Design a Controller Based on Smith Predictor by Direct Synthesis
method for Speed Control DC Motor
ARUN YADAV, MANEESH KUMAR GUPTA
EE Dept., UNSIET, VBS Purvanchal University, Jaunpur, U.P., INDIA
https://orcid.org/0009-0002-0319-0535
Abstract: The modified Smith predictor is designed for a speed control DC motor. A speed controller of a
DC motor by selection of PID parameters using direct synthesis method. Here, the model of a DC motor is
considered as a second-order system for speed control. The PID and PD controller structure reported
recently decouples set-point tracking from disturbance rejection. This work aims to design a speed
controller of a DC motor by selecting of proper PID. Simulation examples show that improved servo and
regulatory performances are achieved by the proposed method as compared to the normal tune PID method
and also checked by perturbed performance. When used for regulatory/servo purposes, a controller
optimized for servo/regulatory application significantly degrades performance.
Keywords: PID Controller, Direct Synthesis Method, Smith predictor, maximum sensitivity, Speed
Control DC Motor.
Received: March 11, 2023. Revised: November 21, 2023. Accepted: December 23, 2023. Published: January 30, 2024.
1. Introduction
DC motors are widely used in industrial
applications requiring adjustable speed, good speed
limits, as well as frequent reversing, braking, and
starting. Rolling mills, paper mills, mine winders,
hoists, machine tools, traction, printing presses, textile
mills, excavators, and cranes are a few examples of
significant applications. Servo motors with fractional
horsepower are frequently used for positioning and
tracking. Despite predictions that AC drives will
eventually replace DC drives, DC drives nevertheless
predominate in variable speed applications today due
to their lower cost, reliability, and ease of control.
There are numerous techniques available for
controlling the speed and position of a DC motor. A
motor speed controller's main function is to take a
signal that represents the desired speed and operate a
motor at that speed. [1]
Because DC motors are single-input, single-output
(SISO) systems, efficient speed control systems may
be constructed with ease. characteristic that enables
precise adjustment control signals to control the
motors across a broad speed range. An armature
current-controlled technique is taken into
consideration for speed control in this study. Due to
principle is to use a control structure that removes the
delay from the feedback loop and permits controller
design based solely on the delay-free portion. [14]
There are two non-integer more changeable constants
in the FOPID controller in addition to the proportional
(Kp), integral (Ki ), and derivative (Kd) integer
constants. parameters: the order of the integral ( ʎ) and
the order of the derivative( µ). Because it is a
generalization of PIDs, this controller technology
retains the benefits of traditional ones while having a
wider design scope. If the FOPID controller
parameters ( Kp , Ki ,Kd ) are properly calibrated, a
better and more reliable performance based on this
novel approach can be obtained. both of PID and
FOPID controllers for the DC motor plant through
obtaining optimum values for their gain parameters.
The proportional gain makes the controller respond to
the error while the integral derivative gain helps to
eliminate steady state error and prevent overshoot
respectively [4]. have provided suitable ranges of the
design parameters thereby making difficult the
selection of a suitable value for the tuning parameter.
The present work is an attempt to propose new tuning
rules for IPTD, IFOPTD, and DIPTD processes for the
general form of the modified Smith predictor reported
in [4] its simplicity and performance qualities, the
proportional-integral-derivative (PID) technique is
used to implement the controller of a speed control
system for a DC motor. [2][3].
International Journal of Electrical Engineering and Computer Science
DOI: 10.37394/232027.2024.6.9
Arun Yadav, Maneesh Kumar Gupta
E-ISSN: 2769-2507
86
Volume 6, 2024
1.2 Speed Control DC Motor
A DC motor with a single rigid rectangular
coil constituted by a single coil where a current flow,
suitably located in a uniform outside magnetic field
(B), then the torque (T) exerted at the coils centre is
given by:
T=ildB (1)
Where is the length of the coil perpendicular to the
magnetic field (m), is the length of a coils edge (m) .
The flux (φ)flowing through the rotor of the DC motor
is proportional to magnetic field B, the above torque
expression can be rewritten as follows:
T = (2)
Where =ld / A . Since in this work, the magnetic
field B is taken to be constant, hence K is constant, and
then the motor torque can be written as:
(3)
Where is a constant for motor torque. The
back electromotive force (EMF) induced in the coil, as
determined by Farady's law, is given
(4)
where is the flux that is moving over a closed coil's
internal surface (Wb). The reverse EMF can be
expressed as follows, which is similar to the cases
examined in (2) and (3).
w (5)
Based on the second Newton's law, the dynamic
system's equation is as follows:
(6)
While the following Kirchhoff's voltage law-based
formulation of the system's electric equation
(7)
where J is the motor's inertia (kg) , B is the motor's
viscous friction coefficient (Nms), w is the motor's
angular velocity (rad/s), and L, R, and an are the
coil's inductance (H) , resistance (Ώ), and voltage ( V),
respectively. The system dynamic equations (6) and
(7) mentioned above can be represented in the s-
domain as follows by using the Laplace transform:
where w is the motor's angular velocity (measured in
m/s) and is the motor's electromotive factor
constant. The motor torque and back emf constants are
equivalent in SI units, that is, Consequently,
both constants are represented by the constant K, as in
K = .
