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welcome to control systems lectures before you started don't think there's a

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few who subscribe to the channel:

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and also whether your control student whose hearing this for the first time

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or a veteran engineer brushing up on the basics appreciate you taking the time to

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watch this video

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and you find these venues hopeful or if you find them not being clear enough

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please leave a comment below and I'll try to address it but now

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on the lecture in this lecture on to describe

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transfer functions and how control engineers use them to model elements

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like filters

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actuators in order to simplify control system design

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and analysis in the simplest description I can find

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a transfer function is the applause transform

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up the impulse response other linear and time-invariant system

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when you set the system initial conditions 20

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now unless you're already familiar with transfer functions

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I'm betting that statement didn't make a whole lot of sense for the time being

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well just think about transfer function as a mysterious black box

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and we imply an input signal into it we had a modified signal out

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and if we've designed this black box correctly

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means pretty good morrow a real physical process

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let's take a completely ridiculous example to explain the benefit of

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transfer functions

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and hope we'll get a better understanding of them in the process see

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you got in trouble at school

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and your ass to rate your name a hundred times on the chalkboard but instead of

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standing right next to the board

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the teacher says you to stand on the opposite side of the room

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and since you can't physically reach the word from where you're standing

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you have to use the strange contraption this consists a

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a flexible stick it has acquired test to the Indic

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and miss clark moves in remote control which drives a pair of linear actuators

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and then moves another flexible stick which is then attached to a piece of

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choc

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which finally is touching the chalkboard and now you can finally start to rate

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your name

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I told you this was ridiculous so just stay with me if we take the simplified

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block diagram

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love this system might look something like this: the input into the whole

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system

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are your hand movements this moves the first flexible stick that all represent

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right now as a black box

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start thinking of these is transfer functions the up into this box is the

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position

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the end of the first ek the position is now the input

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into the remote control remote control another black box converts the position

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into an electrical signal

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in the form a change in voltage this voltage is applied to the linear

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actuators

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which converts the electrical signal the position again this position as the

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input into the final flexible stick

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whose output is the position a piece of choc

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we finally draws on a blackboard and if you've accounted for all the relative

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motions in the system correctly

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you can write your name 100 times and you get to go

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how can you possibly even draw streamlining

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let alone something as complex as your name to such a flexible system

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with so many different transformations the answer

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is by characterizing the behavior each individual part

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and his each black box in combining them to understand how the system behaves as

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a whole

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and the simplest way to do that is with transfer functions

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I'll finish this story in a bit the first thing to practice went back to

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claim

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with some background knowledge and just the smallest amount of math

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we start like most things in classical control theory

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with below-cost transport went to a plus transformed us

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his map a function from the time domain to the esta mi

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I will describe the intricacies of the transform into future lecture

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however for now I'm already here without any mathematical proof as you can

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imagine

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Google+ transform is a complex operation in all but the simplest cases

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likely actually performing this integration is rarely needed

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because the most common transformation between time and yes to main

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are part of software packages like MATLAB or can be easily looked up in

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Tables

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or even just memorized here are some more common ones

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the drop delta function which is a squealing DFT

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the low-cost transform is just more this is also the

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in polls function the head exit teeny

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Google+ transform that is just exit s ex-prime a team which is the first

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derivative

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X is s times exit s minus

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the initial position x0 then there's X double prime which is the second

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derivative 50

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is just s squared times exit s minus as times the initial position

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minus the initial velocity if you watch my video on linear time-invariant

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systems

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you know that any Lt I system can be completely described by its temples

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response

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just as repression all sum it up here if you subject to the LT I system to impose

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function

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est and Rec delta function at time 0

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you can either measure or calculate the output this is called the

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impulse response love the system

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lets the subject this 20 arbitrary input

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how can you determine with the output is that it well you can break that

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arbitrary input up into in infinite number

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love impulses and since this isn't all shy system

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you can steal and shipped through time the impulse responses

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appropriately base all the skilled and shifted impulse

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inputs now finding the system time domain response to the

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arbitrary input is the simplest summing up

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all the incident impulse responses and sense

