1 Table of Contents
List of Figures
List of Tables
List of Abbreviation
DC - Dynamic & Control
CS – Control System
1
2 Abstract
2
1 Introduction
A control system is a system that provides the desired response by controlling the output against
the input which feeds to the system. A typical control system's inputs can be varying by the
certain mechanisms and it helps to get the desired output from the system. These systems could
be a group of electronic or mechanical devices which used control loops to control other
systems or devices and these systems can be automated by using computers. In recent years,
control systems have played a central role in the development and advancement of modern
technology and civilization. Virtually every aspect of our daily lives is affected by more or less
some type of control system. The below figure shows a simple block diagram of the control
system and the control system is represented by a single block.
Figure 1: Simple block diagram of a control system
The main characteristic of the control system is it should be having a clear mathematical
relationship between the input signal and output signal of the controlling system. When the
controlling system having relationship between input and output which represented by linear
and proportionality, the system is called a linear control system. If the control system’s input
and output cannot be represented by a linear and proportionality signals or else input and output
signals are connected by some nonlinear relationships, the system is referred as nonlinear
control system. A high-quality control system’s having several characteristics to uplift its
functionality in a desired manner. Accuracy a one of the most important requirements of a
control system and it is the measuring tolerance of the system or the instrument which
determines the limits of errors that occur when the system is configuring under normal
operating conditions. The control system accuracy can be enhanced by adding feedback
elements into the system. Stability is another important characteristic of a control system. A
linear system is stable if the output is constrained or bounded to each given bounded input. Not
only that, but also a linear system is stable if the transient response converging to zero.
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If a control system generating an unwanted signal is calla as noise. Better performing control
systems could be able to mitigate or reduce the noise. The operating frequency range of a
control system is determined the controlling bandwidth of the control system. If the system is
in a good performing level, the bandwidth should be as large as possible for the frequency
response of the system. Speed is the time the control system takes to achieve steady state
response from transient response. If the control system is performing well the speed should be
high and which having small transient period. The control systems having few oscillation or
steady oscillation output such systems indicates stable control system.
There are various types of control systems and those systems can be used to control the
position, acceleration, temperature, pressure, voltage and current etc. For an example, an air
condition system in a room is cooling the room continuously as long as the power supply is
connected through switch. If the power supply is switched on, the temperature of the room is
gradually drop down and after achieving the required temperature in the room a person can
manually operate and can be switch off the air condition. When the room temperature getting
high due to the impact of the ambient temperature, again the person in the room can switch on
the air condition system and allow the room temperature gets to desired temperature. In such
case the air condition system is operator by manually by the person and this system is a good
example for manual control system.
Assume this system is improved further and now the room temperature is monitoring by a
sensor. Here the sensor measures the temperature difference between the room temperature and
desired temperature. If there are any differences between them, the air conditioner starts to
work until the temperature difference becomes zero or the predetermined level. When the room
temperature becomes to the desired level the air conditioner is again stopping its functioning.
In this case the air conditioner system function, depending upon the difference between, room
temperature and desired temperature. This difference is called the error of the system. This
error signal is feedback to the system to control the input. These control systems are a good
example for automatic control systems.
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2 Typical types of the control systems
According to the functional types, control systems can be classified to two types. These are
open loop control systems and closed loop control systems.
2.1 Open loop control systems
If a system’s controlling action is totally independent of the system output, such systems called
open loop control systems. The manual control systems as explained in above section also an
open loop control system. The open loop control system can be illustrated as below block
diagram.
Figure 2: Block diagram of the open loop control system
Practical examples of control systems are Electric hand drier, Automatic washing machines,
Time-based cloths driers, Stereo system volume controller, Light switches, Bread toaster
machines etc. These systems are runes as long as the system inputs are supplied irrespective to
its generating output signal.
Advantages of open loop systems
Disadvantages of open loop systems
Simple in construction and design
Inaccuracy
Easy to maintain.
Unreliable
Economical.
Input cannot be control against the output
Table 1: Advantages and Disadvantages of the open loop control systems
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2.2 Closed loop control systems
If a system’s controlling action can be governing and rectify by the output in a desired manner,
such systems called closed loop control systems. In such cases the input signal could be
adjusted itself based on the generated output by using the feedback signal. Open loop control
systems also could be converting to closed loop by adding feedback signal to the system. This
system added feedback automatically makes suitable changes in input and eventually it changes
the output of the system into desired manner against to the external disturbance which may
undergo by the system. Therefore, closed loop control systems are automatic control systems.
