robust chaos synchronization

8/22/2019 Robust Chaos Synchronization

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Robust Chaos Synchronization for Chuas Circuits

via Active Sliding Mode Control

Olfa Boubaker1, Rachid Dhifaoui2

National Institute of Applied Sciences and Technology, INSAT Centre Urbain Nord, BP

676 - 1080 Tunis, [email protected] , [email protected]

Abstract. In this paper, we construct, in the presence of uncertainties andexternal disturbances, a robust active sliding controller to achieve master

slave synchronization for Chuas circuit. The master circuit is considered as

a nominal system whereas parameter uncertainties just affect the slave

system. Using a Lyapunov approach and a reaching condition in the sliding

surface, it will be shown that finite time synchronization can be guaranteed

under an explicit relation between control parameters and the level of

uncertainties. Numerical simulations are presented to estimate robustness of

the sliding mode controllers.

Keywords: Chaos synchronization, Active Control, Sliding Mode Control,

Robust Control, Chuas Circuit.

1 Introduction

Chaos theory is a current research area extensively investigated in many application

fields including robotics, non smooth mechanical systems and nonlinear electrical

circuits [1], [2]. Principally, Chaos synchronization [3] is the most important

research field in this area. Synchronization implies that two systems which startfrom two different initial conditions are forced, using a control law, to have

identical dynamics after some transitory time [4, 5]. Control and synchronization of

chaos systems was investigated in many research papers using different control

strategies approaches [6], [7], [8] and [9]. In this research area, the Chuas

oscillator [10, 11], considered as the most famous electrical circuit to exemplify

chaos [12, 13], was used to verify the efficiency of the most proposed approaches.

In the other hand, active control approach [14], [15] is one of the most interesting

control strategies for its simplicity. A generalized design of the active control

strategy is developed in [16] while the adaptive design is investigated in [17]. The

non-adaptive active control strategy is easy to design but cannot be adapted to cases

with unknown parameters, whereas adaptive active control is more powerful in

presence of unknown parameters however the controller is usually expensive and

complex to implement. Since system parameters fluctuate in real experimental

situations by internal and external factors [18], the active sliding mode control


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strategy is a better alternative. Active sliding control is one of the most recent

techniques used in chaos synchronization. It is renowned for its simple design and

robustness in practical applications. Several recent results exist in this field; see for

example [19] and [20]. However, in our best knowledge, active sliding mode

control approach was never applied to synchronize Chuas circuits. The lack ofresults can be justified by the complexities of the dynamics introduced by the

Chuas circuit designed by a piecewise linear system [21, 10]

In this paper, we propose a new procedure to design a robust controller for master

slave chaos synchronization for two modified Chua oscillators via active sliding

mode control strategy. The master circuit is considered as a nominal system

whereas parameter uncertainties affect the slave system. Using a Lyapunovapproach and a reaching condition to the sliding surface and assuming some

conditions on the uncertainties and noise magnitude, we will show that finite time

synchronization can be guaranteed. The paper is organized as follows: The problem

formulation will be exposed in the next section. In section 3, the design procedure

of active sliding mode control is established for synchronizing two identical

modified Chua oscillators with known parameters. Robust synchronization will bedeveloped in section 4. In Section 5, simulation results will be finally presented to

prove the robustness of the proposed approach.

2 Problem Formulation

Let consider the most famous electrical circuit to exemplify chaos: the Chuas

circuit shown in Fig.1 with its well-known nonlinear Chua diode having the resistor

vR [11]. The Chua Circuit can be described by its modified model as [10]:

= + = + = (1.a)where : = + ( ) [| + 1 | | 1|]) (1.b) = is the state vector and , , , are someconstant parameters.

Fig. 1. Nonlinear Chua's circuit

For a specific range of these parameters, it is shown that the modified Chua circuitmodel (1) can have a chaotic behavior [10]. In this case, all state variables are

bounded.


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Consider now, a master system described by the Chua nominal model (1) and a

slave system described by the following uncertain model:

= + + + + + = + + = + + (2.a)

where : = + | + 1| | 1| (2.b)= is the state vector of the slave system, is the externalperturbation and

,

,

some internal perturbations. The objective of the

synchronization problem is to design a control vector = such that chaos synchronization betweensystems (1) and (2) is achieved in a finite time. It is clear that the finite-timesynchronization problem can be transformed into the equivalent problem of the

finite-time stabilization of an error system.

To solve the finite time synchronization problem, let define the error between the

master and slave systems as follows: = (3)Therefore, subtracting (2) from (1), the error dynamics is obtained as follows:

= + y + + + + u = + + u = + u

(4)Definition 1: The master system (1) and the slave system (2) are synchronized in a

finite time if there exist a constant time = 0 such that: = 0 (5)and 0 if .The last problem will be solved under the following assumptions:

A1: The uncertainties and are assumed to be unknown but constant in timeand bounded such that: | | , | | A2: Parameters , , , are chosen such that the slave Chuas system model (1)has a chaotic behavior. Therefore, the state variables , and are allbounded.

A3: In spite of uncertainties, slave Chuas system (2) must also have a chaotic

behavior. Therefore, the state variables

,

and

are all bounded.

