This document discusses nonlinear integral control for DC motor speed control with unknown and variable external torque. It begins with an introduction to DC motors and common speed control techniques. It then provides the basic model of a DC motor and derives the transfer function. It discusses nonlinear control systems and elements like saturation, deadband, and friction. It describes methods for solving nonlinear transient responses, nonlinear system stability, and provides a Simulink model example comparing PI and P controller performance for speed and error. References for DC motor speed control and optimization of PI controllers are also provided.

1. Saif Al-Din Ail Madhi
Department of Mechanical Engineering/ College of Engineering/ University of Baghdad
30/6/2020 1 | P a g e
Nonlinear Integral Control for DC Motor Speed Control
with Unknown and Variable External Torque
University of Baghdad
Name: - Saif Al-din Ali -B-
s.madi1603@coeng.uobaghdad.edu.iq
The fourth stage

2. Saif Al-Din Ail Madhi
Department of Mechanical Engineering/ College of Engineering/ University of Baghdad
30/6/2020 2 | P a g e
TABLE OF CONTENTS
1. Introduction
2. Basic Model of DC Motor
3. NONLINEAR CONTROL SYSTEMS
4. Nonlinear elements
5. Methods of solution nonlinear transient responses
6. nonlinear systems stability
7. Simulink Model

3. Saif Al-Din Ail Madhi
Department of Mechanical Engineering/ College of Engineering/ University of Baghdad
30/6/2020 3 | P a g e
Nonlinear Integral Control for DC Motor Speed Control with
Unknown and Variable External Torque
Abstract: In this research, the control of motors is recognized by the
integration control if the engine recognizes an external load represented by
the torque. The circuit is recognized for the types of controllers and the
identification of non-linear control is controlled by the speed of the motor
1. Introduction
• DC motor converts electrical energy into mechanical energy. It is one of two basic
types of motors: the other type is the alternating current or AC motor. Among DC
motors, there are shunt-wound, series-wound, compound-wound and permanent
magnet motors.
• Basic DC Motor Operation
A DC motor is equipped with magnets, either permanent magnets or electromagnetic
windings, that produce a magnetic field. When current passes through the armature,
also known as the coil or wire, placed between the north and south poles of the magnet,
the field generated by the armature interacts with the field from the magnet and applies
torque. In a DC motor, the magnet forms the stator, the armature is placed on the rotor
and a commutator switches the current flow from one coil to the other. The commutator
connects the stationary power source to the armature through the use of brushes or
conductive rods. Furthermore, DC motors operate at a fixed speed for a fixed voltage
and there is no slip
❖ Speed control techniques in separately excited dc motor:
• By varying the armature voltage for below rated speed.
• By varying field flux should to achieve speed above the rated speed.
❖ Different methods for speed control of DC motor:

4. Saif Al-Din Ail Madhi
Department of Mechanical Engineering/ College of Engineering/ University of Baghdad
30/6/2020 4 | P a g e
• Traditionally armature voltage using Rheostatic method for low power dc
motors.
• Use of conventional PID controllers.
• Neural Network Controllers.
• Constant power motor field weakening controller based on load-adaptive
multi- input multi- output linearization technique (for high speed regimes).
• Single phase uniform PWM ac-dc buck-boost converter with only one
switching device used for armature voltage control.
• Using NARMA-L2 (Non-linear Auto-regressive Moving Average)
controller for the constant torque region
➢ PID (proportional-integral-derivative) control, three types of control: P,
PI and PID.
• P Control. Output power is directly proportional to control error. The
higher the proportion coefficient, the less the output power at the same
control error. Proportional control can be recommended for fast-
response systems with a large transmission coefficient. To adjust the
propotional controller
• PI Control. Output power equals to the sum of proportion and
integration coefficients. The higher the proportion coefficient, the less
the output power at the same control error. The higher the integration
coefficient, the slower the accumulated integration coefficient. PI
control provides zero control error and and is insensitive to interference
of the measurement channel. The PI control disadvantage is slow
reaction to disturbances.