The closed loop control system can be illustrated as below block diagram.
Figure 3: Block diagram of the closed loop control system
Practical examples for the closed loop control systems are Automated electric irons, Water
level controlling systems, Car cooling systems, Automatic volume controllers, Automatic
lighting systems, Servo Voltage Stabilizer controlling systems etc. These system’s inputs are
controlling by the output by the feedback signal until the output is maintain the desired level.
Advantages of closed loop systems
Disadvantages of closed loop systems
More Accurate systems
These systems are costly
This system is less affected by noise
Required more maintenance
Facilitates automation
Complicated to design
Table 2: Advantages and Disadvantages of the Closed loop control systems
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2.3 Introducing of Rectilinear experimental systems and apparatuses
This system is often found in textbooks on dynamics and control and serves as a benchmark
for evaluating control methods. The mechanism features adjustable masses, interchangeable
springs and adjustable air damping. The dynamic properties are generally the rectilinear
equivalent of those of the torsion device with additional parameter adaptability.
Figure 4: Model 210 Rectilinear Plant
The Model 210 has three mass cars that can be loaded with brass weights and connected in
different configurations using springs of different stiffness. The adjustable damper can be used
to dampen the system. A single drive motor drives the system through the first mass carriage,
and position measurements are performed by quadrature encoders.
2.4 Introducing of Torsional experimental systems and apparatuses
This laboratory experiment studies the dynamics of a single and multiple torsion system with
degrees of freedom. The effects of different control configurations are studied in a later part
from the laboratory. The system we use in this experiment is Model 205. Figure 5 shows the
Model 205 Torsion Experiment consisting of three disks supported by a torsional rigid shaft
that is suspended vertically on anti-friction ball bearings.
The shaft is driven by a brushless servo motor connected via a rigid belt (negligible pulling
flexibility) and pulley system with a 3:1 speed reduction ratio. An encoder on the base of the
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shaft measures the angle displacement of the first disk. Two additional encoders measure the
displacements of the other two drives, as shown. The torsion mechanism represents many
physical plants including rigid bodies; flexibility in drive shafts, gears and belts; and coupled
discrete vibrations with actuator at the input of the drive and sensor side by side or at flexibly
coupled output (non-collocated).
Figure 5: Model 205 Torminal Plant
In practice, the system parameters of a device, such as inertia, spring constant, and damping
ratios are often unknown. In this part of the lab, these are unknown parameters are determined
using a process called system identification. this same one parameter will be used later to
implement different controllers.
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2.5 Potential applications of Rectilinear plant systems
Cam attchedements with mechanisms are widely used in many types of modern machines,
heavy machunes, engines and hibrid systems because of their excellent properties for operation
speed, motion accuracy, structural rigidity, and production cost. Below figure 6 illustrates a
typical model of a cam-follower system.
Figure 6: Cam-follower system.
This model consists of two springs, a mass and a dashpot. The rotating cam surface profile
inputs a desired displacement profile x(t) to the follower via a flexible connection modelled by
spring k1. The output mass m models the mass of the follower. Additional flexibility and
damping are modelled by spring k2 and dashpot b. Ideally, the follower response y(t) follows
the desired displacement. However, such flexible systems often react with unwanted vibrations.
Vehicle suspension systems are also an examples
for rectilinear systems. Figure 7 showing a typical
vehicle suspension system and this system also
attched with spring and dashpot to absorb the
vibration whichs generating by wheel while it
moves on uneven surface on the road. This system
supports to keep the desired behaviour on the car
body while the vibration is absorbing by spring and
damping system.
Figure 7: Vehicle suspension system
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2.6 Potential applications of Torsional plant systems
An optical rotary encoder uses optical sensing technology with a rotary code and a pattern on
it. The incremental optical encoder is the most widely used of all rotary encoders due to its low
cost and ability to provide cues that can be easily interpreted to provide movement-related
information, such as: speed or change of position. Figure 8a shows how incremental optical
encoders provide information at the instantaneous position of a rotating shaft through two
output square wave cycles per to produce increase in axis movement. The two output
waveforms are 90 degrees out of phase, which is usually called quadrature signals. In Figure
8b, the quadrature signals are digital signals producing two channels, channel A and channel
B. When the optical encoder rotates clockwise, channel A leads to channel B. If: optical
encoder rotates counterclockwise, channel B leads channel A.
