How do complex numbers actually apply to control systems?

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This video mainly covers the underlying math behind the 'nyquist stability criterion' seen in control systems.
Software Used in Video: www.desmos.com/calculator/
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[@starryfolks]

3blue1brown animation on point

[@matthewadamsteil]

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

[@JulieanGalak]

I didn't discover that fact until long after college. In large part thanks to videos like this.

[@yosefmacgruber1920]

The difference between a blah calculator and a cool calculator. Whether it has *_i_* on one of its buttons. Even if you have to _shift_ to get to it.

[@abhinav3478]

i didnt get why 'i' is so very useful or why 'i' is needed even at college. other than using it in every equation i use :|

[@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👏👏👏

[@johnjacobjinglehimerschmid3555]

Hmm didn't know that. Found it out in my first chemistry course. Worked in a group after class and .... found that as I explained something I knew other stuff made sense and I was able to get more from the class if I helped others by explaining something. Unfortunately, next semester of chemistry nobody wanted to have a study group and man did I struggle. Still was one of the students that was doing well, well enough for the teacher to ask myself and another if we'd tutor in the study lab, didn't feel comfortable for someone to come looking up to me to give them an answer. But I agree ... you want to learn something ... teach what you know other stuff from the fringe will make sense ....

[@eggyrepublic]

Do I find this cool? Yes. Do I understand how it works? Absolutely not.

[@swamihuman9395]

- 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! :)

[@dineshpooja1969]

Eagerly waiting for fourier and Laplace videos !!!!!!!!!!

[@QYong-rq6iw]

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.

[@Embracing_Impermanance]

Same thing happened to me bro I wish this video had come 3 years early

[@bikerfirefarter7280]

Lecturer. They can demonstrate they know how to do something, but they do not explain the underlying thinking. Most are guilty of it. Some just don't realise, some do it to gloat and they don't want competition, and the dumber they keep students the more secure their posts due to demand/'re-takes'; trust me, as a technician I've seen and overheard this often enough.

[@kaustubhsalve9047]

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

[@AnujShahshahmanuj]

Yeah same! A few days ago I had an exam on Control Systems, and they asked the question: "Explain the Nyquist Stability Criterion."

[@thatkindcoder7510]

@@AnujShahshahmanuj Big oof

[@puddleduck1405]

bro same my test was yesterday

[@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

[@zachstar]

Thanks for sticking around man!

[@bikerfirefarter7280]

@@zachstar "previously on complex number stuff". NO BLOODY LINK! Tchah! What's the point of mentioning it if there is NO LINK?

[@shohamsen8986]

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

[@SKARTHIKSELVAN]

Thanks for putting lot of efforts in making these videos.

[@alexramirez5104]

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

[@davidpalmer9780]

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

[@rulisastra]

Unscented kalman filter too, since it uses sigma points

[@kaleevans1692]

Great video. Awesome discussion and your an amazing talent at showing the math graphically and with the right timing and explanation. Keep up the great work!

[@rakshithramesh4333]

Awesome! Really helped me the meaning behind the Nyquist plot to determine stability of closed loop system. Thanks a lot !

[@saifikhan5936]

Superb video 😍 very conceptual. Highly appreciated your efforts ☺️

[@manavrevaprasadu]

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

[@KaustavMajumder]

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

[@atriacharya2967]

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.

[@lifehope4201]

In which college you are...

[@PenguinMaths]

Wonderful video with great visuals.

[@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.

[@lih3391]

Isn't it (a^2-b^2,2ab)?

[@jamesbra4410]

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.

[@shambosaha9727]

3b1b fans: TRIGGERED

[@aaAa-vq1bd]

Triangle inequality should be taught at the very beginning of any analytic geometry course, no?

[@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

[@fgcg4448]

Thanks a lot for this amazing explanation !

[@sujaypoudel4694]

Excellent !! Loved it

[@prajeethkodru1930]

Thanks a lot for making this video!!

[@mridulnatani]

If you had released this video 5 days before, I would have got an A in controls 😅😅

[@aks9545]

It would be cool to have an app where you can put a complexe function and draw your inputs and see the outputs

[@xiaosun9111]

i finally got it! great video, thanks!

[@madaaz6333]

It's a beautiful work!

[@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.

[@DiegoMathemagician]

I love your recent videos

[@tollsimyfpv]

Wow! So illuminating.

[@anandsuralkar2947]

0:51 Me:-tan'1+tan'0.5+tan'0.333

[@irelandtamilandamo]

Wow. ! I can master control systems thru your videos. Thanks man !!

[@bachirblackers7299]

Best video ever !!! Gud job prof !!!

