Stochastic Finite-time Stability Research Articles

Predefined-time stabilization of stochastic nonlinear systems with application to UAVs

The paper presents a new Lyapunov-type predefined-time stabilization control algorithm for stochastic high-order nonlinear systems with asymmetric output constraints. In contrast to stochastic finite-time and fixed-time stabilization, the average value of the settling-time function for stochastic predefined-time stabilization control is independent of both the initial value and the control factors. To mitigate the significant uncertainties arising from the asymmetric output constraint, a tan-type barrier Lyapunov function is formulated. Furthermore, by harnessing the previously mentioned barrier Lyapunov function and integrating the power integrator technique, a controller design strategy is formulated based on the backstepping method. The rigorous analysis in this study proves that the designed controller ensures both the attainment of predefined-time convergence of the system states to the origin in probability and the satisfaction of the output constraint. Finally, an example of a roll angle subsystem for quadrotor UAVs and a numerical illustration are presented to corroborate the theoretical analysis.

Finite-Time Adaptive Fuzzy Backstepping Control for Quadrotor UAV With Stochastic Disturbance

This paper proposes an adaptive fuzzy output feedback control approach for quadrotor unmanned aerial vehicles (QUAV) with stochastic disturbances, besides which we consider the unmeasurable states and unknown nonlinear functions. The QUAV system contains stochastic terms which are not bounded and not differentiable. By combining the It <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\hat{\rm \textbf{o}}$</tex-math> </inline-formula> differential equation with finite-time theory, a novel stochastic finite-time stability controller for QUAV is raised for the first time. The fuzzy logic system (FLS) is utilized to approximate the unknown nonlinear functions in the model. Dynamic surface control technology is introduced to reduce the complexity of differential. Furthermore, an observer with FLS is designed through the adaptive backstepping technique to estimate the immeasurable states. It is proved that this control approach can ensure that all states of the closed-loop QUAV system are semi-global and finite-time stable in probability. Meanwhile, the errors of system outputs and observer converge to a small neighborhood of origin. The simulation results show the effectiveness of the proposed method. <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Note to Practitioners</i> —In practical applications, many systems are often affected by internal and external disturbances, and not all of them can be described by mathematical models, so they are called stochastic disturbances. Taking the QUAV as an example, when the system is disturbed by internal parameters, external environment, system input, sensor error, and other stochastic factors, the deterministic system can no longer accurately describe the controlled object. In this paper, the QUAV is regarded as a stochastic system for the first time, and a novel controller is designed to solve the above problems. In addition, the QUAV needs fast action in some applications, such as emergency collision avoidance, which has high requirements for the convergence performance of the controller. Therefore, studying the finite-time control for QUAV systems with stochastic disturbances is necessary. At the same time, this paper also considers the problem that the state is not measurable and the system contains unknown nonlinear equations.

Finite-step alternating approximately bi-similar symbolic model for Markov jump systems

Establishing a bi-similar abstraction (symbolic model) of control systems has been demonstrated as a viable approach for implementing temporal logic tasks on control systems. Previous research primarily focused on logic tasks over an infinite time horizon. However, practical scenarios often involve control tasks defined within finite time intervals. Hence, this paper addresses the finite-time bi-similar problem specifically for a significant class of stochastic switched systems known as Markov jump systems (MJSs). To achieve this, two novel concepts are introduced: finite-time stochastic incremental stability (FSIS) and finite-step alternating approximate bi-simulation relation. A sufficient condition is firstly established for FSIS of MJSs by leveraging the incremental weak infinitesimal operator. Subsequently, a symbolic model is developed for MJSs by employing gridding techniques. Furthermore, a sufficient condition is derived to ensure that the developed symbolic model exhibits finite-step alternating approximate bi-similarity with the original MJSs. Finally, two numerical examples illustrate the application of the proposed results.

Stability analysis of switched stochastic nonlinear functional systems

AbstractThis article is concerned with stochastic finite time stability (SFTS) and stochastic fixed time stability (SFIXTS) of switched stochastic nonlinear functional systems. By using the Lyapunov–Razumikhin approach, stochastic analysis techniques, and stochastic process theory, several novel theorems on SFTS and SFIXTS are established. Concurrently, the estimations of stochastic settling time are given. Different from the previous general results, this article considers the effects of noise, time‐delays, and switches. The SFTS and SFIXTS conditions are established, which are quite different from the traditional exponential stability or almost sure stability. Furthermore, two examples are given to illustrate the effectiveness of the obtained results.

