Hindawi Publishing Corporation
Discrete Dynamics in Nature and Society
Volume 2011, Article ID 516031, 11 pages
doi:10.1155/2011/516031
Research Article
An FPGA-Based PID Controller Design for Chaos
Synchronization by Evolutionary Programming
Her-Terng Yau,1 Yu-Chi Pu,2 and Simon Cimin Li3
1
Department of Electrical Engineering, National Chin-Yi University of Technology,
Taichung 411, Taiwan
2
Department of Electrical Engineering, Far-East University, No. 49, Zhonghua Rd., Xinshi Cist.,
Tainan 74448, Taiwan
3
Department of Electrical Engineering, National University of Tainan, Tainan 700, Taiwan
Correspondence should be addressed to Her-Terng Yau, pan1012@ms52.hinet.net
Received 7 April 2011; Accepted 26 June 2011
Academic Editor: Marko Robnik
Copyright q 2011 Her-Terng Yau et al. This is an open access article distributed under the Creative
Commons Attribution License, which permits unrestricted use, distribution, and reproduction in
any medium, provided the original work is properly cited.
This paper is concerned with the design of a field programmable gate arrays- FPGAs- based
digital proportional-integral-derivative PID controller for synchronization of a continuous
chaotic model. By using the evolutionary programming EP algorithm, optimal control gains
in PID-controlled chaotic systems are derived such that a performance index of integrated
absolute error IAE is as minimal as possible. To verify the system performance, basic electronic
components, including OPA resistor and capacitor elements, were used to implement the chaotic
Sprott circuits, and FPGA technology was used to implement the proposed digital PID controller.
Numerical and experimental results confirmed the effectiveness of the proposed synchronization
procedure.
1. Introduction
In recent years, chaos synchronization has attracted the interest of researchers in various
fields 1. Chaos synchronization has many potential applications in physics and engineering
and particularly in secure communication 2. The idea of synchronizing two identical chaotic
systems was first introduced by Pecora and Carroll 3. In continuous-time chaotic systems,
synchronization is usually achieved by a master-slave or drive-response approach. Given
a master drive and slave response in a chaotic system, the goal is synchronizing the
behavior of the slave response system to that of the master drive system. To achieve the
synchronization, a nonlinear controller must be designed to obtain signals from the master
and slave systems and to manipulate the slave system. Recently developed control methods
2
Discrete Dynamics in Nature and Society
can achieve chaos synchronization between two identical chaotic systems with different
initial conditions 1. However, none of the proposed methods in our surveyed papers can
obtain an optimal or near-optimal digital controller for synchronizing continuous chaotic
systems according to a performance index specified by an FPGA chip.
Conversely, evolutionary programming EP algorithms have proven effective and
easy to implement for global optimization of complex functions and for solving complex
control problems in engineering 4, 5. Generally, the four steps in the global optimization
algorithm are initialization, mutation, competition, and reproduction. Furthermore, Cao 4
also developed a quasirandom sequence QRS for generating an initial EP population that
avoids formulating clusters around an arbitrary local optimal.
In fact, implementing this technique in digital electronic devices such as field
programmable gate arrays FPGAs can accelerate the development of prototype circuits
such as control and real-time simulation circuits 6. The FPGA comprises thousands of logic
gates, some of which are grouped into a configurable logic block CLB to simplify higherlevel circuit design. Because of its simplicity and programmability, the FPGA is the preferred
option for chip prototype design.
The main objective of this work was to develop an EP-based digital PID control
scheme for solving synchronization problems in FPGA-based chaotic systems. The EP
algorithm derived optimal control gains in PID-controlled chaotic systems such that the
performance index of integrated absolute error IAE was minimized and the master and
slave chaotic systems were synchronized. The optimal architecture for digital controller was
then implemented and simulated by very high-speed description language VHDL and
ModelSim. The developed architectures for each component were then implemented and
tested under the Xilinx Spartan-3 FPGA. Finally, simulation and experimental results were
compared to confirm the effectiveness of the proposed EP-based digital PID scheme for chaos
synchronization.
2. System Description and Problem Formulation
First, consider two single-input single-output SISO master and slave systems described by
the following differential equations:
Master system:
ẋm t ft, xm ,
ym t Cxm .
