Control Bootcamp: Overview
Summary
TLDRIn this bootcamp video lecture, Steve Brenton introduces the fundamentals of optimal and modern control theory, focusing on system descriptions using linear differential equations and controller design. He highlights the importance of closed-loop feedback control over open-loop methods, emphasizing its ability to handle system uncertainties, instabilities, and external disturbances more effectively. The lecture aims to familiarize viewers with major control types and their MATLAB implementation, while addressing the current challenges in control theory.
Takeaways
- 📚 The lecture series is a bootcamp on control theory, focusing on optimal and modern control theory highlights.
- 🔍 The course aims to familiarize students with major types of control theory and their practical applications in MATLAB.
- 🌐 Dynamical systems, modeled by ordinary differential equations, are a successful framework for real-world phenomena like fluid flow, population dynamics, and disease spread.
- 🔧 Control theory goes beyond mere description, aiming to actively manipulate systems to change their behavior through control logic or feedback.
- 🚛 Passive control, like streamlined truck tail sections, is a common form of control that reduces energy expenditure but may not be sufficient for complex systems.
- 🔄 Active control involves pumping energy into a system, with open-loop control being a common form where inputs are pre-planned without feedback.
- 🔄 Open-loop control has limitations, such as constant energy input and inability to adapt to system uncertainties or external disturbances.
- 🔄 Closed-loop feedback control uses sensors to measure system outputs and adjust inputs, providing better performance and stability.
- 🔄 Feedback control can handle system uncertainties, instability, and external disturbances, making it more robust and efficient than open-loop control.
- 🔄 The mathematical architecture involves state-space systems of ordinary differential equations, where feedback can change the system's dynamics and stability.
Q & A
What is the main focus of the bootcamp series on control introduced by Steve Brenton?
-The bootcamp series focuses on rapidly going through the highlights of optimal and modern control theory, including system description, controller design, and estimator design, with an emphasis on practical application using MATLAB.
What are the main goals of the bootcamp series on control theory?
-The main goals are to familiarize participants with major types of optimal and modern control theory, teach them how to use these theories in MATLAB, and provide insights into what is easy and challenging in control theory today.
What is the difference between passive control and active control in the context of control theory?
-Passive control involves designing a system upfront without any energy expenditure, like streamlined tail sections on a truck to reduce drag. Active control, on the other hand, involves pumping energy into the system to actively manipulate its behavior.
What is open-loop control and how does it differ from closed-loop feedback control?
-Open-loop control is a pre-planned control strategy where the input to the system is determined without considering the system's current state. Closed-loop feedback control, however, involves measuring the system's output and using this information to adjust the input in real-time.
Why is closed-loop feedback control considered more effective than open-loop control?
-Closed-loop feedback control is more effective because it can handle uncertainties, change the system's dynamics to improve stability, reject disturbances, and is generally more energy-efficient.
What are some of the benefits of using feedback control in dynamical systems?
-Feedback control can compensate for internal uncertainties, change the system's stability by altering its eigenvalues, reject external disturbances, and is more energy-efficient compared to open-loop control.
How does the bootcamp series plan to address the challenges in control theory?
-The series aims to provide a high-level overview of control theory, highlight the pressing needs, and offer MATLAB examples to help participants understand and overcome these challenges.
What is the significance of eigenvalues in the context of stability in control systems?
-Eigenvalues of the system's dynamic matrix determine the stability of the system. If all eigenvalues have negative real parts, the system is stable; if any have positive real parts, the system is unstable.
How does the bootcamp series approach the topic of system modeling using ordinary differential equations?
-The series starts with linear systems of ordinary differential equations to describe how states interact in a system, using state variables and their derivatives to model various real-world phenomena.
What is the role of the matrix 'B' in the context of control systems described in the script?
-Matrix 'B' represents the actuator's effect on the system's state. It determines how the control input 'U' directly influences the time rate of change of the state variables.
Can you provide an example of how feedback control can be applied to stabilize an inverted pendulum?
