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Accepted Manuscript
Improving the Performance of Networked Control Systems with Time
delay and Data Dropouts Based on Fuzzy Model Predictive Control
Ahmad Sakr , Ahmad M. El-Nagar , Mohammad El-Bardini ,
Mohammed Sharaf
PII:
DOI:
Reference:
S0016-0032(18)30488-5
https://doi.org/10.1016/j.jfranklin.2018.07.012
FI 3557
To appear in:
Journal of the Franklin Institute
Received date:
Revised date:
Accepted date:
15 February 2018
22 July 2018
31 July 2018
Please cite this article as: Ahmad Sakr , Ahmad M. El-Nagar , Mohammad El-Bardini ,
Mohammed Sharaf , Improving the Performance of Networked Control Systems with Time delay and
Data Dropouts Based on Fuzzy Model Predictive Control, Journal of the Franklin Institute (2018), doi:
https://doi.org/10.1016/j.jfranklin.2018.07.012
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ACCEPTED MANUSCRIPT
Improving the Performance of Networked Control
Systems with Time delay and Data Dropouts Based on
Fuzzy Model Predictive Control
Ahmad Sakr a, Ahmad M. El-Nagar b, Mohammad El-Bardini c and Mohammed Sharaf d
Department of Control Engineering, High Institute of Eng., Belbies, Egypt.
b,c,d
b,c,d
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a
Department of Industrial Electronics and Control Engineering
Faculty of Electronic Engineering, Menoufia University, Menouf, 32852, Egypt.
a
b
eng.ahmed.sakr.1988@gmail.com
ahmed.elnagar@el-eng.menofia.edu.eg
c
dralbardini@el-eng.menofia.edu.eg
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d
msharaf597@gmail.com
Abstract
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This paper proposes a fuzzy model predictive control (FMPC) combined with the modified
Smith predictor for networked control systems (NCSs). The network delays and data dropouts
are problems, which greatly reduce the controller performance. For the proposed controller, the
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model of the controlled system is identified on-line using the Takagi – Sugeno (T-S) fuzzy
models based on the Lyapunov function. There are two internal loops in the proposed structure.
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The first is the loop around the FMPC, which predicts the future outputs. The other is the loop
around the plant to give the error between the system model and the actual plant. The proposed
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controller is designed for controlling a DC servo system through a wireless network to improve
the system response. The practical results based on MATLAB/SIMULINK are established. The
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practical results are indicated that the proposed controller is able to respond the networked time
delay and data dropouts compared to other controllers.
Keywords
Networked control systems; Model predictive controller; Smith predictor; DC servo system; T-S
fuzzy model
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1. Introduction
Networked control system (NCS) is a kind of control system in which its components like,
sensor, controller, and actuator are connected through communication channels or a control loop
is closed via a form of communication network [1, 2]. After the fast development of
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communication technology and entering it in many industrial applications as communication
channels, it is easy to monitor the most process at any time and from anywhere around the world
[3]. The advantages of the NCSs over traditional control, which have point to point, are
summarized as: 1) it is easy to reform the controlled system. 2) it reduces the cost and the
number of cables. 3) the installation becomes simple where it reduces the power requirements
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and the time spends for adding new sensors to the controlled system [4, 5]. These advantages of
the NCS made it more common in many applications such as, DC motor [6, 7], mobile robots
[8], manipulators [9, 10], mechanical systems [11], and vehicle engines [12].
The NCS is classified into two parts; the first is the control over the network and the other is
the control of the network. Our research focused on the control over the network, which deals
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with control design and control strategy to reduce the problems of the NCSs like the network
delays and data dropouts [13, 14]. The block diagram of the NCS is shown in Fig. 1, where all
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data transmission between the plant and the controller via communication network may be wired
or wireless. Therefore, the delays in the NCS resultant from two communication channels; the
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first is the communication channel between the controller and actuator, which is established to
send the control signal from the controller to the actuator. This induced delay from this channel
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is known as the forward delay ( Tca ). The other communication channel is between the sensor and
the controller, which is established to send the measured outputs to the controller. The induced
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delay in this channel is known as the feedback delay ( Tsc ). When the network time delays
increase due to the limited capacity and network channels, the performance of the fast plants
decreases and the controlled system may become unstable [4]. A part of the feedback signal from
the sensor and the control signal from the controller may be lost when the data is transmitted via
communication network. This problem is called the data losses or data dropout. There are many
types of research, which are proposed to overcome the problems of the NCS [15 - 19].
Networked predictive control is proposed to overcome the problems of random network delays
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for NCSs [17]. A robust PID controller based on iterative suboptimal algorithms is proposed for
the multivariable NCS [18]. The problem of data losses for the NCSs is handled using networked
predictive control [19].
Model predictive control (MPC) is a form of control, which gives high-quality control
performance compared to traditional controller under changing conditions for many industrial
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problems [20]. The MPC strategy minimizes a predetermined objective function to obtain its
control input at each sampling period. After calculating a sequence of predicted control actions,
the first one is implemented, and the control input is updated after each sampling time with the
new measurements [20, 21]. The MPC has some features that distinguish it from another control
strategy, which makes it spread extensively in industrial systems. In addition, it has the ability to
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handle constraints and the ability to control the MIMO processes [20, 21]. The MPC is proposed
for handling the problems of the NCSs [22 – 30]. The MPC is proposed to overcome the random
delays in the NCSs [22 - 25]. The strategy of predictive control, which known as a dynamic
matrix control (DMC) is proposed for handling the networked time delays [26]. Networked
predictive control (NPC) is proposed for solving the problems of NCSs [27, 28]. The stability
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and optimality for the NCSs based on the MPC are introduced in [29]. The MPC is proposed for
