Embedded intelligent adaptive PI controller for an electromechanical system

Research Article
Embedded intelligent adaptive PI controller for an
electromechanical system
Ahmad M. El-Nagar
Department of Industrial Electronics and Control Engineering, Faculty of Electronic Engineering, Menofia University, Menuf 32852, Egypt
a r t i c l e i n f o
Article history:
Received 20 January 2016
Received in revised form
2 May 2016
Accepted 7 June 2016
Available online 21 June 2016
This paper was recommended for publica-
tion by Prof. A.B. Rad
Keywords:
Adaptive PI controller
Interval type-2 fuzzy neural networks
Nonlinear DC motor
Lyapunov theorem
a b s t r a c t
In this study, an intelligent adaptive controller approach using the interval type-2 fuzzy neural network
(IT2FNN) is presented. The proposed controller consists of a lower level proportional - integral (PI)
controller, which is the main controller and an upper level IT2FNN which tuning on-line the parameters
of a PI controller. The proposed adaptive PI controller based on IT2FNN (API-IT2FNN) is implemented
practically using the Arduino DUE kit for controlling the speed of a nonlinear DC motor-generator system.
The parameters of the IT2FNN are tuned on-line using back-propagation algorithm. The Lyapunov the-
orem is used to derive the stability and convergence of the IT2FNN. The obtained experimental results,
which are compared with other controllers, demonstrate that the proposed API-IT2FNN is able to
improve the system response over a wide range of system uncertainties.
& 2016 ISA. Published by Elsevier Ltd. All rights reserved.
1. Introduction
The traditional PID controllers are still the most generally uti-
lized control structure in a large portion of the modern processes.
This is for the most part, since PID controllers have straight forward
control structures, moderate cost, and adequacy of linear systems
[1–3]. However, it can be difficult to define the suitable PID gains
when the plant to be controlled has a high level of complexity, such
as, modeling nonlinearities, time delay and structural uncertainties.
All these factors could degrade the performance of a PID controller,
which becomes unsatisfactory to guarantee the requirements in
most of the practical systems [4]. For these reasons, many research
papers focused on self-tuning PID controller [5–10], adaptive PID
controller [11–14], self-tuning predictive PID controller [15]. In self-
tuning and adaptive PID controllers, the parameters of the con-
troller were tuned automatically based on the changes of the sys-
tem parameters. However, favorable performance can be obtained,
this adaptive PID controllers are still limited capabilities with
uncertain and nonlinear systems.
The conventional PID controller is combined with a fuzzy logic
controller (FLC) in order to achieve better system performance
over the conventional PID controller such as, the fuzzy PI control
[16], the fuzzy PD control [17] and the fuzzy PID control [18–24].
Chen et al. [25] and Martins et al. [26] proposed an adaptive
algorithm using the neural networks to tuning the parameters of
PID controller. The PID neural network, which the hidden layer
neurons work as the PID controller is proposed by Shu and Pi [27].
Lee and Teng [28] proposed a fuzzy neural network to calculate
the parameters of the PID controller. Despite the significant
improvement of system performance with the fuzzy PID controllers,
PID neural network and adaptive PID controller based on fuzzy
neural network over their conventional counterparts, it should be
noted that they are usually not effective if the system to be con-
trolled has structure uncertainties as the ordinary fuzzy systems
(type-1 FLSs) and fuzzy neural network (Type-1 fuzzy neural net-
work) have limited capabilities to directly handle data uncertainties.
The type-2 fuzzy sets (T2-FSs) that introduced by Zadeh in 1975
are able to handle system uncertainties because their degree of
membership functions are themselves fuzzy. The concept of a T2-
FS is an extension of the concept of an ordinary fuzzy sets (type-1
fuzzy sets; T1-FSs). A T2-FS is characterized by a fuzzy member-
ship function, unlike a T1-FS where the membership grade is a
crisp number in [0, 1] [29]. The conventional PID controller is
combined with an interval type-2 fuzzy logic system (IT2-FLS) in
order to achieve better system performance over the conventional
PID controller with type-1 FLS [30–44]. Although favorable per-
formance can be obtained, these controllers are limited on tuning
the controller parameters related to uncertain systems.
In this paper, the adaptive PI controller using the IT2FNN is
introduced. The proposed controller consists of a lower level PI
controller, which is the main controller and an upper level IT2FNN.
The IT2FNN provides a mechanism for tuning on-line the gains of a
PI controller. The parameters of the IT2FNN are tuned on-line
using the back-propagation algorithm. The Lyapunov theorem is
Contents lists available at ScienceDirect
journal homepage: www.elsevier.com/locate/isatrans
ISA Transactions
http://dx.doi.org/10.1016/j.isatra.2016.06.006
0019-0578/& 2016 ISA. Published by Elsevier Ltd. All rights reserved.
E-mail address: Ahmed_elnagar@menofia.edu.eg
ISA Transactions 64 (2016) 314–327

