How do complex numbers actually apply to control systems?

This video mainly covers the underlying math behind the 'nyquist stability criterion' seen in control systems.

►My Setup:

[matthewadamsteil]

When i got into college i was absolutely shocked at how imaginary number were SO useful! Videos like this brings me inexpressible joy.

[karimkhazaby]

As Feynman said: if u want to master something, teach it. So I think your teaching technique is really awesome. Well done. Thank you so much. Great respect for you and your endeavour👏👏👏

[swamihuman]

- Thx.

- I took controls in college, and studied zeros/poles, and Nyquist - BUT, was presented a largely procedural understanding, though did include concept of stability, of course.

- Years later, I became a math teacher... and I love the underlying insight provided by including mathematical view. And, I think this thinking could/SHOULD have been included in the engineering course.

- BTW, I experienced the same joy of insight when i revisited the Fourier Transform and discovered the idea of an Integral Transform, and how a "spinning" kernel teases out the frequency components - and how a particular kernel results in the Laplace Transform.

- The takeaway: the generality, and insight of math is very powerful! :) <3

[QYong-rqiw]

When I was in a Controls class I couldn't understand or appreciate shits because it was just statements that got thrown at my face. Not sure whether it was the lecturer or myself who was incompetent and brain fogged. Now watching your video finally made me appreciate something, even though I am not bothered yet to really pause and ponder in order to more fully understand the stuff.

[zyzzbodybuilding]

Man ive been watching your vids since you had like 3k subs making those computer science course outlines. You have come a long way and the content keeps getting better. LV you dude, your also a good looking guy

[eggyrepublic]

Do I find this cool? Yes.


Do I understand how it works? Absolutely not.

[shohamsen]

When I learn control theory, we sort of skipped all of this and went directly into the modern theory of control ( at least that's what I think it's called). Everything is dealt with using Lin ear algebra. It's nice to get to know these topics, because it wasn't covered in my control course

[dineshpooja]

Eagerly waiting for fourier and Laplace videos

[kaustubhsalve]

Man you should have put this a week earlier. I had my test back then

[jamesbra]

Your channel is much better and more useful than 3Blue1Brown or any other math/engineering channel. I took complex variables and this video basically substituted the entire class into one it was so comprehensive. Of course minus the residues, complex logarithms, cauchy's formulas, triangle inequality, bounded functions, gamma function, inverse Laplace residues, improper fourier integrals, bessel coefficients etc.

[atriacharya]

Thank you so much, sir. I am a third year electrical engineering student and the last 2 videos have been so very much useful for my broader understanding.

[davidpalmer]

I'd like to see you discuss Kalman Filters, in another video, given their control functionality.

[cosmicvoidtree]

Just for those curious, if you want too do these functions in desmos, the z squared formula is (a^2-b^2, ab) and to add 5 for instance just add a +5 in the x coordinate. For any function, evaluate the the formula then reduce i's (making i^2 into -1) until you have either 1 or 0 i's in each term. For example: a+bi+ci^2+di^3+ei^4 (in this case c and e are being used as variables not as the speed of light and Euler's number), a has no i's so we ignore it, +bi has 1 i in it so we ignore it for now (do remember that it has an i on it as that will be important later), for +ci^2 we cut out the i^2 and make the plus sign a minus sign: +ci^2 --> -c, for di^3 remove 2 of the 3 i's to get -di, and for ei^4 the two pairs of i's cancel to make 1 so it gets converted to e. Taking all of the converted terms you get a+bi-c-di+e, for reasons you'll see next step, lets rewrite it as a-c+e+bi-di. Now put parentheses around it: (a-c+e+bi-di), switch the sign before the first term with an i component with a comma: (a-c+e, bi-di), and completely remove the i's (a-c+e, b-d), don't compensate for them just cut them out entirely. And you've got your point You can substitute any value for the variables, add more or less, make them all only depend on two variables whatever. If you read all this then either: You are a nerd like me, you want to do what I just taught you, or you're someone who is reading this as a challenge or something. Anyway thanks for reading this and later I might add a link to a desmos graph which shows off this.

[TaladrisKpop]

The contour method is simply seeing complex functions as vector fields, but the visualization is really nice. It shows how computers can really helps in mathematics

[hypercomms]

Hello, it's been 40 years since I studied control systems at university, and so was interesting watching your video as way of revising.
I would suggest that to explain how the control system would work, and the application of complex numbers in it, perhaps use a real world example, maybe the operation of a washing machine in terms of maintaining the speed of rotation of the washing machine tub, or maintaining the temperature of the water in the tub.
Another suggestion would be to consider the flyball governor on a steam engine and explain the control system and the Application of complex numbers in that control system..

[alexramirez]

Perfect timing...we are studying this concept right now in my Control Systems class...thanks!

[dakrontu]

The math of complex numbers and Laplace transforms etc is beautiful but I am so glad I took the digital electronics option rather than more control theory.

[manavrevaprasadu]

This guy loves springs so much he puts it in every video

[dozog]

As i said last time. Your script is really good.
The speed may be a bit fast for new learners to pick up all the details, ...
...it is really, really good as an introduction, a refresher or a summary.
(which you don't get if you go slower)

[KaustavMajumder]

Man, I wish I could actual take live lessons from you! You sir, are amazing!