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Controller
Delay
Tca
Network
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handling the problem of data dropout in NCS [30].
Delay
Actuator
Plant
Sensor
Tsc
Fig. 1: Block diagram of a typical NCS.
One of the constraints, which may affect the performance of the MPC as a controller for
nonlinear system is the internal model, which is used with MPC to predict the future output over
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the prediction horizon. If the model is designed offline without considering the possible presence
of disturbance in the process, may lead to inaccurate prediction output and also the decreases the
controller performance. In recent years, the T-S fuzzy model concept has been proposed for
modeling the complex nonlinear systems [31]. In the T-S fuzzy models, the consequents of the
rules are a mathematical expression, which can be any linear functions of any variables [31]. The
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T-S fuzzy model has many successful applications in the industrial process, especially in the
modeling and the control of plant with incomplete knowledge [32 - 34].
In [35], the FMPC is designed for controlling a laboratory tank system where the antecedent
parameters of the T-S fuzzy model are obtained based on the fuzzy clustering method. However,
the consequent parameters are obtained based on the least square method. In [36], the FMPC
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based on the state-space model for the T-S fuzzy model is introduced. The parameters of the T-S
fuzzy model have been estimated using the least square algorithm. In [37], the FMPC is applied
for controlling the distillation column system where the parameters of the T-S fuzzy model are
estimated based on the least square algorithm. The T-S fuzzy model based on the generalized
predictive control (GPC) is proposed for controlling the vehicle system [38]. The parameters of
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the T-S fuzzy models are updated based on the gradient algorithm. In [39], the T-S fuzzy model
predictive control is proposed for controlling the speed of the electrical vehicle where the T-S
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fuzzy model is represented by continuous time state-space model. The parameters of the T-S
fuzzy model are obtained using the local linearization method where the fuzzy system is only an
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approximation of the original nonlinear system. A disadvantage of this method is that there are
no longer guidelines on how to choose the linearization points, or how many linearization points
should be chosen. In [40], the FMPC is proposed for nonlinear system where the initial values of
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the antecedent parameters of the T-S fuzzy model are obtained using the particle swarm
optimization. Then, the gradient algorithm is used for tuning these parameters. However, the
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consequent parameters are obtained based on the sliding-window kernel recursive least square
algorithm. All the previous approaches [35-40] have been used off-line identification based on
various algorithms to identify the nonlinear systems. The stability analysis for the FMPC have
been studied based on various approach such as the linear matrix inequalities [36] and nonquadratic Lyapunov function [39].
In this paper, the FMPC combined with the modified Smith predictor for NCSs is proposed.
The internal model of the MPC is identified on-line based on a T-S fuzzy model. The proposed
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controller is able to respond the problems of the NCS due to the structure of the modified Smith
predictor, which compensates the network delays and improves the system performance with
increasing the networked time delays and data dropouts. The FMPC based on the modified Smith
predictor is designed for controlling the speed of a DC servo system through a wireless network.
The practical results are indicated that the proposed FMPC with the modified Smith predictor is
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able to respond the networked time delays and data dropouts better compared to other
controllers.
The main novelties and contributions of this paper are summarized as: 1) Proposing the
FMPC where the internal model is identified on-line based on the T-S fuzzy model. A new
learning algorithm based on the Lyapunov stability theorem is developed to update the
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antecedent and consequent paramters of the T-S fuzzy model. The optimal value of the learning
rate for the adaption algorithm is obtained based on the Lyapunov function. 2) Studying the
stability analysis of the proposed FMPC based on the Lyapunov stability criterion. 3) The
FMPC is comined with the modified Smith predictor where the model that used in the internal
loops around the plant and the controller is identified on-line based on the T-S fuzzy model. 4)
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The proposed controller is suitable for on-line networked control systems where it implemented
for controlling a DC servo system via a network to overcome the problems of time delays and
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data dropouts. The performance of the proposed controller has achieved better performance
indices than other controllers, as shown in the practical results section.
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The rest of this paper is structured as: the nonlinear system modeling using T-S fuzzy
models is described in section 2. The model predictive control is presented in section 3. The
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modified Smith predictor is presented in section 4. Section 5, presents the FMPC based on the
modified Smith predictor. Section 6, presents the practical results followed by the conclusions
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and relevant references.
2. Nonlinear systems modeling using T-S fuzzy models
This section explains the method of estimating the nonlinear model of the process, since
many processes can be represented by the following model [41]:
y(k )  f y(k  1),, y(k  ny ), u(k  d  1),, u(k  d  nu )  n (k )
5
(1)
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Assume, the unknown plant with a single input u k  and a single output yk  , where n y , nu
are the order of the output and the input, respectively, d is the time delay of the plant and n k 
is a sequence of zero mean Gaussian white noise. In order to find a fuzzy system to model it, it
must be excited by an input while data  uk  , yk  , where k  1, 2, 3, ... is taken from the plant.
form [37]:
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As the plant is unknown, we can attempt to estimate it as a T-S fuzzy model with rules in the
Ri : IF y(k ) is A1i , y(k  1) is A2i and  and y(k  n y  1) is Ani then
y i (k  1)  a i ( z 1 ) y(k )  bi ( z 1 ) u (k  d )   (k ),
Ri represents
i  th
the
a i ( z 1 )  a1i  a2i z 1    ani y z
 ( n y 1)
fuzzy
rule,
N
is
the
number
of
(2)
rules,
and bi ( z 1 )  b1i  b2i z 1    bniu z ( nu 1) . The membership
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where
i  1, 2, 3,, N
function Aij (i th rule, j th universe) is given by:
 1  y (k  j  1)  ci 2 
j
 
