Automatic Control EE 418 “Dr. Ahmed El-Shenawy”
Electric Engineering II
EE 326
Lecture 8
<Dr Ahmed El-Shenawy>
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Automatic Control EE 418 “Dr. Ahmed El-Shenawy”
Time-Domain Analysis
In order to find the time response of a control system, we first need to model
the overall system dynamics and find its equation of motion. The system
could be composed of mechanical, electrical, or other sub-systems. Each
sub-system may have sensors and actuators to sense the environment and
to interact with it.
Next, using Laplace transforms, we can find the transfer function of all the subcomponents and use the block diagram approach or signal flow diagrams to
find the interactions among the system components.
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Automatic Control EE 418 “Dr. Ahmed El-Shenawy”
Depending on our objectives, we can manipulate the system final
response by adding feedback or poles and zeros to the system block
diagram.
Finally, we can find the overall transfer function of the system and, using
inverse Laplace transforms, obtain the time response of the system to a
test input normally a step input.
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Automatic Control EE 418 “Dr. Ahmed El-Shenawy”
TIME RESPONSE OF CONTINUOUSDATA SYSTEMS
The time response of a control system is usually divided into two parts: the
transient response and the steady-state response.
In control systems, transient response is defined as the part of the time
response that goes to zero as time becomes very large.
The steady-state response is simply the part of the total response that
remains after the transient has died out.
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Automatic Control EE 418 “Dr. Ahmed El-Shenawy”
To facilitate the time-domain analysis, the following deterministic test signals
are used.
Ramp-Function Input:
Step-Function Input:
Parabolic-Function Input:
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Automatic Control EE 418 “Dr. Ahmed El-Shenawy”
RESPONSE AND TIME-DOMAIN
SPECIFICATIONS
For linear control systems, the characterization of the transient response is
often done by use of the unit-step function us(t) as the input. The response of a
control system when the input is a unit-step function is called the unit-step response.
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Automatic Control EE 418 “Dr. Ahmed El-Shenawy”
Steady-State Error
the steady-state errors of linear control systems depend on the type of the
reference signal and the type of the system. System error is a signal that
should be quickly reduced to zero, if possible.
where r(t) is the input; u(t), the actuating signal; b(t), the feedback signal;
and y(t), the output.
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Automatic Control EE 418 “Dr. Ahmed El-Shenawy”
Let us assume that the objective of the system is to have the output y(t) track the
input r(t) as closely as possible, and the system transfer functions are
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Automatic Control EE 418 “Dr. Ahmed El-Shenawy”
Type of Control Systems: Unity
Feedback Systems
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Automatic Control EE 418 “Dr. Ahmed El-Shenawy”
Steady-State Error of System with a
Step-Function Input
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Automatic Control EE 418 “Dr. Ahmed El-Shenawy”
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Automatic Control EE 418 “Dr. Ahmed El-Shenawy”
Steady-State Error of System with a
Ramp-Function Input
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Automatic Control EE 418 “Dr. Ahmed El-Shenawy”
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Automatic Control EE 418 “Dr. Ahmed El-Shenawy”
Steady-State Error of System with a
Parabolic-Function Input
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Automatic Control EE 418 “Dr. Ahmed El-Shenawy”
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Automatic Control EE 418 “Dr. Ahmed El-Shenawy”
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Automatic Control EE 418 “Dr. Ahmed El-Shenawy”
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Automatic Control EE 418 “Dr. Ahmed El-Shenawy”
Poles and Zeros of a First-Order
System
a pole exists at s = - 5 , and a zero exists
at -2. These values are plotted on the
complex s-plane
To show the properties of the poles and
zeros, let us find the unit step response
of the system.
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Automatic Control EE 418 “Dr. Ahmed El-Shenawy”
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Automatic Control EE 418 “Dr. Ahmed El-Shenawy”
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Automatic Control EE 418 “Dr. Ahmed El-Shenawy”
PROBLEM: Given the system of Figure, write the output, c(t), in general terms.
Specify the forced and natural parts of the solution.
SOLUTION
Taking the inverse Laplace transform, we get
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Automatic Control EE 418 “Dr. Ahmed El-Shenawy”
First-Order Systems
A first-order system without zeros can be described by the transfer
function shown in Figure
Taking the inverse transform
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Automatic Control EE 418 “Dr. Ahmed El-Shenawy”
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Automatic Control EE 418 “Dr. Ahmed El-Shenawy”
Time Constant
We call l/a the time constant of the response. The time constant can be
described as the time for
to decay to 37% of its initial value
The time constant is the time it takes for the step response to rise to 63%
of its final value
Rise Time, Tr
Rise time is defined as the time for the waveform to go from 0.1 to 0.9 of
its final value.
