Delay-dependent Exponential Stability Research Articles

Exponential synchronization of Markovian jumping neural networks with partly unknown transition probabilities via stochastic sampled-data control

This paper investigates the exponential synchronization for a class of delayed neural networks with Markovian jumping parameters and time varying delays. The considered transition probabilities are assumed to be partially unknown. In addition, the sampling period is assumed to be time-varying that switches between two different values in a random way with given probability. Several delay-dependent synchronization criteria have been derived to guarantee the exponential stability of the error systems and the master systems are stochastically synchronized with the slave systems. By introducing an improved Lyapunov–Krasovskii functional (LKF) including new triple integral terms, free-weighting matrices, partly unknown transition probabilities and combining both the convex combination technique and reciprocal convex technique, a delay-dependent exponential stability criteria is obtained in terms of linear matrix inequalities (LMIs). The information about the lower bound of the discrete time-varying delay is fully used in the LKF. Furthermore, the desired sampled-data synchronization controllers can be solved in terms of the solution to LMIs. Finally, numerical examples are provided to demonstrate the feasibility of the proposed estimation schemes from its gain matrices.

Delay-dependent exponential stability for uncertain neutral stochastic neural networks with interval time-varying delay

This paper is mainly concerned with the problem for the robustly exponential stability in mean square moment of uncertain neutral stochastic neural networks with interval time-varying delay. With an appropriate augmented Lyapunov–Krasovskii functional (LKF) formulated, the convex combination method is utilised to estimate the derivative of the LKF. Some new delay-dependent exponential stability criteria for such systems are obtained in terms of linear matrix inequalities, which involve fewer matrix variables and have less conservatism. Finally, two illustrative numerical examples are given to show the effectiveness of our obtained results.

Research on robust control and exponential stabilization for large scale impulsive hybrid network systems with time-delay

This paper investigates the stability of network control systems, regarded as a class of large-scale hybrid systems, for ubiquitous computing environment. The problem of robust exponential stabilization for the hybrid systems is addressed, which are composed of impulsive subsystems with time-delay and parameter uncertainties. Using the Lyapunov---Krasovskii functional approach and linear matrix inequality method, an adaptive robust controller is designed to stabilize the uncertain continuous subsystems. Then the delay-dependent exponential stability conditions for the whole hybrid system are derived by analyzing the stability of the subsystems. An example is given to show the effectiveness of the proposed design method.

Observer design of discrete-time impulsive switched nonlinear systems with time-varying delays

This paper investigates the problem of observer design for discrete-time impulsive switched nonlinear systems with time-varying delays. Firstly, based on the average dwell time approach and the Lyapunov–Krasovskii functional technique, a delay-dependent exponential stability criterion for the discrete-time impulsive switched nonlinear systems with time-varying delays is derived in terms of a set of linear matrix inequalities (LMIs). Then, sufficient conditions for the existence of an observer that guarantees the exponential stability of the corresponding error system are proposed. Finally, two numerical examples are given to illustrate the effectiveness of the proposed method.

Exponential stability of Markovian jumping stochastic Cohen–Grossberg neural networks with mode-dependent probabilistic time-varying delays and impulses

This paper deals with robust exponential stability of Markovian jumping stochastic Cohen–Grossberg neural networks (MJSCGNNs) with mode-dependent probabilistic time-varying delays, continuously distributed delays and impulsive perturbations. By construction of novel Lyapunov–Krasovskii functional having the triple integral terms, the double integral terms having the positive definite matrices dependent on the system mode and MJSCGNNs system transformation variables, new delay-dependent exponential stability conditions are derived in terms of linear matrix inequalities (LMIs). By establishing a stochastic variable with Bernoulli distribution, the information of probabilistic time-varying delay is considered and transformed into one with deterministic time-varying delay and stochastic parameters. Furthermore, a mode-dependent mean square robust exponential stability criterion is derived by constriction of new Lyapunov–Krasovskii functional having modes in the integral terms, linear matrix inequalities and some stochastic analysis techniques. Finally, two numerical examples are provided to show the effectiveness of the proposed methods.

