Stochastic Stability Research Articles (Page 1)

This study investigates the existence, stability, and controllability of multi-term stochastic fractional impulsive differential equations. By employing the contraction mapping principle, sufficient conditions ensuring stochastic stability are rigorously established. Two classes of systems—linear and nonlinear are analyzed in detail. Furthermore, the controllability of these sys-tems is demonstrated using the Gramian operator approach. To validate the theoretical findings, numerical simulations are performed in MATLAB, illustrating the effectiveness and applicability of the proposed results.

This article investigates the state estimation and security control problem for discrete-time dual-rate cyber-physical systems (CPSs) under denial-of-service (DoS) attacks. The asynchrony predicament between different signals of dual-rate CPSs, exacerbated by the impact of cyber attacks on the sensor-to-controller channel, substantially increases the complexity of state estimation and control processes. Based on the signal-to-interference-plus-noise ratio and two-channel probability descriptions, an improved sample-and-hold (SAH) estimator is applied to dual-rate CPSs, ensuring favorable state estimates while enduring low-frequency sampling and DoS attacks. Furthermore, to solve the performance degradation problem posed by the SAH algorithm, an alternating-sampling-prediction (ASP)-based estimation method is proposed. At each fast-update moment, the predictor generates virtual outputs. The estimator can reconstruct complete state information by alternately using incomplete sampling data and iterative predictive information. Compared with the SAH method, the proposed ASP-based approach significantly enhances the control performance of dual-rate CPSs. Building on two valid estimation methods, the corresponding security control inputs are designed, guaranteeing both ideal control performance and resilience against attacks. Using convex optimization analysis, both estimator and controller gains are calculated to realize the stochastic stability of closed-loop dual-rate CPSs. Finally, the effectiveness and intercomparisons of the two estimation methods are shown by simulating a satellite yaw-angle control system and a quadrotor landing control experiment.

Abstract This paper explores the intersection of stochastic stability and extended dissipative analysis in semi‐Markov jump systems under dynamic adaptive event‐triggering mechanisms, considering nonlinearity and time‐varying delay effects. Although Markov jump systems are widely used in various fields, they suffer limitations due to the exponential distribution of state transition times. To address this, semi‐Markov jump systems have been proposed, accommodating nonexponential distributions and incorporating temporal aspects into state transitions. Four key innovations are presented: a creatively devised dynamic adaptive event‐triggering mechanisms, consideration of time delay and nonlinearity in semi‐Markov jump systems, the adoption of a more simple method to decouple coupling terms, and the use of semi‐Markov jump switching law. Simulation experiments indicate that the achieved theoretical conditions are feasible for realizing extended dissipative and stochastic stability.

In real-world applications, finite time convergence to a desired Lyapunov stable equilibrium is often necessary. This notion of stability is known as finite time stability and refers to systems in which the state trajectory reaches an equilibrium in finite time. This paper explores the notion of finite time stability in probability within the context of nonlinear stochastic dynamical systems. Specifically, we introduce sufficient conditions based on Lyapunov methods, utilizing Lyapunov functions that satisfy scalar differential inequalities involving fractional powers for guaranteeing finite time stability in probability. Then, we address the finite time optimal control problem by developing a framework for designing optimal feedback control laws that achieve finite time stochastic stability of the closed-loop system using a Lyapunov function that also serves as the solution to the steady-state stochastic Hamilton–Jacobi–Bellman equation.

This article addresses the output feedback control problem for a specific class of discrete-time fuzzy singularly perturbed systems subjected to nonuniform sampling and a round-robin protocol. An innovative method for modeling nonuniform sampling periods through nonhomogeneous sojourn probabilities is proposed, offering a more intuitive and adaptable framework for system design and analysis. The round-robin protocol is applied to nonuniformly sampled outputs, optimizing information transmission efficiency and enhancing overall system performance. To tackle potential limitations in state data acquisition, a token-dependent static output feedback controller is developed that addresses the complexities introduced by nonperiodic sampling and asynchronous premise variables. Sufficient conditions are derived to ensure stochastic stability of the closed-loop system. Finally, two simulation examples are presented to validate and demonstrate effectiveness of the theoretical approach.

