Conditions For Finite-time Stability Research Articles

Event-triggered adaptive control for nonlinear systems using time-receding horizons without initial dependence

This paper investigates the full-state constraint event-triggered adaptive control for a class of uncertain strict-feedback systems. The lack of information on the coupling dynamics of virtual variables in backstepping increases the complexity of feedback design. Given this, the requirements of shaping system performance constraints, eliminating initial dependence, and reducing data transfer costs together give rise to an interesting and challenging problem. Constructing the time-receding horizon (TRH) and stitching it with the quadratic Lyapunov function (QLF) is the key to constrained tracking. Specifying TRHs as a set of smooth bounds with fixed-time convergence and forcing the system to stabilize within the constrained region before the prescribed settling time provide a sufficient condition for practical finite-time stability (PFS). For relaxing the initial dependence, a tuning function is designed to match the performance constraints under arbitrary system initial conditions. A dual-channel event-triggered mechanism (ETM) is developed to automatically adjust the controller and estimator data flow updates with less transmission burden. By combining a specific inequality with backstepping, uncertainties are overcome without the “complexity explosion” in recursion steps. Finally, simulations demonstrate the effectiveness of the proposed method.

Finite-time stabilization of nonlinear time-varying systems involving impulsive action

This article investigates the finite-time stabilization of time-varying impulsive systems by time-varying feedback control. Moreover, the stabilizing and destabilizing effects of impulses on the stability of the system are considered, where the stabilizing impulses are used to regulate the unstable continuous dynamics. In addition, it can be found that the local and global characteristics of the finite-time stability may be affected by the stabilizing impulses. By establishing the relationship among the time-varying gain functions, the system structure and impulsive time sequences, some sufficient conditions for finite-time stability of time-varying systems involving impulsive action are proposed. Furthermore, based on the results mentioned above, a new class of finite-time controllers is designed for time-varying affine systems with external noises. The effectiveness of the proposed results is illustrated by three examples.

Stability of Delayed Homogeneous Positive Differential–Difference Systems

Stability theory of linear differential-difference system has been well-established, while fewer results can be found for nonlinear differential-difference systems. In this paper, stability and boundedness of homogeneous differential-difference system with bounded delay is studied based on the system positivity. At first, an exponential stability criterion is obtained, which is an extension of an existing result. Next, another boundary is computed under the previous condition, which proves to be tighter than the first result at least in some cases. Then a finite-time stability condition is achieved for the delay-free system, and an upper bound of the settling time and an explicit boundary of the state are derived. Finally, a numerical example is given to verify the results presented in this paper.

Lyapunov Conditions for Finite-time Stability of Stochastic Functional Systems

Lyapunov Conditions for Finite-time Stability of Stochastic Functional Systems

Finite-time resilient sliding mode control of nonlinear UMV systems subject to DoS attacks

The resilient sliding mode control (SMC) problem of unmanned marine vehicles (UMVs) under the framework of Finite-Time Stability (FTS) is studied in this work, where the network between the UMV and the remote console is supposed to suffer from DoS attacks. The nonlinear UMV is modeled by a T-S fuzzy model, and a novel sliding mode protocol is proposed. The FTS conditions for both the reaching phase and the sliding mode phase are given. It is shown that the FTS of the resulting closed-loop system can be established. A simulation example is finally presented to illustrate the effectiveness of the proposed control scheme.

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.

Finite-Time Stabilization and Optimal Feedback Control for Nonlinear Discrete-Time Systems

Finite-time stability involves dynamical systems whose trajectories converge to an equilibrium state in finite time. Sufficient conditions for finite-time stability have recently been developed in the literature for discrete-time dynamical systems. In this article, we build on these results to develop a framework for addressing the problem of optimal nonlinear analysis and feedback control for finite-time stability and finite-time stabilization for nonlinear discrete-time controlled dynamical systems. Finite-time stability of the closed-loop nonlinear system is guaranteed by means of a Lyapunov function that satisfies a difference inequality involving fractional powers and a minimum operator. This Lyapunov function can clearly be seen to be the solution to a difference equation that corresponds to a steady-state form of the Bellman equation, and hence, guaranteeing both finite-time stability and optimality. Finally, a numerical example is presented to demonstrate the efficacy of the proposed finite-time discrete stabilization framework.

Finite-Time Bounded Tracking Control for a Class of Neutral Systems

In this paper, we investigate finite-time bounded (FTB) tracking control for a class of neutral systems. Firstly, the dynamic equation of the tracking error signal is given based on the original neutral system. Then, we combine it with the equations of the state vector to construct an error system, where the reference signal and the disturbance signal are fused in a new vector. Next, about the error system, we study the input–output finite-time stability problem of the closed-loop system by utilizing the Lyapunov–Krasovskii functional. We also give a finite-time stability condition in the form of linear matrix inequalities (LMIs). Furthermore, the delay-dependent and delay-independent finite-time bounded tracking controllers are designed separately for the original system. Finally, a realistic example is given to show the effectiveness of the controller design method in the paper.