2.1
Slave system:
ẋs t ft, xs But,
ys t Cxs ,
2.2
where xm t xm1 , xm2 , . . . , xmn ∈ Rn and xs t xs1 , xs2 , . . . , xsn ∈ Rn are the state vectors
of master and slave systems, respectively. The f : R × Rn → Rn is a given nonlinear function.
The ym t ∈ R and ys t ∈ R are the outputs of the master and slave systems, respectively. The
B ∈ Rn×1 and C ∈ R1×n are the system matrices. The ut ∈ R is the control input included in
Discrete Dynamics in Nature and Society
3
the slave system 2.2 to synchronize the master and slave systems. Generally, many chaotic
systems can be expressed by 2.1. For example, the Sprott circuit, the modified Chua’s circuit,
the Duffing-Holmes system, the Lorenz system, and the Lu system all belong to the class
defined by 2.1.
Let the error states be e1 xm1 − xs1 , e2 xm2 − xs2 , . . . , en xmn − xsn . The objective
of this study was to use the EP algorithm to design a simple but effective PID controller ut
that can synchronize coupled systems 2.1 and 2.2 under different initial conditions such
that
lim xm t − xs t −→ 0.
t→∞
2.3
The procedure for determining the PID controller u is to first define the output error
signal ye ym − ys and then to define the continuous form of a PID controller with input ye ·
and output u·. The conventional equation is
ut Kp
1
ye t Ti
d
ye tdt Td ye t ,
dt
0
t
2.4
where Kp is the proportional gain constant, Ti is the integral time constant, and Td is
the derivative time constant. When using the FPGA chip to implement this controller, the
continuous-type PID controller 2.4 is reformulated as a digital-type PID controller as shown
below:
Td 1
uk Kp ye k Sk ye k − ye k − 1 .
Ti
T
2.5
Here, uk is the kth sampling output data of the PID controller, Sk ye k is the
error sum, and T is the sampling time constant. Therefore, 2.5 can be rewritten as
uk Kp ye k Ki Sk Kd ye k − ye k − 1 ,
2.6
where Ki Kp 1/Ti is the integral gain constant and Kd Kp Td /T is the derivative gain
constant.
The performance criterion or objective function of a controller design can generally be
defined according to the desired specifications. The two performance criteria typically considered in the EP algorithm are the integrated squared error ISE and the integrated absolute
error IAE. This study uses the IAE index as the objective function OF, which is given as
OF IAE kf
Ek,
2.7
k1
where Ek e1 , e2 , . . . , en , · is the Euclidean norm of a vector, k is the sampling time
point, and kf is the total number of samples. Below, the EP algorithm is used to minimize
4
Discrete Dynamics in Nature and Society
Master
chaotic
system
ym
+
−
ye (t)
PID
u(k)
controller
FPGA
ye (k)
T
ZOH
Slave
chaotic
system
ys
Kp Ki Kd
Evolutionary
programming
algorithm
Figure 1: Block diagram of chaos synchronization system.
the objective function score 2.7 by tuning the digital PID controller and optimizing the gain
parameters.
3. Evolutionary Programming (EP) Algorithm for Solving
the Optimization Problem
Since the EP algorithm is considered an easily implemented and promising technique for the
global optimization of complex functions, this study introduces an EP algorithm for solving
this problem. Figure 1 shows the proposed EP-based PID control system includes synchronized master and slave chaotic systems, a PID controller, and an EP algorithm. The ym is the
output of the master system, ys is the output of the slave system, and u is the control input
generated by the PID controller as defined in 2.5. The parameters of the proposed PID controller are derived by the EP algorithm such that the value of IAE given in 2.7 is minimized.
This section proposes an extended EP algorithm for obtaining the digital PID controller
with optimal gain parameters to minimize the following objective function OF score 2.6.
Let g be the continuously differentiable matrix-valued function defined for g ∈ S, where
S {g ∈ R3 | 0 ≤ gi ≤ Mi , i 1, 2, 3} and Mi is the bounded search space. The optimization
problem is to find g ∗ Kp∗ , Ki∗ , Kd∗ ∈ S such that the OF performance index of the system is
minimized. Mathematically, the optimization problem P1 can be formulated as follows.
P1: To find g∗ ∈ S such that
OF IAE kf
Ek,
for g ∗ ∈ S
3.1
k1
is minimized.
