ECE 461.11 First and Second Order Approximations
Summary
TLDRIn ECE 461 Lecture 11, the focus is on first and second order approximations for control systems. The lecture explains the concept of dominant poles, which are crucial for simplifying high-order systems while preserving their behavior. It covers how to identify these poles and match the DC gain for an accurate model. The importance of settling time, overshoot, and damping ratio is discussed, with methods to determine these parameters from a system's step response. The lecture also touches on the practical use of these concepts in real-world applications, such as car suspensions and missile systems.
Takeaways
- 📚 The lecture introduces first and second order approximations for control systems, emphasizing the impracticality of manually calculating higher order systems like the 250th order Maverick missile model.
- 🔍 The concept of dominant poles is highlighted as crucial in understanding system responses, where one or two poles usually have the most significant impact on the system's behavior.
- 🚗 An example is given using a car's vibrational modes to illustrate how the lump mass of the car represents the dominant pole in its response to road conditions.
- 🔑 The importance of modeling is underscored, with the goal of creating a simpler yet accurate representation of a system by matching the dominant pole and DC gain.
- 📉 Definitions of key terms such as 'dominant pole', 'DC gain', 'two percent settling time', 'overshoot', and 'damping ratio' are provided to establish a foundation for understanding control systems.
- 📈 The process of identifying the dominant pole is explained, often being the pole closest to the origin, and its significance in creating a simplified model of the system.
- 📝 The method of determining system parameters from the step response is demonstrated, showing how the DC gain and settling time can be used to infer the system's characteristics.
- 🔄 The script discusses how to handle complex poles, which require considering both the real and imaginary parts to understand the system's behavior fully.
- 📊 The relationship between the damping ratio and overshoot is detailed, explaining how the damping ratio can be used to predict the system's stability and response to changes.
- 📚 A summary of second order systems is mentioned, which would likely include graphical representations and tables for quick reference in determining system characteristics from response data.
- 🔧 The lecture concludes with an overview of how to use the information about dominant poles and system responses to approximate and understand the behavior of more complex systems.
Q & A
What is the main purpose of identifying dominant poles in control systems?
-The main purpose of identifying dominant poles is to simplify the analysis of a system by focusing on the poles that significantly influence the system's response, allowing for a more manageable and accurate model.
Why is it impractical to find the step response for very high order systems by hand?
-Finding the step response for very high order systems by hand is impractical because it becomes extremely complex and time-consuming, and it can lead to a loss of intuitive understanding of the system's behavior.
What is meant by the term 'DC gain' in the context of control systems?
-The DC gain refers to the gain of a system at low frequencies, specifically at s equals zero, which indicates the system's steady-state response to a constant input.
How is the two percent settling time defined in control systems?
-The two percent settling time is defined as the time it takes for the system's response to reach and stay within 2% of the final steady-state value, which is a standard used for simplicity in calculations.
What is the significance of the damping ratio in control systems?
-The damping ratio is significant as it indicates the amount of overshoot in the system's response to a step input, which is crucial for designing systems with desired stability and performance characteristics.
How can the dominant pole be determined for a system with multiple poles?
-The dominant pole can be determined by analyzing the system's step response or by identifying the pole closest to the origin, as it usually has the largest initial condition and decays the slowest, thus having the most significant impact on the system's response.
What is the relationship between the dominant pole and the system's step response?
-The dominant pole largely dictates the behavior of the system's step response, including the speed of response and the amount of overshoot, making it a key factor in system modeling and analysis.
Why might a simplified model that retains the dominant pole and matches the DC gain be considered 'good enough' for many applications?
-A simplified model that retains the dominant pole and matches the DC gain is considered 'good enough' because it captures the essential behavior of the system, providing an 80-85% accurate representation that is both manageable and useful for most practical purposes.
How can complex poles be handled in the context of system modeling and approximation?
-Complex poles can be handled by ensuring that their complex conjugate is also included in the model, effectively treating them as a second-order system for the purpose of approximation and analysis.
What is the concept of time scaling in control systems, and why is it used?
-Time scaling is the concept of adjusting the time units used in system analysis to align the dominant pole with a more manageable location, typically near minus one, simplifying calculations and making the system's behavior more intuitive.
How can the step response of a system be inferred from its transfer function without explicitly calculating it?
-The step response can be inferred by examining the transfer function's dominant pole and DC gain, as these parameters provide insights into the system's settling time and steady-state behavior, allowing for an estimation of the step response characteristics.
