Stability Of Time-delay Systems Research Articles

새로운 이차 함수 음 조건과 이의 시간 지연 선형 시스템의 안정성에 응용

In this paper, we consider the stability of time-delayed linear systems. First, based on the segmentation of interval and the S-procedure, we derive a new sufficient condition guaranteeing that a quadratic function is negative for a closed interval. Of course, necessary and sufficient conditions exist, but these are computationally burdensome due to too many additional variables, so sufficient conditions with few variables are required. Next, we choose an LKF and find the upper bound of its time derivative along the trajectories of systems. To transform it into the form of LMI, we apply the Bessel-Legendre inequality, the reciprocally convex inequality, and derived quadratic negative condition. Finally, two well-known numerical examples are provided to show that the proposed results are valid and less conservative.

The ISS framework for time-delay systems: a survey

At the occasion of Eduardo D. Sontag’s 70th birthday, we provide here an overview of the tools available to study input-to-state stability (ISS) and related notions for time-delay systems. After a hopefully pedagogical presentation of the main differences with respect to the finite-dimensional theory, we review basic stability concepts for input-free time-delay systems, as well as instruments to guarantee them in practice, including the Lyapunov–Krasovskii, Lyapunov–Razumikhin, and Halanay approaches. We then consider the influence of inputs through the notions of ISS, integral ISS, and input-to-output stability and provide both Lyapunov-like and solutions-based characterizations of these properties. We also show how these notions can be helpful for the stability analysis of interconnected systems, whether in cascade or in feedback form. We finally provide a list of questions which remain open until now.

Memoryless parameter-dependent control strategy of stochastic strict-feedback time delay systems

For a class of stochastic strict-feedback nonlinear systems subject to different time delay states, this paper mainly concerns the problem of global asymptotic stabilization. Two new control strategies that the memoryless parameter-dependent state feedback control and the memoryless parameter-dependent output feedback control are taken into consideration, respectively. By skillfully constructing the Lyapunov-Krasovskii (L-K) functional, taking the proper determined parameter and employing the stochastic nonlinear time delay system (SNTDS) stability theory, the global asymptotic stability of the stochastic closed-loop system can be achieved. The proposed output feedback control scheme is finally utilized for the control design of the one-link manipulator system and two-stage chemical reactor system, which can verify the availability of the control approach.

Event-Triggered Stabilization of Linear Time-Delay Systems via Halanay-Type Inequality

Event-Triggered Stabilization of Linear Time-Delay Systems via Halanay-Type Inequality

Stability test for neutral type delay systems: a piecewise linear approximation scheme*

A necessary and sufficient test for the stability of neutral type linear time-delay systems is presented. Our approach combines the Lyapunov functional expressed in terms of the delay Lyapunov matrix for this class of systems with a piecewise linear approximation of the functional argument. As a result, a stability criterion is given in terms of the non-negative definiteness test of a matrix. An example is presented to validate the obtained result.

Global Stabilization for Uncertain Nonlinear Time-Delay Systems With Saturated Input

In this article, global stabilization for a class of nonlinear uncertain time-delay systems with saturated input is considered. The saturated input is handled by a new auxiliary system with dynamic gain and the linear growth condition with unknown growth rates is removed. A novel adaptive observer-controller coupling design framework is put forward, under which the closed-loop system is globally bounded. Finally, a simulation shows that the proposed technique is effective.

Leader-Following Consensus of Heterogeneous Linear Multiagent Systems With Communication Time-Delays via Adaptive Distributed Observers.

This article investigates the leader-following consensus problem of heterogeneous linear multiagent systems under switching and directed topologies. It is assumed that the communication between agents suffers from time-varying delays and only the neighboring agents of the leader are able to get access to the information of the leader agent, including its agent matrices. A key technical lemma on the input to state stability of time-delayed systems is first established with which the main results of this article can be obtained. An adaptive distributed observer, taking into consideration of communication time delays, is proposed for each follower to estimate the leader's system matrices and its state. Then, a distributed controller based on this adaptive observer is developed. We show that the resulting closed-loop multiagent system achieves the leader-following output consensus. Two examples are finally given to illustrate the effectiveness of the proposed controller.

