Exponential Stability Of Nonlinear Systems Research Articles

Exponential stability of nonlinear systems via event-triggered impulsive control based on partial states

In the framework of partial unknown states, this paper studies the exponential stability problem of nonlinear systems. A novel event-triggered mechanism (ETM) just involving partial known states is designed to guarantee the exponential stability of considered system, where forced impulse with great freedom degree is introduced in ETM. What's more, the information of known states of the system is fetched and applied to supplement the unknown part at impulse instants. With the help of matrix analysis technique, some linear matrix inequalities (LMI) based sufficient conditions are presented from the perspective of the quantity of known and unknown states. Besides, our proposed conditions establish the relationship of event-triggered parameters, impulse interval, and impulse strength. Finally, expository examples and their numerical modeling are appeared to prove the availability of main results.

Exponential stabilization of nonlinear systems under saturated control involving impulse correction

This paper studies the problem of locally exponential stabilization (LES) of nonlinear systems under a class of hybrid control in the framework of actuator saturation, where the limitation of actuator saturation on both continuous state feedback control and impulsive control are fully considered. Based on impulsive control theory and differential inclusion approach, some sufficient conditions for LES are derived, where a novel set inclusion relation is proposed to handle the double saturation nonlinearities. Different from the existing results that each part of saturated hybrid control (SHC) is required to stabilize the system individually, our results relax the requirement by making full use of the correction effect of the impulse. Moreover, the maximum of estimation of domain of attraction is obtained by a convex optimal problem and corresponding algorithm. The result is applied to the robustness for a class of nonlinear systems. Finally, the validity of the results is shown by two examples and their simulations, where the synchronization problem of Chua’s oscillator is illustrated in the framework of actuator saturation.

Event-Triggered Intermittent-Based Impulsive Control for Stabilization of Nonlinear Systems

In this brief, a novel event-triggered intermittent-based impulsive control (ETIIC) strategy is designed to stabilize a class of nonlinear systems, which possesses the strengths of event-triggered control, intermittent control and impulsive control. Particularly, ETIIC allows that impulsive instants appear only at the control intervals. Moreover, the control intervals can be activated by mean of the well-designed event-triggered mechanism that depends explicitly on threshold-value and check-period. A sufficient criterion is provided to ensure exponential stabilization of nonlinear systems via ETIIC, meanwhile, the non-Zeno behavior is confirmed in our results. At last, an example with numerical simulation is presented to illustrate the feasibility of ETIIC and the effectiveness of our results.

Event-triggered control for nonlinear systems with impulse effects

This paper studies the exponential stability of nonlinear systems involving impulse effects based on event-triggered control, where the impulsive sequence is generated by some well-designed event conditions. On one hand, from the impulsive disturbance point of view, some sufficient conditions are derived based on event-triggered sampled-data control with the help of linear matrix inequalities (LMIs) method, where the control effects of sampled-data control and negative effects of impulses are fully considered. On the other hand, from the impulsive control point of view, some LMI-based conditions are presented based on event-triggered hybrid control, i.e., event-triggered sampled-data control and impulsive control, where a class of forced impulses is introduced to reduce the conservatism of the control strategy. The control gains of sampled-data control and impulsive control can be easily designed via our designed LMI conditions. Moreover, the Zeno behaviour can be excluded in our results. Finally, two numerical examples are presented to illustrate the validity of our control strategies.

Some results on the practical α-exponential stability of nonlinear systems

We study the practical exponential stability of nonlinear systems with generalized sufficient conditions. We use Lyapunov’s direct method, hence we extend previous results on the perturbed systems and the time-varying cascade systems. The last part is devoted to the study of the problem of practical α-exponential stabilization for certain classes of nonlinear systems.

Some results on the practical α-exponential stability of nonlinear systems

We study the practical exponential stability of nonlinear systems with generalized sufficient conditions. We use Lyapunov’s direct method, hence we extend previous results on the perturbed systems and the time-varying cascade systems. The last part is devoted to the study of the problem of practical α-exponential stabilization for certain classes of nonlinear systems.

Exponential stability of nonlinear systems with event-triggered schemes and its application

In this paper, we discuss exponential stability for nonlinear systems with sampled-data-based event-triggered schemes. First, a framework is proposed to analyze exponential stability for nonlinear systems under some different triggering conditions. Based on these results, output feedback exponential stabilization is investigated for a class of inherently nonlinear systems under a kind of event-triggered strategies. Finally, the rationality of the theoretical work is verified by numerical simulations.