 ( y (k  j  1))  exp   
i
 
2
j
 

(3)
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i
j
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where c ij and  ij are the mean and standard deviation for the Gaussian membership function,
respectively. Assume, the time delay of the system d is zero and we neglect the white noise  k  .
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The fuzzy system is described as:
yˆ (k  1)  [a11 y (k )    a1ny y (k  n y  1)  b11 u (k )    bn1u u (k  nu  1)]1 (k ) 
CE
[a12 y (k )    an2y y (k  n y  1)  b12 u (k )    bn2u u (k  nu  1)] 2 (k )   
(4)
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[a1N y (k )    anNy y (k  n y  1)  b1N u (k )    bnNu u (k  nu  1)] N (k )
where
i y (k ), y (k  1),, y (k  ny  1)  
i y (k ),, y (k  ny  1) 
  y(k ),, y(k  n
N
i 1
and
6
i
y
 1) 
(5)
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 1  y (k  j  1)  ci 2 
j
 
i y (k ),, y (k  ny  1)    exp   
i
 
2

j 1
j

 

n
(6)
2.1 Learning algorithm
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The antecedent parameters c ij and  ij ; the consequent parameters a ij and b ij for the T-S
fuzzy model are learned on-line using the Lyapunov function. Let, the Lyapunov function is
chosen as:
V (k ) 

1
em (k )2   (k )2
2

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em k   yˆ (k  1)  y(k  1)
(7)
(8)
where V (k ) is a positive definite function, em (k ) is the model error signal as defined in Eq. (8),
yˆ (k  1) is the T-S fuzzy model output and y(k  1) is the desired output.  (k ) represents the
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parameters vector which include all the T-S fuzzy model parameters.
Theorem 1: For the Lyapunov function, V (k ) 


1
em (k )2   (k )2  0 , the condition
2
 (k )  
 em (k ) 

  (k ) 
 (k )  em (k ) 
 e (k ) 

1   m
  (k ) 
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V (k )  0 is satisfied if and only if the parameter adjustment rule is obtained as:
2
(9)
Proof 1: Let the change in the Lyapunov function is defined as:
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V (k )  V (k  1)  V (k )
1
 em (k  1)  em (k ) em (k  1)  em (k )    (k  1)   (k )  (k  1)   (k ) 
2
Let em (k )  em (k  1)  em (k ) and  (k )   (k  1)   (k ) , Eq. (10) can be rewritten as:
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(10)
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  e (k ) 2 

 e (k )  
1
2
   2 (k ) (k )  em (k ) m
 
V (k )   (k )  1   m
   (k )  
2


(
k
)

 




(11)
For a very small change, Eq. (11) can be rewritten as:
  e (k ) 2 

 e (k )  
1
 (k ) 2 1   m
   2 (k ) (k )  em (k ) m
 
   (k )  
2


(
k
)

 




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V (k ) 
(12)
To guarantee the stability, the second condition for the stability is V (k )  0 . Then:
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  e (k ) 2 

 e (k )  
1
2
   2 (k ) (k )  em (k ) m
   0
V (k )   (k )  1   m
   (k )  
2
 (k )  






Eq. (13) can be rewritten as:
  e (k ) 2 

 e (k )  
1
1
2

   2 (k ) (k )  em (k ) m
    z


V (k ) 
 (k ) 1   m


2
 (k ) 
2
  (k )  

 


(13)
(14)
positive value. So, Eq. (14) becomes:
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In order to satisfy the condition in Eq. (13), the value of z in Eq. (14) must have a zero or
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  e (k ) 
1
2

V (k )   (k )  1   m
   (k ) 
2


2


  2 (k ) (k )  e (k ) em (k )    z   0
m
  (k )  



  


(15)
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Consider a general quadratic equation:
a x2  b x  c  0
(16)
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The roots of Eq. (16) are given as:
 b  b 2  4a c
 b  b 2  4a c
r1 
and r2 
2a
2a
(17)
By comparing Eq. (15) with Eq. (16), it is clear that  (k ) acts as x in Eq. (16) and the values
of a, b and c in Eq. (16) are given as:
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1   e (k ) 

a  1   m
2    (k ) 

2


 e (k ) 
1
 , b   (k )  em (k )  m  and c  z
2

  (k ) 

In order to have a single solution for Eq. (16), the term
b 2  4a c must to be equal zero. After
b 2  4a c and squaring both sides, we have
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putting the values of a, b and c in
(18)
2
2

 em (k )     em (k )  
   1  
  z  0
 (k )  em (k )

  (k )      (k )  
So, z is given as:
2
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
 em (k ) 

 (k )  em (k )
  (k ) 

z
2
 e (k ) 

1   m
  (k ) 
Since z  0 , which means
(19)
(20)
2
(21)
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
 em (k ) 

 (k )  em (k )
  (k ) 

0
2
 em (k ) 

1  
  (k ) 
 (k )  
b

2a
AC
CE
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Then, the unique root of Eq. (15) will be given as:
 em (k ) 

  (k ) 
 (k )  em (k )
 e (k ) 

1   m
  (k ) 
2
This completes the proof.
Then, the updating parameters for the T-S fuzzy model are given as:
9
(22)
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 (k  1)   (k )     (k )
  (k )   
 e (k ) 
1   m 
  (k ) 
(23)
2
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where   is the learning rate constant.
 em (k ) 

  (k ) 
 (k )  em (k )
Based on the above theorem, the updating equations for the T-S fuzzy model parameters are
defined as follow:
AN
US
 e (k ) 

c ij (k )  em (k ) mi
 c (k ) 
 j

c ij (k  1)  c ij (k )   c
2
 e (k ) 