(Js+Bw(s)=KI(s)
(8)
Fig 1: Block diagram of a current controlled DC
motor
(Ls+R)I(s)= (9)
Figure 1 in this article depicts the block diagram of
the armature current-controlled DC motor. The open
loop transfer function with the motor voltage E (s) an
as the system input and the motor's rotational velocity
w(s) as the system output is as follows, based on (8)
and (9). [15]
(10)
Applying the realistic DC motor system parameter
values listed in Table 1, the final transfer function of
the DC motor is approximately equal.
(11)
3. Direct Synthesis Method
To control speed of DC motor using direct
synthesis is proposed in this paper. The mathematical
modeling equation are used which used to derive the
transfer function of the dc motor. The closed-loop
transfer function for set-point modifications must be
Parameter
Symbol
Typical Value
Motor inertia
J
0.01kg
Coil
inductance
L
0.5H
Coil resistance
R
1Ώ
Motor
constant
K
0.023Nm/A
Friction
B
0.00003Nms
International Journal of Electrical Engineering and Computer Science
DOI: 10.37394/232027.2024.6.9
Arun Yadav, Maneesh Kumar Gupta
E-ISSN: 2769-2507
87
Volume 6, 2024
specified to determine the modular aspects. Assume
that the process measurement component
(10)
(11)
. The desired transfer function’s numerator is set equal
to the numerator of the obtained transfer function. The
tuning parameters λ and τ represent the desired closed-
loop time constants for servo and regulatory purposes,
respectively. Also, the authors have provided suitable
ranges for selecting these design parameters. [6]
Improved robust performance was achieved as
compared to the tuning method proposed in [7].
Recently, controllers of the above said double two
degree of freedom structure have been designed using
a two-degree of freedom-IMC tuning approach for
processes with a general transfer function in [8]. Speed
control of DC motors has been attracting considerable
interest by many researchers, hence, there are many
studies and research have been published in this issue.
Mickky and Tewari [5] It is observed from the above
literature survey that none of the above-cited works
except [9] and has considered double integrating
processes with time delay for controller design. Also,
most of the published works are based on the direct
synthesis or IMC design approach. It is to be noted that
no guidelines were provided for selecting the tuning
parameters in [8] and [10]. The authors in [11] and
[12] have provided suitable ranges of the design
parameters thereby making difficult the selection of a
suitable value for the tuning parameter. The present
work is an attempt to propose new tuning rules for
IPTD, IFOPTD, and DIPTD processes for the general
form of the modified Smith predictor
Table 1. Typical parameter values for DC motor
3.1 Controller Design:
The modified Smith predictor considered in
the present work is shown in Fig 2
fig 1 Modified smith predictor
where the nominal model of the real process (Gp) that
needs to be regulated is represented by Gm = . The
two controllers utilized for load disturbance rejection
and set point tracking are Gc1 and Gc2. Under nominal
conditions (Gp = Gm), the closed loop
transfer functions between the output and the set point
and the input load disturbance are given by
(14)
(15)
where, respectively, r, y, and d represent for the set
point, controlled variable, and load disturbance at the
plant input. As shown from the mentioned formulas,
y/r only contains Gc1, whereas y/d contains Gc1 and
Gc2. The design of Gc2 to reject the load disturbance
at the plant input comes after Gc1 has been modified
to achieve suitable set point tracking in the present
study.
3.2 Design of Gc1:
The direct synthesis method is used to create
Gc1 and is based on the specification of the desired
closed-loop transfer function for set-point change. The
actual closed loop transfer function is obtained to
specify the intended closed loop transfer function.
The desired transfer function's numerator is set to be
the same as the actual transfer function's numerator.
The number of unidentified controller parameters is
specified as the order of the denominator polynomial
of the intended transfer function. [13] and Gc1 = Kc1
is taken into consideration for the IPTD process
model. For the IFOPTD and DIPTD process models,
Gc1 is assumed to be a PD controller with a transfer
function of Kc1(1 + Td1s).