Finite time stability of linear time varying delay systems using free matrix based integral inequalities

This paper discusses the problem of finite time stability (FTS) for linear delayed systems where the time-varying delay is bounded. First, based on a Lyapunov–Krasovskii Functional (LKF) which contains terms of triple integrals, delay-dependent FTS conditions are provided by introducing some free-weighting matrices. Then, new approximations of the single and multiple quadratic integrals that appear in the LKF derivative are established using integral inequalities, called free-matrix-based integral inequalities. The dimensions of these free matrices integral in our results are less than those obtained in the literature. This reduction in the number of variables does not mean that our method is a particular case but simply that our approach is completely different from the others and therefore more effective. Finally, less conservative design conditions in terms of Linear Matrix Inequalities (LMIs) are proposed and solved by the LMI Tools of MATLAB to show the advantage and effectiveness of the proposed approach.

FINITE-TIME STABILITY OF NONAUTONOMOUS AND AUTONOMOUS LINEAR SYSTEMS

In this research, in view of Lyapunov theory, the finite time stability (FTS) conditions of linear time-delay systems are investigated. Firstly, by using matrix inequality and algebraic inequality methods, the conditions for FTS of nonautonomous and autonomous systems are given respectively. Compared with the existing literature, the judging conditions are easier to verify and have a better conservative type. In addition, by employing the provided FTS theoretical results, several novel criteria for ensuring the stabilization of autonomous delay systems and the stability of impulsive switched nonautonomous time-varying systems are obtained. Eventually, several concrete examples are put forward to validate the theoretical findings.

Formation of a nontrivial finite-time stable attractor in a class of polyhedral sweeping processes with periodic input

We consider a differential inclusion known as a polyhedral sweeping process. The general sweeping process was introduced by J.-J. Moreau as a modeling framework for quasistatic deformations of elastoplastic bodies, and a polyhedral sweeping process is typically used to model stresses in a network of elastoplastic springs. Krejčí’s theorem states that a sweeping process with periodic input has a global attractor which consists of periodic solutions, and all such periodic solutions follow the same trajectory up to a parallel translation. We show that in the case of polyhedral sweeping process with periodic input the attractor has to be a convex polyhedron χ of a fixed shape. We provide examples of elastoplastic spring models leading to structurally stable situations where χ is a one- or two- dimensional polyhedron. In general, an attractor of a polyhedral sweeping process may be either exponentially stable or finite-time stable and the main result of the paper consists of sufficient conditions for finite-time stability of the attractor, with upper estimates for the settling time. The results have implications for the shakedown theory.

Finite-time stability analysis of switched systems with actuator saturation based on event-triggered mechanism

Under the event-triggered scheme, this note focuses on the problem of finite-time stability analysis of the switched system with controller failure and external interference. Firstly, in order to save network communication resources, the mixed event-triggered condition is considered. This mechanism can not only avoid Zeno phenomenon, but also use the sampled information more effectively. In addition, the saturation characteristics of the actuator are fully considered, which can further reduce the number of triggers. According to different event-triggered conditions, an algorithm is proposed to estimate the attraction domain. Then, through the Lyapunov-Krasovskii functional approach and average dwell time technology, the closed-loop switched system is analyzed. The determination condition of finite-time stability is obtained. To illustrate the advantages of our established results, a numerical example is finally given. In addition to saving communication resources, the system stability is guaranteed. The advantages of the proposed method can be verified by simulation.

Lyapunov conditions for finite-time stability of disturbed nonlinear impulsive systems

What we concern in this paper is finite-time control of nonlinear impulsive systems involving external disturbance, where practical finite-time and finite-time stabilization are studied with respect to nonvanishing and vanishing disturbance, respectively. A relationship between the finite settling time and the impulsive frequency is presented to show the stabilizing effect of impulses. It is shown that systems subject to nonvanishing disturbance can enter a disturbance-dependent ultimate bound in a finite-time sense, and a relatively smaller bound of settling time is obtained by utilizing stabilizing impulses. Meanwhile, systems subject to vanishing disturbance can achieve finite-time stabilization at the origin. Moreover, compared with the situation without impulses, the corresponding bound of settling time is also smaller. For the sake of illustrating the validity of proposed results, some examples and their simulations are provided.

Finite-Time Stability for Caputo Nabla Fractional-Order Switched Linear Systems

In this paper, we address the finite-time stability problem of Caputo nabla fractional-order switched linear systems with α∈(0,1). Firstly, the monotonicity of the discrete Mittag-Leffler function is proposed. Secondly, under the per-designed switching rules, the form of the solution for Caputo nabla fractional-order switched linear systems is obtained by using the discrete unit step function. On the above basis, some sufficient conditions of finite-time stability for Caputo nabla fractional-order switched linear systems are proposed, according to the discrete Grönwall inequality and the monotonicity of the discrete Mittag-Leffler function. Finally, simulation verification is carried out via three numerical examples.

