Continuous and Discrete Time PID Control of DC Motor Angular Velocity

This article discusses the strengths and limitations of PID control, examples of PI-controlled system and PID controlled system, modern alternatives to PID control, and continuous and discrete time PID control of DC motor angular velocity.

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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:
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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
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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
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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
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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.
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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.
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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.
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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
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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:
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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
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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).
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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.
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Figure 1. 4. The Good Gain method.
Part d: Reason for discretizing system in modern control.
The system which is digitized it has several advantages over time continuous
system. The flexibility and capability of decision making in the control program is the
chief benefits of digital control systems (Anon., n.d.)among others:
a) High accuracy, since digitized system is represented by 0s and 1s which results to
a very small errors where noise and power supply drift are present
b) Low implementation error
c) High speed
d) Low cost
The modern control applications are using process control where the power of
digital processing techniques are used to perform the desired control tasks (KUO &
HASELMAN, 2014). Although the majority of systems that need to be controlled are
often analog nature, the modern digital control applications are using A/D- and D/A-
conversions as the principal operations to achieve appropriate control of processes.
This has brought the increasing need for discretization in modern control applications
where sampling is carried out by point measurements (OGATA, 2013).
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Part e: motor angular velocity control relevance to complex systems
Motor drives requires a rotor position sensors to correctly perform phase commutations
and current controls. A constant supplies of position data is essential; therefore position
sensors having high resolution, such as a resolver or a shaft encoders, is
characteristically used (Gamazo-Real, et al., July, 2010). For complex systems,
therefore, low-cost Hall-effects sensor are typically used. Moreover, accelerometers or
electromagnetic variables reluctance sensors has widely applied in measuring motor
position and speed (Deka & Haloi, June, 2014). The angular motion sensors based on
magnetic fields sensing principle stand out due to several inherent advantages and
sensing benefits (Deka & Haloi, June, 2014).
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Section 2: Armature-controlled DC motor.
Part a): Transfer Functions of a DC Motor
Figure 2.0 below shows DC motor with an inertial load attached on it. The applied
voltage to the armature and the field and sides of the motor can be signified respectively
as U and V .The inductances and resistances of the armature and the field side the DC
motor are given by: R , L, Rf and Lf where R is the armature resistance, L is armature
inductance, Rf is the field resistance and Lf is the field inductance (Dorf & Bishop, 2001).
Figure 2. 1: The electrical circuit of a DC motor.
The motor torque T m produced by is proportional to armature current and field current,
i.e.
T m ∝ I a If ⟹ Tm=k ia if 2.1
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Where k is constant of proportionality. For a armature-current controlled motors, if is
kept constant, and the armature voltage, U controls the field current I a., thus the motor
torque decreases or increases with the armature current (Dorf & Bishop, 2001). That is
T m=ia Kt ⟹ T m
if
=Kt 2.2
Where Kt is the motor torque. Taking the laplace transform of equation (2.2) gives the
following equation:
Tm ( s )
I f ( s ) =Kt ( s ) 2.3
On the side of the armature of the DC motor the current/ voltage association is given by
Voltage across the resistor plus voltage across the inductor plus the back e.m.f ( eb)
induced by the rotation of the armature windings gives the armature voltage (sailan
& Kuhnert, 2015): That is:
U =V R +V L+ eb
U (t )−eb (t)=ia ( t ) R+ L di (t )
dt 2.4
The back emf is directly proportional to the motor speed:
eb ( t ) =Ke ω ( t ) 2.5
Where Keis the back e.m.f. constant. Taking the Laplace transform of equation (2.5)
gives: The Electromechanical Equations thus can be given as:
Eb ( s )= Ke ω ( s ) 2.6
Taking the Laplace transform of equation (2.4) above and inserting equation (2.6) gives
equation (2.7) below:
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U ( s ) −K e ω ( s ) =Ia ( s ) ( R+sL )U ( s ) −Eb ( s ) =I a ( s ) ( R+ sL ) I a ( s )=[ 1
sL+ R ] [ U ( s )−Eb ( s ) ] 2.7
The Mechanical System Dynamics is given by:
Eb ( s ) = Ke ω ( s )T ( s ) =Ke I a ( s ) 2.8
But T (t) is given by:
T ( t )=J dω ( t )
dt +bω ( t ) 2.9
Where b is the viscous friction coefficient of the motor, and J is the moment of inertia of
the motor. Rewriting the mechanical equation (2.9) as input output equation gives:
T ( s ) = [ sJ +b ] ω ( s ) ⟹ ω ( s ) = [ 1
sJ + b ] T ( s)
Consequently, the motor torque input to rotational speed transfer function changes.