Settling Time, T
Settling time is defined as the time for the response to reach, and stay
within, 2% of its final value
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Automatic Control EE 418 “Dr. Ahmed El-Shenawy”
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Automatic Control EE 418 “Dr. Ahmed El-Shenawy”
Second-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.
Whereas varying a first-order system's parameter simply changes the speed of
the response, changes in the parameters of a second-order system can change
the form of the response.
The unit step response then can be found using C(s) = R(s)G(s), where
R(s) = 1/s, followed by a partial-fraction expansion and the inverse
Laplace transform.
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Automatic Control EE 418 “Dr. Ahmed El-Shenawy”
Overdamped Response
Underdamped Response
the poles that generate the natural response are
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Automatic Control EE 418 “Dr. Ahmed El-Shenawy”
Undamped Response
the two system poles on the imaginary axis at ±j3 generate a sinusoidal
natural response
Critically Damped Response
Critically damped responses are the fastest possible without the overshoot that is
characteristic of the underdamped response.
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Automatic Control EE 418 “Dr. Ahmed El-Shenawy”
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Automatic Control EE 418 “Dr. Ahmed El-Shenawy”
The General Second-Order
System
Natural Frequency,
The natural frequency of a second-order system is the frequency of oscillation
of the system without damping.
Damping Ratio
The ratio that compares the exponential decay frequency of the
envelope to the natural frequency.
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Automatic Control EE 418 “Dr. Ahmed El-Shenawy”
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Automatic Control EE 418 “Dr. Ahmed El-Shenawy”
For each of the systems shown find the value of
response expected.
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and report the kind of
Automatic Control EE 418 “Dr. Ahmed El-Shenawy”
Underdamped Second-Order
Systems
The underdamped second order system, a common model for physical
problems, displays unique behavior that must be itemized; a detailed
description of the underdamped response is necessary for both analysis and
design.
Our first objective is to define transient specifications associated with
underdamped responses. Next we relate these specifications to the pole
location, drawing an association between pole location and the form of the
underdamped second-order response. Finally, we tie the pole location to
system parameters, thus closing the loop: Desired response generates
required system components.
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Automatic Control EE 418 “Dr. Ahmed El-Shenawy”
Let us begin by finding the step response for the general second-order system
The transform of the response, C(s), is the transform of the input times the
transfer function, or
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Automatic Control EE 418 “Dr. Ahmed El-Shenawy”
Taking the inverse Laplace transform, (Assignment)
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Automatic Control EE 418 “Dr. Ahmed El-Shenawy”
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Automatic Control EE 418 “Dr. Ahmed El-Shenawy”
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Automatic Control EE 418 “Dr. Ahmed El-Shenawy”
Rise time, peak time, and settling time yield information about the speed of the
transient response. This information can help a designer determine if the speed
and the nature of the response do or do not degrade the performance of the
system.
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Automatic Control EE 418 “Dr. Ahmed El-Shenawy”
Evaluation of Tp (Peak time)
is found by differentiating c(t) and finding the first zero crossing after t = 0
Completing squares in the denominator, we have
Setting the derivative equal to zero yields
The first peak, which occurs at the peak time, Tp, is found by letting n = 1
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Automatic Control EE 418 “Dr. Ahmed El-Shenawy”
Evaluation of %0S (Percentage overshoot)
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Automatic Control EE 418 “Dr. Ahmed El-Shenawy”
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Automatic Control EE 418 “Dr. Ahmed El-Shenawy”
Evaluation of Ts (Settling Time)
In order to find the settling time, we must find the time for which c(t) reaches
and stays within ±2% of the steady-state value, Cfjnai
the settling time is the time it takes for the amplitude of the decaying sinusoid to
reach 0.02
This equation is a conservative estimate, since we are assuming that
at the settling time
approximation for the settling time
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Automatic Control EE 418 “Dr. Ahmed El-Shenawy”
Evaluation of Tr (Rising Time)
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Automatic Control EE 418 “Dr. Ahmed El-Shenawy”
An explicit formula for the rise time is somewhat hard to calculate. However,
we notice that rise time increases with and decreases with
. The best
linear fit to the curve gives us the approximation
which is reasonably accurate for
A cruder approximation is obtained by finding a best fit curve with
yielding
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Automatic Control EE 418 “Dr. Ahmed El-Shenawy”
Example 1
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Automatic Control EE 418 “Dr. Ahmed El-Shenawy”
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Automatic Control EE 418 “Dr. Ahmed El-Shenawy”
Example 2
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