Discretized LKF method for stability of coupled differential‐difference equations with multiple discrete and distributed delays

Time‐delay systems described by coupled differential‐functional equations include as special cases many types of time‐delay systems and coupled differential‐difference systems with time delays. This article discusses the discretized Lyapunov–Krasovskii functional (LKF) method for the stability problem of coupled differential‐difference equations with multiple discrete and distributed delays. Through independently dividing every delay region that the plane regions consists in two delays to discretize LKF, the exponential stability conditions for coupled systems with multiple discrete and distributed delays are established based on a linear matrix inequality (LMI). The numerical examples show that the analysis limit of delay bound in which the systems are stable may be approached by our result. Copyright © 2011 John Wiley & Sons, Ltd.     

Finite‐time boundedness of switched systems with time‐varying delays via sampled‐data control

In the framework of sampled‐data control, finite‐time boundedness (FTB) of switched systems with time‐varying delays is investigated. Sufficient conditions for FTB of switched systems with time‐varying delays via sampled‐data control are proposed. Moreover, considering the relationship between the sampling period and the mode‐dependent average dwell time, switching signals are designed. In addition, finite‐time weighted L₂‐gain (FTW‐L₂‐gain) of switched systems with time‐varying delays is proposed to measure their disturbance tolerance capacity within a finite‐time interval. Multiple Lyapunov‐Krasovskii functionals are applied to complete subsequent proofs in detail. Simulation results are exemplified to verify the proposed method.     

Delay‐range‐dependent control of nonlinear time‐delay systems under input saturation

This paper describes a delay‐range‐dependent local state feedback controller synthesis approach providing estimation of the region of stability for nonlinear time‐delay systems under input saturation. By employing a Lyapunov–Krasovskii functional, properties of nonlinear functions, local sector condition and Jensen's inequality, a sufficient condition is derived for stabilization of nonlinear systems with interval delays varying within a range. Novel solutions to the delay‐range‐dependent and delay‐dependent stabilization problems for linear and nonlinear time‐delay systems, respectively, subject to input saturation are derived as specific scenarios of the proposed control strategy. Also, a delay‐rate‐independent condition for control of nonlinear systems in the presence of input saturation with unknown delay‐derivative bound information is established. And further, a robust state feedback controller synthesis scheme ensuring L₂ gain reduction from disturbance to output is devised to address the problem of the stabilization of input‐constrained nonlinear time‐delay systems with varying interval lags. The proposed design conditions can be solved using linear matrix inequality tools in connection with conventional cone complementary linearization algorithms. Simulation results for an unstable nonlinear time‐delay network and a large‐scale chemical reactor under input saturation and varying interval time‐delays are analyzed to demonstrate the effectiveness of the proposed methodology. Copyright © 2015 John Wiley & Sons, Ltd.     

New results on stability and L∞‐gain analysis for positive linear differential‐algebraic equations with unbounded time‐varying delays

This article addresses the problems of stability and L_∞‐gain analysis for positive linear differential‐algebraic equations with unbounded time‐varying delays for the first time. First, we consider the stability problem of a class of positive linear differential‐algebraic equations with unbounded time‐varying delays. A new method, which is based on the upper bounding of the state vector by a decreasing function, is presented to analyze the stability of the system. Then, by investigating the monotonicity of state trajectory, the L_∞‐gain for differential‐algebraic systems with unbounded time‐varying delay is characterized. It is shown that the L_∞‐gain for differential‐algebraic systems with unbounded time‐varying delay is also independent of the delays and fully determined by the system matrices. Two numerical examples are given to illustrate the obtained results.     

Novel delay‐derivative‐dependent stability criteria using new bounding techniques

This paper studies the stability of linear systems with interval time‐varying delays. By constructing a new Lyapunov–Krasovskii functional, two delay‐derivative‐dependent stability criteria are formulated by incorporating with two different bounding techniques to estimate some integral terms appearing in the derivative of the Lyapunov–Krasovskii functional. The first stability criterion is derived by using a generalized integral inequality, and the second stability criterion is obtained by employing a reciprocally convex approach. When applying these two stability criteria to check the stability of a linear system with an interval time‐varying delay, it is shown through some numerical examples that the first stability criterion can provide a larger upper bound of the time‐varying delay than the second stability criterion. Copyright © 2012 John Wiley & Sons, Ltd.     

