Closed Loop Transfer Function Video Lecture - Electrical Engineering (EE)

this is the basic control loop that we
have and we call this a closed loop
feedback and we call it feedback because
you're taking an output so here's our
plant and then we're adding some sort of
control component and we have an output
and then we take that output and we feed
it back to earlier part of the chain and
again we take the difference here and we
get the error so the error between the
reference the set point so set point is
say it's set by Valerie this is kind of
our tends to be our human input or our
reference for choosing that and then we
want our system output to look like that
so we are going to use this marker to
derive the stability of the system so
before we're looking at the stability of
the plant and we were looking at just
cutting out all this stuff and just
looking between the input and the output
but now with this closed-loop system our
inputs and outputs are actually a little
bit different so now our input is
actually our reference and our output is
our Y now I'm using y CL here to mean
close the loop so let's actually derive
the transfer function for this closed
loop system and we will use the transfer
function for the plant and we assume
that we know or can model the transfer
function for the controller okay so
let's start over here everything's
linked together so we each have to start
somewhere
we're going to start with Y and Cl so
this is s this would be in the time
domain this is also y CL so of s right
and so well what is that equal to well
something's x GP something x GC and then
we have the error here and the error is
RS minus YC again so let's let's go
backwards then so Y closed loop is equal
to
this transfer function of the plant
that's going to be multiplied by this
transfer function of the control I'm
going to drop all the of s's here but
everything's in a little plus transform
so then we multiply by es but II of s is
what it is RS the reference - what we
already have so why CL the closed-loop
output and what we want to get remember
so we would like to get we're going to
call it t I think the books and few
other places use that so T is going to
be equal to what we want is the closed
loop over the reference okay so let's do
that so we can move that over so expand
that that's what I was looking for our
gpgc and then it's going to be minus
multiplied by that but then we're going
to subtract it and move it over here so
we'll get Y CLF is right on the edge let
me move it over so it's a little bit
clearer so I'll do y CL here and we have
a one from here and then we have these
multiplied together so GP G C okay so
this is pretty easy we know how to do
this we want to move this to the
denominator over here so we'll get Y CL
is equal to r GP G C divided by 1 plus G
P G C so this is almost there to get the
actual transfer function all we need to
do is divide this R so T of s right
what output close with output divided by
the reference is the two plants sort of
and the controller multiplied together
divided by 1 plus those values so this
is our final expression so if we want to
analyze the stability or any other the
the how this system operates the
closed-loop system operates we use this
transfer function to understand its
stability and how it will react alright
so that's a derivation this is important
to know yes oh sorry
gosh always oh hi so this is the correct
equation so it's these two multiplied
together and this is the transfer
function for the closed-loop feedback
system shown here