Outlines
🔍 Simplifying High-Order Systems
The lecture discusses the challenge of finding the step and impulse responses for high-order systems, such as a 250th order system. It introduces the concept of dominant poles, which are the poles that significantly influence a system's response. By focusing on these dominant poles and the DC gain, a simpler yet accurate model can be developed. This method retains the essential behavior of the system while making analysis more manageable.
🖥️ Comparing First and Higher-Order Models
This section compares the step responses of higher-order systems with simplified models that retain the dominant pole and DC gain. The lecture demonstrates that the responses of a complex system and its first-order approximation are nearly identical, except for slight delays. It emphasizes the utility of modeling complex systems with simplified approximations for practical analysis, especially when dealing with systems that include complex poles.
⚙️ Understanding Dominant Poles in System Approximation
Here, the focus is on how dominant poles can be identified and used to approximate a system's behavior. The lecture explains the process of determining the dominant pole by analyzing the step response and how this information can be used to create simplified models. It covers both real and complex poles, detailing how the DC gain, settling time, and frequency of oscillation are crucial parameters in this analysis.
📊 Second-Order Systems and Damping Ratio
The lecture concludes with a detailed explanation of second-order systems, focusing on the damping ratio and its impact on system behavior, including overshoot and resonance. It explains how to derive system parameters from the step response and how to use damping ratio values to predict system behavior. The section also introduces Bode plots and their relevance in control systems, setting the stage for future lectures.
Mindmap
Keywords
💡Control Systems
💡Step Response
💡Impulse Response
💡Dominant Pole
💡DC Gain
💡Settling Time
💡Overshoot
💡Damping Ratio
💡First and Second Order Approximations
💡Transfer Function
💡Time Scaling
Highlights
Introduction to Lecture 11 on first and second order approximations in control systems.
Explanation of why manually finding the step response for high order systems is impractical and the importance of dominant poles.
Dominant poles are the key factors that dictate the system's response, often being the poles closest to the origin.
The concept of modeling to simplify complex systems while retaining accuracy and manageability.
Matching the dominant pole and DC gain for an accurate system model.
Definition of DC gain as the gain at s equals zero.
Discussion on the two percent settling time and its significance in system response.
The impact of overshoot in system design, with examples like egg polishers and car suspensions.
How to determine the dominant pole in a system with multiple poles.
Simplification of a system by keeping the dominant pole and adjusting the DC gain.
Demonstration of how a simplified model can closely resemble a complex system's step response.
Approach to handling complex poles and their conjugates in system modeling.
The process of finding the transfer function from a given step response.
Explanation of how to calculate the settling time and frequency of oscillation from a pole's properties.
Introduction to damping ratio and its role in determining overshoot.
The use of time scaling to normalize dominant pole positions for easier calculations.
Different ways to represent second order systems and how to determine them from system responses.
Summary of the lecture providing insights into the practical applications and theoretical foundations of first and second order system approximations.
Transcripts
welcome to ece 461
control systems lecture number 11 first
and second order approximations
now in the previous lecture we looked at
how to find the step response
and impulse response for a transfer
function
that gets really unwieldy when you start
getting your eighth ninth tenth order
systems
that actually can happen the maverick
missile for example
was modeled as a 250th order system i
don't really want to find the step
response of a 250th order system by hand
plus you lose all intuition if i look at
this transfer function and said
by inspection what's the step response i
really have no idea
so there's the trick when you have a
dynamic system
there's a few poles that really dominate
the response
those the dominant poles what i'd like
to do is take this system
and come up with model which is simpler
but still fairly accurate and he has a
similar step response
keeping the same dominant pole does that
so the concept of what a dominant pole
is for most systems
there's one or two poles that tend to
really dominate the responsive system
for example with my car as you're going
down the road there's vibrational modes
twisting modes
a lot of rattling all those are pulls
the lump mass of the car really
dominates the response however
so if i just modeled that one pole the
lump mass
i'd have a pretty good model for the
system and that's really the purpose of
modeling
how do i come up with a mathematical
model which is simple
meaning useful but still fairly accurate
the idea is if i match the dominant pole
and match the dc gain
i've got a model that's you know really
pretty good you know 80 85 percent
correct
plus it's manageable if when you get a
more accurate model
best read through it throw it into a
computer simulation or matlab
so a couple definitions to start out the
domino pole
is the pole that dominates response
which is kind of a tautology
it's if you look at the step response
there's the overall
response that's uh really dictates how
it behaves
that's the dominant pole usually is the
pole closest to the origin
not always but usually the transfer
function
is the differential equation that
relates the input in the output
the dc gain is the gain at dc at s
equals zero
the two percent settling time is the
differential equation
it'll respond and in theory takes
infinite time to settle out
infinity is not a terribly easy number
to use so what i do
is come up with a way of saying as soon
as i get close to zero
i'll call that the settling time and to
define close
that's kind of arbitrary typically two
percent settling time is used
the reason being is two percent's got a
nice logarithm the log of 0.02 is minus
4.