[@calebsherman886]

I was completely lost this entire video, and at first I was kinda sad because this is something I wish I knew more about. But then I realized it's a good thing because before this video I didn't know what I didn't know.

[@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)

[@housseinnaserdine5874]

very amazing animations

[@WolfrostWasTaken]

Brooooo when you started talking about Control Systems I was like "wtf this is so useful!!!" holy

[@partialorder5596]

"If you want a real-world example of that, just imagine hypothetically that the spring constant were negative." LMAO

[@ayushsharma8804]

thats just an analogy, systems with 'negative spring constants' exist in real life, just try balancing a cone on its tip

[@Topspeedcraft]

I watched this video after watching one named "all you need to know about control theory" by the Matlab channel. I took a complex analysis class in eng major and i never fully understood all the praising my teacher did for the use of complex analysis in control theory. After watching the video i just mentioned, I commented that i would have liked to see some complex analysis there, and now just finishing watching this one. I can say that they complement each other quite nicely so i recommend you watch the one posted by Matlab. It's quite interesting to see how the feedback and feedforward theories relate to the stability and creating new dynamics in the system regarding the graphic insights provided by this video. Also you can see a clear connection with one of the key premises of the "All you need to know about control theory video" which is that you don't need to measure the state of the system to know what inputs would look like if you made a disturbance or shift. You can just observe the behaviour on the simpler open loop system and use nyquist criteria to determine stability of the more complex closed loop system with feedback. This is an amazing video and takes care of a quite interesting topic in a quite digestible way, took me a semester to learn what this video renders just over a dozen minutes. Beautiful work, the dedication and attention to detail shows, this is work well done. Thanks a lot

[@bionic5059]

That refreshed some horrendous memories from the control systems course ...but thanks..i never understood it during my time at the university, I just applied it.

[@TheBeatle49]

This gives me glimmer of understanding what I didn't get as a math major.

[@shiroharra3058]

i love this channel

[@L0j1k]

Hell yeah. 10/10!

[@Christopher._M]

How come it says one wee ago

[@adityams1659]

exactly!! i guess a patreon

[@hansisbrucker813]

Wait 10/10! is a really low score 🤔

[@brogcooper25]

I wish I had these videos when I took controls 10 years ago.

[@tejonBiker]

Pretty neat explanation, zeros AND poles are constant topics and words in control theory, but never covered in this way .

[@andy_lamax]

Wooooow!! I thought I totally understood my control systems class

[@ajaysinha5803]

Well.... As far as i know.... The input which goes throuh a complex function goes into the riemann plane which overlaps the origin 2 times(4D)... Giving an effect of going round the origin 2 times.....plz correct me if i am wrong

[@k.c.sunshine1934]

I studied Electrical Engineering 30 years ago. With this video, I have new intuitive understanding of the stability of control systems, etc. Thank you!!! This presentation video is cool!

[@atriacharya2967]

My god, a 50 yr old man watching technical videos!!!

[@k.c.sunshine1934]

@@atriacharya2967 HaHa. Funny. This time and space in life is one of endless learning.

[@atriacharya2967]

@@k.c.sunshine1934 what do you do for a living, sir?

[@k.c.sunshine1934]

I'm retired from engineering, however, I am still fascinated with technology. My engineering career forced me to constantly read and improve and I maintained the habit into "mid-life." Now that I'm hooked on learning, I will likely continue learning 'til death.

[@atriacharya2967]

@@k.c.sunshine1934 that's great to know, sir. All the best👍💯

[@joyride6062]

Literally did Nyquist criterion in my controls class today and didn't understand a dam thing. You just saved my midterm

[@BinyaminTsadikBenMalka]

Yes, thank you! Go deeper on control systems please!

[@omri9325]

This reminds me of the amazing work in WelchLabs video about complex numbers.

[@kadironsel1913]

Thanks for this Great explanation Zach. at 12:21 you said that the number of times the circles origin are same for both equations. N=Z-P . I am ok with finding "Z" and "P" . But how can we say the "N"s (number of circles around the origin) are same for both equations ? How did we get it?

[@da4mula885]

14:52 nope. Im not in controls class, don't even do anything with math. But you explained it clearly. I honestly got sidetracked to this topic here, but I can see now how I can use it. Gonna use it for feedback signal flow for music programs. Which Gained my interest recently

[@-Rishikesh]

14:42 this is great, but what if my function is not having a unit negative feedback, Should i calculate the correct cltf then reconstruct a oltf with unit negative feedback and then solve?

[@NonTwinBrothers]

Haha, nice use of Desmos!