Based on the simulation results obtained in 4, an extended EP algorithm for solving
the above optimization problem is applied as follows.
Step 1. Generate an initial population P0 p1 , p2 , . . . , pN of size N by randomly initializing
each 3-dimensional solution vector pi ∈ S, i 1, 2, . . . , N according to the quasirandom
sequence QRS.
Discrete Dynamics in Nature and Society
5
0.94
0.92
0.9
0.88
IAE
0.86
0.84
0.82
0.8
0.78
0.76
0.74
0
50
100
150
200
250
300
Iteration
Figure 2: Convergence curve of IAE.
Step 2. Calculate the fitness score objective function fi fpi for each pi , i 1, 2, . . . , N,
where
f
fi pi OF |EkT |.
k
3.2
k1
Step 3. Mutate each pi , i 1, 2, . . . , N, based on the statistical data to double the population
size from N to 2N, and generate piN by using
fi
piN,j pi,j N 0, β
,
fΣ
∀j 1, 2, 3,
3.3
where pi,j denotes the jth element of the ith individual, N0, βfi /fΣ represents a Gaussian
random variable with a mean zero and variance βfi /fΣ , fΣ is the sum of all fitness scores, and
β is a parameter to scale fi /fΣ .
Step 4. Use 3.2 to calculate the fitness score fiN for every piN , i 1, 2, . . . , N. In the
stochastic competition process, pi , i 1, 2, . . . , N randomly competes with pj , j N 1, . . . , 2N. If fi < fj , pi wins; otherwise, pj wins, and pi is replaced by pj . After completing
the competition process, select N winners for the next generation, and let the individual with
the minimum objective function in the winners be p1 .
Step 5. If the value fΣ converges to a minimum value, then let g ∗ p1 be the global optimum
value and g ∗ Kp∗ , Ki∗ , Kd∗ such that the OF performance index of the system is minimized.
Otherwise, return to Step 3.
6
Discrete Dynamics in Nature and Society
25
20
Control gains
15
10
5
0
−5
0
50
100
150
200
250
300
Iteration
Kp
Ki
Kd
Figure 3: Iteration response of Kp , Ki , Kd .
4. Simulation and Experimental Results
This section describes the proposed EP-based digital PID controller design for synchronizing
Sprott circuits, which are the chaotic systems typically studied in the literature 7, 8. Now
consider the following Sprott circuits:
Master:
ẋm1 xm2 ,
ẋm2 xm3 ,
4.1
ẋm3 −1.2xm1 − xm2 − 0.6xm3 2 · signxm1 .
Slave:
ẋs1 xs2 ,
ẋs2 xs3 ut,
4.2
ẋs3 −1.2xs1 − xs2 − 0.6xs3 2 · signxs1 ,
where ẋm and ẋs denote the derivatives of xm and xs , respectively, with respect to time t.
In this example, the initial conditions for master and slave are xm1 0, xm2 0, xm3 0 0.1 0.1 0.1 and xs1 0, xs2 0, xs3 0 −1 − 1 − 1, respectively. Matlab and Simulink are
used to solve the optimization problem P1, where N 30 and β 0.001. The proposed EP
algorithm generates P0 p1 , p2 , . . . , p30 according to the QRS. Several manipulations of the
EP algorithm gets the convergence curve of IAE value versus iteration depicted in Figure 2.
5
4
3
2
1
0
−1
−2
−3
−4
−5
7
Control in action
xm 2 , xs 2
xm 1 , xs 1
Discrete Dynamics in Nature and Society
0
5
10
15
20
25
30
35
40
5
4
3
2
1
0
−1
−2
−3
−4
−5
Control in action
0
5
10
15
Time (s)
xm 1
xs 1
20 25
Time (s)
30
35
40
xm 2
xs 2
a
b
−1
−2
−3
−4
−5
6
Control in action
4
Control in action
e1 , e2 , e3
xm 3 , xs 3
8
5
4
3
2
1
0
2
0
−2
−4
−6
−8
0
5
10
15
20
25
30
35
0
5
10
15
40
20
25
30
35
40
Time (s)
Time (s)
e1
e2
e3
xm 3
xs 3
c
d
20
10
u(t)
0
−10
−20
−30
−40
Control in action
0
5
10
15
20 25
Time (s)
30
35
40
u(t)
e
Figure 4: Time responses using the EP-based PID controller. a xm1 versus xs1 . b xm2 versus xs2 . c
xm3 versus xs3 . d The error states e1 , e2 , e3 . e The control input. The control ut is activated at t 20 sec..
