Final Value Theorem and Steady State Error

[@BrianBDouglas]

Longer than I had originally thought :) From crafting the idea and the flow of the lecture, to drawing (or recording video in some cases), to editing/voice over, and posting to KZbin usually takes me about 10 hours per video spread out over 3 days. And since I also work a full-time job that is why I can only do one video a week. But I really enjoy making them and I've been getting a lot of great responses from people like you so it's easy to continue. Thanks for the comment!

[@rohlay00]

Mate, you are an absolute legend. I'm a second year student studying robotic engineering and you have saved my arse! Seriously. You have made this boring subject (because of how it's taught) really interesting and given me a great intuitive understanding of it. This will have a huge impact for my degree (as control is used in many subjects). Thank you!!

[@youcefyahiaoui2644]

Just ingenious teaching. Brian, you really raised the bar for making control topic not only interesting but enjoyable to watch. Your teaching style needs to be considered as a standard way of teaching as you optimally drive the points home. Kudos!

[@mohamedebrahim4977]

Man it is from 7 years and it really benefits me now in 2020 , so i would like to thank you for your efforts and keep up the good work

[@Pyrohawk]

Every time I encounter a new topic in my controls theory curriculum and I see you've made a video about it, I let out a sigh of relief and smile. I hope you're aware of how impressive and useful these tutorials are.

[@BrianBDouglas]

Thanks for the comment. I'll be address Nyquist after the lead/lag videos ... so hopefully in 4 or 5 weeks.

[@JetNmyFuture]

I now feel like I have a chance of actually learning this topic to the point where I can use it in a practical way. I love your style and presentations. About every 5 minutes, I have a moment where I say "Ooooohhhhhh, yeah, got it now......" Thank you for sharing! fantastic!

[@BrianBDouglas]

I would love to continue this series on through modern control. I have some ideas and concepts I want to finish up in classical control first before moving on so it might be 4-5 months still before I get to topics like state space. But it'll come eventually!

[@BrianBDouglas]

Hi Henning, that is a great list, thanks for the suggestions! I have a plan for the next few videos already but I'm adding these topics to the list for sure. I really like the first topic ... sort of a survey of popular control methods. I'll work on something. I appreciate your comment, thanks again.

[@pacrat90]

I searched KZbin for the topic and hoped that I would find one of your videos, and there it was. Thanks for the great lectures!

[@chenming6940]

You know what? Our department's PhD qualifier exam contained the problem that I found answer in your video!! The type number and its impact on the performance of the system. I didn't have much time to review control theory to prepare for the exam, but I did watched lots of your videos. I only saw type number once in your video and I still remembered that in yesterday's exam. Thank you!! I will recommend all my friends to watch your control video as important preparation for qualifier!

[@GalaxianExplosionGS]

You're awesome!, I have learned a lot from you, rather than one of my "teachers". Thank you for your videos man.

[@deanmillard9853]

these lectures are gospel, better teaching than the lecturers I pay tuition for.

[@mannykrish1]

iam nothing in control systems without brian douglas .thanks to you and you are my lovely gift.

[@mandeadd]

You the real MVP during this time of online classes man.

[@BrianBDouglas]

HI Aaron, if you divide by 's' and replace 's' with zero you are finding the final value of the function to an impulse input. You have to multiply the transfer function by 's' to find the final value to a step response. So for a step response the multiply by 's' and the divide by 's' cancel and you're left with the original transfer function where you set 's' to zero. Then you get 10/5 which is 2. I'm betting the book was solving the final value to a step input.

[@BrianBDouglas]

I have been getting Nyquist requests non stop recently! I'll definitely cover that topic but unfortunately it won't be before your final. Good luck and hopefully you'll still find the videos helpful after your class.

[@jessstuart7495]

The system type comes from characteristics of the error function in a closed-loop control system. Some of the first applications of control system design were motor controls. Type 0 System: The error can go to zero. (motor position control). Type 1 System: The first derivative of the error can go to zero (motor speed control, but not position). Type 2 System: The second derivative of the error can go to zero (motor acceleration control, but not velocity or position).

