Design of fractional - order PID controller based on genetic algorithm

Lingxin Wang1, Chunyang Wang2 , Li Yu3,Yonghua Liu4,Jingxue Sun5

1. School of Electronic and Information Engineering, Changchun University of Science and Technology, Changchun,130022,China
E-mail: 1203620712@qq.com
2. School of Electronic and Information Engineering, Changchun University of Science and Technology, Changchun,130022,China
E-mail: wangchunyang19@163.com(Corresponding Author)
3. School of Electronic and Information Engineering, Changchun University of Science and Technology, Changchun,130022,China
E-mail:yuli_works@163.com
4. School of Electronic and Information Engineering, Changchun University of Science and Technology, Changchun,130022,China
E-mail: 626941720@qq.com
5. School of Electronic and Information Engineering, Changchun University of Science and Technology, Changchun,130022,China
E-mail: 1612851955@qq.com
Abstract: Aiming at the characteristics that dynamic systems can be described by integral and differential equations
involving non-integer orders, this tool is also introduced into fractional-order controllers, that is, controllers with
fractional integral and differential. A modified genetic algorithm The integer order PID controller and the fractional
order controller are used to simulate and compare the controller parameters of the integer and fractional order systems
respectively by using the genetic algorithm, The results show that the fractional-order controller is better than the
integer-order PID controller in the same parameter tuning range.
Key words: genetic algorithm; fractional order PID; parameter tuning

1 INTRODUCTION code string of length n , the total number of

codes that can be generated is 2n , and the
The genetic algorithm is a kind of method which encoding and coding method of the parameter is:
can solve the problem by searching for the
"
000 
= ymin
000
solution space by simulating the most suitable n
(1)

= ymin + δ
process of the biological organism in the natural 000 " 001
 
evolution process of the organism. The variable n

= ymax
value in the domain of the problem to be solved
"
111 111

is the object of study. Optimization Method for n
Optimal Solution. The advantage is not too much Where δ indicates the accuracy of the
dependent on other auxiliary conditions, but only encoding, and the length of the code or the
need to ensure that there is a suitable fitness number of bits n , the value is:
function, in a sufficient number of populations
ymax − ymin
and the number of iterations case, for different δ= (2)
problems can be the optimal solution or 2n − 1
suboptimal Solution, independent of the specific For a parameter value y , if y contains a
problem areas. fractional part and has a precision of m bits,
2 PRINCIPLES OF GENETIC the binary code string shall satisfy:
OPTIMIZATION ALGORITHM 2 n −1 < y − ymin < 2n − 1 (3)
The process of converting the true value
As the application of genetic algorithm is quite
extensive, coding methods are also varied. At y − ymin into a binary number for a point y
present, there is not a set of strict and complete in the interval is the binary coding process of the
convincing guidance theory and standards to parameter.
design appropriate coding methods, commonly For binary digital code string x decoding
used coding methods are: mode of operation is to set its decimal number is
Binary coding: is the most basic encoding, the demical ( x) :
mathematical symbol set is binary
y max − y min
symbols0and1binary symbol set {0,1}, the y = y min + demical ( x ) (4)
length of the code string need to solve the 2n − 1
problem according to the accuracy and Therefore, for a parameters, the encoding
complexity of the problem determine. Assuming length of each parameter is n j , j = 1, 2," , a
that the parameter has a spatial range of
to be optimized, then all parameters are binary
[ ymin , ymax ] , coding the parameter with a binary coded into a binary code whose total length is:

