Conditions For Stochastic Stability Research Articles

Stochastic stabilization based on kinetic-potential energy shaping for stochastic mechanical port-Hamiltonian systems

This paper extends a kinetic-potential energy shaping method to stochastic mechanical port-Hamiltonian systems. The kinetic-potential energy shaping brings a new class of Lyapunov function candidates involving a cross term of the position and the momentum without solving partial differential equations for deterministic port-Hamiltonian systems. However, the conventional kinetic-potential energy shaping does not necessarily work for stochastic port-Hamiltonian systems due to energy increase by stochastic noise. Therefore, we first provide a modification properly compensated by using stochastic generalized canonical transformations. Then, two stochastic stability results are presented. The first result shows a necessary condition for stochastic asymptotic stability for an equilibrium state. The other one shows a necessary condition for stochastic bounded stability for a target state, which is not necessarily an equilibrium point.

Event-triggered interval type-2 fuzzy control of networked systems with extended dissipative under Markovian switching

In this technical note, the problem of the extended dissipative control for interval type-2 fuzzy model with nonlinear network systems in the event-triggered framework is investigated. A novel dynamic event-triggered strategy is developed to ascertain the instants of activation. It has the advantage of reducing data transmission and optimising the utilisation of communication resources. On the basis of the proposed mechanism, the stochastic Lyapunov function including internal dynamic variables is constructed. Based on this function, coupled with the inequality scaling technique and the application of Dynkin's formula, we derive a sufficient condition for the system's stochastic asymptotic stability and extended dissipativity. The design of the dynamic event-triggered extended dissipative controller becomes challenging because the new sufficient condition is nonlinear for some matrix variables. However, utilising the Projection Lemma and the co-design approach, such a controller can be developed based on the obtained analysis result. The methodology is demonstrated through a rigorous numerical simulation case study.

Partial-Information-Based Non-Fragile Intermittent Estimator for Microgrids With Semi-Aperiodic DoS Attacks: Gain Stochastic Float

This paper mainly focuses on the problem of nonfragile intermittent estimate tracking in microgrid systems with denial of service (DoS) attacks and gain fluctuation. Different from the common DC/AC microgrid system, the microgrid system studied in this paper involves large grid control and power line communication technology, which is more in line with the actual operating environment. Simultaneously, aiming at the occurrence of network-induced DoS attacks, an intermittent estimator based on partial measure information under cyber-attacks is designed. More importantly, instead of the existing single aperiodic or periodic DoS attacks, the DoS attacks scheme presented in this paper is semi-periodic DoS attacks whose the attack range is aperiodic and the attack width is certain. Furthermore, a variable obeying the Bernoulli distribution is proposed to characterize the randomly occurring gain fluctuation, in which the no-fragile observation is stochastic uncertain. Then, two sufficient conditions of stochastic stability of observation error system of microgrid are devised with the assistance of two different Lyapunov theories. Finally, the rationality of the developed theoretical method is verified by a numerical example.

Asynchronous Fault Detection for Memristive Neural Networks With Dwell-Time-Based Communication Protocol.

This article studies the asynchronous fault detection filter problem for discrete-time memristive neural networks with a stochastic communication protocol (SCP) and denial-of-service attacks. Aiming at alleviating the occurrence of network-induced phenomena, a dwell-time-based SCP is scheduled to coordinate the packet transmission between sensors and filter, whose deterministic switching signal arranges the proper feedback switching information among the homogeneous Markov processes (HMPs) for different scenarios. A variable obeying the Bernoulli distribution is proposed to characterize the randomly occurring denial-of-service attacks, in which the attack rate is uncertain. More specifically, both dwell-time-based SCP and denial-of-service attacks are modeled by means of compensation strategy. In light of the mode mismatches between data transmission and filter, a hidden Markov model (HMM) is adopted to describe the asynchronous fault detection filter. Consequently, sufficient conditions of stochastic stability of memristive neural networks are devised with the assistance of Lyapunov theory. In the end, a numerical example is applied to show the effectiveness of the theoretical method.

