"2017 Nineteenth International Middle East Power Systems Conference (MEPCON), Menoufia University, Egypt, 19-21 December 2017
A Novel Technique for Online Self-tuning of
Fractional Order PID, Based on Takaji-Sugeno fuzzy
Mahmoud S Gaballa
Abdel-Ghany M Abdel-Ghany
Department of Electric Power & Machines
Helwan University, Faculty of Engineering
Cairo, Egypt
mahmoudsalahgaballa@gmail.com
msg_salah@yahoo.com
Department of Electric Power & Machines
Helwan University, Faculty of Engineering
Cairo, Egypt
ghanymohamed@ieee.org
ghanyghany@hotmail.com
Mohiy Bahgat
Department of Electric Power & Machines
Helwan University, Faculty of Engineering
Cairo, Egypt
dr_mohiy_bahgat@yahoo.com
drmohiybahgat@yahoo.com
Abstract— Many techniques of online self-tuning for controller
parameters have been presented in several studies [12-15]. This
paper presents a new technique for online self-tuning of a
Fractional Order PID (FOPID) controller based on both a type1Fuzzy and a Takaji-Sugeno Fuzzy and deployed to control the
horizontal motion of a dual-axis photo voltaic sun-tracker. A
modification based on Takaji-Sureno Fuzzy (TS-Fuzzy) has been
applied on the Simulink Fractional Order PID block "nipid"
from the toolbox of "ninteger" to become suitable for the
purpose of the online self-tuning. Several experiments have been
carried out on a system model under the control of the modified
FOPID block and the results came very satisfactory.
I. INTRODUCTION
In previous works [1, 2], a design of a dual axis
photovoltaic sun tracker system was presented, see Fig. 1. This
study work presents an implementation of a Fractional Order
PID as the position controller of the horizontal carriage of the
sun tracker which is driven by a Brushless D.C (BLDC) motor.
One contribution in this study is to present a new technique for
FOPID online self-tuning for the five controller parameters (kp,
ki, kd, λ and μ) during simulation time. Another contribution in
this study is to demonstrate a modification on the nipid FOPID
block to be suitable for the online self-tuning.
Index Terms— Sun Tracking System, Takaji-Sugeno, TSFuzzy, Fractional Order PID, Online self-tuning.
NUMENCLATURE
θ
Photovoltaic Panel Angular Position
Photovoltaic Panel Angular Velocity
b
Motor viscous friction constant
Kp Proportional gain of a PID controller
Ki Integral gain of a PID controller.
Kd Derivative gain of a PID controller.
λ
Integral fractional order
μ
Derivative fractional order
J
Moment of inertia
L
Electric induction
Kt Torque constant
Kb Electromotive force
T
Torque
FOPID
Fractional order PID
V
Input voltage signal
R
Motor resistance
BLDC Brushless DC motor
Wx Fuzzy weight of membership x
Fig. 1. Dual-Axis PV Sun Tracker System
The D.C motor control model [3], is shown in Figure 2:
Fig. 2. The equivalent circuit of a D.C motor
978-1-5386-0990-3/17/$31.00 ©2017 IEEE
where: b is the friction coefficient (N.m.s), J : is the
moment of inertia (Kg.m2), R is the electric resistance (Ω), L is
the electric inductance (H), Kt is the torque constant (N.m/A),
and Kb is the electromotive force constant (V.sec/rad). The
torque generated by a D.C motor is proportional to the
magnetic-field strength and the armature current. The D.C
motor used in this study is a permanent magnet D.C motor
(constant magnetic field), therefore, the generated torque T is
only proportional to the armature current (i) as T = K i. The
induced back e.m.f (e) is proportional to the rotor angular
velocity θ as e = K θ , Kb and Kt are equal, therefore K is
used to represent both of them, The current can be derived as :
Jθ+bθ=Ki
(1)
L
+Ri =V
Kθ
(2)
Applying Laplace transform to (1) and (2):
s Js+b θ s =KI s
Ls+R I s =V s
Ksθ s
From (3) and (4) we get :
/
=
(3)
(4)
(5)
An integration has to be applied to the speed to obtain the
transfer function with the position as an output, by simply
multiplying equation (5) by (1/s) as follows :
=
(6)
The state-space representation can be deduced by choosing θ, θ
and i as the state variables, while, V is the input and θ is the
output as follows :
θ
θ
0 1
0
0
θ = 0
i
θ + 0 V
(11)
u s = k + + k .s e s
The fractional-order PID controller is represented as PIλ Dμ,
and is described by the equation shown by Equation 12.
u s = k +
+ k .s
e s
(12)
where, λ: a real number represents the integration fractional
order, µ: a real number represents the derivative fractional
order, kp: the proportional gain of the PID, ki: the integral gain
of the PID and kd: the derivative gain of the PID. Setting λ = 1
and µ = 1, tends to obtain a classical integer PID controller.