Exponential stability of nonlinear systems involving partial unmeasurable states via impulsive control

Abstract This paper investigates the stability problem of partial unmeasurable nonlinear systems under impulsive control. Some sufficient conditions are given to guarantee exponential stability of systems using transition matrix method coupled with dimension expansion technique, where the possibility of the effects of partial unmeasurable states is fully considered. In our proposed method, we not only allow systems to have incomplete states, but also relax restrictions on measurable states, which has a wider range of applications in practice. Finally, two illustrative examples are presented, with their numerical simulations, to demonstrate the effectiveness of main results.

Neural Contraction Metrics for Robust Estimation and Control: A Convex Optimization Approach

This letter presents a new deep learning-based framework for robust nonlinear estimation and control using the concept of a Neural Contraction Metric (NCM). The NCM uses a deep long short-term memory recurrent neural network for a global approximation of an optimal contraction metric, the existence of which is a necessary and sufficient condition for exponential stability of nonlinear systems. The optimality stems from the fact that the contraction metrics sampled offline are the solutions of a convex optimization problem to minimize an upper bound of the steady-state Euclidean distance between perturbed and unperturbed system trajectories. We demonstrate how to exploit NCMs to design an online optimal estimator and controller for nonlinear systems with bounded disturbances utilizing their duality. The performance of our framework is illustrated through Lorenz oscillator state estimation and spacecraft optimal motion planning problems.

Uniform exponential stabilization of nonlinear systems in Banach spaces

This paper presents necessary and sufficient conditions for uniform exponential stabilization of a class of nonlinear systems in Banach state spaces. The stabilization assumptions are formulated in terms of integral estimates involving the control operator and the state of the uncontrolled version of the system at hand. An explicit estimate of the convergence speed is given. Applications to feedback stabilization of affine control systems are given. Illustrative examples are further provided.

A Variational Approach to Local Asymptotic and Exponential Stabilization of Nonlinear Systems

Local asymptotic stabilizability is a topic of great theoretical interest and practical importance. Broadly, if a system \(\dot {x} = f(x,u)\) is locally asymptotically stabilizable, we are guaranteed a feedback controller u(x) that forces convergence to an equilibrium for trajectories initialized sufficiently close to it. A necessary condition was given by Brockett: such controllers exist only when f is open at the equilibrium. Recently, Gupta, Jafari, Kipka, and Mordukhovich considered a modification to this condition, replacing Brockett’s topological openness by the linear openness property of modern variational analysis. In this paper, we show that, under the linear openness assumption, there exists a local diffeomorphism rendering the system exponentially stabilizable by means of continuous stationary feedback laws. Introducing a transversality property and relating it to the above diffeomorphism, we prove that linear openness and transversality on a punctured neighborhood of an equilibrium is sufficient for local exponential stabilizability of systems with a rank-deficient linearization. The main result goes beyond the usual Kalman and Hautus criteria for the existence of exponential stabilizing feedback laws, since it allows us to handle systems for which exponential stabilization is achieved through higher order terms. However, it is implemented so far under a rather restrictive row-rank conditions. This suggests a twofold approach to the use of these properties: a pointwise version is enough to ensure stability via the linearization, while a local version is enough to overcome deficiencies in the linearization.

Exponential Stability of Nonlinear Systems With Delayed Impulses and Applications

We consider a class of nonlinear impulsive systems with delayed impulses, where the time delays in impulses exist between two consecutive impulse instants. Based on the impulsive control theory and the ideas of average dwell time (ADT), a set of Lyapunov-based sufficient conditions for globally exponential stability are obtained. It is shown that the time delay in impulses exhibits double effects (i.e., negative or positive effects) on system dynamics, namely, it may destroy the stability and lead to undesired performance, and conversely, it may stabilize an unstable system and achieve better performance. Then, we apply the theoretical results to synchronization control of chaotic systems and design some impulsive controllers that are formalized in terms of linear matrix inequality and ADT-like conditions. Two numerical examples including chaotic cellular neural network and Chua's oscillator are given to illustrate the applicability of the presented impulsive control schemes. Our examples show that, under some conditions, the impulsive synchronization of chaotic systems can be achieved via the input delays in impulses.

Exponential stability of nonlinear positive systems on time scales

Positive uniform exponential stability of positive nonlinear systems on arbitrary time scales with bounded graininess function is studied. It is proved that linearization preserves positivity of systems. Then it is shown that positive uniform exponential stability of the linear approximation of a nonlinear system implies positive uniform exponential stability of the nonlinear system.

Exponential stability of nonlinear systems with impulsive effects and disturbance input

In this paper, exponential stability of nonlinear systems with impulse time window, disturbance input and bounded gain error is investigated. By means of the above result and the construction of a linear stabilizing feedback controller, another criterion of exponential stability is established. A numerical example is given to demonstrate the effectiveness of the theoretical results.

