Construction Of Lyapunov-Krasovskii Functionals Research Articles

Constructing Lyapunov–Krasovskii functionals for linear positive continuous time difference systems

Abstract In this paper, a linear positive functional difference system with discrete and distributed delays is studied. It is proved that if the considered system is exponentially stable, then it admits both linear and diagonal quadratic Lyapunov–Krasovskii functionals. In addition, new exponential stability conditions for switched linear positive functional difference systems are obtained on the basis of a constructive approach to the design of common Lyapunov–Krasovskii functionals for the associated families of subsystems.

Constructing Lyapunov–Krasovskii functionals for some classes of switched positive systems with infinite delay

Constructing Lyapunov–Krasovskii functionals for some classes of switched positive systems with infinite delay

Input–Output Finite-Time Stabilization of T–S Fuzzy Systems Through Quantized Control Strategy

The issues of input–output finite-time stability and stabilization of Takagi–Sugeno (T–S) fuzzy systems with time-varying state delay and exogenous input signal are studied in this article. For the stabilization process, a controller that does not share the same membership functions as the system is considered and the state feedback is quantized by using a logarithmic quantizer. The desired stability criterion for the addressed fuzzy system is first derived without the control term. It is then extended to the case in which the proposed controller is present. The stability criteria are given in the form of matrix inequalities, which are supported by the augmented Lyapunov–Krasovskii functional, generalized free-weighting-matrix approach and Finsler’s lemma. Moreover, as a special case, an asymptotic stability criterion is obtained and used in a comparative study. As a result, the time-delay interval is much larger than some of the works published very recently, which is due to the augmented Lyapunov–Krasovskii functional construction. In addition, the effectiveness and practicability of the theoretical findings are validated with simulation examples.

Asynchronous finite-time state estimation for semi-Markovian jump neural networks with randomly occurred sensor nonlinearities

This paper addresses the finite-time state estimation problem for semi-Markovian jump neural networks with sensor nonlinearities under the consideration of leakage delay and time-varying delay. The modes of original system and desired estimator are supposed to be asynchronous. Some sufficient conditions are proposed to guarantee the finite-time boundedness as well as mixed H∞ and passive performance of the error system in terms of constructing Lyapunov–Krasovskii functionals. By utilizing affine Bessel-Legendre inequalities, a less conservative result can be achieved. By virtue of linear matrices inequalities approach, the asynchronous state estimator gains are obtained. Two numerical examples are provided to demonstrate the less conservativeness and effectiveness of our method.

An Affine Integral Inequality of an Arbitrary Degree for Stability Analysis of Linear Systems With Time-Varying Delays

This paper is concerned with the stability analysis problems of linear systems with time-varying delays using integral inequalities. To reduce the conservatism of stability criteria obtained with Lyapunov-Kraosvksii approach, there has been a growing tendency to utilize various integral quadratic functions in the construction of Lyapunov-Krasovskii functionals. Consequently, integral inequalities also have played key roles to derive stability criteria guaranteeing the negativity of the Lyapunov-Krasovskii functional’s derivative. Recently, by utilizing first-degree or second-degree orthogonal polynomials, new free-matrix-based integral inequalities have been proposed for integral quadratic functions containing both a system state variable and its time derivative. This paper tries to generalize these inequalities and their stability criteria with a note on the relation among the existing integral inequalities and the proposed one. In this note, it is shown that increasing a degree of the proposed integral inequality only reduces or maintains the conservatism by deriving the hierarchical stability criteria. Four numerical examples including practical systems demonstrate the effectiveness of the proposed methods in terms of allowable upper delay bounds.

Novel Inequalities to Global Mittag-Leffler Synchronization and Stability Analysis of Fractional-Order Quaternion-Valued Neural Networks.