The FO-PID controller provides more flexibility and better
adjustment for the dynamics of control systems [4].
III. FRACTIONAL ORDER PID CONTROLLER ONLINE SELFTUNING
Before going into the deep details of the tuning process, A
real need arouse to have a Simulink block for the Fractional
Order PID controller that enables the designer of changing the
values of all of the five parameters (kp, ki, kd, λ and μ) during
the simulation process.
A. Modified FO-PID nipid block and detailed design steps
The nipid is a visual Simulink block that enables the
utilization of the Fractional Order PID in control. The block
provides a form to set the values of the fractional order
parameters μ and λ as well as the PID coefficients kp, ki and kd
as shown in Fig 3.
(7)
0
i
θ
y= 1 0 0 θ
(8)
i
The output of the motor in (rad) is then passed to a gearbox
with a static gain Kg, and therefore, the output equation can be
accordingly changed to:
θ
(9)
y= K 0 0 θ
i
II. INTEGER AND FRACTIONAL ORDER PID CONTROLLERS
The PID refers to the first letters of the words of
Proportional, Integral and Derivative Controller. PID
controllers are actually the most widely used controllers in
industry. It is considered the basic building block even in
control network that are deployed in sophisticated control
systems. The PID controller equation is shown by Equation 10.
u t = k .e t + k
e t dt + k
(10)
where, u(t):the output signal of the controller, e(t): the error
signal, kp: the proportional gain of the PID, ki: the integral gain
of the PID and kd: the derivative gain of the PID.
The S-Domain PID equation is shown by Equation 11.
Fig. 3. "nipid" Fractional-Order PID block parameters form
The technique, mentioned in this section is aiming to
performing an online self-tuning for the fractional order
parameters μ and λ as well as the PID coefficients kp, ki and kd
during the simulation. A satisfactory solution for this situation
is to use the work space values for the above mentioned five
variables instead of entering them as constant values, but
unfortunately, it's found that the Simulink loads the values of
the variables from the work space into the block once on
running starting and doesn't update them during the simulation.
The above mentioned restriction nessitates the need of having
another fractional-order-PID block that provides external input
channels for the five mentioned parameters. The ideas on
which the design for a modified nipid block that provides
external input channels for the five mentioned parameters are
as follows:
1- Separate the action of the FOPID into three main actions
which are P, I λ and Dμ , with each separate action has a unity
gain and is provided with a subsequent multiplication block to
enable applying external gains for each action (kp, ki and kd),
and finally the outputs of the three separate actions are
summed, see Figure 4.
The 6 triangular memberships are represented by
membership blocks each one gets the λ or μ value as an input
and generates the weight of the membership, see Fig 6.
Fig. 6. TS-Fuzzy weight calculation process of inputs
3- The realization of the TS-Fuzzy formula Equation 13 is
obtained by implementing the model shown in Fig 8 with nipid
blocks are configured as shown in Fig 7.
Fig. 4. Modified nipid block abstract layout by separating the three control
actions
2- Passing the values of μ and λ to the D and I blocks by
means of utilizing TS-Fuzzy technique [6-11] to deploy 6 nipid
sub-blocks for each of the new D and I mentioned block, and
each one of the six nipid blocks has a predefined value for μ or
λ. The input membership functions of both λ and μ value are
chosen to be six triangular functions that are equally distributed
over the range [0.0,1.0], and have their middle vertices placed
at the points {0, 0.2, 0.4, 0.6, 0.8, 1},see Fig 5.
Fig. 7. "nipid" Fractional-Order PID as an integrator with unity gain and
order = 0.4
out =
∑
.
∑
; out =
∑
.