used to derive the stability and convergence of the IT2FNN. The
proposed adaptive PI controller is implemented practically using
the Ardunio DUE Kit for controlling the DC motor – generator set.
The DC motor traditional mathematical model is a second order
linear system that ignores the motor Coulomb friction. Unfortu-
nately, the dead-zone and Coulomb friction can cause the DC
motor system to have nonlinear characteristics. The generator,
which combined with the DC motor is used as an electrical load.
The obtained results show that the ability of the API-IT2FNN
controller to handle the uncertainty in measurement error and
the effect of an electrical load. In addition, the proposed controller
is not designed for a special DC motor-generator system and
consequently, it can be utilized for any DC motor type. Moreover,
the proposed controller can be used to control other mechanical or
electrical systems.
The rest of this paper is organized as follows. The dynamic
model of nonlinear DC motor – generator system is presented in
Section 2. In Section 3, the adaptive PI controller based on the
IT2FNN is included. Section 4, presents the hardware imple-
mentation of the proposed API-IT2FNN. The experimental results
is presented in Section 5. Finally, Section 6 concludes the paper.
2. Dynamic model of the nonlinear DC motor – generator
system
The system described in this study combines a DC motor with
an electrical load (DC generator). The DC motor – generator system
is shown in Fig. 1. The dynamic equations, which represent a DC
motor are given as [45]:
vam ¼ Ramiam þLam
diam
dt
þeam ð1Þ
eam ¼ kemωm ð2Þ
Tm ¼ ktmiam ð3Þ
Jm
dωm
dt
¼ Tm ÀBmωm ÀTs ð4Þ
where the parameters of the motor are described in Table 1.
The dynamic equations representing a DC generator are given
as [45]:
Jg
dωg
dt
¼ Ts ÀBgωg ÀTg ð5Þ
Tg ¼ ktgiag ð6Þ
eag ¼ kegωg ¼ Ragiag þLag
diag
dt
þRLiag ð7Þ
Ts ¼ ksðθm ÀθgÞþBsðωm ÀωgÞ ð8Þ
with
dθm
dt
¼ ωm;
dθg
dt
¼ ωg ð9Þ
where the parameters of the generator are described in Table 1.
The linear model lacks reliability and accuracy because the
Coulomb friction and the dead zone appear as hard nonlinearities
when the system rotates in both directions [45]. A general model
including Coulomb friction is [46]:
Tf ðωÞ ¼ β0sgnðωÞþβ1eÀ β2 ωj j
sgnðωÞ ð10Þ
where β0; β1; β2 are constants. The function sgn is defined as [46]:
sgnðωÞ ¼
1 ω40
0 ω ¼ 0
À1 ωo0
8
><
>:
ð11Þ
Eqs. (4) and (5) are modified in order to represent the nonlinear
model as follows:
Jm
dωm
dt
¼ Tm ÀBmωm ÀTs ÀTf ðωmÞ ð12Þ
Jg
dωg
dt
¼ Ts ÀBgωg ÀTg ÀTf ðωgÞ ð13Þ
3. Adaptive PI controller based on IT2FNN
In this section, the adaptive PI controller using the IT2FNN is
presented as shown in Fig. 2. It consists of a lower level classical
controller and an upper level intelligent system. The upper level
system provides a mechanism to tuning the gains of a classical PI
controller, whereas the lower level controller is the main con-
troller. In the proposed controller structure, a conventional PI
controller is selected for the lower level and the IT2FNN is used in
the upper level.
Fig. 1. Diagram of the DC motor – generator set.
Table 1
Parameters of the DC motor – generator system.
Parameter Description Unit
iam and iag The motor and generator armature current A
Ram and Rag The motor and generator armature coil resistance Ω
Lam and Lag The motor and generator armature coil inductance H
eam and eag The motor and generator back electromotive-force
voltage
V
ωm and ωg The motor and generator angular speed rad=s
Tm and Tg The motor and generator torque Nm
kem and keg The motor and generator electrical constant Vs=rad
ktm and ktg The motor and generator mechanical constant
Jm and Jg The moment of motor and generator inertia kg m2
Bm and Bg The coefficient of viscous motor and generator
friction
Nm=ðrad=sÞ
θm and θg The motor angle and generator angle displacements rad
Ts The transmitted shaft torque Nm
ks The shaft elasticity Nm=rad
Bs The inner damping coefficient of the shaft Nm=ðrad=sÞ
RL The load resistance Ω
vam The motor armature applied voltage V
A.M. El-Nagar / ISA Transactions 64 (2016) 314–327 315

