Analysis Of Stochastic Systems Research Articles

Finite-time stability in probability of stochastic delay systems via generalized Halanay inequality

This paper presents novel advancements in the analysis and control of stochastic systems with time-varying delays, focusing on the concept of finite-time stability. Firstly, a new kind of generalized Halanay inequality is established. With this crucial approach, two essential stochastic Lyapunov theorems on the finite-time stability in probability of stochastic delay systems with bounded time delays and unbounded time delays, respectively, are derived. In addition, upper bound estimate for the stochastic settling time is given. Furthermore, a type of nonlinear stochastic delay systems is investigated and a delay-independent controller is proposed to achieve finite-time stability in probability. The effectiveness of the results is demonstrated through two illustrative examples.

Dynamical analysis and fault detection application of a time-delayed multi-stable stochastic resonance system driven by white correlated noises.

Bearing fault diagnosis is a crucial means to ensure the normal operation of machinery, reduce maintenance costs, enhance equipments safety and extend the service life of bearings. However, the noise interference in input signals often affects the effective extraction of bearing fault signals. In this paper, a time-delay multi-stable stochastic resonance (SR) system driven by white correlated noises is proposed. Firstly, the expressions of the steady-state probability density (SPD) function and the mean first passage time (MFPT) is derived. Simultaneously, we delve into the influences of various system parameters, including multiplicative noise intensity, additive noise intensity, noise correlated strength, time-delay, and time-delay feedback intensity, on the dynamical behavior exhibited by particles within the system. Then, according to the signal-to-noise ratio (SNR) formula, the paper investigates the impact of system parameters on the SNR. It is found that the ease of SR occurrence is directly related to the system parameters. Finally, the experimental results demonstrate that the proposed method exhibits superior performance in detecting faults compared with the classical bistable SR system and the quad-stable SR system.

Event‐Triggered Generalized Extended State Observer‐Based Control for Nonlinear Networked Systems Under Gain Variation and Multi‐Channel Attacks

ABSTRACTThis paper investigates the event‐triggered generalized extended state observer‐based estimation issue for a class of nonlinear networked control systems with unmodeled dynamics, external disturbances and multi‐channel attacks. In order to resist different forms of attack threats on multiple communication channels from sensors to the observer, Markov chain is introduced to describe the stochastic switching or jumping behavior between different attack modes. Considering the limited network resources, the measured outputs are transmitted to the observer only when the triggering conditions are met. Moreover, the parameters modeled by the Bernoulli process are adopted to help analyze potential random gain variations of the generalized extended state observer. By employing Lyapunov stability theory and stochastic systems analysis, sufficient conditions are derived to ensure that the augmented system is exponentially bounded in mean square, and the expected observer gains are further determined through linear matrix inequalities. Finally, a numerical example and a simulation related to the RLC series circuit system are conducted, illustrating the proposed method's effectiveness.

Deterministic analysis of stochastic FHN systems based on Gaussian decoupling

This paper systematically explores the deterministic characteristics of FitzHugh-Nagumo (FHN) systems’ response to synaptic noise in statistical sense. With the help of Gauss decoupling approximation, two substitute systems of FHN systems’ response to synaptic noise are established by ignoring the cumulants higher than the second order, and the feasibilities of using the substitute systems for response analysis are demonstrated through error analysis. Then, the deterministic analyses of FHN systems with synaptic noise are carried out by means of the two substitute systems. Numerical results show that whether it is the neuronal system or the network system, the synaptic noise can effectively regulate its dynamics, and induce the mode transitions of discharge activity. Based on the class II excitability of FHN neurons, the system activity has a nonlinear dependence on the noise parameter and the other variables of interest. Particularly, the synaptic noise not only makes the neuronal system transition from low-level to high-level narrow-amplitude oscillation by competing with the input signal, but also contributes to the response or detection of this system to weak input signals. This study reveals the deterministic characteristics of FHN systems with synaptic noise, which can provide a reference for the large-scale analysis.

