INTRODUCTION
Chaos synchronization has gained a lot of attention among scientists
from variety of research fields over the last few years (Chen and Dong,
1998). Chaos synchronization can be applied in the vast areas of physics
and engineering science, especially in secure communication (Kocarev and
Parlitz, 1995; Murali and Lakshmanan, 1998). The idea of synchronizing
two identical chaotic systems was first introduced by Carroll and Pecora
(1991). In the most synchronization approaches used for the continuoustime
chaotic systems, masterslave or driveresponse formulism is employed
(Pecora and Carroll, 1990). Let us call a particular chaotic system the
master (drive) and another one the slave (response). The goal is to synchronize
the slave (response) system behavior to the master (drive) one. In order
to achieve the synchronization, a nonlinear controller that obtains signals
from the master and slave systems and manipulates the slave system should
be designed. Recently, many control methods have been developed to achieve
chaos synchronization between two identical chaotic systems with different
initial conditions (Yau et al., 2005, 2006; Yau, 2004; Lin et
al., 2005). However, in the earlier research, none discusses how to
obtain optimal or near optimal digital controller to synchronize continuous
chaotic systems according to a specified performance index. As we know,
the Evolutionary Programming Algorithm (EPA) has been considered as a
useful technique for finding the global optimization solutions for certain
complicated functions and also has been applied to solve difficult problems
in the field of control engineering (Cao, 1997). Generally speaking, the
EP algorithm for global optimization contains four parts: initialization,
mutation, competition and reproduction. Furthermore, a Quasi Random Sequence
(QRS) is used to generate an initial population for EP algorithm to avoid
causing clustering around an arbitrary local optimum. On the other hand,
a great majority of industrial processes are still controlled by means
of proportionalintegralderivative (PID) controller due to its simplicity
in architecture and acceptable performance. However, it is extremely difficult
to find the optimal set of PID gains for nonlinear dynamical systems.
Since the PID controller gains play an important role in determining the
behavior of the dynamical system, many tuning schemes for linear systems
have been proposed in the literature (Chien et al., 1952; Cameron
and Seborg, 1983; David et al., 2006). The objective of this study
is to present a simple but effective digital PID controller to implement
the mutual synchronization of two identical Sprott chaotic circuits. The
EP algorithm is used for determining the optimal control gains of digital
PID controller. An optimization problem is then well defined and an EP
algorithm is presented to solve the optimization problem such that the
cost function of masterslavesystem is minimized as possibly. The numerical
and experimental results are used to demonstrate the proposed controller
in this research.
PROBLEM FORMULATION
The one type of Sprott circuits is defined by David et al.
(2006):
where, the dots above the variable x means time derivatives (first, second
and third) and sign (·) is the sign function. A state representation
of system (1) can be obtained by defining Now
define a master system, denoted with the subscript m, given in the state
form:
Also define a slave system with the same form, denoted with the subscript
s:
Now let us define the state errors between the master system and slave system as:
One main objective of this work is to present a simple but effective
PID controller based on EP algorithm to achieve the synchronization of
two identical chaotic systems with different initial conditions. The term
u in (3) is a PID controller obtained via EP algorithm to guarantee the
synchronization performance. The procedure to determine the PID controller
u is to first define the output error signal y_{e} = y_{s}y_{m},
then the transfer function of a digital PID controller, from input e(z)
to output u(z) in zdomain as shown in Fig. 1, is generally
given by:
where, T is the sampling time, k_{p} is the proportional gain,
k_{i} is the integral time constant and k_{d} is the derivative
time constant.

