Time-varying Delay Systems Research Articles

Stability Analysis of Time-Delay Systems via a Delay-Derivative-Partitioning Approach

This paper is devoted to the study of delay-dependent stability of time-varying delay systems. A delay-derivative-partitioning approach is proposed. By constructing an augmented Lyapunov functional that contains two delay-product-dependent terms and using the delay-derivative-partitioning approach, an improved stability condition is established. The derived condition is described as a set of linear matrix inequalities, which can be readily implemented. Finally, four examples are carried out so as to attest the effect and merit of the presented approach.

Further results on stability analysis of time-varying delay systems via novel integral inequalities and improved Lyapunov-Krasovskii functionals

<abstract><p>This work develops some novel approaches to investigate the stability analysis issue of linear systems with time-varying delays. Compared with the existing results, we give three innovation points which can lead to less conservative stability results. Firstly, two novel integral inequalities are developed to deal with the single integral terms with delay-dependent matrix. Secondly, a novel Lyapunov-Krasovskii functional with time-varying delay dependent matrix, rather than constant matrix is constructed. Thirdly, two improved stability criteria are established by applying the newly developed Lyapunov-Krasovskii functional and integral inequalities. Finally, three numerical examples are presented to validate the superiority of the proposed method.</p></abstract>

Further results on finite-time stability of neutral nonlinear multi-term fractional order time-varying delay systems

In this paper, the finite-time stability for nonlinear neutral multi-term fractional order systems with time-varying input and state delays is investigated. By use of the generalized Gronwall inequality and extended form of the generalized Gronwall inequality, new sufficient conditions for finite-time stability of such systems are obtained. Finally, numerical examples are given to illustrate the effectiveness and applicability of the proposed theoretical results.

Finite-Time Boundedness of Linear Uncertain Switched Positive Time-Varying Delay Systems with Finite-Time Unbounded Subsystems and Exogenous Disturbance

The problem of finite-time boundedness for a class of linear switched positive time-varying delay systems with interval uncertainties and exogenous disturbance is addressed. This characteristic research is that the studied systems include the finite-time bounded subsystems and finite-time unbounded subsystems. Both a slow mode-dependent average dwell time and a fast mode-dependent average dwell time switching techniques are utilized reasonably. And by applying a copositive Lyapunov-Krasovskii functional, novel delay-dependent sufficient criteria are derived to guarantee such systems to be finite-time bounded concerning the given parameters and designed switching signal. Furthermore, new finite-time boundedness criteria of the systems without interval uncertainties are also obtained. Finally, the efficiency of the theoretical results is presented in two illustrative examples.

Stability Analysis of Time-Varying Delay Systems with an Improved Type of LKF

This paper further investigates the problem of stability for a general linear system with time-varying delays. Firstly an improved type of Lyapunov–Krasovskii functional is introduced with integral and nonintegral terms and time-correlation terms. Referring a few existing papers, some valid inequalities mathematical analysis techniques are used in this paper in order to reduce the conservatism of the system. Finally, two examples are presented to demonstrate the advantages of the proposed tactics in this paper.

An improved exponential stability criterion and distributed H ∞ filter design for spatially interconnected discrete-time systems with time-varying delay

This paper proposes an improved exponential stability analysis criterion and designs a distributed filter for a spatially interconnected discrete-time time-varying delay (SIDTD) system. Firstly, combining the Wirtinger-based inequality with reciprocally convex combination inequality, a new and less conservative exponential stability criterion, which guarantees the well-posedness, exponential stability, and performance of the SIDTD system, is reduced to standard linear matrix inequalities. Moreover, we design a distributed structure-preserving filter based on the new stability result. The structure-preserving filter and the delay-free filter can be obtained by employing the Finsler lemma. Finally, the validity and advantages of the derived results are shown by a numerical example and a practical example.

Estimates for Solutions to a Class of Nonlinear Time-Varying Delay Systems

Estimates for Solutions to a Class of Nonlinear Time-Varying Delay Systems

Observer-Based Stabilization of Linear Discrete Time-VaryingDelay Systems

Abstract This paper presents the design of an observer-based stabilizing controller for linear discrete-time systems subject to interval time-varying state-delay. In this work, the problem has been formulated in convex optimization framework by constructing a new Lyapunov–Krasovskii (LK) functional to derive delay-dependent stabilization criteria. The summation inequality and the extended reciprocally convex inequality are exploited to obtain a less conservative delay upper bound in linear matrix inequality (LMI) framework. The derived stability conditions are delay-dependent and thus ensure global asymptotic stability in presence of any time-delay less than the obtained delay upper bound. Numerical examples are included to demonstrate the usefulness of the developed results.

