Part of Lecture Series on Automatic Control Systems delivered by me to Final year Diploma in Engg. Students. Equally useful for higher level. Easy language and step by step procedure for drawing Bode Plots. Three illustrative examples are included.
Root locus is a graphical representation of the closed-loop poles as a system parameter is varied.
It can be used to describe qualitatively the performance of a system as various parameters are changed.
It gives graphic representation of a system’s transient response and also stability.
We can see the range of stability, instability, and the conditions that cause a system to break into oscillation.
Part of Lecture Series on Automatic Control Systems delivered by me to Final year Diploma in Engg. Students. Equally useful for higher level. Easy language and step by step procedure for drawing Bode Plots. Three illustrative examples are included.
Root locus is a graphical representation of the closed-loop poles as a system parameter is varied.
It can be used to describe qualitatively the performance of a system as various parameters are changed.
It gives graphic representation of a system’s transient response and also stability.
We can see the range of stability, instability, and the conditions that cause a system to break into oscillation.
Poles and Zeros of a transfer function are the frequencies for which the value of the denominator and numerator of transfer function becomes zero respectively
This presentation explains about the introduction of Nyquist Stability criterion. It clearly shows advantages and disadvantages of Nyquist Stability criterion and also explains importance of Nyquist Stability criterion and steps required to sketch the Nyquist plot. It explains about the steps required to sketch Nyquist plot clearly. It also explains about sketching Nyquist plot and determines the stability by using Nyquist Stability criterion with an example.
This presentation explains about the introduction of Polar Plot, advantages and disadvantages of polar plot and also steps to draw polar plot. and also explains about how to draw polar plot with an examples. It also explains how to draw polar plot with numerous examples and stability analysis by using polar plot.
This presentation gives complete idea about time domain analysis of first and second order system, type number, time domain specifications, steady state error and error constants and numerical examples.
Poles and Zeros of a transfer function are the frequencies for which the value of the denominator and numerator of transfer function becomes zero respectively
This presentation explains about the introduction of Nyquist Stability criterion. It clearly shows advantages and disadvantages of Nyquist Stability criterion and also explains importance of Nyquist Stability criterion and steps required to sketch the Nyquist plot. It explains about the steps required to sketch Nyquist plot clearly. It also explains about sketching Nyquist plot and determines the stability by using Nyquist Stability criterion with an example.
This presentation explains about the introduction of Polar Plot, advantages and disadvantages of polar plot and also steps to draw polar plot. and also explains about how to draw polar plot with an examples. It also explains how to draw polar plot with numerous examples and stability analysis by using polar plot.
This presentation gives complete idea about time domain analysis of first and second order system, type number, time domain specifications, steady state error and error constants and numerical examples.
ME-314 Introduction to Control Engineering is a course taught to Mechanical Engineering senior undergrads. The course is taught by Dr. Bilal Siddiqui at DHA Suffa University. This lecture is about time response of systems derived by inspection of poles and zeros. First and second order systems are considered, along with higher order and nonminimum phase systems
Here's the continuation of the report:
3.2.1 Parallel Plate Capacitor (continued)
As the IV fluid droplets move between the plates of the capacitor, the capacitance increases due to the change in the dielectric constant, resulting in the observation of a peak in capacitance.
3.2.2 Semi-cylindrical Capacitor
The semi-cylindrical capacitor consists of two semi-cylindrical conductors (plates) facing each other with a gap between them. The gap between the plates is filled with a dielectric material, typically the IV fluid.
When a potential difference is applied across the plates, electric field lines form between them. The dielectric material between the plates enhances the capacitance by reducing the electric field strength and increasing the charge storage capacity.
3.2.3 Cylindrical Cross Capacitor
The cylindrical cross capacitor is composed of two cylindrical conductors (rods) intersecting at right angles to form a cross shape. The space between the rods is filled with a dielectric material, such as the IV fluid.