3.1. PI controller
A typical classical PI controller can be expressed as [43]:
uðtÞ ¼ kpeðtÞþki
Z
eðtÞdt
eðtÞ ¼ yrðtÞÀymðtÞ ð14Þ
where uðtÞ is the control signal at time t. kp and ki are the pro-
portional and the integral gains, respectively. The set-point, the
measured output and the error signal are denoted as yrðtÞ, ymðtÞ
and eðtÞ, respectively. In the classical PI controller, the values of kp
and ki in Eq. (14) are adjusted by the operator according to the
changes in the process condition.
3.2. Adaptive PI controller
The Eq. (14) can be rewritten as:
uðtÞ ¼ kpðtÞeðtÞþkiðtÞ
Z
eðtÞdt ð15Þ
where
kpðtÞ ¼ k
0
p þρ1ðtÞ
kiðtÞ ¼ k
0
i þρ2ðtÞ
The parameters k
0
p and k
0
i to be pre-tuned are time invariant
constants while the PI controller is working. The parameters ρ1ðtÞ
and ρ2ðtÞare time varying and updated in real time. Thus the
adaptation of ρ1ðtÞ and ρ2ðtÞ means the adaptation of the kpðtÞ and
kiðtÞ. The adaptation parameters ρ1ðtÞ and ρ2ðtÞ are calculated
using the IT2FNN. By developing the IT2FNN, these parameters can
be updated on-line, according to the changes in the process con-
dition and further it will enhance the conventional controller
performance over a wide operating range.
The structure of upper-level IT2FNN is shown in Fig. 3. It has
two inputs (error signal eðkÞ and the change in error signal
ΔeðkÞ ¼ eðkÞÀeðkÀ1Þ) and two outputs (ρ1 and ρ2); where k
denotes the number of iterations.
Fig. 4 shows the interval type-2 Gaussian membership function
(MF) with a ﬁxed mean, m, and an uncertain standard deviation in
½σ1; σ2Š. This MF can be described as Eq. (16) [47]:
μ~A ðxÞ ¼ exp À
1
2
xÀm
σ
2
!
; σA½ σ; σŠ ð16Þ
The IT2-FS is limited by an upper MF (UMF) and a lower MF
(LMF), which are denoted μ~A ðxÞ and μ ~A
ðx0
Þ, respectively. The regionFig. 2. Adaptive PI controller based on IT2FNN.
Fig. 3. Structure of IT2FNN.
A.M. El-Nagar / ISA Transactions 64 (2016) 314–327316

between them is called the footprint of uncertainty. The IF-THEN
rule for the IT2FNN can be described as:
Rin : IF x1
1 is ~M
n
i1; and x1
2 is ~M
n
i2; then ρi is w4
in; _w_
4
in
h i
ð17Þ
where n ¼ 1; 2; ::::; N is the rule number; i ¼ 1; 2, x1
1 ¼ eðkÞ and x1
2
¼ ΔeðkÞ are the inputs of the IT2FNN; ~M
n
i1 and ~M
n
i2 are the IT2-FSs
of the antecedent part; ½w4
in; w4
in
Šis a centroid set of the output MF,
which can be called weighting interval set. The IT2FNN is intro-
duced as the following:
Input Layer: The input and output for every node s are
described as:
net1
s ðkÞ ¼ x1
s ; x1
s ¼ f
1
s ðnet1
s ðkÞÞ ¼ net1
s ðkÞ; s ¼ 1; 2 ð18Þ
Membership Layer: Each node executes an interval type-2
fuzzy MF, for the jth node.
v2
ijðkÞ ¼ ~M
j
sðx2
s Þ ¼ f
2
ijðnet2
ijðkÞÞ
¼ expðnet2
ijðkÞÞ ¼
v2
ijðkÞ as σsj ¼ σsj
_v_
2
ijðkÞ as σsj ¼ _σ_sj
; j ¼ 1; :::; 6 and i ¼ 1; 2
8

:
ð19Þ
where net2
ijðkÞ ¼ Àð1=2Þðx2
s ÀmsjÞ2
=ðσsjÞ2
; f is the number of MFs in
each input node. The output of this layer v2
ijðkÞ is also denoted as
½v2
ij
ðkÞ; v2
ijðkÞŠ.
Rule Layer: For the Nth rule node
net3
inðkÞ ¼ ∏
j
w3
jnx3
ijðkÞ
u3
inðkÞ ¼ f
3
inðnet3
inðkÞÞ ¼ net3
inðkÞ
v3
inðkÞ ¼ ∏
n
j ¼ 2
w3
jnv2
ij