Robustness Analysis of Global Exponential Stability of Nonlinear Systems With Deviating Argument and Stochastic Disturbance

Robust performance of nonlinear systems has attracted phenomenal worldwide attention. It is well known that deviating argument and stochastic disturbance may derail the evolution properties of nonlinear systems. Then the following issue has become a major bottleneck: for a given globally exponentially stable nonlinear system, the perturbed nonlinear system can sustain how much the length of the interval of the deviating function and the noise intensity so that the perturbed nonlinear system in the presence of deviating argument and stochastic disturbance may remain to be exponentially stable. In this paper, theoretical investigation has been made on the robustness of global exponential stability of nonlinear systems with deviating argument and stochastic disturbance. The allowable upper bounds of the length of interval of deviating function and the noise intensity are derived for the perturbed nonlinear systems to remain exponentially stable. It is also proven that, if the length of interval of deviating function and the noise intensity of perturbed nonlinear systems are lower than the upper bounds derived herein, the nonlinear systems infected by deviating argument and stochastic disturbance are still exponentially stable. Finally, we give several simulation examples to demonstrate the efficacy of the proposed results.

Quasi sure exponential stabilization of nonlinear systems via intermittent <inline-formula><tex-math id="M1">\begin{document}$ G $\end{document}</tex-math></inline-formula>-Brownian motion

This paper focuses on the quasi sure exponential stabilization of nonlinear systems. By virtue of exponential martingale inequality under \begin{document}$ G $\end{document} -framework and intermittent \begin{document}$ G $\end{document} -Brownian motion (in short, \begin{document}$ G $\end{document} -ISSs), we establish the sufficient conditions to guarantee quasi surely exponential stability. The efficiency of the proposed results is illustrated by the memristor-based Chua's oscillator.

Robustness of Global Exponential Stability of Nonlinear Systems With Random Disturbances and Time Delays

The robust stability of nonlinear systems has been studied extensively. It is well known that time delays and additive noises may derail the stability of nonlinear systems. This paper presents theoretical results on the robustness of the exponential stability of nonlinear systems in the presence of time delays and random disturbances. For a given exponentially stable (ES) nonlinear system, it is interesting to know how much time delay and noise intensity there are so that the system may remain to be ES when the system is subject to delay and noise. Upper bounds of allowable delays and noise intensities are derived for nonlinear systems to keep their exponential stability. It is proven that if the noise and delay of ES nonlinear systems are lower than the upper bounds derived herein, the nonlinear systems infected by noises and delays are ensured to be ES. Three numerical examples are given to substantiate the efficacy of the results.

Robust nonlinear sampled‐data system analysis based on Fridman's method and S‐procedure

SummaryThis paper is devoted to the evaluation of sampling interval providing robust exponential stability of nonlinear system with sector‐bounded nonlinearities. It extends our previous results (R. E. Seifullaev, A. L. Fradkov. Sampled‐data control of nonlinear oscillations based on LMIs and Fridman's method. In 5th IFAC International Workshop on Periodic Control Systems, 95‐100. Caen, France. 2013). The proposed approach exploits E. Fridman's method for linear systems based on a general time‐dependent Lyapunov–Krasovskii functional. With classical results of V. A. Yakubovich about S‐procedure, the problem is reduced to feasibility analysis of linear matrix inequalities. The results are illustrated by example: the pendulum system with friction and sector‐bounded multiple nonlinearities. Copyright © 2015 John Wiley & Sons, Ltd.

Robustness analysis of global exponential stability of non‐linear systems with time delays and neutral terms

The global stability of non-linear dynamical systems has been investigated extensively in recent decades. It is well known that time delay and neutral term could derail the stability of non-linear systems. This study presents new results on the robustness of the global exponential stability of non-linear systems with respect to time delay and neutral term. Given globally exponentially stable non-linear systems, the problems to be addressed herein are how much time delay and neutral term contraction coefficient are allowed so that the non-linear systems can remain to be globally exponentially stable, in the presence of time delay and neutral term. Upper bounds of allowable time delay and neutral term contraction coefficient will be derived for non-linear systems to sustain their global exponential stability. A numerical example is provided to illustrate the results.

A new development for the Tikhonov Theorem in nonlinear singular perturbation systems

This paper deals with the exponential stability of nonlinear perturbation systems under a new condition. A novel criterion of exponential stability of nonlinear systems is firstly given in a general form. In this criterion, a new kind of characteristic value is introduced, which makes the exponential stability measurable. Based on this criterion, a new development for the Tikhonov Theorem in nonlinear singular perturbation systems is presented.