When a potential difference is applied between the rods, electric field lines form between them. The dielectric material between the rods enhances the capacitance by reducing the electric field strength and increasing the charge storage capacity, similar to the semi-cylindrical design.
3.3 Advantages of Capacitive Sensing Approach
Capacitive sensing for IV fluid monitoring offers several advantages over other automated monitoring methods:
1. Non-invasive operation: The sensors do not require direct contact with the IV fluid, reducing the risk of contamination or disruption to the therapy.
2. High sensitivity: Capacitive sensors can detect minute changes in capacitance, enabling precise tracking of IV fluid droplets.
3. Low cost: The sensors can be constructed using relatively inexpensive materials, making them a cost-effective solution.
4. Low power consumption: Capacitive sensors typically have low power requirements, making them suitable for continuous monitoring applications.
5. Ease of implementation: The sensors can be easily integrated into existing IV setups without significant modifications.
6. Stable measurements: Capacitive sensors can provide stable and repeatable measurements across different IV fluid types.
Chapter 4: Experimental Setup and Results
4.1 Description of Experimental Setup
To evaluate the performance of capacitive sensors for IV fluid monitoring, an experimental setup was constructed. The setup included various capacitive sensor designs, such as parallel plate, semi-cylindrical, and cylindrical cross capacitors, positioned around an IV drip chamber.
The sensors were connected to a capacitance measurement circuit, which recorded the changes in capacitance as IV fluid droplets passed through the sensor's electric field. Multiple experiments were conducted using different IV fluid types and flow rates to assess the sensors' accuracy, repeatability, and sensitivity.
4.2 Measurements with
ELEG 421 Control Systems Transient and Steady State .docxtoltonkendal
ELEG 421
Control Systems
Transient and Steady State
Response Analyses
Dr. Ashraf A. Zaher
American University of Kuwait
College of Arts and Science
Department of Electrical and Computer Engineering
Layout
2
Objectives
This chapter introduces the analysis of the time response of different
control systems under different scenarios. Only first and second order
systems will be considered in details using analytical and numerical
methods. Extension to higher order systems will be developed. Both
transient and steady state responses will be evaluated. Stability analysis
will be analyzed for different kinds of feedback, while investigating the
effect of both proportional and derivative control actions on the
performance of the closed-loop system. Finally systems types and
steady state errors will be calculated for unity feedback.
Outcomes
By the end of this chapter, students will be able to:
evaluate both transient/steady state responses for control systems,
analyze the stability of closed-loop LTI systems,
investigate the effect of P and I control actions on performance, and
understand dominant dynamics of higher order systems.
Dr. Ashraf Zaher
Introduction
3
Test signals
Transient response
Steady state response
Analytical techniques, and
Numerical (simulation) techniques.
Stability (definition and analysis methods),
Relative stability, and
Effect of P/I control actions on stability and performance.
Summary of the used systems:
First order systems,
Second order systems, and
Higher order systems.
Dr. Ashraf Zaher
Test Signals
4 Dr. Ashraf Zaher
Impulse function:
Used to simulate shock inputs,
Laplace transform: 1.
Step function:
Used to simulate sudden disturbances,
Laplace transform: 1/s.
Ramp function:
Used to simulate gradually changing inputs,
Laplace transform: 1/s2.
Sinusoidal function(s):
Used to test response to a certain frequency,
Laplace transform: s/(s2+ω2) for cos(ωt) and ω/(s2+ω2) for sin(ωt).
White noise function:
Used to simulate random noise,
It is a stochastic signal that is easier to deal with in the time domain.
Total response:
C(s) = R(s)*TF(s) = Ctr(s) + Css(s) → c(t) = ctr(t) + css(t)
Fundamentals
5 Dr. Ashraf Zaher
Definitions:
Zeros (Z) of the TF
Poles (P) of the TF
Transient Response (Natural)
Steady State Response (Forced)
Total Response
Limits:
Initial values
Final values
Systems (?Zs):
First order (one P)
Second order (two Ps)
Higher order!
