This document discusses the damping ratio of unit step responses in control systems. It defines damping ratio as the ratio of the actual damping coefficient to the critical damping coefficient. It describes the different types of damping including underdamped, overdamped, and critically damped systems. It discusses using a unit step function as a common test input and analyzing the step response to identify system properties. MATLAB coding examples are provided to simulate step responses and the document discusses applications in identification from step response testing.

1. Design and Analysis of Damping
Ratio of Unit Step Response
Submitted by: Akshat Jain

2. Table Of Contents
• Introduction
• Definition
• Damping ratio
• Oscillating Cases
• Step response
• Principle
• Working Procedure
• Damping type
• First Order System
• Second Order System
• MATLAB coding
• Application
• Advantages and Limitations
• Recent trend And Future Scope
• Conclusion
• Reference

3. Introduction
• Assuming that the system is asymptotically stable, and then the system
response in the long run is determined by its steady state component only.
For control systems it is important that steady state response values are as
close as possible to desired ones (specified ones) so that we have to study
the corresponding errors, which represent the difference between the
actual and desired system outputs at steady state, and examine conditions
under which these errors can be reduced or even eliminated.
• Modern day control engineering is a relatively new field of study that
gained significant attention during the 20th century with the advancement
of technology. It can be broadly defined or classified as practical
application of control theory. Control engineering has an essential role in a
wide range of control systems, from simple household washing machines
to high-performance F-16 fighter aircraft. It seeks to understand physical
systems, using mathematical modeling, in terms of inputs, outputs and
various components with different behaviors, use control systems design
tools to develop controllers for those systems and implement controllers
in physical systems employing available technology

4. Definition
Damping Ratio:
• The damping ratio is a parameter, usually denoted by ζ
(zeta) that characterizes the frequency response of a
second order ordinary differential equation. It is
particularly important in the study of control theory. It is
also important in the harmonic oscillator.
• The damping ratio provides a mathematical means of
expressing the level of damping in a system relative to
critical damping. For a damped harmonic oscillator with
mass m, damping coefficient c, and spring constant k, it can
be defined as the ratio of the damping coefficient in the
system's differential equation to the critical damping
coefficient:

5. Step response
• The step response of a system in a given initial
state consists of the time evolution of its
outputs when its control inputs are Heaviside
step functions. In electronic engineering and
control theory, step response is the time
behavior of the outputs of a general system
when its inputs change from zero to one in a
very short time. The concept can be extended
to the abstract mathematical notion of a
dynamical system using an evolution
parameter.

6. Working Procedure• Working Procedure
• Using the natural frequency of a harmonic oscillator
•
• and the definition of the damping ratio above, we can rewrite this as:
• This equation can be solved with the approach.
• where C and s are both complex constants. That approach assumes a solution that is oscillatory
and/or decaying exponentially. Using it in the ODE gives a condition on the frequency of the
damped oscillations,
• Undamped: Is the case where corresponds to the undamped simple harmonic oscillator, and in
that case the solution looks like
• , as expected.
• Underdamped: If s is a complex number, then the solution is a decaying exponential combined with
an oscillatory portion that looks like
• . This case occurs for , and is referred to as underdamped.
• Overdamped: If s is a real number, then the solution is simply a decaying exponential with no
oscillation. This case occurs for , and is referred to as overdamped.
• Critically damped: The case where is the border between the overdamped and underdamped
cases, and is referred to as critically damped. This turns out to be a desirable outcome in many
cases where engineering design of a damped oscillator is required (e.g., a door closing
mechanism).
• One of the most common test inputs used is the unit step function,

7. Coding
step(sys)
step(sys,Tfinal)
step(sys,t)
step(sys1,sys2,...,sysN)
step(sys1,sys2,...,sysN,Tfinal)
step(sys1,sys2,...,sysN,t)
y = step(sys,t)
[y,t] = step(sys)
[y,t] = step(sys,Tfinal)
[y,t,x] = step(sys)
[y,t,x,ysd] = step(sys)
[y,...] = step(sys,...,options)

