I paid an exorbitant amount of money to listen to a professor rush through Impulse Invariance, Bilinear Transform, and Pre-Warped Bilinear Transform all in one lecture and understood none of it. I watched a couple of your videos for less than an hour and I have a much much deeper understanding of these concepts. I have a feeling I'm going to ace this exam now. Thank you so much for your help, Mr. Douglas, your explanations are succinct, and intuitive.
Just a small sign mistake: at 2:47 the green equation: the denominator has to be '2 - j' instead of '2 + j', the yellow one checks out with the blue one again. It doesn't influence the rest of the video. Thanks by the way for all the videos on your channel it helps this student studying for his control theory exam.
Im just halfway through the video and i have to say you're an excellent teacher. Good pacing, explanations and the way you show everything makes me understand the theory behind this a lot more than reading a book. Great job!
Been stuck on trying to understand the intiuition of z-transform for a while now, but this video pretty nuch made it as clear as possible. Thank you for the excellent and understandable explanation.
*Black background and coloured notation* Me: "Oh... So relaxing and pleasing" Brian: "Let me show you in Matlab" *Opens a completely white UI* Me: "Ah! My eyes! Too much light! I can't see!" XD
@2:47 it should be j/(2-j). Just a small mistake in a very good video. For those who can't figure out last step. j/(2-j) was multiplied by a complex conjugate. Thank you
Thank you for using the name 'trapezoidal integration'. George Ellis' book 'Control system design guide' uses it and you were the only mention I could find and understand!
Nice, this video is worth rewatching. I loved the way you juxtaposed the duality. The child in me loves the colors, its like like sprinkles on a trapezoidally integrated ice cream.
It might also be worth mentioning Padé approximation. For example this would also allow you to approximate a continues system with a time delay with a normal transfer function (from this it can also be shown that a time delay makes your system nonminimum phase). PS: Do you know if the other methods also have a unique transformation from discrete to continue as well?
I'm not sure I can fit Padé approximations into the next video without going too far off on a tangent :( I'll give the subject a proper showing when I make a video on continuous systems with time delays.
With an unique transformations I am referring to whether or not you would be end up at the same system if you would first transform a continuous system to discrete and then back to continuous again, as you showed in this video for the Tustin's method. For zoh I think this should hold as well, since in state space the first can be obtained using expm([A B; 0 0]*T), so using logm() would allow you to revert back to the same continuous system. For matched the transform I think should also be able to go both ways for similar reasons, but this time the exponents and logarithms are taken from scalar values. For the impulse and foh method I am not so sure.
Less then a year ago I also used the 1/1 Padé approximation for an observer for a continuous system with delay in simulink (however in a more efficient implementation that would have been much easier to do as a discrete system). A lot of continuous systems that are sampled by a sensor will practically experience at least half a sample time delay, so continuous systems with delay are really common. I do however find it annoying that MATLAB's tfest() does not allow you to fit the delay of the system as well.
Thank you for the nice explanation. One calcification required. At @10:30 it should be ∆x((y(xk-1)-y(xk))/2 as ai think instead of ∆x((xk-1)-(xk))/2. Please calrify it.
Hi Brian, just a small and possibly silly doubt, but usually we say that transforming from s-domain to z-domain we usually lose some information thus z to s domain transformation does not usually yield the same result. Now, bilinear transformation is another approximated method of determining z domain representation of a system from s-domain. Then how @4:14 are we able to obtain the same equation in s domain as we started with?
I want to thank you in advance before asking the question. :) I get how to derive the equations but there's a point I think I'm missing. Why did you say G(s) = 1/s at 10.54 of the video? Couldn't it be something else i.e. 2nd order or higher?