A stochastic analysis of particle systems with pairing

Motivated by a general principle governing regulation mechanisms in biological cells, we investigate a general interaction scheme between different populations of particles and specific particles, referred to as agents. Assuming that each particle follows a random path in the medium, when a particle and an agent meet, they may bind and form a pair which has some specific functional properties. Such a pair is also subject to random events and it splits after some random amount of time. In a stochastic context, using a Markovian model for the vector of the number of paired particles, and by taking the total number of particles as a scaling parameter, we study the asymptotic behavior of the time evolution of the number of paired particles. Two scenarios are investigated: one with a large but fixed number of agents, and the other one, the dynamic case, when agents are created at a bounded rate and may die after some time when they are not paired. A first order limit theorem is established for the time evolution of the system in both cases. The proof of an averaging principle of the dynamic case is one of the main contributions of the paper. The impact of dynamical arrivals of agents on the level of pairing of the system is discussed.

Stability analysis of damped fractional stochastic differential systems with Poisson jumps: an successive approximation approach

This paper aims to investigate a new class of fractional stochastic differential systems under the influence of damping and Poisson jumps. First, the existence and uniqueness of mild solutions for the proposed system are investigated under the non-Lipschitz conditions. The results are formulated and proved by using the ( β , δ ) -regularised family, Grönwall's inequality, and successive approximation technique, which is different from the fixed point approach. Moreover, novel stability criteria for the considered system are obtained by utilising the corollary of the Bihari inequality. Finally, the correctness of the obtained results is verified by example.

Stability analysis for neutral stochastic time-varying systems with delayed impulses

This article investigates the stability of stochastic neutral delayed systems (SNDS) with delayed impulses, which includes the stability in input-to-state exponentially stable in pth moment (p-ISES), integral ISES in pth moment (p-iISES) and eλt-weighted ISES in pth moment (eλt-p-ISES). Compared with the existing works, we allow the neutral term, delayed impulses, time-varying coefficients in the diffusion condition, bounded time-varying delay (BTVD) to be in a stochastic system, which arise the difficulty. By utilizing the techniques of Lyapunov-Krasovskii (L-K) function, generalized delay integral inequality and average impulsive interval (AII), some desired results are obtained by surmounting the underlying difficulty. Finally, an example is given to show the validity of our work.

Controller design and energy analysis for stochastic differential systems with one-sided polynomial growth under finite-time attraction constraint

This article presents a novel controller design approach for stochastic nonlinear systems exhibiting one-sided polynomial growth. The objective is to achieve finite-time attractiveness and estimate control energy consumption. By constructing an appropriate controller and employing inequality techniques, sufficient conditions for achieving finite-time attractiveness in stochastic nonlinear systems are established. Moreover, the article introduces the concept of expected energy consumption for stochastic systems and provides upper bounds on control time and energy consumption. A comprehensive trade-off analysis between control time and energy consumption is conducted to identify the optimal control intensity. The proposed approach is validated through numerical examples, demonstrating its effectiveness in practice.

Stability analysis for stochastic nonlinear systems with state-dependent switching and time-delay

The stability of stochastic nonlinear systems with time-delay and state-dependent switching is studied in this paper. Firstly, system maximal solution is constructed and further extended as the global solution under certain mild conditions. Secondly, several Krasovskii-type stability and Razumikhin-type theorems are proposed by uniformly asymptotically stable functions (UASFs). These stability criteria overcome some stringent constraints associated with Lyapunov functional/functions. Finally, the theoretical findings are validated through two simulation examples.

Small-Gain and Small-Angle Conditions for Feedback Stability Analysis of Linear Stochastic Systems

Small-Gain and Small-Angle Conditions for Feedback Stability Analysis of Linear Stochastic Systems

Averaging principles for multiscale stochastic Cahn–Hilliard system

In this paper, we will establish averaging principles for the multiscale stochastic Cahn–Hilliard system. The stochastic averaging principle is a powerful tool for studying qualitative analysis of stochastic dynamical systems with different time-scales. Under suitable conditions, two kinds of averaging principle (the autonomous case and the nonautonomous case) are proved, and as a consequence, the multiscale system can be reduced to a single stochastic Cahn–Hilliard equation (averaged equation) with a modified coefficient, the slow component of multiscale stochastic Cahn–Hilliard system towards to the solution of the averaged equation in moment (the autonomous case) and in probability (the nonautonomous case).