More:
Stability and relative stability
Steady state errors (unity feedback)
First Order Systems
6 Dr. Ashraf Zaher
TF:
T: time constant
Unit Step Response:
1
1
)(
)(
+
=
TssR
sC
)/1(
11
1
1
1
11
)(
TssTs
T
sTss
sC
+
−=
+
−=
+
=
Ttetc /1)( −−=
632.01)( 1 =−== −eTtc
T
e
Tdt
tdc Tt
t
11)( /
0
== −
=
01)0( 0 =−== etc
11)( =−=∞= −∞etc
First Order Systems.
HW3 – Nichols plots and frequency domain specifications FORM.docxsheronlewthwaite
HW3 – Nichols plots and frequency domain specifications
FORMULAE FOR SECOND ORDER SPECIFICATIONS:
If the higher order closed-loop system has two dominant poles (and no dominant zeros) it can be approximated by a second order
system. Here are the formulae for transient specifications for second order systems or approximate second order systems:
1. Natural frequency: 𝜔𝑛 (distance of pole to origin)
2. Damping ratio: 𝜁 (related to angle of the pole)
3. The closed-loop dominant poles: 𝑝1, 𝑝2 = −𝜎 ± 𝑗𝜔𝑑 = −𝜔𝑛𝜁 ± 𝑗𝜔𝑛√1 − 𝜁
2
4. Angle of the poles with negative real axis: 𝛽 = arccos 𝜁 = cos−1 𝜁
5. The damped frequency of oscillations is the imaginary part of the roots: 𝜔𝑑
6. The exponential time constant is the reciprocal of the real part of the roots: 𝜏 =
1
𝜎
=
1
𝜔𝑛𝜁
7. Distance of the pole from origin is √𝜎2 + 𝜔𝑑
2 = 𝜔𝑛 = natural frequency.
8. Settling time: 𝑇𝑠 =
4
𝜎
=
4
𝜔𝑛𝜁
9. Peak time: 𝑇𝑝 =
𝜋
𝜔𝑑
10. Maximum overshoot: 𝑀𝑝 = 𝑒
−
𝜋𝜁
√1− 𝜁2
= 𝑒−𝜋 cot 𝛽 = 𝑒
−𝜋
𝜎
𝜔𝑑
11. To get percent overshoot multiply 𝑀𝑝 with 100.
Similar characteristics in the frequency domain are:
1. Resonant frequency of the closed loop system: 𝜔𝑟 = 𝜔𝑛√1 − 2𝜁
2
2. Resonant peak amplitude the closed loop system: 𝑀𝑟 =
1
2𝜁√1−𝜁2
Note that there is no resonant frequency or peak when the damping ratio is greater than
1
√2
= 0.707
3. Cutoff frequency the closed loop system: 𝜔𝑐 = 𝜔𝑛√1 − 2𝜁
2 + √1 + (1 − 2𝜁2)2
Cutoff frequency tells about the bandwidth and is related to speed of the closed loop system.
Higher cut-off frequency → higher bandwidth → lower settling time, rise-time, peak time.
4. Damping ratio of the closed loop system 𝜁 is related to the phase margin of the open loop system:
𝜙 = tan−1
2
√√4+
1
𝜁4
−2
(𝑖𝑛 𝑟𝑎𝑑𝑖𝑎𝑛𝑠) ≈ 100𝜁 (𝑖𝑛 𝑑𝑒𝑔𝑟𝑒𝑒𝑠)
The approximate formula for phase margin, 100𝜁, works only for phase margins up to about 60 degrees.