Stability and Feedback Control of Volterra Type Systems with Time Delay

New results for stability and feedback control of time delay systems were proposed. These results were obtained by using Lyapunov Razumikhin method to approximate the stability of the uncontrolled Volterra type system with delay and designing a state feedback controller using a model transformation technique, the Lyapunov matrix equation and the Razumikhin approach for the stabilization of the controlled Volterra type system with delay. Examples are given to illustrate the effectiveness of the theoretical results.

A Semiglobal Approach for Stabilizing Nonlinear Systems With State Delays by Memoryless Linear Feedback

This article investigates the problem of how to control nonlinear systems with delays in the state by memoryless linear feedback. The notions of semiglobal asymptotic stabilization and sublevel sets are introduced in the context of time-delay systems. With the aid of Razumikhin theorem, we develop a semiglobal design method for the construction of Lyapunov functions, associated sublevel sets, and delay-free linear state feedback laws, step-by-step, achieving semiglobal asymptotic stabilization for time-delay nonlinear systems in a lower triangular form. In contrast to the global stabilization of nonlinear systems with delays in the state, which is usually achieved by <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">dynamic state</i> feedback (Lin and Zhang, 2020), the significance of this work is to point out that a tradeoff of the control objectives, e.g., semiglobal versus global stabilization, makes it possible to control a class of time-delay nonlinear systems by <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">static linear</i> feedback. Extensions to nonlinear systems with globally asymptotically locally expoentially stable (GALES)-like inverse dynamics are also included in this article.

Stabilization of nonlinear time-delay systems: Flexible delayed impulsive control

This paper studies the stabilization of nonlinear time-delay systems under flexible delayed impulsive control. Some sufficient conditions are provided for establishing stability property in terms of exponential Lyapunov-Razumikhin functions. It is shown that the size of delay in continuous dynamics can be flexible. Specially, it can be smaller or larger than the impulsive intervals, and there is no magnitude relationship between the delay in continuous flow and impulsive delay. In most existing results, from the impulsive control point of view, the Lyapunov functions were based on the assumption that there was a common threshold at every impulse point. In this study, utilizing the proposed method of average impulsive estimation (AIE), the rate coefficients are flexible, and the impulsive delay can be integrated to guarantee the effect of stabilization of impulses. As an application, the theoretical results are applied to the synchronization of a chaotic neural network, and the impulsive control input is formalized in terms of linear matrix inequalities (LMIs). The efficiency of the derived results is illustrated by two numerical examples.

Potential Impacts of Delay on Stability of Impulsive Control Systems

In this article, the exponential stability of impulsive control systems with time delay is studied. By using the average impulsive interval method, some sufficient Lyapunov-based conditions are established for the stability of impulsive time-delay systems, and the impacts of delay on the stability analysis method are further revealed. It is interesting to show that some unstable impulsive time-delay systems may be stabilized by increasing the time delay in continuous dynamics. More interestingly, it is proved that along with the increase of the delay within a certain range, the convergence rate of such impulsive time-delay systems also increases correspondingly. Further, a strict comparison principle for impulsive control systems with delay is established. Then by utilizing this comparison principle, it can be shown that for some stable impulsive systems with delay, under certain conditions, the stability is robust against any large but bounded delay. Compared with the previous results on delay-free impulsive systems, some potential impacts of delay on the stability are investigated. Particularly, the obtained results are extended to the case of impulsive control systems with hybrid impulses, which contain both stabilizing impulses and destabilizing impulses. Three illustrative examples are presented to reveal the potential impacts of delay on the stability of impulsive control systems.

Analytical Solution of the Fractional Linear Time-Delay Systems and their Ulam-Hyers Stability

We introduce the delayed Mittag-Leffler type matrix functions, delayed fractional cosine, and delayed fractional sine and use the Laplace transform to obtain an analytical solution to the IVP for a Hilfer type fractional linear time-delay systemD0,tμ,νzt+Azt+Ωzt−h=ftof order1&lt;μ&lt;2and type0≤ν≤1, with nonpermutable matricesAandΩ. Moreover, we study Ulam-Hyers stability of the Hilfer type fractional linear time-delay system. Obtained results extend those for Caputo and Riemann-Liouville type fractional linear time-delay systems with permutable matrices and new even for these fractional delay systems.

