Intro to Control - 5.3 Understanding Linearization

we're here back talking about linearization and I really want to cover how it works and so that you really understand how linearization works on what you need to do to linearize a function so we're going to review that again and I'm going to start out with just a function here and so here we just have an input of Y and some function f of Y and what we want to do in our process is eventually we want to do three things first one we want to do is one is to find the equilibrium point point alright and in any system what this really means is we want to find where our function f of y equals zero so f of y equals zero so that's the first thing we need to do and we do this because we want to linearize the function around this point so in this one it's pretty easy to visualize F y equals zero find that point here we are and we'll generally call that point in control we'll call that Y and I'll be using little subscript e so we put a little subscript e it's the input that makes the total function zero okay so we found that point that's the first thing we need to do the second thing kind of comes in two parts two and three are actually very related second thing we need to do is to recenter the axes for our function around the equilibrium point so we're going to say center axes axis at equilibrium yeah okay so we're going to Center it at equilibrium and the reason that we have to do this is because all of our linear functions have to go through zero so if you recall from our linear and nonlinear discussion a linear function always has to go through zero essentially it's going to look like a line of some sort going through the origin this equation we need it to go through this point so this is the equilibrium so we need to Center that on that point and the way that we do this is that we rename our axes so I covered this before but essentially what we do instead of having Y here we define something called Delta Y so we say Y is equal to whatever this equilibrium point was or is ye and then we're going to define our new axes based on here so we're going to define a little Delta Y here so this is equal to Y e the equilibrium point plus Delta Y so all we're doing is changing our variables we're moving our axes around the equilibrium point and then calling our new variable Delta Y and this is the perturbation signal okay so that's the first thing that we do and I guess the second thing the third thing that we do now is to linearize non-linear Oh technically that's not a - usually non-linear um components will say and this is where the Taylor series expansion comes in very important so Taylor series expansion and I just going to write the generic form of that so if you aren't not as familiar with this you can go back to some other videos math videos that explain how the Taylor expansion works I'm going to write the general form and then we're going to apply it to our function so usually they say f is equal to X it's F of Y o they use f of X so okay so a a here in the general expression is the equilibrium point or the point that you're linearizing around so we'll call equilibrium glib point spelling is not my strong point sorry all right so then you have to take the first derivative DT of F evaluated at a so that's the equilibrium point right and then you have X minus a X being your normal variable minus equilibrium point over one factorial and then you do the next term you have the second derivative DT squared of s a evaluated a and this time it's X minus a squared over two factorial and this goes on infinitely this an infinite series and if you have the full infinite then it is equal to your original nonlinear function here for controls we completely ignore anything above the first-order so first derivative so ours is really an approximation of that so what we're trying to do now is to define a new function and because we're only looking at the first order it will be a line so we're essentially trying to make a linear approximate here and before I call that F Prime some people got confused so I'm just going to call this G this time so this will be G of Delta Y the Euler new function that we're approximating around the equilibrium point and that will be a linear function all right so let's try to apply this so this we ignore we've outlined the basic steps for linearization and now let's take an example so let's say that we have y double dot double derivative plus y 1 dot first derivative and we're going to make it equal to negative cosine of Y okay so to do this now we need to first find the equilibrium point okay so when we do that we assume that the dynamics of the system are zero so that means that y double dot or y dot are zero if there's an input we also assume that that's zero in this case this is a single variable so we just assume these two are 0 which means so if we do step one on the equilibrium we would get negative cosine of Y is equal grim it's equal to zero okay and if we think back to what our cosine graph looks like a negative cosine there would actually be multiple equilibrium points here but we are going to assume so here a negative cosine would look like this right so we're going to we'll assume this equilibrium point which is going to be PI over 2 okay so we're going to say ye is equal to PI over 2 so let's find the linearize around this point okay so we found the eleven point great next now we need the center it around the equilibrium point so really we're just saying we're plugging this value into here so this is going to be our next equation so we look back at our equation and we say well hey there's um two different sides let's look at each side independently so if we look at this we actually see that the derivative components themselves are already linear and if you don't believe me then you have to go back to superposition let's do an example with this real quick so let's say that efg we're going to make a new function H we're going to call this H momentarily so H of guess Y so if we have h of y1 equals D DT squared of y1 and then we have h of y2 equals d squared DT of y2 so double derivative of both if we add these together so if we do actually if we add y1 plus y2 and then put it into our function so H of Y 1 plus y 2 would be equal to derivative double derivative mr. squared here would a y1 plus y2 we know that we can distribute the derivative here so we would end up with double derivative of both which is equivalent to these two so you get H of Y 1 plus h of like you so if you got confused on you know this is already a linear function this is superposition you can go through and do the test for ma Jenaya T it will work out so these derivatives are already linear functions so we actually don't have to linearize this this part because they're already linear so all we have to do on this one is actually apply this equation so our moving of the axes to this side and this part of the equation will already be linear so if we do that and erase this what we can do on the left side then is do I will D DT squares a double derivative instead of Y now we have Y E minus DT y plus BG Delta online Delta Y look what I'm writing sorry if I misspeak and then we can do the same thing with this component so the first derivative now of no longer Y but now ye plus d y ye plus Delta Y and then we're going to linearize that side later if you look at this so if you look at just this term well we have Y E which is actually a constant value right it's not a variable its equilibrium value so if we take the double derivative of a constant it's zero so this really just becomes our double derivative I'm going to write your hand do you guys are ok with that just the drive a double derivative of this so we get two dots on top of this variable Delta Y the same thing happens here the first derivative of a constant is zero so we just get the dynamics first derivative around Delta Y so then we can do Delta Y and take the first derivative with time so we see is that really we took our original and we just were able to replace it with the Delta Y because the derivatives of the constant of Y E is zero so this is already linear and so we don't actually have to linearize it because it's already done and now we can go to the other side and look at that here we see that this is not linear this part so we do need to linearize it using the Taylor series expansion okay so let's do that so let's remember what the Taylor expansion is here now let's try to we'll bring these put it back into our terms now so we'll take the Taylor expansion of this just directly and see what happens well first we need to take F of a in this case a is our equilibrium value so we'll take cosine negative cosine of our equilibrium value okay and we chose it to be zero so here it will be PI over two which will be zero so this will go to zero and that will always happen if this does not go away there's something wrong with your your equilibrium value so that's a good check next we go to the next part so we say okay well we'll take the derivative of our function here and evaluate it at a which is the equilibrium point okay so derivative of cosine is negative sine but we have two negatives so we get sine of our Y E which again is PI over two and then let's F of a throw to f of a and then we have this thing so X minus a and a is our y e we see that if we rewrite this we can write it as Delta Y is equal to Y minus y e which is the same as Y minus a here so we can actually directly put this Delta Y into there so we're kind of already we're linearizing it and changing the axis at the same time and then the 1 factorial is just 1 so this is our equation so almost there this went to 0 we just need to find the sine of PI over 2 if we think back to our what a sine equation is like that PI over 2 is going to be 1 so this will be 1 and we get 1 Delta Y so this would be our linearized version of this of our equation here and this is one of the endpoints so you could say this is a full linearized version of our system at the equilibrium point if this had inputs and outputs then you could take the Laplace transform and figure out the transfer function but in this example we only did a single dimensional so we'll do look at some more using multi variable but I'll stop here for now I apologize that I fail at spelling because I filled is completely wrong it's equilibrium spelling is important

Channel: katkimshow

Views: 58,799

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Keywords: Introduction to Controls, UNIST, Linearization, System Modeling

Id: 0SGnWCroLxs

Length: 15min 59sec (959 seconds)

Published: Sat Sep 27 2014