Two general integral inequalities and their applications to stability analysis for systems with time‐varying delay

Integral inequalities have been widely used in stability analysis for systems with time‐varying delay because they directly produce bounds for integral terms with respect to quadratic functions. This paper presents two general integral inequalities from which almost all of the existing integral inequalities can be obtained, such as Jensen inequality, the Wirtinger‐based inequality, the Bessel–Legendre inequality, the Wirtinger‐based double integral inequality, and the auxiliary function‐based integral inequalities. Based on orthogonal polynomials defined in different inner spaces, various concrete single/multiple integral inequalities are obtained. They can produce more accurate bounds with more orthogonal polynomials considered. To show the effectiveness of the new inequalities, their applications to stability analysis for systems with time‐varying delay are demonstrated with two numerical examples. Copyright © 2016 John Wiley & Sons, Ltd.     

New delay range–dependent stability criteria for interval time‐varying delay systems via Wirtinger‐based inequalities

Stability conditions for time‐delay systems using the Lyapunov‐based methodologies are generically expressed in terms of linear matrix inequalities. However, due to assuming restrictive conditions in deriving the linear matrix inequalities, the established stability conditions can be strictly conservative. This paper attempts to relax this problem for linear systems with interval time‐varying delays. A double‐integral inequality is derived inspired by Wirtinger‐based single‐integral inequality. Using the advanced integral inequalities, the reciprocally convex combination techniques and necessary slack variables, together with extracting a condition for the positive definiteness of the Lyapunov functional, novel stability criteria, have been established for the system. The effectiveness of the criteria is evaluated via 2 numerical examples. The results indicate that more complex stability criteria not only improve the stability region but also bring computational expenses.     

Delay‐Dependent Stability Criterion for Discrete‐Time Systems with Time‐Varying Delays

The stability analysis problem is considered for linear discrete‐time systems with time‐varying delays. A novel summation inequality is proposed, which takes the double summation information of the system state into consideration. The inequality relaxes the recently proposed discrete Wirtinger inequality and its improved version. Based on construction of a suitable Lyapunov‐Krasovskii functional and the novel summation inequality, an improved delay‐dependent stability criterion for asymptotic stability of the systems is derived in terms of linear matrix inequalities. Numerical examples are given to demonstrate the advantages of the proposed method.     

New absolute stability results for Lurie systems with interval time‐varying delay based on improved Wirtinger‐type integral inequality

This work focuses on the absolute stability problem of Lurie control system with interval time‐varying delay and sector‐bounded nonlinearity. Firstly, we present a refined Wirtinger's integral inequality and establish an improved Wirtinger‐type double integral inequality. Secondly, a modified augmented Lyapunov‐Krasovskii functional (LKF) is constructed to analyze the stability of Lurie system, where the information on the lower and upper bounds of the delay and the delay itself are fully exploited. Based on the proposed integral inequalities and some bounding techniques, the upper bound of the derivative of the LKF can be estimated more tightly. Accordingly, the proposed absolute stability criteria, formulated in terms of linear matrix inequalities, are less conservative than those in previous literature. Finally, numerical examples demonstrate the effectiveness and advantage of the proposed method.     

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.     

Delay‐dependent conditions for finite time stability of linear time‐varying systems with delay

This paper investigates the problem of finite time stability of linear time‐varying system with delay. By constructing an augmented time‐varying Lyapunov functional and using the Wirtinger‐type inequality deductively, delay‐dependent finite time stability conditions are derived and presented in terms of differential linear matrix inequalities (DLMIs). Then, the DLMIs are transformed into a series of recursive linear matrix inequalities (RLMIs) by discretizing the time interval into equally spaced time distances, and an algorithm is given to solve the RLMIs. Examples illustrate the feasibility and effectiveness of the proposed method.     

Bessel summation inequalities for stability analysis of discrete‐time systems with time‐varying delays

This paper is concerned with the stability analysis problems of discrete‐time systems with time‐varying delays using summation inequalities. In the literature focusing on the Lyapunov‐Krasovskii approach, the Jensen integral/summation inequalities have played important roles to develop less conservative stability criteria and thus have been widely studied. Recently, the Jensen integral inequality was successfully generalized to the Bessel‐Legendre inequalities constructed with arbitrary‐order Legendre polynomials. It was also shown that general inequality contributes to the less conservatism of stability criteria. In the case of discrete‐time systems, however, the Jensen summation inequality are hardly extensible to general ones since there have still not been general discrete orthogonal polynomials applicable to the developments of summation inequalities. Motivated by such observations, this paper proposes novel discrete orthogonal polynomials and then successfully derives general summation inequalities. The resulting summation inequalities are discrete‐time counterparts of the Bessel‐Legendre inequalities but are not based on the discrete Legendre polynomials. By developing hierarchical stability criteria based on the proposed summation inequalities, the effectiveness of the proposed approaches is demonstrated via three numerical examples for the stability analysis of discrete‐time systems with time‐varying delays.     