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something an infinite number of signals is impossible to do a one-time

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mathematicians came up with what is called the convolution integral

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always plane using this block diagram notation if you have an arbitrary input

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you with T

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in you apply it to impulse response GFT

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Danielle Kawai T is equal to the convolution a few

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Angie rain here in general format or

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if you prefer shorthand using this star but these two representations

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are equivalent so in this representation

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GFT is the impulse response and used he is

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any arbitrary input here's what's great about the applause transform

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if you think a low-cost transform love the input

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which becomes you s and you keep along plus transformer the impulse response

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GS then the output wire s

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is just multiplication overview Angie

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NGN s low-cost transform up the impulse response

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which is called the transfer function so we have reduced it difficult convolution

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in a whole

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with a much simpler multiplication step or more accurately

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below-cost transforms taking care of the convolution for you

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to make sure that the sayings 10 was walk to this one more time

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but this time with an example take a simple harmonic oscillator

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with mass him and spring constant k which is subjected to an input forcing

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function you ft

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it can be shown that the equation of motion of such as system

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is am times the acceleration class K

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times the position and that said equal to the input you ft

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and find the impulse response in the system we say UT to the direct delta

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function

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and now it's all this differential equation using the low-cost transform

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to you that we take a look lost transformer the left side which is him

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times will cost transform the acceleration which we go back to our

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table

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we can find X double price remember that there were no initial conditions for

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this problem so we can set these 20

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and you're left with is s squared times Xmas

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so we go back down can replace external prime ft with esquire

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Xmas now we can do the same with the 2nd ter

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go back up 21 find X-eighty is just X

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s it seems simple to get K

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X a vast now you can take full cost transforming the right hand side

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which is just the impost function which has a low-cost transform

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of what

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from here's just a few outbreak steps to solve for x

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s which is the impulse response up the system

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in the esta mi which turns out is one over and that squared plus que

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but what if we want this is the time to me all we have to do is take the

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in personal plus transform and this is one that I have memorized

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so I had to look this up in a table and it turns out is one of the square root

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km

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times the sign the square root of came over him times t

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or signs lead as you'd expect or if the input wasn't an impulse but a ramp

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function

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you have tea crostini how would you go about solving with responses to this

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input

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well if you're in the time domain you do this to convolution

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or UNC would convey all teeny with that sign is horrible

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and this is a pretty difficult integration to do

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likely we can do this in the S two main by taking the low-cost transform up the

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ramp

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which is one over a squared been defined the

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response to this ramp all we have to do is multiply

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one over a square I want over ms squared plus que

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and remember also that one over and ask where class K

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is the imports response to the system in the S two main

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so this is the transfer function and notice that we got this transfer

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function by taking the low-cost transform love your question motion

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we never need to go to the time to me snow we know that let's get back to our

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example above

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if we were able to write the equations of motion for each of these individual

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processes

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then by the method described below we would have the transfer functions

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which we can insert into all these unknown black boxes

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for example the sticks can be modeled as an inertial is spring and damper

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this has a transfer function of 10 for CS plus que

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the Remo can be modeled as a second order process as it converts mechanical

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position and electrical voltage

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in a linear actuators to be modeled as a mass

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snow would have been a near-impossible Senate into cross we're trying to do

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this in time domain with convolution

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combining his transfer functions in the S two main is as simple as multiplying

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them all together

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and the result is the transfer function up the entire planet

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where the input into the plant is your hand motion still

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and the output is the chart position

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so hopefully you can see how important transfer functions art modeling

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control system in lectures on frequency response an open-loop and closed-loop

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performance

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you'll see that there are many other ways we use transfer functions

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the last thing I want to leave you with is just something to think about

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this is actually a closed-loop system that you the troubled students part

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the reference signal is your name this is where you are written on the

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chalkboard

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you sense the current state of the system with your eyes in you compare

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what you're seeing to what you want written

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your brain interprets these errors and generates commands to move your hands

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well what if you want to remove yourself from this controlled in turn it into an

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automatic control system

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we'll discuss that in future lectures

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