978-1-5090-4657-7/17/$31.00 2017
c IEEE 808
 N = n1 + n2 + " + na (5) probability of the way the father of individual
selection to the offspring. Commonly used
Floating-Point Coding: Each individual
methods are roulette methods, random
corresponds to a different floating-point number
competition, the best choice to retain, no
within the solution space. The number of
playback random selection , this selection of
floating-point numbers is equal to the number of
roulette methods.
variables to be solved. The coding method is
The Roulette Wheel Selection method is the
mainly used to solve some multidimensional and
operation of selecting the current solution in a
high precision continuous function optimization
put-back fashion according to the randomly
problems. The advantage of this method is that it
generated probability after converting the
can maintain better population diversity in
existing solution to probability. Assuming that
crossover operation and mutation operation,
especially in the larger solution space. , And the the population of the current solution is N and
higher the accuracy of the lower complexity. the fitness value of the i th individual is F (i ) ,
Specific steps are as follows: the probability of the individual being selected
(1) The parameters are encoded is:
From the solution space of the problem to be F (i)
solved, N initial values are randomly selected as Pi = N
(7)
the parent and constitute the initial population. In ¦ F (i )
i =1
order to avoid the time occupation and precision
The N probability values are arranged in order
loss caused by the conversion between binary
and floating point (decimal), this paper adopts from small to large. In the selection process,
the floating-point encoding method to encode the randomly generated numbers between [0,1] are
M +1 M
N initial values with easy-to-implement binary
encoding and floating-point encoding method. between ¦i =1
pi and ¦ p , (M < n) ,
i =1
i then the
(2) Select the appropriate fitness function
Since the probability of choosing the parent to Mth individual is selected to the next generation
the next generation is related to the individual until N individuals are selected until. The
fitness value, it is also a probabilistic problem. number of groups is limited and the random
Therefore, this method can not guarantee that selection of the solution operation and other
individuals with high fitness values can select reasons, each solution in the actual operation
the next generation, and individuals with low was selected to the next generation of the
fitness values Can not be selected for the next number of its selected number of times between
generation. The fitness function has a great n

influence on the optimization ability and the expected value of n * F (i ) / ¦ F (i ) there is

i =1
computation time of the genetic algorithm, and it
a certain error, This error is relatively large, the
needs to transform the fitness function in
result is even more adaptable individuals are not
different stages according to the needs. In the
necessarily selected, should find a way to solve
control theory, a commonly used fitness function
this loss because the optimal solution can not be
is often a combination of system error, control of
the optimal solution of the plight.
the form, one of the forms shown below, is also
Finally, you need to determine the location of the
used in this paper.
∞ intersection. Crossover operations include
2 pairing of individuals, determining the location
J = ³ ( w1 e(t ) + w2u 2 (t ))dt + w3tr (6) of the intersection and the number of crossovers.
0
For a population of N individuals, a total of
Where e(t ) is the error of the system and [ N / 2] pairs can be dubbed, and [ X ] is the
u (t ) is the controller quantity. The weighted largest integer that does not exceed X . Many
integrals of these two items reflect the energy of scholars have proposed various methods for
the system, to a certain extent, the stability of the crossover operation, such as a single point of
system, w1 , w2 is the weight, representing the intersection, multi-point crossover, etc., which
are well suited for binary coding cross operations,
error and the controller The importance of output and arithmetic crossover for floating point
to affect the outcome. tr represents the rise encoding.
A= (aa
1 2a3"aa i i+1 | ai+2ai+3 "aN ) A= (aa
1 2a3"aa i i+1 | bi+2bi+3 "bN )
time of the system, and w3 is the →
B = (bbb
1 2 3 "bb
i i+1 | b b
i+2 i+3 "bN ) B = (bbb
1 2 3 " i i+1 | ai+2ai+3"aN )
bb
corresponding weight. Figure 1 Single point crossover
(3) genetic manipulation process
Selection operation, also known as reproduction
operation, according to each generation of the
fitness function value converted to the

