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This paper proposes the generalized projective synchronization for chaotic heavy symmetric gyroscope systems versus external disturbances via sliding rule-based fuzzy control. Because of the nonlinear terms of the gyroscope, the system exhibits complex and chaotic motions. Based on Lyapunov stability theory and fuzzy rules, the nonlinear controller and some generic sufficient conditions for global asymptotic synchronization are attained. The fuzzy rules are directly constructed subject to a common Lyapunov function such that the error dynamics of two identical chaotic motions of symmetric gyros satisfy stability in the Lyapunov sense. The proposed method allows us to arbitrarily adjust the desired scaling by controlling the slave system. It is not necessary to calculate the Lyapunov exponents and the Eigen values of the Jacobian matrix. It is a systematic procedure for synchronization of chaotic systems. It can be applied to a variety of chaotic systems no matter whether it contains external excitation or not. It needs only one controller to realize synchronization no matter how much dimensions the chaotic system contains, and the controller is easy to be implemented. The designed controller is robust versus model uncertainty and external disturbances. Numerical simulation results demonstrate the validity and feasibility of the proposed method.

Dynamic chaos is a very interesting nonlinear effect which has been intensively studied during the last three decades. Chaotic phenomena can be found in many scientific and engineering fields such as biological systems, electronic circuits, power converters, and chemical systems [

Since the synchronization of chaotic dynamical systems has been observed by Pecora and Carroll [

On the other hand, the dynamics of a gyro is a very interesting nonlinear problem in classical mechanics. The gyro has attributes of great utility to navigational, aeronautical, and space engineering [

The goal of this paper is to synchronize two chaotic heavy symmetric gyroscope systems versus external disturbances. To achieve this goal, sliding rule-based fuzzy control is applied. In addition, the results of this paper may be extended to synchronize many classes of nonlinear chaotic systems.

This paper is organized as follows. In Section

The symmetric gyroscope mounted on a vibrating base is shown in Figure

A schematic diagram of a symmetric gyroscope.

This gyro system exhibits complex dynamics and has been studied by [

Time series of

In the next section, the chaos synchronization problem has been explained.

Consider two coupled, chaotic gyroscope systems, where the master and slave systems are denoted by

Defining the generalized synchronization errors between the master and slave systems as follows:

In order to simplify the following procedure, a nonlinear function is defined as follows:

The objective of the synchronization problem is to design the appropriate control signal

The scheme of GPS of chaotic gyroscope systems versus disturbances via the fuzzy system based on sliding mode control is shown in Figure

The scheme of generalized projective synchronization of chaotic gyroscopes versus disturbances via the sliding rule-based fuzzy control.

Using the sliding mode control method for GPS of chaotic gyroscope systems, involves two basic steps:

selecting an appropriate sliding surface such that the sliding motion on the sliding manifold is stable and ensures

establishing a robust control law which guarantees the existence of the sliding manifold

The sliding surfaces are defined as follows [

The rate of convergence of the sliding surface is governed by the value assigned to parameter

In this study, define a sliding surface as

A set of the fuzzy linguistic rules based on expert knowledge are applied to design the control law of fuzzy logic control. To overcome the trail-and-error tuning of the membership functions and rule base, the fuzzy rules are directly defined such that the error dynamics satisfies stability in the Lyapunov sense. The basic fuzzy logic system is composed of five function blocks [

The fuzzy rule base consists of a collection of fuzzy if-then rules expressed as the form: if

In this study, the FLC is designed as follows: the signal

If

For given input values of the process variables, their degrees of membership

Table

Using

To solve the control problem presented in (

Differentiating (

Substituting (

The corresponding requirement of Lyapunov stability is [

Rule table of FLC.

Rule | Antecedent | Consequent |

1 | ||

2 | ||

3 |

If

Equation (

Let us choose the control input as follows such that (

If

Let us choose the control input as follows such that (

If

Equation (

Let us choose the control input as follows such that (

Therefore, all of the rules in the FLC can lead to Lyapunov stable subsystems under the same Lyapunov function (

In this section, numerical simulations are given to demonstrate GPS of the chaotic gyros versus disturbances via the sliding rule-based fuzzy control. The parameters of nonlinear chaotic gyroscope systems are specified in Section

The external disturbance

Notice that, to reduce the system chattering, the sign functions are substituted with the saturation functions.

The time responses of the master and the slave system for GPS with

Time responses of the master and slave systems (

Synchronization errors for GPS with

Synchronization error (

In addition, the control input and sliding surface for GPS with

The sliding surface and input control (

Time responses of the master and slave systems (complete synchronization

Synchronization error (complete synchronization

The sliding surface and input control (complete synchronization

Time responses of the master and slave systems (antisynchronization

Synchronization error (antisynchronization

The sliding surface and input control (antisynchronization

The simulation results of GPS via the sliding rule-based fuzzy control have good performances and confirm that the master and the slave systems achieve the synchronized states, when external disturbance occurs. Also, these results demonstrate that the synchronization error states are regulated to zero asymptotically. It is observed that the proposed method is capable to GPS, when disturbances occur.

In this paper, generalized projective synchronization of chaotic gyroscope systems with external disturbances via sliding rule-based fuzzy control has been investigated. Based on Lyapunov stability theory and fuzzy rules, the nonlinear controller and some generic sufficient conditions for global asymptotic synchronization are attained. To achieve GPS, it is clear that the proposed method is capable for creating a full-range GPS of all state variables in a proportional way. It also allows us to arbitrarily adjust the desired scaling by controlling the slave system. The advantages of this method can be summarized as follows:

it is a systematic procedure for GPS of chaotic gyroscope system;

the controller is easy to be implemented;

it is not necessary to calculate the Lyapunov exponents and the eigenvalues of the Jacobian matrix, which makes it simple and convenient;

the controller is robust versus external disturbances.

Simulations results have verified the effectiveness of this method for GPS of chaotic gyroscope systems.

Since the gyro has been utilized to describe the mode in navigational, aeronautical, or space engineering, the generalized projective synchronization procedure in this study may have practical applications in the future.