New results on stability analysis for neutral stochastic linear system

In this study, the problem for the robustly exponential stability in mean square moment of uncertain neutral stochastic linear system with mixed time-varying delays is considered. By constructing an augmented Lyapunov–Krasovskii functional and utilising a convex polyhedron method, some new delay-dependent exponential stability criteria for such system are established in forms of linear matrix inequalities, which have less conservatism. Finally, three illustrative numerical examples are given to show the effectiveness of the obtained results.

Exponential [formula omitted] output tracking control for positive switched linear systems with time-varying delays

The problem of exponential L1 output tracking control for positive switched linear systems with time-varying delays is addressed in this paper. The exponential L1-gain performance index is introduced to study such a problem. By resorting to the average dwell time approach, and also by constructing an appropriate piecewise co-positive type Lyapunov–Krasovskii functional, a new delay-dependent exponential stability criterion is developed, and the exponential L1-gain performance is analyzed. Based on the results obtained, a state feedback controller is constructed such that the exponential L1 output tracking performance is satisfied. A numerical example is given to demonstrate the effectiveness of the proposed method.

Delay-dependent exponential stability criteria for neutral systems with interval time-varying delays and nonlinear perturbations

This paper investigates the problem of global exponential stability for neutral systems with interval time varying delays and nonlinear perturbations. It is assumed that the state delay belongs to a given interval, which means that both the lower and upper bounds of the time-varying delay are available. The uncertainties under consideration are norm-bounded. Based on the Lyapunov–Krasovskii stability theory, delay-partitioning technique and lower bounds lemma, less conservative delay-dependent exponential stability criteria are derived in terms of linear matrix inequalities (LMIs) with fewer decision variables than the existing ones. Numerical examples are given to show the effectiveness of the proposed method.

Delay‐dependent robust H ∞ filter design for uncertain neutral stochastic system with time‐varying delay

This study is concerned with the problem of the robust H ∞ filter design for uncertain neutral stochastic system with time-varying delay. With an appropriate augmented Lyapunov-Krasovskii functional formulated and a convex polyhedron method utilised, some novel delay-dependent exponential stability criteria for uncertain neutral stochastic system with time-varying delay are obtained in terms of linear matrix inequalities. Based on the obtained stability criteria, some sufficient conditions for the solvability of the robust H ∞ filter design are derived. The desired H ∞ filter is designed to guarantee the robustly exponential stability for the filtering error system with a prescribed H ∞ performance level. The novelty is that the desired results are obtained with less conservatism. Finally, two illustrative numerical examples are given to demonstrate the effectiveness of the authors obtained results.

Delay-dependent exponential stability for Markovian jumping stochastic Cohen-Grossberg neural networks with p-Laplace diffusion and partially known transition rates via a differential inequality

In this paper, new stochastic global exponential stability criteria for delayed impulsive Markovian jumping p-Laplace diffusion Cohen-Grossberg neural networks (CGNNs) with partially unknown transition rates are derived based on a novel Lyapunov-Krasovskii functional approach, a differential inequality lemma and the linear matrix inequality (LMI) technique. The employed methods are different from those of previous related literature to some extent. Moreover, a numerical example is given to illustrate the effectiveness and less conservatism of the proposed method due to the significant improvement in the allowable upper bounds of time delays.

Exponential stability analysis of linear systems with multiple successive delay components

A general class of linear systems with multiple successive delay components is considered in this article. The delays are assumed to vary in intervals, and delay-dependent exponential stability conditions are derived in terms of linear matrix inequalities. To reduce conservativeness, a new Lyapunov–Krasovskii functional is designed to contain more complete state information, so that a derivation procedure with time-varying delays treated as uncertain parameters can be adopted. Usage of slack variables and inequalities are refrained as much as possible when bounds on the Lyapunov derivative are sought. The stability criteria are tested by two popular numerical examples, with less conservative results obtained in all the checked cases. Besides, a practical application of the derived conditions is illustrated.