Observer‐based sampled‐data control for networked systems with consecutive packet dropouts

AbstractThis paper investigates the observer‐based sampled‐data control problem for networked systems with consecutive packet dropouts. A new consecutive packet dropout model is established to describe the phenomenon of packet dropout in both the sensor‐controller (S‐C) and controller‐actuator (C‐A) channels. First, the considered system with observer‐based control scheme is studied in the discrete‐time domain by using equivalent discretization transformation. In order to simplify subsequent analysis, matrix exponential computation is employed to transform the discrete‐time system into a more tractable form. Then, the proposed new consecutive packet dropout model is taken into consideration to establish the stochastic mean square exponential stability condition. Furthermore, a novel two‐step synthesis approach is used to design the observer‐based controller. Finally, the effectiveness of the proposed observer‐based sampled‐data control approach is verified using a numerical example. In addition, the advantages of the proposed approach are validated through comparisons.

Non-fragile guaranteed cost control for networked nonlinear Markov jump systems under multiple cyber-attacks

This paper is concerned with the problem of non-fragile guaranteed cost control (GCC) for networked nonlinear Markov jump systems subject to multiple cyber-attacks, which are characterized by Takagi–Sugeno (T–S) fuzzy model with time-varying delay. Specifically, a variety of cyber-attacks, including deception attacks and Denial-of-Service (DoS) attacks, are considered, which occur in the forward and feedback communication links, respectively. To achieve stochastic stability under guaranteed cost function (GCF), the paper proposes a Lyapunov–Krasovskii (L–K) function approach. The approach derives sufficient conditions for stochastic stability, and obtains non-fragile controller gains and the uniform upper bound of the GCF using linear matrix inequalities (LMIs) technique. Finally, the effectiveness of the proposed algorithm is evaluated by simulation experiment.

Bifurcation Analysis of a Wind Turbine Generator Drive System with Stochastic Excitation Under Both Displacement and Velocity Delayed Feedback

In this paper, a wind turbine generator drive system with stochastic excitation under both displacement and velocity delayed feedback is considered. Firstly, the center manifold method is used to approximate the delay term of the system, so that the Itô-stochastic differential equation can be obtained by random average method. Through the maximal Lyapunov exponential method, the local stochastic stability and random D-bifurcation conditions of the system are obtained. Secondly, it is verified that the increase of noise intensity and delay value induces the occurrence of random P-bifurcation of the system through Monte Carlo numerical simulations. In addition, the theoretical chaos threshold of the system is derived by the random Melnikov method. The results show that the chaos threshold decreases as the noise intensity increases, and the increase in time delay leads to a delay in the chaotic behavior of the system. Finally, the correctness and effectiveness of the chaos-theoretic analysis are verified based on the one-parameter bifurcation diagrams and the two-parameter bifurcation diagrams.

Stabilization and Reconstruction of Sampled-Data Boolean Control Networks Under Noisy Sampling Interval

In this paper, we consider stabilization and reconstruction of sampled-data Boolean control networks (BCNs) under noisy sampling interval. A sampled-data BCN under noisy sampling interval is first converted into a probabilistic Boolean network (PBN). We then obtain some necessary and sufficient conditions for global stochastic stability of the considered sampled-data BCN under two types of noisy sampling intervals. However, in analyzing the stochastic stability of large-scale sampled-data BCNs under noisy sampling interval, using the above necessary and sufficient conditions leads to huge computational cost. Therefore, for a large-scale sampled-data BCN, we have to transform it into a size-reduced probabilistic logical network. Then by studying the stochastic stability of the probabilistic logical network, some sufficient conditions for global stochastic stability of the large-scale sampled-data BCN are obtained. Moreover, based on the given steady-state probabilities of the transformed PBN, the reconstruction problem of sampled-data BCNs under noisy sampling interval can be well solved as a linear programming (LP) problem.