so back in the days of slide rules
logarithms were painful to use
so they use something that has got a
nice logarithm and that standard is
stuck
overshoot if i have a step response is
how much it overshoots
sometimes that's really important for
example if i want to have an egg
polisher
i want to have no overshoot because i'll
crack the egg if your car hits a pothole
you want it to bounce about three times
for about 30 of your shoot
typically when you specify how the
system should behave you specify the
overshoot and the settling time
from that i've got to translate it to
control systems terms
the damping ratio is the angle of the
pole cosine of the angle is your zeta
the damping ratio damping ratio tells
you the overshoot
uh so if i have a system i've got to
figure out which one is the dominant
pole
as i mentioned it's the one closest to
the origin and you can see that here
suppose i have three poles one minus one
minus ten minus one hundred
which one's dominant well one way to do
that is take the step response
i'll apply a step input do your partial
fraction expansion
and it says here's your step response
the 2 is the forced response
that comes from my step input these are
the transients
and if you notice this term in the
transient that first one
it really dominates it starts out 10
times larger than
anyone else plus it lasts 10 times
longer
so this is called the dominant pole
almost invariably it's the pole closest
s equals zero
the reason being is this input this
forcing function excites these poles
the pull closest to it has the largest
excitement
largest initial condition and it decays
the slowest
kind of gives you a double whammy the
largest initial condition flash the
longest
is the dominant pole
with that i can do things and come up
with a simplified model
if i were to say this guy is too
complicated i want to simplify the model
for my analysis
well if i keep the dominant pull keep
the dc gain
it's almost the same system
for example the dominant pulls at minus
one the dc gain here is two
so this has the same dc gain same
dominant pole it's almost the same
system
and you can check that throw this in
matlab find the step response i get the
red curve
find the step response to the first
order system i get the blue curve
and if you notice the two are almost
identical
there's a little bit of delay on the
third order system
sometimes they'll say it's a first order
system plus a delay that delay
models all the polls that i ignored it
takes into account that slight
shift in there
dominant poles also work when you have
complex poles for example here i've got
a fourth order system
our fifth order actually the pole close
to the origin
is just pull up minus one plus j2 if you
have a complex pole
in this class you have to have its
complex conjugate so that's where you
get a second order
system uh
to come up with a model port keep the
same dominant pole
and match the dc gain so again plug in s
equals zero find the gain
plug in s equals zero the numerator is
whatever it takes to make the
dc gains match i get 4.507
now take the two systems find the step
response here's the fifth order system
in red
the second order systems in blue and
notice they're almost the same
i've got the same dc gain same overshoot
same frequency of oscillation
there's a slight delay in the fifth
order system the second order system
doesn't model
but it's pretty close when you get more
accurate i can say second order system
plus a delay
kind of a side light in this class most
the systems we're looking at
like these guys have a dominant pull
right around minus one
the reason for that it makes the math a
lot easier one's got nice numerical
properties
one squared is one one cubed is one one
of the fourth is one
um and what that kind of applies is time
scaling
if my pull is not at minus one so it
pulls that minus a thousand
if i time scale it so that my x axis
instead of being seconds is milliseconds
now the dominant pole is at minus one so
if you see
all your systems having poles near minus
1 at this class that kind of assumes
time scaling
they've been talking about economies
where it takes months for the economy to
to settle out
my time unit might be months or years
if it's something quicker my time unit
might be milliseconds
regardless typically the dominant pulls
right around one
whatever your time unit is at least in
this class
i'm not good that's just kind of a side
light so let's go on with first and
second order approximations
since i've got dominant poles you
typically have either a real dominant
pole
or a complex dominant pole if it's a
real dominant pole i'll just have a
single dominant pole
if you have a first order system say a
single dominant pole
there's really only two degrees of
freedom a generic first order systems
can be written in the form of a over s
plus b if i can tell you two things
about the system
i can tell you what the system is i've
only got two degrees of freedom
one piece of information is the dc gain
plug in s equals zero i get a over b
second piece of information is the
settling time this decays is e to the
minus bt
for that to decay down to 0.02 that's
the 2
settling time take the log of both sides
log of 0.02
is minus 4 actually minus 3.97 close to
minus 4.
