Continuous & Discrete Time PID Control of DC Motor Angular Velocity

This report focuses on the implementation and analysis of Proportional-Integral-Derivative (PID) control for Direct Current (DC) motors, specifically addressing the control of angular velocity in both continuous and discrete time domains. The report begins by outlining the strengths and limitations of PID control, comparing it with modern alternatives, and justifying the need for system discretization in contemporary control applications. It provides examples of PI and PID controlled systems, such as car cruise control and DC servo motors, respectively. The second section delves into the transfer functions of an armature-controlled DC motor, performs root locus analysis, and discusses PID tuning methods, evaluating the performance and robustness of the designed controllers. The final section involves discretizing the DC motor system, determining open and closed-loop transfer functions in the Z-domain, conducting root locus analysis, and applying PID control to the discretized system, ultimately comparing the results with those obtained in the continuous-time domain. MATLAB simulations are employed throughout the report to validate the theoretical analysis and demonstrate the effectiveness of the proposed control strategies. Desklib provides access to a wealth of similar solved assignments and study resources for students.

Control_and_Instrumentation_2
Continuous and Discrete Time PID Control of DC Motor Angular
Velocity
BY (STUDENT/ AUTHOR NAME)
CLASS (COURSE) NAME:
TUTOR (PROFESSOR):
SCHOOL NAME:
THE CITY/STATE:
DATE:
1

Control_and_Instrumentation_2
Contents
List of Figures and Tables.................................................................................................3
List of Figures.................................................................................................................3
List of Tables..................................................................................................................4
Introduction........................................................................................................................5
Section 1: PI and PID controllers.......................................................................................7
Part a: The strengths and limitations of PID control......................................................7
Part b: Examples of PI-controlled system and PID controlled system........................10
Part c: Modern alternatives to PID control...................................................................14
Part d: Reason for discretizing system in modern control...........................................15
Part e: motor angular velocity control relevance to complex systems.........................16
Section 2: Armature-controlled DC motor.......................................................................17
Part a): Transfer Functions of a DC Motor...................................................................17
Part b: The root locus performance.............................................................................20
Part c: PID tuning.........................................................................................................23
Part d. Performance and robustness of controller.......................................................26
Part e: Conclusions......................................................................................................28
Section 3: Discretizing the DC motor in section 2...........................................................28
Part a: Determining the open loop and closed loop Z-domain transfer functions of the
discretized system........................................................................................................28
2

Control_and_Instrumentation_2
Part b: The root locus analysis.....................................................................................29
Part c: Applying PID on the discretized system...........................................................32
Part d: Comparison between the s-domain PID and z-Domain PID............................35
Part e: Conclusions......................................................................................................35
Conclusions.....................................................................................................................36
References.......................................................................................................................38
List of Figures and Tables
List of Figures
Figure 1. 1: The electric circuit of the armature and the free-body diagram of the rotor. .5
Figure 1. 2: Car cruise control with PI controller.............................................................11
Figure 1. 3: Ziegler-Nichols method...................................................................................14
Figure 1. 4. The Good Gain method....................................................................................15
Figure 2. 1: The electrical circuit of a DC motor..............................................................17
Figure 2. 2: The system block diagram...........................................................................20
Figure 2. 3: The root locus of DC motor..........................................................................22
Figure 2. 4: DC motor PID control block diagram............................................................24
Figure 2. 5: The Ziegler-Nichols tuning method..............................................................25
Figure 2. 6: The Chien-Hrones-Reswick tuning method step response.........................26
Figure 3. 1: Root locus of discretized system with Ts=0.0001 s.....................................31
3

Control_and_Instrumentation_2
Figure 3. 2: Root locus of discretized system with Ts=0.001 s.......................................32
Figure 3. 3: Root locus of discretized system with Ts=0.01 s.........................................32
Figure 3. 4: Block representation of discrete PID controller (Fadali & Visioli, 2013)......33
List of Tables
Table 1. 1Factors affecting PID controllers.......................................................................8
Table 1. 2. The block diagram of PID Servo Control.......................................................12
Table 2. 1: Control parameters for the Ziegler-Nichols tuning method...........................25
Table 2. 2: Control parameters for the Chien-Hrones-Reswick tuning method..............26
Table 2. 3: The Ziegler-Nichols tuning method robustness and performance of the DC
motor................................................................................................................................27
Table 2. 4: The Chien-Hrones-Reswick tuning method robustness and performance of
the DC motor....................................................................................................................28
4