Stochastic robustness analysis of wing-aileron flutter suppression systems

In this work a stochastic robustness analysis approach is used to compare the performances of two non-linear flutter suppression systems designed for stabilization of a wing-aileron lifting structure. A reduced order modelling approach based on an alternative aeroelastic beam finite element is presented and used for the flutter suppression systems design, which are driven by a simple adaptive controller and a sliding mode controller, respectively. A heuristic algorithm is used to determine the simple adaptive controller invariant parameters at the design point while a simple parametric study is performed to tune the sliding mode controller. The performances of each closed loop system are evaluated taking into account mass, stiffnesses, and aerodynamics uncertainties of the aeroelastic plant also considering the presence of discrete gust disturbances.

Stability analysis of stochastic differential systems with state‐dependent delay subject to average‐delay impulses

In this paper, we study the stochastic weak local exponential stability of stochastic differential systems with state‐dependent delay (SDD) subject to average‐delay impulses by using Lyapunov–Krasovskii functional and stochastic analytical skill. Unlike time‐dependent delay discussed in the previous literature, we introduce SDD in impulsive stochastic differential systems, where SDD is a stochastic variable associated with system state, consequently leading to an indeterminate delay boundary. Our findings reveal that when destabilizing delayed impulses satisfying specific conditions are generated, the stability of stochastic differential systems with destabilizing delayed impulses and stable continuous stochastic dynamics can still be maintained. Additionally, if stable delayed impulses satisfy certain conditions, stability of the systems under consideration can also be achieved, irrespective of the stable status on the continuous stochastic dynamics. Finally, two examples are given to demonstrate the validity of theoretical results.

Stability analysis for nonautonomous impulsive hybrid stochastic delay systems

In this paper, the problems on the existence and uniqueness, and the input-to-state stability for the global solution of nonlinear nonautonomous impulsive stochastic delay systems with Markovian switching are considered. The existence and uniqueness for the global solution of such systems is studied by using the Lyapunov function, the theory of stochastic analysis, and the function of the impulsive density. In order to overcome the difficulty stemming from the simultaneous presence of the indefinite derivative for the Lyapunov function, the bounded external disturbances, and a nonnegative and bounded time-varying delay, one impulsive generalized Halanay inequality of integral version is established. The input-to-state stability, the integral input-to-state stability, the stochastic input-to-state stability and the eλt(λ>0)-weighted input-to-state stability for the global solution of nonlinear nonautonomous impulsive stochastic delay systems with Markovian switching are investigated, respectively. One example is provided to illustrate the effectiveness of the theoretical results obtained.

A domain decomposition solver for spectral stochastic finite element: an approximate sparse expansion approach

Spectral stochastic finite element (SSFE) has been widely used in the uncertainty quantification of real-life problems. However, the prohibitive computational burden prevents the application of the method in practical engineering systems because an enormous augmented system has to be solved. Although the domain decomposition method has been introduced to SSFE to improve the efficiency for the solution of the augmented system, there still exist significant challenges in solving the extended Schur complement (e-SC) system from domain decomposition method. In this paper, we develop an approximate sparse expansion-based domain decomposition solver to generalize the application of SSFE. An approximate sparse expansion is first presented for the subdomain-level augmented matrix so that the computational cost in each iteration of the preconditioned conjugate gradient is greatly alleviated. Based on the developed sparse expansion, we further establish an approximate sparse preconditioner to accelerate the convergence of the preconditioned conjugate gradient. The developed approximate sparse expansion-based domain decomposition solver is then incorporated in the context of SSFE. Since the difficulties of solving the e-SC system have been overcome, the developed approximate sparse expansion-based solver greatly improves the computational efficiency of the solution of the e-SC system, and thereby, the SSFE is capable of dealing with large-scale engineering systems. Two numerical examples demonstrate that the developed method can significantly enhance the efficiency for the stochastic response analysis of practical engineering systems.

