Delay-dependent Exponential Stability Research Articles

Delay-dependent stability conditions for differential–difference equations with small commutators in a banach space

Let [Formula: see text] and [Formula: see text] be bounded operators in a Banach space. For the equation [Formula: see text] we derive sharp explicit delay-dependent exponential stability conditions via the commutator [Formula: see text]. The stability conditions in terms of the commutator enable us to construct a feedback with delay, which stabilizes a system. Gives us a possibility construct Stabilization of Systems with Retarded Feedback in a Banach Space. Our results are new even in the finite-dimensional case.

Delay-dependent Exponential Stability of Linear Stochastic Neutral Systems with General Delays Described by Stieltjes Integrals

Delay-dependent Exponential Stability of Linear Stochastic Neutral Systems with General Delays Described by Stieltjes Integrals

Delay-Dependent Stability of Impulsive Stochastic Systems with Multiple Delays

This paper associates with stability analysis of linear impulsive stochastic delay systems (ISDSs). Although many conclusions about the stability of ISDSs have been obtained based on Lyapunov’s method, relatively few research theories about delay-dependent stability with less conservativeness have been established. Therefore, we introduce an appropriate Lyapunov-Krasovskii functional (LKF) to work out this problem, and a novel delay-dependent exponential stability theorem is first deduced. On the other hand, when mean-square stability is considered, we present delay-dependent stability conditions, it is of interest to note that the proposed conditions do not depend on the size of delays in the diffusion term, which solves the problems of determining the mean-square stability of ISDSs for which the diffusion term delays are not available. In the end, two numerical examples are carried out to verify the feasibility of our conclusions.

On stability of SDOF systems with delayed position and velocity feedback

In this paper, we study the stability problem of a single-degree-of freedom system with delayed position and velocity feedback, which can be found in the modeling of several engineering problems in control and vibration. For this class of delay system, we derive some necessary and sufficient conditions for delay-dependent exponential stability. These conditions determine a region in the feedback gains space for assuring the delay-dependent exponential stability. We show how the results can be applied to control and delayed resonator (DR) problems.

Exponential Stability of Discrete-Time Neural Networks With Large Delay.

We study the exponential stability of discrete-time neural networks (NNs) with a time-varying delay which contains a few intermittent large delays (LDs). By modeling the considered discrete-time NN as a discrete-time switched NN which contains two subsystems and one of them may be unstable over the LD periods (LDPs), switching techniques are employed to analyze the problem. Delay-dependent exponential stability conditions to check the frequency and the length of the LDs allowed for guaranteeing the exponential stability are proposed by applying a novel Lyapunov-Krasovskii functional (LKF) with LDP-based terms, Wirtinger-based summation inequality, and reciprocally convex combination technique. Based on these conditions, associated evaluation algorithms are developed. Finally, two numerical examples are provided to demonstrate the effectiveness of the proposed method.

Delay-Dependent Stability Conditions for Non-autonomous Functional Differential Equations with Several Delays in a Banach Space

Abstract Let Bj (t) (j = 1,..., m) and B(t, τ) (t ≥ 0, 0 ≤ τ ≤ 1) be bounded variable operators in a Banach space. We consider the equation u ′ ( t ) = ∑ k = 1 m B k ( t ) u ( t - h k ( t ) ) + ∫ 0 1 B ( t , τ ) u ( t - h 0 ( τ ) ) d τ ( t ≥ 0 ) , u'\left( t \right) = \sum\limits_{k = 1}^m {{B_k}\left( t \right)u\left( {t - {h_k}\left( t \right)} \right)} + \int\limits_0^1 {B\left( {t,\tau } \right)u\left( {t - {h_0}\left( \tau \right)} \right)d\tau \,\,\,\,\left( {t \ge 0} \right),} where hk (t) (t ≥ 0; k = 1, ..., m) and h 0(τ) are continuous nonnegative bounded functions. Explicit delay-dependent exponential stability conditions for that equation are established. Applications to integro-differential equations with delay are also discussed

Stability analysis for nonlinear Markov jump neutral stochastic functional differential systems

Recently, the asymptotic stability for Markov jump stochastic functional differential systems (SFDSs) was studied, whose stability criteria relied on the intervals lengths of continuous delays. Whereas, so far all the existing references require the rigorous global Lipschitz condition for the delay parts of the drift coefficients and do not consider the challenging factors of exponential decay and neutral issue. Motivated by the aforementioned considerations and the advantages of the degenerate functionals, this paper aims to weaken the global Lipschitz condition for the delay parts of the drift coefficients and investigate the delay-dependent exponential stability and asymptotic boundedness for highly nonlinear Markov jump neutral SFDSs with the method of multiple degenerate functionals. Of course, the delay-independent assertions are as well derived here.

