Cases Of Time-varying Delays Research Articles

Delay-Dependent Exponential Stabilization for Linear Distributed Parameter Systems With Time-Varying Delay

This paper extends the framework of Lyapunov–Krasovskii functional to address the problem of exponential stabilization for a class of linearly distributed parameter systems (DPSs) with continuous differentiable time-varying delay and a spatiotemporal control input, where the system model is described by parabolic partial differential-difference equations (PDdEs) subject to homogeneous Neumann or Dirichlet boundary conditions. By constructing an appropriate Lyapunov–Krasovskii functional candidate and using some inequality techniques (e.g., spatial integral form of Jensen's inequalities and vector-valued Wirtinger's inequalities), some delay-dependent exponential stabilization conditions are derived, and presented in terms of standard linear matrix inequalities (LMIs). These stabilization conditions are applicable to both slow-varying and fast-varying time delay cases. The detailed and rigorous proof of the closed-loop exponential stability is also provided in this paper. Moreover, the main results of this paper are reduced to the constant time delay case and extended to the stochastic time-varying delay case, and also extended to address the problem of exponential stabilization for linear parabolic PDdE systems with a temporal control input. The numerical simulation results of two examples show the effectiveness and merit of the main results.

Impulsive control of unstable neural networks with unbounded time-varying delays

This paper considers the impulsive control of unstable neural networks with unbounded time-varying delays, where the time delays to be addressed include the unbounded discrete time-varying delay and unbounded distributed time-varying delay. By employing impulsive control theory and some analysis techniques, several sufficient conditions ensuring μ-stability, including uniform stability, (global) asymptotical stability, and (global) exponential stability, are derived. It is shown that an unstable delay neural network, especially for the case of unbounded time-varying delays, can be stabilized and has μ-stability via proper impulsive control strategies. Three numerical examples and their simulations are presented to demonstrate the effectiveness of the control strategy.

Neuronal State Estimation for Neural Networks With Two Additive Time-Varying Delay Components.

This paper is concerned with the state estimation for neural networks with two additive time-varying delay components. Three cases of these two time-varying delays are fully considered: 1) both delays are differentiable uniformly bounded with delay-derivative bounded by some constants; 2) one delay is continuous uniformly bounded while the other is differentiable uniformly bounded with delay-derivative bounded by certain constants; and 3) both delays are continuous uniformly bounded. First, an extended reciprocally convex inequality is introduced to bound reciprocally convex combinations appearing in the derivative of some Lyapunov-Krasovskii functional. Second, sufficient conditions are derived based on the extended inequality for three cases of time-varying delays, respectively. Third, a linear-matrix-inequality-based approach with two tuning parameters is proposed to design desired Luenberger estimators such that the error system is globally asymptotically stable. This approach is then applied to state estimation on neural networks with a single interval time-varying delay. Finally, two numerical examples are given to illustrate the effectiveness of the proposed method.

Effects of Mixed-Modes on the Stability Analysis of Switched Time-Varying Delay Systems

A phenomenon, known as mixed-mode , occurs naturally in switched delay systems, and it makes the stability analysis for such systems a challenge, especially for the case of time-varying delays. This note aims to investigate the effects of mixed-modes on the stability analysis of switched time-varying delay systems, which are motivated from networked control applications. By constructing a Lyapunov-Krasovskii functional and using the estimation of quadrature , as well as the average dwell time approach, new exponential stability conditions are given for the switched time-varying delay systems with both stable and unstable subsystems. This can be seen as a generalization of existing results that mainly focus on the cases of constant delay and all stable subsystems. Finally, simulation examples are presented to illustrate the developed results.

Sampled-Data Stabilization for Fuzzy Genetic Regulatory Networks with Leakage Delays.

This paper deals with the sampled-data stabilization problem for Takagi-Sugeno (T-S) fuzzy genetic regulatory networks with leakage delays. A novel Lyapunov-Krasovskii functional (LKF) is established by the non-uniform division of the delay intervals with triplex and quadruplex integral terms. Using such LKFs for constant and time-varying delay cases, new stability conditions are obtained in the T-S fuzzy framework. Based on this, a new condition for the sampled-data controller design is proposed using a linear matrix inequality representation. A numerical result is provided to show the effectiveness and potential of the developed design method.