Further Results on Robust Exponential Stabilization for Time-varying Delay Saturating Actuator Systems with Delay-dependence

The problem of delay-dependent exponential robust stabilization for a class of uncertain saturating actuator systems with time-varying delay is investigated. Novel exponential stability and stabilization criteria for the system are derived using the Lyapunov-Krasovskii functional combined with Leibniz-Newton formula. The issue of exponential stabilization for time-varying delay systems with saturating actuator using generalized eigenvalue problem (GEVP) approach remains open, which motivates this paper. The designed controller is dependent on the time-delay and its rate of change. All the conditions are presented in terms of linear matrix inequalities (LMIs), which can solved efficiently by using the convex optimization algorithms. A state feedback control law is also given such that the resultant closed-loop system is stable for admissible uncertainties. Two numerical examples are given to demonstrate the efficiency of the obtained results.

Delay-dependent robust $$H_\infty $$ control for 2-D discrete nonlinear systems with state delays

This paper investigates the problem of robust $$H_\infty $$ control for a class of 2-D (two-dimensional) discrete state delayed systems with sector nonlinearity described by a model of Roesser type. Firstly, a delay-dependent sufficient condition of robust exponential stability for such 2-D discrete systems is derived in linear matrix inequalities (LMIs) form. Secondly, a delay-dependent exponential stability criterion with $$H_\infty $$ performance for the considered systems is also proposed. Then a state feedback $$H_\infty $$ controller is constructed based on the above results. Finally, numerical examples are given to illustrate the effectiveness of the proposed method.

Exponential Stability of Uncertain T-S Fuzzy Switched Systems with Time Delay

This paper discusses the delay-dependent exponential stability of a class of uncertain T-S fuzzy switched systems with time delay. The method is based on Lyapunov stability theorem and free weighting matrices approach. Two illustrative examples are given to demonstrate the effectiveness of the proposed method.

Improved delay-dependent exponential stability criteria for time-delay system

This paper considers the problem of time delay-dependent exponential stability criteria for the time-delay linear system. Utilizing the linear inequality matrices (LMIs) and slack matrices, a novel criterion based on the Lyapunov–Krasovskii methodology is derived for the exponential stability of the time-delay system. Based on the criteria condition we concluded that the upper bound of the exponential decay rate for the time-delay system can be easily calculated. In addition, an improved sufficient condition for the robust exponential stability of uncertain time-delay system is also proposed. Numerical examples are provided to show the effectiveness of our results. Comparisons between the results derived by our criteria and the one given in Liu (2004) [1], Mondie and Kharitonov (2005) [2], and Xu et al. (2006) [3] show that our methods are less conservative in general. Furthermore, numerical results also show that our criteria can guarantee larger exponential decay rates than the ones derived by Liu (2004) [1] and Mondie and Kharitonov (2005) [2] in all time delay points we have tested and in some of time delay points obtained by Xu et al. (2006) [3].

Delay-Dependent Exponential Optimal Synchronization for Nonidentical Chaotic Systems via Neural-Network-Based Approach

A novel approach is presented to realize the optimal exponential synchronization of nonidentical multiple time-delay chaotic (MTDC) systems via fuzzy control scheme. A neural-network (NN) model is first constructed for the MTDC system. Then, a linear differential inclusion (LDI) state-space representation is established for the dynamics of the NN model. Based on this LDI state-space representation, a delay-dependent exponential stability criterion of the error system derived in terms of Lyapunov's direct method is proposed to guarantee that the trajectories of the slave system can approach those of the master system. Subsequently, the stability condition of this criterion is reformulated into a linear matrix inequality (LMI). According to the LMI, a fuzzy controller is synthesized not only to realize the exponential synchronization but also to achieve the optimal performance by minimizing the disturbance attenuation level at the same time. Finally, a numerical example with simulations is given to demonstrate the effectiveness of our approach.

Novel Global Exponential Stability Criterion for Recurrent Neural Networks with Time-Varying Delay

The problem of global exponential stability for recurrent neural networks with time-varying delay is investigated. By dividing the time delay interval [0,τ(t)] intoK+1dynamical subintervals, a new Lyapunov-Krasovskii functional is introduced; then, a novel linear-matrix-inequality (LMI-) based delay-dependent exponential stability criterion is derived, which is less conservative than some previous literatures (Zhang et al., 2005; He et al., 2006; and Wu et al., 2008). An illustrate example is finally provided to show the effectiveness and the advantage of the proposed result.

