Design a Controller Based on Smith Predictor by Direct Synthesis
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 a 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. The aim of this work is to design a speed controller of a DC motor by selection
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 check 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: July 21, 2022. Revised: October 19, 2023. Accepted: November 20, 2023. Published: December 31, 2023.
1. Introduction
DC motors are widely used in industrial
applications that require an adjustable speed and 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 often used for positioning and
tracking. Despite predictions that AC drives will
eventually replace DC drives, DC drives nonetheless
predominate in variable speed applications today due
to their lower cost, reliability, and ease of control.
There are numerous techniques available to control 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 motors
across a broad speed range. An armature current
controlled technique is taken into consideration for
speed control in this study. Due to principle, it is
possible 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 noninteger more changeable constants
in the FOPID controller in addition to the integer
constants proportional (Kp), integral (Ki ), and
derivative (Kd). 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 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 speed control
system for a DC motor. [2][3].
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 the outside magnetic
Engineering World
DOI:10.37394/232025.2023.5.22
Arun Yadav, Maneesh Kumar Gupta
E-ISSN: 2692-5079
194
Volume 5, 2023
field (B), then the torque (T) exerted at the coils center
is given by:
T=ildB (1)
Where is the length of the coil perpendicular to the
magnetic field (m), is the length of the edge of a coil
(m). The flux (φ)flowing through the rotor of the DC
motor is proportional to the 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 law, the dynamic
system's equation is as follows:
(6)
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 values of the parameters of the
DC motor system listed in Table 1, the final transfer
function of the DC motor is approximately equal.
(11)
3. Direct Synthesis Method
Controlling the speed of the DC motor using
direct synthesis is proposed in this paper. The
mathematical modeling equation are used which used
to derive the transfer function of dc motor. The closed-
loop transfer function for set-point modifications must
be specified to determine the modular aspects. Assume
that the process measurement component is
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
Engineering World
DOI:10.37394/232025.2023.5.22
Arun Yadav, Maneesh Kumar Gupta
E-ISSN: 2692-5079
195
Volume 5, 2023
(10)
(11)
. The numerator of the desired transfer function 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. In addition, the
authors have provided suitable ranges for selecting
these design parameters. [6] Improved robust
performance was achieved compared to the tuning
method proposed in [7]. Recently, controllers of the
above-mentioned double-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 motor has
been attracting a considerable interest by many
researchers, hence, many studies and researches have
been published on this issue. Mickky and Tewari [5] It
is observed from the above literature survey that none
of the works cited above except [9] and has considered
double integration processes with time delay for
controller design. In addition, most of the published
work is based on the direct synthesis or IMC design
approach. It should 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, 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. Values of typical parameters for DC motor
4. Controller Design
The modified Smith predictor considered in
the present work is shown in Figure 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 utilised 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 follows after Gc1 has been modified
to achieve a suitable set-point tracking in the present
study.
4.1 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 in
order 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 account 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)
Engineering World
DOI:10.37394/232025.2023.5.22
Arun Yadav, Maneesh Kumar Gupta
E-ISSN: 2692-5079
196
Volume 5, 2023
(17)
(18)
=
(19)
=
(20)
=
(21)
Desire a closed-loop system
=
(22)
Ti = ʎ( 3-)
(23)
=
(24)
4.2 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)
5. Simulation and Results
Using the Matlab tool, speed motor control
system controllers based on PID techniques are
developed. The controller parameters are tuned with
direct synthesis method in maximum sensitivity 1.2,
then I have got = 1.38 for 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 the the speed control of DC motor
in integral square error (ISE 6.39) and integral
absolute error (IAE 14.45) in the tuning of 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
Engineering World
DOI:10.37394/232025.2023.5.22
Arun Yadav, Maneesh Kumar Gupta
E-ISSN: 2692-5079
197
Volume 5, 2023
Fig. 3 step response of DC motor for servo
The maximum sensitivity 1.2 in without
disturbance normal PID (ISE 5.223, IAE 10.06) and
the Smith predictor (ISE 0.8512 IAE 1.914) in graph
and the 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 1.2 in the normal PID
without input step (ISE 1.4, IAE 4.91) and smith
predictor (ISE 1.277, IAE 3.395) in graph and the good
response smith predictor in regulatory speed control
DC motor
PID controller for speed control of DC motor
is change in 30% and -30 %.
Process model
ISE
IAE
Smith
Predictor
2
5.195
+30%change
in K
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 the DC motor
The perturbation +30% change in the maximum
sensitivity of K T1 and T2 1.2 for normal PID (ISE
6.625, IAE 14.13) and the smith predictor (ISE 2, IAE
5.195) step response speed control of the DC motor.