This article is concerned with the problem of the global Mittag-Leffler synchronization and stability for fractional-order quaternion-valued neural networks (FOQVNNs). The systems of FOQVNNs, which contain either general activation functions or linear threshold ones, are successfully established. Meanwhile, two distinct methods, such as separation and nonseparation, have been employed to solve the transformation of the studied systems of FOQVNNs, which dissatisfy the commutativity of quaternion multiplication. Moreover, two novel inequalities are deduced based on the general parameters. Compared with the existing inequalities, the new inequalities have their unique superiorities because they can make full use of the additional parameters. Due to the Lyapunov theory, two novel Lyapunov-Krasovskii functionals (LKFs) can be easily constructed. The novelty of LKFs comes from a wider range of parameters, which can be involved in the construction of LKFs. Furthermore, mainly based on the new inequalities and LKFs, more multiple and more flexible criteria are efficiently obtained for the discussed problem. Finally, four numerical examples are given to demonstrate the related effectiveness and availability of the derived criteria.

Delay-partitioning-based reachable set estimation of Markovian jump neural networks with time-varying delay

The objective of this paper is to estimate the reachable set for a class of delayed neural networks (NNs) subject to Markovian jump parameters and bounded disturbance. First, by virtue of delay-partitioning method, the time-varying delay is divided into some delay components. With suitably constructing Lyapunov–Krasovskii functionals (LKFs), a less conservative delay-dependent condition of finding an ellipsoid-like set to contain all state trajectories that start from the origin is derived in terms of linear matrix inequalities (LMIs). Then the integrally free-matrix-based inequality approach together with the extended reciprocally convex technique is employed to further reduce the conservatism on characterizing bounds of some integral terms. Thanks to a group of free-connection weighting matrices, the proposed reachable set estimation approach is extended to the case that transition probabilities are partially known. Finally, numerical simulations indicate that the derived results are effective and less conservative.

Lyapunov–Krasovskii functionals for input‐to‐state stability of switched non‐linear systems with time‐varying input delay

This study investigates the input-to-state stability issue and the Lyapunov–Krasovskii functional construction for a class of switched non-linear systems with time-varying input delay under mode-dependent average dwell-time scheme. The authors assume that an exponential stabilising state feedback controller has been predesigned for the nominal system and the Lyapunov function for the nominal closed-loop system is available. When input delay and disturbance are involved, the closed-loop system may be unstable under the average dwell time for the nominal system. In allusion to this instance, by introducing a positive definite relaxation matrix and constructing a novel piecewise Lyapunov–Krasovskii functional based on the known Lyapunov function, they provide sufficient conditions and the upper bound of the input delay such that the closed-loop system is input-to-state stable with respect to the disturbance. Finally, numerical examples are given to illustrate the effective of the proposed method.

Lyapunov–Krasovskii functionals for predictor feedback control of linear systems with multiple input delays

This paper is concerned with the Lyapunov–Krasovskii functional construction of linear control systems with multiple input delays. By transforming the predictor feedback control systems into a delay-free linear system with external inputs, a Lyapunov–Krasovskii functional is constructed in terms of a set of linear matrix inequalities (LMIs). It is shown that the solvability of this set of LMIs is equivalent to the asymptotic stability of the delay-free linear system induced from the predictor feedback control system. The proposed Lyapunov–Krasovskii functional is also found to be an ISS Lyapunov–Krasovskii functional for the predictor feedback control systems. An example is worked out to validate the effectiveness of the proposed method.

Relaxed passivity conditions for neural networks with time-varying delays

This paper revisits the problem of passivity analysis for neural networks with time-varying delays. A new delay-dependent criterion is obtained in terms of linear matrix inequalities, guaranteeing that the input and output of the considered neural network satisfy a prescribed passivity-inequality constraint. This newly presented criterion does not require all the symmetric matrices involved in the employed quadratic Lyapunov–Krasovskii functional to be positive definite. This feature is remarkable since it sheds new light on the traditional ideas for constructing Lyapunov–Krasovskii functionals. More importantly, the conservatism of delay-dependent passivity conditions can be reduced due to the relaxation on the positive-definiteness of every Lyapunov matrix. It is shown both theoretically and numerically that the passivity criterion proposed in this paper is truly less conservative than some of the latest results in the literature.