Exponential Stability of Delayed Generalized Neural Networks With Intermittent Large-Delay Periods

This article investigates the exponential stability of generalized neural networks (GNNs) with a time-varying delay. Different from the literatures on the similar topic, the considered time delay contains a few intermittent large-delay periods (LDPs). A new approach is proposed to determine how frequent and how long the LDPs are allowed for guaranteeing the exponential stability by using switching techniques. The GNN is first modeled as a switched time-delay system which may include an unstable subsystem. Then, based on a novel Lyapunov–Krasovskii functional with LDP-based terms, a delay-dependent exponential stability criterion and associated evaluation algorithm are developed. Finally, two numerical examples are provided to show the effectiveness of the proposed method.

Non‐fragile control of periodic piecewise linear time‐varying systems with time delay

In this study, the stability and stabilisation issues of a class of periodic piecewise linear time-varying systems with time delay are studied. Based on a lemma with respects to the negative definite matrix polynomial, a sufficient delay-dependent exponential stability condition is obtained via constructing a Lyapunov–Krasovskii functional with periodic matrix functions. Then, for two types of norm-bounded uncertainties of time-varying controller gains, the non-fragile stabilising controllers under the multiplicative and additive gain perturbations are designed to stabilise the closed-loop system. The obtained controller gains could be solved with linear matrix inequalities directly instead of using the iterative algorithm. That is valued in engineering applications. Numerical examples are presented to demonstrate the effectiveness of the proposed methods.

Exponential H ∞ output tracking control for coupled switched systems with states of different dimensions

This paper is concerned with the output tracking control for complex switched systems with non-diffusively delayed coupling. The states' dimensions are different in the coupled switched systems and reference system. On the base of the average dwell time and piecewise Lyapunov functional methods, a delay-dependent exponential stability criterion is derived for the switched systems under the QUAD condition which is less conservative than the Lipschitz condition. The state feedback controller is employed in the design of output tracking, and the tracking performance is guaranteed by solving a linear matrix inequality. Finally, three numerical examples are provided to show the advantage and effectiveness of the proposed theoretical results.

Exponential stability criterion of the switched neural networks with time-varying delay

In this paper, the delay-dependent stability problem of the switched neural networks with time-varying delay is considered. By taking advantage of the average dwell time method and Lyapunov–Krasovskii functional (LKF) method, and using free-matrix-based integral inequality and the extended reciprocally convex matrix inequality, a less conservative delay-dependent exponential stability criterion in linear matrix inequalities (LMIs) is developed. Two numerical examples are given to demonstrate the benefits of the proposed criterion.

Extended dissipative estimator design for uncertain switched delayed neural networks via a novel triple integral inequality

This paper addresses the problem of extended dissipative estimator design for uncertain switched neural networks (SNNs) with mixed time-varying delays and general activation functions. Firstly, for dealing with triple integral term, a new integral inequality is derived. Secondly, based on the theory of convex combination, we propose a novel flexible delay division method and corresponding modified Lyapunov–Krasovskii functional (LKF) is established. Thirdly, a switching estimator design approach is contributed, which ensures that the resulting augmented system is extended dissipative. Combining the extended reciprocally convex technique with Wirtinger-based integral inequality, improved delay-dependent exponential stability criterion is obtained. Finally, a example with two cases is provided to illustrate the feasibility and effectiveness of the developed theoretical results.

Exponential stability and extended dissipativity criteria for generalized discrete-time neural networks with additive time-varying delays

This paper is concerned with exponential stability and extended dissipativity criteria for generalized discrete-time neural networks (GDNNs) with additive time-varying delays. The generalized dissipativity analysis combines a few previous results into a framework, such as l2−l∞ performance, H∞ performance, passivity performance, strictly (Q,S,R)−γ−dissipative and strictly (Q,S,R)−dissipative. The definition of exponential stability for GDNNs is given with a new and more appropriate expression. A novel augmented Lyapunov-Krasovskii functional (LKF) which involves more information about the additive time-varying delays is constructed. By introducing more zero equalities and using a new double summation inequality together with Finsler’s lemma, an improved delay-dependent exponential stability and extended dissipativity criterion are derived in terms of convex combination technique (CCT). Finally, numerical examples are given to illustrate the usefulness and advantages of the proposed methods.