2017 29th Chinese Control And Decision Conference (CCDC) 809

 A = (a1a2 a3 " ai ai +1 | ai + 2 ai +3 " ak −1 | ak ak +1 " aN ) Due to the certainty of mutation manipulation, it
B = (b1b2b3 " bi bi +1 | bi + 2bi +3 " bk −1 | bk bk +1 " bN ) is possible to destroy the best gene, so a
relatively low probability of mutation is used for
→ the best gene and a higher probability of
A = (a1a2 a3 " ai ai +1 | bi + 2bi +3 " bk −1 | ak ak +1 " aN ) mutation for a gene of lower fitness in order to
B = (b1b2b3 " bi bi +1 | ai + 2 ai +3 " ak −1 | bk bk +1 " bN ) expect a better Of the gene. In this paper, we do
Figure 2 Two-point crossover not use the fixed mutation probability, but adopt
In the above two figures, ai , bi (i < n) represents the method of probability linear decreasing.
a binary number of 0 and 1, and a vertical line 3 DESIGN OF INTEGER ORDER
(" | ") represents a cross position. Multipoint CONTROLLER
crossover is similar to this. For a single-point
The general structure of an integer-order PID
crossover operation, each code has N − 1
controller is as follows:
cross-point, if you need two-point crossover,
each lattice encoding ( N − 1)( N − 2) / 2 cross
kp

r e u y
points. For a single point of intersection, the ki p( s)
method of operation is, for a string of code, -
kd
resulting in a probability of random Pc ∈ [0,1] ,
then N * Pc represents the location of the cross,
Figure 3 integer order PID controller structure
until the N individual operation is completed. The control law is:
Two-point crossover and multipoint crossover de(t )
are similar to this method except that the u (t ) = k p e(t ) + ki ³ e(t ) dt + kd ˄10
corresponding number of probabilities need to be dt
generated to determine the position at which to ˅
cross. 4 DESIGN OF FRACTIONAL
For paired chromosomes A, B that encode ORDER CONTROLLER
parameters as floating-point numbers, pairs of
The structure of the fractional-order controller is
pairs of chromosomes are often manipulated
shown below:
using arithmetic crossovers
Ai +1 = α B i + (1 − α ) A i
kp
(8)
B i +1 = α Ai + (1 − α ) B i r e
ki s − λ
u
p(s)
y

Where Ai , Bi is the crossed chromosome i ,

kd s μ
Ai +1 , B i +1 is the chromosome i+1 , α ∈ (0,1)
can be set as a constant or set to a random value
as needed. Figure 4 FOPID controller structure
Mutation operation itself is a random operation The mathematical expression of fractional order
algorithm. One method of floating-point controller is:
encoding variant operations is: u(t ) = k p e(t ) + ki I −λ e(t ) + kd D μ e(t ) (11)
A = Amin + ( Amax − Amin ) × α ˄9˅
5 PARAMETER TUNING
Where Amin , Amax denotes the minimum and
ALGORITHM OF INTEGER
maximum values of the code and α ∈ (0,1) is a
ORDER COTROLLER BASED ON
random number.
Genetic algorithm in the probability of GENETIC ALGORITHM
occurrence of mutation operation is very low, First, we should determine the appropriate
which is similar to the biological variation in fitness function, the fitness function selected in
nature, so the mutation operator usually values this paper for the type (6), the parameters taken
between 0.001 to 0.1, far below the probability
in this paper ω1 = 0.9 ˈ ω2 = 0.1 ˈ ω3 = 2 .The
of crossover operation.
It should be noted that, although the probability parameter range is limited to
of mutation is too large, the mutation probability k p ∈ [0,100], k d ∈ [0, 50], ki ∈ [0,50] . Population
may cause the system to converge to the local size = 40. The crossover probability Pc = 0.7 ,
optimum instead of the global optimum, and the and the mutation probability is linearly
smaller the crossover, the mutation probability is decreasing. Pm = 0.1 − [Size : −1 : 1]* 0.01/ Size ,
likely to cause the slow convergence of the
algorithm, therefore, must be reasonable Select where the constant 0.01 can have high mutation
probability, but not 0, the parameter can be set as
crossover, mutation probability.