v3
in
ðkÞ ¼ ∏
n
j ¼ 1
w3
jnv2
ij
n ¼ 1; :::; N
8

:
ð20Þ
where x3
ij represents the j th input to the node of this layer; w3
jn are
weights between the membership layer and the rule layer. These
weights are set to be unity to simplify the implementation of the
IT2FNN for the real-time applications. The output of this layer; v3
in
ðkÞ is denoted as ½v3
in
ðkÞ; v3
inðkÞŠ.
Type-Reduction Layer: This layer is used to perform the TR
using our simplified TR method [48]. The process of this layer is
introduced as the following:
net4
ilðkÞ ¼
PN
n ¼ 1
w4
inv3
inðkÞ
PN
n ¼ 1
v3
inðkÞ
ð21Þ
v4
ilðkÞ ¼ f
4
ilðnet4
ilðkÞÞ ¼ net4
ilðkÞ ¼
v4
il ¼
XN
n ¼ 1
w4
inv3
inðkÞ
XN
n ¼ 1
v3
inðkÞ
¼ WT
i V
v4
il
¼
XN
n ¼ 1
w4
inv3
in
ðkÞ
XN
n ¼ 1
v3
in
ðkÞ
¼ WT
i V
; l ¼ 1
8

:
ð22Þ
where w4
in A½w4
in w4
in
Š is the centriod of the output MF; w4
in ¼ ðw4
in
þw4
in
Þ=2 is the average value between the two endpoints of the
centriod; Wi ¼ w4
i1 w4
i2…w4
in
Â ÃT
.
V ¼
v
3
i1ðkÞ
XN
n ¼ 1
v3
inðkÞ
v
3
i2ðkÞ
XN
n ¼ 1
v3
inðkÞ
⋯ v
3
inðkÞ
XN
n ¼ 1
v3
inðkÞ
2
6
4
3
7
5
T
and
V ¼
v3
i1
ðkÞ
XN
n ¼ 1
v3
in
ðkÞ
v3
i2
ðkÞ
XN
n ¼ 1
v3
in
ðkÞ
⋯
v3
in
ðkÞ
XN
n ¼ 1
v3
in
ðkÞ
2
6
4
3
7
5
T
Output Layer: This layer performs the output of the network
as:
ρi ¼
v4
il þv4
il
2
¼
1
2
WT
i V þWT
i V

¼
1
2
WT
i VðX1
; m; σÞ i ¼ 1; 2 ð23Þ
where αi is the output of the IT2FNN, V ¼ V V
Â ÃT
, X1
¼ x1
1 x1
2
Â Ã
,
m ¼ m11…m1s m21 …m2s½ Š, and σ ¼ σ11 …σ1s σ21…σ2s½ Š:
3.3. Learning algorithm
The back-propagation algorithm is used to updating the
weights of the IT2FNN. So, the target function is represented as:
J ¼
1
2
yrðkÞÀymðkÞ
À Á2
ð24Þ
where yrðkÞ is the desired output, and ymðkÞ is the measured out-
put.
w4
inðkþ1Þ ¼ w4
inðkÞþΔw4
inðkÞ ð25Þ
Δw4
in ¼ Àη1
∂J
∂w4
in
¼ Àη1
∂J
∂ym
:
∂ym
∂u
:
∂u
∂ρi
∂ρi
∂w4
in
ð26Þ
Δw4
inðkÞ ¼
1
2
ηw
ðyrðkÞÀymðkÞÞ :
δu : δρi :
1
2
v3
inðkÞ
PN
n ¼ 1
v3
inðkÞ
þ
1
2
v3
in
ðkÞ
PN
n ¼ 1
v3
inðkÞ
0
B
B
B
@
1
C
C
C
A
; n ¼ 1; 2; :::; N ð27Þ
where ηw
is the learning rate for tuning the weighting interval
factor of the IT2FNN, Δw4
in is the increment of weight w4
in, δu is the
change of the output towards the change of the input which
defined as follows:
δu ¼
∂ym
∂u
ffi
ymðkÞÀymðkÀ1Þ
uðkÞÀuðkÀ1Þ
ð28Þ
Fig. 4. Interval type-2 fuzzy set with uncertain mean.