Finite-time stochastic integral input-to-state stability and its applications

In this paper, finite-time stabilization is investigated in depth for stochastic nonlinear systems with stochastic inverse dynamics. The contributions of this work are characterized by the following novel features: (i) Motivated by the concept of stochastic integral input-to-state stability (SiISS) and the existing stochastic finite-time stability theorem, a new concept of finite-time stochastic integral input-to-state stability (FT-SiISS) using Lyapunov function is first introduced, and an inclusion relationship among three types of stochastic stability, i.e., FT-SiISS, FT-SISS and SISS, is rigorously established. On the basis of FT-SiISS, we develop two new FT-SiISS small-gain conditions and discuss their relationship. Then, to analyze finite-time stability of stochastic nonlinear systems with FT-SiISS inverse dynamics, a stochastic finite-time stability theorem is improved. (ii) As the application of (i), we propose a unified finite-time control method, which can simultaneously handle stochastic nonlinear systems with FT-SiISS or FT-SISS inverse dynamics, and guarantee that the closed-loop system has an almost surely continuous solution, all the closed-loop signals are bounded almost surely, and its trivial solution is stochastically finite-time stable.

Stochastic finite-time stability and stabilization for stochastic nonlinear systems with impulse effects

In this paper, some unified stochastic finite-time stability (SFTS) criteria are derived for the impulsive stochastic nonlinear systems by using the stochastic analysis theory, stopping time technology, Lyapunov approach and limit average dwell time (LADT) technology when the impulse effect is destabilizing impulse or stabilizing impulse. Meanwhile, in order to achieve stochastic finite-time stabilization for stochastic nonlinear systems, the impulse control approach is also obtained, which can be divided into two cases: (1) A stochastic finite-time instable (SFTI) stochastic system can become SFTS under the impulse control strategy; (2) A SFTS stochastic system can keep the SFTS under the impulsive disturbance. Furthermore, by using the impulse dependent average dwell time (IDADT) technology, the results are also established for the multiple impulse effects, i.e. destabilizing impulse and stabilizing impulse exist in the system simultaneously. Finally, two useful examples are provided.

Stochastic finite-time stabilization for stochastic multi-links coupled systems with time delays

Stochastic finite-time stabilization for stochastic multi-links coupled systems with time delays

Finite-time stability and asynchronous H∞ control for highly nonlinear hybrid stochastic systems

In this paper, the pth moment finite-time stochastic stability (p-MFTS stability) and asynchronous H∞ control for highly nonlinear hybrid stochastic systems (HNHSSs) are investigated. Firstly, the definition of p-MSFT stability for HNHSSs is proposed, and the stability criteria are given by the mode-dependent average dwell time (MDADT) method. Secondly, the asynchronous H∞ controller is designed to ensure that HNHSSs are pth moment finite-time stochastically stabilizable (p-MFTS stabilizable) and satisfy H∞ performance. When designing the asynchronous H∞ controller, deterministic asynchrony and stochastic asynchrony are considered simultaneously. Moreover, since the coefficients of HNHSSs have highly nonlinear structures without satisfying the linear growth condition (LGC), the Lyapunov function U and the operator LU are bounded by polynomials of degree higher than p, which means that weaker conditions of p-MFTS stability are proposed. Finally, examples are presented to illustrate the availability of the results.

Event-triggered finite-time fault detection for delayed networked systems with conic-type nonlinearity and deception attacks

The paper is concerned with the finite-time fault detection (FTFD) problem for a class of delayed networked systems subject to conic-type nonlinearity and randomly occurring deception attacks (RODAs) via dynamic event-triggered mechanism (DETM). The nonlinear function with the conic-type constraint is limited to a known hypersphere with uncertain center. Moreover, a variable governed by Bernoulli distribution is introduced to characterize the RODAs phenomenon. In order to reduce unnecessary communication transmissions, a DETM is considered in the design of finite-time fault detection filter (FTFDF) for the addressed networked systems with time-delays. This paper focuses on the design of an FTFDF via the DETM to ensure the finite-time stochastic stability of error dynamics system with satisfactory the prescribed H∞ performance. Moreover, the desired FTFDF parameter matrices are obtained by solving linear matrix inequalities. In the end, a simulation example is employed to illustrate the validity of the proposed FTFD method.