810 2017 29th Chinese Control And Decision Conference (CCDC)

needed, you can also make Pm = 0.1 , this time
for a fixed probability of mutation, the following
procedures are used in the method.
Let MinX (1) = 0 be the lower limit of K p ,
MinX (1) = 0 be the lower limit of K i , and
MinX (1) = 0 be the lower limit of Kd .
MinX (1) = 100 is the upper limit of
Kp , MinX (1) = 50 is the upper limit
of Ki , MinX (1) = 50 is the upper limit of K d .
Then the initial value is set to random value Figure 7 Controller parameter curve
matrix: Figure 5 fitness curve, abscissa 400 is the
K pid (:,1) = MinX (1) + (MaxX (1) − MinX (1)) * rand (Size,1) number of genetic algorithm cycle, we can see
that with the increase in the number of cycles,
,
K pid (:,2) = MinX ( 2) + ( MaxX ( 2) − MinX ( 2)) * rand ( Size,1)
the fitness curve decreased rapidly, while the
control parameters of the controller is gradually
, increased, the fitness value of the final trend At
K pid (:,3) = MinX (3) + ( MaxX (3) − MinX (3)) * rand ( Size,1)
0.468241307735691, the internal subgraph is the
. local magnification. Step response from the
The fitness function controls the response of the system diagram, you can see in a very short
object with these three variables as the controller period of time, the output signal reaches a steady
parameters. When the input is the step-over state, the final value of 0.996887643653813, the
signal, it is the step response of the system. The error is small. The final parameter values are
same is true for future fitness functions. The K p = 39.625679938736, K i =12.2521791402685,
simulation results are as follows:
K d =6.99889953738961.
6 PARAMETER TUNING
ALGORITHM OF FRACTIONAL-
ORDER COTROLLER BASED ON
GENETIC ALGORITHM
For the fractional-order controller, the parameter
range is k p ∈ [0,100] , k d ∈ [0,50] ,
k i ∈ [0,50] , λ ∈ [0,1.2], μ ∈ [0,1.2] ,the
population Size=40, the crossover probability is
Figure 5 GA algorithm integer fitness curve Pc = 0.7 , and the mutation probability is
Pm = 0.1 . Using the above
MinX (1) = zeros (1), MaxX (1) = 100 * ones (1) is the
range of K p ,
MinX ( 2) = zeros (1), MaxX ( 2) = 50 * ones (1) is the
range of K i ,
MinX (3) = zeros (1), MaxX (3) = 50 * ones (1) is the
range of K d ,
MinX ( 4) = zeros (1), MaxX (4) = 1.2 * ones (1) is the
range of λ ,
MinX (5) = zeros (1), MaxX (5) = 1.2 * ones (1) is the
range of μ .
Figure 6 System step response

Using the random value of the way, the initial

value is:
K pid (:,1) = MinX(1) + (MaxX(1) − MinX(1))* rand(Size,1)
,
K pid (:,2) = MinX (2) + (MaxX(2) − MinX (2)) * rand(Size,1)
,
K pid (:,3) = MinX (3) + (MaxX (3) − MinX (3)) * rand( Size,1)
,

2017 29th Chinese Control And Decision Conference (CCDC) 811

 K pid (:,4) = MinX(4) + (MaxX(4) − MinX(4)) * rand(Size,1) μ = 0.997339567500875 . The results of the two
, runs have been very close.
K pid (:,5) = MinX (5) + (MaxX (5) − MinX (5)) * rand(Size,1)
7 SIMULATION RESULTS AND
The simulation results are as follows:
ANALYSIS
The following is a simulation of fractional-order
PI λ D μ controller and integer-order PID
controller simulation, the object is a
second-order transfer function:
400
G(s) = 2
s + 50
The following simulation results are obtained:

Figure 8 GA algorithm to set the fractional fitness curve

Figure 11 Integer order controller step response

rin,yout

Figure 9 System step response

Figure 12. Step response of a fractional-order controller

The simulation results show that the fractional
order control system can effectively improve the
stability of the system when the rise time is too
long, and it has greater stability margin than the
ordinary integer PID control system.
8 CONCLUSION
Fig.10 Curve of parameter change of fractional-order
controller The simulation results show that the fractional
The final output of the system order PI λ D μ controller can obtain better
0.997036230395526, than the integer order
control effect than the IOPID controller in the
controller error is smaller, the curve changes
fractional order system with the same parameter
smoothly, because the reading is too small, it is
setting range.
difficult to show in the figure, but in the work
space is easy to see, the controller five
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812 2017 29th Chinese Control And Decision Conference (CCDC)

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