In many practical cases the derivative in Eq. (28) can be easily
estimated or replaced þ1 or -1 [49]. From Eq. (15), δρi can be
defined as:
δρ1 ¼
∂u
∂ρ1
¼ ΔeðkÞ; and δρ2 ¼
∂u
∂ρ2
¼ eðkÞ ð29Þ
Through the same principle, it can change the weight Δ_w_
4
in as
follows:
w4
in
ðkþ1Þ ¼ w4
in
ðkÞþΔw4
in
ðkÞ ð30Þ
Δw4
in
ðkÞ ¼
1
2
ηw
ðyrðkÞÀymðkÞÞ:δu:
δρi :
1
2
v3
inðkÞ
PN
n ¼ 1
v3
inðkÞ
þ
1
2
v3
in
ðkÞ
PN
n ¼ 1
v3
inðkÞ
0
B
B
B
@
1
C
C
C
A
; n ¼ 1; 2; :::; N ð31Þ
Also, the mean and standard deviation of the IT2FNN can be
updated as follow:
m1ijðkþ1Þ ¼ m1ijðkÞþ
1
2
ηm
ðyrðkÞÀymðkÞÞ:
δu δρi
PN
n ¼ 1
u3
LNðkÞ
:
Xjf
n ¼ 1þhf
Xf
v ¼ 1
w4
in
ðkÞ:v2
1ij:v2
2iv:
eðkÞÀm1ijðkÞ
σ1ij
2
ðkÞ
j ¼ 1; 2; :::; f ; h ¼ 0; 1; 2; :::; f À1 and i ¼ 1; 2 ð32Þ
m2ijðkþ1Þ ¼ m2ijðkÞþ
1
2
ηm
δu δρi
PN
n ¼ 1
_v_
3
in
ðkÞ
:
X
n ¼ j
Xf
v ¼ 1
_w_
4
inðkÞ:v2
1iv:v2
2ij:
ΔeðkÞÀm2ijðkÞ
σ2ij
2
ðkÞ
j ¼ 1; 2; :::; f ; n ¼ j; jþf ; jþ2f ; :::; jþf ðf À1Þ and i ¼ 1; 2 ð33Þ
σ1ijðkþ1Þ ¼ σ1ijðkÞþ
1
2
ησ
δu δρi
PN
n ¼ 1
v3
inðkÞ
:
Xjf
n ¼ 1 þ if
Xf
v ¼ 1
w4
in
ðkÞ:v2
1ij:v2
2iv:
ðeðkÞÀσ1ijðkÞÞ2
σ1ij
3
ðkÞ
j ¼ 1; 2; :::; f ; h ¼ 0; 1; 2; :::; f À1 and i ¼ 1; 2 ð34Þ
σ1ij
ðkþ1Þ ¼ σ1ij
ðkÞþ
1
2
ησðyrðkÞÀymðkÞÞ:
δu δρi
Pn
N ¼ 1
v3
inðkÞ
:
Xjf
n ¼ 1 þ if
Xf
v ¼ 1
w4
inðkÞ:v2
1ij
:v2
2iv
:
ðeðkÞÀσ1ij
ðkÞÞ2
σ1ij
3ðkÞ
j ¼ 1; 2; :::; f ; h ¼ 0; 1; 2; :::; f À1 and i ¼ 1; 2 ð35Þ
σ2ijðkþ1Þ ¼ σ2ijðkÞþ
1
2
δu δρi
PN
n ¼ 1
_u_
3
in
ðkÞ
:
X
n ¼ j
Xf
v ¼ 1
_w_
4
inðkÞ:v2
1iv:v2
2ij:
ðΔeðkÞÀσ2ijðkÞÞ2
σ2ij
3
ðkÞ
j ¼ 1; 2; :::; f ; n ¼ j; jþf ; jþ2f ; :::; jþf ðf À1Þ and i ¼ 1; 2 ð36Þ
_σ_2ijðkþ1Þ ¼ _σ_2ijðkÞþ
1
2
δu δρi
PN
n ¼ 1
u3
inðkÞ
:
X
n ¼ j
Xf
v ¼ 1
w4
inðkÞ:_v_
2
1iv:_v_
2
2ij:
ðΔeðkÞÀ_σ_2ijðkÞÞ2
_σ_2ij
3ðkÞ
j ¼ 1; 2; :::; f ; N ¼ j; jþf ; jþ2f ; :::; jþf ðf À1Þ and i ¼ 1; 2 ð37Þ
where ηm
and ησ are the learning rate parameters of mean and
standard deviation of the IT2FNN.
3.4. Convergence analysis
In the learning procedures of the IT2FNN, the update para-
meters require a proper choice of learning rates η. In this section,
the approach of selecting properly η is developed. The Lyapunov
function can be defined as:
VðkÞ ¼
1
2
e2
ðkÞ ð38Þ
where eðkÞ is the control error. The change in the Lyapunov func-
tion is obtained by:
ΔVðkÞ ¼ Vðkþ1ÞÀVðkÞ ¼
1
2
e2
ðkþ1ÞÀe2
ðkÞ
Â Ã
ð39Þ
The error difference of the IT2FNN can be represented by
[50,51]:
eðkþ1Þ ¼ eðkÞþΔeðkÞ ¼ eðkÞþ
∂eðkÞ
∂W
!T
ΔW ð40Þ
where W is the parameter vector of the IT2FNN weighting vector,
W ¼ ½ w4
in w4
in
m1ij m2ij σ1ij σ1ij
σ2ij σ2ij Š and ΔW repre-
sent the corresponding change of this weight. In the IT2FNN, the
weight change is obtained from the update rules of Eqs. (25)–(37)
as:
ΔWðkÞ ¼ ÀηW
eðkÞ
∂eðkÞ
∂W
% ηW
eðkÞ
∂ρiðkÞ
∂W
ð41Þ
where ηW
is the learning rate of corresponding weight component
in the IT2FNN.
Theorem: Let ηW
¼ ηw
ηm
ησ
h i
be the learning rates of the
IT2FNN and define HW
max as HW
max ¼ max k‖HW
ðkÞ‖, where
HW
ðkÞ ¼ ∂ρiðkÞ=∂W, and ‖:‖ is the Euclidean norm. Then, the
asymptotic convergence of the IT2FNN controller is guaranteed if
ηW
are chosen to satisfy:
0oηW
o
2
HW
max
2
ð42Þ
in which Wis the weights of the IT2FNN parameters w; m; σ.
Proof: From Eqs. (39 and 40), the change in the Lyapunov
function is:
ΔVðkÞ ¼
1
2
e2
ðkþ1ÞÀe2
ðkÞ
Â Ã
¼ ΔeðkÞ eðkÞþ
1
2
ΔeðkÞ
!
ð43Þ
ΔVðkÞ ¼
∂eðkÞ
∂W
!T
ηW
eðkÞ
∂ρi
∂W
: eðkÞþ
1
2
∂eðkÞ
∂W
!T
ηW
eðkÞ
∂ρi
∂W
( )
ð44Þ
Since for the IT2FNN, ∂eðkÞ=∂W ¼ À∂ρi=∂W, and let
HW
ðkÞ ¼ ∂ρiðkÞ=∂W, HW
max ¼ max k‖HW
ðkÞ‖, we obtain
Fig. 5. Experimental setup for DC motor – generator system.