-In the case of an inverted pendulum, feedback control can be used by measuring the pendulum's angle and angular velocity (states), and then applying a control input 'U' that is proportional to these states (e.g., U = -KX), which can stabilize the pendulum by adjusting the base's position or motor voltage in real-time.
Outlines
📚 Introduction to Control Theory Bootcamp
Steve Brenton introduces a bootcamp series on control theory, aiming to provide a rapid yet comprehensive overview of optimal and modern control theory. The lecture will cover system descriptions using linear differential equations, controller design, and estimators like the Kalman filter. The focus is on high-level concepts rather than in-depth mathematical treatments, with the goal of familiarizing the audience with major types of control theory and their applications in MATLAB. Brenton emphasizes the importance of understanding both the easy and challenging aspects of control theory today.
🔧 Types of Control: Passive and Active
The script discusses the distinction between passive and active control systems. Passive control, such as the aerodynamic design of a truck to reduce drag, requires no energy input once designed. In contrast, active control involves continuous energy input to manipulate a system's behavior, exemplified by the open-loop control of an inverted pendulum. The lecture highlights the limitations of open-loop control, such as constant energy expenditure and lack of adaptability to uncertainties or disturbances, and introduces closed-loop feedback control as a more efficient and robust alternative.
🔄 Benefits of Closed-Loop Feedback Control
This paragraph delves into the advantages of closed-loop feedback control over open-loop control. It addresses the ability of feedback control to handle system uncertainties, change the system's dynamics to improve stability, and reject external disturbances. The efficiency of feedback control is also highlighted, as it can achieve system stabilization with minimal energy input. The paragraph sets the stage for further exploration of feedback control mechanisms and their implementation in various systems.
🛠️ Control System Dynamics and Feedback Effects
The final paragraph introduces the mathematical framework of state-space systems of ordinary differential equations, which form the basis of control theory. It explains how feedback control can alter the system's dynamics by changing the eigenvalues of the closed-loop system, thus affecting its stability. The paragraph also discusses the design of control laws and the importance of system controllability and actuation. The goal is to enable the audience to understand how to manipulate system behavior using feedback control, with the potential to stabilize an originally unstable system like an inverted pendulum.
Mindmap
Keywords
💡Control Theory
💡System Description
💡Controllers
💡Estimators
💡Passive Control
💡Active Control
💡Open-Loop Control
💡Closed-Loop Feedback Control
💡Uncertainty
💡Stability
💡Eigenvalues
💡Efficiency
Highlights
Introduction to a bootcamp series on control theory by Steve Brenton.
The series will cover optimal and modern control theory at a high level.
Discussion on writing system descriptions using linear differential equations.
Designing controllers to manipulate system behavior is a key topic.
Introduction to designing estimators like the Kalman filter for sensor data.
The course aims to familiarize students with major types of control theory.
Teaching the practical application of control theory using MATLAB.
Highlighting the ease and challenges in control theory today.
Perspective on modeling dynamical systems for real-world phenomena.
The importance of going beyond description to actively manipulate systems.
Differentiating between passive and active control methods.
Explanation of open-loop control and its limitations.
Advantages of closed-loop feedback control over open-loop.
The role of feedback in handling system uncertainty and instability.
Feedback control's ability to reject disturbances in the system.
Efficiency of feedback control in energy usage.
Fundamental change in system dynamics and stability with feedback control.
The mathematical framework of state-space systems and ordinary differential equations.
How feedback control can change the eigenvalues of a system for stability.
The importance of controllability and the design of effective control laws.
The series aims to provide a quick understanding of control theory with MATLAB examples.