5. Saif Al-Din Ail Madhi
Department of Mechanical Engineering/ College of Engineering/ University of Baghdad
30/6/2020 5 | P a g e
• PID Control. Output power equals to the sum of three coefficients:
proportional, integral and differential. The higher the proportion
coefficient, the less the output power at the same control error. The
higher the integration coefficient, the slower the accumulated
integration coefficient. The higher the differentiation coefficient, the
greater the response of the system to the disturbance. The PID
controller is used in inertial systems with relatively low noise level of
the measuring channel.
Fig (1) PID (proportional-integral-derivative) control
2. Basic Model of DC Motor
The basic principle behind DC motor speed control is that the output speed of DC
motor can be varied by controlling armature voltage for speed below and up to rated
speed keeping field voltage constant. The output speed is compared with the reference
speed and error signal is fed to speed controller. Controller output will vary whenever
there is a difference in the reference speed and the speed feedback. The output of the
speed controller is the control voltage Ec that controls the operation duty cycle of (here
the converter used is a IGBT) converter. The converter output give the required Va
required to bring motor back to the desired speed. The Reference speed is provided

6. Saif Al-Din Ail Madhi
Department of Mechanical Engineering/ College of Engineering/ University of Baghdad
30/6/2020 6 | P a g e
through a potential divider because the voltage from potential divider is linearly
related to the speed of the DC motor. The output speed of motor is measured by Tacho-
generator and since Tacho voltage will not be perfectly dc and will have some ripple.
So, we require a filter with a gain to bring Tacho output back to controller level. The
basic block diagram for DC motor speed control is show below:
Fig (2)Closed Loop System Model for Speed Control of DC Motor
Fig (3) Separately Excited DC motor
The armature equation is shown below:
Va =Eg+ IaRa+ La (dIa/dt)
The description for the notations used is given below:
1. Va is the armature voltage in volts.
2. Eg is the motor back emf in volts.
3. Ia is the armature current in amperes.
4. Ra is the armature resistance in ohms
5. La is the armature inductance in Henry.
Now the torque equation will be given by:
Td = Jdω/dt +Bω+TL
Where:
1. TL is load torque in Nm.
2. Td is the torque developed in Nm.
3. J is moment of inertia in kg/m².
4. B is friction coefficient of the motor.
5. ω is angular velocity in rad/sec.
Assuming absence (negligible) of friction in rotor of motor, it will yield:
B=0
Therefore, new torque equation will be given by:

7. Saif Al-Din Ail Madhi
Department of Mechanical Engineering/ College of Engineering/ University of Baghdad
30/6/2020 7 | P a g e
Td = Jdω/dt + TL --------- (i)
Taking field flux as Φ and (Back EMF Constant) Kv as K. Equation for back emf of motor
will be:
Eg = K Φ ω --------- (ii)
Also, Td = K Φ Ia --------- (iii)
From motor’s basic armature equation, after taking Laplace Transform on both sides, we will
get: Ia(S) = (Va – Eg)/(Ra + LaS)
Now, taking equation (ii) into consideration, we have:
=> Ia(s) = (Va – KΦω)/ Ra(1+ LaS/Ra )
And, ω(s) = (Td - TL )/JS = (KΦIa - TL ) /JS
Also, The armature time constant will be given by:
(Armature Time Constant) Ta = La/Ra
Fig (4) Model of Separately Excited DC Motor
After simplifying the above motor model, the overall transfer function will be as given
below:
ω (s) / Va(s) = [KΦ /Ra] /JS(1+TaS) /[ 1 +(K²Φ² /Ra) /JS(1+TaS)]
Further simplifying the above transfer function will yield:
ω(s) /Va(s) = (1 /kΦ) /{ 1 +(k²Φ² /Ra) /JS(1+TaS)} ---------------- (iv)
Assuming, Tm = JRa / (kΦ) ² as electromechanical time constant [1]. Then the above
transfer function can be written as below:
ω(s)/Va(s) = (1/kΦ)/ [STm (1+STa)+1] --------(v)
Let us assume that during starting of motor, load torque TL = 0 and applying full voltage Va
Also assuming negligible armature inductance, the basic armature equation can be written as:
Va = KΦω(t) + IaRa
At the same time Torque equation will be:
Td = Jdω/dt = KΦIa ----- (vi)
Putting the value of Ia in above armature equation:
Va=KΦω(t)+(Jdω/dt)Ra/ KΦ
Dividing on both sides by KΦ,
Va/KΦ=ω(t)+JRa(dω/dt)/(KΦ)² ------------------------(vii)
Va/KΦ is the value of motor speed under no load condition.
Therefore,
ω(no load)=ω(t)+JRa(dω/dt)/(KΦ)² = ω (t) + Tm (dω/dt)
Where, KΦ = Km(say) And, Tm=JRa/(KΦ)²=JRa/(Km)²