Figure 8: Optical incremental encoder
Another example is Hard Disk Driver. The major
components in a modern Hard Disk includes device
enclosure, which usually consists of a base plate and
a cover to provide supports to the spindle, actuator,
and electronics card. Disk stack assembly, where
several disks are stacked on the spindle motor shaft
and rotate at up to 15,000 rotations per minute
(RPM) in high end 3.5-inch drives.
Figure 9: Hard Disk Drive head position control
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3 System modelling
3.1 One degree of freedom rectilinear plant dynamic model
A linear system is a system that has two mathematical properties: Homogeneity and Additivity.
while using electricity we can correct a linear system by using resistors, capacitors, chokes. In
rectilinear factories, there is also another method that can correct a linear system using a
transfer function. With this method, the transfer function in the mechanical system can be
obtained in terms of force-displacement (i.e. force is written in terms of displacement)
Figure 9: Single degree of freedom mass spring damper system
F- Applied Force
k- Spring Constant
c- Damper Coefficient
m- Trolley mass
t- Force applying time
x- Displacement of the trolley

Assuming the trolley doesn’t undergo an any other friction forces.
By Apply Newton’s 2nd Low to the system, ⃗⃗⃗⃗F = ma⃗
dx(t)
d2 x(t)
F(t) − c.
− k. x(t) = m.
dt
dt 2
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Hence,
d2 x(t)
dx(t)
F(t) = m.
+
c.
+ k. x(t)
dt 2
dt
By transferring the equation time domain to Laplace domain,
F(s) = m. s2 . X(s) + c. s. X(s) + k. X(s)
F(s) = X(s)(m. s2 + c. s + k)
Multiplying the both sides of the equation by 1/m constant,
F(s)⁄ = X(s) (s2 + (c. s)⁄ + k⁄ )
m
m
m
Rearranging the equation by cross multiplying the equation to get standard form,
X(s)
1/m
= 2
F(s)
s + (c/m) . s + (k/m)
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3.2 Torsional One degree of freedom plant model
The model of the torsion system can be obtained by using three elements, moment of inertia
(J) of the mass, damping coefficient of viscous friction (c), torsion spring with stiffness (K).
When a torque was applied to a torsion system, it was counteracted by an opposing torque due
to friction, moment of inertia and elasticity of the system.
Figure 10: One DOF Torsional Plant schematic representation
 - disk angle of rotation,
J - disk moment of inertia,
c – Coefficient of viscous friction
T(t) – torque produced by the electric motor
By Apply Newton’s 2nd Low to the system, ⃗⃗⃗⃗F = ma⃗
dθ(t)
d2 θ(t)
T(t) − c.
= J.
dt
dt 2
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Hence,
d2 θ(t)
dθ(t)
T(t) = J.
+
c.
dt 2
dt
By transferring the equation time domain to Laplace domain,
T(s) = J. s2 . θ(s) + c. s. θ(s)
Multiplying the both sides of the equation by 1/J constant,
T(s)⁄ = θ(s) (s2 + (c. s)⁄ )
J
J
Rearranging the equation by cross multiplying the equation to get standard form,
θ(s)
1/J
= 2
T(s)
s + (c/J) . s
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4 Simulation Analysis
4.1 Open loop Tranfe function block diagram in MATLAB simulation for
rectilinear plant dynamic
4.1.1 Open loop transfer function output graph
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4.2 Closed loop proportional control transfer function block diagram in
MATLAB simulation for rectilinear plant dynamic
4.2.1 Closed loop proportional control tranfer function step input condisiones
4.2.2 Closed loop proportional control transfer function block diagram
Below shows MATLAB Simulink block diagram for Kp=0.05, the same block diagram was
used for Kp=0.2 and Kp=4 gain values as well.
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4.2.3 Closed loop proportional control transfer function output graph
Kp=0.05
According to the graphe the Overshoot is - 0.496% , Rise time – 219.391ms and stady state
error is 8.687.
Kp=0.2
As per graphe the Overshoot is– 0.487% , Rise time –91.43ms and stady state error is 3.84.
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Kp=4
As per graphe the Overshoot– 30.921% , Rise time – 6.506ms stady state error is 0.2569.
4.2.4 Observation sheet with MATLAB Closed loop proportional control transfer
function
Simulation
No.
1
2
3
Kp Gain
Value
0.05
0.2
4
Rise Time
(ms)
219.391
91.43
6.506
Overshoot
0.496
0.487
30.921
Steady
State Error
8.687
3.84
0.2569
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4.3 PID Conroller Closed loop transfer function block diagram
4.3.1 PID controller and Closed loop tranfer function step input condisiones
4.3.2 PID controller and Closed loop transfer function block diagram
Below shows MATLAB Simulink block diagram for (Kp=0.08, Kd=0.002, Ki=0.8) values
and the same block diagram was used for other PID gain values as well.