[@mrjohn6021]

you're such a king man, I have a professor who graduated from MIT but you still explain way better!!

[@BrianBDouglas]

Hi vibhu, a zero in a transfer function is just an 's' in the numerator which corresponds to a derivative. However, you have to be careful because adding a derivative term to your plant G(s) won't actually help solve this tracking problem in the feedback system. That's because the system transfer function is U/(1+G) and so there's no way to actually ONLY add a zero to this closed loop transfer function by ONLY changing G. At least one way to get around this is by adding a feed forward path.

[@timothdev]

i cant just thank...... Some day i will teach someone something valuable like this .......... Just saying thanks is not the best way to be grateful, it should come from the heart and that becomes action......... i ll for sure do something valuable for nothing.....just like you do !

[@frankcollins6580]

with modern software's ability to solve differential equations easily, process control can be taught almost entirely in the time domain. Laplace is hanging on only because its engrained in process control texts. Excellent teaching! Perhaps you could do a series for understanding process control within t.

[@hanbiniwa]

I had that "aha" moment (following details closely, but then feeling like all of a sudden thrown above to the sky for a bird's eyes view) at: "We can find final value of output, but who cares?" -> leading into steady state error as the meaningful measurement of system performance... That seems really obvious now, but somehow completely missed it while reading a textbook. Thanks for kindling the intuition Brian!

[@BrianBDouglas]

Basically, the feed forward path would predict how to change the output so that it could follow an accelerating input and then the feedback path would attempt to reduce the error to zero. In this case the error would come from the differences between the feed forward model and the real system. Unless you're familiar with what I just said already I don't think this explanation will help you much :( I'll address feed forward in the future and hopefully it'll clear things up then.

[@markkerollos]

Wow that's a lot of work. From a viewer's point of view though, it definitely pays off. I'm currently doing controls as a subject and it's probably the most interesting subject. Keep up the good work, and hopefully you get some sponsorship or something for your vids to push you forward.

[@lazio20roe]

your videos are amazing, my lecturers should be ashamed, they make this content seem impossible to understand, you make it seem fairly easy. Thank you so much, i dont know if you get anything back from this, in that case you are truly selfless. In my eyes selflessness is the most important and most often neglected human trait.

[@mdnafiskhan9336]

Thanks for providing us with great lectures for free

[@LKodiak]

You should be an axual teacher. So many hard to grasp topics in one playlist, explained nicely and cleanly while my teacher didn't manage to do the same in half a year of lectures

[@Thewowhahahawow]

Damn, this is gold. Beautiful and concise, you make learning a lot faster and easier!

[@zoen2862]

You've just saved my bacon for my 1st assignment doing my masters degree... thank you so much!

[@pimpingbuss]

like something like this Question: a) Plot this Nyquist diagram: GH(s)= 50 / s(s+2)(s+4) b) Investigate the stability of the system based on Nyquist diagram. c) Use Routh-Hurwitz stability criterion to validate the stability Thank you sooo much!

[@BrianBDouglas]

Andy, I don't follow any particular book when creating these lectures so I can't recommend one single book to use to follow along with. However, my all time favorite controls book is "Control Theory" by JR Leigh. If you're looking for a book that explains the topics in straightforward language this is the book. It's not heavy on math but if you're looking for more depth the reference section will point to other books. Also, you can read a lot of it for free on google books so that's great too!

[@adrianrevolotaipe3855]

what i great videos , all this serie of videos will help me a lot in the unversity ........

[@OmarSiddiqi__]

Great Video! However, I'm wondering if you made a video to explain what one could do to analyze the SS error of a system with poles in the right half plane - or is that even possible? Thanks!

[@BrianBDouglas]

Hi Abu, I'll continue to touch on system ID in my Control System Lab videos which I'm trying to put out every other week. But in the lecture series I think I need to address so of the other topics first before hand which is why I've switched topics so abruptly.