1   mi
 c (k ) 
 j

where
(24)
ED
M




i
 y (k  j  1)  c i (k ) 
em (k ) 
yˆ (k  1)  yˆ (k  1)
j
 (25)

 N
. i  y (k ),, y (k  n  1) 
2
i
i




c j (k )

(
k
)
j


  i  y (k ),, y (k  n  1)  
 i 1

where
CE
PT
 ij (k  1)   ij (k )   


 em (k ) 

i



(
k
)
 j

2
 e (k ) 

1   mi
  (k ) 
 j

 ij (k )  em (k )
(26)
AC




i
 y (k  j  1)  ci (k ) 2 
em (k ) 
yˆ (k  1)  yˆ (k  1)
j
 (27)

 N
. i  y (k ),, y (k  n  1) 
3
i
i




 j (k )

(
k
)
j


  i  y (k ),, y (k  n  1)  
 i 1


10



ACCEPTED MANUSCRIPT
 e (k ) 

a ij (k )  em (k ) mi
 a (k ) 
j
i
i
a


a j (k  1)  a j (k )  
2
 e (k ) 

1   mi
 a (k ) 
 j

(28)
CR
IP
T
where




em (k )  i  y (k ),, y (k  n  1)  

y (k  j  1)

a ij (k )  N



y
(
k
),

,
y
(
k

n

1
)
 i

 i 1

(29)
AN
US
 e (k ) 

b ij (k )  em (k ) mi
 b (k ) 
 j

b ij (k  1)  b ij (k )   b
2
 e (k ) 

1   mi
 b (k ) 
 j

M
where
(30)
PT
ED




em (k )  i  y (k ),, y (k  n  1)  

u (k  j  1)

bij (k )  N
  i  y (k ),, y (k  n  1)  
 i 1

(31)
CE
2.2 Convergence of the T-S fuzzy model
In this section, some convergence theorems are developed to derive appropriate learning
AC
rates adaptively for the T-S fuzzy model. Assume a Lyapunov function can be defined as:
1 2
v(k )  em (k )
2
(32)
The change in the Lyapunov function is obtained as:
v(k )  v(k  1)  v(k ) 
11

1 2
2
em (k  1)  em (k )
2

(33)
ACCEPTED MANUSCRIPT
The error difference of T-S fuzzy model can be represented as:
 e (k ) 
em (k  1)  em (k )  em (k )  em (k )   m  
  
T
a ij
bij ] and the parameter vector change (  ) is obtained from the update
rules of Eq. (23) as:
 ( k )    em ( k )
CR
IP
T
where   [cij  ij
(34)
 em (k ) 

  ( k ) 
 (k )  em (k ) 
2
(35)
AN
US
 e ( k ) 

1   m
  ( k ) 
 y ( k ) 

 (k )  em (k ) 
  ( k ) 
   em ( k )
M
Let S  (k ) is defined as:
 y ( k ) 

1  
  ( k ) 
PT
 y (k ) 

  (k ) 
 (k )  em (k ) 
ED
S  (k ) 
2
 y (k ) 

1  
  (k ) 
2
(36)
CE
Eq. (35) can be rewritten as:

AC
Theorem 2 : Let     c  
 (k )    em (k ) S  (k )
(37)
 a  b  be the learning rates of the T-S fuzzy model. Then, the
asymptotic convergence of the T-S fuzzy model is guaranteed if   are chosen to satisfy:
0   
2
 y (k )  
   S (k )
T
in which  is the T-S fuzzy model parameters c ij ,  ij , a ij and b ij .
12
(38)
ACCEPTED MANUSCRIPT
Proof 2: From Eqs. (33) and (34), the change in the Lyapunov function is:
1


v(k )  em (k ) em (k )  em (k )
2


(39)
CR
IP
T
T
T


1  e (k ) 


 e (k ) 
v(k )   m    em (k ) S  (k ).em (k )   m    em (k ) S  (k )
2   


  


(40)
Since for the T-S fuzzy model em (k )    y(k )  , we obtain:
T
T


1
 y (k )  
  y ( k ) 



v(k )  em (k ) 
S
(
k
)
e
(
k
)

e
(
k
)

S
(
k
)
m
m







2








1 2
v(k )   em (k ) 
2
AN
US

T
 y(k )   

   S (k )  2  

where
1
 y(k ) 
T

T
 y(k )   
2
   S (k )    em (k )

 y(k ) 
T

(42)
(43)
ED
M
   
S  (k ) 2    
S  (k ) 




2










(41)
If   0 , then v(k )  0 is satisfied. Thus, the asymptotic convergence of the T-S fuzzy model
PT
is achieved, and Eq. (38) is obtained. This completes the proof.
3. Model predictive control
CE
3.1 Predictive control law
The MPC depends primarily on the accuracy of the process model selection in one of the
AC
different ways, such as system identification or fuzzy models [42]. So that it can predict the
future results of the output properly and give a much better system response. Therefore, it is able
to solve many of the problems that appear in the industry, clearly the most important one that this
research cares, network delay and data dropouts. The T-S fuzzy model that of the form in Eq. (2)
can be rewritten as follows:
A( z 1 ) y(k )  B( z 1 ) u(k  1)  n(k )
13
(44)
ACCEPTED MANUSCRIPT
where
A( z 1 )  1  A1 z 1  An y z
n y
,
B( z 1 )  1  B1 z 1  Bnu z  nu
and
n(k )
is the
disturbances model that described by:
n(k )  C ( z 1 )
e( k )