Giving equation 2.9 below:
ω ( s )
Tm ( s ) =
1
J
s + b
J
2.9
Equation (2.9) is a first order system. The block diagram for equations (2.7), (2.8) and
(2.9) is given in figure 2.2 below:
I a ( s ) =[ 1
sL+R ] [ U ( s ) −Eb ( s ) ] ; T ( s ) =Ke Ia ( s ) ;ω ( s ) = [ 1
sJ + b ] T ( s)
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Figure 2. 2: The system block diagram.
From figure 2.2, the dc motor has an inherent feedback from the CEMF. This improves
system stability by adding an electromechanical damping (Anon., 2017). To get the
transfer function from the input voltage of armature to the output speed of the motor, the
feedback formula is used to reduce the block diagram as:
ω ( s )
U ( s ) = G ( s )
1+G ( s ) H ( s )
Where:
G ( s )=Kt . [ 1
sL+ R ]. [ 1
sJ +b ]= kt
( sL+ R ) ( sJ +b ) H ( s )=Ke
Thus, the transfer function is given as:
G ( s ) = ω ( s )
U ( s ) =
kt
( sL+R ) ( sJ +b )
1+ [ kt
( sL+R ) ( sJ +b ) ] Ke
¿ kt
( sL+ R ) ( sJ +b ) +Kt Ke
2.10
Equation (2.10) is the transfer function of a DC motor.
Part b: The root locus performance
Equation (2.10) can also be written as:
G ( s ) = Kt
JL s2 + ( Lb+ JR ) s+ ( Rb+ Kt Ke ) 2.11
The physical parameters for the assignment are:
J=0.1 kg . m2 :b=0.02 NMs : K e=0.02 V
rea /sec: K t=0.02 Nm / Amp : R=3.5Ohm : L=0.95 H
Inputting the given values in equation (2.11) gives the transfer function. From the
MATLAB the transfer function is given by:
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G ( s )= 0.02
0.095 s2+0.369 s+ 0.0704
The following MATLAB code finds the root locus of the system described by equation
(2.11)
Figure 2.3 below shows the root locus of the DC motor control given above:
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Figure 2. 3: The root locus of DC motor
The root locus designs are used in prediction of closed-loop responses from the root
locus plots which describes possible closed-loop poles location and are drawn from the
open-loop transfer functions (Thomas & Poongod, 2009). Thus, modification of root
locus is impossible by adding poles and zeros through the controller for achieving the
desired closed-loop responses of the system (Melkin, 2017). From the root locus of the
system are
p1=−3.6830
p2=−0.2012
The poles are on negative, thus the system is stable. Also, both open-loop poles are
closely located to the left, thus the poles affects the closed-loop dynamics.
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Part c: PID tuning.
PID controllers are generic control loop feedbacks mechanisms used widely in
industrial control systems which are majorly usually used feedback controllers. PID
controllers calculates the error values that exist between measured process variables
and the desired responses. The PID controller tries to minimize the errors by correcting
the process control inputs. The controller algorithms has three (3) parameters which
includes: (I) the proportional K p constant which depends on the present error. In DC
motor K pis used for increasing the response speed system and reducing the steady-
state error. (II). the integral Ki= K p
T i s depends on the accumulation of past errors, it is
used for elimination of steady-state error at all integral time constants (Owen, 2012). (III)
the derivative Kd =K p [ T d s ]value that predicts the future error, depending on the current
rate of changes. It is used for reducing the system response overshoot (Phillips &
Harbor, 2000) (Abdulameer, et al., 2016). The time constant formulae for PID controller
is given in equation (2.12) below (Abdulameer, et al., 2016):
Gc= (1+ 1
T i s +T d s )K p 2.12
Figure 2.4 below illustrates the block diagram of DC motor control system (Abdulameer,
et al., 2016)
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Figure 2. 4: DC motor PID control block diagram
Using MATLAB, the following code is used for PID tunning methods for a step
response to the input.
The output response of the DC motor is given in figure 2.5 below for the Ziegler-Nichols
tuning method.
Figure 2. 5: The Ziegler-Nichols tuning method
The control parameter of the Ziegler-Nichols tuning method are given in table 2.1 below.
Tuned Baseline
Kp 9.7632 5.7046
Ti 2.7178 1.8629
Td 0.24888 1.28
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Table 2. 1: Control parameters for the Ziegler-Nichols tuning method.
The output response of the DC motor is given in figure 2.6 below for the Chien-Hrones-
Reswick tuning method
Figure 2. 6: The Chien-Hrones-Reswick tuning method step response
The control parameter of the Chien-Hrones-Reswick tuning method are given in table
2.1 below.