Control_and_Instrumentation_2
Introduction
Many industrial sectors today, direct current motors (also called DC motors) are used in
different ways from automobiles to robotics small and medium-sized driving applications
regularly features DC motors for the wide range of functionalities. A DC motor can be
defined as an electric motor which runs on direct current. Common actuators in control
systems are Direct Current motors. The DC motors provide direct rotary motion and,
coupled with cables and drums or wheels, providing translational motion. The electric
circuit of the armature and the free-body diagram of the rotor are shown in figure 1.1
below (Melkin, 2017):
:
Figure 1. 1: The electric circuit of the armature and the free-body diagram of the rotor
In this assignment, it deals with Continuous and discrete time PID control of DC motor
angular velocity. The major control system considered in this assignment is the PID
controller, which is simulated using MATLAB software. The system is then discretized
and compared with the continuous time control.
5

Control_and_Instrumentation_2
Section 1: PI and PID controllers.
Part a: The strengths and limitations of PID control.
PID (Proportional+ Integral+ Derivative) controller provide a range of
amendments because it contains three (3) key controls which includes P-control, I-
control and D-control which may be altered. PID Controller control and handles system
characteristics like settling time, percentage overshoot, stability, steady-state error, rise
time, etc. Even if there are three control elements in the controller, it still has some
disadvantage, because the implementation complexity increases in the system (Abu-
Khalaf, et al., 2009). Though, each control element has different functions, the elements
are exclusively dependent to each other; since single element can be varied by
changing another element. Consequently, PID design is complex as compared to the
designing P- controller, PD- controller or PI- controller (Anon., 2016). In this part, the
strength and disadvantages of PID controller in terms of implementation of the
controller, stabilization requirements, performances, robustness, energy consumptions
and steady state errors
i. Implementation of the controller.
During implementation of PID controller, one strength on implementing the PID
controller is that it is easier to construct and design. The PID controllers can be
an analogue circuit or a logic gate circuit or MCU or inductors and resistor
circuit. Conversely, PID controller needs acceptable and a better sampling time
for implementing which requires to be very accurate
ii. Stabilization requirements.
6

Control_and_Instrumentation_2
The PID controllers to be stable it needs several factor such as Kp, Ki, and Kd.
When the needs are all met the PID controller is more stable. The following
table 1.1 illustrate how these element affect the PID percentage overshoot, rise
time and steady state error.
Elements Effect on Rise
time
Effect on
overshoot
Effect on steady
state error
Kp Reduces increases Reduces
Ki Reduces increases Eliminates
Kd No/small chage Reduces No effect
Table 1. 1Factors affecting PID controllers
In obtaining a very accurate PID controlled system, these requirements
indicated in table 1.1 above must be met to be able to withstand external
disturbances like noise, vibrations, etc. Failure to meet the requirement, the
system becomes unstable.
iii. Performances.
The performances of PID control systems is evaluated by its ability to overcome
the disturbances effects referred to as the disturbance rejection of the control
systems.
A small value for derivative value is required since it might result into unstable
system due to the high sensitivity to disturbance such as noise and vibration.
High value of derivative will result to oscillation of the system, thus unstable
system.
7

Control_and_Instrumentation_2
The rise time or responses time of PID controller is required to be less than 2
percent of the system output and having a stable state. Moreover, speed of the
peak time is required to be considerably faster in reaching the peak values for
the given system.
iv. Robustness
The robustness of a system can be attained when the stability and the
performance of PID Controllers are not affected by a smaller differences in plant
or the operating condition. The advantage of the PID controller is that, the
robustness is achieved for system with less robust.
v. Energy consumptions.
The PID controller are designed to consume less power, for a system which is
unstable it may consume a lot of power. But when PID contlol are introduce the
system gains its stability, thus less energy is dissipated resulting to less power
consumption
vi. steady state errors
Steady state errors can be well-defined as the difference values between the
exact output produced by the system and the desired output of the same
system. PID controllers are used to minimize the steady state error in the control
systems over time and the error rate (Novotecknik, 2009). For a PID, it reduces
the error rate and sse of the system, When sse is zero, that means the desired
output of the system is met. The integral components (Ki) sums the error term
over time. The integral components increase continuously if there exist a small
error. The phenomenon in which the integral component continue to increase is
8