Asynchronously finite‐time fault detection for switched systems with unstable modes and intermittent measurements

This study proposes a novel fault detection (FD) filter design approach for switched systems with unstable modes and intermittent measurement; the FD filter design approach considers asynchronous switching and ensures finite-time stability (FTS). First, a model of switched systems with undesirable communication and unstable modes is established. The FD filter is developed using the observer technique, and the conditions of FTS and H∞ performance for the above systems are proposed and verified. The multiple discontinuous Lyapunov function (MDLF) method and the mode-dependent average dwell time (DT) method are used to obtain the switching strategy, which includes fast and slow switching to address the stability of unstable and stable modes, respectively. Compared with conventional approaches, the proposed method can realize less conservative stability conditions and a more flexible design framework, providing an efficient approach for improving system performance. Finally, based on the determined FTS criteria and H∞ performance, the FD filter for switched systems is presented. Simulation results are provided to verify the effectiveness of the proposed method.

Finite-time mixed [formula omitted] and passivity analysis for Takagi–Sugeno fuzzy system with time-varying delays and actuator faults

This paper comes up with an novel augmented Lyapunov–Krasovskii functional (LKF) approach to process finite-time passivity-based H∞ non-fragile state feedback control issue of Takagi–Sugeno fuzzy system (TSFS) via taking the effects of both time-varying delays and actuator faults. Designing a non-fragile controller to guarantee that TSFS is finite-time bounded (FTB) and meet finite-time mixed H∞ and passive performance is the focus of this paper. On the basis of the new augmented LKF and improved boundary techniques, sufficient delay-dependent conditions are inferred to acquire the desired results of the TSFS. Furthermore, the derived finite-time stability conditions are derived in terms of linear matrix inequalities (LMIs) and the controller gain matrices are confirmed through solving LMIs. Subsequently, three examples are addressed to certify the applicability and advantage of the put forward methodology.

Global finite-time stability of delayed quaternion-valued neural networks based on a class of extended Lyapunov-Razumikhin methods.

In this paper, a class of global finite-time stability problem for quaternion-valued neural networks with time-varying delays are investigated by adopting an extended modification Lyapunov-Razumikhin (L-R) method and a new upper bounds estimation of system solution in terms of convergence rate was obtained. Firstly, a new extended method of L-R is proposed to solve the general difficulty to find a proper Lyapunov functional. Then, a new suitable controller is designed, the new conditions of inequalities global finite-time stability are obtained via combining with the former proposed L-R method in the separated real-valued system. Finally, for purpose of verifying the availability of the theorem presented, two given illustrative examples are shown.

Robust Finite-time H∞ Control for Discrete-time Nonlinear Uncertain Singular Systems with Time-delay

This paper deals with the problems of finite-time H∞ control for a class of discrete-time nonlinear singular systems subject to uncertainties and external disturbance. Firstly, the finite-time stability problem is discussed for discrete-time nonlinear uncertain singular systems with time-delay. The sufficient conditions of finite-time stability of discrete-time nonlinear singular systems are established. Then, by using the Lyapunov functional method, a criterion is established to ensure that the closed-loop system with external disturbance is finite-time bounded. We design the controller gain matrix. Finally, we provide a numerical example to illustrate the effectiveness of the proposed results.

New conditions for finite‐time stability of impulsive dynamical systems via piecewise quadratic functions

In this paper, the use of time-varying piecewise quadratic functions is investigated to characterize the finite-time stability of state-dependent impulsive dynamical linear systems. Finite-time stability defines the behavior of a dynamic system over a bounded time interval. More precisely, a system is said to be finite-time stable if, given a set of initial conditions, its state vector does not exit a predefined domain for a certain finite interval of time. This paper presents new sufficient conditions for finite-time stability based on time-varying piecewise quadratic functions. These conditions can be reformulated as a set of Linear Matrix Inequalities that can be efficiently solved through convex optimization solvers. Different numerical analysis are included in order to prove that the presented conditions are able to improve the results presented so far in the literature.

Robust Finite-Time Stability for Uncertain Discrete-Time Stochastic Nonlinear Systems with Time-Varying Delay.

The main concern of this paper is finite-time stability (FTS) for uncertain discrete-time stochastic nonlinear systems (DSNSs) with time-varying delay (TVD) and multiplicative noise. First, a Lyapunov–Krasovskii function (LKF) is constructed, using the forward difference, and less conservative stability criteria are obtained. By solving a series of linear matrix inequalities (LMIs), some sufficient conditions for FTS of the stochastic system are found. Moreover, FTS is presented for a stochastic nominal system. Lastly, the validity and improvement of the proposed methods are shown with two simulation examples.