(45)
CR
IP
T
The plant parameters are obtained every instance with the T-S fuzzy model. Therefore, the
model of predictive control is not estimated off-line, but on-line using the T-S fuzzy model,
which makes the model is more accurate. Assume the polynomial C ( z 1 )  1 and   1  z 1 . Eq.
(44) can be rewritten as:
e(k )
1  z 1
AN
US
A( z 1 ) y(k )  B( z 1 ) u (k  1) 
(46)
where e(k ) is the disturbance or noise with zero mean.
To obtain the j step prediction of y(k  j ) , the following identity must be considered:
M
1  E j ( z 1 ) A( z 1 )  z 1Fj ( z 1 ), deg E j  j  1, deg Fj  ny ,
Gj  E j B
(47)
ED
where E j and F j are polynomials uniquely defined and the prediction interval j is given.
Multiplying Eq. (46) by E j  z j and after some mathematical operations, we obtain the multi-
PT
step ahead predictor of y(k ) as:
AC
where
CE
yˆ (k  j | k )  G j ( z 1 ) u(k  j  1)  Fj ( z 1 ) y(k )
G j ( z 1 )  E j ( z 1 ) B( z 1 ) 
B( z 1 )[1  z  j Fj ( z 1 )]
A( z 1 )
(48)
(49)
The main target of an objective function is to make future outputs ( y(k  j ) ) on considered
horizon follows the reference signal ( wr (k ) ). Therefore, there are different cost functions for
obtaining the control law, which depend on various MPC algorithms. The GPC algorithm
depends on minimizing the objective function J over receding prediction horizon:
14
ACCEPTED MANUSCRIPT
J ( N1 , N 2 , N u ) 
N2
Nu
l  N1
l 1
 (l )[ yˆ (k  l )  wr (k  l )]2   (l )[u(k  l  1)]2
(50)
where N1 , N 2 and N u are the minimum, maximum prediction horizons, and control horizon,
respectively,  (l ) and (l ) are the weighting sequence, where the choice of these parameters is
described in details in [43 - 45]. Longer horizons tend to produce more aggressive control action,
CR
IP
T
more overshoot, and faster response. As presented in [46], when  (l ) =1 is the case where there
are no constraints. Then, the j th steps ahead predictor y(k  j ) can be expressed in the vector
form as:
AN
US
y(k  j)  G u(k )  F y(k )  H u(k 1)
(51)
and the corresponding control law is given by [43, 45]:
u(k )  d T (wr  y0 )
(52)
where wr  wr (k  1)  wr (k  N2 ) , y0  y0 (k  1 k )  y0 (k  N2 k ),
 g0
 g
 1
G 

 
 g N 1
 2
PT
ED
d T  1,0,0(GT G  I )1 GT and
M
T

g N 2 2











g N 2  Nu 
0
0

g0
CE
Here, the step response coefficient is g j for the model and if the future control increments are
AC
all zeros then y 0 (k  j k ) is the free response [30].
3.2 Stability analysis
The T-S fuzzy model has been shown stable where the updating parameters are derived via
the Lyapunov stability theorem as shown in Theorem 1 and 2 (Section 2). The obtained control
low (Eq. (52)) has zero steady-state error due to   1  z 1 which introduced into the Eq. (46) in
order to provide an integral action and so, the steady-state error is eliminated. The following
theorem state that the closed loop system based on the FMPC is stable.
15
ACCEPTED MANUSCRIPT
Theorem 3: Let that the T-S fuzzy model (Eq. (4)) is controlled by Eq. (52), the estimated
parameters c ij ,  ij , a ij and b ij are bounded, all set-points are constant ( wr (k )  wr ) and the
steady-state error has zero ( lim j E{wr  y(k  j )}  0) . Then the closed loop is stable.
Proof 3: Assume a Lyapunov function is chosen as:
n
(k )  lim n  E{wr  y(k  j  1)}
CR
IP
T
2
j 1
The change in the Lyapunov function is defined as:
(k )  (k  1)  (k )
n
n
 lim n  E{wr  y (k  j )}  lim n  E{wr  y (k  j  1)}
2
(54)
j 1
AN
US
j 1
2
(53)
Eq. (54) can be rewritten as:
n
(k )  lim n  E{wr  y (k  j  1)}  E{wr  y (k )}  lim j E{wr  y (k  j )}
2
j 1
n
2
 lim n  E{wr  y (k  j  1)}
j 1
  wr  E{ yˆ (k )   (k )}
  ( wr  yˆ (k )) 2
(55)
ED
2
M
2
2
PT
From Eq. (55), we find that (k )  0 is satisfied. Thus, the closed loop system based on
CE
the FMPC is stable.
4. Smith predictor
In this section, we introduce the structure of the Smith predictor. The simplified block
AC
diagram of the NCS is shown in Fig. 2. X (s) is the input signal and C (s) is the output signal.
Tca is the networked time delay between the controller and the actuator, Tsc is the networked
time delay between the controller and the sensor, Gc (s) is the controller transfer function (TF)
and the GP (s) is the real system TF without delay time. The overall TF for the closed loop NCS
is obtained as: (where Tc  Tca  Tsc )
16
ACCEPTED MANUSCRIPT
E ( s)  X ( s)  C ( s) e
Tsc s
(56)
T s
Gc ( s) G p ( s) e ca
C ( s)

X ( s) 1  Gc ( s) G p ( s) e  (Tca Tsc ) s
(57)
T s
(58)
CR
IP
T
Gc ( s) G p ( s) e ca
C ( s)