(16)
(17)
International Journal of Electrical Engineering and Computer Science
DOI: 10.37394/232027.2024.6.9
Arun Yadav, Maneesh Kumar Gupta
E-ISSN: 2769-2507
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Volume 6, 2024
(18)
=
(19)
=
(20)
=
(21)
Desire close loop system
=
(22)
Ti = ʎ( 3-)
(23)
=
(24)
3.3 Design of Gc2:
The characteristic equation comprises two
elements, which can be seen as (1 +) and (1
+). Substitute the and in the control
equation ( and replace with even
, because of required . rules for
PI/PID controllers with the following transfer
function:
=
(25)
=
(26)
=
=0 (27)
Characteristics for equation
=
(28)
=
(29)
=
(30)
3.4 Simulation and Results
Using the Matlab tool, speed motor control
system controllers based on PID techniques are
developed. The parameters of controllers are tuned in
the direct synthesis method in maximum sensitivity
1.2, then I have got ʎ =1.38 for the first controller and
µ= 0.8 for the second controller.
Table .2 Step servo and regulatory
Fig 2 Step response of DC motor
The step response of speed control of DC motor in
integral square error (ISE 6.39) and integral absolute
error (IAE 14.45) in tuning PID and Smith predictor
(ISE 2.127, IAE 5.309)
Step response of PID controller for speed control of
DC motor
Process model
ISE
IAE
For full
system
Smith
Predictor
2.12
7
5.309
Tuning
PID
6.39
14.45
Servo
Smith
Predictor
0.8512
1.914
TuningPID
5.223
10.06
Regulatory
Smith
Predictor
1.277
3.395
Tuning
PID
1.4
4.91
International Journal of Electrical Engineering and Computer Science
DOI: 10.37394/232027.2024.6.9
Arun Yadav, Maneesh Kumar Gupta
E-ISSN: 2769-2507
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Volume 6, 2024
Fig 3 step response of DC motor for servo
The Maximum sensitivity of 1.2 in without
disturbance normal PID (ISE 5.223, IAE 10.06) and
Smith predictor (ISE 0.8512 IAE 1.914) in the graph
and good response Smith predictor in servo speed
control DC motor
Step response servo of PID controller for speed
control of DC motor
Fig 4 Step response of DC motor for regulatory
The Maximum sensitivity of 1.2 in without input step
normal PID (ISE 1.4, IAE 4.91) and smith predictor
(ISE 1.277, IAE 3.395) in graph and good response
smith predictor in regulatory speed control DC motor
PID controller for speed control of the DC
motor is changed in 30% and -30 %.
Process model
ISE
IAE
+30%change
in K
Smith
Predictor
2
5.195
Tuning
PID
6.625
14.13
-30%change
in -
30% in K
Smith
Predictor
2.382
5.579
Tuning
PID
6.596
16
Table 3: Performance of Perturbation
Fig 5 step response perturbation+30 of DC motor
The perturbation +30% change in K T1 and T2
maximum sensitivity 1.2 for normal PID (ISE 6.625,
IAE 14.13) and smith predictor (ISE 2, IAE 5.195)
step response speed control of DC motor.
Step response +30 change in K T1&T2 of PID
controller for speed control of DC motor
Fig 6 step response of perturbation -30 % of DC motor
The perturbation -30% change in T1 and T2 -30%
change in K maximum sensitivity 1.2 response normal
PID (ISE 6.596, IAE 16) and smith predictor (ISE
2.382, IAE 5.579) in better performance smith
predictor in speed control of DC motor
Step response -30 change in T1&T2 -30 in K of PID
controller for speed control of DC motor
International Journal of Electrical Engineering and Computer Science
DOI: 10.37394/232027.2024.6.9
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E-ISSN: 2769-2507
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Volume 6, 2024
4. Conclusion
This research presents an investigation into the
development of a speed control system for DC motor.
The set point tracking controller is tuned using a direct
synthesis approach, whereas a PID controller is used
for rejecting the load disturbance. The system's closed-
loop performance is implied by the tuning parameters
for servo and regulatory purposes, which are specified
to achieve maximum sensitivity equal to 1.2
Compared with the normal tuned PID we got smith
predictor best PID/PD control gives a better response
with normal tuned PID control of the best performance
smith predictor speed control of DC motor, the rotor
performance of the proposed tuning strategy is also
improved. The simulation results show the proposed
method improves the system's overall performance
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Contribution of Individual Authors to the
Creation of a Scientific Article (Ghostwriting
Policy)
The authors equally contributed in the present
research, at all stages from the formulation of the
problem to the final findings and solution.
Sources of Funding for Research Presented in a
Scientific Article or Scientific Article Itself
No funding was received for conducting this study.
Conflict of Interest
The authors have no conflicts of interest to declare
that are relevant to the content of this article.
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(Attribution 4.0 International, CC BY 4.0)
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International Journal of Electrical Engineering and Computer Science
DOI: 10.37394/232027.2024.6.9
Arun Yadav, Maneesh Kumar Gupta
E-ISSN: 2769-2507
91
Volume 6, 2024