Stability analysis and stabilization of Markovian jump systems with time‐varying delay and uncertain transition information

The paper investigates the problems of stability and stabilization of Markovian jump systems with time‐varying delays and uncertain transition rates matrix. First, the stochastic scaled small‐gain theorem is introduced to analyze the stability of the Markovian jump system. Then, a new stability criterion is proposed by using a new Lyapunov‐Krasovskii functional combined with Wirtinger‐based integral inequality. The proposed stability condition is demonstrated to be less conservative than other existing results. The merit of the proposed approach lies in its reduced conservatism, which is made possible by a new precise triangle inequality and a new Lyapunov‐Krasovskii functional. Moreover, a controller design criterion is presented according to the stability criterion. Furthermore, the transition rate matrix is treated as partially known and with uncertainty, and the relevant stability and stabilization criteria are proposed. Finally, 3 numerical examples are provided to illustrate the superior result of the stability criteria and the effectiveness of the proposed controller design method.     

Distributed Consensus of Multi‐Agent Systems with Input Faults and Time‐Varying Delays

This work studies the consensus problem of multi‐agent systems with input faults and time‐varying delays. The assumed faults in the system are loss of effectiveness of actuator and nonlinear additive term mixed with nominal input. For system faults that are nonlinear, constraints of constant norm‐bounded, sector nonlinearity, and unbounded nonlinearity with known basis functions are considered. Employing a Lyapunov–Krasovskii functional method, delay dependent consensus criteria are established to show the exponential behavior of the system. Finally, one simulation example is solved to demonstrate the advantage of the obtained results.     

Output Feedback Stabilization for a Class of Stochastic High‐Order Feedforward Nonlinear Systems with Time‐Varying Delay

This paper considers the output feedback control problem for a class of stochastic high‐order feedforward nonlinear systems with time‐varying delay. Compared with existing works, the features of our system include different bounded time‐varying delays, more general high‐order power and homogeneous feedforward growth conditions. Firstly, we use the adding one power integrator technique to construct an output feedback controller without nonlinearities. Then, by introducing a scaling gain into the controller and choosing an appropriate Lyapunov–Krasovskii functional, the closed‐loop system can be rendered globally asymptotically stable in probability. A simulation example is provided to illustrate the effectiveness of the designed controller.     

Robust filter design for piecewise discrete‐time systems with time‐varying delays

A novel delay‐dependent filtering design approach is developed for a class of linear piecewise discrete‐time systems with convex‐bounded parametric uncertainties and time‐varying delays. The time‐delays appear in the state as well as the output and measurement channels. The filter has a linear full‐order structure and guarantees the desired estimation accuracy over the entire uncertainty polytope. The desired accuracy is assessed in terms of either ??_∞‐performance or ??₂–??_∞ criteria. A new parametrization procedure based on a combined Finsler's Lemma and piecewise Lyapunov–Krasovskii functional is established to yield sufficient conditions for delay‐dependent filter feasibility. The filter gains are determined by solving a convex optimization problem over linear matrix inequalities. In comparison to the existing design methods, the developed methodology yields the least conservative measures since all previous overdesign limitations are almost eliminated. By means of simulation examples, the advantages of the developed technique are readily demonstrated. Copyright © 2009 John Wiley & Sons, Ltd.     

Relaxed results on reachable set estimation of time‐delay systems with bounded peak inputs

This paper is concerned with the problem of reachable set estimation (RSE) for linear systems with time‐varying delays and bounded peak inputs. The purpose is to find an ellipsoid that contains the system state in presence of all bounded peak inputs. First, the RSE problem for nominal time‐delay systems is studied based on a relaxed Lyapunov–Krasovskii functional which does not require all the involved symmetric matrices to be positive definite. Delay‐dependent and delay‐rate‐dependent conditions for the existence of a desired ellipsoid are obtained. Second, the RSE problem for time‐delay systems with time‐varying polytopic uncertainties is investigated. Under the assumption that the uncertain parameters are differentiable and their derivatives are bounded by known scalars, parameter‐rate‐dependent conditions for the existence of a desired ellipsoid are derived by using a parameter‐dependent Lyapunov–Krasovskii functional. When the differentiability of the uncertain parameters is not taken into account, a common Lyapunov–Krasovskii functional is employed to tackle the addressed problem, and parameter‐rate‐independent conditions are presented. All the obtained conditions are given in terms of matrix inequalities, which become linear matrix inequalities when only one non‐convex scalar is prescribed. Finally, the reduced conservatism of the obtained results in comparison with recent ones in the literature is shown through numerical examples. Copyright © 2015 John Wiley & Sons, Ltd.     