X ( s) 1  Gc ( s) G p ( s) e  Tc s
It is clear from Eq. (58), there is a time delay in the denominator of the TF, which affects on the
performance of the system.
E (s)
Gc (s)
-
e
Tca s
G p (s)
C (s)
AN
US
X (s) +
Tsc s
M
e
Fig. 2: Simplified block diagram of NCS.
ED
4.1 Traditional Smith predictor
The traditional Smith predictor for the NCSs is shown in Fig. 3, which contains an internal
PT
loop around the TF of the controller. This internal loop is the main part of the Smith predictor,
which makes compensation of time delays [47, 48]. G pm (s) is the nominal model of the system
CE
without the delay time. The TF of the overall system ( Gs (s) ) when the Smith predictor is
AC
involved, is obtained as:
E(s)  X (s)  C2 (s)  C1 (s)  C (s) e
Tsc s
(59)
If G pm (s)  G p (s) , Tca  Tsc and Tcom  Tc , then:
E ( s)  X ( s)  C ( s)
17
(60)
ACCEPTED MANUSCRIPT
Gc ( s)
1  (1  e
) Gc ( s) G pm ( s)
(61)
G p ( s) Gc ( s) Tca
C ( s)

e
X ( s ) 1  G p ( s ) Gc ( s )
(62)
Gs ( s) 
Tcom
CR
IP
T
From Eqs. (58) and (62), the Smith predictor is able to remove the time delay. In this
traditional Smith predictor, there are some constraints like Tca  Tsc and Tc  Tcom which must be
achieved to improve the controller stability and compensate the delays. These constraints make
the controller must know the delays in the network and need an extra part to compensate these
X (s) +
-
+
+
C2 ( s)
AN
US
delays like an extra network.
E (s) Controller
Gc (s)
e Tcom
C1 ( s)
e
Tca s
G p (s)
C (s)
Delay
G pm (s)
e
Tsc s
Delay
ED
M
Delay
U (s)
Fig. 3: Traditional Smith predictor for NCSs.
PT
4.2 Modified Smith Predictor
CE
Fig. 4, shows the block diagram of the modified Smith predictor. It contains two internal
loops; the first is around the controller and the other with delay Tpm around the real plant. This
AC
structure improves the function of the Smith predictor where there is no need to know the
network delay. Therefore, there are no constraints for forward delay Tca and feedback delay Tsc .
The closed loop TF for the plant with delay T p is given as follows:
G p ( s) Gc ( s) Tca s T p s
C ( s)

e
e , G p ( s)  G pm ( s) and Tp  Tpm
X ( s) 1  G p ( s)Gc ( s)
18
(63)
ACCEPTED MANUSCRIPT
where the prediction value of T p is represented by Tpm . Eq. (63) shows that the modified Smith
predictor is able to respond the networked time delay. As shown in Fig. 4, the controller is
dependent on the feedback signal, which obtained from the system model G pm (s) at the internal
loop around the controller. So, when there is a data dropout in the measured output C (s) , the
CR
IP
T
controller will respond to this data dropout due to the internal loop around the controller.
The modified Smith predictor is dependent on the model that obtained for the controlled
system. So, in this study the system model around the plant and around the controller is
identified on-line based on the T-S fuzzy model that described in section 2.
-
+
C 2 ( s)
E (s) Controller
Gc (s)
C1 ( s)
U (s)
e
Tca s
G p ( s )e
AN
US
X (s) +
T p s
C (s)
Delay
G pm (s)
G pm ( s)e
M
e
T pm s
-
+
Tsc s
Delay
ED
Fig. 4: Modified Smith predictor for NCSs.
PT
5. FMPC based on modified Smith predictor
Fig. 5, shows the overall structure of the proposed controller, which combines the FMPC,
CE
which described in sections 2 and 3 with the modified Smith predictor, which described in
section 4. In Fig. 5, the reference signal is wr (k ) , u (k ) is the manipulated variable, y(k ) is the
AC
measured output from the controlled system, yˆ1 (k ) and yˆ 2 (k ) are predicted outputs of the
controlled system model and cm (k ) is the result of Smith predictor where:
cm (t )  y(k )  yˆ 2 (k )  yˆ1 (k )
(64)
The FMPC block has two inputs; wr (k ) and cm (k ) and one output u (k ) . There is an
internal loop with the model of the system around the FMPC. There is another loop around the
system to give the error between the model of the system and the actual system. As shown in Fig.
19
ACCEPTED MANUSCRIPT
5, the system model at the internal loops is identified on-line based on the T-S fuzzy model. If
there are no disturbances, or noise and the model of the system are obtained accurately, the error
between the model and actual system will equal to zero ( y(k )  yˆ 2 (k )  0 ). It is clear that the
input signal cm (k ) for the proposed controller, which combines the modified Smith predictor
with the FMPC is not affected with network delay and data dropout. In addition, it depends on
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the plant model, which is designed using the T-S fuzzy model. This structure makes the
controller like double predictor one with Smith predictor and the other in the FMPC block.
FMPC
wr (k )
u (k )
MPC
cm (k )
+
Adaptive T-S fuzzy
model
R1
+
yˆ 2 (k )
-
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cˆ(k )
T-S Fuzzy
Model
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yˆ1 (k )
y(k )
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Rn
-
System
T-S Fuzzy
Model
On-line
Adaption
+
Tca
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+
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Fig. 5: Overall structure of the proposed controller (FMPC with modified Smith predictor).