Exponential stability of integral time-varying delay system

This paper investigates exponential stability problem of integral time-varying delay system. Based on a novel exponential stability theorem, sufficient conditions for exponential stability of integral time-varying delay system are obtained in the form of coupled linear matrix inequalities (LMIs). These sufficient stability conditions cover some previous results as special cases when the integral time-varying delay system reduce to the integral time-invariant delay system. These sufficient conditions cannot be obtained directly from Theorem 1 given in Li, Zheng, and Wang [(2016). Exponential stability analysis of integral delay systems with multiple exponential kernels. Journal of the Frankline Institute, 353, 1639–1653. https://doi.org/10.1016/j.jfranklin.2015.12.016] due to the presence of time-varying delays. This is the main motivation for the research being carried out in this paper. Four example s are provided to show the effectiveness and advantages of the proposed method.

Adaptive consensus-based distributed [formula omitted] filtering with switching topology subject to partial information on transition probabilities

This article is dedicated to designing distributed state estimators for time-varying delay systems. The filtering network communication topology is time-varying, and it is presumed that the switching mechanism follows a homogeneous Markov chain, with partially unknown information in the probability transition matrix. An adaptive event-triggered mechanism is embedded in the sensor network, which reduces unnecessary communication data and saves communication resources. By utilising the mode-dependent Lyapunov function, sufficient conditions are deduced to ensure the H∞-consensus performance for the filtering error dynamics. Then, some intensive analysis is performed to obtain the explicit expressions of the distributed H∞ state estimator by introducing slack variables and free-connection weighting matrices. Finally, two numerical simulations are proposed to demonstrate the effectiveness of theoretical results.

Robust finite-time sliding mode control for discrete-time singular system with time-varying delays

This paper explores the finite-time bounded issue for discrete-time singular time-varying delay system via sliding mode control method. A suitable discrete-time sliding mode control law is constructed to drive the state trajectories onto the specified sliding surface in a given finite time interval. Meanwhile, sufficient conditions for finite-time bounded to the closed-loop delayed system are provided in both reaching phase and sliding motion phase. In addition, the finite-time sliding mode controller gain matrix can be solved by using the linear matrix inequalities approach. Finally, three numerical examples are illustrated to demonstrate the superiority and practicability of presented results.

Hierarchical stability conditions of systems with time-varying delay

This paper studies the stability problem of time-varying delay systems. Firstly, By constructing a delay-product Lyapunov-Krasovskii functional, a stability criterion is established, where the advantages of convexity properties are being exploited. This stability criterion in terms of linear matrix inequality (LMI) can be checked exactly using an improved negativity condition of quadratic polynomials. It is demonstrated in the numeric examples that the proposed method can greatly reduce the conservativeness without leading to increase the number of decision variables.

Composite slack-matrix-based integral inequality and its application to stability analysis of time-delay systems

This paper focuses on the delay-dependent stability problem of time-varying delay linear systems. A composite slack-matrix-based integral inequality (CSMBII) is presented in delay-product types. It overcomes the limitation of reciprocal convexity in Bessel–Legendre integral inequality such that CSMBII is more convenient to cope with time-varying delay systems. And it also overcomes the shortcoming that the slack matrices in previous works are independent of time-varying delay. Based on CSMBII, a new delay-dependent stability criterion is derived for time delay systems. A numerical example is given to illustrate the effectiveness of the stability criterion.

Robust [formula omitted] observer-based sliding mode control for uncertain Takagi–Sugeno fuzzy descriptor systems with unmeasurable premise variables and time-varying delay

This work focuses on the observer-based sliding mode control (OBSMC) synthesis for non-linear T-S fuzzy descriptor time-varying delay systems, which subject to uncertainties and unmeasurable premise variables (UPVs). Due to the state variables are difficult to completely accessible, a sliding mode observer (SMO) is designed to estimate the unmeasurable variables, and the nonlinear systems are represented to an equivalent multiple model (MM) forms under the MM formulation with UPVs. Then two sliding surfaces are proposed for the observer system and error system respectively to ensure stabilization, which can effectively deal with the nonlinearity and uncertainties as well. By the utilization of the linear matrix inequality (LMI) technique, we present the stability criterions such that the sliding mode dynamics with H∞ performance are proved to be admissible. Moreover, the SMC is obtained so we can drive the system trajectories to the predefined sliding surface in a finite time. At last, three simulation examples are provided to certificate the availability of our approach.

