Wirtinger-based Double Integral Inequality Research Articles

A Novel Delay-Product-Type Functional Method to Extended Dissipativity Analysis for Markovian Jump Neural Networks

This paper studies the problem of extended dissipativity analysis for Markovian jump neural networks (MJNNs) with time-varying delay. Combining Wirtinger-based double integral inequality and S-procedure lemma, a novel double integral-based delay-product-type (DIDPT) Lyapunov functional is constructed in this paper, which avoids the incomplete components in the existing works. Then, based on parameter-dependent reciprocally convex inequality (PDRCI) and the novel DIDPT, a new extended dissipativity condition is obtained for MJNNs. A numerical example is employed to illustrate the advantages of the proposed method.

Sampled data Control for Takagi-Sugeno Fuzzy Systems with Actuator Saturation

This paper deals with the problem of sampled data control for a class of Takagi-Sugeno fuzzy systems with actuator saturation. On the basis of the Lyapunov-Krasovskii functional (LKF) and Wirtinger-based double integral inequality, a set of sufficient conditions are derived in terms of linear matrix inequalities (LMIs) which can be easily solved by using some standard numerical packages, then, a state feedback controller is constructed for guaranteing the stability of the considered system. Eventually, the effectiveness of the proposed method have been validated through numerical simulations.

Event triggered reliable synchronization of semi-Markovian jumping complex dynamical networks via generalized integral inequalities

This paper focuses on the synchronization problem of semi-Markovian jumping complex dynamical networks with time-varying coupling delays against actuator failures. In an aim to shrink the treatment of network resources event triggered control strategy is proposed to achieve the synchronization criteria. By constructing Lyapunov–Krasovski functional, some delay dependent criteria that assures the synchronization of CDN are derived with the help of the general integral inequalities. It should be noted that the general integral inequality used here is general than that of Jensen inequality, the Wirtinger-based inequality, the Bessel-Legendre inequality, the Wirtinger-based double integral inequality, and the auxiliary function-based integral inequalities. The resulting LMIs can be easily verified with the help of the available softwares. Finally, simulation results are proposed to verify the effectiveness of the general integral inequality and designed control law.

Stability and stabilization of T–S fuzzy systems with time delay via Wirtinger-based double integral inequality

Abstract This paper concerns the issue of stabilization and stability analysis for Takagi–Sugeno (T–S) fuzzy systems with time delay. A new type of Lyapunov–Krasovskii functional (LKF), including Non-Quadratic Lyapunov functional and triple integral term, is introduced to obtain stability conditions of fuzzy time-delay systems. A Wirtinger-based double integral inequality is used to estimate integral term, and the free weighting variable technique is also employed for controller syntheses and stability analysis. Additionally, less conservative and newer stability conditions with delay-dependent are proposed in the form of linear matrix inequalities (LMIs). Furthermore, Several examples are provided to illustrate how effective the suggested approaches are.

Further results on dissipativity analysis for Markovian jump neural networks with randomly occurring uncertainties and leakage delays

This paper is concerned with the mixed $$H_\infty$$ and dissipativity performance for Markovian jump neural networks with time delay in the leakage term and randomly occurring uncertainties. The randomly occurring uncertainties are assumed to be mutually uncorrelated Bernoulli-distributed white noise sequences. By introducing a triple-integrable term in the Lyapunov functional, the Wirtinger-based double-integral inequality is utilized to bound the derivative of the triple-integral term and then a sufficient condition is derived to ensure that the considered neural networks to be strict $$({\mathcal {Q,S,R}}) - \gamma$$ -dissipative and passive. These conditions are presented in terms of linear matrix inequalities, which can be easily solved by using standard numerical software. Finally, numerical examples are given to show the effectiveness and the potential significance of the proposed results.

Synchronization for memristive chaotic neural networks using Wirtinger-based multiple integral inequality

This paper investigates the synchronization problem for a class of memristive chaotic neural networks with time-varying delays. Based on te Wirtinger-based double integral inequality, two novel inequalities are proposed, which are multiple integral forms of the Wirtinger-based integral inequality. Next, by applying the reciprocally convex combination approach for high order case and a free-matrix-based inequality, novel delay-dependent conditions are established to achieve the synchronization for the memristive chaotic neural networks. The results are based on dividing the bounding of activation function into two subintervals with equal length. Finally, a numerical example is provided to demonstrate the effectiveness of the theoretical results.

Vibration control of structural systems via robust non-fragile sampled-data control scheme

This paper investigates the dissipative non-fragile output feedback sampled-data control problem for uncertain structural systems. In the proposed system, uncertainties are assumed to be time-varying and bounded, which are considered in the linear fractional transformation form. The main objective of this paper is to design a non-fragile sampled-data controller such that the resulting closed-loop system is strictly (Q,S,R)-α-dissipative. On the basis of a suitable Lyapunov–Krasovskii functional and linear matrix inequality technique, a new set of sufficient conditions is established to achieve the required result for both nominal and uncertain systems. In particular, an output feedback dissipative sampled-data controller is designed by solving a set of matrix inequalities. More precisely, Schur complement lemma, free weighting matrix approach and Wirtinger-based double integral inequality are utilized to substantially simplify the derivation of the main results. Finally, simulations based on structural systems are conducted to illustrate the effectiveness of the proposed control scheme.

