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I have been used SciLab for a day from now and I really excited to develop so called "H_{\infty}" controllers.

Normaly I have used LQG-controllers with integral actions, but I have heard that $H_{\infty}$ controllers is much better in real life because a $H_{\infty}$ controller do not require a perfect model of the system, as LQG-controller want to have.

If you look at this link: https://help.scilab.org/docs/6.0.0/en_US/section_64a8529216e858b335b0e6c058385350.html

You will find this:

  • ccontrg — Central H-infinity continuous time controller
  • dhinf — H_infinity design of discrete-time systems
  • dhnorm — discrete H-infinity norm
  • gamitg — H-infinity gamma iterations for continuous time systems
  • h2norm — H2 norm of a continuous time proper dynamical system
  • h_cl — closed loop matrix
  • h_inf — Continuous time H-infinity (central) controller
  • h_inf_st — static H_infinity problem
  • h_norm — H-infinity norm
  • hinf — H_infinity design of continuous-time systems
  • linf — infinity norm
  • linfn — infinity norm
  • macglov — Continuous time dynamical systems Mac Farlane Glover problem
  • nehari — Nehari approximant of continuous time dynamical systems
  • parrot — Parrot's problem

My question is: What is important of this, if I want to develop a controller $K$ which have guaranteed stability margins - robustness.

I have two books about robust control, but I don't recognize anything when I read the list. What should I use? What should I focus on? The CACSD library for SciLab look like a mess.

Let's look at:

  • h_inf - Continuous time H-infinity (central) controller

Ok! A $H_{\infty}$ controller!

Or should I use:

  • ccontrg — Central H-infinity continuous time controller

My books tells me nothing about central controllers. So I guess that I have too start with the "ccontrg"....

Ok! The formula or code...what ever..tells me this:

[K]=ccontrg(P,r,gamma);

Ok! The $P$ is the agumented state space model, r..I don't know but gamma needs to be some kind of limit I guess?

The definition of $H_{\infty}$ is:

$$ ||F(P, K)||_{\infty} = max_{\omega}\bar{\sigma}(F(P, K)(i\omega))$$

Which means that our maximum singular value $E_{11}$(singular = positive value) of the

$$[U, E, V] = svd(P)$$

Cannot be larger that $\gamma$. So

$$max_{\omega}\bar{\sigma}(F(P, K)(i\omega)) < \gamma$$

Is the definition of $H_{\infty}$ controllers.

As you see...I know what I'm doing, but I still feel very unsure about this $H_{\infty}$ part of library. None of those functions asking the user about weighting matrices.

A $H_{\infty}$ controller look like the block diagram above. In this case $K = R$.

