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‫Welcome back.

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‫So I hope you thought about what use of X, use of Y and use of Z could be.

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‫You have constants here that are multiplied by Y and then D, why the X?

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‫And the result is the second derivative of Y with respect to X.

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‫And so you have the key constants here and also here and also here, and you multiply them by error

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‫and the change of error with respect to time and here as well, error and then the derivative of error

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‫with respect to time.

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‫The first one was in the examination.

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‫Now this one is in the Y dimension and then this one here is in the Z dimension like this.

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‫So look, a constant minus Q times Y plus another constant minus P times the Y, the X equals the second

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‫derivative of Y with respect to X and then here K times the error plus another K times the error derivative

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‫just like here.

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‫So what do you think what this use of X is.

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‫Well if you have a second derivative here, then this use of X will also be a second error derivative

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‫with respect to time.

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‫And in this case it would be in the X dimension.

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‫Then here you would have the second time derivative of the error in the Y dimension, and then here

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‫you will have it in the Z dimension.

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‫So if we forget about the dimension for a moment and just think about the error variables, then it

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‫will look like this error double that equals K one times error plus key two times error dot.

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‫And you can also write it like this error double dot.

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‫And then you put these terms on the other side of the equation sine so you will have for example, minus

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‫K two times the error that and minus K one times the error equals zero.

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‫Or you can also write it down with a different notation.

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‫For example, this could be the second derivative of error with respect to time.

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‫Then this will be this term here and this will be this term here and all that equals to zero.

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‫So you see your control laws are in fact homogeneous.

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‫LTI second order differential equations like here.

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‫And so you would have one differential equation for the X there.

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‫Mention one for the Y dimension and then one for the Z dimension.

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‫So you would in total have three homogeneous LTI second order differential equations.

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‫The only difference here is that your dependent variables are the errors in the X, Y and Z dimension.

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‫Now these are your dependent variables.

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‫The errors and your independent variable is time.

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‫And that's why you see the change of error with respect to time you see.

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‫And then the second derivative of error with respect to time.

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‫So that's your independent variable now.

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‫And also you had plus signs here and now.

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‫Here you have minus signs.

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‫But that doesn't matter.

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‫I mean, you can also write this thing like this minus and then minus P, and then you will have plus

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‫P and the same thing here, minus and minus Q and then you would have positive Q here.

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‫In the end they are just constants.

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‫The important thing is to realize that your control laws are homogeneous.

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‫LTI second order differential equations and that is awesome because now we can do some interesting things

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‫to drive our errors to zero.