Tuned Baseline
s 0.93546 5.7046
Ti 2.6683 1.8629
Td 0.66707 1.28
Table 2. 2: Control parameters for the Chien-Hrones-Reswick tuning method.
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Part d. Performance and robustness of controller.
Table 2.4 below illustrate the robustness and performance of the PID controller as
depicted from MATLAB for Ziegler-Nichols tuning method
Tuned Baseline
Rise time 2.73 seconds 4.44 seconds
Settling time 1.11 seconds 15.5 seconds
Overshoot 8.07 % 5.87 %
Peak 1.08 1.06
Gain margin Inf dB @ NaN rad/s Inf dB @ NaN rad/s
Phase Margin 74.1 deg. @ 0.58 rad/s 74.1 deg. @ 0.366 rad/s
Closed-loop stability Stable Stable
Table 2. 3: The Ziegler-Nichols tuning method robustness and performance of the DC
motor
Table 2.4 below illustrate the robustness and performance of the PID controller of the
DC motor as depicted from MATLAB for Chien-Hrones-Reswick tuning method
Tuned Baseline
Rise time 19.1 seconds 4.44 seconds
Settling time 28.6 seconds 15.5 seconds
Overshoot 0.0405 % 5.87 %
Peak 1.0 1.06
Gain margin Inf dB @ NaN rad/s Inf dB @ NaN rad/s
Phase Margin 78.0 deg. @ 0.0919 rad/s 74.1 deg. @ 0.366 rad/s
Closed-loop stability Stable Stable
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Table 2. 4: The Chien-Hrones-Reswick tuning method robustness and performance of
the DC motor
Part e: Conclusions.
In this part of the assignment, it deals with the control system of armature-controlled DC
motor. The transfer function is given with the parameter, the system is found to be
stable since the poles were found to be on the left hand side near the origin. For the PID
turning, the results illustrates that each technique has its specific merits as compared to
the other method. Taking the specified DC motor speed control transfer function in
equation (2.11), it shows that the Ziegler-Nichols method produces a faster response of
the system having acceptable overshoot whereas Chien-Hrones-Reswick method of
tuning the PID has a smaller overshoot having suitable system transient responses.
Section 3: Discretizing the DC motor in section 2.
Part a: Determining the open loop and closed loop Z-domain transfer
functions of the discretized system.
The system discussed in section 2, can be discretized into open loop and closed
loop Z-domain transfer functions. Here, the transfer function in equation (2.11) is
changed from the continuous Laplace S-domain to the discrete z-domain by use of
MATLAB. The software is used to attain the mentioned conversion via use of the c2d
command. The MATLAB command needs three (3) parameters: a system model (given
by equation 2.11), the type of hold circuit and the sampling time. In this part of
assignment, since the type of hold circuit is not given, then ZOH (Zero Order Hold) will
be assumed.
The Matlab code for is given below.
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From the MATLAB, and assuming that the sampling time is 0.001 seconds, the open
loop discrete time transfer fiction is given in equation 3.1 below as:
Gopen ( Z )= 1.051e-07 z +1.05e-07
z2−1.966 z +0.9961 3.1
And for the closed loop discrete time transfer fiction is given in equation 3.2 below:
Gclose ( Z )= 1.051e-07 z +1.05e-07
z2 −1.966 z +0.9961 3.2
It can be observed that, the z-domain of a closed loop and open loop is the same, since
equation (3.1) is equal to equation (3.2).
Part b: The root locus analysis.
Using three (3) different sapling time e.g. Ts1=0.0001, Ts2=0.001 and ts3=0.01
seconds, the following code performs the root locus of each time and finds the pole
locations.
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Figure 3.1 illustrates the root locus of discretized system with sampling time of 0.0001
seconds.
Figure 3. 1: Root locus of discretized system with Ts=0.0001 s
The pole location is: p1=1.000, p2=0.9996
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Figure 3.2 llustrates the root locus of discretized system with sampling time of 0.001
seconds.
Figure 3. 2: Root locus of discretized system with Ts=0.001 s
The pole location is: p1=0.9998, p2=0.9963.
Figure 3.3 illustrates the root locus of discretized system with sampling time of 0.01
seconds.
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Figure 3. 3: Root locus of discretized system with Ts=0.01 s
The pole location is: p1=1.000, p2=0.9996
Part c: Applying PID on the discretized system.