A novel control algorithm for the self-organized fission behavior of flocking system with time delay

This paper studies the self-organized fission control problem for flocking system with time delay. Both constant and time-varying time delay cases are considered. Firstly, a novel information coupling degree (ICD) based fission control algorithm, which is able to split a coherent flock into multiple sub-groups under conflict external stimuli, is proposed. Then, for the case of constant time delay, the sufficient conditions for the fission control algorithm is derived using Lyapunov-Razumikhin theorem. For the case of time-varying time delay, Lyapunov-Krasovskii functional method is adopted to obtain the sufficient conditions for the fission control algorithm in terms of linear matrix inequalities (LMIs). Finally, numerical simulations are provided to illustrate the effectiveness of the theoretical results.

Decentralized attitude synchronization tracking control for multiple spacecraft under directed communication topology

This paper studies the attitude synchronization tracking control of spacecraft formation flying with a directed communication topology and presents three different controllers. By introducing a novel error variable associated with rotation matrix, a decentralized attitude synchronization controller, which could obtain almost global asymptotical stability of the closed-loop system, is developed. Then, considering model uncertainties and unknown external disturbances, we propose a robust adaptive attitude synchronization controller by designing adaptive laws to estimate the unknown parameters. After that, the third controller is proposed by extending this method to the case of time-varying communication delays via Lyapunov–Krasovskii analysis. The distinctive feature of this work is to address attitude coordinated control with model uncertainties, unknown disturbances and time-varying delays in a decentralized framework, with a strongly connected directed information flow. It is shown that tracking and synchronization of an arbitrary desired attitude can be achieved when the stability condition is satisfied. Simulation results are provided to demonstrate the effectiveness of the proposed control schemes.

Delay-dependent Stability Analysis of Microgrid with Constant and Time-varying Communication Delays

This article describes the procedure to develop a cyber physical model of a microgrid considering the influence of constant and time-varying communication delays. The microgrid central controller transmits information through an open communication network that imposes certain constraints. One of the major constraints is communication delays, which affect the dynamic performance of the system. These delays may be constant or time varying. The impact of these time delays on the stability is addressed by finding an index called the time-delay margin. This article proposes an analytical method to determine the delay margin of a distributed generation based microgrid with constant and time-varying delays. In the case of constant time delays, the transcendental characteristic equation of the proposed microgrid model is transformed into a polynomial. The real roots of the polynomial coincide exactly with the imaginary roots of the characteristic equation, which decides the delay-dependent stability of the system. For the case of time-varying delays, a sufficient condition that verifies the asymptotic stability of the system is presented in this article. The delay margin values are theoretically calculated for different sets of microgrid central controller gains, and their accuracy is validated by simulation studies.

Stability Criteria for Nonlinearly Perturbed Load Frequency Systems With Time-Delay

In this paper, new criteria are presented for ascertaining delay-dependent stability of networked multi-area load frequency control (LFC) systems with feedback loop delay in presence of unknown and time-varying exogenous load disturbance. The proposed stability criteria are based on Lyapunov-Krasovskii (LK) functional approach, and they are expressed in linear matrix inequality (LMI) framework. Since the unknown exogenous load disturbance in the power system (cause) ultimately affects the evolution of the state variables of the system (effect), in the reported stability criteria, the load disturbance is mathematically modelled as a norm-bounded nonlinear time-varying function of current and delayed state vectors, and subsequently incorporated into the delay-dependent stability analysis. Two cases of time-delays are considered in the stability analysis: time-varying single feedback-loop delay element and time-invariant multiple feedback-loop delay elements. In addition, to realize less conservative stability criteria with minimal number of decision variables, in the stability analysis, tighter integral inequalities, viz., reciprocal convex combination lemma (for time-varying delay case) and Jenson integral inequality (for time-invariant delay case) are employed. The proposed results are illustrated on standard benchmark LFC systems, and supported suitably by simulation results.