Control_and_Instrumentation_2
referred to as integral windup, which occurs when the integral actions reach to
saturations and does not reduce the error to zero. The importance for the
integrator are the ant windup operations for saturating the actuator (Owen,
2012).
Part b: Examples of PI-controlled system and PID controlled system.
PI Controller:
PI controllers are mostly used in eliminating the steady state error which may result
from P-controllers. An example is the cruise control systems control which is used to
control the speed/velocity of the vehicle, similarly by regulating the throttle positions. In
fact cruise controls actuate the throttle valves by a cable connection to actuators in
place of pressing the car pedals. Figure 1.2 below shows the pi cruise control (Deka &
Haloi, June, 2014).
Figure 1. 2: Car cruise control with PI controller
The aim of cruise control systems are maintaining constant car speed in spite of having
external disturbances such as change of road grade or wind. The control is
accomplished by computing the car speed, therafter the speed is compared with the
reference/desired speed and automatically, the throttle is adjusted in accordance with
control law (Deka & Haloi, June, 2014).
PID Controller:
9

Control_and_Instrumentation_2
PID controllers have the best control dynamic counting zero steady state errors, faster
responses (shorter rise-time), higher stability and no oscillations as compared to other
controllers such as PI-controller. An example used in PID control in many industries is a
DC servo motor. The elementary components of normal servo motion systems are
illustrated in figure 1.2 below. Yy use of standard Laplace notations. In the figure 1.2,
servo drive close a current loops and are made simply as linear transfer functions G(s) .
Obviously, the servo drive contains a peak current limit, thus the linear models are not
completely accurate; nevertheless, it provides a sensible representations for the
analysis. Basically, servo drives receives voltage commands that represent the
preferred motor currents. The shaft torque, of the motor, T m is directly associated to
motor current, I by torque constant, Kt . Equation (1.1) below shows the mentioned
above relationships.
T m=Kt I (1.1)
The transfer function of the current/torque regulators can be estimated as unity for
relative lower motion frequency which is needed.
Table 1. 2. The block diagram of PID Servo Control
The servo-motors are made as torque constant,Kt a viscous damping term, b, and lump
inertia, J . The lump inertia terms contains the servo-motor and inertia of the load. There
10

Control_and_Instrumentation_2
is an assumption that the loads are firmly coupled in such a way that the torsional
rigidity passes the natural mechanical resonance points further than the servo controller
bandwidths. In this case, it become easier to model the total systems inertia as the sum
of load inertia and the motor for a frequency that can be controlled. Slightly added
complicated models are required if coupler dynamic is integrated.
The real motor positions, q (s ), typically estimated by by a resolve or an encoder
couples directly to the motor shaft. Once more, the assumption made above also
assumes that the feedback devices are firmly mounted in a way that mechanical
resonant frequency can be ignored without any effect. Disturbances from external shaft
torque, Td , is added to the generated torque by the current of the motor to give the
torque available for accelerating the total inertia, J (Ziegler & Nichols, 2000).
There exist three (3) gains for adjusting in PID controllers, Kd , Ki∧K pwhich acts on the
position errors given in equation 1.2 below. The superscript * denotes a commanded
values (Ziegler & Nichols, 2000):
error ( t )=θ¿ ( t )−θ ( t ) 1. 2
The outputs of PID controllers are torque signals. The mathematical expressions in time
domain is illustrated in equation 1.3 below (Ziegler & Nichols, 2000):
PI Doutput ( t ) =Kd
d
dt ( error ( t ) )+ Ki∫ ( error ( t ) ) dt+ K p ( error ( t ) ) 1.3
Part c: Modern alternatives to PID control
The two modern alternative to PID controls for slow process and system with uncertain
parameters are Ziegler-Nicholas method and good gain control method. These two are
lab methods used in tuning PID controller (WILLIAMSON, 2015).
11

Control_and_Instrumentation_2
The Ziegler–Nichols methods are exploratory methods whereby PID controllers are
tuned through setting the D (derivative) and I (integral) gains to zero (WILLIAMSON,
2015). The "P" (proportional) gain, K p is raised till it attains the final gain. This is the
point at which the outputs of the control loops has consistent and stable oscillations.
The maximum gain attained and the oscillation period are used to set the derivative, D,
promotional, P and integral, I gains which depends on controller type used. This method
can be used for simulations and it is also probably the most common to use in real life.
Figure 1. 3: Ziegler-Nichols method
The Good Gain method is used to give better stability to the control loop better stability than that
of Ziegler-Nichols' methods (OGATA, 2013). The Good Gain method, as simple as it is, can be
used both on real processes (without any knowledge about the processes to be controlled), and in
simulated systems. This method gives better stability and does not need the control loop to get
into oscillations when tuning (OGATA, 2013). These are two benefits of this method as
compared with the Ziegler-Nichols’ methods.
12