A neural dynamic event-triggered mechanism for adaptive sliding mode control of nonlinear networked Markovian jump systems.

We prove that infinitely renormalizable contracting Lorenz maps with bounded geometry or the so-called a priori bounds satisfy the slow recurrence condition to the singular point c at its two critical values c 1 − and c 1 + . As the first application, we show that the pointwise Lyapunov exponent at c 1 − and c 1 + equals 0. As the second application, we show that such maps are stochastically stable.

Stochastic stabilization and destabilization of nonlinear switched system by periodic stochastic controls based on Lévy jump noise

Stochastic perturbation and stability analysis of a reduced order model of natural circulation loops

Stochastic stability of an elastically constrained wheelset system under additive and multiplicative color noise excitations

In this paper, the H ∞ control of nonlinear networked systems under denial-of-service (DoS) attacks based on event-triggered encryption observer is studied. To begin with, the nonlinear problem is solved by Takagi-Sugeno fuzzy model. In addition, considering the utilization of network resources, a novel adaptive event-triggered mechanism is reported to better determine when the signal is sent. Under the framework of network security, an encrypted observer is given, and the impact of DoS attacks is considered in the network channel from actuator to observer, where DoS attacks are described by random variables satisfying Bernoulli distribution. Then, based on the above considerations, sufficient conditions for the observer-based controller to guarantee the stochastic stability and H ∞ performance of the closed-loop system are given. The theoretical results of the observer-based controller parameters are obtained by solving linear matrix inequalities. Lastly, the feasibility of the reported method is verified by a mechanical motion system.

This study presents a stochastic predator-prey model in a three-patch ecosystem, motivated by cage-based fish farming. Each patch hosts prey and predator populations, with inter-patch prey migration and unbounded variations in the population represented by stochastic terms. The model integrates logistic prey growth, predation, and mortality within a coupled system of stochastic differential equations. We assess stochastic stability using stochastic Lyapunov function methods. Numerical simulations confirm that when predator efficiency ei < 1, the total population remains bounded, indicating stability. However, for ei > 1, the system becomes unstable. The model also demonstrates that prey populations remain viable under low harvesting rates (ν1 = ν2 = ν3 = 0.02) and moderate noise intensities (0.10 ≤ σ ≤ 0.90). This work contributes to sustainable resource management by offering a robust framework for modeling predator-prey interactions in multi-patch environments.

ABSTRACT This paper investigates the trajectory tracking control problem of underactuated surface vessel (USV) systems affected by slowly varying environmental disturbances. A stochastic disturbance observer (SDO) is constructed to estimate the slowly varying environmental disturbances with random terms, and meanwhile, considering the characteristics of extensive computing of traditional backstepping methods in designing of the virtual controller, dynamic surface control technology (DSC) is introduced to solve the problem. Based on the SDO and DSC, a tracking controller is constructed to enable ships to accurately track reference trajectories by integrating the disturbance observer-based control (DOBC) method with the DSC strategy. Finally, the stability analysis process of closed-loop systems is provided using stochastic stability theory, which can ensure that all signals are asymptotically bounded in mean square. The effectiveness of this proposed scheme is verified by Matlab simulations, which the results demonstrate the USV can precisely track the reference trajectory.

Stochastic Stability Analysis Method for DC Microgrids Based on Theories of Moment Stability and Stochastic Integral