Non‐Fragile Extended Dissipativity Control Design for Generalized Neural Networks with Interval Time‐Delay Signals

The problem of non‐fragile extended dissipative control design for a class of generalized neural networks (GNNs) with interval time‐delay signals is investigated in this paper. By constructing a suitable Lyapunov‐Krasovskii functional (LKF) with double and triple integral terms, and estimating their derivative by using the Wirtinger single integral inequality (WSII) and Wirtinger double integral inequality (WDII) technique respectively, and that is mixed with the reciprocally convex combination (RCC) approach. A new delay‐dependent non‐fragile extended dissipative control design for GNNs are expressed in terms of the linear matrix inequalities (LMIs). Then, the desired non‐fragile extended dissipative controller can be obtained by solving the linear matrix inequalities (LMIs). Furthermore, a non‐fragile state feedback controller is designed for GNNs such that the closed‐loop system is extended dissiptive. Thus, the non‐fragile extended dissipative controller can be adopted to deal with the non‐fragile performance, non‐fragile performance, non‐fragile passive performance, non‐fragile mixed and passivity performance, and non‐fragile dissipative performance for GNNs by selecting the weighting matrices. Finally, simulation studies are demonstrated for showing the feasibility of the proposed method. Among them, one example was supported by the real‐life application of the quadruple tank process system.     

Improved Robust Stability Criteria for Time‐Delay Lur'e System

This paper deals with the problem of the robustly absolute stability for neutral‐type Lur'e systems with mixed time‐varying delay. By combining the piecewise analysis theory with the reciprocally convex method and Wirtinger‐based inequality technology, some new delay‐dependent stability criteria are proposed via a modified Lyapunov‐Krasovskii functional (LKF) approach. The stability conditions can be solved by using standard linear matrix inequality (LMI) convex optimization solvers. The criteria are less conservative than some previous ones. Three numerical examples are presented to show the effectiveness of the proposed approach.     

Polynomials‐based summation inequalities and their applications to discrete‐time systems with time‐varying delays

This paper proposes a novel summation inequality, say a polynomials‐based summation inequality, which contains well‐known summation inequalities as special cases. By specially choosing slack matrices, polynomial functions, and an arbitrary vector, it reduces to Moon's inequality, a discrete‐time counterpart of Wirtinger‐based integral inequality, auxiliary function‐based summation inequalities employing the same‐order orthogonal polynomial functions. Thus, the proposed summation inequality is more general than other summation inequalities. Additionally, this paper derives the polynomials‐based summation inequality employing first‐order and second‐order orthogonal polynomial functions, which contributes to obtaining improved stability criteria for discrete‐time systems with time‐varying delays. Copyright © 2017 John Wiley & Sons, Ltd.     

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.     

Exponential Stability and Stabilization for Quadratic Discrete‐Time Systems with Time Delay

In this note, we deal with the exponential stability and stabilization problems for quadratic discrete‐time systems with time delay. By using the quadratic Lyapunov function and a so called ‘Finsler's lemma', delay‐independent sufficient conditions for local stability and stabilization for quadratic discrete‐time systems with time delay are derived in terms of linear matrix inequalities (LMIs). Based on these sufficient conditions, iterative linear matrix inequality algorithms are proposed for maximizing the stability regions of the systems. Finally, two examples are given to illustrate the effectiveness of the methods presented in this paper.     

On reachable set estimation of two‐dimensional systems described by the Roesser model with time‐varying delays

In this paper, the problem of reachable set estimation of two‐dimensional (2‐D) discrete‐time systems described by the Roesser model with interval time‐varying delays is considered for the first time. New 2‐D weighted summation inequalities, which provide a tighter lower bound than the commonly used Jensen summation inequality, are proposed. Based on the Lyapunov‐Krasovskii functional approach, and by using the 2‐D weighted summation inequalities presented in this paper, new delay‐dependent conditions are derived to ensure the existence of an ellipsoid that bounds the system states in the presence of bounded disturbances. The derived conditions are expressed in terms of linear matrix inequalities, which can be solved by various computational tools to determine a smallest possible ellipsoidal bound. Applications to exponential stability analysis of 2‐D systems with delays are also presented. The effectiveness of the obtained results are illustrated by numerical examples.     

Delay‐dependent stability and stabilization of neutral time‐delay systems

This paper is concerned with the problem of stability and stabilization of neutral time‐delay systems. A new delay‐dependent stability condition is derived in terms of linear matrix inequality by constructing a new Lyapunov functional and using some integral inequalities without introducing any free‐weighting matrices. On the basis of the obtained stability condition, a stabilizing method is also proposed. Using an iterative algorithm, the state feedback controller can be obtained. Numerical examples illustrate that the proposed methods are effective and lead to less conservative results. Copyright © 2008 John Wiley & Sons, Ltd.     