There is no doubt that you always do a good work on your videos, but I need to point out 2 little (and not misconceptual) corrections: 1. At 8:40, you wrote in your last equation: z ≈ (1+sT/2) / (1-sT/2) which is right. However, at 9:01, the same equation somehow turned into z ≈ (1+sT) / (1-sT) during the page scrolling; 2. At 10:40, in your last 2 equations, you used the notation f(xᵢ) as xᵢ (the index i here is generic), but you also used the notation x for Δx, so the parameter x in y(x₁) equation can be interpreted as both a parameter in yₖ equation and the functions xₖ₋₁ and xₖ of that parameter at index k. So I guess a more appropriate notation for yₖ would be: yₖ ≈ Δx [f(xₖ₋₁) + f(xₖ)]/2 + yₖ₋₁ where I used the same f(·) notation for the function that is being integrated on both equations and also used the approximate sign (≈) instead of the equal sign (=). Anyway, regardless my points here, most people probably will not notice or don't bother with those problems. Congrats for making such an amazing channel and for making the only well made didactic control videos on UA-cam. You earned it!
OK, you are amazing! I wrote the wrong equation at first, and rather than go back and redraw the entire thing I decided to do a little final cut magic and fix it in post. I got a bit lazy and thought, "surely, no one will notice the error during the scrolling" and opted not to change it for those few frames. Your second point is well taken. I'll add a mention of this in the errata section in the description. Thanks for such a thorough review!
That's no problem at all. You are the one that is amazing dedicating your time and effort in favor of doing science the right way: sharing your knowledge! Thank you, also, for the quick response. I didn't expect that. =) Keep being awesome!!
Clarification please if I am not following... and for people who where confused by this too Running through these I kept thing fractions where flipped, in particular I kept reading 2/T was T/2 and the addition and subtraction was getting flipped on the other fraction... However, running through all of the derivations of the formula several times, I fixed a bunch of algebra errors. I eventually came up with Top notch video +1 like dominates all, hence T(z+1) in the denominator. It's awkward, really weird, and soundz like a promo, hence the negative for the second equation's denominator (with Z=). It fixed my mistakes. Hope it helps someone. ----------------------------------------- Not important At 2:53 I got j/(2+j) = 0.2+2j
i've watched the whole thing once and got the hang of it, had to watch it a second time to be able to verify all the math and i got one concern. in 2:48 isn't for s=-1+j and T=2 sec z supposed to be j/(2-j) instead of j/(2+j) or did i get something wrong?
Thank you very much.I am a Chinese graduate student. If I buy books from your website, can you mail them to China?I think my friends and I will need it.
Hi, thank you very much for your videos, you make them really interesting and didactic. I'm watching them parallel to my lessons (Erasmus student, so I searched for external sources) .If I remember correctly in the first or second video of the series you said you would explain deeper the reasoning behind z=e^st, but i think you have not done it yet, could you? Edit: It was on the second video (minute 16:12). Oh, and also the explanation to the definition of the Z transform. Thanks!
Steve Gergetz i had the same question before :D write the same question in UA-cam search, it has good answers. Make sure to add the keywords (digital, or analog) i.e. what is an analog filter
Knowledge Dosage has a more immediate answer. I'll touch on it a bit in the next video to give some context and then maybe do an entire video series dedicated to filtering.
I paid an exorbitant amount of money to listen to a professor rush through Impulse Invariance, Bilinear Transform, and Pre-Warped Bilinear Transform all in one lecture and understood none of it.
I watched a couple of your videos for less than an hour and I have a much much deeper understanding of these concepts. I have a feeling I'm going to ace this exam now. Thank you so much for your help, Mr. Douglas, your explanations are succinct, and intuitive.
Just a small sign mistake: at 2:47 the green equation: the denominator has to be '2 - j' instead of '2 + j', the yellow one checks out with the blue one again. It doesn't influence the rest of the video. Thanks by the way for all the videos on your channel it helps this student studying for his control theory exam.
thanks for pointing it out, because I was relally doubting my math skills here
Im just halfway through the video and i have to say you're an excellent teacher. Good pacing, explanations and the way you show everything makes me understand the theory behind this a lot more than reading a book. Great job!
You are my hero Brian!!!! Your contribution to our Homeworks and knowledge will be unforgetable
Been stuck on trying to understand the intiuition of z-transform for a while now, but this video pretty nuch made it as clear as possible. Thank you for the excellent and understandable explanation.
You are awesome. Thank you for making me understand what a bilinear transform is.