∑
;
where:λ , μ ∈ 0.0, 0.2, 0.4, 0.6, 0.8, 1.0 ;
W is the weight of λ ;
W is the weight of μ ;
Fig. 5. Input Membership of the variables λ and μ
F is the output of nipid whose λ value is λ
(13)
F is the output of nipid whose μ value is μ
To validate the matching between the responses of both
modified and normal nipid as integrators, another experiment
has been carried out using the two blocks as fractional
integrators and the two responses were then compared, see
Fig 10.
Fig. 10. Block diagram and responses of the modified nipid and normal nipid
as integrators with lamda = 0.45
Utilizing the modified nipid as a fractional-order derivative
The modified nipid block will be used as a derivative with a
unit ramp input signal and μ=0.45, see Fig 11. Generation of
the output signal goes through the following steps:
1- Calculate the weights of the memberships.
From Fig 6, WSM = 0.75 and WLM = 0.25, the rest of the
weights all equal to zero.
Fig. 8. Modified-nipid integrator part basing on old nipid and TS-Fuzzy
Utilizing the modified nipid as fractional-order integrator:
The modified nipid will be used as an integrator with input
signal equals to 1 and λ = 0.45, see Fig 9. The generation of the
output signal goes through the following steps:
1- Calculate the weights of the memberships. From Fig 6, WSM
= 0.75 and WLM = 0.25, the rest of the weights all equal to zero.
2- Consider the outputs of the two nipid blocks that have λ's
corresponding to non-zero weights i.e. μ ϵ {SM , LM} ≡ μ ϵ
{0.4, 0.6}, see Fig 6.
3- The output signal F, see Fig 11, is then calculated as :
F=
.
.
= 0.75 F
+ 0.25 F
;
where FSM is the output signal of the nipid with μ = 0.4 and
FLM is the output signal of the nipid with μ = 0.6
2- Consider the outputs of the two nipid blocks that have λ's
corresponding to non-zero weights i.e. λ ϵ {SM , LM} ≡ λ ϵ
{0.4, 0.6}, see Fig 6.
3- The output signal F, see Fig 9, is then calculated as :
F=
.
.
= 0.75 F
+ 0.25 F
; where
FSM is the output signal of the nipid with λ = 0.4 and FLM is the
output signal of the nipid with λ = 0.6
Fig. 11. Block diagram and response of the modified-nipid as a fractionalorder integrator at λϵ{0.4, 0.6} and the response of modified-nipid as a
fractional integrator with λ = 0.45
To validate the matching of the responses of the modified
and the normal nipid, another experiment was done using the
two blocks as fractional derivatives having the same
parameters, the two responses were compared, see Fig 12.
Fig. 9. Output signals of normal nipid as a fractional integrator at λϵ{0.4 ,
0.6}and that of modified-nipid with λ = 0.45
tuning and the modified nipid fractional order PID was used
as the loop controller, see Fig 15.
Fig. 12. The modified and normal nipid as derivatives with Mu=0.45
In the two above examples, comparing the responses of the
normal and modified nipid showed a small difference between
the two curves. Actually, this is very acceptable for two
reasons, the first reason is that the difference is very small
(about 1.2%), and the second reason is that fuzzy blending is,
by nature, an approximate solution. However, a solution that
actually minimized the mentioned difference is also tried here
just by increasing the number of membership functions to be
11 instead of 6. The responses of the two above experiments
with the new memberships is shown in Fig 13.
Fig. 15. Deploying FLC and modified-nipid for FOPID parameter tuning
The values of the parameters after tuning will be Kp2, Ki2,
Kd2, μ2 and λ2, such that : Kp2 = Kp * Kp1 ; Ki2 = Ki * Ki1 ; Kd2
= Kd * Kd1 ; μ2 = μ * μ1 and λ2 = λ * λ1; where Kp1, Ki1, Kd1, μ1
and λ1 are the outputs of the Fuzzy Logic controller and Kp, Ki,
Kd, μ and λ are the initial values of the parameter before tuning.
The structure of fuzzy logic control goes through three main
stages (fuzzification, rule base and defuzzification).
1- Fuzzification:
In this stage the input values are converted into linguistic
form. As demonstrated in Figure 16-A the controller has two
inputs: error and error change signals.
Fig. 13. Normal and modified nipid with 11 rules as integrators and
derivatives
B. Results of online self-tuning of Fractional Order PID
controller
The controlled model considered in this study is given in
Fig 14.