solve for t i get the settling time is
four over the pull
or conversely the pull is four of the
settling time
so that means by inspection depending
where the pole is on what the settling
time is
as an example here's the tenth order
system
again finding the inverse laplace
transform by hand is going to be really
really painful
i don't have to do that if i take the
system
plug in s equals zero i get a dc gain of
one that's one piece of information
factor this i get 10 poles the dominant
pole is right here
the one closest to zero the dominant
pole is at 0.02
meaning the settling time is 4 of your
pull 179 seconds
and if you take the step response sure
enough that's what you get
the dc gain is 1 and the settling time
is right around here
179 seconds so again by inspection
i can look at this system and tell you
what the step response is
i can also go backwards given the step
response what's the system
again dc gain is one settling time is
179 seconds
give or take it's hard to be real
accurate with a graph
which means that the pull is at 0.02
and the numerator is whatever it takes
to make the dc gain 1.
and notice when you take the go
backwards
what you capture is the dominant pole
the other poles are really hard to see
they are there the other poles are right
here that information
really hard to pick out usually just get
the dominant pole
hence the name dominant
that's for a single pole if i have a
complex pole
i'll also have a complex conjugate and
i'll have a second order system
or second order approximations for
second order system there's a couple
ways to write it
i've got three unknowns three parameters
however you do it so if i could pull
three pieces of information up a graph
i can tell you what the system is
uh the at plug in s equals zero
that gives you the dc gain in this case
it's going to be a
if i look at the real part of the pole
that's decaying exponential the real
part tells you the two percent settling
time
the complex part of the pole omega d
tells the frequency of oscillation
so if i can tell you the settling time i
can tell you the frequency of
oscillation
i can tell you the dc gain i can tell
you what the system is
for example for this system the
dc gain is 0.5
the real part is minus 2 said the
settling time is 4 for 2
2 seconds and the frequency of
oscillation is 20 radians per second
about 3 hertz
another way to write it is in polar form
if i write the poles in polar form
multiply it out i get s squared plus two
zeta
omega naught plus omega n squared omega
n is the amplitude of the pole
zeta is cosine of the angle it's called
the damping ratio
the damping ratio tells you the upper
shoot that's kind of important a lot of
times control systems
the overshoot is what i want to specify
for example an egg polisher
battleship guns when i have no overshoot
a jet engine has a damping ratio 0.8 two
percent overshoot
wanting to throttle forward the engine
can speed up a little bit about two
percent of our shoot then settle out
on a car i'm going to have three
oscillations meaning right around 50
over shoot depending upon the system i
specify the overshoot
the overshoot tells you the dipping
ratio the damping ratio tells you the
angle
for example suppose i had this system
i want to sketch the step response can
the trick just find the dominant pole
that's right here minus one plus j two
the settling time will be four over the
real part four seconds
the frequency of oscillation is two
radians per second
the angle of the pull the angle 63
degrees
cosine of the angle is your damping
ratio 0.44
the overshoot is 20
so the step response looks like this
and i can also go backwards given the
step response find the transfer function
give the step response the dc gain is
0.94
the two percent settling time is about
four seconds
and the upper shoot is twenty percent uh
twenty percent overshoot tells the zetas
point is 0.45
it tells you the angle and if i know the
real part under the angle
i can tell the complex part using some
trig so there's my system
picking to match the dc gain and you got
your system
notice again i just picked off the
dominant pole the other poles are really
hard to see
there's a summary go on bison academy
there is a summary of second order
systems
what that looks like is this
uh this graph right here is the step
response
versus damping ratio so damping ratio
one
you've got no overshoot at point one
i've got about seventy percent over
shoot
the angle is the damping ratio right
here on the real axis stamping versus
one
and on the j omega axis the damping
ratio is zero
when you get to frequency domain time
and frequency are related
if i have a pole on the real axis
damping ratio is one the gain just drops
off with frequency
butterworth filters at 0.7 that's the
maximum
flat gain below that i get chebyshev
filters
the equations for damping ratio
overshoot time to peak so on
or probably more useful the thing i like
using
on the homework sets and tests you'll
have this on the tests
i just use the table if i have 20
overshoot
that means the damping ratio is between
point four and point five color point
four five
it's actually point four five five nine
if i have ten percent over shoot
the damping ratio is between point six
and point five it's actually about point
six one
using the table i can translate
overshoot to damping ratio
uh let's see what else and when we get
to bode plots i can also tell you
daping ratio versus resonance how much
resonance i can tolerate tells what the
poles are
but that will be towards the end of the
semester
so that's lecture number 11 for ece 461
control systems
first and second order approximations
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