Errors specifications:
1. For a typical system with a plant G and controller K.
The error is 𝐸(𝑠) =
1
1+𝐺𝐾
𝑅
The steady state error is 𝑒(∞) = lim
𝑠→0
𝐸(𝑠)𝑠 = lim
𝑠→0
𝑠𝑅
1+𝐺𝐾
2. Steady state error for a step input R(s) = 1/s is:
𝑒(∞) = lim
𝑠→0
1
1 + 𝐺𝐾
=
1
1 + lim
𝑠→0
𝐺𝐾
=
1
1 + 𝑑𝑐 𝑔𝑎𝑖𝑛 𝑜𝑓 𝐺𝐾
=
1
1 + 𝐾𝑝
3. Steady state error for a ramp input R(s) = 1/s2 is:
𝑒(∞) = lim
𝑠→0
1
𝑠(1 + 𝐺𝐾)
=
1
lim
𝑠→0
𝑠𝐺𝐾
=
1
𝐾𝑣
If GK has no integrator 𝐾𝑣 is 0 (ramp error is infinite). If GK has 1 integrator 𝐾𝑣 is finite, it is the x-intercept (frequency
intercept) of the asymptote line from the left half of the bode(GK). If GK has more integrators 𝐾𝑣 is infinite.
4. To reduce the error due to reference command (and disturbance dy) of frequency 𝜔 by a factor of F: |𝐺(𝑗𝜔)| > 𝐹 + 1 .
5. To reduce the error due to a sinusoidal noise command of frequency 𝜔 by a factor of F: |𝐺(
Democratizing Fuzzing at Scale by Abhishek Aryaabh.arya
Presented at NUS: Fuzzing and Software Security Summer School 2024
This keynote talks about the democratization of fuzzing at scale, highlighting the collaboration between open source communities, academia, and industry to advance the field of fuzzing. It delves into the history of fuzzing, the development of scalable fuzzing platforms, and the empowerment of community-driven research. The talk will further discuss recent advancements leveraging AI/ML and offer insights into the future evolution of the fuzzing landscape.
Immunizing Image Classifiers Against Localized Adversary Attacksgerogepatton
This paper addresses the vulnerability of deep learning models, particularly convolutional neural networks
(CNN)s, to adversarial attacks and presents a proactive training technique designed to counter them. We
introduce a novel volumization algorithm, which transforms 2D images into 3D volumetric representations.
When combined with 3D convolution and deep curriculum learning optimization (CLO), itsignificantly improves
the immunity of models against localized universal attacks by up to 40%. We evaluate our proposed approach
using contemporary CNN architectures and the modified Canadian Institute for Advanced Research (CIFAR-10
and CIFAR-100) and ImageNet Large Scale Visual Recognition Challenge (ILSVRC12) datasets, showcasing
accuracy improvements over previous techniques. The results indicate that the combination of the volumetric
input and curriculum learning holds significant promise for mitigating adversarial attacks without necessitating
adversary training.
Courier management system project report.pdfKamal Acharya
It is now-a-days very important for the people to send or receive articles like imported furniture, electronic items, gifts, business goods and the like. People depend vastly on different transport systems which mostly use the manual way of receiving and delivering the articles. There is no way to track the articles till they are received and there is no way to let the customer know what happened in transit, once he booked some articles. In such a situation, we need a system which completely computerizes the cargo activities including time to time tracking of the articles sent. This need is fulfilled by Courier Management System software which is online software for the cargo management people that enables them to receive the goods from a source and send them to a required destination and track their status from time to time.
Sachpazis:Terzaghi Bearing Capacity Estimation in simple terms with Calculati...Dr.Costas Sachpazis
Terzaghi's soil bearing capacity theory, developed by Karl Terzaghi, is a fundamental principle in geotechnical engineering used to determine the bearing capacity of shallow foundations. This theory provides a method to calculate the ultimate bearing capacity of soil, which is the maximum load per unit area that the soil can support without undergoing shear failure. The Calculation HTML Code included.
Cosmetic shop management system project report.pdfKamal Acharya
Buying new cosmetic products is difficult. It can even be scary for those who have sensitive skin and are prone to skin trouble. The information needed to alleviate this problem is on the back of each product, but it's thought to interpret those ingredient lists unless you have a background in chemistry.