Robust [formula omitted] and [formula omitted] memory filter design for linear uncertain discrete-time delay systems

This paper is concerned with the problems of robust full-order H2 and H∞ filter design for linear uncertain discrete-time systems with multiple state delays. The uncertain parameters affecting the matrices of the system are supposed to be time-invariant and to belong to a polytopic domain. The main novelty is the fact that the filter contains an arbitrary number of past states and past output measures of the system, yielding a filtering system with memory. Linear matrix inequality relaxations based on polynomially parameter-dependent Lyapunov matrices and slack variables are proposed for the H2 and H∞ filter design. Due to the extra dynamics introduced through the delayed states, the robust memory filter is able to provide less conservative results in terms of the H∞ and the H2 performance when compared to the memoryless case. Throughout the paper, the multiple delays are considered to be fixed and time-invariant, but an extension of the conditions to cope with unknown delays belonging to a given interval is also presented for both time-varying and time-invariant delay cases. Numerical examples are given to demonstrate the improvements of the proposed approach with respect to other methods from the literature.

Exponential stabilization of non-autonomous delayed neural networks via Riccati equations

This paper concerns with the problem of exponential stabilization for a class of non-autonomous neural networks with mixed discrete and distributed time-varying delays. Two cases of discrete time-varying delay, namely (i) slowly time-varying; and (ii) fast time-varying, are considered. By constructing an appropriate Lyapunov–Krasovskii functional in case (i) and utilizing the Razumikhin technique in case (ii), we establish some new delay-dependent conditions for designing a memoryless state feedback controller which stabilizes the system with an exponential convergence of the resulting closed-loop system. The proposed conditions are derived through solutions of some types of Riccati differential equations. Applications to control a class of autonomous neural networks with mixed time-varying delays are also discussed in this paper. Some numerical examples are provided to illustrate the effectiveness of the obtained results.

Exponential stabilization of linear systems with time-varying delayed state feedback via partial spectrum assignment

We consider the problem of controlling a linear system when the state is available with a known time-varying delay (delayed-state feedback control) or the actuator is affected by a delay. The solution proposed in this paper consists in partially assigning the spectrum of the closed-loop system to guarantee the exponential zero-state stability with a prescribed decay rate by means of a finite-dimensional control law. A non conservative bound on the maximum allowed delay for the prescribed decay rate is presented, which holds for both cases of constant and time-varying delays. An advantage over recent and similar approaches is that differentiability or continuity of the delay function is not required. We compare the performance of our approach, in terms of delay bound and input signal, with another recent approach.

On designing decentralized impulsive controllers for synchronization of complex dynamical networks with nonidentical nodes and coupling delays

This paper investigates the problem of designing decentralized impulsive controllers for synchronization of a class of complex dynamical networks (CDNs) about some prescribed goal function. The CDNs are allowed to possess nonidentical nodes and coupling delays. Two cases of time-varying coupling delays are considered: the case where the coupling delays are uniformly bounded, and the case where the derivatives of the coupling delays are not greater than 1. The synchronization analysis for the first case is performed by applying a time-varying Lyapunov function based method combined with Razumikhin-type technique, while the synchronization analysis for the second case is conducted based on a time-varying Lyapunov functional based method. For each case, by utilizing a convex combination technique, the resulting synchronization criterion is formulated as the feasibility problem of a set of linear matrix inequalities (LMIs). Then, sufficient conditions on the existence of a decentralized impulsive controller are presented by employing these newly obtained synchronization criteria. The local impulse gain matrices can be designed by solving a set of LMIs. Finally, two representative examples are given to illustrate the correctness of the theoretical results.

New Global Synchronization Analysis for Complex Networks with Coupling Delay

Global synchronization analysis for complex networks with coupling delay is investigated. Firstly the constant time delay is analyzed and then the case for time-varying delay is considered. Sufficient conditions for network synchronization are given based on Lyapunov functional, linear matrix inequality, and Kronecker product technique. The unknown variables in the sufficient conditions are fewer than those in the recent reference. Moreover, for the time-varying delay case, we find that the conditions are dependent on the bounds of both time delay and its derivative, and the derivative of the time-varying delay can be any value in the bounds. Finally, numerical examples are given to validate the effectiveness of the obtained results.

Delay-dependent coupling for a multi-agent LTI consensus system with inter-agent delays

Delay-dependent coupling (DDC) is considered in this paper in a broadly studied linear time-invariant multi-agent consensus system in which agents communicate with each other under homogeneous delays, while attempting to reach consensus. The coupling among the agents is designed here as an explicit parameter of this delay, allowing couplings to autonomously adapt based on the delay value, and in order to guarantee stability and a certain degree of robustness in the network despite the destabilizing effect of delay. Design procedures, analysis of convergence speed of consensus, comprehensive numerical studies for the case of time-varying delay, and limitations are presented.

