Imaginary Geometry in Control Systems

[brobertson89]

So I have a control systems mid-semester exam coming up and the lecturer has posted up a formula sheet for us. However it is different to past years exams and has a geometry section with the following equations:

e^(±jθ)=cos(θ)±jsin(θ)
cos(θ)=(e^jθ+e^-jθ)/2
sin(θ)=(e^jθ-e^-jθ)/2j

Now I've seen these equations once before, however not in this course. In fact I have gone over every lecture, every tutorial, every practical and even the textbook looking for where these equations might be used and I can't find anything. So I was just wondering if anyone has any ideas on what I should be ready for.

Cheers,

Brandon

 

[viscousflow]

oscillatory systems?

 

[wsabol]

Maybe root locus. I know that is in the imaginary plane, but I don't ever remember using those equtions, althought its entirely possible to do so.

 

[viscousflow]

Maybe root locus. I know that is in the imaginary plane, but I don't ever remember using those equtions, althought its entirely possible to do so.

Those equations are the fundamental basis of oscillatory systems theory. Over damped systems have a real solution only. Under damped systems have a real and imaginary solution. How does an imaginary solution create an oscillatory system? See the OP's equations. Those equations are basic equations found in any Diff Eq course and should be fundamental to any engineer (control systems especially).

 

[wsabol]

Those equations are the fundamental basis of oscillatory systems theory. Over damped systems have a real solution only. Under damped systems have a real and imaginary solution. How does an imaginary solution create an oscillatory system? See the OP's equations. Those equations are basic equations found in any Diff Eq course and should be fundamental to any engineer (control systems especially).

Yes, I know that. But specifically for root locus problems, I don't recall using those equations.

I doubt his prof will require a FULL derivation of simple harmonic motion from the EOMs. I think the presumption that

A*e^jwt + B*e^-jwt
equals
C*sin(wt) + D*cos(wt)

is pretty standard. You don't have to show the gory details.

 

[brobertson89]

So it turns out that it was not too hard in the end, I was given a transfer function with a quadratic equation in the denominator and then asked to place it into partial fraction form. I was then asked to find the Laplace Transform of the function and as the roots of the quadratic were imaginary numbers I used those equations.

 

[brobertson89]

Thank you for the help though, I appreciate it.