[@realcygnus]

superb

[@altuber99_athlete]

Another example of a unstable system is an electric circuit in which the Laplace transform of the output has poles with positive real part (i.e. they're located in the right-half side of the complex plane.) This is just like the example MajorPrep gave in the end, but I actually undertand it more: if the L.T of the output has a pole with real part, then the I.L.T. (i.e. the output in the time domain) will have a term with an exponential factor (to see this, recall the table of L.P. pairs and the partial fraction expansion.) Thus, if the pole has *positive* real part, then the exponential will have a positive exponent, for example y(t)=3e^2t amps. Notice that, as t increases, so does the output. What does it mean? After some time, the output current in the circuit will be huge! And we all know what high current can do.

[@devrajyaguru2271]

Really helped me to get the best knowledge about imaginary numbers

[@yvesdelombaerde5909]

Plan complexe et courant triphasé: de grands souvenirs!

[@happyhayot]

Any recommendations for MOOCs in control systems?

[@MusicEngineeer]

nice . looking forward to fourier and laplace transforms! ...maybe z-transforms also? (i'm a dsp guy :-))

[@The-Devils-Advocate]

3:34 can we say that since specific pairs of input points correspond to a single output in a square transformation, thus “halving” our number of “points” and “distance,” and causing the time needed to traverse the output contour to be half of the time needed to traverse the input contour?

[@devrimturker]

Excellent, everything in nutshell. May be a relation between mandelbrot set and nyquist added, nyquist seems shaved mandelbrot, leaving its fractals ;)

[@ajaysinha5803]

These types of vids really boosts my curiosity to the next lvl

[@thinkfirst6374]

This problem was in my highschool exam(12th grade) it was so easy, Greetings from Syria.

[@crawfordrhoderick2942]

what calculator is the best to use for engineering.

[@usmansaeed3662]

What software do you use for plotting the contours?

[@adrien5568]

Can you put the link for the desmos graph ?

[@bhanusri3732]

At 12:20 what does it mean the number of rotations for f(z) + 1 around 0 is equal to number of rotations for f(z) around -1

[@mathscienceonlineacademy4512]

Complex numbers is first chapter in our textbook of Mathematics.

[@pasangtsering2380]

Which software you are using to show the graph???

[@jackwang862]

Complex number gets both elegancy and pracitcal value.Awesome with inexpressible joy.

[@danielmalo1753]

Don't understand half the concepts here but it's nice

[@theperipateticaccrescent7685]

Can you make a video on comparing different types of majors and how much money each makes. I want to choose engineering for money and want to earn a lot of money (by business,start-up, entrepreneurship....etc). But am a bit unclear about how to go about it.

[@univicta7212]

Yes, please make a video on, Top Earning engineering professions (✯ᴗ✯)

[@shambosaha9727]

Money? AAAAAAAA...

[@ganaieabdullah6713]

Can you make vedios on how to make audrino project

[@rulisastra]

You're a genius. Why am I even here?

[@nirgadish6573]

I'm confused: the plot at the snapshot of the video - the green shaded region, that is the image of the interior of a rectangle surrounding both a zero, and a pole has 4 corners. They are the images of the 4 corners of the rectangle. BUT, why are the plotted corners not at right angles?! The image of a right angle should be a right angle. So what's going on with that plot??

[@elkhayder]

The first time i've seen those function and there plots in the systems control class ( asservissement ) I was like "they did not get enough with real functions, now they are drawing doodles"

[@user-gt2th3wz9c]

Love u)

[@jewulo]

How did it get from 10i to 10exp((i * π)/2) @ 1:43? Sorry for my ignorance.

[@happyhinoka8722]

using Euler's formula r(e^ix) = r(cos x + i sin x) cmiiw :)

[@williamhensley8322]

You need to foil the complex numbers in the video... The ones in the form of a+bi. There are 3 of them. You have to know how the imaginary component of the complex numbers are multiplied.

[@jewulo]

@@williamhensley8322 What do you mean by foil the complex numbers?

[@williamhensley8322]

@@jewulo if a complex number can be defined as a + bi, then if you had to multiply 2 complex numbers, it would be in the form of (a1+b1i)(a2+b2i). Then you would multiply these two complex numbers in the same way you would multiply (x+1)(2x-2), multiplying the first terms, outside terms, inside terms, and the last terms and sum the products.

[@williamhensley8322]

Keeping in mind that an i^2 term is equal to -1.

[@puddleduck1405]

Im in highschool rn, and we've just finished the complex number topic. I didnt understand how they are useful in real life but this is so cool!

