‫Welcome back.
‫So I hope you thought about what use of X, use of Y and use of Z could be.
‫You have constants here that are multiplied by Y and then D, why the X?
‫And the result is the second derivative of Y with respect to X.
‫And so you have the key constants here and also here and also here, and you multiply them by error
‫and the change of error with respect to time and here as well, error and then the derivative of error
‫with respect to time.
‫The first one was in the examination.
‫Now this one is in the Y dimension and then this one here is in the Z dimension like this.
‫So look, a constant minus Q times Y plus another constant minus P times the Y, the X equals the second
‫derivative of Y with respect to X and then here K times the error plus another K times the error derivative
‫just like here.
‫So what do you think what this use of X is.
‫Well if you have a second derivative here, then this use of X will also be a second error derivative
‫with respect to time.
‫And in this case it would be in the X dimension.
‫Then here you would have the second time derivative of the error in the Y dimension, and then here
‫you will have it in the Z dimension.
‫So if we forget about the dimension for a moment and just think about the error variables, then it
‫will look like this error double that equals K one times error plus key two times error dot.
‫And you can also write it like this error double dot.
‫And then you put these terms on the other side of the equation sine so you will have for example, minus
‫K two times the error that and minus K one times the error equals zero.
‫Or you can also write it down with a different notation.
‫For example, this could be the second derivative of error with respect to time.
‫Then this will be this term here and this will be this term here and all that equals to zero.
‫So you see your control laws are in fact homogeneous.
‫LTI second order differential equations like here.
‫And so you would have one differential equation for the X there.
‫Mention one for the Y dimension and then one for the Z dimension.
‫So you would in total have three homogeneous LTI second order differential equations.
‫The only difference here is that your dependent variables are the errors in the X, Y and Z dimension.
‫Now these are your dependent variables.
‫The errors and your independent variable is time.
‫And that's why you see the change of error with respect to time you see.
‫And then the second derivative of error with respect to time.
‫So that's your independent variable now.
‫And also you had plus signs here and now.
‫Here you have minus signs.
‫But that doesn't matter.
‫I mean, you can also write this thing like this minus and then minus P, and then you will have plus
‫P and the same thing here, minus and minus Q and then you would have positive Q here.
‫In the end they are just constants.
‫The important thing is to realize that your control laws are homogeneous.
‫LTI second order differential equations and that is awesome because now we can do some interesting things
‫to drive our errors to zero.