Fig. 6. Response of the DC motor – generator system for step change.
Fig. 7. MAE for task 1.
Fig. 8. Response of the DC motor – generator system for uncertainty in measurement error (20%).

Fig. 9. Response of the DC motor – generator system for uncertainty in measurement error (40%).
Fig. 10. MAE for task 2 (20%).

ΔVðkÞ ¼ ÀeðkÞ‖HW
ðkÞ‖2
ηW
eÀ
1
2
eηW
‖HW
ðkÞ‖2

ð45Þ
ΔVðkÞ ¼ À
1
2
e2
ðkÞ‖HW
ðkÞ‖2
ηW
2ÀηW
‖HW
ðkÞ‖2

Àβe2
ðkÞ: ð46Þ
where
β ¼
1
2
‖HW
ðkÞ‖2
ηW
2ÀηW
‖HW
ðkÞ‖2

ð47Þ
If β40, then ΔVðkÞo0 is satisﬁed. Thus, the asymptotic con-
vergence of the proposed control system is guaranteed, and from
Eq. (47), we obtain Eq. (42). This completes the proof.
4. Embedded adaptive PI controller
In this section, the proposed API-IT2FNN is implemented practi-
cally using an Arduino DUE microcontroller kit for controlling the DC
Fig. 12. Response of the DC motor – generator system for uncertainty due to the load (23%).
Fig. 13. Response of the DC motor – generator system for uncertainty due to the load (46%).

Fig. 16. Response of the DC motor – generator system (Task 4).

motor – generator system. An Arduino DUE kit has many features
such as: a 84 MHz crystal frequency, a 32 bit CortexM3 ARM micro-
controller, 4 universal asynchronous receiver transmitter (UART), 512
KB ﬂash memory and 96 KB random access memory (RAM). The
proposed controller program is done using Arduino Integrated
Development Environment (Arduino IDE) with version 1.6.5 software.
Fig. 5 shows the experimental setup for the DC motor – gen-
erator system. The speed is measured using the tachometer and
the DC motor drive circuit is 3 Amp. H-Bridge, which used to drive
the DC motor – generator system. The electric lamps are used as an
electrical load for the DC motor. The implementation procedure of
the proposed API-IT2FNN for the DC motor – generator system
using an Arduino kit is summarized as follows:
– Step 1: Measure the speed of the motor using the tachometer
and sent it to the Arduino kit via analog input.
– Step 2: Compute the input variables (input layer) for the IT2FNN
using Eq. (18).
– Step 3: Compute the output of membership layer for the IT2FNN
using Eq. (19).
Fig. 18. Updating the controller gains for the API-IT2FLS.
Fig. 19. Updating the controller gains for the API-IT2FNN.

