Stability Of Time-delay Systems Research Articles

Research on Stability of Time-delay Force Feedback Teleoperation System Based on Scattering Matrix

Abstract The successful implementation of the robot teleoperation system has greatly improved the ability of remote robots. The force feedback technology is applied to the teleoperation system. On the one hand, the main operator manipulates the operator’s related movement information to the target robot; on the other hand, the main hand feeds back the force situation of the target robot in the remote control to the operator. From the robot under the control of the system can more accurately complete the task, with a higher work efficiency. Teleoperation system with force feedback function generally adopts bilateral control, the master and slave end of the system have a certain delay in the transmission process, it makes the stability of the system destroyed. This paper uses a two-port network to simulate the communications link of the teleoperation system as a lossless transmission line, solves the stability problem of teleoperation systems with force feedback in the presence of time delay effectively. Experimental simulation results show that this method can guarantee a higher stability of the system..

Reduced Order and Observer-Based Reset Control Systems with Time Delays

This paper establishes a new mechanism to stabilize plants using reduced order reset controllers. The proposed method uses state feedback to change the dynamics of plants to guarantee oscillation behavior instead of stability, then the reset mechanism will lead to stability. We show that the base system could be unstable while the reset mechanism drives the states to the equilibrium point. The order of the reset controller equals the rank of the plant’s input matrix. We show that the controller dynamics force some states to converge to the equilibrium point within a finite time. The behavior of the rest of the plant’s states depends greatly on the selection of the state feedback gain which can be selected by any appropriate conventional method. Moreover, the stability of reset time-delay systems is addressed based on a similar theorem of the Lyapunov-Krasovskii theory. Sufficient conditions are given in terms of linear matrix inequalities to guarantee asymptotic stability of the overall dynamics. Simulation results are presented to demonstrate the effectiveness of the proposed reset approaches.

Stabilization for time-delay nonlinear systems with unknown time-varying control coefficients

A control scheme based on dynamic gains is proposed for the time-varying nonlinear time-delay systems with unknown control coefficients. A class of Nussbaum functions are introduced to deal with the problem of unknown control directions. Dynamic gains technique and Lyapunov–Krasovskii functional are developed to handle the time delays in nonlinear system. It is shown that the system state is regulated to origin asymptotically, and the boundedness of all closed-loop signals is guaranteed. Simulation results are provided to demonstrate the effectiveness of the proposed methodology.

Practical fixed-time ISS of neutral time-delay systems with application to stabilization by using delays

The concept of practical fixed-time input-to-state stability for neutral time-delay systems with exogenous perturbations is introduced. Lyapunov–Krasovskii theorems are formulated in explicit and implicit ways. Further, the problem of static nonlinear output-feedback stabilization of a linear system with parametric uncertainties, external bounded state and output disturbances by using artificial delays is considered. The constructive control design consists in solving linear matrix inequalities with only four tuning parameters to be chosen. It is shown both, theoretically and numerically, that the system governed by the proposed controller converges faster to the given invariant set than in the case of using its linear counterpart.

Lyapunov–Krasovskii Characterizations of Integral Input-to-State Stability of Delay Systems With Nonstrict Dissipation Rates

In this article, we provide several characterizations of integral input-to-state stability (iISS) of time-delay systems. These characterizations differ in the way the considered Lyapunov–Krasovskii functionals (LKFs) dissipate along the solutions of the system. This dissipation can involve the LKF itself, as in existing iISS characterizations, but can alternatively involve the instantaneous value of the solution’s norm (pointwise dissipation), the supremum norm of the state history (historywise dissipation), or a mix of the two ( <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">${\mathcal K}{\mathcal L}$</tex-math></inline-formula> dissipation). We show that all of them guarantee iISS. By relying on a recent converse result by Y. Lin and Y. Wang, we show that most of them are also necessary for iISS. These relaxed dissipation rates simplify the iISS analysis of time-delay systems and contribute to uniforming iISS theory with that of input-free systems. Proofs rely on several results for time-delay systems that may be of interest on their own, including a novel characterization of global asymptotic stability and the fact that iISS is equivalent to global asymptotic stability of the input-free system plus a uniform bounded energy-bounded state property.