[@ES-qe1nh]

Yeah it's a bit of a jump from high school complex (de moivres, Euler's formula) to stuff like this (Cauchy, residue, integrals in C --> C) I'd say take calculus before you dabble in this if you want to absorb more knowledge

[@puddleduck1405]

@@ES-qe1nh ahh yes, in the UK we dont get a choice haha, we have to do calculus. ive never heard of Cauchy b4, it sounds interesting tho I might look into it!

[@ES-qe1nh]

@@puddleduck1405 Cauchy... Isn't a concept. I just quickly referenced it because of Cauchy's residue theorem in complex analysis. Obviously, Cauchy was a french mathematician and engineer.

[@puddleduck1405]

@@ES-qe1nh oh cool

[@fritschdavid]

I loved this so much. i have to see more of your world. This relationship of the untransformed vs. transformed make me think of a concept i am trying to understand: monads. which show up in leibnitz' philosopy, mathematics and as programming concept to e.g. deal better with empty values as far as i understood so far. I was shocked to see you are promoting nord vpn in another video. I intuitively know that this driven by Snodens former breadgiver. Nobody talks about the issue that using a vpn you make your internet access company unable to see the traffic, but you give whatever provider you choose full access to your stream of activity in all detail. i trust (at least a german) telecom much more to run their business according to the laws of data protection.. which of course is a european luxury. do you resonate with that?

[@Avighna]

Here's how I solved the first angle question before watching the video. The sum of angles (via the pythagorean theorem / analytical geometry, whichever you prefer), which I'm gonna call S, is: S = atan(1) + atan(1/2) + atan(1/3) S = 45° + atan(1/2) + atan(1/3) S = 45° + atan(tan(atan(1/2)+atan(1/3))) S = 45° + atan( [ (tan(atan(1/2)) + tan(atan(1/3)) ] / [1 - tan(atan(1/2)) * tan(atan(1/3))] ) S = 45° + atan( [ (1/2 + 1/3 ] / [1 - 1/2*1/3] ) S = 45° + atan( (5/6) / (5/6) ) S = 45° + atan(1) S = 45° + 45° = 90°

[@JCisHere778]

How about complex potentials and conformal mapping to solve potential flow problems

[@saido3278]

What software do you use for animation?

[@zachstar]

desmos

[@luxshokk]

About the angles in the rectangle at the beginning: To be completely rigorous, wouldn't we have to acknoledge that technically, the phase of the resulting number could also be 90° plus any multiple of 360°? By adding angles together, you may, in general, indeed get a result over 360°. Is there a nice argument to show that it is indeed 90° and not 450°?

[@nomechomath5644]

Good question! If α, β and γ are all greater than 0° and less than 90°, then: α+β+γ > 0° + 0° + 0° α+β+γ > 0° and α+β+γ < 90° + 90° + 90° α+β+γ < 270° thus their sum is greater than 0° and less than 270°. That's how you can show that they must add 90° and not any other angle like 450°.

[@adityams1659]

hii Could you please please make a video on how you animate with desmos pleaseeeeeeeee!!!

[@zachstar]

Not my next video but the following one is going to be involve a lot of animations that will be useful to those in a linear algebra class. I'm gonna make a separate video after that where I explain how I made that and I'll be providing the template I use so you guys can play around with it yourselves.

[@charlesallan6978]

We can use these theorem to mathematically transform the word 'though' into a useful form whence the g is pronounced instead of remaining silent and the ou becomes ow. Thence the h remains silent.

[@jakoblenke3012]

Interesting how the bottom circle at 15:07 crosses the imaginary axis at sqrt(2)....

[@ckzslmr1405]

very nice explanation

[@mnada72]

Another great video. I was hapless to understand Nyquist Criterion but now the idea started to sink

[@spazticle8752]

can you talk about engineering physics major

[@tafiqulrafi2569]

I thing I wanna ask u is why did u leave ur job as an electrical engineer? Well, I m in class 12 and next year I'll do undergrad in electrical engineering

[@rikadomez8201]

Whomever came up with the idea of square-rooting the negative 1 is simply genius.

[@anuzis]

Can you do a video on everyday applications of convex optimization? (not huge things like power grid optimization, but small every-day situations where humans could benefit from convex optimization if they were capable of doing it well)

[@vatsan2483]

Hey dude.. can u make a playlist on channel that has all the videos for electrical engineers who have this one mandatory for their course.. this would be appreciated

[@vatsan2483]

Like can u please group into one playlist.. with Laplace, Nyquist and alll..for easy accessibility. Thanks in advance 👍👍👍

[@zachstar]

Here you go! I'll add more as they come. kzbin.info/aero/PLi5WqFHu_OJN7Mb2aAgqcao2R3s9wbBEr

[@shimadabr]

I didn't undertand anything. Nice video tho