Robust Stability and Stabilization Conditions for Nonlinear Networked Control Systems With Network-Induced Delay via T–S Fuzzy Model

Robust stability and stabilization problems for a class of discrete-time nonlinear systems are studied in this article. The control signal is transmitted to the nonlinear systems through a lossy communication channel in which time-varying network-induced delay occurs. A Takagi–Sugeno (T–S) fuzzy model is used to represent the nonlinear systems. Using membership function boundary information, a novel membership-function-dependent (MFD) analysis approach is presented. Unlike most previous results, the proposed MFD analysis approach does not need to increase the number of slack variables to reduce conservatism. Combining auxiliary matrices and this approach, we propose some new fuzzy Lyapunov–Krasovskii functionals (FLKFs) that do not require the restrictions of positive definiteness imposed on Lyapunov matrices. Using the FLKF and MFD analysis approach, a sufficient condition in the form of linear matrix inequalities (LMIs) is developed to guarantee the stability of T–S fuzzy time-varying delay systems. Based on the results for stability, less conservative stabilization conditions are also derived. Finally, numerical examples show that less conservative results can be obtained by the proposed approach. The inverted pendulum system with state delay or network-induced delay is also studied to verify the effectiveness of the suggested approach.

Universal adaptive control for a class of nonlinear time-varying delay systems with unknown output function

This paper considers the stabilization problem for a class of nonlinear time-varying delay systems with unknown homogeneous growth rates and unknown output function. By introducing the dynamic gain into the system, an adaptive observer is designed. Then, combined with the Lyapunov–Krasovskii functional, an adaptive controller is proposed to guarantee the global stabilization of the closed-loop system. Two examples are provided to verify the effectiveness of the designed controller.

Exponential stability of time delay systems based on intermediate polynomial-based weighted functions

We investigate exponential stability of time-varying delay systems in this article. By considering intermediate polynomials and introducing slack variables, we develop new intermediate polynomial-based weighted functions (IPWFs). Then an exponential stability criterion, in terms of linear matrix inequalities (LMIs), is obtained by choosing an appropriate Lyapunov–Krasovskii functional (LKF) and some novel techniques on IPWFs. Numerical results show that the criterion obtained in this article is less conservative than some previous results.

Sufficient Stability Conditions for Time-varying Networks of Telegrapher's Equations or Difference-Delay Equations

We give a sufficient condition for exponential stability of a network of lossless telegrapher's equations, coupled by linear time-varying boundary conditions. The sufficient conditions is in terms of dissipativity of the couplings, which is natural for instance in the context of microwave circuits. Exponential stability is with respect to any $L^p$-norm, $1\leq p\leq\infty$. This also yields a sufficient condition for exponential stability to a special class of linear time-varying difference delay systems which is quite explicit and tractable. One ingredient of the proof is that $L^p$ exponential stability for such difference delay systems is independent of $p$, thereby reproving in a simpler way some results from [3].

Adaptive Observer for Systems with Distributed Output Delay

Abstract We are considering the problem of sampled-data observer design for nonlinear time-varying delay systems that are state- and parameter-affine. The novelty is that the system is subject to both distributed delay and parameter uncertainty. A Kalman-like observer is developed to deal with both state and parameter estimation. Its main components are: (i) a time-varying-gain state-estimator involving both output and parameter rate injections; (ii) a distributed-nature adaptive output-predictor that compensates for delay and output sampling delay; (iii) an optimized parameter-estimator that copes with parameter uncertainty, optimization is intended in the sense that we make use of all available information, unlike similar existing estimators. The resulting observer is shown to be exponentially convergent, for small delays and sampling intervals, provided the input signal is sufficiently exciting. The analysis is performed using a Lyapunov-Krasovskii functional, Halanay’s lemma, Wirtinger’s inequality and other tools.

Stability of neural-network based train cruise advisory control with aperiodical measurements

Abstract In this paper, a neural-network based driver advisory train cruise control system is considered. The controller assists the train driver with advisory signals by considering train and driver’s actual state information (attention and fatigue) measured by dedicated sensors. Considering delays in sensor measurements, this paper aims to assess closed-loop stability of driver-in-the-loop advisory train cruise control. For this purpose, the driver model is considered as a time-varying system, the train model includes rolling and aerodynamic resistance forces and the advisory control is considered to be a sampled-data based three layer multi-layer perceptron. Further, the aperiodic measurement problem is approached as stability analysis of time-varying delayed system. Based on recent developments on the design of augemented Lyapunov Krasovskii Functional (LKF) using Bessel-Legendre inequality for time-varying delays, sufficiency conditions for the existence of L2 stability of the driver-train system in terms of solvable Linear Matrix Inequalities are provided. Further a case study is presented to illustrate the effectiveness of the proposed method.