The computations at each iteration k can be summarized as the following steps:
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Step 1: Set wr (k ) , n y , nu , d , N , N1 , N 2 and N u as described in section 3.
Step 2: Measure the current system output y(k ) , obtain the output of the T-S fuzzy model
and the difference between them as described in section 3 and 4.
Step 3: Update c ij ,  ij , a ij and b ij of the FMPC using Eqs. (24), (26), (28) and (30) as
described in section 2.
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Step 4: Define the learning rate based on Eq. (38) as described in section 2.
Step 5: Determine the change of control signal u (k ) from Eq. (52). Then send the control
signal u (k ) to the controlled system.
Step 6: Repeat steps 2-5.
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6. Practical results
In this section, the proposed FMPC with modified Smith predictor is designed for
controlling a DC servo system through a wireless network.
6.1 Model validation
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In this section, we show the model validation using the T-S fuzzy model based on the
Lyapunov function, which described in Section 2 to clarify the robustness of the proposed T-S
fuzzy model to identify the DC servo system. Fig. 6, shows the input test signal, which applied
practically to the DC servo system and also applied to the T-S fuzzy model. Fig. 7, shows the
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measured output and the T-S fuzzy model output. It is clear that the T-S fuzzy model based on
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the Lyapunov function is able to identify the DC servo system.
Fig. 6: Input signal.
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Fig. 7: Measured and T-S fuzzy outputs.
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6.2 Experimental setup
Fig. 8, shows the experimental setup used in this study. The network consists of two PCs.
The PC1 is used to implement the proposed FMPC with the modified Smith predictor that shown
in Fig. 5. The implementation for the proposed controller is done based on the
MATLAB/SIMULINK as shown in Fig. 9. As shown earlier in Fig. 5, the internal loop around
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the DC servo system is implemented using the PC2. The implementation of the internal loop is
done based on the MATLAB/SIMULINK as shown in Fig. 10. The PC2 sends and receives the
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signal to and from the controller (PC1) through the network and deliver the signal to the servo
system. The Arduino Mega Kit is connected to PC2 through USB port and works as interface
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module between the DC servo system and the PC2. The PC2 receives the tacho signal through
analog pin (A0) of the Arduino Kit and sends the feedback signal to the PC1 through the
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network. In addition, it receives the control signal from PC1 and delivers it to the servo system
through PWM pin (A3) of the Arduino Kit. The communication network between the two PCs is
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executed through Wi-Fi.
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Fig. 8: Experimental setup.
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Fig. 9: Implementation of the FMPC with modified Smith predictor (PC1).
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Fig. 10: Implementation of internal loop around the DC servo system (PC2).
6.3 Experimental tasks
In order to show the improvements of the proposed FMPC with modified Smith predictor,
the practical results are compared with the results of the FMPC. The sample period for the DC
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servo system is 0.01sec, the control horizon is 2, the prediction horizon is 10, the weight of
control signal is 7 and the weight of output is 1. The reference speed is 1000 RPM. The main
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objective of the proposed controller is improving the performance of the system that connected
via a wireless network to reduce the effect of the network time delays and the data dropouts.
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The practical results are performed based on the practical network consists of two PCs as
shown in Fig. 8. In all experimental tasks, there are data dropouts due to the practical
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communication network, where the packet loss occurs when one or more packets of data
traveling across a computer network fail to reach their destination. Five practical tasks are
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considered to show the effect of delays and load to the system. Three performance indices are
employed as quantitative measures for comparing the proposed controller and the FMPC; the
mean absolute errors (MAE), integral of square error (ISE) and the root mean square of errors
(RMSE), which are defined as [49]:
MAE 
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N
1
| e(k ) |

