Problem Of Exponential Stability Research Articles

Exponential stability of nonlinear switched systems with hybrid delayed impulses

AbstractThis article addresses the exponential stability problem of switched systems with hybrid delayed impulses that depend on both current states and historical states. Some sufficient conditions for exponential stability based on Lyapunov functions are obtained. In particular, the potential stabilizing and destabilizing effects of time delays in impulses are investigated. It is shown that when continuous dynamics are stable, the primal stabilizing impulses may destabilize the system if there exist delays in the impulses. Interestingly, under certain conditions, such delayed impulses cannot destroy the stability of the whole system for all impulsive time sequences with arbitrarily bounded impulsive intervals, and the sizes of the delays in the impulses can theoretically be arbitrarily finite. Finally, to illustrate the validity of the derived results, some examples and simulations are provided.

Stability analysis of cyclic switched nonlinear systems via event-triggered method

In this paper, the exponential stability problem is addressed for a class of cyclic switched nonlinear systems with unstable modes. A novel event-triggered cyclic switching scheme (ETCSS) is proposed to generate a cyclic switching signal that exponentially stabilizes the considered system for any given initial configuration. Some easily verifiable stability criteria for switched nonlinear and linear systems are established, and the effects of event parameters on the convergence rate are also analyzed. Different from the existing studies, here the developed switching scheme is not only cyclic but also event-triggered and thereby the switching frequency is relatively low. Finally, a numerical example is given to verify the efficiency of the method.

Exponential stability and stabilization of positive T‐S fuzzy delayed systems

AbstractThe analysis of positive nonlinear delayed systems is of great importance for many real‐world applications. Such systems' stability and stabilization assessment is still an open topic, and there is limited literature on this field. Moreover, further convergence conditions should be considered for many experimental processes, such as exponential stability analysis, which is highly important. Considering the above, we deal in this study with the problem of exponential stability and stabilization of nonlinear fuzzy positive systems with delay. We establish exponential stability criteria using Lyapunov–Krasovskii functional (LKF) and a delay bi‐decomposition approach for bounded and time‐varying delayed systems. The obtained results are then extended to the exponential stabilization case. The control law is designed using Parallel distributed compensation (PDC). The proposed approach, formulated in terms of linear matrix inequalities (LMIs), allows reducing the conservativeness of the delay‐dependent conditions. A comparative study is presented to illustrate the superiority of our method. Moreover, simulation results for the two tanks process show the advantages of the proposed control design.

Exponential stability of neural networks with a time-varying delay via a cubic function negative-determination lemma

The problem of global exponential stability of neural networks with a time-varying delay is studied in this article. Firstly, to fully utilize the cross-term relationships among state variables, an improved augmented delay-product-type Lyapunov-Krasovskii functional, including an extra double integral state, is established for the stability analysis. Accordingly, this augmented LKF derivative is a higher-order function of the time-varying delay. Then, three state vectors are considered to reduce the order of the function to cubic. So, to obtain the feasible negative-definiteness condition of this LKF derivative of non-convexity, a negative-determination lemma for cubic functions is employed to handle this problem. As a result, a novel stability criterion is obtained. Two well-known numerical examples illustrate the effectiveness of the criterion.

ON EXPONENTIAL STABILITY OF A CLASS OF DELAY DIFFERENTIAL EQUATIONS IN NEOCLASSICAL GROWTH MODELS

In this paper, the problem of global exponential stability of delay differential equations in a neoclassical growth model is investigated by comparison techniques via differential inequalities.

Rapid exponential stabilization of nonlinear continuous systems via event-triggered impulsive control

Abstract This article investigates the problem of rapid exponential stabilization for nonlinear continuous systems via event-triggered impulsive control (ETIC). First, we propose a trigger mechanism that, when triggered by a predefined event, causes the closed-loop system exponentially stable. Then, the exponential stabilization is achieved by the designed ETIC with or without data dropout. The case where there are delays in the ETIC signals is also studied, and the exponential stabilization is proved. Finally, a numerical study is presented, along with numerical illustrations of the stability results.

An exponential stabilization of random impulsive control systems and its application to chaotic systems

This paper addresses the problem of pth moment exponential stability and stabilization for random impulsive control systems. Some novel pth moment exponential stability and synchronization approaches are established based on the method of maximum and minimum eigenvalues by using the Lyapunov functions and Razumukhin technique. Finally, we show that the stability and synchronization behavior of random impulses are faster than the fixed time impulses. Some physical examples are given to verify the validity and usefulness of the results obtained.

Exponential stabilisation analysis of a class of delayed inertial memristive neural networks

This paper focuses on the exponential stabilisation problem of inertial memristive neural networks (IMNNs) with unbounded discrete time-varying delays. The considered IMNNs are modelled by second-order derivatives equations by introducing the inertial terms. By using nonsmooth analysis, Lyapunov stability theory, inequality techniques and integral-differential of Lyapunov functional method, a feedback controller is designed to guarantee pth moment exponential stabilisation of the addressed IMNNs under the framework of Filippov solutions. It is worth noticing that the considered time delays of IMNNs can be unbounded. Finally, a numerical example is presented to illustrate the effectiveness of the main theoretical result.