Instead of buying and hoping for the best, we can use data science to help us predict which products may be good fits for us. It includes various function programs to do the above mentioned tasks.
Data file handling has been effectively used in the program.
The automated cosmetic shop management system should deal with the automation of general workflow and administration process of the shop. The main processes of the system focus on customer's request where the system is able to search the most appropriate products and deliver it to the customers. It should help the employees to quickly identify the list of cosmetic product that have reached the minimum quantity and also keep a track of expired date for each cosmetic product. It should help the employees to find the rack number in which the product is placed.It is also Faster and more efficient way.
Vaccine management system project report documentation..pdfKamal Acharya
The Division of Vaccine and Immunization is facing increasing difficulty monitoring vaccines and other commodities distribution once they have been distributed from the national stores. With the introduction of new vaccines, more challenges have been anticipated with this additions posing serious threat to the already over strained vaccine supply chain system in Kenya.
TECHNICAL TRAINING MANUAL GENERAL FAMILIARIZATION COURSEDuvanRamosGarzon1
AIRCRAFT GENERAL
The Single Aisle is the most advanced family aircraft in service today, with fly-by-wire flight controls.
The A318, A319, A320 and A321 are twin-engine subsonic medium range aircraft.
The family offers a choice of engines
Saudi Arabia stands as a titan in the global energy landscape, renowned for its abundant oil and gas resources. It's the largest exporter of petroleum and holds some of the world's most significant reserves. Let's delve into the top 10 oil and gas projects shaping Saudi Arabia's energy future in 2024.
About
Indigenized remote control interface card suitable for MAFI system CCR equipment. Compatible for IDM8000 CCR. Backplane mounted serial and TCP/Ethernet communication module for CCR remote access. IDM 8000 CCR remote control on serial and TCP protocol.
• Remote control: Parallel or serial interface.
• Compatible with MAFI CCR system.
• Compatible with IDM8000 CCR.
• Compatible with Backplane mount serial communication.
• Compatible with commercial and Defence aviation CCR system.
• Remote control system for accessing CCR and allied system over serial or TCP.
• Indigenized local Support/presence in India.
• Easy in configuration using DIP switches.
Technical Specifications
Indigenized remote control interface card suitable for MAFI system CCR equipment. Compatible for IDM8000 CCR. Backplane mounted serial and TCP/Ethernet communication module for CCR remote access. IDM 8000 CCR remote control on serial and TCP protocol.
Key Features
Indigenized remote control interface card suitable for MAFI system CCR equipment. Compatible for IDM8000 CCR. Backplane mounted serial and TCP/Ethernet communication module for CCR remote access. IDM 8000 CCR remote control on serial and TCP protocol.
• Remote control: Parallel or serial interface
• Compatible with MAFI CCR system
• Copatiable with IDM8000 CCR
• Compatible with Backplane mount serial communication.
• Compatible with commercial and Defence aviation CCR system.
• Remote control system for accessing CCR and allied system over serial or TCP.
• Indigenized local Support/presence in India.
Application
• Remote control: Parallel or serial interface.
• Compatible with MAFI CCR system.
• Compatible with IDM8000 CCR.
• Compatible with Backplane mount serial communication.
• Compatible with commercial and Defence aviation CCR system.
• Remote control system for accessing CCR and allied system over serial or TCP.
• Indigenized local Support/presence in India.
• Easy in configuration using DIP switches.
Welcome to WIPAC Monthly the magazine brought to you by the LinkedIn Group Water Industry Process Automation & Control.
In this month's edition, along with this month's industry news to celebrate the 13 years since the group was created we have articles including
A case study of the used of Advanced Process Control at the Wastewater Treatment works at Lleida in Spain
A look back on an article on smart wastewater networks in order to see how the industry has measured up in the interim around the adoption of Digital Transformation in the Water Industry.