[@widdalightsout]

You are amazing.. Please don't ever stop teaching :)

[@mattkozienski2879]

9 years on youre helping me with my biomed engineering classes xd

[@OliverFoote]

I think you're about to save my life. Thank you

[@ar0568]

So THATS WHAT THOSE SYSTEM TYPES ARE?!?! thanks man took me way too long trying to figure something so simple out.

[@smiwabiva6657]

I find this very very relaxing. This is music to my ears

[@AJDification]

Exam in 3 hours, cheers dude that was super helpful :)

[@isagumus1]

The best explanation ever so far ! Appreciated

[@yejinlee9721]

Thanks you for your correct explains on FVT &Steady State Error ,wich is so informative to understand control engineering easily.

[@sashamuller9743]

you make control theory fun to learn thank you

[@hanans7527]

YOU MADE EVERYTHING EASIER. THANKS DUDE!

[@yash_jivrajani]

The best way to learn control systems!!

[@markkerollos]

Brian your videos are ridiculously good. You somehow manage to teach what would be more than 1 hour of lecture time in less than 15 minutes and yet I get more out of these than any lecture. Out of curiosity, how long does each one of these take you to produce?

[@pieceworth]

Thank you, Mr Douglas. You save my life

[@BrianBDouglas]

Of course I spoke too soon and in haste! :-( You can add a zero to your error transfer function by adding a pole at the origin in your controller. Essentially if you add a 1/s (or integrator) to your controller this is increasing the type by one. If you work out how that affects the error transfer function you'll see that it adds a zero at the origin which will cancel the additional pole added by the input function. Sorry for the confusion.

[@valeriuok]

Just wanted to add a note to that final statement. The system type has to be increased (more poles at the origin) in order to follow an input of a higher type. Basically, to follow an input of type n with 0 steady-state error, we need a system of type n.

[@BrianBDouglas]

Hello Fidelis, I posted a video showing how I make a control system video lust last week. I explain there which hardware and software I use. You should be able to create a very similar style video using the tablet and the software you mention. As for the drawing it took me about a week of practice to learn how to write on the tablet! It's not like writing on paper. I kept writing up at an angle or just really sloppy. Just stay with it and you'll get it. Good luck with your project.

[@alishahabi6274]

Very Nice Brian! I really enjoy you the way you explain the concepts. Thank you. Keep up the awesome videos!

[@pimpingbuss]

WOW! man I love your videos! please keep up with your good work! wish we had a prof at the university like you!

[@gustavvingborgnielsen9707]

"But we're control engineers..." Me as a 2nd year mechanical engineering student trying to learn all of this in a 5 ects course: *sobbing Thank you for great explanations, they help a lot!

[@ahmedkhaledmohamedmohamed746]

Best one I heard teaching control really Thank You

[@kekeo1-o4b]

2024 and this vid helps me a lot , thanks bru

[@BrianBDouglas]

@younesshah, where would the 2 come from? If we were integrating in the time domain then there would be some coefficient, but we are integrating in the S-domain. In the S-domain integration is just 1/s. So to integrate an impulse function (1 in the S-domain) once would be 1*1/s = 1/s or a step. Then just keep multiplying by 1/s to get higher orders, 1/s^2 for ramp and 1/s^3 for parabolic. By the way, congrats on taking your last exam for your degree! I'm sure it feels great to be done!

[@kevinklose]

What I'm confused about is that I read in multiple sources that a type 2 system achieves perfect tracking of a ramp input... what does that mean? I can see how a type 1 achieves perfect tracking of a step input, but not the type two for ramp :$

[@joshuapatterson5095]

Mate!!!! Absolutely love your videos!!!! You're like the Sal Khan of Control Systems :-)

[@asim_alameen]

''The impulse response of an integrator will be the integral of the impulse'' Can you elaborate more? How the integral of the impulse became a step function? I know it's a simple question, but it's not exactly clear for at the moment. By the way, you are explanation is just excellent!

[@BrianBDouglas]

If I ever see you in real life than I shall take you up on your offer!

[@Hypnodr0ne]

Very descriptive and helpful. Thank you!