Optimal Exponential Synchronization of Chaotic Systems with Multiple Time Delays via Fuzzy Control

This study presents an effective approach to realize the optimal <path id=x1D43B d=M865 650q-1 -4 -4 -14t-4 -14q-62 -5 -77 -19.5t-29 -82.5l-74 -394q-12 -61 -0.5 -77t75.5 -21l-6 -28h-273l8 28q64 5 82 21t29 76l36 198h-380l-37 -197q-11 -64 0.5 -78.5t79.5 -19.5l-6 -28h-268l6 28q60 6 75.5 21.5t26.5 76.5l75 394q13 66 2 81.5t-77 20.5l8 28 h263l-6 -28q-58 -5 -75.5 -21t-30.5 -81l-26 -153h377l29 153q12 67 2 81t-74 21l5 28h268z /> <path id=x221E d=M983 225q0 -112 -67 -174.5t-150 -62.5q-91 0 -154.5 43.5t-113.5 129.5q-49 -85 -104 -129t-138 -44q-98 0 -158.5 66t-60.5 154q0 59 21 106.5t54.5 75.5t70.5 43t73 15q90 0 152.5 -43.5t112.5 -128.5q48 84 104.5 128t140.5 44q93 0 155 -65t62 -158zM478 196 q-27 49 -47 80t-50 67t-64 54t-73 18q-48 0 -81.5 -47t-33.5 -128q0 -96 37.5 -157.5t99.5 -61.5q68 0 117.5 47t94.5 128zM889 204q0 91 -35.5 151t-99.5 60q-68 0 -119 -47t-95 -127q27 -49 47 -80.5t50 -67.5t65 -54t74 -18q113 0 113 183z /> exponential synchronization of multiple time-delay chaotic (MTDC) systems. First, a neural network (NN) model is employed to approximate the MTDC system. Then, a linear differential inclusion (LDI) state-space representation is established for the dynamics of the NN model. Based on this LDI state-space representation, this study proposes a delay-dependent exponential stability criterion of the error system derived in terms of Lyapunov’s direct method to ensure that the trajectories of the slave system can approach those of the master system. Subsequently, the stability condition of this criterion is reformulated into a linear matrix inequality (LMI). Based on the LMI, a fuzzy controller is synthesized not only to realize the exponential synchronization but also to achieve the optimal performance by minimizing the disturbance attenuation level. Finally, a numerical example with simulations is provided to illustrate the concepts discussed throughout this work.

Global Exponential Stability Criteria for Bidirectional Associative Memory Neural Networks with Time-Varying Delays

The global exponential stability for bidirectional associative memory neural networks with time-varying delays is studied. In our study, the lower and upper bounds of the activation functions are allowed to be either positive, negative, or zero. By constructing new and improved Lyapunov-Krasovskii functional and introducing free-weighting matrices, a new and improved delay-dependent exponential stability for BAM neural networks with time-varying delays is derived in the form of linear matrix inequality (LMI). Numerical examples are given to demonstrate that the derived condition is less conservative than some existing results given in the literature.

New delay-dependent exponential stability for discrete-time recurrent neural networks with mixed time-delays

The existence, uniqueness and globally exponential stability is investigated for a class of discrete-time recurrent neural networks with discrete and bounded distributed delays. The activation functions are required to be neither bounded nor monotonic. By introducing triple-sum terms, a new Lyapunov-Krasovskii functional is constructed. After using the homeomorphism mapping principle, discrete Jensen inequality and generalised discrete Jensen inequality, a linear matrix inequality LMI approach is developed to establish delay-dependent sufficient conditions for exponential stability of the discrete-time neural networks. As the obtained conditions are expressed in terms of LMIs, the feasibility can be easily checked by using the numerically efficient MATLAB LMI toolbox. Two numerical examples are also given to show the effectiveness of our results.