8. Application
• Identification from step response is one of the most significant topics in process control.
Although there is a concern about its persistency of excitation, the step input might be the
most commonly used excitation signal for identification in process industries. These methods
are commonly used in industrial applications because they involve minimal computation. Due
to the limitations of the graphical techniques in terms of their applicability and accuracy,
more computationally involved methods have been developed subsequently. An important
development in identification from step response is the introduction of the integral equation
approach for simultaneous estimation of the delay proposed by Wang and Zhang. Based on
this procedure, over the last few years a number of new methods have been proposed to
deal with the different practical issues in step response based identification
• In evaluating the uncertainty of the standard measuring system for lightning-impulse high
voltage, which is composed of the standard voltage divider, the digital recorder and the
calibrators, step-response tests of the standard voltage divider may be useful. In this paper,
convolution algorithm is employed to calculate output impulse-voltage waveforms from
measured step-response waveforms.. Therefore, for the standard voltage divider, step-
response parameters, i.e. experimental response time, partial response time, settling time,
and overshoot have almost nothing to do with the error of the virtual front time.

9. Advantages
• Using a computer to control a process has a number of
important advantages over controlling the same process
manually.
• Computer systems respond more quickly than humans. A
computer system can take readings from sensors and
turn devices on and off many thousands of times a second.
• Once the initial purchase cost has been paid, control systems
are usually reasonably cheap to run. Most computer control
systems have lower operating costs than similar systems
which are manned by humans.
• Computer control systems are very reliable. Unlike a human a
control system will not lose concentration. Computer systems
can continue to operate reliably twenty four hours a day.

10. Recent Trend & Future Scope
• Originally, control engineering was all about continuous systems.
Development of computer control tools posed a requirement of
discrete control system engineering because the communications
between the computer-based digital controller and the physical
system are governed by a computer clock. The equivalent
to Laplace transform in the discrete domain is the Z-transform.
Today, many of the control systems are computer controlled and
they consist of both digital and analog components.
• Therefore, at the design stage either digital components are
mapped into the continuous domain and the design is carried out in
the continuous domain, or analog components are mapped into
discrete domain and design is carried out there. The first of these
two methods is more commonly encountered in practice because
many industrial systems have many continuous systems
components, including mechanical, fluid, biological and analog
electrical components, with a few digital controllers.

11. Conclusion
• Since modern small microprocessors are so cheap (often less than $1 US),
it's very common to implement control systems, including feedback loops,
with computers, often in an embedded system. The feedback controls are
simulated by having the computer make periodic measurements and then
calculate from this stream of measurements (see digital signal processing,
sampled data systems).
• Computers emulate logic devices by making measurements of switch
inputs, calculating a logic function from these measurements and then
sending the results out to electronically controlled switches.
• Logic systems and feedback controllers are usually implemented with
programmable logic controllers which are devices available from electrical
supply houses. They include a little computer and a simplified system for
programming. Most often they are programmed with personal computers.
• Logic controllers have also been constructed from relays, hydraulic and
pneumatic devices as well as electronics using both transistors and
vacuum tubes (feedback controllers can also be constructed in this
manner).

12. Reference
• https://en.wikipedia.org/wiki/Step_response
• http://lpsa.swarthmore.edu/Transient/TransInputs/TransStep.html
• http://in.mathworks.com/help/control/ref/step.html?requestedDomain=in.mat
hworks.com
• http://ocw.mit.edu/courses/mathematics/18-03sc-differential-equations-fall-
2011/unit-iii-fourier-series-and-laplace-transform/unit-step-and-unit-impulse-
response/MIT18_03SCF11_s25_2text.pdf
• https://www.youtube.com/watch?v=_g-lzZ5e0h0
• http://in.mathworks.com/help/control/ref/damp.html
• http://www.electrical4u.com/transient-state-and-steady-state-response-of-
control-system/
• https://www.maplesoft.com/content/EngineeringFundamentals/11/MapleDocu
ment_11/Block%20Diagrams,%20Feedback%20and%20Transient%20Response%2
0Specifications.pdf
• https://en.wikipedia.org/wiki/Control_engineering
• http://adsabs.harvard.edu/abs/2011IJTPE.131..455M