Wirtinger-based multiple integral inequality approach to synchronization of stochastic neural networks

This paper investigates the synchronization problem for a class of chaotic neural networks with discrete and unbounded distributed time delays under stochastic perturbations. Firstly, based on the Wirtinger-based double integral inequality, two novel inequalities are proposed, which are multiple integral forms of the Wirtinger-based integral inequality. Next, by applying the Jensen-type integral inequality for stochastic case and combining the Jensen integral inequality with the reciprocally convex combination approach, a delay-dependent criterion is developed to achieve the synchronization for the stochastic chaotic neural networks in the sense of mean square. In the case of no stochastic perturbations, by applying the reciprocally convex combination approach for high order case and a free-matrix-based inequality, novel delay-dependent conditions are established to achieve the synchronization for the chaotic neural networks. All the results are based on dividing the bounding of activation function into two subintervals with equal length. Finally, two numerical examples are provided to demonstrate the effectiveness of the theoretical results.

Robust finite time stabilization analysis for uncertain neural networks with leakage delay and probabilistic time-varying delays

This paper investigates the problem of robust finite time stabilization for a uncertain neural networks with leakage delay and probabilistic time-varying delays. By introducing a stochastic variable which satisfies Bernoulli distribution, the information of probabilistic time-varying delay is equivalently transformed into the deterministic time-varying delay with stochastic parameters. The main objective of this paper is to design a memoryless state feedback control such that the resulting proposed system is robustly finite time stable with admissible uncertainties. Based on a suitable Lyapunov–Krasovskii functional, model transformation technique and Wirtinger-based double integral inequality, the general framework is obtained in terms of linear matrix inequalities to determine the finite time stability and to achieve the control design. Finally, three numerical examples are presented to validate the effectiveness and less conservatism of the proposed method.

Stability of Markovian Jump Generalized Neural Networks With Interval Time-Varying Delays

This paper examines the problem of asymptotic stability for Markovian jump generalized neural networks with interval time-varying delays. Markovian jump parameters are modeled as a continuous-time and finite-state Markov chain. By constructing a suitable Lyapunov-Krasovskii functional (LKF) and using the linear matrix inequality (LMI) formulation, new delay-dependent stability conditions are established to ascertain the mean-square asymptotic stability result of the equilibrium point. The reciprocally convex combination technique, Jensen's inequality, and the Wirtinger-based double integral inequality are used to handle single and double integral terms in the time derivative of the LKF. The developed results are represented by the LMI. The effectiveness and advantages of the new design method are explained using five numerical examples.

Stability analysis of time-delay systems via free-matrix-based double integral inequality

ABSTRACTBased on the free-weighting matrix and integral-inequality methods, a free-matrix-based double integral inequality is proposed in this paper, which includes the Wirtinger-based double integral inequality as a special case. By introducing some free matrices into the inequality, more freedom can be provided in bounding the quadratic double integral. The connection of the new integral inequality and Wirtinger-based double one is well described, which gives a sufficient condition for the application of the new inequality to be less conservative. Furthermore, to investigate the effectiveness of the proposed inequality, a new delay-dependent stability criterion is derived in terms of linear matrix inequalities. Numerical examples are given to demonstrate the advantages of the proposed method.

Robust finite-time sampled-data control of linear systems subject to random occurring delays and its application to Four-Tank system

This work studies the problem of robust finite-time sampled-data control of linear systems subject to random occurring delays and its application to Four-Tank system. The input time delays are described by Bernoulli distributed white sequences. By employing a newly augmented Lyapunov–Krasovskii functional with Jensen’s inequality, free-weighting matrix method and Wirtinger-based double integral inequality approach, some less conservative delay-dependent conditions for finite-time sampled-data control law are obtained. Finally, to reveal the proposed technique, an application to a Four-Tank system is addressed to illustrate the effectiveness of developed techniques.

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

ABSTRACTThis 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.

Robust stabilization of uncertain neural networks with additive time-varying delays∗∗This work was supported by SERB project DST, New Delhi, India under the F. No: YSS/2014/000447 dated 20.11.2015.

Abstract This study is mainly concerned with the problem of robust stabilization of neural networks with two additive time-varying delays via Wirtinger-based double integral inequality. The main purpose of this paper is to design a memoryless state feedback controller which ensuring that the global asymptotic stability of closed loop system. By constructing a suitable Lyapunov-Krosovskii functionals, utilizing Wirtinger-based double integral inequality, the sufficient criteria are derived in terms of linear matrix inequalities technique which ensuring the global asymptotic stability of the proposed neural networks with norm bounded uncertainties. The desired controllers can be calculated by solving the linear matrix inequalities with the help of some standard numerical packages. Finally, the numerical examples are given to demonstrate the effectiveness of the theoretical results.

New passivity criteria for memristive uncertain neural networks with leakage and time-varying delays

In this paper, the problem of passivity analysis is studied for memristor-based uncertain neural networks with leakage and time-varying delays. By combining differential inclusions with set-valued maps, the system of memristive neural networks is changed into the conventional one. By adding a triple quadratic integral and relaxing the requirement for the positive definiteness of some matrices, a proper Lyapunov–Krasovskii functional is constructed. Based on the establishment of the novel Lyapunov–Krasovskii functional, the new passivity criteria are derived by mainly applying Wirtinger-based double integral inequality, S-procedure and so on. Moreover, the conservatism of passivity conditions can be reduced. Finally, four numerical examples are given to show the effectiveness and less conservatism of the proposed criteria.

Stability of time-delay systems via Wirtinger-based double integral inequality

Based on the Wirtinger-based integral inequality, a double integral form of the Wirtinger-based integral inequality (hereafter called as Wirtinger-based double integral inequality) is introduced in...

Stability of time-delay systems via Wirtinger-based double integral inequality

Based on the Wirtinger-based integral inequality, a double integral form of the Wirtinger-based integral inequality (hereafter called as Wirtinger-based double integral inequality) is introduced in this paper. To show the effectiveness of the proposed inequality, two stability criteria for systems with discrete and distributed delays are derived within the framework of linear matrix inequalities (LMIs). The advantage of employing the proposed inequalities is illustrated via two numerical examples.