Stability and robust stability for systems with a time-varying delay

To concern the stability and robust stability criteria for systems with time-varying delays, this note uses not only the time-varying-delayed state x ( t - h ( t ) ) but also the delay-upper-bounded state x ( t - h ¯ ) to exploit all possible information for the relationship among a current state x ( t ) , an exactly delayed state x ( t - h ( t ) ) , a marginally delayed state x ( t - h ¯ ) , and the derivative of the state x ˙ ( t ) , when constructing Lyapunov–Krasovskii functionals and some appropriate integral inequalities, originally suggested by Park (1999. A delay-dependent stability criterion for systems with uncertain time-invariant delays. IEEE Transactions on Automatic Control, 44(4), 876–877). Two fundamental criteria are provided for the cases where no bound of delay derivative is assumed and where an upper bound of delay derivative is assumed. Examples show the resulting criteria outperform all existing ones in the literature.

Stability of Systems With Uncertain Delays: A New “Complete” Lyapunov–Krasovskii Functional

Stability of linear systems with uncertain time-varying delay is considered in the case, where the nominal value of the delay is constant and nonzero. Recently a new construction of Lyapunov-Krasovskii functionals (LKFs) has been introduced: To a nominal LKF, which is appropriate to the system with nominal delays, terms are added that correspond to the perturbed system and that vanish when the delay perturbations approach 0. In the present note we combine a complete nominal LKF, the derivative of which along the trajectories depends on states and their derivatives, with the additional terms depending on the delay perturbation. The new method is applied to the case of multiple uncertain delays with one nonzero nominal value. Numerical examples illustrate the efficiency of the method.

ON ROBUST STABILITY VIA COMPLETE LYAPUNOV KRASOVSKII FUNCTIONAL

Stability of linear systems with norm-bounded uncertainties and uncertain time-varying delays is considered. The delays are supposed to be bounded and fast-varying (without any constraints on the delay derivative). Sufficient stability conditions are derived via complete Lyapunov-Krasovskii functional (LKF). A new LKF construction, which was recently introduced for systems with uncertain delays, is extended to the case of norm-bounded uncertainties: to a nominal LKF, which is appropriate to the system with the nominal value of the coefficients and of the delays, terms are added that correspond to the perturbed system and that vanish when the uncertainties approach 0. Numerical examples illustrate the efficiency of the method.

Passivity analysis of neural networks with time delay

The passivity conditions for delayed neural networks (DNNs) are considered in this paper. We firstly derive the passivity condition for DNNs without uncertainties, and then extend the result to the case with time-varying parametric uncertainties. The proposed approach is based on a Lyapunov-Krasovskii functional construction. The passivity conditions are presented in terms of linear matrix inequalities, which can be easily solved by using the effective interior-point algorithm. Numerical examples are also given to demonstrate the effectiveness of the theoretical results.

On the passivity of linear delay systems

This note presents some sufficient conditions for a linear system with (point or distributed) delay in the state to be passive. Passivity based (memoryless) stabilization of linear delay systems is also addressed. The proposed approach is based on a Lyapunov-Krasovskii functional construction. Numerical examples are also included.

An optimization method for constructing Lyapunov-Krasovskii functionals in stationary lag systems

A method is proposed for numerical construction of Lyapunov-Krasovskii functionals for analyzing the stability of linear stationary lag systems. The functional is constructed in the form of the sum of two quadratic sums. Positive-definite matrices of quadratic forms are found as a solution of a nonsmooth optimization problem on a convex set. Bibliography: 8 titles.

A method of construction of Lyapunov-Krasovskii functionals for linear systems with deviating argument

A method of construction of Lyapunov-Krasovskii functionals for linear systems with deviating argument