enter image description here

euraad
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  • Just a minor comment: Don't expect to see any revolutionary improvements in robustness when going to $H_{\infty}$ compared to LQG. It is robust in the particular sense the robustness problem is stated, with nice math, but in practice that does not say much. There's a reason $H_{\infty}$ basically is unheard of in practice. If it was better it would have been adopted much more. The paper Robust, fragile or optimal by Keel et. al might be relevant. – Johan Löfberg Aug 19 '17 at 18:59
  • @JohanLöfberg So you mean that LQG is much more used that H-inf?Isn't H-inf controller the new and better "LQG controller"? Because Doyle's paper said that LQG controllers have no guaranteed stability margins, which makes the LQG very weak for disturbances? In pratice the LQG controller can be far more robust that H-inf? – euraad Aug 19 '17 at 19:35
  • @JohanLöfberg If you had to choose...LQG with integral action, or H-inf with integral action? – euraad Aug 19 '17 at 19:42
  • LQG everyday of the week. Natural tuning and clear separation of control and estimation. – Johan Löfberg Aug 19 '17 at 19:47
  • Doyles paper was seminal and opened up a whole new field of control. Academically fascinating and theoretically better does not mean better in practice. I cannot think of real case where $H_{\infty}$ actually was used (there are many of course, but it just doesn't compare to LQG) – Johan Löfberg Aug 19 '17 at 19:50
  • Doyle showed that if you tune your LQG really really bad, you will get crappy margins. It took 20 years for anyone to find such an example... – Johan Löfberg Aug 19 '17 at 19:52
  • @JohanLöfberg Okej. But if LQG is more used in pratice, why are H-inf even used then? I studied at Umeå Universitet and that university teach out H-inf every day. – euraad Aug 19 '17 at 20:01
  • I have a new book, called "Multivariable Feedback control - Analysis and design" from Sigurd Skogestad. That book teach robust control in practice and showing some industrial examples too. Great book. The first chapters are free to read. – euraad Aug 19 '17 at 20:06
  • We teach $H_{\infty}$ to our students too, but in practice the claimed/possible benefits of $H_{\infty}$ simply isn't enough for most cases. – Johan Löfberg Aug 19 '17 at 20:15
  • Why is it even used? You could just as well ask why LQG is used when 99% of all control problems can be solved using PID. Why use LQG when you can use MPC instead and handle constraints in a structured way? – Johan Löfberg Aug 19 '17 at 20:17
  • @JohanLöfberg OK. I though H-inf was the future. Maybe I should stick to LQG/LTR with integral action then. There is an old saying "If you know it, you do it, if you don't know it, you don't do it". Example: If everybody knows PID and LQG and LQG is +1 better, but PID can solve the problem too. LQG is used. I think that's the reason why people do research in control theory - nice math. So I don't know if I should only stick to LQG controllers, or I should continiue to learn H-inf control (Robust control). My goal is learn to develop a robust controller for heavy industry. – euraad Aug 19 '17 at 20:24
  • But I know that PID can be used to solve the heavy industry problem too. – euraad Aug 19 '17 at 20:25
  • $H_{\inf}$ is probably not the future. If it was, it would have been much more adopted by now. But if your goal is to learn, implement and see what it gives you and draw your own conclusions. – Johan Löfberg Aug 19 '17 at 20:33
  • @JohanLöfberg What controller do you think is the future? Adaptive controller? – euraad Aug 19 '17 at 20:39
  • There's no such thing, but MPC is one of the few successful developments in control which has gained momentum in practice. I might be biased though... – Johan Löfberg Aug 19 '17 at 20:51
  • Yes, I have heard that MPC is used well in the industry. But when I read your comments, it's seems that is not worth to read control theory at all ;) Because 99% of all control problems can be solved by a PID. – euraad Aug 19 '17 at 20:55
  • The final 1% is why we do research and where the fun problems are. – Johan Löfberg Aug 19 '17 at 20:57
  • So can you name one specific area when robust control comes in handy? Spacecreaft? Flights? Robotics? Or are all that controlled by optimal controllers? – euraad Aug 19 '17 at 21:01
  • By the way. You should definitely read the book "Multivariable feedback control - Analysis and design", by Sigurd Skogestad, if you what to have pratical applications about H-inf controllers. The book teaches diffrent ways to comput the most robust contol law. :) – euraad Aug 19 '17 at 21:04
  • Robust control is always important. Whether it is designed using $H_{\infty}$ is another issues. $H_{\infty}$ has not gained any significant impact in any domain as far as I know. – Johan Löfberg Aug 19 '17 at 21:04
  • According to my book, H-inf are used well in chemical industry and areas where you only have one chance and cannot take any mistakes. – euraad Aug 19 '17 at 21:07
  • Trust me, I've read that book. I've teached using it too. – Johan Löfberg Aug 19 '17 at 21:07
  • Ok. What do you think about that book? Pratical, or non-pratical? – euraad Aug 19 '17 at 21:09
  • Depends on what level you are talking about. For our Phd students we use Robust and optimal control by Zhou et al as the theoretical book and Skogestad as a more practical complement. A good book nevertheless – Johan Löfberg Aug 19 '17 at 21:13
  • The level I talking about it to create robust multivariable controllers for industrial use. When I mean industrial, I focus on excavators, crushers, robot arms, tank shocks etc. Not holding 60 bar pressure for a hydraulic LS-pump or holding the temperature 20 dC in a classroom. – euraad Aug 19 '17 at 21:18
  • Sure, as it has lots of practical examples as far as I remember – Johan Löfberg Aug 19 '17 at 21:20
  • Ok. Thank you! I think that all my questions has been answered. Now I know some more opinions about robust control. – euraad Aug 19 '17 at 21:23
  • Gentlemen, I would like it very much, if either of you could post a summary of your discussion as an answer. @Johan? Looks like you reached some kind of a conclusion, so it would be a nice way to wrap up the thread. No pressure, of course. Only if you find the time. I only showed up here because the length of the comment exchange raised an automatic flag that I'm supposed to handle :-) – Jyrki Lahtonen Aug 20 '17 at 08:05
  • Johan gavel me very improtant information about H-inf controller that I could not miss. This kind of information, you cannot find in books. – euraad Aug 20 '17 at 09:36
  • I figured out that much, Daniel. It is also ok for you to post a summary of what you learned. That way you may still get more feedback. I quite undestand if you want to give Johan the courtesy of road, though. And I'm wondering whether I should show this thread to my bridge teammates :-) (two of which are at regleringsteknikavd. i Åbo Akademi) – Jyrki Lahtonen Aug 20 '17 at 18:06
  • I havent learned so much about H-infinity control because i still looking for how to build one. For example, my lastest question, i looking for stabilizable and detectable. Finding no information at all. I was reading some papers at Åbo Akademi Reglertekniksavdelning http://users.abo.fi/htoivone/advcont.html Did not found any help there, even there are very good articles there. So the summary about what I have learned here is that H-inf is not a good answer due to 99% of the control problems are solved with a PID controller. – euraad Aug 20 '17 at 18:14

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