8. Saif Al-Din Ail Madhi
Department of Mechanical Engineering/ College of Engineering/ University of Baghdad
30/6/2020 8 | P a g e
Therefore, J = Tm (Km) ²/ Ra --------- (viii)
From motor torque equation, we have:
ω(s) = KmIa(s)/JS - TL/JS -------- (ix)
From equation (viii) and (ix), we have:
Now, Replacing KΦ by Km in equation (v), we will get:
ω(s)/Va(s)=(1/Km) / (1+STm+S²TaTm) ------------ (x)
Since, the armature time constant Ta is much less than the electromechanical time constant
Tm, (Ta << Tm)
Simplifying, 1 + STm + S²TaTm ≈ 1 + S (Ta+Tm) + S²TaTm = (1 + STm)(1 + STa)
The largest time constant will play main role in delaying the system when the transfer
function is in time constant form. To compensate that delay due to largest time constant we
can use PI controller as speed controller. It is because the zero of the PI controller can be
chosen in such a way that this large delay can be cancelled. In Control system term a time
delay generally corresponds to a lag and zero means a lead, so the PI controller will try to
compensate the whole system . Hence, the equation can be written as:
ω(s)/Va(s) = (1/Km)/((1 + STm)(1 + STa)) -----(xi)
Tm and Ta are the time constants of the above system transfer function which will determine
the response of the system. Hence the dc motor can be replaced by the transfer function
obtained in equation (xi) in the DC drive model shown earlier.
3.NONLINEAR CONTROL SYSTEMS
All physical systems are nonlinear. In the some area of dynamic
characteristic is possible make a linearization. Same of nonlinearities in
the system are undesirable, there are parasite nonlinearities e.g. friction,
backlash, saturation, etc. Another are putting to the system wittingly, e.g.
relay types nonlinearities – simple control loops with the costless two
step controllers or three step controllers.
Dynamic characteristics of nonlinear systems are described by nonlinear
differential equations. Axiom of superposition is not valid and no
transfer functions. Practical systems are solved approximately or are
solved by numerical and graphical methods. The static characteristic of
nonlinear system is a general curve, which is determined by analytic
expressions, graphs or values tables. In nonlinear systems is possible
generation: self-excitation oscillations or step resonances.

9. Saif Al-Din Ail Madhi
Department of Mechanical Engineering/ College of Engineering/ University of Baghdad
30/6/2020 9 | P a g e
Base differences between linear and nonlinear systems :
• input/output relationships are not linear
• nonlinear systems are described by nonlinear differential equations,
• static characteristic is not linear,
• definition of stability is more complicated,
• axiom of superposition is not valid ,
• relations between various modes of descriptions are complicated (from
state space description is not possible a simple receive transfer function),
• nonlinear system is composed by linear and nonlinear parts.
4.Nonlinear elements
There are described static characteristics of nonlinear elements than are important
in control technologies:
• Saturation or Clipping (Fig.5) – Some devices have natural maximum
values, such as voltage or pressure limitations caused by a regulated supply.
Typically occurs at sensors, amplifiers, mechanical end stops, speed
limitation, etc
Fig 5: Nonlinear static characteristic of saturation
• Deadband (Fig. 6) - The region where the applied voltage has no effect is
called the deadband e.g. sensors, friction in all components e.g.
servomotors.
Fig 6: Nonlinear characteristic with deadband

