Stability Of Linear Time-delay Systems Research Articles

Robust Stabilization of Linear Time-Delay Systems under Denial-of-Service Attacks.

This research examines new methods for stabilizing linear time-delay systems that are subject to denial-of-service (DoS) attacks. The study takes into account the different effects that a DoS attack can have on the system, specifically delay-independent and -dependent behaviour. The traditional proportional-integral-derivative (PID) acts on the error signal, which is the difference between the reference input and the measured output. The approach in this paper uses what we call the PID state feedback strategy, where the controller acts on the state signal. Our proposed strategy uses the Lyapunov-Krasovskii functional (LKF) to develop new linear matrix inequalities (LMIs). The study considers two scenarios where the time delay is either a continuous bounded function or a differentiable and time-varying function that falls within certain bounds. In both cases, new LMIs are derived to find the PID-like state feedback gains that will ensure robust stabilization. The findings are illustrated with numerical examples.

Regional Stabilization for Linear Time-delay Systems Under Amplitude and Rate Saturations of Actuators

Regional Stabilization for Linear Time-delay Systems Under Amplitude and Rate Saturations of Actuators

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

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

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.

Event-triggered Stabilization of Linear Time-delay Systems by Static Output Feedback Control

In this paper, we study the stabilization problem for a class of linear systems with a time-varying state delay. An event-triggered static output feedback controller is proposed, such that the sampling frequency and the update time of the controller are both lowered. First, we present a novel event-triggering mechanism depending not only on the output but also on an exponential term, with which less sampling is required and Zeno behavior can be excluded at the same time. Some sufficient conditions are then obtained, under which an exponential convergence can be achieved by means of the comparison principle approach. It is further shown that the parameters design can be easily given if the case reduces to the state feedback control. Moreover, two examples are presented to show the effectiveness of the results.

New versions of Bessel–Legendre inequality and their applications to systems with time-varying delay

This paper is concerned with the stability of linear time-delay systems by applying the Lyapunov–Krasovskii (L–K) functional method. Two versions of Bessel–Legendre (B-L) inequality are developed that are more suitable to deal with the stability problem of systems with a time-varying delay. Meanwhile, in order to take full advantage of the interest of the new versions of B-L inequality, a novel L–K functional is properly tailored by integrating the integral information of the state into quadratic and integral terms. As a result, more relaxed stability conditions are obtained, whose effectiveness is illustrated by two commonly-used numerical examples.

Reversals in stability of linear time-delay systems: A finer characterization

In most of the numerical examples of time-delay systems proposed in the literature, the number of unstable characteristic roots remains positive before and after a multiple critical imaginary root (CIR) appears (as the delay, seen as a parameter, increases). This fact may lead to some misunderstandings: (i) A multiple CIR may at most affect the instability degree; (ii) It cannot cause any stability reversals (stability transitions from instability to stability). As far as we know, whether the appearance of a multiple CIR can induce stability is still unclear (in fact, when a CIR generates a stability reversal has not been specifically investigated). In this paper, we provide a finer analysis of stability reversals and some new insights into the classification: the link between the multiplicity of a CIR and the asymptotic behavior with the stabilizing effect. Based on these results, we present an example illustrating that a multiple CIR’s asymptotic behavior is able to cause a stability reversal. To the best of the authors’ knowledge, such an example is a novelty in the literature on time-delay systems.

A survey on Lyapunov-based methods for stability of linear time-delay systems

Recently, stability analysis of time-delay systems has received much attention. Rich results have been obtained on this topic using various approaches and techniques. Most of those results are based on Lyapunov stability theories. The purpose of this article is to give a broad overview of stability of linear time-delay systems with emphasis on the more recent progress. Methods and techniques for the choice of an appropriate Lyapunov functional and the estimation of the derivative of the Lyapunov functional are reported in this article, and special attention is paid to reduce the conservatism of stability conditions using as few as possible decision variables. Several future research directions on this topic are also discussed.

A General Complex Dynamical Network With Time-Varying Delays and Its Novel Controlled Synchronization Criteria

In this paper, a general complex dynamical network with time-varying delays is introduced. Then, the synchronization problem of the complex dynamical network is further investigated. Most studies make use of linear matrix inequalities (LMIs) to obtain synchronization criteria. However, it is difficult to construct the Lyapunov function using LMIs. To avoid constructing the Lyapunov function, the stability of the synchronization state is investigated based on the theory of the asymptotic stability of linear time-delay systems in this paper. Then, we have obtained two new general stability criteria of the synchronization state in complex dynamical networks in light of the stability theory of the linear time-delay system. Hence, these criteria are easy to be verified. Moreover, these synchronization conditions are applicable to networks with different topologies and sizes. In the end, numerical simulations are provided to verify the effectiveness and feasibility of the developed theorems.

Quadratic stabilization for linear time-delay systems with a logarithmic quantizer

In this paper, we study quantized and delayed state feedback control of the linear time invariant (LTI) systems by considering the logarithmic quantizer and the quantized control input. The concepts of quadratic stabilization and control Lyapunov functionals (CLFs) are extended to the LTI systems with delays and quantized signals. After generalizing the sector bounded approach into LTI systems, we propose that a LTI time-delay system via the linear state feedback and quantization is quadratic stabilization if and only if the corresponding uncertain auxiliary system is quadratic stabilization under the case of the single control input or the multiple control inputs. Two examples are provided to illustrate the effectiveness of the proposed methods.