Consensus of multiagent systems with nonlinear dynamics and time‐varying communication delays

This paper deals with the local consensus of multiagent systems with nonlinear dynamics communication delays simultaneously. By introducing a weighted average state and applying the properties of the Laplacian matrix eigenvalues, the system is decoupled into several subsystems, firstly, to reduce complexity of theory analysis. Then, a new augmented vector containing single and double integral terms is constructed and the corresponding Lyapunov functional with triple integral terms is introduced. Meanwhile, in order to improve the estimating accuracy of the derivatives of constructed Lyapunov functional, single integral inequalities and double integral inequalities via auxiliary functions, an extended relaxed integral inequality and an reciprocally convex approach are used, as a result, stability criterion with less conservatism is derived, which guarantees the local consensus of the considered systems. Finally, numerical examples are provided to check the improvement of the proposed method over the existing works.     

Finite‐time stability and stabilization of semi‐Markovian jump systems with time delay

Semi‐Markovian jump systems are more general than Markovian jump systems in modeling practical systems. On the other hand, the finite‐time stochastic stability is also more effective than stochastic stability in practical systems. This paper focuses on the finite‐time stochastic stability, exponential stochastic stability, and stabilization of semi‐Markovian jump systems with time‐varying delay. First, a new stability condition is presented to guarantee the finite‐time stochastic stability of the system by using a new Lyapunov‐Krasovskii functional combined with Wirtinger‐based integral inequality. Second, the stability criterion is further proved to guarantee the exponential stochastic stability of the system. Moreover, a controller design method is also presented according to the stability criterion. Finally, an example is provided to illustrate that the proposed stability condition is less conservative than other existing results. Additionally, we use the proposed method to design a controller for a load frequency control system to illustrate the effectiveness of the method in a practical system of the proposed method.     

Wirtinger-based integral inequality: Application to time-delay systems

In the last decade, the Jensen inequality has been intensively used in the context of time-delay or sampled-data systems since it is an appropriate tool to derive tractable stability conditions expressed in terms of linear matrix inequalities (LMIs). However, it is also well-known that this inequality introduces an undesirable conservatism in the stability conditions and looking at the literature, reducing this gap is a relevant issue and always an open problem. In this paper, we propose an alternative inequality based on the Fourier Theory, more precisely on the Wirtinger inequalities. It is shown that this resulting inequality encompasses the Jensen one and also leads to tractable LMI conditions. In order to illustrate the potential gain of employing this new inequality with respect to the Jensen one, two applications on time-delay and sampled-data stability analysis are provided.     

GENERALIZED QUADRATIC STABILIZATION FOR DISCRETE‐TIME SINGULAR SYSTEMS WITH TIME‐DELAY AND NONLINEAR PERTURBATION

This paper discusses a generalized quadratic stabilization problem for a class of discrete‐time singular systems with time‐delay and nonlinear perturbation (DSSDP), which the satisfies Lipschitz condition. By means of the S‐procedure approach, necessary and sufficient conditions are presented via a matrix inequality such that the control system is generalized quadratically stabilizable. An explicit expression of the static state feedback controllers is obtained via some free choices of parameters. It is shown in this paper that generalized quadratic stability also implies exponential stability for linear discrete‐time singular systems or more generally, DSSDP. In addition, this new approach for discrete singular systems (DSS) is developed in order to cast the problem as a convex optimization involving linear matrix inequalities (LMIs), such that the controller can stabilize the overall system. This approach provides generalized quadratic stabilization for uncertain DSS and also extends the existing robust stabilization results for non‐singular discrete systems with perturbation. The approach is illustrated here by means of numerical examples.     

H∞ state estimation of generalised neural networks with interval time-varying delays

This paper focuses on studying the H_∞ state estimation of generalised neural networks with interval time-varying delays. The integral terms in the time derivative of the Lyapunov–Krasovskii functional are handled by the Jensen’s inequality, reciprocally convex combination approach and a new Wirtinger-based double integral inequality. A delay-dependent criterion is derived under which the estimation error system is globally asymptotically stable with H_∞ performance. The proposed conditions are represented by linear matrix inequalities. Optimal H_∞ norm bounds are obtained easily by solving convex problems in terms of linear matrix inequalities. The advantage of employing the proposed inequalities is illustrated by numerical examples.     

New less‐conservative stability results for uncertain stochastic neural networks with fewer slack variables

The exponential stability problem is investigated for a class of uncertain stochastic neural networks with discrete and unbounded distributed time delays. Two types of uncertainty are considered: one is time‐varying structured uncertainty, whereas the other is interval uncertainty. With the application of the Jensen integral inequality and constructing appropriate Lyapunov–Krasovskii functional based on delay partitioning, several improved delay‐dependent criteria are developed to achieve the exponential stability in mean square in terms of linear matrix inequalities. It is established theoretically that two special cases of the obtained criteria are less conservative than some existing results but including fewer slack variables. As the present conditions involve fewer free weighting matrices, the computational burden is largely reduced. Three numerical examples are provided to demonstrate the effectiveness of the theoretical results. Copyright © 2012 John Wiley & Sons, Ltd.