Exponential Stabilization for a Class of Strict-Feedback Nonlinear Time Delay Systems via State Feedback Control Scheme

This paper considers the exponential stabilization problem for a class of strict-feedback nonlinear systems with multiple time-varying delays, whose nonlinear terms satisfy the linear growth condition. The state feedback controller that relies on a positive parameter to be determined is constructed to deal with nonlinear terms. By tactfully introducing the Lyapunov–Krasovskii functional with an exponential function and selecting the applicable parameter to be determined, the implementable state feedback controller can be obtained to guarantee that the closed-loop system is exponentially stable. The proposed state feedback control scheme is finally applied to the control design of two-stage chemical reactor system, which illustrates the effectiveness of the control method.

Formula omitted] stabilization problem for memristive neural networks with time-varying delays

This work explores memristive neural networks’ (MNNs) stability and stabilization problems by considering the time-varying delay and external disturbance. First, the MNNs were transformed into a tractable model by defining the logical switched functions, paving the way to utilize the robust analysis method for the associated connection weights. Second, by proposing a new Lyapunov–Krasovskii functional constructed by the delay-partitioning approach and the free weight matrices, the exponential stability (ES) problem of the transformed MNNs model is investigated. Third, the state feedback controller’s design scheme is devised to ensure the ES of the overall closed-loop system with a prescribed H∞ disturbance attenuation performance level γ. The results are formed in terms of linear matrix inequalities. Finally, two suitable examples express the efficacy of the proposed results.

Exponential stabilisation of delayed Lur'e systems with multi-rate sampled-data control

In this paper, we focus on the multi-rate sampled-data based exponential stabilisation problem of delayed Lur'e systems. First, a merging time sequence is constructed from the sampling instants of multiple sensors to form the so-called analysis interval and the closed-loop system is represented by a system with multiple time-varying delays over the constructed analysis interval. Then, a useful estimation about the norm of the system state is established. On this basis, a stabilisation criterion is proposed by constructing a two-sided looped-functional. The proposed functional drops positive definite restrictions over the whole analysis interval . Therefore, it can reduce the conservatism of the derived criterion. And, the feasible solution of the multi-rate sampled-data controller gain can be provided by solving a set of LMIs. Finally, the given example shows the advantage of the main results.

Exponential stability of switched linear impulsive time-varying system and its application

This paper aims at the problem of exponential stability for switched linear impulsive time-varying system. By constructing two different switched discretized Lyapunov functions, some new sufficient conditions ensuring the global exponential stability of switched linear impulsive time-varying system are provided, which can be employed to the case when all subsystems are unstable. Furthermore, we apply theoretical results to the consensus of multi-agent system with switching topologies. Finally, numerical examples demonstrate the effectiveness of given results.

Fault-tolerant control for T-S fuzzy systems with an aperiodic adaptive event-triggered sampling

This article investigates the exponential stabilization problem for Takagi-Sugeno (T-S) fuzzy systems with actuator faults and aperiodic sampling. First, the fault-tolerant control is designed for T-S fuzzy systems under the given actuator fault model. Second, unlike the existing periodic mechanisms, an improved aperiodic adaptive event-triggered (AET) control scheme is established. The threshold is updated adaptively during the sampling process, which greatly reduces the amount of information transmitted. Then, based on Lyapunov-Krasovskii functional (LKF) method, two sufficient exponential stability conditions are derived. In order to derive the less conservative result, a two-sided looped-functional is constructed. In addition, according to stability conditions and fault-tolerant mechanism, the corresponding sampled-data controller is designed. Finally, a numerical example and a simplified gas turbine system are employed to demonstrate the virtue and effectiveness of application of the proposed scheme.

New criteria of almost sure exponential stability and instability of nonlinear stochastic systems with a generalization to stochastic coupled systems

This paper focuses on the almost sure exponential stability and instability problems for nonlinear stochastic systems. In contrast with previous works, combining with Lyapunov’s second method and inequalities techniques, two classes of less conservative criteria are obtained, depending on piecewise continuous scalar functions (PCSFs). It is highly desirable that the PCSF can not only replace the constant coefficient of the upper bound estimation which belongs to the diffusion operator of a Lyapunov function in available literature but also even be unbounded. Especially, when PCSFs are assumed to be periodic, our theoretical results can be transformed into the case of almost sure exponential stability and instability of nonlinear systems with intermittent noises, and corresponding criteria are derived. Furthermore, the sufficient criteria we present are popularized to stochastic coupled systems taking advantage of graph theory and Lyapunov method. As an application of theoretical results, we put forward an exploration to the synchronization analysis of a class of single-link robot arms system via intermittent noise control. Finally, several numerical examples consisting of a time-varying stochastic system and a stochastic coupled oscillators system as well as single-link robot arms systems are provided to illustrate the feasibility of the stated theoretical results.

