Concept Of Finite-time Stability Research Articles

Stability of Caputa– Hadmmard Fractional Differential Nonlinear ControlSystem with Delay Riemann −Katugampola

This research examines the finite-time stability of a specific category of neural networks with fractional order. Using modified Gronwall inequality and estimations of Mittag–Leffler functions, we provide adequate requirements to guarantee the finite-time stability of neural models with Caputo fractional derivatives. In addition, we have also established insights about the asymptotic stability of fractional-order neural models ,this research examines the Finite-time stability of a specific fractional-order neural network using the "generalized Grunwald inequality". This study introduces the concept of finite-time stability for neural models with fractional derivatives. By utilizing the generalized Gronwall inequality and estimates of Mittag-Leffler functions, sufficient conditions are derived to guarantee the finite-time stability of these models. The study also establishes results regarding the asymptotical stability of fractional-order neural models,our results suggest that our conclusions are more precise than the current literature on stability requirements, and we provide examples demonstrating the proposed theory's significance

Robust limit cycle control for finite-time generation of sustained oscillations in nonlinear systems with mixed dead-zone and saturation

This paper suggests a novel limit cycle control technique, based on the finite-time stability scheme, for producing sustained oscillations in a class of nonlinear complex systems. The considered systems are constrained to input dead-zone and saturation simultaneously and suffer from non-vanishing perturbations due to model uncertainties, and external disturbances. For this purpose, an innovative sliding manifold is proposed based on the topology of the wanted limit cycle which provides the conditions for developing the finite time stability concept for limit sets. With the aid of the finite-time command filter and the cooperation of the sliding mode control (SMC) method with the backstepping technique, a robust controller is constructed which guarantees the finite-time convergence of phase trajectories of the closed-loop system toward the wanted limit cycle. This leads to the creation of sustained oscillations, with the desirable amplitude and frequency, in the system's output. The stability analysis is proved based on the Lyapunov approach through the set stability concept. Finally, the effectiveness of the suggested methodology is confirmed by a variety of simulations.

Scrutinization of finite time stability of fractional impulsive neutral model with disturbance

Finite time stability practically examines the trajectories of a system which converge to equilibrium state in a short period of time. This notion requires predefined bounds on system parameters and bounded time interval. Considering the idea that many practical system often operates over time interval being finite rather than infinite, we explore the finite time stability concept of damped fractional system with neutral conditions and impulsive effects. The desired bounds for the stability of the system is derived by implementing Gronwall’s inequality conditions. Further, the finite time stability conditions of the proposed fractional linear model is extended to nonlinearity term with disturbance. Finally, numerical simulations are given to show the effectiveness of the derived results.

Finite-time stability of hybrid systems with continuous and Boolean dynamics

The hybrid systems with continuous and discrete variables can be used to describe many real-world phenomena. In this paper, by generalizing the mathematical form of gene regulatory networks, a novel class of hybrid systems consisting of continuous and Boolean dynamics is investigated. Firstly, the new hybrid system is introduced in detail, and a concept of finite-time stability (FTS) for it is proposed. Next, the existence and uniqueness of solutions are proved by fixed point theory. Furthermore, based on Lyapunov functions and the semi-tensor product (STP), i.e., Cheng product, some sufficient conditions of FTS for the hybrid systems are presented. The main results are illustrated by two numerical examples.

Asymptotical Stability and Stabilization of Continuous-Time Probabilistic Logic Networks

Discrete-time probabilistic logic networks (DT-PLNs), of which probabilistic Boolean networks (PBNs) are a special type, are an important qualitative model for gene regulatory networks (GRNs). Although a DT-PLN can predict the long-term behavior of a GRN, using it to describe the transient kinetics at the microtimescale level remains inconvenient. In this article, we investigate the problems associated with the stability and stabilization of continuous-time probabilistic logic networks (CT-PLNs). First, we demonstrate that the concept of finite-time stability for DT-PLNs cannot be extended to CT-PLNs owing to the nonsingularity of transitional probability matrices. Thus, we introduce the concept of asymptotical stability, which is defined as the convergence in distribution of the network state. Second, by developing the theory of invariant subsets for CT-PLNs, a necessary and sufficient condition for asymptotical stability with respect to a subset is proposed, which is expressed in terms of the transition rate matrix of probability. Third, for a CT-PLN with input nodes, termed a continuous-time probabilistic logic control network, we discuss the subsets that are invariant under piece-wise constant input. Based thereupon, we propose a necessary and sufficient condition under which asymptotically stabilizing sampled-data feedback exists. A method for designing sampled-data feedback is proposed. Last, Monte Carlo simulation algorithms are proposed to efficiently simulate a CT-PLN in the time domain. Examples are provided to demonstrate the methods proposed in this article.