8
Discrete Dynamics in Nature and Society
R23
10 k
R24
10 meg
R27
1 meg
3
R25
10 k
C4
200 pF
U12
UA741
VCC
1
−
R35
1 meg
4
+
R42 VEE
10 meg
C6
200 pF
1 meg
6
8
−
+
10 k
+
R31
16
40 k
R32
1k
R33
10 k
17
18
D4
D3
1N4148 1N4148
15
−
+
U16
UA741
R34
VCC
10 k
VCC
VEE
19
VEE
R38
10 k
R39
R40
10 k
− V
CC
20
VEE
VEE
U18
UA741
VCC
U19
UA741
VCC
R45
+
R37
6k
R29
14
VEE
U15
UA741
VCC
VCC
VEE
−
R41
V2
12
10 k
VEE
xm 2
VEE
+
−V
CC
+
−V
CC
V1
12
U13
UA741
VCC
13
U14
UA741
VCC
+
xm 1
R26
12 12
k
2
R30 V
EE
10 meg
C5
200 pF
5
+
VCC
VEE
VEE
R28
10 k
VCC
VEE
+
−
11
VCC
U11
UA741
VCC
VEE
10 k
U21
UA741
V CC
21
VEE
−V
R46
10 k
VCC
V EE
R47
10 k
VEE
CC
U20
UA741
VCC
−V
CC
22
+
VEE
+
VEE
VEE
xm 3
23
VEE
a
R24
10 meg
R25
10 k
11
C4
200 pF
R27
1 meg
3
1
R23
10 k
U11
UA741
VCC
−V
CC
+
U12
UA741
V
−V CC
CC
VEE
2
VEE
VEE
R28
10 k
+
xs 3
u(t)
R12
10 k
R13
10 k
R30 V
EE
10 meg
C5
200 pF
R1
10 k
U21
UA741
VCC
R35
−V
1 meg
24
CC
VEE
5
4
+
xs 2
R37
6k
−V
CC
VEE
1 meg
8
6
−V
CC
+
VEE
VEE
VEE
U15
UA741
R32
V
− VCC CC 1 K
VEE
15
+
VEE
+
U19
UA741
VCC
R45
U13
UA741
R29
VCC
−
10 k
13 VCC
VEE
14
U18
R41
UA741
10 k
R40
VCC
−
10 k
20 VCC
VEE
21
R42 VEE
10 meg
C6
200 pF
VEE
R46
10 k
7
R47
10 k
V1
12
12 R26
12 k
+
U14
UA741
VCC
+
VEE
VCC
+
xs 1
U20
UA741
VCC
−
xs 3
22 VCC
V EE
23
+V2
12
VEE
16
R31
40 k
U16
UA741
R34
V CC
−
10 k
17
18 VCC
VEE
D4
D3
19
1N4148 1N4148 +
R33
10 k
VEE
R38
10 k
R39
10 k
U21x
UA741
VCC
−V
CC
VEE
+
VEE
+
VEE
b
Figure 5: Electronic implementations of Sprott circuits. a master system; b slave system.
Discrete Dynamics in Nature and Society
u(k)
9
ys
Slave
chaotic
system
DAC
−
FPGA
+
ADC
ye (k)
ym
Master
chaotic
system
PID control system
a
b
Figure 6: The Black diagram a and the photograph b of the proposed chaos synchronization system.
Figure 2 clearly shows that convergence occurs after about 170 iterations, and the final value
of IAE is fg ∗ 0.7592. The corresponding PID control gains are g ∗ Kp∗ , Ki∗ , Kd∗ 20 0.0018 20. Figure 3 also shows the Kp , Ki , and Kd trajectories during the evolutionary
procedure. Figure 4 shows the output response when using the resulting PID control gain z∗ .
The simulation results confirm that the EP algorithm and proposed PID controller effectively
synchronize Sprott chaotic systems.