Covariance-analytic performance criteria, Hardy-Schatten norms and Wick-like ordering of cascaded systems

This paper is concerned with robust performance analysis of linear stochastic systems acting as an integral operator on a standard Wiener process at the input and producing a stationary Gaussian random process at the output. We propose a performance criterion which uses the trace of an analytic function of the spectral density of the output process. This class of “covariance-analytic” cost functionals includes the usual mean square and risk-sensitive criteria as particular cases. Due to the “cost-shaping” analytic function, the covariance-analytic performance criterion depends on higher-order Hardy-Schatten norms of the system transfer function. We discuss the links of these norms with the asymptotic behaviour of cumulants of finite-horizon quadratic functionals of the system output and their variational properties pertaining to system robustness to statistically uncertain inputs which differ from the standard Wiener process. In the case of strictly proper finite-dimensional systems, governed in state space by linear stochastic differential equations, we develop a method for recursively computing the Hardy-Schatten norms through a recently proposed technique of rearranging cascaded linear systems, which resembles the Wick ordering of mixed products of noncommuting annihilation and creation operators in quantum mechanics. This computational procedure, which is one of the main results of the paper, involves a recurrence sequence of solutions to algebraic Lyapunov equations and represents the covariance-analytic cost as the squared H2-norm of an auxiliary cascaded system. These results are also compared with an alternative approach, which uses higher-order derivatives of stabilising solutions of parameter-dependent algebraic Riccati equations, and illustrated by a numerical example.

Comparative analysis of classical and stochastic Maccari system of nonlinear equations

In this paper, the exact solutions of classical and stochastic Maccari system is constructed. The exact comparative solutions are examined and plotted. Interesting results in the case of multiplicative noise are formulated and graphically elaborated. The applications of the stochastic Maccari system are added for the physical purpose. The existence of results for the real part of underlying system are discussed first time for a priori estimates. The perturbations, which disturbed the formation of Langmuir waves, are geometrically expressed in this article. Due to the presence of multiplicative noise term, our system brings a real flavor to the dynamics of the problem.

Nonlinear Optimal Control for Stochastic Dynamical Systems

This paper presents a comprehensive framework addressing optimal nonlinear analysis and feedback control synthesis for nonlinear stochastic dynamical systems. The focus lies on establishing connections between stochastic Lyapunov theory and stochastic Hamilton–Jacobi–Bellman theory within a unified perspective. We demonstrate that the closed-loop nonlinear system’s asymptotic stability in probability is ensured through a Lyapunov function, identified as the solution to the steady-state form of the stochastic Hamilton–Jacobi–Bellman equation. This dual assurance guarantees both stochastic stability and optimality. Additionally, optimal feedback controllers for affine nonlinear systems are developed using an inverse optimality framework tailored to the stochastic stabilization problem. Furthermore, the paper derives stability margins for optimal and inverse optimal stochastic feedback regulators. Gain, sector, and disk margin guarantees are established for nonlinear stochastic dynamical systems controlled by nonlinear optimal and inverse optimal Hamilton–Jacobi–Bellman controllers.

The nonlinear dynamics analysis of stochastic delay Jeffcott rotor-seal system with the elastic support

The establishment of stochastic delay Jeffcott rotor-seal system not only considers the track irregularity caused by random noise and the possible influence of stochastic parameter excitation, but also considers the time-delay characteristics of sealing force. The rotation of the rotor shaft causes the gas in the chamber to produce a dynamic effect, resulting in a rotating force, which delays the feedback on the rotating shaft. Firstly, the one-dimensional average Itoˆ differential equation is obtained by simplifying the infinite dimensional stochastic delay differential equation with the perturbation method. Secondly, the global and local stability of the rotor system are obtained by analyzing the singular boundary theory and the maximum Lyapunov exponent. Then, the conditions and types of stochastic bifurcation of rotor system are obtained by analyzing the steady-state joint probability density function. Finally, numerical simulation verifies the accuracy of the theoretical analysis, showing that the time delay affects rotor system to reach the stable critical speed for the first time, and the noise disturbance has a certain stability effect on the rotor system.

Stochastic Geometry-based Analysis of Cell-Free Massive MIMO Systems with Aerial Users

Stochastic Geometry-based Analysis of Cell-Free Massive MIMO Systems with Aerial Users