Complete LKF approach to stabilization for linear systems with time-varying input delay

This paper is concerned with stability analysis and stabilization for linear time-delay systems with interval time-varying input delay. By using an augmented complete Lyapunov–Krasovskii functional (LKF) and introducing appropriate terms in dealing with the positiveness of the LKF, we establish new stability and stabilization criteria in terms of linear matrix inequalities (LMIs). The present method leads to some significant improvements over existing results. Moreover, the main feature of this work lies in that the present results are applicable for time-delay systems with unstable delay-free case. Three numerical examples are given to show the effectiveness and merits of the present results.

Sufficient conditions for stability of linear time-delay systems with dependent delays

According to the delay property, linear time-delay (LTD) systems can be classified as LTD systems with dependent delays (LTD.DD) and LTD systems with independent delays (LTDID). This paper reveals that the stability condition for LTDID systems can be applied to LTD.DD systems, a sufficient stability condition for LTD.DD systems is derived from it, while only fewer of the LTD.DD systems can satisfy the stability condition due to the very strict limitation for the delays of the LTD.DD systems. To solve the problem, based on two-dimensional (2-D) hybrid polynomials, some sufficient conditions for stability of LTD.DD systems are proposed. Examples show that the proposed stability test algorithms are simple and valid.

Delay dependent stability of linear time-delay systems

This paper deals with the problem of delay dependent stability for both ordinary and large-scale time-delay systems. Some necessary and sufficient conditions for delay-dependent asymptotic stability of continuous and discrete linear time-delay systems are derived. These results have been extended to the large-scale time-delay systems covering the cases of two and multiple existing subsystems. The delay-dependent criteria are derived by Lyapunov's direct method and are exclusively based on the solvents of particular matrix equation and Lyapunov equation for non-delay systems. Obtained stability conditions do not possess conservatism. Numerical examples have been worked out to show the applicability of results derived.

Semi-global stabilization of linear time-delay systems with control energy constraint

This note is concerned with the problem of semi-globally stabilizing a linear system with an input delay and a constraint on the energy of its input. Under the condition of null controllability with vanishing energy, the parametric Lyapunov equation based L2 low gain feedback is adopted to solve the problem. The proposed approach is applied to the linearized model of the relative motion in the orbit plane of a spacecraft with respect to another spacecraft in a circular orbit around the Earth to validate its effectiveness.

On the conservatism of the sum-of-squares method for analysis of time-delayed systems

In this paper, we present a converse Lyapunov result which shows that using polynomial Lyapunov functionals to prove the stability of linear time-delay systems is not conservative. This result motivates the sum-of-squares approach to stability analysis of linear time-delay systems. This main result is based on an extension of the Weierstrass approximation theorem to show that linear varieties of polynomials can be used to approximate linear varieties of the space of continuous functions.

Semi-Global Stabilization of Linear Time-Delay Systems with Input Energy Constraint

AbstractThis paper is concerned with semi-global stabilization of linear systems with actuator delay and energy constraints. Under the condition of null controllability by vanishing energy, the parametric Lyapunov equation based L2 low gain feedback is adopted to solve the problem. If the delay in the system is exactly known, a delay-dependent controller is designed and if the delay in the system is either time-varying or not exactly known, a delay-independent controller is established. The proposed approach is used in the linearized model of the relative motion in the orbit plane of a spacecraft with respect to another in a circular orbit around the Earth to validate its effectiveness.

On Stability of Linear Time-Delay Systems with Multiple Time-Varying Delays

In this letter, delay-dependent stability criterion for linear time-delay systems with multiple time varying delays is proposed by employing the Lyapunov-Krasovskii functional approach and integral inequality. By the N-segmentation of delay length, we obtain less conservative results on the delay bounds which guarantee the asymptotic stability of the linear time-delay systems with multiple time varying delays. Simulation results show that the proposed stability criteria are less conservative than several other existing criteria.

Parameterization approach to stability and feedback stabilization of linear time-delay systems

This paper develops a new parameterized approach to the problems of delay-dependent analysis and feedback stabilization for a class of linear continuous-time systems with time-varying delays. An appropriate Lyapunov-Krasovskii functional is constructed to exhibit the delay-dependent dynamics. The construction guarantees avoiding bounding methods and effectively deploying injecting parametrized variables to facilitate systematic analysis. Delay-dependent stability provides a characterization of linear matrix inequalities (LMIs)-based conditions under which the linear time-delay system is asymptotically stable with a γ-level £2 gain. By delay-dependent stabilization, a state-feedback scheme is designed to guarantee that the closed-loop switched system enjoys the delay-dependent asymptotic stability with a prescribed γ-level £2 gain. It is established that the methodology provides the least conservatism in comparison with other published methods. Extension to systems with convex-bounded parameter uncertainties in all system matrices is also provided. All the developed results are tested on representative examples.

Synchronization Stability in Weighted Complex Networks with Coupling Delays

Realistic networks display not only a complex topological structure, but also a heterogeneous distribution of weights in connection strengths. In addition, the information spreading through a complex network is often associated with time delays due to the finite speed of signal transmission over a distance. Hence, the weighted complex network with coupling delays have meaningful implications in real world, and resultantly gains increasing attention in various fields of science and engineering. Based on the theory of asymptotic stability of linear time-delay systems, synchronization stability of the weighted complex dynamical network with coupling delays is investigated, and simple criteria are obtained for both delay-independent and delay-dependent stabilities of synchronization states. The obtained criteria in this paper encompass the established results in the literature as special cases. Some examples are given to illustrate the theoretical results.

Positive Forms and Stability of Linear Time-Delay Systems

We consider the problem of constructing Lyapunov functions for linear differential equations with delays. For such systems it is known that exponential stability implies the existence of a positive Lyapunov function which is quadratic on the space of continuous functions. We give an explicit parameterization of a sequence of finite-dimensional subsets of the cone of positive Lyapunov functions using positive semidefinite matrices. This allows stability analysis of linear time-delay systems to be formulated as a semidefinite program.