‫Welcome back.
‫In the world of differential equations, there is a type of differential equations called a linear time
‫invariant second order differential equation, so an LTI second order differential equation.
‫And this is its general form.
‫That's how it looks like.
‫Now, before we go any further, let's first take a simpler example.
‫Let's say that we have a typical equation in which Y equals F as a function of X, and let's say that
‫F as a function of X equals three X squared plus two.
‫So that means that Y equals three X squared plus two.
‫It's a typical function, right.
‫And so your ex is an independent variable and your wife is a dependent variable, so your Y is a function
‫of X, your Y depends on X, you put some kind of number into the X variable and then that will give
‫you a value for your Y variable.
‫A differential equation, however, is an equation in which the dependent variable Y not only depends
‫on the independent variable X, but it also depends on the change of Y with respect to X.
‫DUI, the ex and the derivative can be first order or even a second order, or it can even be a third
‫order or a fourth order, et cetera.
‫Right.
‫The important thing is to have at least one term that contains the derivative of Y with respect to X,
‫regardless of its order, whether its first order or second or third order, it doesn't matter.
‫And then it's a differential equation.
‫You can also rewrite this general differential equation form like this queue times y, which is this
‫one here, equals F as a function of X minus P, D, Y, the X, which is this one that you put on the
‫other side of the equation sine and then minus the second derivative of Y with respect to X and by the
‫way, the P and Q letters, they're simply random constants.
‫So there's some kind of numbers, they are not functions but there some kind of constants.
‫And now you take this.
‫Q And you put it here, then also here and then also here, because now you can express y variable in
‫terms of everything else.
‫And if our F as a function of X equals three times X squared plus two, then that means that Y equals
‫three times X squared plus two divided by Q then minus P divided by Q times this derivative of Y with
‫respect to X and then minus one over Q times the second derivative of Y with respect to X..
‫So you see that in this case Y is a function of X, Y, the X and the second derivative of Y with respect
‫to X and because of this and this, you know that it's a differential equation since in our case the
‫high is derivative, is the second derivative, then we call it an LTI second order differential equation.
‫It's second order because the highest derivative is the second derivative.
‫Now if F as a function of X equals a zero, not this function any more, three times X squared plus
‫two, but instead it equals zero, then.
‫Your differential equation will look like this.
‫This would be its general form, and in that case, why only depends on its own derivatives.
‫And in that case, we say that this differential equation is a homogeneous LTI second order differential
‫equation.
‫And now what does it mean when we talk about solving a differential equation?
‫Solving a differential equation means that you go from this form here to this form here.
‫All right.
‫So you have y as a function of X, D.U.I., D, X, and then the second derivative of Y with respect
‫to X, and that equals something.
‫You have some kind of equation just like here.
‫And if you solve a differential equation, then your solution will be a function and that function will
‫get rid of these derivative independent variables.
‫And then you will have y as a function of X equals something, some kind of function that will only
‫contain the X independent variable.
‫It will not contain the derivatives as independent variables anymore.
‫So remember, the solution for a differential equation is not just a number, it is a function.
‫You get rid of the derivatives and you express y only in terms of X, and that's what solving differential
‫equations is all about.
‫And now let's go back to our control inputs now that you know how a homogeneous LTI second order differential
‫equation looks like, which can be written like this, let's rewrite it like this.
‫Now we're going to keep the second derivative of Y with respect to X on one side of the equation sine
‫and the rest will be on the other side of the equation sine.
‫So I've put y here as the first term and then I've put the X here as the second term and then why is
‫multiplied now by negative Q and the Y?
‫The X is multiplied by negative P because we put them on the other side of the equation sine and now
‫I'm going to write our control loss.
‫So you X equals this equation K one times X plus K two times E dot x then this is for use of Y and this
‫is for use of Z.
‫And so now I ask you, what are you X, you Y and you Z.
‫If you look at this differential equation here and and these three control laws, then tell me what
‫are you up X use of Y and use up Z.
‫So let it be your exercise.
‫Think about it and I will see you in the next video.
‫Thank you very much.