COLLEGE BUS MANAGEMENT SYSTEM PROJECT REPORT.pdfKamal Acharya
The College Bus Management system is completely developed by Visual Basic .NET Version. The application is connect with most secured database language MS SQL Server. The application is develop by using best combination of front-end and back-end languages. The application is totally design like flat user interface. This flat user interface is more attractive user interface in 2017. The application is gives more important to the system functionality. The application is to manage the student’s details, driver’s details, bus details, bus route details, bus fees details and more. The application has only one unit for admin. The admin can manage the entire application. The admin can login into the application by using username and password of the admin. The application is develop for big and small colleges. It is more user friendly for non-computer person. Even they can easily learn how to manage the application within hours. The application is more secure by the admin. The system will give an effective output for the VB.Net and SQL Server given as input to the system. The compiled java program given as input to the system, after scanning the program will generate different reports. The application generates the report for users. The admin can view and download the report of the data. The application deliver the excel format reports. Because, excel formatted reports is very easy to understand the income and expense of the college bus. This application is mainly develop for windows operating system users. In 2017, 73% of people enterprises are using windows operating system. So the application will easily install for all the windows operating system users. The application-developed size is very low. The application consumes very low space in disk. Therefore, the user can allocate very minimum local disk space for this application.
1. Control Systems (CS)
Ali Raza
Assistant Professor
Indus University Karachi, Pakistan
email: ali_raza@faculty.muet.edu.pk
Lecture-14-15
Time Domain Analysis of 2nd Order Systems
1
2. Introduction
• We have already discussed the affect of location of poles and zeros on
the transient response of 1st order systems.
• Compared to the simplicity of a first-order system, a second-order system
exhibits a wide range of responses that must be analyzed and described.
• Varying a first-order system's parameter (T, K) simply changes the speed
and offset of the response
• Whereas, changes in the parameters of a second-order system can
change the form of the response.
• A second-order system can display characteristics much like a first-order
system or, depending on component values, display damped or pure
oscillations for its transient response. 2
3. Introduction
• A general second-order system is characterized by the
following transfer function.
22
2
2 nn
n
sssR
sC
)(
)(
3
un-damped natural frequency of the second order system,
which is the frequency of oscillation of the system without
damping.
n
damping ratio of the second order system, which is a measure
of the degree of resistance to change in the system output.
4. Example#1
42
4
2
sssR
sC
)(
)(
• Determine the un-damped natural frequency and damping ratio
of the following second order system.
42
n
22
2
2 nn
n
sssR
sC
)(
)(
• Compare the numerator and denominator of the given transfer
function with the general 2nd order transfer function.
sec/radn 2
ssn 22
422 222
ssss nn
50.
1 n
4
6. Introduction
• According the value of , a second-order system can be set into
one of the four categories:
1
1
2
2
nn
nn
1. Overdamped - when the system has two real distinct poles ( >1).
-a-b-c
δ
jω
6
7. Introduction
• According the value of , a second-order system can be set into
one of the four categories:
1
1
2
2
nn
nn
2. Underdamped - when the system has two complex conjugate poles (0 < <1)
-a-b-c
δ
jω
7
8. Introduction
• According the value of , a second-order system can be set into
one of the four categories:
1
1
2
2
nn
nn
3. Undamped - when the system has two imaginary poles ( = 0).
-a-b-c
δ
jω
8
9. Introduction
• According the value of , a second-order system can be set into
one of the four categories:
1
1
2
2
nn
nn
4. Critically damped - when the system has two real but equal poles ( = 1).
-a-b-c
δ
jω
9
11. Time-Domain Specification
11
• The delay (td) time is the time required for the response to
reach half the final value the very first time.
12. Time-Domain Specification
• The rise time is the time required for the response to rise from 10%
to 90%, 5% to 95%, or 0% to 100% of its final value.
• For underdamped second order systems, the 0% to 100% rise time is
normally used. For overdamped systems, the 10% to 90% rise time is
commonly used.
