This paper investigates the aperiodic sampled-data control for a chaotic system. Firstly, Takagi–Sugeno (T-S) fuzzy models for the chaotic systems are established. The lower and upper bounds of the sampling period are taken into consideration. Then, the criteria for mean square exponential stability analysis and aperiodic sampled-data controller synthesis are provided by means of linear matrix inequalities. And the real sampling patterns can be fully captured by constructing suitable Lyapunov functions. Finally, an illustrative example shows that the proposed method is effective to guarantee that the system’s states are stable with aperiodic sampled data.

Recently, the chaotic system has gradually become one of the hot topics in the field of nonlinear system. The chaotic system has wide application in several areas such as information processing, chemical reaction, power conversion, secure communication biological system, and other aspects. Thus, the control problem for the chaotic system has attracted considerable attention (see [

On the other hand, the chaotic system can be described with T-S fuzzy models (TSFM) [

In the past few years, the sampled-data system has become an important topic because modern control systems widely used the digital computers to control continuous-time systems (see [

It is noted that the sampling periods of the existing references for the chaotic system are ideally assumed to be constant. However, the sampling period is aperiodic due to the aging of sensors and the interference of noise environment. Hence, to consider nonperiodicity sampling for designing sampled-data controller is practically significant. Besides, in these papers, the lower bound of the sampling period is often considered to be 0 which will lead to considerable conservatism because the value of the variable period may change in a range. Therefore, the lower bound and upper bound of the sampling period should be both considered. Finally, LKF has room for improving to fully capture the real sampling patterns.

Motivated by the above, in this paper, the issue about aperiodic fuzzy sampled-data control of the chaotic system is discussed. Firstly, TSFM is represented for chaotic systems. Then, both lower and upper bounds of the variable period are taken into consideration. In terms of LMI approach, Lyapunov theorem is involved for the stability analysis which can fully capture the sampling patterns. Then, the designed method of fuzzy sampled-data controller is introduced. Finally, a simulation of a chaotic system is conducted to verify the effectiveness of the given strategy.

Consider a chaotic system as follows:

Mode Rule i: IF

Assume that the state variables of chaotic systems are measured in

The framework of the sampled-data chaotic system is given in Figure

The framework of the sampled-data chaotic system.

Then, based on parallel distributed compensation, the fuzzy sampled-data controller is designed.

Controller Rule i: IF

Substituting (

The paper’s purpose is designing a fuzzy sampled-data controller (FSDC) to satisfy that

System (

A longer sampling period is achieved.

In this section, the sufficient stability criteria for system (

For scales

The novel LKF is proposed:

It is noted that

Taking the derivative of

For any free matrix

Then, combining (

From (

Then

The mean square exponential stability criteria for system (

For the purpose of reducing the conservatism, more relaxed constraint matrices are introduced in LKF (

Furthermore, the sampled-data controller (

For scales

Then, the controller gain matrix

Let

Define

Pre- and post-multiply (

Note that the lower and upper bounds of the variable period are both considered in Theorem

In the section, a simulation example for a chaotic system will be used to verify the effectiveness of given methods.

Consider a chaotic system as follows [

Assume that

Mode rule 1: IF

Mode rule 2: IF

where

Then, the membership functions are

and the trajectories of system (30) are exhibited in Figure

Trajectory of the chaotic system.

Let

The maximum upper bound for

Method | [ | [ | [ | Theorem |
---|---|---|---|---|

0.0016 | 0.0022 | 0.0031 | 0.0037 |

Then, we verify the effectiveness of the given method. The initial state is

Then, from (

Under controller (

Responses of _{1}(

Responses of _{2}(

Responses of _{3}(

This paper discusses the aperiodic sampled-data control problem for a chaotic system with TSFM. And both lower and upper bounds of the sampling period are considered in the paper. Then, the criteria of mean square exponential stability are given. By constructing an appropriate LKF, the sampling patterns are fully captured and less conservative result is obtained. The simulation result is used to verify that the proposed fuzzy aperiodic sampled-data control strategy is effective.

No data were used to support this study.

The authors declare that they have no conflicts of interest.

This study was supported by the National Natural Science Foundation of China (51579114 and 51879119), Natural Science Foundation of Fujian Province (2018J01484 and 2020J01660), and Youth Innovation Foundation of Xiamen (3502Z20206019).