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4.3.3 Closed loop transfer function output graphs
Simulation No.1
Simulation No.2
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Simulation No.3
Simulation No.4
21
Simulation No.5 & 6
Simulation No.7
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4.3.4 Observation sheet with MATLAB Simulink under PID condolers
Simulation
No.
1
2
3
4
5
6
7
Kp Gain
Value
0.08
0.08
0.08
0.08
0.08
0.08
0.08
Kd value
Ki Value
0.002
0.4
1
2
0.6
0.6
0.6
0.8
0.007
0.007
0.007
0.002
0.002
0.004
Rise Time
(ms)
209.494
7.005
7.415
7.224
7.224
7.419
Overshoot
29.221%
0.521%
0.538%
0.526%
0.526%
0.515%
Steady
State Error
2.979
9.635
10
9.658
9.763
9.763
9.873
4.4 Proportional-based position control Simulink analysis for torsional
system
4.4.1 Proportional-based position control Simulink step input condisiones
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4.4.2 Proportional-based position control Simulink analysis for torsional system transfer
function block diagram
Below shows MATLAB Simulink block diagram for Kp=0.004 value and the same block
diagram was used for other Kp gain values as well.
4.4.3 Proportional-based position control Simulink analysis for torsional system output
graphs
Simulation No.1
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Simulation No.2
Simulation No.3
4.4.4 Observation sheet with MATLAB Simulink under Proportional-based position
control Simulink analysis
Simulation
No.
1
2
3
Kp Gain
Value
0.004
0.008
0.012
Rise Time
(s)
8.587
7.953
7.792
Overshoot
6.989%
8.152%
4.737%
Steady
State Error
4.712
4.712
4.712
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4.5 PD-Based position control Simulink analysis for torsional system
4.5.1 PD-Based position control Simulink step input condisiones
4.5.2 PD-Based position control Simulink system transfer function block diagram
Below shows MATLAB Simulink block diagram for Kp=0.02 and Kd=0.008 value and the
same block diagram was used for other Kp and Kd gain values as well.
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4.5.3 PD-Based position control Simulink system output graphs
Simulation No.1
Simulation No.2
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Simulation No.3
4.5.4 Observation sheet with MATLAB Simulink under PD-Based position control
Simulink system analysis
Simulation
No.
1
2
3
Kp Gain
Value
0.02
0.02
0.02
Kd Gain
Value
0.008
0.004
0.006
Rise Time
(s)
7.237
7.104
7.201
Overshoot
0.510%
0.515%
0.510%
Steady
State Error
4.712
4.712
4.712
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4.6 Proportional feedback-based Speed Controller torsional System
4.6.1 Proportional feedback-based Speed Controller torsional System step input
condisiones
4.6.2 Proportional feedback-based Speed Controller torsional System transfer function
block diagram
Below shows MATLAB Simulink block diagram for Kp=0.02 value and the same block
diagram was used for other Kp gain values as well. Since the Torque value is not giving much
changed to the final output diagrams under given Kp values, the torque (𝜏 = 1) value taken as
1.
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4.6.3 Proportional feedback-based Speed Controller torsional System output graphs
Simulation No.1
Simulation No.2
30
Simulation No.3
4.6.4 Observation sheet with MATLAB Simulink under Proportional feedback-based
Speed Controller torsional System
Simulation
No.
1
2
3
Kp Gain
Value
0.02
0.2
2
Rise Time
(ms)
568.865
527.926
527.926
Overshoot
68.644%
62.727%
62.727%
Steady
State Error
0.1745
0.1747
0.1747
5 Discussion
By investigating these rectilinear and torsional systems modelling and experimenting with
them with the help of MATLAB software Simulink gives vital advantage to observe and
understand these systems behaviour and to be able to analysing them by controlling the PI, PD
and PID gain values. This entire assessment is undergo based on one degree of freedom plants
dynamic modelling and MATLAB Simulink upon them with PI, PD and PID condoling.
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In first simulation, the
For the first Proportional- Based position Control system different values of increasing Kp
gains, rise time gone decrease and overshoot was increase. Overall steady state error was very
high when the Kp Gain Value taken smaller.