k 1
N
(65)
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
ISE   e 2 (t ) . dt
(66)
0
RMSE 
N
1
(e(k ))2

k 1
N
(67)
Task 1: Variable delays in NCSs and load effect.
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This task shows the influence of variable time delays and an external load. The time-varying
delay is performed using a block in MATLAB/SIMULINK, which have two parameters;
minimum and maximum, which added to the real network delay in order to simulate the
increasing of real network delay. The external load is presented using the additional part in our
system called the magnetic load. This magnetic load has a potentiometer to adjust the value of
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the load and keep the load level. In this task, there is a variable delay with values between (0.01
and 0.32) sec. and a 25% load effect, which is inserted at 6 sec. The response of the proposed
controller and the FMPC for this task is shown in Fig. 11. The performance of the proposed
FMPC with modified Smith predictor has a lower overshoot and settling time than that obtained
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for the FMPC.
Task 2: Normal Network delays in NCSs with a drop in the network and load effect.
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This practical experiment shows the influence of the normal network delays with drop or cut
in the network between the two PCs form instant 4 sec to instant 6.5 sec where the external load
is inserted at instant 6 sec before the network reconnected between the two PCs. Fig. 12, shows
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the response of the DC servo system for this task. The response of the proposed FMPC with the
modified Smith predictor has achieved a good performance without an overshoot after the
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connection backs between the two PCs because there is an internal loop with a system model
around the controller. The response for the FMPC has an overshoot after the connection backs
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between the two PCs.
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Fig. 11: Response of the DC servo system in NCS (Task 1).
Fig. 12: Response of the DC servo system in NCS (Task 2).
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Task 3: Normal network delays in NCSs with a cut in the network.
This practical task shows the effect of the normal network delays with a cut in the network
between the two PCs from the starting point to 0.35 sec. Fig. 13, shows the response of the DC
servo system for this task. In this case, we display two control signals; the first is the control
signal at PC1 (before sending to the system via the network) and the other is the controller signal
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at PC2 (after sending to the system via the network). The delay between two control signals is
due to the cut in the network. It is clear that the response of the proposed FMPC with the
modified Smith predictor has a settling time and an overshoot lower than that of the FMPC.
Therefore, the FMPC with the modified Smith predictor is able to respond the problem of data
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dropouts in the NCSs better than the FMPC.
Fig. 13: Response of the DC servo system (Task 3).
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Task 4: Variable delays in NCSs with data dropouts.
This practical task shows the effect of variable delays in NCSs with weakening the
connection between the two PCs to increase the dropout of the signal. This can be achieved by
increasing the distance between the two PCs from 8 sec to 14 sec. The variable delays are
between (0.01 and 0.34) sec. Fig. 14, shows the response of the DC servo system for this task. It
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is clear that the performance of the FMPC with the modified Smith predictor has a small
overshoot and settling time. However, the response for the FMPC has a large settling time, an
overshoot and an oscillation around the set-point after weakening the connection between the
two PCs. Therefore, the proposed FMPC with the modified Smith predictor is superior to
Fig. 14: Response of the DC servo system (Task 4).
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respond the effect of the data dropouts rather than the FMPC with increasing network delay.
Task 5: Variable delays in NCSs under variable set-point
This practical task shows the effect of the step change with a variable delay. For the first
step, the amplitude is 1000 RPM and the variable delay is between (0.01 and 0.2) sec. For the
second step, the amplitude is 800 RPM. For the final step, the amplitude is 1200 RPM. Fig. 15,
shows the response of the DC servo system for this task. The response of the proposed FMPC
with the modified Smith predictor is better than the FMPC with variable delay and step change.
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Therefore, the FMPC with the modified Smith predictor is superior to respond the set-point
changes rather than the FMPC.
As shown above in all experimental tasks, the proposed FMPC with the modified Smith
predictor is able to respond to the problems of the NCSs due to the following: 1) the structure of
the modified Smith predictor, which dependent on the T-S fuzzy model. This structure is able to
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compensate the network time delay and the data dropouts. 2) the internal model of the FMPC,
which is identified on-line based on the T-S fuzzy model using the Lyapunov function. To show
the ability of the proposed controller to respond the problems of the NCSs, the comparisons
between the proposed controller and other controllers based on other structures of the Smith
predictor are done. Tables 1 - 3 list the MAE, RMSE, and ISE values for the FMPC, PID
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controller with the Smith predictor [50, 51], the fuzzy PI (FPI) controller with the Smith
predictor [51] and the proposed FMPC with the modified Smith predictor for all the above
experimental tasks. The values of the MAE, RMSE, and ISE for the proposed FMPC with the
modified Smith predictor are lower than that obtained for other controllers. Therefore, the
proposed FMPC with the modified Smith is preferable than other controllers where it improves
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the performance of the system and it is able to overcome the problems of networked time delays
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and data dropouts.
Fig. 15: Response of the DC servo system (Task 5).
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Table 1: MAE values for all practical tasks.
Task 2
Task 3
Task 4
Task 5
FMPC
0.1194
0.1265
0.3076
0.1122
0.085
PID with Smith [47]
0.1061
0.1166
0.2313
0.0812
0.0611
FPI with Smith [51]
0.1033
0.1121
0.2231
0.0787
0.0581
MPC with Smith
0.1011
0.0998
0.2077
0.0678
0.0551
Proposed controller
0.0925
0.0957
0.2064
0.0611
0.0530
Task 4
Task 5
0.5340
0.5781
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Task 1
Task 1
Task 2
Task 3
FMPC
0.4953
0.4518
0.8377
PID with Smith [47]
0.4711
FPI with Smith [51]
0.4621
MPC with Smith
0.4431
Proposed controller
0.4200
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Table 2: RMSE values for all practical tasks.
0.7822
0.4612
0.5314
0.4233
0.7512
0.4551
0.5193
0.4112
0.7231
0.4038
0.5032
0.3878
0.6887
0.3480
0.5029
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0.4411
Table 3: ISE values for all practical tasks.
Task 2
Task 3
Task 4
Task 5
2.4557
2.0430
3.5160
4.5660
5.0171
PID with Smith [47]
2.2301
1.8123
2.8320
2.4155
4.2113
FPI with Smith [51]
2.0311
1.7909
2.6540
2.1909
4.0480
MPC with Smith
1.8541
1.6055
2.6048
2.0145
3.8210
1.7658
1.5052
2.3763
1.9387
3.7962
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FMPC
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Task 1
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Proposed controller
7. Conclusions
This paper proposed the FMPC with the modified Smith predictor for NCSs. The proposed
controller is able to respond and overcome the problems of the NCSs; the time delays and data
dropouts. The structure of the modified Smith predictor compensates the network time delays.
The internal model for the FMPC is identified on-line based on the T-S fuzzy models using the
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Lyapunov function. In addition, the convergence analysis of the T-S fuzzy model is studied to
find the appropriate range for the learning rate. The stability of the FMPC is studied based on the
Lyapunov theorem to achieve the stability of the closed loop system. The structure of the
modified Smith predictor has two internal loops; the first is around the FMPC and the other is
around the system. The proposed FMPC with the modified Smith predictor is applied practically
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for controlling a DC servo system through a wireless network. The practical results are
established to show the robustness of the proposed controller. The proposed controller is
evaluated using five practical tasks including variable time delay with load effect, normal
network delays with a drop in the network and the load effect, normal network delays with a cut
in the network at the starting point, variable time delays with data dropouts and variable time
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delays with step-point change. To show the robustness of the proposed controller, the
experimental tasks are compared with the results of the PID controller with Smith predictor, a
fuzzy PI controller with Smith predictor, MPC with Smith predictor and the FMPC. The MAE,
ISE, and RMSE are measured for all the controllers to show the controller performance. The
obtained values for the proposed FMPC with modified Smith predictor are lower than that
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obtained for other controllers, which indicate that the proposed controller is able to overcome the
problems of the networked time delays and data dropouts. This method proved with the practical
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experimental superiority and the ability to ingress in many industrial applications that need
networks.
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