Finite-time extended state observer based nonsingular fast terminal sliding mode control of autonomous underwater vehicles

In this paper, a finite-time extended state observer-based nonsingular fast terminal sliding mode control (NFTSMC) is proposed for the trajectory tracking of autonomous underwater vehicles (AUVs) with various hydrodynamic uncertainties and external disturbances. First, a proportional–integral velocity variable based third-order fast finite-time extended state observer is designed to estimate the lumped uncertainties and their first derivatives. The concepts of finite-time stability and a new proposition are utilized to prove the finite-time uniformly ultimately boundness of the proposed observer. Then, based on the disturbance estimation, an NFTSMC is developed for AUVs. The inverse tangent function and linear control term are used to provide robustness against estimation error and avoid chattering in the thrusters. The position and attitude trajectories are proved to be fast finite-time stable via the Lyapunov theorem. To verify the theoretical analysis, comparative numerical simulations are provided to demonstrate the effectiveness of the proposed control scheme.

Finite-time stability of uncertain fractional difference equations

Uncertain fractional difference equations may preferably describe the behavior of the systems with the memory effect and discrete feature in the uncertain environment. So it is of great significance to investigate their stability. In this paper, the concept of finite-time stability almost surely for uncertain fractional difference equations is introduced. A finite-time stability theorem is then stated by Mittag–Leffler function and proved by a generalized Gronwall inequality on a finite time. Some examples are finally presented to illustrate the validity of our results.

Stochastic finite-time stability of reaction-diffusion Cohen–Grossberg neural networks with time-varying delays

This paper investigates the problem of the stochastic finite-time stability of reaction-diffusion Cohen-Grossberg neural networks with time varying delays using the Dirichlet boundary condition. The concept of finite-time stability for the system is first derived by using the stochastic conditions. By constructing a new Lyapunov–Krasovskii functional and utilizing Jensen’s inequality, Wirtinger’s type inequality technique, the Gronwall inequality and the linear matrix inequality (LMI) frame work, conditions are obtained which guarantee the stochastically finite-time stability of Cohen–Grossberg neural networks. Finally, two numerical examples are given to show the effectiveness of the proposed results.

Continuous Finite-Time Attitude Control for Rigid Spacecraft Based on Angular Velocity Observer

This paper addresses the problem of designing an angular velocity observer and an output feedback attitude controller with finite-time convergence and disturbances for spacecraft. First, two new concepts of finite-time stability are proposed and defined as the local fast-finite-time stability and the fast-finite-time uniformly ultimately boundness, which can be seen as the extensions of the traditional fast-finite-time stability. Then, based on these two concepts of stability, a fast-finite-time observer is designed to estimate the unknown angular velocity. Next, based on the estimation of the angular velocity, a nonsingular and continuous attitude control algorithm is proposed to achieve the finite-time stability or finite-time boundness. With consideration of the observation errors and disturbances in the closed-loop system, a rigorous analysis of the proposed strategy is provided through Lyapunov approach. It shows that the observation errors and the spacecraft attitude will converge to a region of zero in finite time. Numerical simulation studies are presented to illustrate the effectiveness of the proposed observer-based attitude control scheme.

Finite-Time Stability and Control of 2D Continuous–Discrete Systems in Roesser Model

This study is conducted to investigate stability and control problems within a finite-time interval for a 2D continuous–discrete system in Roesser model. The concepts of finite-time stability (FTS) and finite-time boundedness (FTB) are naturally extended to the 2D continuous–discrete system. Recursive relations between system states are first obtained, then sufficient conditions for FTS and FTB in the system are derived, and a finite-time controller is supplied to the system. Sufficient conditions for finite-time stabilization are also provided for the linear repetitive process. Examples of metal rolling operation are presented to illustrate the proposed method.

Finite-time fault-tolerant control of neutral systems against actuator saturation and nonlinear actuator faults

This paper deals with the problem of finite-time fault-tolerant control design for a class of neutral systems subject to actuator saturation in which the time-varying delay appears in both the state and the state derivative. Specifically, this is the first attempt to consider the concept of finite-time stability for neutral systems under nonlinear fault-tolerant controller. In particular, the fault model which consists of both linear and nonlinear parts is implemented through the design of controller. By applying the Lyapunov technique and some integral inequalities, a set of sufficient conditions is derived in the form of linear matrix inequalities to ensure the finite-time stability of the addressed neutral systems. Finally, two numerical examples with simulation results are given to illustrate the effectiveness of the proposed method.

Finite‐time stability and stabilisation of distributed parameter systems

This study addresses the problems of finite-time (FT) stability and stabilisation for distributed parameter systems. First, the authors extend the concepts of FT stability and FT stabilisation to the distributed parameter systems. Then, sufficient conditions of the L 2-FT stability and W 1,2-FT stability for the distributed parameter systems are established in terms of linear matrix inequalities. Based on these sufficient conditions, the authors design the state feedback controllers which guarantee the closed-loop distributed parameter systems to be L 2-FT stable and W 1,2-FT stable, respectively. Finally, numerical examples are given to illustrate the proposed results.