[@DrunkinHulk]

Such a great Video, However I am still stuck. @11:53 you missed an S but didn't specify where the S would go making the equation useless am me stuck. I have to do a unit step and then a unit ramp and I found the unit step somewhere else. I'm hoping to reverse that and find the S so I can do the ramp. Otherwise great video and I learned a lot. Thanks

[@thewayoftheroll]

Can you go into further detail about the end when you design G(s) on the bottom? I'm lost there

[@doktoren99]

You are very good, my man!

[@dandyremix2360]

Simple and to the point. Thank you sir.

[@TheBirdMan]

Today I understood what FVT is really ! In my full course I didn't knew what this really means lolz :D

[@lolitahaze6470]

That is so clear and awesome! Great video

[@BreadG69]

This dude is a saint

[@aregayohannes9973]

You are special! Thanks Brian!

[@nikkiegargi]

12 odd minutes of awesomeness!!!nerds be turned on like whaaaaaaaaaat!! *claps*

[@fisfisarenskanal]

So helpful!! Greetings from LiU Sweden =)

[@lazybites7286]

How do you find the final value if there is a single pole on the RHS? Example the transfer function is 3 / (s^2+2s-3) and we have to find the final value of system with a step input.

[@himym910]

Thank you sir from Waterloo University, Canada !

[@milan_shah]

Hi Brian, thanks for the video Both the links you provided in the description is not working. So, if possible, please update them so that we can go through that. Another thing that I would like to ask you, in the video at 5:20 you said final value theorem would give mean value of the oscillations. Don't you think it's wrong as the step response the system having poles only on imaginary axis will be oscillatory. But this oscillations are not symmetric to the Time axis, in fact these oscillations are above Time axis so the mean value of their amplitudes won't be zero. While for these type of systems FVT gives 0. Sorry if I am wrong !

[@Dipsiner]

+Milan Shah The reason you are getting an incorrect value is because FVT is not applicable. He said the final value is undefined if the poles (they come in conjugate pairs) lie on the imaginary axis because they are continuous sinusoids. If the poles lie on the imaginary axis, the system is undamped and thus the oscillations will continue forever. So, using the Final Value Theorem is not applicable. However, if you use the FVT even though it is not applicable then the corresponding value will give you the mean value of the oscillations, which is not undefined. In the example prior to the one you mention he shows that if you use FVT on a system with poles in the RHP (unstable) the FVT will yield an incorrect response as well. The important part of the video is to understand Final Value Theorem and when it is applicable. I hope this response helped.

[@mdasrafulkabir3659]

What will happen of steady state error if there is any disturbance function add at system. what will be formula or procedure?

[@eliadkidkid]

we are using the lhp because we are moving from the S domain to the time domain, so the integral is for positive e^st ?

[@gypsycathunter]

You are my friend. Keep telling me facts.

[@stereodrive]

First I would like to thank you very much for your effort for producing this videos in such a amazing quality. It helps me really much during my studies! I have a little question, I'm apparently bad in mathematics. Around minute 9:00 you start to explain how the final value can be calculated for the different system types. My question is when I have the transfer function 1 / (s² + s) , do I have to factorize it , or can I just simple put all the s's to 0 and get my answer as 1 / (0 ²+ 0 ) = 1. In the following example for a system with a step input, we are using lim s ->0 s * 1/s * 1 / ( s² + s). Because I can short out the s * 1/s all left is the TF from the previous example 1 / (s² + s) (System 1) but now it equals infinity? Shouldn't it equal = 1 as well ?

[@xMrJanuaryx]

I have a question. What if I add a disturbance to the last system and I want to find the steady state error. Is it going to be the superposition of e(s)/u(s) + e(s)/d(s) where d(s) is my disturbance?

[@alessandrotrujillo2605]

What would happen if not only we get roots on the y axis but also on the right hand side? Would it be oscillatory? or would it go to infinity? Thanks in advance

[@ramonsantiago4573]

what about disturbance steady-state error? I'm a little confused, do we just do: Ess = lim (s ->0) (sT(s)W(s)) ? where W(s) is the disturbance input and T(s) is the transfer function of the system without input reference R(s).