Transcripts
hi everyone I'm Steve Brenton and this
is the first video lecture on a series
I'm calling a bootcamp on control where
I'm going to rapidly go through the
highlights of optimal and modern control
theory so this is going to include how
to write down a system description of a
control system with inputs and outputs
in terms of a system of linear
differential equations and now how to
design controllers to manipulate the
behavior of that system how to design
estimators like the common filter so the
diffuse limited sensors you could
reconstruct various aspects of that
system this is not meant to be an
exhaustive in-depth treatment of the
subject but really kept at a high level
and my goal is to first of all get you
familiar with the major types of optimal
and modern control theory I want to
teach you how to use these in MATLAB to
actually work with a real system and
what I also want to give you a feeling
for is what in control theory is easy
and what's still quite challenging today
so that you can get up to speed on the
real pressing needs of control theory
today okay and again this is not
exhaustive so you know if this is really
important to you and you want to you
know you like control theory and you
want to go more into depth there's
deeper treatments both on the math side
and only applied design side okay and so
I want to give you just a little bit of
perspective I think about the world in
terms of dynamical systems so systems of
ordinary differential equations in terms
of the state of your system and this has
been an extremely successful viewpoint
for modeling real world phenomenon okay
so we model the fluid flow over a wing
or the population dynamics in a city or
the spread of a disease or the stock
market climate planets moving around the
solar system all of these are modeled as
dynamical systems and this has been a
very very successful framework to take
in data from the real world and build
models that you can use for prediction
but often we want to go beyond just
describing the system of interest and we
want to actually manipulate the system
actively to change its behavior
and so that could be just imposing some
control logic just setting inputs into
the that system in a certain pre-planned
way to manipulate it or you could
actually measure that system and make
decisions based on how the system is
responding to what you're doing okay and
so that's kind of the overarching view
and control theory is that you have some
dynamical system and interest maybe it's
a pendulum or a crane that you want to
make more stable you write down the
system of equations and then you design
some control policy that changes the
behavior of your system to be more
desirable okay so that's what we're
going to talk about and so I want to
begin by just talking about the various
types of control that there are so
there's lots of control that goes around
all around us every day that is not
active it's called passive control so
I'm going to draw just a diagram of the
different types of control so one type
that's very common you see all the time
is passive control okay so for example
if you see a large 18-wheeler transport
truck going down the highway and it has
those streamlined tail sections that's a
form of passive control that's passively
causing the air around the truck to
behave in a favorable way to reduce drag
and if you can get away with passive
control of your system that's actually
great because you just have to design an
up front and then there's no energy
expenditure and hopefully you get the
desired effective for example minimizing
drag on a truck but passive control is
typically not enough and so oftentimes
we need to do something like active
control and so active control
essentially just means that this is
control when we're actually pumping
energy into the system to actively
manipulate its behavior okay and there's
lots and lots of different types of
active control so one that I'm going to
tell you about is open-loop this is
probably the most common form of active
control where essentially you have your
system of interest
and I'm just going to actually draw this
as a block here so you have some system
and the system has some input so I'm
going to call them variable U and it has
some outputs that are variable Y okay
and so what open loop control does is it
essentially reverse designs your system
and inverts the dynamics to figure out
exactly what is the perfect input u to
get a desired output Y okay and so if I
take something like a inverted pendulum
so we know that if I if I am very
careful I can stabilize this inverted
pendulum but it in physics you'll learn
that if you just pump this pendulum up
and down at a high enough frequency it
will naturally stabilize the dynamics
okay so if my base just oscillates at a
high frequency sine wave then the
dynamics of this pendulum so the base is
you may be why is the angle of this
pendulum and my desired control is to
make this pendulum essentially stay at
vertical okay and so if I pump in energy
in a pre-planned way I just make my hand
go up and down in the sinusoid I can put
in a sinusoid
Y that I want okay and essentially that
is open with control it's very commonly
used essentially you think about your
system you pre plan a trajectory and you
just enact that control law okay but the
downside of open-loop control is that
you're always putting in energy to this
to this u so in the inverted pendulum
example I constantly have to be pumping
this thing up and down sinusoidally and
the minute I stop the stability it
becomes unstable and it falls okay and
so the idea is that what we can do is
called closed-loop feed feedback control