– Step 4: Compute the rule layer output using Eq. (20).
– Step 5: Perform the type-reduction to calculate the output of the
IT2FNN (updating factors) using Eqs. (21)–(23).
– Step 6: Compute the control signal using Eq. (15) and sent it to
the drive circuit via PWM output.
– Step 7: Calculate the target function using Eq. (24) and updating
the parameters of the IT2FNN using Eqs. (25)–(37), go to step 1.
5. Practical results
In this section, the experimental results for the DC motor – gen-
erator system using the embedded API-IT2FNN are presented. In order
to clarify the improvements of the proposed API-IT2FNN, the practical
results with the adaptive PI controller based on the IT2FLS (API-
IT2FLS) which is proposed previously [43] are also presented. Six
different experimental tasks are considered where the desired speed is
500 rev= min. Three performance indices are measured for comparing
the API-IT2FNN controller and the API-IT2FLS controller; the mean
absolute errors (MAE), the root mean square of errors (RMSE) and the
integral of square of errors (ISE) criteria which are defined as:
MAE ¼
1
kf
Xkf
k ¼ 1
eðkÞ

ð48Þ
RMSE ¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1
kf
Xkf
k ¼ 1
ðeðkÞÞ2
v
u
u
t ð49Þ
ISE ¼
Z 1
0
e tð Þ½ Š2
dt ð50Þ
where eðtÞ is the error signal and kf is the number of iterations
The type-2 MFs are initialized as follows:
m11 ¼ m14 ¼ m21 ¼ m24 ¼ À0:5; m12 ¼ m15 ¼ m22
¼ m25 ¼ 0; m13 ¼ m16 ¼ m23 ¼ m26 ¼ 0:5; σij ¼ 0:25; σij
¼ 0:15.

The initial values for the layer weights of the IT2FNN network
are chosen randomly between [À0.5, 0.5]. After the initial values
of the controller parameters are chosen based on the limitations of
the universe of discourse, the online learning algorithm shown in
Eqs. (25)–(37) is updated with controller parameters. Any other
values of the universe of discourse are acceptable because on-line
learning algorithm is updated with the ﬁnal values, which guar-
antee the system stability.
Table 2
ISE values for the controllers.
Task 1 Task 2
(20%)
Task 2
(40%)
Task 3
(23%)
Task 3
(46%)
Task 4 Task 5 Task 6
Classical
PI
729.77 86.94 93.87 91.57 95.18 102.99 63.07 95.21
STPIC 510.20 44.05 52.79 47.01 48.61 54.97 51.88 83.52
API-IT2FLS
[43]
490.50 43.00 50.46 44.87 45.97 51.03 47.01 78.03
API-
IT2FNN
314.75 37.95 41.17 38.99 37.73 40.98 31.98 51.43
Table 3
RMSE values for the controllers.
Task 1 Task 2
(20%)
Task 2
(40%)
Task 3
(23%)
Task 3
(46%)
Task 4 Task 5 Task 6
Classical
PI
0.7798 0.1474 0.1532 0.1513 0.1543 0.1605 0.1025 0.0976
STPIC 0.3571 0.1049 0.1149 0.1084 0.1102 0.1172 0.0930 0.0914
API-
IT2FLS
[43]
0.2022 0.1037 0.1123 0.1059 0.1072 0.1129 0.0885 0.0883
API-
IT2FNN
0.1620 0.0974 0.1014 0.0987 0.0971 0.1012 0.0730 0.0717