Using MATLAB, the PID controller can be used to discretize a continuous-time
PID controllers or by creating a discrete-time PID controllers straight. With the pid
commands, the methodologies used for discretizing the integral terms and the derivative
terms can be independently specified. In MATLAB command, pidstd() is used to create
a discrete-time controllers. The discrete time controller is given by the following
equation 3.3 below (MathWorks, 1994-2018):
C ( z ) =
( 1+ 1
Ti
IF ( z ) + Td
T d
N +DF ( z ) ) K p
Where IF(z) and DF(z) are respectively integrator formula for the integrator and
integrator formula for derivative filter given by:
DF ( z )=IF ( z ) = T s
z−1
In this section different discrete integrators formula are chosen by use of the DFormula
and IFormula input (MathWorks, 1994-2018). In MATLAB software design and
simulation sampled-data control systems can be done as shown in figure 3.4 below.
The frequency domain consideration is used to design. This design leads to a pole-
cancellation PID controls technique.
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Figure 3. 4: Block representation of discrete PID controller (Fadali & Visioli, 2013).
The following MATLAB code is used in application of PID controller in discrete time with
different sampling time as used in part b above.
From MATLAB, and
When Ts=0.0001 seconds, the followings can be deducted:
P ( z )=1.0525e-09 (z +1)
z( z−1)( z −1)
Sample time: 0.0001 seconds.
Discrete-time zero/pole/gain model
Zeros, z=-0.9999
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Poles: p1=1.0000; p2= 0.9996; p3=0
kd = 1.0525e-09
When Ts=0.001 seconds, the followings can be deducted:
P ( z ) =1.0513e-07 (z +0.9987)
z ( z−1)(z−0.9963)
Sample time: 0.001 seconds.
Discrete-time zero/pole/gain model
Zeros, z=-0.9987
Poles: p1=.9998; p2= 0.9963; p3=0
kd = 1.0513e-07
When Ts=0.01 seconds, the PID controller does not work together with the system,
since it becomes unstable. PI term in this part is essential for achieving the steady state
zero error necessity while the PD term is used for accelerating the systems response
(phillips, 2012).
Part d: Comparison between the s-domain PID and z-Domain PID.
In both controllers, they offer pneumatic or electronic functionality. PID controllers such
as auto-tuning functions, MATLAB software structure selections whether to use
interactive, parallel, series, cascaded, etc, and remotely networks configuration. In both
cases, the controllers’ implementations has to cope with performance, robustness and
stability of the system. In discrete Transform (or z-domain), algorithms has to compact
with sufficient sampling in the digital forms, process wind-ups, avoidance of proportional
and derivative kicks, velocity and positional implementations, bumpless parameter
tuning, quantization effect in integral actions, etc. when these factors are met in discete
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form, the performance and robustness is higher than that of s-domain (ResearchGate,
2015). Here I esed adaptive PID control algorithm for comparison
By use of tic and toc command in MATLAb, it is obvous that the disct=rete form tkes
less thie as compared to continuous time simulation of the given armature-controlled
DC motor.
Part e: Conclusions.
Any modification required for the PID-controllers may be implemented with discrete PID-
controllers. From part d above, it can be seen that the significant modifications for the
integrator is the ant-windup-operations for saturating the actuator, a soft mode changes
when switching from automatic to manual operations and/or manual to automatic,
bumpless parameters value varies in adaptive PID algorithm (Also for self-tuning
algorithms will behave the same) (Aalto, 2010). For example, when actuators saturate
integrator grows continuously to high values. Therefore, when designing discrete PID
controller some parameters are considered, although this type of PID is faster and more
stable. For the armature-controlled Dc motor system provided, the performance is high
and is stable as depicted in part c above.
Conclusions
The assignment consisted three (3) parts. In the first part, I was able to understand the
major benefits and drawback of PID controllers. Some applications of PI and PID
controller in day-todays life is discussed. Furthermore in this part, the advantages of
using digital systems over continuous time systems. It is found that, discrete system are
more proffered than continuous time system since discrete system has low error rate,
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low cost, high speed, etc. Also, we found how motor angular velocity control can be
relevant to complex systems in day today life.
In part two (2), the control of angular velocity of a DC motor shaft by varying an input
voltage u is illustrated. Simulation of continuous time PID control system of a DC motor
is done using MATLAB software. The system stability, robustness, response time and
other characteristics of system are simulated. A tuning PID controller is then simulated
using different metric to determine which metrics is better and preferred to control the
DC motor. It is found that, the Ziegler-Nichols method produces a faster response of the
system having acceptable overshoot while Chien-Hrones-Reswick method of tuning the
PID has a smaller overshoot having suitable system transient responses.
In the last part of the system, the DC motor is discretized and simulated in MATLAB to
compare with the continuous time system. Different metrics of discrete system is used
to determine the system response, robustness, stability and easiness of implementation
and the error rate of the system. It is found that, using discrete system, the DC motor is
more stable, more robust, less response time and less error rate as compared to
continuous time system. Although, in both continuous and discrete system, choosing
the PID controller should be chosen carefully to provide a stable and robust system
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