Dwell‐time‐based control synthesis of switched positive systems with all unstabilizable subsystems

AbstractThis article investigates the robust exponential stabilization of a class of switched positive systems with uncertainties by co‐designing controllers for subsystems and dwell time switching strategy. The uncertainties refer to interval uncertainties. One of the distinguishing features is that none of the forced individual subsystems is assumed to be stabilized. The other feature is that the stabilization for the unforced switched system can not be solvable by designing dwell time switching signal. First, for switched positive systems without uncertainties, a type of multiple time‐varying co‐positive Lyapunov functions is used, and the computable sufficient conditions on the state feedback controller for each subsystem are derived in the framework of dwell time strategy. Moreover, the exponential stabilization problem is solved by confining the lower and upper bounds of the dwell time, restricting the upper bound of derivative of considered Lyapunov function of the active subsystem, and decreasing the Lyapunov function values at successive switching instants of the overall switched system. Then, the results are extended to the robust exponential stabilization case for switched positive systems with uncertainties, and sufficient conditions are given in the same framework of the dwell time for that of the forced switched systems by further designing state feedback controllers. Finally, illustration examples are given to illustrate the effectiveness of the proposed methods.

Rapid exponential stabilization by boundary state feedback for a class of coupled nonlinear ODE and $ 1-d $ heat diffusion equation

<p style='text-indent:20px;'>In this paper, we solve the problem of rapid exponential stabilization for coupled nonlinear ordinary differential equation (ODE) and <inline-formula><tex-math id="M2">\begin{document}$ 1-d $\end{document}</tex-math></inline-formula> unstable linear heat diffusion. The control acts at a boundary of the heat domain and the heat equation enters in the ODE by Dirichlet connection. We show that the infinite dimensional backstepping transformation introduced recently for stabilization of coupled linear ODE-PDE can deal with a nonlinear ODE and obtain a global stabilization result. Our result is innovative and no similar result can be found in the literature as it combines the three following factors, i) nonlinear term in the ODE subsystem, ii) unstable PDE subsystem and iii) mixed boundary condition. Not only this, the techniques used in this work can provide answers to fundamental questions, such as the stabilization of coupled systems where both subsystems may contain nonlinear terms.</p>

Exponential Stability of Impulsive Cascaded Systems and its Application in Robot Control

In this paper, we study the problem of exponential stability of impulsive cascaded systems. In particular, we provide some sufficient conditions that guarantee the exponential stability of the cascaded systems, provided that the two subsystems are also exponentially stable. The proof of the stability of the cascaded systems is based on the second Lyapunov method and the existence of converse theorems for the stability of impulsive systems. Finally, the usefulness of our results is illustrated by its application to the problem of trajectory tracking for a wheeled robot.

Absolute Exponential Stability for Switching Positive Systems With Time-Delay Based on the Φ-Dependent ADT Strategy

The absolute exponential stability problem for the continuous- and discrete-time, respectively, switching positive nonlinear time-delay system is investigated under the nonlinearity of systems satisfying a certain sector condition in this paper. Two multiple co-positive Lyapunov-Krasovskii functionals (MCLKF) of such systems are constructed for the continuous- and discrete-time contexts respectively. By using the MCLKF and the Φ-dependent average dwell time switching strategy, more general stability conditions compatible with some existing works are established. Furthermore, the obtained results are generalized to other types of systems. A numerical example, finally, implies the validity and significance of the results proposed.

Simultaneous exponential stabilization for stochastic port-controlled Hamiltonian systems with actuator saturations, fading channels and delays

This paper is concerned with the simultaneous exponential stabilization problem for a set of stochastic port-controlled Hamiltonian (PCH) systems. Due to the limited bandwidth of the channels, the phenomena of fading channels and transmission delays which are described by a time-varying stochastic model always occur in the communication channels from the controller to the actuator. Meanwhile, actuator saturation constraint is taken into account. On the basis of dissipative Hamiltonian structural and saturating actuator properties, those stochastic PCH systems are combined to generate an augmented system. By utilizing the stochastic analysis theory, sufficient criterions are given for the simultaneous stabilization controller design ensuring that the closed-loop system is simultaneously exponentially mean-square stable (SEMSS). For the case that there exist external disturbances in the systems, some results on stability analysis and controller design are given. The developed controller design scheme is proved by a three-helicopter model simulation example.

Robust Exponential Stabilization for a Class of Nonlinear Uncertain Systems with Time-varying Delays

This paper is concerned with the problem of robust exponential stabilization for a class of nonlinear uncertain systems with time-varying delays. By using appropriately chosen Lyapunov-Krasovskii functional, together with the Finsler’s lemma, sufficient conditions for exponential stability of nonlinear uncertain systems with time-varying delays are proposed in terms of linear matrix inequality (LMI). Then, novel sufficient conditions are developed to ensure the nonlinear uncertain system with time-varying delay is robust exponentially stabilizable in terms of linear matrix inequality with state feedback control. Finally, a numerical example is given to illustrate the efficiency of proposed methods.