Output feedback control for a class of stochastic high‐order nonlinear systems with time‐varying delays

This paper discusses the problem of output feedback stabilization for a more general class of stochastic high‐order nonlinear systems with time‐varying delays. On the basis of a subtle homogeneous observer and controller construction, and the homogeneous domination approach, the closed‐loop system is globally asymptotically stable in probability, by choosing an appropriate Lyapunov–Krasovskii functional. An example is given to illustrate the effectiveness of the proposed design procedure. Copyright © 2013 John Wiley & Sons, Ltd.     

Stability analysis of discrete‐time systems with time‐varying delays: generalized zero equalities approach

This paper suggests a generalized zero equality lemma for summations, which leads to making a new Lyapunov–Krasovskii functional with more state terms in the summands and thus applying various zero equalities for deriving stability criteria of discrete‐time systems with interval time‐varying delays. Also, using a discrete‐time counter part of Wirtinger‐based integral inequality, Jensen inequality, and a lower bound lemma for reciprocal convexity, the forward difference of the Lyapunov–Krasovskii functional is bounded by the combinations of various state terms including not only summation terms but also their interval‐normalized versions, which contributes to making the criteria less conservative. Numerical examples show the improved performance of the criteria in terms of maximum delay bounds. Copyright © 2016 John Wiley & Sons, Ltd.     

Resilient decentralized stabilization of interconnected time‐delay systems with polytopic uncertainties

Decentralized delay‐dependent local stability and resilient feedback stabilization methods are developed for a class of linear interconnected continuous‐time systems. The subsystems are time‐delay plants which are subjected to convex‐bounded parametric uncertainties and additive feedback gain perturbations while allowing time‐varying delays to occur within the local subsystems and across the interconnections. The delay‐dependent local stability conditions are established at the subsystem level through the construction of appropriate Lyapunov–Krasovskii functional. We characterize decentralized linear matrix inequalities (LMIs)‐based delay‐dependent stability conditions by deploying an injection procedure such that every local subsystem is delay‐dependent robustly asymptotically stable with an γ‐level ??₂‐gain. Resilient decentralized state‐feedback stabilization schemes are designed, which takes into account additive gain perturbations such that the family of closed‐loop feedback subsystems enjoys the delay‐dependent asymptotic stability with a prescribed γ‐level ??₂‐gain for each subsystem. The decentralized feedback gains are determined by convex optimization over LMIs. All the developed results are tested on representative examples. Copyright © 2010 John Wiley & Sons, Ltd.     

Robust stability of nonlinear time‐delay systems with interval time‐varying delay

This paper deals with the problem of obtaining delay‐dependent stability conditions and L₂‐gain analysis for a class of nonlinear time‐delay systems with norm‐bounded and possibly time‐varying uncertainties. No restrictions on the derivative of the time‐varying delay are imposed, though lower and upper bounds of the delay interval are assumed to be known. A Lyapunov–Krasovskii functional approach is proposed to derive novel delay‐dependent stability conditions which are expressed in terms of linear matrix inequalities (LMIs). To reduce conservatism, the work exploits the idea of splitting the delay interval in multiple regions, so that specific conditions can be imposed to a unique functional in the different regions. This improves the computed bounds for certain delay‐dependent integral terms, providing less conservative LMI conditions. Examples are provided to demonstrate the reduced conservatism with respect to the available results in the literature. Copyright © 2010 John Wiley & Sons, Ltd.     

Observer‐based H ∞ output tracking control for networked control systems

This paper is concerned with observer‐based H _∞ output tracking control for networked control systems. An observer‐based controller is implemented through a communication network to drive the output of a controlled plant to track the output of a reference model. The inputs of the controlled plant and the observer‐based tracking controller are updated in an asynchronous way because of the effects of network‐induced delays and packet dropouts in the controller‐to‐actuator channel. Taking the asynchronous characteristic into consideration, the resulting closed‐loop system is modeled as a system with two interval time‐varying delays. A Lyapunov–Krasovskii functional, which makes use of information about the lower and upper bounds of the interval time‐varying delays, is constructed to derive a delay‐dependent criterion such that the closed‐loop system has a desired H _∞ tracking performance. Notice that a separation principle cannot be used to design an observer gain and a control gain due to the asynchronous inputs of the plant and the controller. Instead, a novel design algorithm is proposed by applying a particle swarm optimization technique with the feasibility of the stability criterion to search for the minimum H _∞ tracking performance and the corresponding gains. The effectiveness of the proposed method is illustrated by an example. Copyright © 2013 John Wiley & Sons, Ltd.     

