Final Value Theorem and Steady State Error

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The Final Value Theorem is a way we can determine what value the time domain function approaches at infinity but from the S-domain transfer function. This is very helpful when we're trying to find out what the steady state error is for our control system, or to easily identify how to change the controller to erase or minimize the steady state error.
Two proofs of the Final Value Theoerm:
www.ee.kth.se/~tn/.../Basic.../Initial_and_Final_Value_Theorems_uk.pdf
renyi.ece.iastate.edu/zhengdao/initial-value-theorem.pdf
Errata:
7:55 I wrote "If all poles are in LHP then type 1 and FV=0" and it should be "If all poles are in the LHP then type 0 and FV=0"
11:53 I left the 's' off the final value theorem equation. It should be the limit as s approaches 0 of 's' times the transfer function.
Don't forget to subscribe! I'm on Twitter @BrianBDouglas!
If you have any questions on it leave them in the comment section below or on Twitter and I'll try my best to answer them.
I will be loading a new video each week and welcome suggestions for new topics. Please leave a comment or question below and I will do my best to address it. Thanks for watching!

[@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.

[@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!

[@GalaxianExplosionGS]

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

[@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!

[@Thewowhahahawow]

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

[@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]

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

[@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.

[@alishahabi6274]

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

[@deanmillard9853]

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

[@isagumus1]

The best explanation ever so far ! Appreciated

[@mrjohn6021]

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

[@mandeadd]

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

[@hanans7527]

YOU MADE EVERYTHING EASIER. THANKS DUDE!

[@mannykrish1]

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

[@widdalightsout]

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

[@ahmedkhaledmohamedmohamed746]

Best one I heard teaching control really Thank You

[@pieceworth]

Thank you, Mr Douglas. You save my life

[@ar0568]

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

[@mdnafiskhan9336]

Thanks for providing us with great lectures for free

[@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.

[@Hypnodr0ne]

Very descriptive and helpful. Thank you!

[@mattkozienski2879]

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

[@dandyremix2360]

Simple and to the point. Thank you sir.

[@doktoren99]

You are very good, my man!

[@wilerjrxd]

Great lecture! Congrats! :)

[@lolitahaze6470]

That is so clear and awesome! Great video

[@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

[@AJDification]

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

[@adrianrevolotaipe3855]

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

[@pimpingbuss]

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

[@imahugeman]

This is exceptional. Thank you so much!

[@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).

[@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.

[@aregayohannes9973]

You are special! Thanks Brian!

[@javoncook2441]

That was fire!! Great job!!

[@OliverFoote]

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

[@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

[@zoen2862]

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

[@OzzzyKane]

thank you so much for making these.!!!

[@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.

[@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!

[@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?

[@sarvariabhinav]

your videos are sick bro...thanks

[@R0kmyS0X]

Thank you. Your video helped alot!

[@joshuapatterson5095]

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

[@yash_jivrajani]

The best way to learn control systems!!

[@fisfisarenskanal]

So helpful!! Greetings from LiU Sweden =)

[@randomandspontaneous]

Great video Brian. Helped alot! thanks!:)

[@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.

[@Contarius9]

You're amazing. Thank you so much!

[@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.

[@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 !

[@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.

[@arjenwassenaar9963]

This is just amazing!

[@smiwabiva6657]

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

[@switchology_io]

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

[@himym910]

Thank you sir from Waterloo University, Canada !

[@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.

[@vijayn2443]

Thanks a lot...this was helpful

[@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!

[@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 :$

[@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.

[@systemf290]

really good video thx!

[@BreadG69]

This dude is a saint

[@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!

[@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?

[@JackST95]

Brilliant Video

[@gypsycathunter]

You are my friend. Keep telling me facts.

[@alexanderali2971]

great video thank you so much

[@mdasrafulkabir3659]

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

[@mahmutagdere5438]

You are perfect man ! :)

[@ashishdayal058]

thanks a lot..it really helped

[@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!

[@thewayoftheroll6358]

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

[@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 ?

[@Hafidtsh]

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

[@TheBirdMan]

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

[@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.

[@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

[@pyrojamin]

so helpful thanks

[@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.

[@laurastevenson8141]

Thanks SO much!

[@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!

[@elimarpineda6496]

THANK YOU!

[@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

[@UDAYKUMAR-zk2vt]

Super sir. awesome

[@13shamas]

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

[@josephgandisannoh7341]

Thank you

[@lazio20roe]

do you have another video with a Ess example with numbers? thanks again

[@SajjadAli-yr1lq]

Thank you sir..

[@aleksandaryanarov7141]

Awesome! And how can I calculate the gain of a system to reach a SSe of 0.2?

[@MrRObot-bin]

Awesome!

[@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??

[@serkutaktas577]

This is perfect