The moment Lyapunov exponent in stochastic stability theory serves as a critical metric for investigating flutter stability in turbulence. However, theoretical analysis of the moment Lyapunov exponent remains confined to two-degree-of-freedom aeroelastic systems. For stochastic stability problems involving multi-modal coupling effects, current research still necessitates reliance on the Monte Carlo simulation, an extremely time-consuming approach, to determine the moment Lyapunov exponent. Multi-modal coupling effects and the unsteady characteristic of aerodynamic self-excited forces in flutter of flexible structures, such as long-span bridges, prove non-negligible. Theoretical expressions for the moment Lyapunov exponent of multi-modal coupled aeroelastic systems in turbulence remain unestablished in contemporary stochastic stability theory. This paper presents, for the first time, the asymptotic expansion for the moment Lyapunov exponent of the multi-modal coupled flutter system incorporating unsteady aerodynamic forces. Based on the asymptotic expansion, a novel analytical method is proposed for assessing the flutter stability of multi-modal coupled aeroelastic systems in turbulence. The proposed method is validated through a finite element model case study of a long-span suspension bridge. The numerical results demonstrate that the proposed method achieves comparable accuracy to Monte Carlo simulations in determining the moment Lyapunov exponents for a high-dimensional stochastic dynamic system of multi-modal coupled flutter, with computational efficiency improved by four orders of magnitude. This study proposes an accurate and efficient computational method for multi-modal coupled flutter analysis of aeroelastic systems in turbulence while establishing a transferable framework for determining the moment Lyapunov exponent in high-dimensional stochastic dynamic systems.

Stochastic asymptotical stability of the predator-free and positive equilibrium for food chain model with vigilance under stochastic perturbations

Adaptive event-triggered dissipative filtering for interval type-2 fuzzy semi-Markov jump systems with quantization and sensor failures.

ABSTRACTThis paper discusses the adaptive control problem for a class of stochastic nonlinear systems with uncertain nonlinear functions and uncertain measurement functions. First, a series of neural network functions are used to estimate the unknown nonlinear terms. Then reasonable assumptions about the unknown powers 's are raised based on the notion of the homogeneity with monotone degrees. By recursively constructing twice continuous differential Lyapunov functions, the adaptive feedback controller is designed to deal with the unknown coefficients. Based on the stochastic stability theorem, it is proved that all signals in the closed‐loop system are bounded in probability. Furthermore, the effectiveness of the proposed control approach is verified by a practical example and a numerical simulation.

ABSTRACTThis article presents a novel observer‐based control design for time‐delay systems under random nonlinearities and probabilistic disturbances. The time‐delay is considered to be a variant to model real‐world applications. The system is subjected to randomly varying input nonlinearity. The input nonlinearity which exhibits randomness is characterized by a stochastic variable with uncertain probabilities. The incorporation of generalized energy functions and the fulfillment of orthogonal conditions play a fundamental role in comprehending the error function, controlling input, and observing the output of delayed nonlinear systems. The main goal of the article is to derive tractable conditions for stability analysis with reduced conservatism. By the Lyapunov–Krasovskii approach, a robust observer‐based control strategy is proposed so that conditions which ensure stochastic stability of the system are obtained. An improved set of less conservative sufficient conditions is established by resorting to delay fractionizing and matrix inequality techniques, which are also dependent on the size of the delay fraction. Finally, numerical examples which provide evidence of desired results with the chosen control system and opted techniques are illustrated.

ABSTRACTRecent research provided proof‐of‐concept that the randomness of lead times in inventory systems can be exploited to achieve large—potentially unlimited—performance improvements, compared to the case of constant lead time. Specifically, the Generalized Base Stock (GBS) policy serves as such proof‐of‐concept—it can deliver unlimited improvements within a certain class of models, when the ratio of the minimum lead time to the mean lead time can be arbitrarily small. In this paper, we explore what improvements are actually achievable under practical system constraints, most importantly—in discrete‐time systems, where the minimum‐to‐mean lead time ratio is lower bounded by a positive constant; and also, which policies both allow significant improvements and are attractive for practical use. We consider a discrete‐time version of GBS and introduce two new discrete‐time policies, labeled ADAPTIVE and PIPELINE. We prove the stochastic stability and finiteness of average inventory level under GBS, ADAPTIVE, and PIPELINE policies, in the important special case of bounded lead time. We use simulations to evaluate the performance of the three policies and their dependence on lead time distributions. We observe that the performance improvements, provided by our policies under practical constraints, can indeed be very significant, and they are larger when the lead time “randomness” (say, variance) is larger. It also appears that the PIPELINE policy typically has the best performance and is robust from the practical use point of view, in the sense that it applies to a wide range of practical scenarios and does not require careful parameter tuning.