Step response +30 change in K T1&T2 of the PID
controller for speed control of the DC motor
Fig 6 step response of the 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
Engineering World
DOI:10.37394/232025.2023.5.22
Arun Yadav, Maneesh Kumar Gupta
E-ISSN: 2692-5079
198
Volume 5, 2023
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 the
PID controller for speed control of the DC motor
6. Simulation study
If the value of this present work, the closed-
loop performance of the suggested method is
compared with the performance of recently described
solutions. All tuning options are compared using the
controller settings that produce a maximum sensitivity
equal to 2. By cascading a first-order low-pass filter
with a time constant equal to 0.1 times the derivative
time constant, the pure derivative sections of Gc1 and
Gc2 are implemented. The performance metrics ISE,
IAE, settling time (st) are used to compare the
effectiveness of the various tuning techniques. Fast
set-point tracking and disturbance rejection are
implied by a small value of ISE/IAE. ISE and IAE in
the controlled variable are denoted mathematically by.
ISE=
(31)
IAE =
(32)
where e(t) is the difference between the set-point
input and the controlled close-loop transfer function.
The settlement time is the time it takes for the step
response to maintain its ultimate value within 2%
7. Conclusion
This research presents an investigation of the
development of speed control system for the DC
motor. The set-point tracking controller is tuned using
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 and compare with the normal tuned PID we got
smith predictor best PID/PD control give 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 improve. The simulation results show
that the proposed method improves the system's
overall performance.
References
[1]. Akhilesh Kumar Mishra1,” Speed Control of
DC Motor Using Artificial Bee Colony a
Optimization Technique” Universal Journal
of Electrical and Electronic Engineering
1(3): 68- 75, 2013 A.
[2]. Rajasekhar, Sh. Das and A. Abraham, 2013,
“Speed Control of Dc Motor Using Particle
Swarm Optimization’. In Proceeding of
IEEE World Congress on Nature and
Biologically Inspired Computing (NaBIC),
259-266
[3]. Ahmad, A., Saad, Z., Osman, M., Isa I.,
Sadimin, S. and Abdullah, S., 2010. ‘Control
of the magnetic levitation system using fuzzy
logic control’, in Proceedings of the IEEE
Second International Conference on
Computational Intelligence, 51–56.
[4]. Ercin, O., and Coban, R., 2011. ‘Comparison
of the Artificial Bee Colony and the Bees
Algorithm for PID Controller Tuning’, in
Proceedings of the IEEE Conference on
Innovations in Intelligent Systems and
Applications (INISTA). 595-598
[5]. . Lin, M.G., Lakshminarayanan, S., &
Rangaiah, G.P. (2008). ‘A comparative study
of recent/popular PID tuning rules for
stable, first-order plus dead time, single
input, single output processes’ Industrial and
Engineering Chemistry Research, 47,344–
368.
[6]. Moina Ajmeria & Ahmad Alia, ‘Direct
synthesis-based tuning of the parallel control
structure for integrating processes’
International Journal of Systems Science20
Dec 2013.
[7]. Lu, X., Y.-S. Yang, Q.-G. Wang, and W.-X.
Zheng, ‘A double two-degree-of-freedom
control scheme for improved control of
unstable delay processes’, J. Process
Control, Vol. 15, pp. 605–614 (2005).
[8]. Tan, W., ‘Analysis and design of a double
two-degree of-freedom control scheme’, ISA
Trans., Vol. 49, pp.311–317 (2010).
[9]. Liu, T., YZ Cai, D. Y. Gu, and W. D. Zhang,
‘New modified Smith predictor scheme for
integrating and unstable processes with time
delay’, IEE Proc. Control Theory Appl., Vol.
152, No. 2, pp. 238–246 (2005).
[10]. Chen, D. and D. E. Seborg, “PI/PID
controller design based on direct synthesis
and disturbance rejection’, Ind. Eng. Chem.
Res., Vol. 41, pp.4807–4822 (2002)
[11]. Rao A. S., V. S. R. Rao, and M.
Chidambaram, “Set point weighted modified
Smith predictor for integrating and double
integrating processes with time delay”, ISA
Trans., Vol. 46, pp. 59–71 (2007).
[12]. Mataušek, M. R., and A. I. Ribić,
“Control of stable, integrating and unstable
Engineering World
DOI:10.37394/232025.2023.5.22
Arun Yadav, Maneesh Kumar Gupta
E-ISSN: 2692-5079
199
Volume 5, 2023
processes by the modified Smith predictor,”
J. Process Control, Vol. 22, pp. 338–343
[13]. Chen, D., and D. E. Seborg, “PI/PID
controller design based on direct synthesis
and disturbance Rejection,” Ind. Eng. Chem.
Res., Vol. 41, pp. 4807–4822 (2002).
[14]. J. E. Normey-Ricoa and E. F.
Camacho, ‘Dead-time compensators: A
survey, Control Engineering Practice, vol.
16, pp. 407–428, 2008.
[15]. Rinku, S., Subhransu, P. and
Gagandeep, K. Kamyad, “Design of
Fractional Order PID Controller for Speed
Control of DC Motor”, International Journal
of Scientific and Research Publications,
2012, 2(6), 51-60.
.
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.
Creative Commons Attribution License 4.0
(Attribution 4.0 International, CC BY 4.0)
This article is published under the terms of the
Creative Commons Attribution License 4.0
https://creativecommons.org/licenses/by/4.0/deed.en
_US
Engineering World
DOI:10.37394/232025.2023.5.22
Arun Yadav, Maneesh Kumar Gupta
E-ISSN: 2692-5079
200
Volume 5, 2023
