounds for robust performance are obtained. Then, a sufficient and necessary condition for guaranteeing robust performance is obtained. All the results are given in terms of the structured singular value. Finally, numerical examples are presented to show the effectiveness of the proposed results.

Adaptive event‐triggered resilient stabilization for nonlinear semi‐Markov jump systems subject to DoS attacks

AbstractThis article investigates the problem of adaptive event‐triggered resilient stabilization for nonlinear semi‐Markov jump systems with DoS attacks. In this article, both DoS attacks and event‐triggered transmission strategy (ETTS) are considered. The DoS attacks may disturb the event‐triggered strategy, which implies that the original inequality condition of ETTS will no longer be satisfied. Different from the traditional ETTS, the adaptive ETTS is designed to reduce bandwidth usage. Based on the adaptive ETTS and resilient controller, the sufficient condition for stochastic stability of the closed‐loop system is proposed. Then, the calculation method of controller and observer parameters is given in the form of linear matrix inequality. Finally, the proposed adaptive event‐triggered resilient controller is applied to a single‐link robot arm model, which can show effectiveness of the proposed method.

A stochastic two-dimensional intelligent driver car-following model with vehicular dynamics

The law of vehicle movement has long been studied under the umbrella of microscopic traffic flow models, especially the car-following (CF) models. These models of the movement of vehicles serve as the backbone of traffic flow analysis, simulation, autonomous vehicle development, etc. Two-dimensional (2D) vehicular movement is basically stochastic and is the result of interactions between a driver’s behavior and a vehicle’s characteristics. Current microscopic models either neglect 2D noise, or overlook vehicle dynamics. The modeling capabilities, thus, are limited, so that stochastic lateral movement cannot be reproduced. The present research extends an intelligent driver model (IDM) by explicitly considering both vehicle dynamics and 2D noises to formulate a stochastic 2D IDM model, with vehicle dynamics based on the stochastic differential equation (SDE) theory. Control inputs from the vehicle include the steer rate and longitudinal acceleration, both of which are developed based on an idea from a traditional intelligent driver model. The stochastic stability condition is analyzed on the basis of Lyapunov theory. Numerical analysis is used to assess the two cases: (i) when a vehicle accelerates from a standstill and (ii) when a platoon of vehicles follow a leader with a stop-and-go speed profile, the formation of congestion and subsequent dispersion are simulated. The results show that the model can reproduce the stochastic 2D trajectories of the vehicle and the marginal distribution of lateral movement. The proposed model can be used in both a simulation platform and a behavioral analysis of a human driver in traffic flow.

Robust control of uncertain time-delay Markovian jump quaternion-valued neural networks subject to partially known transition probabilities: direct quaternion method.

This paper addresses the issue of robust stochastic stabilization and control of uncertain time-delay Markovian jump quaternion-valued neural networks (MJQVNNs) subject to partially known transition probabilities. First, the direct quaternion method is proposed to analyse the MJQVNNs, which is different from some conventional methods in that the former is without any decomposition for systems. After that, in order to estimate the upper bound of the derivative of the constructed Lyapunov-Krasovskii functional(LKF) more accurately, the real-valued convex inequality is extended to quaternion domain. Then, by designed the mode-dependent state feedback controllers, the robust stochastic stabilization conditions of MJQVNNs are given for the admissible uncertainties, and reduce the influence of input disturbance on the controlled output to a specified performance level. Lastly, two numerical examples are given to illustrate the effectiveness of the proposed method.

Sampled-Data Synchronization of Stochastic Markovian Jump Neural Networks With Time-Varying Delay.

In this article, sampled-data synchronization problem for stochastic Markovian jump neural networks (SMJNNs) with time-varying delay under aperiodic sampled-data control is considered. By constructing mode-dependent one-sided loop-based Lyapunov functional and mode-dependent two-sided loop-based Lyapunov functional and using the Itô formula, two different stochastic stability criteria are proposed for error SMJNNs with aperiodic sampled data. The slave system can be guaranteed to synchronize with the master system based on the proposed stochastic stability conditions. Furthermore, two corresponding mode-dependent aperiodic sampled-data controllers design methods are presented for error SMJNNs based on these two different stochastic stability criteria, respectively. Finally, two numerical simulation examples are provided to illustrate that the design method of aperiodic sampled-data controller given in this article can effectively stabilize unstable SMJNNs. It is also shown that the mode-dependent two-sided looped-functional method gives less conservative results than the mode-dependent one-sided looped-functional method.