so closed-loop feedback control
and essentially what this means is that
we take sensors bring my pendants drying
out
we take sensors sensor measurements of
what the system is actually doing and
then somehow we build a controller I'm
just going to call this a controller and
we feed that back into our our input
signal that can manipulate the system so
for example in that inverted pendulum
example as a human if I had a tall
enough pendulum so it's slow enough I
could actually measure with my eyes if
it's starting to wobble and I could do
much more subtle control so if you have
ever played around with as a kid with a
broomstick cat trying to stabilize it
you know that you can actually get
pretty good at it so that with very low
energy input or very small hand motions
you can stabilize this thing so that it
doesn't fall okay and so that's the
basic idea is that by measuring the
output you can often do much much better
than just feeding in kind of a
pre-planned control law okay so sensor
based feedback measuring the output and
then feeding that back as the input is
basically going to be the entire subject
of what we're going to talk about in
this control boot camp so closed-loop
feedback control is the name of the game
and that's that's most of what we're
going to talk about now that's not to
say that if you can design a good open
loop or a good passive control there you
know there are some times you would do
that but in the systems we're going to
be interested in closed loop feedback
based on sensors is going to give
dramatically better performance okay and
so I want to talk a little bit about why
you would have feedback so I just want
to make a quick list why feedback
because this is a very very important
important topic in control theory so I
want to motivate again just maybe in
more concrete terms
why would I actually measure the system
and feed it back instead of just
ignoring any measurements and using
open-loop so why feedback over open-loop
control okay so this is a question I
always ask my class and I let them think
for a little bit why would you actually
want to have
the sensors feeding back into your
system okay so one answer that I get
most often is maybe my system has some
inherent uncertainty okay so if my
system is uncertain
so uncertainty is one of the main and
enemies of open-loop control right so if
I have this pendulum and I perfectly
pre-planned what I want to do let's say
that the pendulum is one centimeter
taller or it's a little bit heavier or
there's wind blowing or something like
that then any kind of uncertainty in
that system is going to make it so that
my pre-planned trajectory is going to be
suboptimal
but if I measure the outputs and I
realize that it's not doing what I want
it to do I can adjust my control law
even if I don't have a perfect model of
my system okay so uncertainty is a big
one
another really important one is
instability so with open-loop control I
can never fundamentally change the
behavior of the system itself so in the
pendulum example I could pump in an
amount of energy with the sinusoidal
based motion that would force the system
to kind of correct itself up to vertical
but I'm not actually changing the
systems dynamics itself the system still
is unstable and has an unstable
eigenvalue but when I have feedback
control I can directly manipulate the
actual dynamics of this closed-loop
system and I can change the the dynamic
properties I can change the eigenvalues
of this closed-loop system okay and I'm
going to show you that as the last
example in this overview so the third
thing that I think is really really neat
is that with feedback control you can
also reject disturbances in your system
so let's say that I have some external
disturbance D that's coming into my
system and this happens all of the time
so so for example let's say in my
pendulum example there's a gust of wind
so that's a
disturbance that would be very hard for
me to predict or model or measure so
there's this gust of wind that comes and
if I had an open lead strategy
essentially it might not be able to
correct for that gust of wind where is
that gust of wind will pass through the
system dynamics will be measurable
through some sensor and if my feedback
control is good enough I can actually
correct for that disturbance so I think
of uncertainty as internal system
uncertainty kind of disturbances to my
model and I think if disturbances as
external or exogenous forcing of the
system that may be too difficult or too
costly or too complicated to to model or
predict or measure okay and feedback
essentially handles all of those basic
issues that can handle disturbances that
can handle uncertainty and it can
fundamentally change the stability of
your system to make it more or less
stable by actually changing the
eigenvalues of this closed-loop system
and unfortunately open-loop can't do any
of those things which is a huge drawback
and I guess the fourth one is energy or
efficiency
so I'll just say efficient control so
again in the case of the pendulum in the
open-loop case I constantly had to pump
this thing up and down so I was always
putting energy in but in the case of
sensor-based or elegant feedback control
you can picture yourself trying to
stabilize this broomstick if you're
doing a really good job if you have a
really good controller this thing is
barely moving at all and so you almost
have to put no energy in to correct it
so effective sensor based feedback
control is also much more efficient
which is really really important in lots
of applications so if you're going to
send a rocket somewhere you better have
an efficient controller because you