New stability criteria for singular systems with time-varying delay and nonlinear perturbations

This paper concerns the stability analysis for singular systems with time-varying delay and nonlinear perturbations. Two cases of time-varying delay, which is differentiable (Case 1) or not differentiable (Case 2), are considered. The considered nonlinear perturbations includes the norm-bounded uncertainties as a special case. Some delay-dependent stability criteria are derived by using a delay decomposition approach. In the delay decomposition approach, the entire delay interval is divided into multiple sub-intervals for which different energy functions are defined for building new Lyapunov–Krasovskii functional. Some numerical and practical examples are given to show the effectiveness and significant improvement of the proposed method.

New stability criteria of linear singular systems with time-varying delay

Most of the existing results on the stability problem of delayed singular systems only pertain to the case of constant delay. This is due to the fact that time-varying delay makes it hardly possible to explicitly express the fast variables. In this paper, aiming at dealing with the case of time-varying delay, we create a way to prove the stability by using a perturbation approach. Rather, we first get the decay rate for slow variables by using Lyapunov functional approach and, furthermore, guarantee that the fast variables fall into decay by characterising their effect on the derived decay rate. Also, we present a convexity technique in computing the constructed Lyapunov functional which contributes to the elimination of the possible conservatism caused by the varying rate of delay. Finally, we provide two numerical examples to demonstrate the effectiveness of the method.

Synchronization for coupled neural networks with interval delay: a novel augmented Lyapunov-Krasovskii functional method.

This paper is concerned with the synchronization problems for an array of neural networks with hybrid coupling and interval time-varying delay. First, a novel augmented Lyapunov-Krasovskii functional (LKF) method is proposed to develop delay-dependent synchronization criteria for the networks, which makes use of more relaxed conditions by employing the new type of augmented matrices with Kronecker product operation. The proposed method can handle a multitude of Kronecker product operations in the LKF and alleviates the requirements of the positive definiteness of some conditional matrices which are usually considered in the existing methods for complex networks. This leads to a significant improvement in the performance of the synchronization criteria, i.e., less conservative synchronization results can be obtained. Meanwhile, the case of fast time-varying delay can also be handled by the proposed method. Furthermore, based on the derived criteria, a robust synchronization criterion is obtained for the system with uncertainties both in coefficient and coupling matrix terms. Since an expression based on linear matrix inequality is used, the proposed criteria can be easily checked in practice. Finally, numerical examples are provided to show the effectiveness of the proposed method.

Partial Spectrum Assignment for Systems with Delayed State Feedback

We consider the problem of controlling a linear system when the state is available with a non-negligible delay (delayed-state-feedback control). In such conditions, the resulting closed-loop system is always a time-delay-system. The solution proposed in this paper consists in partially assigning the spectrum of the closed-loop system while ensuring the exponential zero-state stability with a prescribed decay rate. The proposed approach is appealing for its simplicity and can be applied in both cases of constant and time-varying delays. Sufficient stability conditions, that in some cases are also necessary, are provided. Such conditions allow to easily compute a lower bound, and in some cases the exact value, of the maximum delay that ensures the prescribed closed-loop behavior (Partial Spectrum Assignment with prescribed exponential stability).

New results on H ∞ filter design for nonlinear systems with time-delay through a T-S fuzzy model approach

H ∞ filter design for nonlinear systems with time-delay via a T-S fuzzy model approach is investigated based on a piecewise analysis method. Two cases of time-varying delays are fully considered: one is the time-varying delay being continuous uniformly bounded while the other is the time-varying delay being differentiable uniformly bounded with delay-derivative bounded by a constant. Based on a piecewise analysis method, the variation interval of the time delay is first divided into several subintervals, then the convexity property of the matrix inequality and the free weighting matrix method are fully used in this article. Some novel delay-dependent H ∞ filtering criteria are expressed as a set of linear matrix inequalities, which can lead to much less conservative analysis results. Finally, a numerical example is given to illustrate that the results in this article are more effective and less conservative than some existing ones.