14. Time-Domain Specification
14
The maximum overshoot is the maximum peak value of the
response curve measured from unity. If the final steady-state
value of the response differs from unity, then it is common to
use the maximum percent overshoot. It is defined by
The amount of the maximum (percent) overshoot directly
indicates the relative stability of the system.
15. Time-Domain Specification
15
• The settling time is the time required for the response curve
to reach and stay within a range about the final value of size
specified by absolute percentage of the final value (usually 2%
or 5%).
16. S-Plane
δ
jω
• Natural Undamped Frequency.
n
• Distance from the origin of s-
plane to pole is natural
undamped frequency in
rad/sec.
16
17. S-Plane
δ
jω
• Let us draw a circle of radius 3 in s-plane.
3
-3
-3
3
• If a pole is located anywhere on the circumference of the circle the
natural undamped frequency would be 3 rad/sec.
17
25. Example-2
• Determine the natural frequency and damping ratio of the poles from the
following pz-map.
-4 -3.5 -3 -2.5 -2 -1.5 -1 -0.5 0
-1.5
-1
-0.5
0
0.5
1
1.5
0.220.420.60.740.840.91
0.96
0.99
0.220.420.60.740.840.91
0.96
0.99
0.511.522.533.54
Pole-Zero Map
ImaginaryAxis(seconds-1
)
25
26. Example-3
• Determine the natural
frequency and damping ratio of
the poles from the given pz-
map.
• Also determine the transfer
function of the system and state
whether system is
underdamped, overdamped,
undamped or critically damped.
-3 -2.5 -2 -1.5 -1 -0.5 0
-3
-2
-1
0
1
2
3
0.420.560.7
0.82
0.91
0.975
0.5
1
1.5
2
2.5
3
0.5
1
1.5
2
2.5
3
0.140.280.420.560.7
0.82
0.91
0.975
0.140.28
Pole-Zero Map
ImaginaryAxis(seconds-1
)
26
27. Example-4
• The natural frequency of closed
loop poles of 2nd order system is 2
rad/sec and damping ratio is 0.5.
• Determine the location of closed
loop poles so that the damping
ratio remains same but the natural
undamped frequency is doubled.
42
4
2 222
2
sssssR
sC
nn
n
)(
)(
-2 -1.5 -1 -0.5 0
-3
-2
-1
0
1
2
3
0.280.380.5
0.64
0.8
0.94
0.5
1
1.5
2
2.5
3
0.5
1
1.5
2
2.5
3
0.080.170.280.380.5
0.64
0.8
0.94
0.080.17
Pole-Zero Map
Real Axis
ImaginaryAxis
27
28. Example-4
• Determine the location of closed loop poles so that the damping ratio remains same
but the natural undamped frequency is doubled.
-8 -6 -4 -2 0 2 4
-5
-4
-3
-2
-1
0
1
2
3
4
5
0.5
0.5
24
Pole-Zero Map
ImaginaryAxis
28
30. Step Response of underdamped System
222222
2
21
nnnn
n
ss
s
s
sC
)(
• The partial fraction expansion of above equation is given as
22
2
21
nn
n
ss
s
s
sC
)(
2
2 ns
22
1 n
222
1
21
nn
n
s
s
s
sC )(
22
2
2 nn
n
sssR
sC
)(
)(
22
2
2 nn
n
sss
sC
)(
Step Response
30
31. Step Response of underdamped System
• Above equation can be written as
222
1
21
nn
n
s
s
s
sC )(
22
21
dn
n
s
s
s
sC
)(
2
1 nd• Where , is the frequency of transient oscillations
and is called damped natural frequency.