The proposed PID controller was then tested in an actual system. Figure 5 8 shows
the experimental results when 4.1 and 4.2 were applied in a circuit connected in a
master/slave configuration. Figure 6 shows how the controller was implemented with Xilinx
Spartan-3 FPGA and AD/DA converters and a 1289 kHz sampling rate. Figure 7 shows that
the slave circuit response was synchronized to the master circuit response after the control
was activated at t 20 second. The experimental results of error dynamics in Figure 7d
show the convergence to a very small synchronization error.
5. Conclusions
A simple and successful digital PID controller for an FPGA chip was proposed for using
an evolutionary programming algorithm to synchronize two chaotic systems. The derived
10
Discrete Dynamics in Nature and Society
Tek
Stop
M Pos: 0.000 s
xm 1
CH1
BW limit
Off
200 MHz
1
xs 1
2
Tek
Stop
M Pos: 0.000 s
xm 2
Coupling
DC
BW limit
Off
200 MHz
1
Volts/Div
Coarse
Probe
1x
Voltage
xs 2
Invert
Off
CH1 0.00 V
CH1 2.00 V CH2 2.00 V M 500 ms
28-May-10 21:24 <10 Hz
b
M Pos: 0.000 s
Stop
xm 3
CH1
xs 3
Tek
M Pos: 0.000 s
Stop
Coupling
DC
BW limit
Off
200 MHz
1
Volts/Div
Coarse
1
Voltage
e2
2
e3
c
BW limit
Off
200 MHz
Volts/Div
Coarse
Probe
1x
3
Voltage
Invert
Off
Invert
Off
CH1 0.00 V
CH1 2.00 V CH2 2.00 V M 500 ms
28-May-10 20:52 < 10 Hz
CH1
Coupling
DC
e1
Probe
1x
2
Probe
Voltage
a
Tek
Volts/Div
Coarse
1x
2
Invert
Off
CH1 2.00 V CH2 2.00 V M 500 ms
CH1 0.00 V
28-May-10 20:02 <10 Hz
CH1
Coupling
DC
CH1 0.00 V
CH1 5.00 V CH2 5.00 V M 500 ms
CH3 5.00 V
28-May-10 21:24 <10 Hz
d
Figure 7: Experimental results of synchronization. a xm1 versus xs1 . b xm2 versus xs2 . c xm3 versus
xs3 . d The error states e1 , e2 , e3 .
EP algorithm efficiently obtains the gains for three PID controllers by solving specified
optimization problems. Moreover, the effectiveness of the proposed EP-based PID scheme
was then demonstrated in a chaotic circuit system. The simulation and experimental results
confirm that the proposed methods are effective and practical. Compared to existing methods
of chaotic synchronization, the proposed FPGA-based optimal PID controller is not only
effective, but also simple in terms of design and implementation.
Acknowledgment
The authors would like to thank Y.-C. Pu for his contribution to this paper.
References
1 G. Chen and X. Dong, From Chaos to Order: Methodologies, Perspectives and Applications, World Scientific
Publishing Company, Singapore, 1998.
2 L. Kocarev and U. Parlitz, “General approach for chaotic synchronization with applications to
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3 L. M. Pecora and T. L. Carroll, “Synchronization in chaotic systems,” Physical Review Letters, vol. 64,
no. 8, pp. 821–824, 1990.
4 Y. J. Cao, “Eigenvalue optimization problems via evolutionary programming,” Electronics Letters, vol.
33, no. 7, pp. 642–643, 1997.
Discrete Dynamics in Nature and Society
11
5 W. D. Chang and J. J. Yan, “Optimum setting of PID controllersbased on using evolutionary
programming algorithm,” Journal of the Chinese lnstitute of Engineers, vol. 27, no. 3, pp. 439–442, 2004.
6 Z. Zhou, T. Li, T. Takahashi, and E. Ho, “FPGA realization of a high-performance servo controller for
PMSM,” in Proceedings of the 9th IEEE Applied Power Electronics Conference and Exposition, vol. 3, pp.
1604–1609, 2004.
7 J. C. Sprott, “A new class of chaotic circuit,” Physics Letters, Section A, vol. 266, no. 1, pp. 19–23, 2000.
8 D. I. R. Almeida, J. Alvarez, and J. G. Barajas, “Robust synchronization of Sprott circuits using sliding
mode control,” Chaos, Solitons & Fractals, vol. 30, no. 1, pp. 11–18, 2006.
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