‫Welcome back.
‫So now you have learned how to go from London one and London two to the Constance Kay one and Kate two,
‫and now let's look at the same thing.
‫But from a different point of view, suppose that again, this is your differential equation, this
‫one right here with your key one and key to values.
‫The thing with differential equations is that they are often used to represent systems.
‫That means that they could also be converted into state space equations.
‫And also note that I have been using the word LTI or linear time invariant a lot in front of our differential
‫equations.
‫Well, let's put our differential equation in this space form.
‫I can rewrite this equation like this error dot equals error dot and then error double dot equals K
‫one times error plus key two times error dot.
‫And it's a valid statement, right?
‫There is nothing wrong with that.
‫In the second equation, I simply put this term and this term to the other side of the equation, sine
‫and well error dot equals zero dot.
‫It's just true, right.
‫It's like five equals five.
‫But of course you might be asking why would I write error dot equals error dot.
‫What's the purpose of it?
‫Well, the reason why I wrote it like this is because now I can represent this system in a vector matrix
‫form.
‫I can have a vector here so I can have error dot here then error double dot here, then I have some
‫kind of matrix here and then I have another vector which is error and then error dot.
‫You see it has two elements here, error and error dot.
‫So thanks to this error that equals error dot equation, I can now fill in the first row of this matrix.
‫So if error that equals error dot, then that means that you will have zero here and you will have one
‫here.
‫And now if you look at the second equation here, this one.
‫Then you can see that you will have a key one here and key to right over here.
‫So tell me now, does it seem familiar to you?
‫Well, it does to me, because to me what it looks like is this x dot as a vector equals and then a
‫matrix times the state vector.
‫So this is the state time derivative vector equals eight times the state vector.
‫So this matrix here, that's your A matrix.
‫And so if you assume that your key values K one and key to our constants, then this system here in
‫fact is a linear time invariant or an LTI system.
‫So you see I've taken this differential equation and I have converted it into an LTI states based system,
‫and that is why I have been using the word linear time invariant in front of our differential equation.
‫I have always been saying homogeneous, linear time invariant or LTI second order differential equation.
‫And the way I know that this differential equation is an LTI differential equation is because if I transform
‫it into a state based form, then I will have an LTI state based form in which this a matrix is a constant
‫matrix.
‫It's time invariant.
‫It doesn't change with time.
‫And another thing that you might have noticed is that you only have an A matrix here.
‫There is no B matrix here.
‫That's because this differential equation here is homogeneous.
‫Let's take a look at a non homogeneous differential equation.
‫So this is our differential equation here and this one is not a homogeneous one because it has a term
‫here on the other side of the equation sine that is not zero.
‫And so as an exercise, try to put this differential equation into your state based form, try doing
‫it yourself and then we'll look at it in the next video.
‫Thank you very much.