* For the second PID-based position control system approach can observe that seven different
values of the Ki and Kd, for 0.04 Kp stable Proportional gain. Smallest rise time can be taken
by Kd = 2 and the Ki=0.007, and the Highest Rise time can be taken as Kd=1 and Ki=0.007,
there for can observe basically two different rise time simulate with the Derivative value of Kd
of the system. Peek overshoot percentage were high at the fourth simulation when the value of
rise time reach to its lowest value. For the Overshoot was lowest stable at the range of 0.8 of
Kd value and this can be observe that overshoot depend with this Kd value. By taking values
compare to the proportional Position control system there were small steady state errors can be
observe in this PID based position Control system varies with the integral controller (Ki).
*Third option for a torsional system used of proportional based position control system below
analysis can be obtain. Lowest rise time goes with the higher Kp gain value and the maximum
Overshoot can be taken by the lowest Kp value, and Overshoot takes miners value. Specially
there can be obtained no Steady state errors while increasing and decreasing Kp gains.
* For the PD based Position control torsional system different values of simulation draft can
be explored some narrow increase between the step input and end with steady state. Narrow
increase varies with the different Kd values. Here Kp value was set to be as 0.02 and varied by
Kd values thus the rise time gone high at the lowest Kd value and the different lower value of
a rise time can be obtained with the 0.004=Kd value. Overshoot varies with the Kd value and
80% maximum can obtained with Higher Kd values. And here also no steady state error. And
here can obtained some damping in the system.
*Speed Control for a torsional system here used derivative before the error signal goes to the
Kp proportional Compared with a step move of 0.174 Radian/s. With the different Kp values.
For a High value of Kp gain rise time taken slow and the overshoot taken as higher percentage.
With Simulink draft steady state errors can be neglected. By summarizing this observation
following characteristics of P, I and D controllers can be briefly discussed.23 A proportional
Controller (Kp) have the effect of reducing the rise time of the control system. But that was not
effected directly to the steady state error. An integral control (Ki) have the effect of reducing
the stability of the system, by reducing the overshoot, also rise time will be effect as when the
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Ki increase but improvement of total transient response can be obtained. A derivative Control
(Kd) can increased the stability of the system by reducing the overshoot. And also can improve
the transient response with the little effect of rise time. A PD Controller can be effect of the
damping of the system, But the steady state response not be effected. A PI Controller overall
settling time can observe as high, when the use of PI could improve the system stability and
reduce steady state error of the system at same time. For a PID decrease the whole system
settling time and transient response of a system can control for a stable position from the PID.
And this can be eliminate the steady state error of the system. When taking a PD controller has
option to decreased the system settling time considerably, However to control the steady state
error, ant Kd value must be high and this led to be decrease the response times of the system,
so this is more suitable for the position control system rather than using PID control system.
Without integral controller PD is a type zero system hence it have a finite steady state error to
a unit step Input. Effect of PID Controller
 It improved stability as well as to decrease Steady State Error.
 It adds a pole at origin which increases type of system which result into reduction of steady
state error.
 It adds 2 zeroes in LHP, one finite zero to avoid effect on stability & other zero to improve
stability of system.
For the use of PID controller for an actual plant should be consider about some matters that
can affect weather PID can Used or not The performance of the PID Controller in nonlinear
system is variable because PID controllers are linear. The derivative term Kd is susceptible to
Noise disturbance. A small amount of measurement or process noise can cause large amounts
of change in the output. It is often helpful to filter the measurements with a low-pass filter in
order to remove higherfrequency noise components For HVAC system a system which heats
much faster than it cools can’t handle because it’s non-symmetric system24
6 Conclusion and Future work
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Using of a PID Controller can be concluded as for both second order linear and torsional
systems can be control by different controlling system methods and PID based Controller
system is found to better result for linear system compared to a proportional system. For a
Torsional system PD controller can yield still better results by employing different position
control approach. Fast reaction on change of the controller input (Kd), increase in control signal
to lead error to zero (Ki) and suitable action inside control error area to eliminate oscillations
(Kp
7 Reference
1)R.C.Dorf and R.H.Bishoop “Modern Control Systems”,9th Edition;Prentice Hall, 2000.
2)B.C. Kuok “Automatic Control Systems”, John Wiley & Sons, Inc., 7thEdition, 1995.
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3)Gene Franklin, J.D. Powell, Abbas Emami-Naeini “Feedback Control ofDynamic
Systems”, Prentice Hall, 2005
5)Katsuhiko Ogata “Matlab for Control Engineers”, Prentice Hall, 2007.
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