Finite-Time Control for Discrete-Time Positive Systems Subject to Event-Triggered Scheme and Markov Jump Parameters

This brief investigates the finite-time control problem for discrete-time positive systems subject to event-triggered scheme and Markov jump parameters. Switching transition probability is used to extend the fixed transition probability, which is more suitable to describe the performance of Markov jump systems (MJSs). On the basis of the 1-norm, a developed event-triggered scheme is proposed to be associated with the sample signal and the actual signal. Furthermore, a linear Lyapunov function and an average dwell time approach are adopted to obtain sufficient conditions for guaranteeing stochastic finite-time stability of the underlying systems. Additionally, a matrix decomposition strategy is constructed to design a finite-time event-triggered control law such that the corresponding MJSs are positive and stochastically finite-time bounded. All conditions are proposed on the basis of linear programming. Finally, a local railway transportation model is provided to verify the practicability of the strategy.

Adaptive fuzzy fault-tolerant containment control for nonlinear multiagent systems based on modified distributed observer

This paper investigates the finite-time fault-tolerant containment control problem for stochastic nonlinear multiagent systems with intermittent actuator faults. A modified distributed observer is designed to estimate unmeasured states in systems. Compared with the existing result, the estimation precision of the modified observer is significantly enhanced in the presence of various uncertainties in the control channel. Moreover, this paper designs a compensation mechanism for intermittent faults to suppress the adverse effects caused by each fault. Then a finite-time fuzzy fault-tolerant containment control scheme is proposed to make all followers converge to the convex hull formed by leaders in a finite time. It is proved that all signals of closed-loop systems are semi-globally finite-time stable in probability based on the finite-time stochastic Lyapunov stability theory, and containment errors can converge to a small neighborhood of the origin. Finally, simulation studies are provided to illustrate the effectiveness of the proposed control scheme.

Finite-time impulsive control for stochastic T-S fuzzy systems: A waiting-time-based event-triggered method

The finite-time impulsive control problem is studied for stochastic T-S fuzzy systems with parameter uncertainties. In order to reduce numbers of impulsive control and information transmission, we proposed event-triggered impulsive control (ETIC) scheme involving the co-design of event-triggered mechanisms and impulsive gains to ensure stochastic finite-time stability (SFTS) of considered systems. For ETIC scheme, impulsive inputs are adopted to systems only at event-triggering instants. The waiting-time is introduced into ETIC scheme to avoid Zeno behavior and save communication resources further. Compared with the time-triggered impulsive control, a theoretical analysis indicates that ETIC not only removes the restriction of upper bound for impulsive intervals, but also significantly reduce the number of impulsive control while ensuring the same control performance. Two examples illustrated the advantage and validity of ETIC scheme.

On the stochastic finite‐time stability for switched stochastic nonlinear systems

AbstractIn this paper, the stochastic finite‐time stability (SFTS) criteria for switched stochastic nonlinear systems (SSNS) with nonlocal Lipschitz coefficients are established. The main tools that we used here are mode‐dependent limiting average dwell time (MD‐LADT), stopping time technology and the stochastic analysis theory. The established criteria show that the diffusion coefficient, the drift coefficient and the switching law play crucial roles in obtaining the SFTS for the SSNS. Besides, our criteria provide some approaches on how to design the switching law to achieve the SFTS if a switched stochastic system contains SFTS subsystems and stochastic finite‐time instable (SFTI) subsystems simultaneously. Finally, an example is given to illustrate the importance and usefulness of the theoretical results.

Output feedback stabilization of stochastic planar nonlinear systems with output constraint

This paper is concerned with the problem of output feedback stabilization of stochastic planar nonlinear systems with output constraint. By introducing some new nonlinear mappings, adding a power integrator and homogeneous domination techniques, according to different value ranges of the homogeneous degree in the same form of assumptions, we establish two new design and analysis frames of output feedback stabilization to achieve respectively stochastic asymptotic stability (or asymptotical stability in probability) and stochastic finite-time stability of the trivial solution of the closed-loop system while guaranteeing the requirements of almost sure boundedness of the closed-loop signals and the symmetric output constraint.