Less conservative stability criteria for linear systems with interval time‐varying delays

The problem of the stability of a linear system with an interval time‐varying delay is investigated. A new Lyapunov–Krasovskii functional that fully uses information about the lower bound of the time‐varying delay is constructed to derive new stability criteria. It is proved that the proposed Lyapunov–Krasovskii functional can lead to less conservative results than some existing ones. Based on the proposed Lyapunov–Krasovskii functional, two stability conditions are developed using two different methods to estimate Lyapunov–Krasovskii functional's derivative. Two numerical examples are given to illustrate that the two stability conditions are complementary and yield a larger maximum upper bound of the time‐varying delay than some existing results. Copyright © 2013 John Wiley & Sons, Ltd.     

Improved stability and H∞ performance criteria for linear systems with interval time-varying delays via three terms approximation

This paper investigates the problem of delay dependent stability and H_∞ performance for a class of linear systems with interval time-varying delay. A new model transformation is first proposed by employing a three-terms approximation of delayed state variables, for a better approximation of delayed state. By using scaled small gain theorem and a simple Lyapunov–Krasovskii functional, new stability and H_∞ performance criteria are proposed in terms of linear matrix inequalities, which can be easily solved by using standard numerical packages. Finally, numerical examples are presented to illustrate the effectiveness of the proposed method.     

H∞ Non‐Fragile Synchronous Guaranteed Control of Uncertainty Complex Dynamic Network with Time‐Varying Delay

A kind of H_∞ non‐fragile synchronization guaranteed control method is put forward for a class of uncertain time‐varying delay complex network systems with disturbance input. The network under consideration includes unknown but bounded nonlinear coupling functions f(x) and the coupling term and node system with time‐varying delays. The nonlinear vector function f(x) need not be differentiable but should satisfy the norm bound. A non‐fragile state feedback controller of the gain with sufficiently large regulation margin is designed. It is ensured that the parameters of the controller could still be effective under small perturbation. The sufficient conditions for the existence of H_∞ synchronous non‐fragile guaranteed control of this system have been obtained by constructing a suitable Lyapunov‐Krasovskii functional, adopting matrix analysis, using the theorem of Schur complement and linear matrix inequalities (LMI). These conditions can guarantee robust asymptotic stability for each node of network with disturbance as well as achieve a prescribed robust H_∞ performance level. Finally, the feasibility of the designed method is verified by a numerical example.     

Robust delay‐probability‐distribution‐dependent stability of uncertain stochastic genetic regulatory networks with random discrete delays and distributed delays

This study is concerned with the problem of robust delay‐probability‐distribution‐dependent stability of uncertain stochastic genetic regulatory networks with mixed time‐varying delays. The parameter uncertainties are modeled as having a structured linear fractional form. Besides, we consider that the derivatives of the discrete time delays have different upper bounds in various delay intervals. Moreover, less conservative conditions are obtained by choosing an augmented novel Lyapunov–Krasovskii functional and using the lower bound lemma together with the Jensen inequality lemma. Furthermore, the criteria can be applicable to both fast and slow time‐varying delays. Finally, numerical examples are presented to illustrate the effectiveness of the theoretical results. Copyright © 2013 John Wiley & Sons, Ltd.     

Delay‐range‐dependent H∞ control for Markovian jump systems with mode‐dependent time delays

This paper considers mean‐square exponential stability and H_∞ control problems for Markovian jump systems (MJSs) with time delays which are time‐varying in an interval and depend on system mode. By exploiting a novel Lyapunov‐Krasovskii functional which takes into account the range of delay, and by making use of some techniques, new delay‐range‐dependent stability result and bounded real lemma for MJSs are obtained, where the introduction of the lower bound of delay is shown to be advantageous for reducing conservatism. Moreover, a sufficient condition for the solvability of the H_∞ control problem is derived in terms of linear matrix inequalities. Finally, illustrative examples are presented to show the advantage and effectiveness of the proposed approaches. Copyright © 2010 John Wiley and Sons Asia Pte Ltd and Chinese Automatic Control Society