Protocol-Based Control for Discrete-Time Positive Markovian Switching Models With Deception Attacks

This brief investigates the problem of protocol-based control for positive Markovian switching models subject to actuator faults and deception attacks. Bernoulli distribution is adopted to depict random deception attacks. Considering the positivity of Markovian switching models, an event-triggered protocol is constructed based on a 1-norm to be related to the error signal and the state signal. In order to handle actuator faults and random cyber attacks, exponentially stochastic stability conditions are established under the event-triggered protocol by developing a linear copositive Lyapunov function approach. Furthermore, a non-fragile control law combined with the event-triggered protocol is proposed such that exponential stochastic stability of the corresponding system is achieved on the basis of matrix decomposition strategy and linear programming. Finally, a data communication network model is provided to demonstrate the effectiveness of the proposed controller design.

Moment Lyapunov exponents and stochastic nonstability of a circular cylindrical shells

The paper investigates the stochastic stability conditions of circular cylindrical shells, compressed by time-dependent stochastic membrane forces. The main contribution of the paper is the determination of the moment Lyapunov exponents of the linear cylindrical shell for the first time. The moment and almost-sure stochastic stability of a three-degree-of-freedom coupled continuous system, under parametric excitation of white noise, are investigated. It is assumed that the system possesses a small damping. This type of a shell structure model requires a specific scheme for determining the moment Lyapunov exponents. The system of stochastic differential equations of motion is decoupled by using the contact transformation method with the obtained appropriate symplectic matrix of the system. Then a scheme for determining the moment Lyapunov exponents is presented which contains an analytical procedure. The largest Lyapunov exponent is calculated through its relation with the moment Lyapunov exponent. The moment and almost-sure stability boundaries are obtained analytically. These results are important and useful in engineering applications as an example of a procedure for dynamic analysis of mechanical systems with coupled stochastic oscillators.

Stability analysis for time-varying positive systems with stochastic impulses

AbstractThis article addresses the stochastically exponential stability and mean stability of positive time-varying systems with stochastic impulses. The term ‘stochastic impulse’ means the randomness of impulsive densities or intensities. More specifically, the impulsive maps are not unique and the impulsive intensities are independent random variables with different distributions. The occurrence instants of impulses are restricted by several different processes, e.g. a mode-dependent average impulsive interval, a Markov chain, a Poisson process and a renewal process. Using a time-varying copositive Lyapunov function and stochastic analysis theory, several stochastic stability conditions are given. Finally, an example with four cases is presented to show the effectiveness of the proposed results.

On the driver's stochastic nature in car-following behavior: Modeling and stabilizing based on the V2I environment

<abstract> <p>The driver's stochastic nature is one of the important causes of traffic oscillation. To better describe the impact of the driver's stochastic characteristics on car-following behavior, we propose a stochastic full velocity difference model (SFVDM) considering the stochastic variation of the desired velocity. In order to mitigate traffic oscillation caused by driving stochasticity, we further propose a stable speed guidance model (S-SFVDM) by leveraging vehicle-to-infrastructure communication. Stochastic linear stability conditions are derived to demonstrate the prominent influence of the driver's stochasticity on the stability of traffic flow and the improvement of traffic flow stability by the proposed guidance strategy, respectively. We present numerical tests to demonstrate the effectiveness of the proposed models. The results show that the SFVDM can capture the traffic oscillation caused by the driver's stochastic desired velocity and reproduce the same disturbance growth pattern as in the field experiment. The results also indicate that the S-SFVDM can significantly expand the stable area of traffic flow to decrease the negative impact on traffic flow stability caused by the driver's stochastic nature.</p> </abstract>

Adaptive control of T-S fuzzy systems with Markov switching parameters through observer-based sliding mode approach

This paper addresses the problem of observer-based adaptive sliding mode control (SMC) for nonlinear Markov jump systems (MJSs), using a Takagi-Sugeno (T-S) fuzzy model with the premise variables of fuzzy rules depend on system state. Firstly, an integral sliding surface is designed based on fuzzy observer system; Secondly, sufficient conditions for stochastic stability and $ H_{\infty} $ perturbation attenuate level are provided for the obtained sliding mode dynamics and error systems using linear matrix inequality techniques. Furthermore, an adaptive SMC law is combined with the observed state variables to ensure finite time reachability of the sliding surface. Lastly, the theory is validated with simulations based on a practical example.