Fig. 1: 
The digital controlled system 
In general, for a controller design, the performance criterion or objective
function can be defined according to our desired specifications. Two kinds
of performance criteria usually considered in the EP algorithm are the
Integrated Squared Error (ISE) and the integrated absolute error (IAE).
In this study, the IAE index is used as the Objective Function (OF), which
is given as:
where, E(k)T = [e_{1} (kT) e_{2} (kT) e_{3} (kT)]
and · is the Euclidean norm of a vector and k is referred to
as the sampling time point and k_{f} is the total numbers of sampling.
In the following, based on using EP algorithm, we will develop a tuning
method for the digital PID controller with optimal gain parameters to
minimize the objective function score (6).
SOLVING THE OPTIMIZATION PROBLEM VIA EPA
Here, an extended EP algorithm is proposed to obtain the digital
PID controller with optimal gain parameters to minimize the following
Objective Function (OF) score (6). Let g be the continuously differentiable
matrixvalued function defined for gεS, where S = {gεR^{3}0<=g_{i}<=M_{i},
I = 1,2,3}, M_{i} is the searching space and is bounded. The optimization
problem involves finding
such that the OF performance index of the system is minimized. More accurately,
the optimization problem (P1) is stated mathematically as:
(P1): To find g*εS such that
is minimized.
Based on the results shown in Cao (1997), an extended EP algorithm for
solving the above optimization problem is described as following steps:
Step 1: Generate an initial population p_{0} = [p_{1},
p_{2}, K, pN] of size N by randomly initializing each 3dimensional
solution vector p_{i} ε S, i = 1,2,..., N, according to the
Quasi Random Sequence (QRS).
Step 2: Calculate the fitness score (objective function) f_{i}
= f(p_{i}) for every p_{i}, i = 1,2,K, N, where
Step 3: Mutate every pi, i = 1,2,..., N, based on the statistics
to double the population size from N to 2N and generate p_{i}+N
in the following:
where, p_{i}, j denotes the jth element of the ith individual,
represents a Gaussion random variable with a mean zero and variance
is the sum of all fitness scores and β is a parameter to scale
Step 4: Calculate the fitness score f_{i}+N for every
pi, N, i = 1,2,..., N, by using Eq. 8. By the stochastic competition process,
pi, i = 1,2,..., N, competes with p_{j}, j = N+1,...,2N randomly
each against the other. If f_{i}<f_{j}, p_{i}
is the winner and survives; otherwise, p_{j} is the winner and
p_{i} is replaced by p_{j}. After the competition process,
we select the N winners for the next generation and let the individual
with the minimum objective function in the winners be p_{1}.
Step 5: If the value f_{Σ} converges to a minimum
value, then g* = p_{1} is the global optimum value and such
that the OF performance index of the system is minimized as possibly.
Otherwise, return to Step 3.
SIMULATION AND EXPERIMENTAL RESULTS
The initial value conditions (x_{1m}, x_{2m}, x_{3m})
= (0.1, 0.1, 0.1) and (x_{1s}, x_{2s}, x_{3s})
= (1, 1,1) are used in this numerical example. From Fig.
2, it can be seen that the master system exists a complex trajectory
in the phase plane. Figure 3 reveals that the corresponding
maximum Lyapunov exponent has a positive value and thus it can
be inferred that the master system is in a state of chaotic motion (Chen and
Dong, 1998).

Fig. 2: 
The phase plane trajectory of a Sprott circuit 

Fig. 3: 
The maximum Lyapunov exponent of x_{1m} plot
as a function of the number of driving cycles 
We solved the optimization problem (P1) with N = 70 and β = 0.01
by using the control toolbox of Matlab and Simulink. According to the
proposed EP algorithm, we generated P_{0} = [p_{1}, p_{2},
. . ., p_{70}] according to the QRS. It can be easily observed
from Fig. 4 that it converges after about 100 iterations
and its final value of IAE is f(z*) = 0.1623. Correspondingly, the PID
control gains areThe trajectories of k_{p}, k_{i} and k_{d} during
the evolutionary procedure are also shown in Fig. 5.
For reference, the output response using the resulting PID control gains
z* is then shown in Fig. 6. The numerical simulation
results show that the proposed PID controller via EP algorithm is viable
for synchronization of chaotic systems.
In order to verify the proposed PID controller in practical system, an
electronic circuit implementing Eq. (23)
is shown in Fig. 7, shows the experimental results when these circuits are connected in a master/slave configuration.

Fig. 4: 
Convergence curve of IAE 

Fig. 5: 
Trajectories of k_{p}, k_{i} and k_{d} 

Fig. 6: 
Numerical time responses of chaos synchronization in
Sprott circuits: master and slave system outputs are x_{1m},
x_{2m}, x_{3m} (——)
and x_{1s}, x_{2s}, x_{3s} (  ), respectively 

Fig 7: 
Master (a) and slave (b) Sprott circuits 
The
control was implemented on an industrial computer with a sampling rate
of 2000 Hz and shown in Fig. 8. From Fig. 9, it
can be seen that the slave circuit response is synchronized to the master
circuit response after the control is action at t = 15 sec. The experimental
results of error dynamics in Fig. 10 show a convergence
to a very small synchronization error and a continuous control input is
obtained.

Fig. 8: 
A photograph of the proposed Sprott chaos synchronization
system 

Fig. 9: 
Experimental time responses of chaos synchronization
in Sprott circuits: master and slave system outputs are x_{1m},
x_{2m}, x_{3m} (——)
and x_{1s}, x_{2s}, x_{3s} (  ), respectively 

Fig. 10: 
Experimental results. Syncronization errors e_{1},
e_{2}, e_{3} and control signal (u) 
CONCLUSION
In this study, using the evolutionary programming algorithm, a simple
and effective digital PID controller has been proposed for synchronization
of two Sprott circuits. Three gains of digital PID controller can be directly
obtained by solving a specified optimization problem as defined above
by performing Steps 15. Compared with the existing reports for chaotic
synchronization, the proposed digital PID controller based on EP algorithm
is not only effective but also simple in architecture to implement in
a digital base controller.