*Black background and coloured notation*
Me: "Oh... So relaxing and pleasing"
Brian: "Let me show you in Matlab"
*Opens a completely white UI*
Me: "Ah! My eyes! Too much light! I can't see!"
XD
Francesco Bucciantini ha! I think that every time.
Matlab really needs a dark mode!
From QT to COntrol systems great tuts my friend
@2:47 it should be j/(2-j). Just a small mistake in a very good video. For those who can't figure out last step. j/(2-j) was multiplied by a complex conjugate. Thank you
thanks for the hint!
Thank you for blowing my mind. Your control lectures built the fundamentals to get me through my graduate controls courses.
Thank you for using the name 'trapezoidal integration'. George Ellis' book 'Control system design guide' uses it and you were the only mention I could find and understand!
Without question, this is an excellent video. Thank you.
Some people were born to be teachers. Great video.
Excellent refresher! Thank you very much!
Nice, this video is worth rewatching. I loved the way you juxtaposed the duality. The child in me loves the colors, its like like sprinkles on a trapezoidally integrated ice cream.
Wow Brian what an incredible video. Thank you so much for the beautiful explanation
you are a hero brian!
Probably by next week I’ll complete all lectures we want more I’ love it 😍
Really great explanation
This guy is a beast.
You are the best man! Charged by your videos!
Congratulations from Brazil.
this is sooo nicely explained
Thank you very much!
Thanks man! you are awesome! and don't forget the state space video :D
I hope to see this comment on every video I make until I create the state space series! Don't let me forget :)
Brian Douglas sure bro ! I am happy to know that the over asking for state space doesn't bother you! :D :D god bless you man
Great work, thanks for posting!
These videos are gold. Keep on going man!
It might also be worth mentioning Padé approximation. For example this would also allow you to approximate a continues system with a time delay with a normal transfer function (from this it can also be shown that a time delay makes your system nonminimum phase).
PS: Do you know if the other methods also have a unique transformation from discrete to continue as well?
The Padé approximation is a good suggestion. I'll see if I can work it into the flow. What did you mean by your question about unique transformations?
I'm not sure I can fit Padé approximations into the next video without going too far off on a tangent :( I'll give the subject a proper showing when I make a video on continuous systems with time delays.
With an unique transformations I am referring to whether or not you would be end up at the same system if you would first transform a continuous system to discrete and then back to continuous again, as you showed in this video for the Tustin's method. For zoh I think this should hold as well, since in state space the first can be obtained using expm([A B; 0 0]*T), so using logm() would allow you to revert back to the same continuous system. For matched the transform I think should also be able to go both ways for similar reasons, but this time the exponents and logarithms are taken from scalar values. For the impulse and foh method I am not so sure.
Less then a year ago I also used the 1/1 Padé approximation for an observer for a continuous system with delay in simulink (however in a more efficient implementation that would have been much easier to do as a discrete system). A lot of continuous systems that are sampled by a sensor will practically experience at least half a sample time delay, so continuous systems with delay are really common. I do however find it annoying that MATLAB's tfest() does not allow you to fit the delay of the system as well.
Thank you for the nice explanation.
One calcification required. At @10:30 it should be ∆x((y(xk-1)-y(xk))/2 as ai think instead of ∆x((xk-1)-(xk))/2.
Please calrify it.
Hi Brian, just a small and possibly silly doubt, but usually we say that transforming from s-domain to z-domain we usually lose some information thus z to s domain transformation does not usually yield the same result. Now, bilinear transformation is another approximated method of determining z domain representation of a system from s-domain. Then how @4:14 are we able to obtain the same equation in s domain as we started with?
Maybe I am wrong. Isn't bilinear transformation the same as the FOH methods? Overall, they are both using ramps to simulate the waveform.
Thank you! Very understandable.
I want to thank you in advance before asking the question. :) I get how to derive the equations but there's a point I think I'm missing. Why did you say G(s) = 1/s at 10.54 of the video? Couldn't it be something else i.e. 2nd order or higher?