H∞ control problem of linear periodic piecewise time-delay systems

ABSTRACTThis paper investigates the H∞ control problem based on exponential stability and weighted L2-gain analyses for a class of continuous-time linear periodic piecewise systems with time delay. A periodic piecewise Lyapunov–Krasovskii functional is developed by integrating a discontinuous time-varying matrix function with two global terms. By applying the improved constraints to the stability and L2-gain analyses, sufficient delay-dependent exponential stability and weighted L2-gain criteria are proposed for the periodic piecewise time-delay system. Based on these analyses, an H∞ control scheme is designed under the considerations of periodic state feedback control input and iterative optimisation. Finally, numerical examples are presented to illustrate the effectiveness of our proposed conditions.

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.

Novel delay-dependent exponential stabilization criteria of a nonlinear system with mixed time-varying delays via hybrid intermittent feedback control

In this paper, we investigate the problem of exponential stabilization criteria for a nonlinear system with mixed time-varying delays, including discrete interval and distributed time-varying delays. The time-varying delays are not necessarily differentiable. The exponential stabilization criteria of the nonlinear system are proposed via hybrid intermittent feedback control. Based on the improved Lyapunov-Krasovskii functionals with Leibniz-Newton’s formula, Jensen’s inequality and the reciprocal convex combination technique, the novel delay-dependent sufficient condition is derived in terms of linear matrix inequalities (LMIs). The obtained LMIs can be efficiently solved by standard convex optimization algorithms. A numerical example is given to demonstrate the effectiveness of the obtained result. Moreover, the results in this article generalize and improve the corresponding results of the recent works.

Optimal fuzzy control of exponential synchronisation via genetic algorithm

ABSTRACTIn this study, a novel approach via GA-based fuzzy control is proposed to realize the exponential optimal H∞ synchronisation of MTDC systems. A robustness design of model-based fuzzy control is first presented to overcome the effect of modelling errors between the MTDC systems and T-S fuzzy models. Next, a delay-dependent exponential stability criterion is derived in terms of Lyapunov's direct method to guarantee that the trajectories of the slave system can approach those of the master system. Subsequently, the stability conditions of this criterion are reformulated into LMIs. According to the LMIs, a fuzzy controller is then synthesised to exponentially stabilise the error systems. Moreover, the capability of GA in random search for near-optimal solutions, the lower and upper bounds of the search space based on the feedback gains via LMI approach can be set so that the GA will seek better feedback gains of fuzzy controllers to speed up the synchronisation. Additionally, an IGA was proposed to overcome both the shortcomings of premature convergence of GA and local search. According to the IGA, a fuzzy controller is synthesised not only to realise the exponential synchronisation but also to achieve the optimal H∞ performance by minimising the disturbance attenuation level.

New delay-range-dependent exponential stability criteria for certain neutral differential equations with interval discrete and distributed time-varying delays

In this research, we investigate the problem of delay-dependent exponential stability analysis for certain neutral differential equations with discrete and distributed time-varying delays. The time-varying delays are continuous functions belonging to the given interval delays, which mean that the lower and upper bounds for the time-varying delays are available. The restrictions on the derivative of interval time-varying delays are needed. Based on a class of novel augmented Lyapunov-Krasovskii functionals, a model transformation, the decomposition technique of constant coefficients, the Leibniz-Newton formula, and utilization of a zero equation, new delay-range-dependent exponential stability criteria are derived in terms of the linear matrix inequality (LMI) for the equations considered. Numerical examples suggest for the results given to illustrate the effectiveness and improvement over some existing methods.

A novel approach to exponential stability of continuous-time Roesser systems with directional time-varying delays

This paper addresses the problem of exponential stability analysis of two-dimensional (2D) linear continuous-time systems with directional time-varying delays. An abstract Lyapunov-like theorem which ensures that a 2D linear system with delays is exponentially stable for a prescribed decay rate is exploited for the first time. In light of the abstract theorem, and by utilizing new 2D weighted integral inequalities proposed in this paper, new delay-dependent exponential stability conditions are derived in terms of tractable matrix inequalities which can be solved by various computational tools to obtain maximum allowable bound of delays and exponential decay rate. Two numerical examples are given to illustrate the effectiveness of the obtained results.

New Approach to Exponential Stability Analysis and Stabilization for Delayed T-S Fuzzy Markovian Jump Systems

This paper is concerned with delay-dependent exponential stability analysis and stabilization for continuous-time T-S fuzzy Markovian jump systems with mode-dependent time-varying delay. By constructing a novel Lyapunov-Krasovskii functional and utilizing some advanced techniques, less conservative conditions are presented to guarantee the closed-loop system is mean-square exponentially stable. Then, the stabilization conditions are derived and the fuzzy controller can be obtained by solving a set solutions of LMIs. The upper bound of time-delay that the system can be stabilized is given by using an optimal algorithm. Two examples are presented to illustrate the effectiveness and potential of our methods.