Quantitative Mean Square Exponential Stability and Stabilization of Linear Itô Stochastic Markovian Jump Systems Driven by Both Brownian and Poisson Noises

In this paper, quantitative mean square exponential stability and stabilization of Itô-type linear stochastic Markovian jump systems with Brownian and Poisson noises are investigated. First, the definition of quantitative mean square exponential stability, which takes into account the transient and steady behaviors of the system, is presented. Second, the relationship between general finite-time mean square stability, finite-time stochastic stability, and quantitative mean square exponential stability is proposed. Subsequently, some sufficient conditions for the existence of state feedback and observer-based controllers are derived, and an algorithm is offered to solve the matrix inequalities resulting from quantitative mean square exponential stabilization. Finally, the effectiveness of the proposed results is illustrated with the numerical example and the practical example.

Stochastic Finite-Time Stability for Stochastic Nonlinear Systems with Stochastic Impulses

In this paper, some novel stochastic finite-time stability criteria for stochastic nonlinear systems with stochastic impulse effects are established. The results in this paper blackgeneralized the related results in from two aspects: 1. the model in is the deterministic systems, which means that the noise effect that can be described as a symmetric Markov process Brownian motion is considered in our models; 2. the stochastic finite-time stability criterion is established in this paper, not the asymptotic stability and the input-to-state stability that are studied in the form literature. Finally, an example is given to show the significance blackand usefulness of our results.

Finite Time Stability and Optimal Finite Time Stabilization for Discrete-Time Stochastic Dynamical Systems

In this paper, we address finite time stability in probability of discrete-time stochastic dynamical systems. Specifically, a stochastic comparison lemma is constructed along with a scalar system involving a generalized deadzone function to establish almost sure convergence and finite time stability in probability. This result is used to provide Lyapunov theorems for finite time stability in probability for It&#x00F4;-type stationary nonlinear stochastic difference equations involving Lyapunov difference conditions on the minimum of the Lyapunov function itself along with a fractional power of the Lyapunov function. In addition, we establish sufficient conditions for almost sure lower semicontinuity of the stochastic settling-time capturing the average settling time behavior of the discrete-time nonlinear stochastic dynamical system. Furthermore, a stochastic finite-time optimal control framework is developed by exploiting connections between Lyapunov theory for finite time stability in probability and stochastic Bellman theory. In particular, we show that finite time stability in probability of the closed-loop nonlinear system is guaranteed by means of a Lyapunov function that can clearly be seen to be the solution to the steady state form of the stochastic Bellman equation, and hence, guaranteeing both stochastic finite time stability and optimality.

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.

Command filter-based adaptive neural finite-time control for stochastic nonlinear systems with time-varying full-state constraints and asymmetric input saturation

This article studies a tracking control problem for a class of stochastic nonlinear systems with time-varying full-state constraints and asymmetric input saturation. Firstly, the Gauss Error Function is introduced to solve the difficulty arising from the saturation nonlinearity. Meanwhile, to overcome the problem of calculating explosion caused by the repeated differentiation of the virtual control signals, a finite-time command filter with a compensation mechanism is developed. Combining the neural networks' approximation ability and the backstepping technique, an adaptive neural finite-time control strategy is proposed for the considered system by constructing the time-varying barrier Lyapunov function. Under the proposed control strategy, it is guaranteed that all signals are bounded in a sense of mean square, the output of the system can track the reference signal within a finite time and all states will not violate the constraints. Furthermore, the stability of the closed-loop system is analysed based on the stochastic finite-time stability theory. Finally, a simulation example verifies the effectiveness of the proposed control strategy.

Finite-Time Event-Triggered Control for Semi-Markovian Switching Cyber-Physical Systems With FDI Attacks and Applications

This paper addresses the finite-time event-triggered control problem for nonlinear semi-Markovian switching cyber-physical systems (S-MSCPSs) under false data injection (FDI) attacks. Compared with the traditional time-triggered mechanism, the proposed event-triggered scheme (ETS) can effectively avoid network resource waste. Considering the network-induced delay in the modeling, a closed-loop system model with time delay is established in the unified framework. By the use of a mode-dependent piecewise Lyapunov-Krasovskii functional (LKF), stochastic finite-time stability (SFTS) criteria are established for the resultant closed-loop system. Then, some solvability conditions are established for the desired finite-time controller in light of a linear matrix inequality framework. Finally, an application example of vertical take-off and landing helicopter model (VTOLHM) is provided to demonstrate the effectiveness of the theoretical findings.