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This paper is concerned with non-fragile sliding mode control of uncertain chaotic systems with external disturbance. Firstly, a new sliding surface is proposed, and sufficient conditions are derived to guarantee that sliding mode dynamics is asymptotically stable with a generalized

Chaotic behavior is a seemingly random phenomenon of a deterministic system that is characterized by sensitive deaspendence on initial conditions. Many electronic, mechanical, and chemical systems exhibit chaotic dynamics. Therefore, chaos is a very interesting nonlinear phenomenon, and control of chaotic systems has been paid much attention by researchers since the pioneering work of Ott et al. [

Among the above-mentioned methods, sliding mode control is a very effective approach to control chaos because of its attractive features such as fast response, good transient response, and insensitivity to variations in system parameters and external disturbances [

Generally speaking, accurate controllers are required to stabilize control systems, so all of the controller coefficients are exact values in designing a desired controller. However, in practice, the uncertainty is not avoided, and it may be caused by many reasons, such as finite word length in digital systems, the imprecision inherent in analog systems, and the need for additional tuning of parameters in the final controller implementation [

In this paper, the problem of non-fragile sliding mode control of uncertain chaotic systems with external disturbance is considered. Firstly, a new integral-type sliding surface is proposed, and the reaching phase is eliminated, and the proposed sliding surface function includes a norm-bounded parameter uncertainty; this term is added not only to make the design of sliding surface more relaxed, but also to guarantee that the sliding mode dynamics has better robust performance. For both the cases with additive and multiplicative uncertainties, sufficient conditions are derived to make sliding mode dynamics stable with a generalized

The following notations will be used throughout this paper.

Consider the following chaotic system:

The block diagram of the system (

The block diagram of the system (

The following lemmas are necessary for future discussion.

For a given matrix

Let

For system (

For system (

In this paper, the following two classes of parameter uncertainty will be considered:

From (

Let

Sliding mode dynamics (

When

Sliding mode dynamics (

Consider the system (

Consider the following Lyapunov functional candidate:

From (

In order to construct the generalized

From (

The proof is completed.

For sliding mode dynamics (

Consider the system (

Pre- and postmultiplying (

From (

The proof is completed.

Consider the system (

The proof is similar to Theorem

The input

Consider the system (

Consider a Lyapunov function candidate as follows:

Substituting (

In this section, we use Genesio's chaotic system to verify the effectiveness of the method. Genesio's system with additional input is as follows:

Without loss of generality, we consider

Then, solving LMIs (

The initial value is

The phase curves of uncontrolled Genesio's system.

State

The control input

Sliding mode

Figure

In this paper, the problem of non-fragile sliding mode control of uncertain chaotic systems with external disturbance is investigated. A new sliding surface is proposed, and sufficient conditions are derived for asymptotic stability with a generalized

The authors are grateful for the support of the National Natural Science Foundation of China under Grants nos. 61074003 and 60904023.