don't want to be wasting fuel okay so
the last thing I want to show you is
just this idea of why you can change the
fundamental system dynamic dynamics and
change the stability with feedback
control okay so the basic property that
we're going to or the basic mathematical
architecture we're going to be working
with in this class is going to be a stay
space system of ordinary differential
equations so we're going to have a state
variable X X as a vector that describes
all of the quantities of interest in my
system so for example in my pendulum it
could be the angle and angular velocity
it could be two states if I have you
know an airplane going through the sky
it could be the three the position
vector XY and Z and also its its
rotation angles and their derivatives
okay so it could be like a six degree of
freedom or twelve state twelve component
vector X and so what we're going to look
at is the system X dot equals ax so
we're going to start with linear systems
of equations that describe how those
states interact with each other okay and
so I'm going to assume that we're all
pretty comfortable with this linear
systems of OD e so for example we know
that the solution of this is X of T
equals e to the matrix say T times X at
time x zero okay so we know how the
system behaves we know that if a has any
eigenvalues with a positive real part
then the system will be unstable and if
all of the eigen values have negative
real part then these have stable
dynamics that they go to zero as time
goes to infinity but what we're going to
do in control theory is we're going to
add plus B U so we're going to add this
ability to actuate or manipulate our
system okay so we're going to say that U
is our actuator it's the thing we can
its our control knob okay so it could be
in the case of the pendulum it could be
the position of the base or it could be
the voltage onto a motor that controls
something but this is the knob that we
get to turn to try to stabilize our
system and B tells you how this control
knob directly affects the time rate of
change of my state okay and down the
road we're going to look at another
extension where we're actually going to
measure only certain aspects of the
state so we're going to measure so
linear combination of the state X and
this might actually be a limited set of
measurements we might not measure all of
this the state of its high-dimensional
and we might only have access to those
few sensor measurements in Y but for now
let's just talk about the top equation
so if I assume that I can measure
everything in the system and in this
case of the pendulum as a human I have a
pretty good estimate of its vertical
position and how fast it's moving so
let's say I can measure all of X then we
can develop a control law let's say u
equals minus some matrix K times X okay
so I'm just going to say let's posit a
basic control law that my control input
U is going to be some matrix times X
just some constant constant times the
components of X when I plug this in so
this is this is really sensor based
feedback where y equals x okay in this
case we're assuming that y equals x we
can measure all of our state and we're
going to feed that back into a control
law which is minus K u equals minus K
times X and we're going to try to modify
the dynamics so if you plug u equals
minus KX into our dynamics we basically
get and let's make another color here we
basically get X dot equals ax and then
minus B K X okay so B is maybe a tall
vector the same or set of vectors the
same height as X K it's kind of the
transpose size of that and so this is a
matrix of size n by n if X is an
n-dimensional state and so this equals a
minus BK times X so notice that by by
measuring the state in this case we're
measuring the full state X and feeding
that back to the control u through this
law u equals minus
a X we're able to actually change the
dynamic matrix so now we have a new
dynamical system X dot equals a minus BK
times X and so it's actually the
eigenvalues and of this matrix that tell
you if the system is stable so I can
have a really originally unstable system
like this inverted pendulum and by
measuring the state and feeding it back
to my control knobs I get to move I can
stabilize the dynamics I can actually
make the system asymptotically stable
okay and so figuring out when you can do
this so this doesn't work for all
systems and for all measurements and for
all actuators so figuring out when the
system is controllable and how to design
this case so that it is well controlled
are going to be the subjects of the next
couple of lectures okay but really
really important feedback solve all of
these fundamental problems if I have an
uncertainty in my system I can
compensate for it by measuring what's
actually happening and feeding that back
if I have an instability in my system I
can actually change the dynamics with
this feedback and you can't really do
that with open-loop I can also account
for external disturbances like a gust of
wind that might have been really hard to
measure and could totally throw off your
pre-planned trajectory but if you
measure what's happening you can account
for and correct for that
and finally feedback control is
efficient if you're doing effective
feedback control to stabilize a system
then the more effective you are the less
energy you have to put in okay
so this should be a really exciting set
of lectures I'm really hoping to get you
up to speed quickly and with MATLAB
examples so that you can control these
systems you can design controllers to
actually manipulate your system to do
what you want it to do okay thank you
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