Application of Double Integral Inequality for the Stability Analysis of the Singular Time-Varying Delay System

This paper aims to establish and analyze the stability of continuous-time linear descriptor systems with a time-varying delay, based on the famous method Lyapunov KRASOVSKI function and the Linear Matrix Inequality (LMI) technique. The founded theorem, not only allows the stability, but also the admissibility, so that the regularity and the impulse free of the studied descriptor system. The established criterion has been met using a neutral approach, which is the system transformation that checks singular time delayed system stability via a less complicated computation. Additionally to this new form, a major integral inequality lemma has also been used, due to its conservative results. The novelty of this idea is represented in the exploitation of the new double integral inequality proposed, in the admissibility analysis of singular time-delayed system via a neutral approach, which has been used to deliver an alternative delay dependent criterion in term of (LMI). The purpose of this idea is to check this admissibility of the said system through linear matrix inequality founded by programming it on the software numeric computing platform MATLAB. The originality of this article has been obtained by the combination of the double inequality technique and the neutral approach. At the end of this article, a numerical example has been provided using the LMI toolbox, in order to prove the validity and the efficiency of the proposed method. A comparison table will be presented to show the developed results in some other works compared with the proposed theorem, noting that the simulations of this example have been also performed in the MATLAB environment.

Positivity and Stability of Fractional-Order Linear Time-Delay Systems

Positivity and Stability of Fractional-Order Linear Time-Delay Systems

Positive Solutions and Exponential Stability of Nonlinear Time-Delay Systems in the Model of BAM-Cohen-Grossberg Neural Networks

Positive Solutions and Exponential Stability of Nonlinear Time-Delay Systems in the Model of BAM-Cohen-Grossberg Neural Networks

Improved Design of Nonlinear Control Systems with Time Delay

It is well known that time delay in nonlinear control systems may lead to the deterioration or even destabilization of the closed-loop systems. Therefore, specific analysis techniques and design methods are needed to be developed for nonlinear control systems in the presence of time delay. This chapter aims to give a broad overview of the stability and control of nonlinear time-delay systems. Firstly, we present some motivations and a comprehensive survey for the study of time-delay systems. Then, a brief overview of some control approaches is provided, specifically, the Lyapunov-Krasoviskii functional method for high-order polynomial uncertainties nonlinear time-delay systems, and nonlinear time-delay systems with nonlinear input, the Lyapunov-Razumikhin method for triangular structure nonlinear time-delay systems, dynamic gain control for full state time-delay systems. Finally, an application in chemical reactor systems is provided and some related open problems are discussed.

Stability of time-delay systems via the Razumikhin method

Stability of time-delay systems via the Razumikhin method

Integral input-to-state stable time-delay systems in cascade

We consider the cascade interconnection of two nonlinear time-delay systems, each of which being integral input-to-state stable (iISS). We provide an explicit growth condition on the dissipation rate of the driving system and the input rate of the driven system under which the overall cascade is itself iISS. Building upon recent iISS characterizations of time-delay systems, the method allows to consider Lyapunov–Krasovskii functionals (LKF) that dissipate in a point-wise manner along solutions, which simplifies its applicability as compared to dissipation rates involving the whole LKF itself. Another key feature of our approach is that the growth condition is imposed only on the driving state variables that actually appear in the LKF analysis of the driven subsystem: this feature happens to be new also in a delay-free context and helps reducing the conservatism of existing approaches.

Stability analysis of time-delay incommensurate fractional-order systems

In this paper, the stability of time-delay incommensurate fractional-order systems is investigated. Using the Nyquist Theorem, a non-conservative condition is obtained that guarantees the stability of these systems. A formula is proposed which shows the delay values that break this stability condition. Using this formula associated with the assumption that the zero-delay system is stable, one can find the maximum allowable values for delay that guarantee the stability. The proposed method can be used for incommensurate fractional-order systems without any restrictions on fractional-orders or the system dimension. The effectiveness of the proposed method is investigated by applying it on the gene regularity network model.

Stability and Stabilization of Stochastic Neutral-type Markovian Jump Time-delay Systems with Two Delays

Stability and Stabilization of Stochastic Neutral-type Markovian Jump Time-delay Systems with Two Delays

A new criterion of asymptotic stability for Hopfield neural networks with time-varying delay

The objective of this paper is to analyze the stability of Hopfield neural networks with time-varying delay. For the system to operate in a steady state, it is important to guarantee the stability of Hopfield neural networks with time-varying delay. The Lyapunov-Krasovsky functional method is the main method for investigating the stability of time-delayed systems. On the basis of this method, the stability of Hopfield neural networks with time-varying delay is ana-lysed. It is known that due to such factors as communication time, limited switching speed of various active devices, time delays often arise in various technical systems, which significantly degrade the performance of the system, which can in turn lead to a complete loss of stability. In this regard, a Lyapunov-Krasovsky type delay-product functional was con-structed in the paper, which allows more information about the time delay and reduces the conservatism of the method. Then a generalized integral inequality based on the free matrix was used. A new criterion for asymptotic stability of Hop-field neural networks with time-varying delay, which has less conservatism, was formulated. The effectiveness of the proposed method is illustrated. Thus an asymptotic stability criterion for Hopfield neural networks with time-varying delay was formulated and justified. The expanded Lyapunov-Krasovsky functional is constructed on the basis of delay and quadratic multiplicative functional, and the derivative of the functional is defined by a matrix integral inequality with free weights. The effectiveness of the method is illustrated by a model example.