10. Saif Al-Din Ail Madhi
Department of Mechanical Engineering/ College of Engineering/ University of Baghdad
30/6/2020 10 | P a g e
• Friction (Fig. 7) – Friction is common in less expensive motors, and when
a motor is driving a mechanical system. In systems there are two type of
friction that must be considered. The static friction, ’stiction’, will prevent
initial motion. If the system breaks free and starts turning, the kinetic friction
will provide a roughly constant friction.
Fig.7: Nonlinear static characteristic of friction.
5. Methods of solution nonlinear transient responses
➢ Analytic methods
• Approximation of nonlinear characteristic by analytic expression,
• Approximation of nonlinear characteristic by linear partitions,
• Approximation of nonlinear characteristic by tangent line or secant line
in the working point – linearization in praxis.
➢ Estimate analytic methods
➢ Graphic and graphic-analytic methods
• Isoclines method,
• Phase-space method
➢ Numeric methods of nonlinear systems solution
➢
6.nonlinear systems stability
Stability of linear systems is attribute of the system, which don’t depend on the
instant state or on the input signal. Several definitions of the stability exist for
nonlinear systems. At nonlinear systems are defined steady (stationary) states, which
are following:
• Static steady state, which is represented by a point in the phase space, it is
the singular point.

11. Saif Al-Din Ail Madhi
Department of Mechanical Engineering/ College of Engineering/ University of Baghdad
30/6/2020 11 | P a g e
• Periodic steady state, which is represented by a curve in the phase space,
it is the limit cycle.
Every of steady states can be a stabile steady state (after deflection from the
steady state has ability to stabilize) or a non-stabile steady state (after deflection from
the steady state has not ability to stabilize). Stability of nonlinear systems depends
on initial values of states.
➢ Stability of nonlinear systems are examined by following methods:
• First Ljapunov method
• Second Ljapunov method
• Popov criterion of the stability
• Equivalent transfers method
7. Simulink Model
In this part of the report, an issue is resolved on the MATLAB 19
program, and we find the amount of speed and the amount of error
rates for both types of control (pi - p).
Fig (8) Simulink Model

12. Saif Al-Din Ail Madhi
Department of Mechanical Engineering/ College of Engineering/ University of Baghdad
30/6/2020 12 | P a g e
Fig (9) Model
Fig (10) Error
We note the amount of fluctuation in the error during control by Integrator until it reaches a state
of stability, unlike PI, the processing is quick

13. Saif Al-Din Ail Madhi
Department of Mechanical Engineering/ College of Engineering/ University of Baghdad
30/6/2020 13 | P a g e
Fig (11) Speed
We note the control responsiveness of the type pi and that the amount of time needed to reach the
tie is 15 seconds

14. Saif Al-Din Ail Madhi
Department of Mechanical Engineering/ College of Engineering/ University of Baghdad
30/6/2020 14 | P a g e
REFERENCES
1. Optimal Tuning of PI Controller for Speed Control of DC motor drive using Particle
Swarm Optimization
Rohit G. Kanojiya, Student, Y.C.C.E, and P. M. Meshram, Associate professor, Y.C.C.E
2. The Speed Control of DC Motor under the Load Condition using PI and PID
Controllers
Muhammed Reşit Corapsiz1,a) and Hakan Kahveci2
3. PI controller for DC motor speed realized with Arduino and Simulink
Mario Gavran*, Mato Fruk** and Goran Vujisić**
4. PI Controller of Speed Regulation of Brushless DC Motor Based on Particle
Swarm Optimization Algorithm with Improved Inertia Weights
Wei Xie,1 Jie-Sheng Wang ,1,2 and Hai-Bo Wang3
5. NONLINEAR INTEGRAL CONTROL DESIGN FOR DC MOTOR
SPEED CONTROL WITH UNKNOWN AND VARIABLE
EXTERNAL TORQUE
Dr. Shibly Ahmed Al-Samarraie1, *Bashar Fateh Midhat2, Ivan Isho Gorial3
6. Steady-State Integral Proportional Integral Controller for PI Motor Speed
Controllers
Choon Lih Hoo*, Sallehuddin Mohamed Haris*, Edwin Chin Yau
Chung**, and Nik Abdullah Nik Mohamed***
7. Modern Control Systems ThirTeenTh edition Richard C. Dorf University
of California, Davis Robert H. Bishop The University of South Florida