In this paper, the initial values of the PI controller gains are set
by trial and error method to make the transient response of the
system as good as possible. It is possible to use well-defined
methods such as Ziegler – Nichols method to set the initial values.
5.1. Task 1: step change in the set-point
To test the tracking of the DC motor – generator system, the
set-point of the speed is changed from 500 rev= min to À500
rev= minas shown in Fig. 6. It is clear that the response of the DC
motor speed for the proposed API-IT2FNN is made significantly
better than the API-IT2FLS. Fig. 7 shows that the MAE for the
proposed API-IT2FNN is lower than that obtained for the API-
IT2FLS. Therefore, the proposed API-IT2FNN is able to respond the
Coulomb friction and the dead zone nonlinearity better than the
API-IT2FLS.
5.2. Task 2: uncertainty due to the measurement error
Figs. 8 and 9 show the response of the controlled system when
there is an uncertainty due to the measurement error with 20%
and 40%, respectively at the time (t ¼ 20 s). It is clear that the
response of the proposed API-IT2FNN has lower settling time after
adding the uncertainty value compared with API-IT2FLS. Figs. 10
and 11 show that the MAE for the proposed API-IT2FNN is less
than that obtained for the API-IT2FLS. Therefore, the proposed
API-IT2FNN is able to reduce the effect of the uncertainty due to
the measurement error compared with the API-IT2FLS.
5.3. Task 3: uncertainty due to an electric load
Figs. 12 and 13 show the response of the controlled system
when there is an uncertainty in the load with 23% and 46%,
respectively at the time (t ¼ 20 s). It is clear that the response of
the proposed API-IT2FNN is made significantly better than the API-
IT2FLS. Figs. 14 and 15 show that the MAE for the proposed API-
IT2FNN is less than that obtained for the API-IT2FLS. Therefore, the
proposed API-IT2FNN is able to respond the uncertainty due to an
electric load on the DC motor compared with the API-IT2FLS.
5.4. Task 4: uncertainty due to the measurement error and the
electric load
In this task, 20% measurement error and 23% electric load at the
time (t¼20 s) are applied to the DC motor. Fig. 16 shows the
response of the controlled system for this task. Fig. 17 shows that
the MAE for the proposed API-IT2FNN is less than that obtained for
the API-IT2FLS. So, the response of the proposed API-IT2FNN is
able to respond the effect of the uncertainty due to the measure-
ment error and an electric load on the DC motor – generator
system. Figs. 18 and 19 show the varying of the proportional and
integral gains for the API-IT2FLS and the API-IT2FNN, respectively.
It’s clear that the proportional and integral gains for the PI con-
troller at the transient period and at the effect of uncertainty (20 s)
are tuned on-line using the adaption mechanisms (IT2FLS and
IT2FNN). The IT2FNN is tuned the PI controller gains larger than
the IT2FLS which make the proposed controller is able to respond
to the system uncertainties compared with the API-IT2FLS.
5.5. Task 5: input disturbance
Fig. 20 shows the effect of adding random disturbance for the
control signal at time equal 20 s. for both controllers. It is clear that
the performance of the DC motor – generator system for the
proposed API-IT2FNN have made significantly better than the
performance for the API-IT2FLS. Fig. 21 shows that the MAE for the
proposed controller is lower than that obtained for the API-IT2FLS.
5.6. Task 6: simulated neglected dynamics
In this simulation task, we show the effect of neglected
dynamics by adding uncertainty values for the system parameters
to show the robustness of the proposed controller. Figs. 22 and 23
show the performance of the proposed API-IT2FNN and API-IT2FLS
for this task. It is clear that the performance of the proposed API-
IT2FNN is able to reduce the effect of neglected dynamics better
than the API-IT2FLS.
Tables 2 and 3 list the ISE and the RMSE values, respectively, for
the classical PI controller, the self-tuning PI controller (STPIC), the
API-IT2FLS and the proposed API-IT2FNN for all the above practical
results. As shown in Tables 2 and 3, the values of the ISE and the
RMSE for the proposed API-IT2FNN are lower than that obtained
for other controllers. So, the proposed API-IT2FNN is superior to
respond the uncertainties in the DC motor – generator system
rather than other controllers.
The computational complexity of the adaptive algorithm is
depended on the following: 1) the number of the updated para-
meters. 2) The structure of the adaptive algorithm (IT2FNN) (such
as the number of membership in each input). 3) The TR method,
which used in the layer 4. In this paper, we have reduced the
computational complexity of the adaptive algorithm by: 1)
selecting the lower level of the structure PI controller instead of
PID controller. 2) Selecting the simplified TR method, which is less
computational cost comparing to other methods [48].
6. Conclusion
In this paper, the adaptive PI controller based on the IT2FNN is
proposed. The proposed controller consists of a lower level PI
controller, which is the main controller and an upper level IT2FNN.
The IT2FNN provides a mechanism for tuning on-line the gains of a
PI controller. Back-propagation algorithm is used to train IT2FNN.
The Lyapunov theorem is used to derive the stability and con-
vergence of the IT2FNN. The proposed API-IT2FNN is implemented
using the Ardunio DUE Kit for controlling the DC motor – gen-
erator system. Six experimental tasks have been performed
including the step change in DC motor speed, the uncertainty due
to the measurement error in the motor speed, the uncertainty due
to an electric load, input disturbance and the effect of neglected
dynamics. The experimental results of the proposed API-IT2FNN
are compared with the results of the adaptive PI based on the
IT2FLS. The proposed controller based on the IT2FNN is able to
varying the PI gains larger than the IT2FLS. Three performance
indices; MAE, ISE and RMSE are measured for the proposed API-
IT2FNN and the API-IT2FLS for all experimental tasks and these
indices establish the superiority of the proposed API-IT2FNN to
respond to system uncertainties compared with the API-IT2FLS.
Thus, the methodology proposed in this study can be used to
realize a robust, practically realizable, adaptive PI controller cap-
able of controlling a real-life plant with system uncertainties with
acceptable closed-loop response.
The main contributions of this study are summarized as: 1)
proposing adaptive PI controller based on the IT2FNN. 2) Imple-
menting practically the proposed controller using the Arduino
DEU kit for controlling the nonlinear DC motor – generator system.

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