Study on Hopf Branch of Stability of Time-Delay Unified System and Performance Evaluation Based on Deep Learning

Deep learning is a breakthrough in machine learning research. It aims to establish a deep network structure that can simulate the human brain for analysis and learning, interpret data through the mechanism of layer-by-layer abstract feature representation, and has excellent feature learning capabilities. According to the input-output performance evaluation data of colleges and universities, three experiments are done. First, the feature expression ability of RBM, the basic building block of deep learning, is studied, and compared with PCA, the results show that RBM-fine-tuning has better performance than PCA-expressed classifier; the reconstruction error can be used to judge the hidden layer. As the number of RBM layers increases, the classification accuracy gradually increases, indicating the feasibility of the RBMs feature extractor. Second, the model in this study has a higher prediction accuracy than other classification models and clarifies the effectiveness of the modular deep learning model based on RBMs from the perspectives of network convergence analysis and network output analysis. The ability is stronger than DBN, and the obtained abstract feature representation is more conducive to classification. Although the classification accuracy rate of the model in this study has been improved, the model has certain limitations. The network initialization is still set based on experiments and experience, and the prediction accuracy rate is only 88.3%, which needs to be improved. The parameter training algorithm of RBMs can be further studied. To improve the more accurate reference basis for the performance evaluation of colleges and universities. Third, in the research of dynamical systems, the stability of the time-delay unified system at zero equilibrium and positive equilibrium is studied, and the conditions for generating the Hopf branch are given. At the same time, some conclusions are obtained through theoretical analysis. Numerical simulations further verify the validity of the theoretical results.

Razumikhin and Krasovskii approaches for safe stabilization

This paper studies the stabilization and safety problems of nonlinear time-delay systems. Following both Razumikhin and Krasovskii approaches, we propose novel control Lyapunov functions/functionals for the stabilization problem and novel control barrier functions/functionals for the safety problem. The proposed control Lyapunov and barrier functions/functionals extend the existing ones from the delay-free case to the time-delay case, and allow for designing the stabilizing and safety controllers in closed-form. Since analytical solutions to time-delay optimal control problems are hard to be achieved, a sliding mode control based approach is developed to merge the proposed control Lyapunov and barrier functions/functionals. Based on the sliding surface functional, a feedback control law is established to investigate the stabilization and safety objectives simultaneously. In particular, the properties of the sliding surface functional are analyzed, and further how to construct the sliding surface functional is discussed. Finally, the proposed approaches are illustrated via two numerical examples from the connected cruise control problem of automotive systems and the synchronization problem of multi-agent systems.

Multiplicity–Induced–Dominancy property for second–order neutral differential equations with application in oscillation damping

This paper addresses the exponential stability of linear time-delay systems of neutral type. In general, it is quite a challenge to establish conditions on the parameters of the system in order to guarantee such a stability. Recent works emphasized the link between maximal multiplicity and dominant roots. Indeed, conditions for a given multiple root to be necessarily dominant are investigated, this property is known as Multiplicity-Induced-Dominancy (MID). The aim of this paper is to explore the effect of multiple roots with admissible multiplicities exhibiting, under appropriate conditions, the validity of the MID property for second-order neutral time-delay differential equations with a single delay. The ensuing control methodology is summarized in a five-steps algorithm that can be exploited in the design of higher-order systems. The main ingredient of the proposed method is the dominancy proof for multiple spectral values based on frequency bounds established via integral equations. As an illustration, the stabilization of the classical oscillator benefits from the obtained results.

New finite-time stability for fractional-order time-varying time-delay linear systems: A Lyapunov approach

The primary goal of this paper is to examine the finite-time stability and finite-time contractive stability of the linear systems in fractional domain with time-varying delays. We develop some sufficient criteria for finite-time contractive stability and finite-time stability utilizing fractional-order Lyapunov-Razumikhin technique. To validate the proposed conditions, two different types of dynamical systems are taken into account, one is general time-delay fractional-order system and another one is fractional-order linear time-varying time-delay system, furthermore the efficacy of the stability conditions is demonstrated numerically.

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.