Very well and understandable Video, thank you :)
Pure genius 😊😊
There is no doubt that you always do a good work on your videos, but I need to point out 2 little (and not misconceptual) corrections:
1. At 8:40, you wrote in your last equation:
z ≈ (1+sT/2) / (1-sT/2)
which is right. However, at 9:01, the same equation somehow turned into
z ≈ (1+sT) / (1-sT)
during the page scrolling;
2. At 10:40, in your last 2 equations, you used the notation f(xᵢ) as xᵢ (the index i here is generic), but you also used the notation x for Δx, so the parameter x in y(x₁) equation can be interpreted as both a parameter in yₖ equation and the functions xₖ₋₁ and xₖ of that parameter at index k. So I guess a more appropriate notation for yₖ would be:
yₖ ≈ Δx [f(xₖ₋₁) + f(xₖ)]/2 + yₖ₋₁
where I used the same f(·) notation for the function that is being integrated on both equations and also used the approximate sign (≈) instead of the equal sign (=).
Anyway, regardless my points here, most people probably will not notice or don't bother with those problems.
Congrats for making such an amazing channel and for making the only well made didactic control videos on UA-cam. You earned it!
OK, you are amazing! I wrote the wrong equation at first, and rather than go back and redraw the entire thing I decided to do a little final cut magic and fix it in post. I got a bit lazy and thought, "surely, no one will notice the error during the scrolling" and opted not to change it for those few frames. Your second point is well taken. I'll add a mention of this in the errata section in the description. Thanks for such a thorough review!
That's no problem at all. You are the one that is amazing dedicating your time and effort in favor of doing science the right way: sharing your knowledge! Thank you, also, for the quick response. I didn't expect that. =)
Keep being awesome!!
Clarification please if I am not following... and for people who where confused by this too
Running through these I kept thing fractions where flipped, in particular I kept reading 2/T was T/2 and the addition and subtraction was getting flipped on the other fraction...
However, running through all of the derivations of the formula several times, I fixed a bunch of algebra errors.
I eventually came up with Top notch video +1 like dominates all, hence T(z+1) in the denominator.
It's awkward, really weird, and soundz like a promo, hence the negative for the second equation's denominator (with Z=).
It fixed my mistakes.
Hope it helps someone.
-----------------------------------------
Not important
At 2:53 I got j/(2+j) = 0.2+2j
Brian can you make videos for QFT design or suggest the resources for learning
thanks!! :)
Thanks for such videoes
Ah ive actually done some trapezoidal integration already, so that helps that you inform us of this name.
i've watched the whole thing once and got the hang of it, had to watch it a second time to be able to verify all the math and i got one concern. in 2:48 isn't for s=-1+j and T=2 sec z supposed to be j/(2-j) instead of j/(2+j) or did i get something wrong?
saw that too
Thank you very much.I am a Chinese graduate student. If I buy books from your website, can you mail them to China?I think my friends and I will need it.
You are great!
Thanks a lot.
insane!
thank you sir
Is this valid for converting any s plane transfer function to Z plane?
mind blowing awesome！
Hi, thank you very much for your videos, you make them really interesting and didactic. I'm watching them parallel to my lessons (Erasmus student, so I searched for external sources) .If I remember correctly in the first or second video of the series you said you would explain deeper the reasoning behind z=e^st, but i think you have not done it yet, could you?
Edit: It was on the second video (minute 16:12).
Oh, and also the explanation to the definition of the Z transform. Thanks!
Yes! I still have to get to these topics. Maybe two videos from now I'll create one on the z-transform and z=e^st. Thanks for the reminder :)
Greats !
nice it help me
2:16 nah I trust you bru
What is a filter and/or what does it do?
Steve Gergetz i had the same question before :D write the same question in UA-cam search, it has good answers. Make sure to add the keywords (digital, or analog) i.e. what is an analog filter
Knowledge Dosage has a more immediate answer. I'll touch on it a bit in the next video to give some context and then maybe do an entire video series dedicated to filtering.
Thanks KD & BD.
Vengo por el profe de Control, Rafael Torres
15 mins video help me figure out the question bother me 15 hours
Unbelievable
"Lets start with a quick refresher of trapezoidal integration"
Meanwhile I'm sat here having never heard of it before