Discrete Wirtinger-based Inequality Research Articles

Discrete-State Decomposition Technique of Dissipativity Analysis for Discrete-Time Singular Systems With Time-Varying Delays.

This article is concerned with the problem of dissipativity for discrete-time singular systems with time-varying delays. First, the discrete-state decomposition technique is proposed after performing the restricted equivalent transformation for singular systems. To reduce the use of decision variables, the state-decomposed Lyapunov function is established based on the decomposed state vectors. Second, to obtain the condition with less conservatism, the two zero-value equations, especially concerning difference subsystems and algebraic ones, the discrete Wirtinger-based inequality and the extended reciprocally convex inequality are employed to bound the forward difference of the Lyapunov function. Then, the less conservative dissipativity criteria with lower computational complexity are obtained. Finally, simulation results are provided to demonstrate the superiority of the proposed technique.

Observer-based state estimation for discrete-time semi-Markovian jump neural networks with round-robin protocol against cyber attacks

This paper investigates an observer-based state estimation issue for discrete-time semi-Markovian jump neural networks with Round-Robin protocol and cyber attacks. In order to avoid the network congestion and save the communication resources, the Round-Robin protocol is used to schedule the data transmissions over the networks. Specifically, the cyber attacks are modeled as a set of random variables satisfying the Bernoulli distribution. On the basis of the Lyapunov functional and the discrete Wirtinger-based inequality technique, some sufficient conditions are established to guarantee the dissipativity performance and mean square exponential stability of the argument system. In order to compute the estimator gain parameters, a linear matrix inequality approach is utilized. Finally, two illustrative examples are provided to demonstrate the effectiveness of the proposed state estimation algorithm.

Exponential stability of discrete‐time delayed neural networks with saturated impulsive control

This paper examines the problem of the locally exponentially stability for impulsive discrete-time delayed neural networks (IDDNNs) with actuator saturation. By fully considering the delay information of the state of the considered system, a new delay-dependent polytopic representation within a discrete-time framework is obtained. Based on the delay-independent polytopic representation approach, the saturation term is expressed as a delay-dependent convex combination. In order to obtain some less conservative stability conditions and estimate a larger of the domain of attraction, a novel type of Lyapunov–Krasovskii function (LKF) dependent on the delay information and the impulses instant is proposed, which is called time-dependent LKF. Then, by combining with the proposed LKF, a discrete Wirtinger-based inequality, an extended reciprocally convex matrix inequality and some novel analysis techniques, several new exponential stability criteria dependent on the bounds of the delay are presented. Moreover, when saturation constraints are not considered in the impulsive controller, the stability of the system is also discussed. Finally, two examples are given to confirm the applicability of the proposed results.

Discrete Legendre polynomials-based inequality for stability of time-varying delayed systems

This paper proposes Discrete Legendre Polynomial(DLP)-based inequality by solving the best weighted approximation of a given time series. The proposed inequality could significantly reduce the conservativeness in stability analysis of systems with constant or interval time-varying delays. Also former well-known integral inequities, such as discrete Jensen inequality, discrete Wirtinger-based inequality, are both included in the proposed DLP-based inequality as special cases with lower-order approximation. Stability criterion with less conservatism is then developed for both constant and time-varying delayed systems. Several numerical examples are given to demonstrate the effectiveness and benefit of the proposed method.

Regional Stabilization for Discrete Time-Delay Systems With Actuator Saturations via A Delay-Dependent Polytopic Approach

This paper is concerned with the regional stabilization problem for discrete time-delay systems with actuator saturations. Different from some existing delay-independent techniques handling the saturation nonlinearity, a delay-dependent polytopic approach is first proposed in a discrete-time framework. Then, by combining with an augmented Lyapunov function, a discrete Wirtinger-based inequality and some novel analysis techniques, improved delay-dependent regional stabilization conditions are established by means of linear matrix inequalities. Finally, it is shown via two numerical examples that the obtained results are less conservative than some existing ones yet providing a larger estimate of an initial condition set.

Finite-Time $H_{\infty}$ State Estimation for Discrete Time-Delayed Genetic Regulatory Networks Under Stochastic Communication Protocols

This paper investigates the problem of finite-time $H_{\infty }$ state estimation for discrete time-delayed genetic regulatory networks under stochastic communication protocols (SCPs). The network measurements are transmitted from two groups of sensors to a remote state estimator via two independent communication channels of limited bandwidths, and two SCPs are utilized to orchestrate the transmission orders of sensor nodes with aim to avoid data collisions. The estimation error dynamics is modeled by a Markovian switching system with two switching signals. By constructing a transmission-order-dependent Lyapunov–Krasovskii functional and utilizing an up-to-date discrete Wirtinger-based inequality together with the reciprocally convex approach, sufficient conditions are established to guarantee the stochastic finite-time boundedness for the estimation error dynamics with a prescribed $H_{\infty }$ disturbance attenuation level. The parameters of the state estimator are designed by solving a convex optimization problem which minimizes the disturbance attenuation level subject to several inequality constraints. The repressilator model is utilized to illustrate the effectiveness of the design procedure of the proposed state estimator.

State Estimation for Discrete Time-Delayed Genetic Regulatory Networks With Stochastic Noises Under the Round-Robin Protocols.