Delay-Dependent Sliding Mode Variable Structure Control of Vehicle Magneto-Rheological Semi-Active Suspension

The vehicle semi-active suspension with Magneto-Rheological Damper (MRD) has been a hot research topic of this decade, featuring the challenging task of the robust control with actuator time delay considerations. In this study, a delay dependent sliding mode variable structure control, based on the Linear Matrix Inequality (LMI), is proposed to suppress the vibration of the Magneto-Rheological Semi-Active Suspension (MRSS) control system. In accordance with the nonlinear characteristics of MRD, a dynamic model of automotive semi-active suspension system, considering time delay, is established. By defining a parameter-dependent Lyapunov switching functional, the conditions for asymptotic stability of closed-loop time delay system are derived, while the sliding mode variable structure control with reduced conservatism is designed. According to the method of LMI, the asymptotic stability problem of sliding mode is transformed into a feasibility problem, which can be solved by the solver ‘feasp’ in LMI toolbox. In addition, the calculation of the critical time delay of MRSS is expressed as a generalized eigenvalue optimization problem. For comparison purposes, three representative controllers, including a conventional sliding mode controller, a delay dependent controller, and a smith compensation, are studied. Simulation and real vehicle testing on bump and random road responses show that, the designed delay dependent controller can ensure the stability of the suspension system, weaken the influence of time delay on the control performance and effectively improve the ride comfort of the vehicle.

Robust Stability Analysis of Time-Varying Delay Systems

The generalized Myshkis problem consists in analysis of robust stability of time-delay system under variation of its parameters within a set in an infinite-dimensional space. In our paper, we consider a linear system, where the delay is a bounded function under an additional restriction on its growth rate. Some sufficient conditions of solvability of the generalized Myshkis problem are expressed in the form of quadratic convex optimization task. The new approach is based on a modification of the Lyapunov-Razumikhin method.

Order of Legendre-LMI conditions to assess stability of time-delay systems

This paper investigates the stability analysis of time-delay systems through Lya-punov arguments. Using the existence of a complete Lyapunov-Krasovskii functional and relying on the polynomial approximation theory, our main goal is to approximate the complete Lyapunov functional and to take profit of the supergeometric convergence rate of the truncated error part. Necessary and sufficient conditions in the linear matrix inequality (LMI) framework for sufficiently large approximated orders are consequently proposed. Moreover, an estimation of the necessary order is provided analytically with respect to system parameters.

Global output feedback stabilization for nonlinear fractional order time delay systems

Global output feedback stabilization for nonlinear fractional order time delay systems

Extremum seeking via a time-delay approach to averaging

In this paper, we present a constructive approach to extremum seeking (ES) by using a time-delay approach to averaging. We consider gradient-based ES of static maps in the case of one and two variables, and we study two ES methods: the classical one and a more recent bounded ES method. By transforming the ES dynamics into a time-delay system where the delay is the period of averaging, we derive the practical stability conditions for the resulting time-delay system. The time-delay system stability guarantees the stability of the original ES plant. Under assumption of some known bounds on the extremum point, the extremum value and the Hessian, the time-delay approach provides a quantitative calculation on the lower bound of the frequency and on the upper bound of the resulting ultimate bound. We also give a bound on the neighborhood of the extremum point starting from which the solution is ultimately bounded. When the extremum value is unknown, we provide, for the first time, the asymptotic ultimate bound in terms of the frequency in the case of bounded ES. Moreover, our explicit bound on the seeking error of ES control systems allows to select appropriate tuning parameters (such as dither frequency, magnitude, and control gain). Two numerical examples illustrate the efficiency of our method. Particularly, our quantitative bounds are more efficient for the classical ES than for the bounded one. However, the latter bounds correspond to a more general case with unknown extremum value.

Stability Analysis and Robust Stabilization of Uncertain Fuzzy Time-Delay Systems

New sufficient conditions for delay-independent and delay-dependent robust stability of uncertain fuzzy time-delay systems based on uncertain fuzzy Takagi-Sugeno (T-S) models are presented by using the properties of matrix and norm measurements. Further sufficient conditions are formulated, in terms of the linear matrix inequalities (LMIs) of robust stabilization, and are developed via the technique of parallel distributed compensation (PDC), and then the simplification of the conditions for the controller design of uncertain fuzzy time-delay systems. The proposed methods are simple and effective. Some examples below are presented to illustrate our results.