A4: The non linear uncertainty

is bounded such that:

+ A5: p is assumed to be a white noise: The external perturbation is then boundedsuch as: | | , , , , are some positive constants.


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3 Active sliding mode control synchronization

Active sliding mode control is a discontinuous control strategy which will be

designed in two design stages. The first stage is to select an appropriate active

controller in order to overcome the nonlinearities in the error dynamical model

between the master and the slave systems. The second stage is to design a sliding

mode controller to achieve the synchronization.

In this section, design procedure of the active sliding mode controller is established

for synchronizing two identical modified Chua oscillators with known parameters.

The internal and external perturbations are then not considered here. So, the

dynamical error system (4) can be written as:

= + + = + + = + (6)3.1. First stage: Active controller Design

To overcome the nonlinearities in the error dynamical model (6), let define the

active control vector as: = (7) and are respectively defined by : = (8)where:

=

+ 01 1 10

= y

00

Substituting (7) in (6) we get:

= (9)where:

= 0 00 00 0 For all constants , , > 0 , the error system (6) is asymptotically stable for theactive control vector (7).

3.2. Second stage: Active sliding mode control design

Let, now, define the active sliding mode control vector as (7) where: = + (10)where = and is a bounded input scalar. Substituting(10) in (6) we get:

= + = + = +

(11)For = = = , the system (11) can be written as:


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= + (12)where is the identity matrix. Let now impose to the input vector toverify [22]: = < 0 0 (13)where is a sliding surface defined by: = (14)where = . Furthermore, impose to the sliding surface to satisfythe reaching condition [23]: = (15)Substituting (12) in the derivative of (14), we obtain:

e = C = + (16)

Equating (15) to (16), we obtain the expression of the input vector such that: = (17)Theorem 1. For all scalars > 0 and > 0, the system (12) is finite time stablefor the active sliding mode control (17) using the sliding surface (14) and its

trajectories converge to the equilibrium 0 in a finite time.Proof:

Impose to the error dynamics (12) to have a Lyapunov function:=0.5 (18)The derivative of the Lyapunov function is then given by: = = (19)Using (16) and (17) in (19), the derivative of the Lyapunov function can be written

as:

= (20)

Using (14), the derivative of the Lyapunov function is given by: = (21)For all scalar > 0 and > 0 we have always < 0 for 0. The computationof the reaching time can be made using the reaching condition (15).

4 Robust chaos synchronization

Theorem 2:If the assumptions A1 to A5 are fulfilled, the master system (1) and the slave system

(2) are synchronized in a finite time for the control law: = (22)where :

= + 01 1 10 = 00 using the sliding surface defined by (14) whatever the constant vectors = , = and if there exist a positive definite scalar

satisfying for all positive definite scalars r and q:| | < / (23)


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Proof:

Impose to the error dynamics (4) to have the Lyapunov function (18). Using (14)

and (17) for the error dynamics (4), the derivative function (19) can be written as:

= + (24)where:

= + + p0 (25)Using (14), the derivative of the Lyapunov function (24) is given by: = + (26)or equivalently:

= (27) < 0 is always guaranteed if: < 0 > 0 > 0 < 0 (28)or equivalently: | | < /| | (29)where is some positive scalar constant satisfying: (30)The constant can be computed using assumptions A1 to A5 .

5 Application

To prove the efficiency of the proposed approach, simulation results are conductedusing the parameters , , , =10,18,1.28,0.69). For the nominal mastersystem (1) and the uncertain slave system (2). Uncertainties in the slave Chuascircuit are fixed as : = 0.1 , = 0.1 , =0.1 , | | =0.1. Thechaotic dynamics of the modified Chua's circuit are shown by Fig.2 whereas the

attractor is point up by Fig.3. Robust synchronization is achieved using the theorem

2 for the parameters:= 100 100 100; = 1 1 1; =100; = 10The error dynamics between master and slave systems are displayed by Fig.4

whereas the sliding surface is given by Fig.5. Simulation results prove that robust

chaos synchronization is well achieved in a finite time.

Fig.2. Chaotic dynamics of the Chua's circuit Fig.3. Chua's circuit attractor

0 10 20 30 40 50 60 70 80 90 100-5

0

5

Time (s)

x1(t)

0 10 20 30 40 50 60 70 80 90 100-1

0

1

Time (s)

x2(t)

0 10 20 30 40 50 60 70 80 90 100-5

0

5

Time (s)

x3(t)

-4 -3 -2 -1 0 1 2 3 4-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

x1(t)

x2(t)


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Fig.4. Error dynamics Fig.5. Finite time reaching time

Furthermore, when comparing our approach to related works, we can confirm that

the synchronization problem of the Chuas systems (1) and (2) cant be solved

using the approaches proposed in [19] and [20].

6 Conclusion

Robust chaos synchronization problem is solved for master slave Chuas circuits

using an active sliding mode control approach. An explicit relation between control

parameters and the level of uncertainties was derived for which the robust

controllers ensure finite time stabilization of error dynamics between the master

and the slave Chuas circuits.

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