Control-Aware Transmission Scheduling for Industrial Network Systems Over a Shared Communication Medium

In this article, we consider the design of dynamic transmission scheduling policies for the industrial network systems sharing scarce communication resources. Only a few subsystems can obtain channel access for information updates to close their control loops at each time step, which highlights the necessity of designing optimal transmission scheduling schemes to achieve a minimum average linear quadratic cost of the industrial network systems. We first propose a greedy state-error-dependent scheduling (SES) policy based on the one-step expected profit and discuss its stability employing the Lyapunov function method. After formulating the scheduling optimization as a Markov decision process problem and relaxing with a soft constraint, we develop a heuristic near-optimal solution that guarantees the optimality of certainty equivalent controllers, namely, Whittle’s index-inspired error-dependent scheduling (WIES). A stochastic stability condition of WIES is further given based on f-ergodicity. Due to low computational complexity and ease of implementation, the proposed schemes are suitable for large-scale heterogeneous industrial network systems. Finally, simulation results show that the proposed policies outperform the existing round-robin, holding-time-prioritized, and error-aware scheduling schemes.

Composite event-triggered disturbance-observer-based control and passive control for semi-Markovian jump nonlinear systems with actuator saturation

In this paper, the composite control problem of continuous time semi-Markovian jump system with actuator saturation, multiple disturbances and unknown nonlinearity is investigated. A stochastic stability condition and a lower bound of the time between two consecutive events are derived to avoid Zeno behaviour, based on the combination of disturbance-observer-based control and event-triggered control. In addition, the passive control of the system based on the disturbance observer is studied. Finally, numerical simulation proves that the designed composite controller can effectively save communication resources, and can maintain the system to meet the passive performance indicators of the obtained closed-loop system under the influence of multiple disturbances and actuator saturation.

Resilient sliding mode control of multiple autonomous underwater vehicles under stochastic DoS attack

This paper is concerned with the security sliding mode-based leader-following control of multiple autonomous unmanned underwater vehicles (AUVs) system subject to stochastic DoS attacks. Firstly, a Bernoulli variable is used to describe the phenomenon of stochastic DoS attacks between controllers and actuators, and the attack probability is known for controller design. Secondly, the closed-loop system is established by taking the integral sliding mode control scheme and stochastic DoS attacks into account. Then, the conditions of stochastic stability for system are given by using the stochastic systems analysis method and the model decomposition technique. In the simulation part, a group of AUVs models is used to verify the validity of the proposed state consensus control scheme. The simulation example shows that all the state can achieve consensus even in presence of randomly launched DoS attack.

Decentralized dynamic event‐triggered control for networked control systems under unreliable communication network and actuator saturation

AbstractIn this paper, the stochastic stability of networked control systems (NCSs) with nonlinear perturbation and parameter uncertainties is studied. A decentralized dynamic event triggering scheme (DDETS) is introduced to reduce energy consumption and network transmission burden. Each channel can decide whether to transmit signals sampled by corresponding sensors through this mechanism. Due to the unreliability of communication network, a new mathematical model of NCSs based on multi‐channel fading and hybrid cyber attacks is established, and the actuator is constrained by saturation. A sufficient condition for stochastic stability based on static controller is derived through Lyapunov stability theory, and two numerical examples are given to demonstrate the effectiveness of the method.

Stochastic replicator dynamics and evolutionary stability.

To develop the concept of evolutionary stability in a stochastic environment, we investigate the continuous-time dynamics of a two-phenotype linear evolutionary game with generally correlated random payoffs in pairwise interactions. By using the Gram-Schmidt orthogonalization procedure and Itô's formula, we deduce a stochastic differential equationfor the phenotype frequencies that extends the replicator equation, called the stochastic replicator equation. We give conditions for stochastic stability of a fixation state or a constant interior equilibrium point with respect to the stochastic dynamics of the two phenotypes. We show that, if a fixation state is stochastically stable, then the pure strategy corresponding to this fixation state must be stochastically evolutionarily stable with respect to mixed strategies. However, this is not the case for a mixed strategy that corresponds to a stochastically stable constant interior equilibrium point with respect to the two phenotypes.