[@johnbowman9734]

Do all these rules for steady state error still apply for a system that is NOT unity feedback?

[@sajjankumarhello]

Hey Brian! If we have a pole on Origin along with pole on Imaginary axis, can we still apply the Final Value theorem ? what will be the system type ? and Thanks for you videos.

[@magedelsharkawy1366]

sajjan kumar you cannot apply Final Value Theorem to a system if it has 1 pole on the imaginary axis or the RHP. Even if you have 1 pole in the LHP you still cannot apply FVT.

[@sivonparansun]

I am interested in getting a copy of your book! whats the best way to do that

[@phillipmaser132]

Brian, you are on top of this. Great learning from you. I have a control problem I want to share and find a way for matlab to solve and find the coefficients and two sensor system and one output. Do you have or use fmincon in mathlab?

[@fidelischimombe2800]

I liked your video Brian, i failed to understand this concept for three weeks, but you made me understand. I am working on a project whereby i am making videos for high school kidz in my country Zimbabwe, what tools do you use. I have bought a bamboo tablet, and i am using smoothdraw, and a screen capture, but i can't write anything meaning ful on my bamboo tablet

[@wilerjrxd]

Great lecture! Congrats! :)

[@13shamas]

In ess do you just take the plant into equation or the whole system ( controller, sensor)

[@Zuwwar]

Is system stability related to its final value? Like for example I have a system 1/s^2(s+1), now this is of course a stable system but when I find its final value its equal to infinity. So what is the meaning of this that my system is stable but at the same time its final value is infinity?

[@brainblue2352]

+Zuwwar Khan Jadoon are you sure your illustration is a stable system?In your system,it has 2 poles in origin,which represent a ramp respond,and the very respond is not steady,isn't it?

[@Zuwwar]

That is exactly what I mean that the response of a ramp function is not steady which says that it is not stable but when we look at the poles location then there isn't any pole in the right half plane then how is the system not stable if there is no pole in the RHP??

[@alphasatari]

When at least one pole lie on origin or on imaginary axis then system is termed as marginally stable or more precisely unstable.

[@javoncook2441]

That was fire!! Great job!!

[@imahugeman]

This is exceptional. Thank you so much!

[@sarvariabhinav]

your videos are sick bro...thanks

[@switchology_io]

Your Videos really help my understanding! Thank's a lot! Are you going to do some vids on state space representation?

[@benjaminjohannesson2095]

When taking the limit to obtain the steady state error would it be s * (U(s)/(1+G(s))?

[@theredbobcat]

According to my teachings at University, the steady state error should be multiplied by a factor of s. How does this change things?

[@OzzzyKane]

thank you so much for making these.!!!

[@davidfof13]

Excellent. Brian can you do a video on Nyquist analysis, Nyquist diagram and Nyquist criterion? I want to understand the whole thing. Cheers if you can

[@MexterO123]

So the final value of a system changes depending on the input since depending on the input you could change the system type? correct?

[@merrickcaptain871]

If we have a pole on Imaginary axis along with one in LHP, the response will eventually settle. Can we use Final value theorem for this case?

[@BrianBDouglas]

Merrick Captain If you only have a single pole on the imaginary axis then it must be at the origin (at least for real physical systems). If that is the case then you can use FVT on this because a pole at the origin just changes the system Type. i.e. 1/(s(s+1)). But if you have a pair of poles on the imaginary axis not at the origin then the response won't settle and you can't use FVT on it. Try plotting 1/((s+1)(s^2+1)) and you'll see.

[@vishnunarayanv526]

In order for the system to respond to a constantly accelerating input, we change the system type by adding a POLE at the origin, so that an extra s comes in the numerator and the error term goes to zero for s =0, right? Towards the end of the lecture, it was told to be a zero. We add a pole so that the transfer function itself helps in integrating the input thus making the output to accelerate as fast as input??

[@Hafidtsh]

Its so hard for me, ah damn , thanks Mr 👍