Membership-Function-Dependent Design of Quantized Fuzzy Sampled-Data Controller for Semi-Markovian Jump Systems With Actuator Faults

This article concerns a fuzzy sampled-data controller for nonlinear semi-Markovian jump systems (SMJS) via the fuzzy dependent stochastic looped-Lyapunov functional (FSLF) approach. To do this, with the help of the Takagi–Sugeno (T–S) method, the stochastic jump parameters, actuator faults, and logarithmic quantizer are all considered in a unified framework. The stochastic parameters are generated by the semi-Markov process, whose transition rates depend upon the sojourn time. The nonlinear SMJS depicts real-time models subject to unpredictable structural changes, such as single-link robotic arm (S-LRA) systems. A key issue under consideration is how to design a fuzzy sampled-data controller to ensure stochastic stability for the nonlinear SMJS in the presence of actuator faults and state quantization. For this purpose, by using the weak infinitesimal operator of constructed FSLF, sufficient conditions are determined to realize the stochastic stability of the proposed nonlinear systems. Then, the stochastic stability conditions for the proposed sampled-data controller scheme can be derived under a linear matrix inequality framework. The validity of the theoretical findings is verified by Rossler’s chaotic systems. The S-LRA model is demonstrated to illustrate the applicability and efficacy of the proposed controller scheme.

Stochastic fuzzy predictive fault‐tolerant control for time‐delay nonlinear system with actuator fault under a certain probability

AbstractOwing to the characteristics of time‐delay, fault randomness, uncertainty, nonlinearity and unknown interference in industrial production processes, a stochastic fuzzy predictive fault‐tolerant control algorithm is put forward based on the traditional fault‐tolerant control algorithm. The main idea of this algorithm is to integrate actuator fault into the established Takagi‐Sugeno (T‐S) model to solve actuator fault under a certain probability for the nonlinear industrial processes with time‐varying delays by combining stochastic control theory and relevant theorems. First, the actuator fault under a certain probability is considered as a T‐S model, which can be used as a description of the fault situation in the nonlinear industrial processes. Afterwards, the augmented state space model is established through integrating state deviation and output tracking error. Second, the stochastic fuzzy predictive fault‐tolerant control law can be designed on the basis of the augmented model. Meanwhile, the actuator control mode for different faults is given. If the actuator fault is under a small probability, the control mode is switched to normal control; if the actuator fault is under a large probability, the control mode is switched to fault‐tolerant control. To this end, this control algorithm can reduce energy consumption and raw material consumption. On this basis, the designed control law can be solved by using the given stochastic stability conditions in terms of linear matrix inequality. Finally, the temperature control process of a strongly nonlinear continuous stirred tank reactor is selected as a simulation object to prove the feasibility and effectivity of this algorithm.

Stochastic Stability of Markovian Neural Networks With Generally Hybrid Transition Rates.

This article studies the problem of the stability for Markovian neural networks (MNNs) with time delay. The transition rate is considered to be generally hybrid, which treats those existing ones as its special cases. The introduced generally hybrid transition rates (GHTRs) make these systems more general and practical. Apropos of the GHTRs, a double-boundary approach rather than the traditional estimation method is introduced to make full use of the error information in GHTRs. In order to fully capture system information, a parameter-type-delay-dependent-matrix (PTDDM) approach is proposed, in which the PTDDM approach removes some zero components on slack matrices in previous works. Thus, the PTDDM approach can fully link the relationship among time delay and state-related vectors. Based on these ingredients, a novel stochastic stability condition is proposed for MNNs with GHTRs. A numerical example is illustrated to demonstrate the effectiveness of the proposed approaches.

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.