This paper investigates the problem of state estimation for discrete time-delayed genetic regulatory networks with stochastic process noises and bounded exogenous disturbances under the Round-Robin (RR) protocols. The network measurement outputs obtained by two groups of sensors are transmitted to two remote sub-estimators via two independent communication channels, respectively. To lighten the communication loads of the networks and reduce the occurrence rate of data collisions, two RR protocols are utilized to orchestrate the transmission orders of sensor nodes in two groups, respectively. The error dynamics of the state estimation is governed by a switched system with periodic switching parameters. By constructing a transmission-order-dependent Lyapunov-like functional and utilizing the up-to-date discrete Wirtinger-based inequality together with the reciprocally convex approach, sufficient conditions are established to guarantee the exponentially ultimate boundedness of the estimation error dynamics in mean square with a prescribed upper bound on the decay rate. An asymptotic upper bound of the outputs of the estimation errors in mean square is derived and the estimator parameters are then obtained by minimizing such an upper bound subject to linear matrix inequality constraints. The repressilator model is utilized to illustrate the effectiveness of the designed estimator.

Further Results on Reachable Set Bounding for Discrete-Time System with Time-Varying Delay and Bounded Disturbance Inputs

This paper investigates the problem of reachable set bounding for discrete-time system with time-varying delay and bounded disturbance inputs. Together with a new Lyapunov-Krasovskii functional, discrete Wirtinger-based inequality, and reciprocally convex approach, sufficient conditions are derived to find an ellipsoid to bound the reachable sets of discrete-time delayed system. The main advantage of this paper lies in two aspects: first, the initial state vectors are not necessarily zero; second, the obtained criteria in this paper do not really require all the symmetric matrices involved in the employed Lyapunov-Krasovskii functional to be positive definite. Finally, two numerical examples are provided to illustrate the effectiveness of the proposed method.

L2−l∞ state estimation for discrete-time switched neural networks with time-varying delay

Abstract This paper is concerned with the l 2 − l ∞ state estimation problem for discrete-time switched neural networks with time-varying delay. The main objective is to design a mode-dependent state estimator such that the error dynamics is exponentially stable with a weighted l 2 − l ∞ performance level. By incorporating the novel l 2 − l ∞ performance analysis approach, the augmented piecewise Lyapunov-like functionals, the discrete Wirtinger-based inequality and the average-dwell-time switching, less conservative sufficient conditions are proposed by means of linear matrix inequalities. A numerical example is given to illustrate the effectiveness and benefits of the obtained results.

Robust Stability of Discrete-Time Randomly Switched Delayed Genetic Regulatory Networks with Known Sojourn Probabilities

This study investigates stability problems related to discrete-time randomly switched genetic regulatory networks (GRNs) with time-varying delays. A new discrete-time randomly switched GRN model with known sojourn probabilities is proposed. By utilizing the discrete Wirtinger-based inequality and a newly proposed constraint condition on the feedback regulatory function, which have not been fully used in stability analysis of discrete-time GRNs, we establish delay-dependent stability and robust stability criteria. These criteria possess the sojourn probabilities of randomly switched GRNs. Two numerical examples are provided to demonstrate the effectiveness of the established results.

New robust stability condition for discrete-time recurrent neural networks with time-varying delays and nonlinear perturbations

In this paper, the robust delay-dependent stability problem is investigated for discrete-time recurrent neural networks (DRNNs) with time-varying delays and nonlinear perturbations. A novel summation inequality is proposed, which takes information on the double summation of system state into consideration and further extends the discrete Wirtinger-based inequality. By utilizing technique of the novel inequality and Lyapunov–Krasovskii functionals, a sufficient condition on robust stability of DRNNs with time-varying delays and nonlinear perturbations is obtained in terms of linear matrix inequality. The numerical example is included to show that the proposed method is effective and provides less conservative results.

Finite-time stability for discrete-time system with time-varying delay and nonlinear perturbations

In this paper, the problem of finite-time stability for discrete-time system with time-varying delay and nonlinear perturbations is investigated. By constructing a novel Lyapunov–Krasovskii functional and employing a new summation inequality named discrete Wirtinger-based inequality, reciprocally convex approach and zero equality, the improved finite-time stability criteria are derived to guarantee that the state of the system with time-varying delay does not exceed a given threshold when fixed time interval. Furthermore, the obtained conditions are formulated in forms of linear matrix inequalities which can be solved by using some standard numerical packages. Finally, three numerical examples are given to show the effectiveness and less conservatism of the proposed method.

Discrete inequalities based on multiple auxiliary functions and their applications to stability analysis of time-delay systems

This paper presents new discrete inequalities for single summation and double summation. These inequalities are based on multiple auxiliary functions and include the Jensen discrete inequality and the discrete Wirtinger-based inequality as special cases. An application of these discrete inequalities to analyze stability of linear discrete systems with an interval time-varying delay is studied and a less conservative stability condition is obtained. Three numerical examples are given to show the effectiveness of the obtained stability condition.

Improved Stability and Stabilization Criteria for Uncertain T–S Fuzzy Systems with Interval Time-Varying Delay via Discrete Wirtinger-Based Inequality

This paper introduces a discrete Wirtinger-based inequality to investigate the problem of delay-dependent stability and stabilization of uncertain T–S fuzzy systems with interval time-varying delays. By constructing a novel Lyapunov functional and applying the discrete Wirtinger-based inequality to deal with the sum terms in the derivation of the results, two delay-dependent stability criteria and a stabilization criterion are obtained in terms of matrix inequalities. Numerical examples show that the derived stability and stabilization criteria can provide a larger allowable upper delay bound than some existing results while depending on relatively less scalar decision variables.

Discrete Wirtinger-based inequality and its application

In this paper, we derive a new inequality, which encompasses the discrete Jensen inequality. The new inequality is applied to analyze stability of linear discrete systems with an interval time-varying delay and a less conservative stability condition is obtained. Two numerical examples are given to show the effectiveness of the obtained stability condition.