Stochastic evolutionary stability in matrix games with random payoffs.

Evolutionary game theory and the concept of an evolutionarily stable strategy have been not only extensively developed and successfully applied to explain the evolution of animal behavior, but also widely used in economics and social sciences. Recently, in order to reveal the stochastic dynamical properties of evolutionary games in randomly fluctuating environments, the concept of stochastic evolutionary stability based on conditions for stochastic local stability for a fixation state was developed in the context of a symmetric matrix game with two phenotypes and random payoffs in pairwise interactions [Zheng et al., Phys. Rev. E 96, 032414 (2017)2470-004510.1103/PhysRevE.96.032414]. In this paper, we extend this study to more general situations, namely, multiphenotype symmetric as well as asymmetric matrix games with random payoffs. Conditions for stochastic local stability and stochastic evolutionary stability are established. Conditions for a fixation state to be stochastically unstable and almost everywhere stochastically unstable are distinguished in a multiphenotype setting according to the initial population state. Our results provide some alternative perspective and a more general theoretical framework for a better understanding of the evolution of animal behavior in a stochastic environment.

STABILITY OF CONTROLLED STOCHASTIC DYNAMIC SYSTEMS OF RANDOM STRUCTURE WITH MARKOV SWITCHES AND POISSON PERTURBATIONS

Lyapunov’s second method is used to study the problem of stability of controlled stochastic dynamical systems of random structure with Markov and Poisson perturbations. Markov switches reflect random effects on the system at fixed points in time. Poisson perturbations describe random effects on the system at random times. In both cases there may be breaks in the phase trajectory of the first kind. The conditions for the coefficients of the system are written, which guarantee the existence and uniqueness of the solution of the stochastic system of a random structure, which is under the action of Markov switches and Poisson perturbations. The differences between these systems and systems that do not contain internal perturbations in the equation, which cause a change in the structure of the system, and external perturbations, which cause breaks in the phase trajectory at fixed points in time, are discussed. The upper bound of the solution for the norm is obtained. The definition of the discrete Lyapunov operator based on the system and the Lyapunov function for the above-mentioned systems is given. Sufficient conditions of asymptotic stochastic stability in general, stability in l.i.m. and asymptotic stability in the l.i.m. for controlled stochastic dynamic systems of random structure with Markov switches and Poisson perturbations are obtained. A model example that reflects the features of the stability of the solution of a system with perturbations is considered: the conditions of asymptotic stability in the root mean square as a whole are established; the conditions of exponential stability and exponential instability are discussed. For linear systems, the necessary and sufficient stability conditions are determined in the example, based on the generalized Lyapunov exponent.

Dynamic event‐triggered control for Markovian jump linear systems with time‐varying delay

AbstractIn this article, the problem of dynamic event‐triggered control (ETC) for Markovian jump linear systems (MJLSs) with time‐varying delay is investigated. First, the original system is transformed into an interconnection system with input–output form. Then, the dynamic ETC is proposed to reduce the transmission frequency. Meanwhile, the conditions of stochastic stability are proposed by constructing a novel stochastic Lyapunov–Krasovskii functional and using the scaled stochastic small‐gain theorem. Further, the controller design method is presented to stabilize the closed‐loop MJLSs with time‐varying delay. The merit of this article lies in that the conservatism for stability results is reduced and the transmission efficiency is improved by reducing the event‐triggered times, which is made possible by designing dynamic ETC. Finally, two numerical examples are provided to demonstrate the advantages and effectiveness of the proposed methods.