Fig. 14. The model considered for the implementation of PID and FOPID
controllers
The initial values for kp, ki and kd were deduced from the
tuning technique of Zeigler-Nichols, and the two fractional
order coefficients were firstly set to 1.0, the value of the
friction coefficient b was then manually increased to simulate
a real increase in rotor friction, and accordingly the response
got slower. The goal now is to perform a parameter tuning
process so that the new response is improved. A 25-rule base
Fuzzy-logic controller was deployed to perform the online
Fig. 16. A-Input memberships of e and Δe, B-Output memberships of FLC
outputs
The universe of discourse for both the two inputs is chosen
to be in the range of [-1,1], and the linguistic labels are NB,
NM, NS, Z, PS, PM and PB. The linguistic labels of the
outputs are Z, MS, S, M, B, MB and VB. Figure 16-B shows
the memberships of outputs of fuzzy logic control.
TABLE V. RULE BASE MATRIX FOR Μ1
2- Rule Base
A human decision process is simulated via a decision logic
which is described by the rule base technique. Both error and
change of error were assigned 7 linguistic labels, resulting in
having 7 × 7 = 49 rule base. A rule simplification to 25 rule
base was presented in [5] shown in Table 1, 2, 3, 4 and 5.
TABLE I. RULE BASE MATRIX FOR KP1
∆e/e
NB
NS
ZE
PS
PB
NB
VB
B
ZE
B
VB
NS
VB
B
ZE
B
VB
ZE
VB
B
MS
B
VB
PS
VB
MB
S
MB
VB
PB
VB
VB
S
VB
VB
TABLE II. RULE BASE MATRIX FOR KI1
∆e/e
NB
NS
ZE
PS
PB
NB
M
S
MS
S
M
NS
M
S
MS
S
M
ZE
M
S
ZE
S
M
PS
M
S
MS
S
M
PB
M
S
MS
S
M
∆e/e
NB
NB
ZE
NS
MS
ZE
M
PS
MB
PB
VB
NS
MS
S
M
B
MB
ZE
M
M
M
M
M
PS
MB
B
M
B
MB
PB
VB
MB
M
MB
VB
3- Defuzzification
In this stage, the fuzzy outputs are converted to crisp
output. The parameter tuning was carried out via three
combinations:(1)Tuning μ and λ only while keeping kp, ki and
kd unchanged, (2)Tuning kp, ki and kd while keeping μ and λ
unchanged and (3) Tuning all of the five parameters kp, ki, kd, μ
and λ. A comparison between the time responses of the above
three techniques in addition to the response of the system
without tuning was made. The used model and the output
responses were shown in Fig 17.
TABLE III. RULE BASE MATRIX FOR KD1
∆e/e
NB
NB
ZE
NS
S
ZE
M
PS
MB
PB
VB
NS
S
B
MB
VB
VB
ZE
M
MB
MB
VB
VB
PS
B
VB
VB
VB
VB
PB
VB
VB
VB
VB
VB
TABLE IV. RULE BASE MATRIX FOR Λ1
∆e/e
NB
NB
ZE
NS
MS
ZE
M
PS
MB
PB
VB
NS
MS
S
M
B
MB
ZE
M
M
M
M
M
PS
MB
B
M
S
MS
PB
VB
MB
M
MS
ZE
Fig. 17. Several parameter combinations for tuning FOPID
From Fig. 17, it was found that the best performance is
achieved when all of the five parameters of the controller are
tuneable during the simulation.
IV. CONCLUSION
This study presented an application of Fractional Order PID
as the controller of a PV sun tracker horizontal dc motor. A
technique for online self-tuning for all of the five parameters of
the FOPID using a Fuzzy Logic Controller was introduced. A
technical constraint aroused during the work, that is the used
FOPID Simulink bock of "nipid" doesn't enable changing the
controller parameters during the simulation time, therefore, a
new tuneable FOPID block was designed basing of the old
nipid and the TS-Fuzzy technique, the new FOPID block
enables the parameters changing during the simulation. The
tuning process was carried out via making several experiments
changing the combinations of the tuned parameters in each
experiments and the responses of the experiments were
compared. Finally it was found that the best performance is
achieved when all of the five parameters of the controller are
tuneable..
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