Finite-Time Stability and Controller Design of Continuous-Time Polynomial Fuzzy Systems

Finite-time stability and stabilization problem is first investigated for continuous-time polynomial fuzzy systems. The concept of finite-time stability and stabilization is given for polynomial fuzzy systems based on the idea of classical references. A sum-of-squares- (SOS-) based approach is used to obtain the finite-time stability and stabilization conditions, which include some classical results as special cases. The proposed conditions can be solved with the help of powerful Matlab toolbox SOSTOOLS and a semidefinite-program (SDP) solver. Finally, two numerical examples and one practical example are employed to illustrate the validity and effectiveness of the provided conditions.

Finite-Time Stability of Discrete Switched Singular Positive Systems

The finite-time stability problem of discrete switched singular positive systems (DSSPSs) is investigated in this paper. First, the concept of finite-time stability for DSSPSs is proposed, and a necessary and sufficient condition of finite-time stability for DSSPSs under arbitrary switching is obtained. Second, based on the mode-dependent average dwell time approach, by constructing the quasi-linear Lyapunov function, a sufficient stability criterion of finite-time stability for DSSPSs is derived in terms of a set of linear matrix inequalities. Finally, a numerical example is given to show the effectiveness of the proposed techniques.

Finite-time control of interconnected impulsive switched systems with time-varying delay

In this paper, the problem of finite-time control is addressed for a class of interconnected impulsive switched systems with time-varying delay and dynamical disturbance. The concepts of finite-time stability and finite-time boundedness are extended to interconnected impulsive switched systems, respectively. By resorting to the average dwell time approach and constructing an appropriate Lyapunov–Krasovskii functional, some sufficient criteria are provided to cope with the problems of finite-time boundedness and finite-time stability. Then, based on the decentralized switched control scheme, a state-feedback controller is established to ensure the finite-time stability and the finite-time boundedness. Finally, an example and simulation result are demonstrated to illustrate the correctness and effectiveness of the proposed method.

Finite-time stability analysis of discrete-time fuzzy Hopfield neural network

The finite-time stability analysis of discrete-time fuzzy Hopfield neural networks is studied in this paper. Firstly, the concept of finite-time stability is generalized to the fuzzy neural networks. And then by the Lyapunov approach and linear matrix inequality technique, a sufficient condition for the system to be finite-time stable is proposed, based on which, the finite-time stability condition for the system with norm bounded uncertainties is also given. Finally a numerical example is given to illustrate the effectiveness of the proposed approach.

正脉冲切换系统的有限时间稳定性分析

本文解决了正脉冲切换系统的有限时间稳定性。首先把有限时间稳定的概念推广到正脉冲切换系统，从解的存在性入手，用求解微分方程组的方法，系统的解若满足一些条件，即可证明系统的稳定性。最后给出正脉冲切换系统稳定的充分条件。 This paper addresses the finite-time stability of positive impulsive switched systems. First, the concept of finite-time stability is extended to positive impulsive switched systems, starting from the existence of solutions and solving differential equations. If the solution of the systems satisfies some conditions, then we can prove the stability of the system. Finally, the sufficient conditions of positive impulsive switched systems are given.

Robust Finite-Time Stability of Fractional Order Linear Time-Varying Impulsive Systems

This paper considers the robust finite-time stability (FTS) of fractional order linear time-varying impulsive systems. First, using the fractional order Lyapunov function and generalized Gronwall inequality, some sufficient conditions are given to verify the robust FTS of fractional order linear time-varying systems. Then, the concept of FTS is extended to fractional order impulsive systems. A sufficient condition is given to verify the robust FTS of fractional order linear time-varying impulsive systems by combining the method of average dwell time with fractional order Lyapunov function. Finally, two numerical examples are provided to illustrate the theoretical results.

Finite Time Stability of Stochastic Hybrid Systems

This paper considers the finite time stability of stochastic hybrid systems, which has both Markovian switching and impulsive effect. First, the concept of finite time stability is extended to stochastic hybrid systems. Then, by using common Lyapunov function and multiple Lyapunov functions theory, two sufficient conditions for finite time stability of stochastic hybrid systems are presented. Furthermore, a new notion called stochastic minimum dwell time is proposed and then, combining it with the method of multiple Lyapunov functions, a sufficient condition for finite time stability of stochastic hybrid systems is given. Finally, a numerical example is provided to illustrate the theoretical results.

Finite-Time Stability Analysis for a Class of Continuous Switched Descriptor Systems

Finite-time stability has more practical application values than the classical Lyapunov asymptotic stability over a fixed finite-time interval. The problems of finite-time stability and finite-time boundedness for a class of continuous switched descriptor systems are considered in this paper. Based on the average dwell time approach and the multiple Lyapunov functions technique, the concepts of finite-time stability and boundedness are extended to continuous switched descriptor systems. In addition, sufficient conditions for the existence of state feedback controllers in terms of linear matrix inequalities (LMIs) are obtained with arbitrary switching rules, which guarantee that the switched descriptor system is finite-time stable and finite-time bounded, respectively. Finally, two numerical examples are presented to illustrate the reasonableness and effectiveness of the proposed results.