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This paper considers the problem of the robust stability for the nonlinear system with time-varying delay and parameters uncertainties. Based on the

In general, in all engineering systems, such as communication system, electronics, neural networks, and control systems, disturbance usually emerges in many kinds of situation and then affects the performances of systems. It deserves to be mentioned that, in some special cases, disturbance can reform the instant condition to improve system performances. For examples, the proper local noise power can increase the signal-to-noise ratio of a noisy bistable system and optimize the stochastic resonance of coupled systems, such as small-world and scale-free neuronal networks. [

In these two decades, sliding mode control (SMC) has been a useful and distinctive robust control strategy for many kinds of engineer systems. Depending on the proposed switching surface and discontinuous controller, the trajectories of dynamic systems can be guided to the fixed sliding manifold. The proposed performance on request can be satisfied. In general, there are two main advantages of SMC which are the reducing order of dynamics from the purposed switching functions and robustness of restraining system uncertainties. Many studies have been conducted on SMC [

On the other hand, in the past, most of papers set the control gain in advance to achieve the sufficient condition of the stability. It is not a suitable and accurate way to define the parameters of the system although the parameters are gotten by trial and error. Therefore, in this paper, we use the linear matrix inequality (LMI) theorem to optimize the quasi-sliding mode control gain. By using the computer software MATLAB, quasi-sliding mode controller gain can be found. Based on the

Throughout this paper,

This paper is organized as follows. In Section

In this paper, we consider the stability of a nonlinear uncertain neutral system with time-varying delayed state and disturbance. At first, we consider the system without the uncertain segment. Then, a simple extension of nonlinear systems with uncertain part is considered at the next narration. As mentioned above, we firstly consider the nonlinear system (

Before presenting the main result, we need the following lemma and definition.

Let

For a given matrix

Under the control input

Under the zero initial conditions, the performance index

This paper aims at proposing the

Then, the equivalent controller

Then, the second step is to design the proposed

Consider nonlinear dynamics system (

Define the Lyapunov function:

In order to solve the

Consider nonlinear dynamics system (

Define Lyapunov function:

Pre-multiplying and post-multiplying the

If matrix (

At last, we consider the nonlinear system (

The following result is obtained from Theorem

Consider nonlinear dynamics system with the uncertainty part (

Then, the nonlinear dynamic system with the uncertainty part (

From condition (

Consider the following nonlinear system with the nonautonomous noise, time-varying delay, and parameters uncertainty described as follows:

Time responses of Gauss white noise

Based on Theorem

Time responses of states of the controlled nonlinear system without disturbances.

Time responses of switching surface

Time responses of states of the controlled nonlinear system under effects of disturbances.

Time responses of switching surface

This study investigated the robust stability of the nonlinear system with time-varying delay and parameters uncertainties. Based on the

The authors declare that there is no conflict of interests regarding the publication of this paper.

The authors thank the Ministry of Science and Technology, Taiwan, for supporting this work under Grants NSC 102-2221-E-269-021 and NSC 102-2622-E-269-003-CC3. The authors also wish to thank the anonymous reviewers for providing constructive suggestions.