welcome to control systems lectures before you started don't think there's a
few who subscribe to the channel:
and also whether your control student whose hearing this for the first time
or a veteran engineer brushing up on the basics appreciate you taking the time to
watch this video
and you find these venues hopeful or if you find them not being clear enough
please leave a comment below and I'll try to address it but now
on the lecture in this lecture on to describe
transfer functions and how control engineers use them to model elements
like filters
actuators in order to simplify control system design
and analysis in the simplest description I can find
a transfer function is the applause transform
up the impulse response other linear and time-invariant system
when you set the system initial conditions 20
now unless you're already familiar with transfer functions
I'm betting that statement didn't make a whole lot of sense for the time being
well just think about transfer function as a mysterious black box
and we imply an input signal into it we had a modified signal out
and if we've designed this black box correctly
means pretty good morrow a real physical process
let's take a completely ridiculous example to explain the benefit of
transfer functions
and hope we'll get a better understanding of them in the process see
you got in trouble at school
and your ass to rate your name a hundred times on the chalkboard but instead of
standing right next to the board
the teacher says you to stand on the opposite side of the room
and since you can't physically reach the word from where you're standing
you have to use the strange contraption this consists a
a flexible stick it has acquired test to the Indic
and miss clark moves in remote control which drives a pair of linear actuators
and then moves another flexible stick which is then attached to a piece of
choc
which finally is touching the chalkboard and now you can finally start to rate
your name
I told you this was ridiculous so just stay with me if we take the simplified
block diagram
love this system might look something like this: the input into the whole
system
are your hand movements this moves the first flexible stick that all represent
right now as a black box
start thinking of these is transfer functions the up into this box is the
position
the end of the first ek the position is now the input
into the remote control remote control another black box converts the position
into an electrical signal
in the form a change in voltage this voltage is applied to the linear
actuators
which converts the electrical signal the position again this position as the
input into the final flexible stick
whose output is the position a piece of choc
we finally draws on a blackboard and if you've accounted for all the relative
motions in the system correctly
you can write your name 100 times and you get to go
how can you possibly even draw streamlining
let alone something as complex as your name to such a flexible system
with so many different transformations the answer
is by characterizing the behavior each individual part
and his each black box in combining them to understand how the system behaves as
a whole
and the simplest way to do that is with transfer functions
I'll finish this story in a bit the first thing to practice went back to
claim
with some background knowledge and just the smallest amount of math
we start like most things in classical control theory
with below-cost transport went to a plus transformed us
his map a function from the time domain to the esta mi
I will describe the intricacies of the transform into future lecture
however for now I'm already here without any mathematical proof as you can
imagine
Google+ transform is a complex operation in all but the simplest cases
likely actually performing this integration is rarely needed
because the most common transformation between time and yes to main
are part of software packages like MATLAB or can be easily looked up in
Tables
or even just memorized here are some more common ones
the drop delta function which is a squealing DFT
the low-cost transform is just more this is also the
in polls function the head exit teeny
Google+ transform that is just exit s ex-prime a team which is the first
derivative
X is s times exit s minus
the initial position x0 then there's X double prime which is the second
derivative 50
is just s squared times exit s minus as times the initial position
minus the initial velocity if you watch my video on linear time-invariant
systems
you know that any Lt I system can be completely described by its temples
response
just as repression all sum it up here if you subject to the LT I system to impose
function
est and Rec delta function at time 0
you can either measure or calculate the output this is called the
impulse response love the system
lets the subject this 20 arbitrary input
how can you determine with the output is that it well you can break that
arbitrary input up into in infinite number
love impulses and since this isn't all shy system
you can steal and shipped through time the impulse responses
appropriately base all the skilled and shifted impulse
inputs now finding the system time domain response to the
arbitrary input is the simplest summing up
all the incident impulse responses and sense
something an infinite number of signals is impossible to do a one-time
mathematicians came up with what is called the convolution integral
always plane using this block diagram notation if you have an arbitrary input
you with T
in you apply it to impulse response GFT
Danielle Kawai T is equal to the convolution a few
Angie rain here in general format or
if you prefer shorthand using this star but these two representations
are equivalent so in this representation
GFT is the impulse response and used he is
any arbitrary input here's what's great about the applause transform
if you think a low-cost transform love the input
which becomes you s and you keep along plus transformer the impulse response
GS then the output wire s
is just multiplication overview Angie
NGN s low-cost transform up the impulse response
which is called the transfer function so we have reduced it difficult convolution
in a whole
with a much simpler multiplication step or more accurately
below-cost transforms taking care of the convolution for you
to make sure that the sayings 10 was walk to this one more time
but this time with an example take a simple harmonic oscillator
with mass him and spring constant k which is subjected to an input forcing
function you ft
it can be shown that the equation of motion of such as system
is am times the acceleration class K
times the position and that said equal to the input you ft
and find the impulse response in the system we say UT to the direct delta
function
and now it's all this differential equation using the low-cost transform
to you that we take a look lost transformer the left side which is him
times will cost transform the acceleration which we go back to our
table
we can find X double price remember that there were no initial conditions for
this problem so we can set these 20
and you're left with is s squared times Xmas
so we go back down can replace external prime ft with esquire
Xmas now we can do the same with the 2nd ter
go back up 21 find X-eighty is just X
s it seems simple to get K
X a vast now you can take full cost transforming the right hand side
which is just the impost function which has a low-cost transform
of what
from here's just a few outbreak steps to solve for x
s which is the impulse response up the system
in the esta mi which turns out is one over and that squared plus que
but what if we want this is the time to me all we have to do is take the
in personal plus transform and this is one that I have memorized
so I had to look this up in a table and it turns out is one of the square root
km
times the sign the square root of came over him times t
or signs lead as you'd expect or if the input wasn't an impulse but a ramp
function
you have tea crostini how would you go about solving with responses to this
input
well if you're in the time domain you do this to convolution
or UNC would convey all teeny with that sign is horrible
and this is a pretty difficult integration to do
likely we can do this in the S two main by taking the low-cost transform up the
ramp
which is one over a squared been defined the
response to this ramp all we have to do is multiply
one over a square I want over ms squared plus que
and remember also that one over and ask where class K
is the imports response to the system in the S two main
so this is the transfer function and notice that we got this transfer
function by taking the low-cost transform love your question motion
we never need to go to the time to me snow we know that let's get back to our
example above
if we were able to write the equations of motion for each of these individual
processes
then by the method described below we would have the transfer functions
which we can insert into all these unknown black boxes
for example the sticks can be modeled as an inertial is spring and damper
this has a transfer function of 10 for CS plus que
the Remo can be modeled as a second order process as it converts mechanical
position and electrical voltage
in a linear actuators to be modeled as a mass
snow would have been a near-impossible Senate into cross we're trying to do
this in time domain with convolution
combining his transfer functions in the S two main is as simple as multiplying
them all together
and the result is the transfer function up the entire planet
where the input into the plant is your hand motion still
and the output is the chart position
so hopefully you can see how important transfer functions art modeling
control system in lectures on frequency response an open-loop and closed-loop
performance
you'll see that there are many other ways we use transfer functions
the last thing I want to leave you with is just something to think about
this is actually a closed-loop system that you the troubled students part
the reference signal is your name this is where you are written on the
chalkboard
you sense the current state of the system with your eyes in you compare
what you're seeing to what you want written
your brain interprets these errors and generates commands to move your hands
well what if you want to remove yourself from this controlled in turn it into an
automatic control system
we'll discuss that in future lectures