Improved H∞ deconvolution filter design for Lur’e singular Markovian jump systems based on sector bounded condition

This paper investigates the deconvolution filter for Lur’e time-varying delays singular Markovian jump systems. Firstly, by establishing mode-dependent Lyapunov–Krasovskill functional and considering sector bounded conditions, stochastic stability conditions and H∞ performance index are obtained for Lur’e singular Markovian jump systems. Secondly, both regularity and impulse-freeness are acquired by singular value decomposition technique. Thirdly, deconvolution filter is realized by virtue of linear matrix inequalities. Ultimately, the utility of this present method is validated by a numerical example and an oil catalytic cracking process.

Finite-time Stochastic Stability and Stabilization for Uncertain Discrete-time Stochastic Systems with Time-varying Delay

This paper deals with the problems of finite-time stochastic stability and stabilization for discrete-time stochastic systems with parametric uncertainties and time-varying delay. Using the Lyapunov-Krasovskii functional method, some sufficient conditions of finite-time stochastic stability for a class of discrete-time stochastic uncertain systems are established in term of matrix inequalities. Then, a new criterion is proposed to ensure the closed-loop system is finite-time stochastically stable. The controller gain is designed. Finally, two numerical examples are given to illustrate the effectiveness of the proposed results.

Event‐triggered feedback control for delayed singular jump systems based on sampled observer and exponential detector

AbstractThis paper is intended to investigate the problem of event‐triggered feedback control for time‐varying delays singular jump systems based on sampled observer and exponential detector. By virtue of the designed event‐triggered control strategies, the state estimate signals are transmitted to the controller under the networked communication environment and aperiodic sampling scheme. The driving threshold is devised by exponentially decreasing function, which is used to guarantee not only the validity of event‐triggered strategy but also the admissibility of the delayed singular jump systems. Neoteric stochastic admissibility, stabilization and performance conditions are obtained in terms of linear matrix inequalities. Simulation examples including an oil catalytic cracking process are utilized to illustrate the validity and superiority of the proposed event‐triggered feedback control technique.

Local stochastic stability of SIRS models without Lyapunov functions

Most of stability analysis for stochastic epidemiological models involve Lyapunov functions. This work shows how sufficient conditions for the local stochastic asymptotic stability of a nonlinear system can be derived from the stability analysis of an ordinary linear system. In the particular stochastic SIR/SIRS models proposed here to illustrate the technique, the stability study of the obtained ordinary systems reduces to calculate the spectrum of the governing matrix.

A stochastic SEIHR model for COVID-19 data fluctuations.

Although deterministic compartmental models are useful for predicting the general trend of a disease’s spread, they are unable to describe the random daily fluctuations in the number of new infections and hospitalizations, which is crucial in determining the necessary healthcare capacity for a specified level of risk. In this paper, we propose a stochastic SEIHR (sSEIHR) model to describe such random fluctuations and provide sufficient conditions for stochastic stability of the disease-free equilibrium, based on the basic reproduction number that we estimated. Our extensive numerical results demonstrate strong threshold behavior near the estimated basic reproduction number, suggesting that the necessary conditions for stochastic stability are close to the sufficient conditions derived. Furthermore, we found that increasing the noise level slightly reduces the final proportion of infected individuals. In addition, we analyze COVID-19 data from various regions worldwide and demonstrate that by changing only a few parameter values, our sSEIHR model can accurately describe both the general trend and the random fluctuations in the number of daily new cases in each region, allowing governments and hospitals to make more accurate caseload predictions using fewer compartments and parameters than other comparable stochastic compartmental models.

Aperiodic sampled‐data controller design for stochastic Markovian jump systems and its application

AbstractIn this paper, aperiodic sampled‐data controller design problem is solved for stochastic Markovian jump systems (SMJSs). By introducing mode‐dependent one‐sided loop‐based Lyapunov functional and mode‐dependent two‐sided loop‐based Lyapunov functional and using Itô formula, two different stochastic stability conditions for SMJSs under aperiodic sampled‐data control are presented, respectively. Based on these two stability conditions, corresponding mode‐dependent aperiodic sampled‐data controller is designed for SMJSs. Finally, practical aircraft model are applied to illustrate the effectiveness of the method. It is also shown that mode‐dependent two‐sided loop‐based Lyapunov functional can obtain a larger upper bound of the sampling interval than mode‐dependent one‐sided loop‐based Lyapunov functional.