• The inverse Laplace transform of above equation can be obtained
easily if C(s) is written in the following form:
2222
1
dn
n
dn
n
ss
s
s
sC
)(
31
32. Step Response of underdamped System
2222
1
dn
n
dn
n
ss
s
s
sC
)(
22
2
2
22
1
11
dn
n
dn
n
ss
s
s
sC
)(
22222
1
1
dn
d
dn
n
ss
s
s
sC
)(
tetetc d
t
d
t nn
sincos)(
2
1
1
32
33. Step Response of underdamped System
tetetc d
t
d
t nn
sincos)(
2
1
1
ttetc dd
tn
sincos)(
2
1
1
n
nd
2
1
• When 0
ttc ncos)( 1
33
40. Time Domain Specifications
n
s Tt
4
4
n
s Tt
3
3 100
2
1
eM p
2
1
n
d
pt
2
1
n
d
rt
Rise Time Peak Time
Settling Time (2%)
Settling Time (4%)
Maximum Overshoot
40
41. Example#5
• Consider the system shown in following figure, where
damping ratio is 0.6 and natural undamped frequency is 5
rad/sec. Obtain the rise time tr, peak time tp, maximum
overshoot Mp, and settling time 2% and 5% criterion ts when
the system is subjected to a unit-step input.
41
47. Example#6
• For the system shown in Figure-(a), determine the values of gain K
and velocity-feedback constant Kh so that the maximum overshoot
in the unit-step response is 0.2 and the peak time is 1 sec. With
these values of K and Kh, obtain the rise time and settling time.
Assume that J=1 kg-m2 and B=1 N-m/rad/sec.
47
49. Example#6
Nm/rad/secandSince 11 2
BkgmJ
KsKKs
K
sR
sC
h
)()(
)(
12
22
2
2 nn
n
sssR
sC
)(
)(
• Comparing above T.F with general 2nd order T.F
Kn
K
KKh
2
1 )(
49
50. Example#6
• Maximum overshoot is 0.2.
Kn
K
KKh
2
1 )(
20
2
1
.ln)ln(
e
• The peak time is 1 sec
d
pt
2
45601
1413
.
.
n
2
1
1413
1
n
.
533.n
50
53. Example#7
When the system shown in Figure(a) is subjected to a unit-step input,
the system output responds as shown in Figure(b). Determine the
values of a and c from the response curve.
)( 1css
a
53
54. Example#8
Figure (a) shows a mechanical vibratory system. When 2 lb of force
(step input) is applied to the system, the mass oscillates, as shown in
Figure (b). Determine m, b, and k of the system from this response
curve.
54
55. Example#9
Given the system shown in following figure, find J and D to yield 20%
overshoot and a settling time of 2 seconds for a step input of torque
T(t).
55
58. Example # 10
• Ships at sea undergo
motion about their roll axis,
as shown in Figure. Fins
called stabilizers are used to
reduce this rolling motion.
The stabilizers can be
positioned by a closed-loop
roll control system that
consists of components,
such as fin actuators and
sensors, as well as the ship’s
roll dynamics.
58
59. Example # 10
• Assume the roll dynamics, which relates the roll-angle output, 𝜃(𝑠),
to a disturbance-torque input, TD(s), is
• Do the following:
a) Find the natural frequency, damping ratio, peak time, settling
time, rise time, and percent overshoot.
b) Find the analytical expression for the output response to a unit
step input.
59
60. Step Response of critically damped System ( )
• The partial fraction expansion of above equation is given as
2
2
n
n
ssR
sC
)(
)(
2
2
n
n
ss
sC
)(
Step Response
22
2
nnn
n
s
C
s
B
s
A
ss
2
11
n
n
n sss
sC
)(
teetc t
n
t nn
1)(
tetc n
tn
11)(
1
60
62. 62
Example 11: Describe the nature of the second-order system
response via the value of the damping ratio for the systems with
transfer function
Second – Order System
128
12
)(.1 2
ss
sG
168
16
)(.2 2
ss
sG
208
20
)(.3 2
ss
sG
Do them as your own
revision
63. Example-12
• For each of the transfer
functions find the locations of
the poles and zeros, plot them
on the s-plane, and then write
an expression for the general
form of the step response
without solving for the inverse
Laplace transform. State the
nature of each response
(overdamped, underdamped,
and so on).
63