# What is $H_2$ and $H_{\infty}$ control? [closed]

I can create a Linear Quadratic Gaussian Integral(LQGI) controller very easy by using GNU Octave. LQGI is in the area of Optimal Control Theory.

But there is something called Robust Control Theory. And every time I'm searching for that I got these in my search field:

• $H_2$
• $H_{\infty}$

Questions:

1. What are they?
2. How do I apply them to control the processes?
3. Are they good for control systems?
4. Are they better that PID and LQGI?
5. Are they easy?
6. Are they for nonlinear system?
7. Can they have a Extended Kalman Filter?
8. What's the difference between them both?
9. Can you show me an easy example how to apply them?
10. Other?

## closed as too broad by Marcus Müller, MBaz, jojek♦Aug 2 '17 at 19:01

Please edit the question to limit it to a specific problem with enough detail to identify an adequate answer. Avoid asking multiple distinct questions at once. See the How to Ask page for help clarifying this question. If this question can be reworded to fit the rules in the help center, please edit the question.

• Hi! can you please refer each of your (many!) questions to en.wikipedia.org/wiki/H-infinity_methods_in_control_theory and what you don't see answered by that? It's very complicated to describe complex theory from scratch, but it's easier to discuss potential misunderstandings when one has a common "base of understanding". – Marcus Müller Jul 31 '17 at 19:56
• Well, I don't require that I want the whole explanation. If you say that it's very complicated to describe complex theory from scratch. Then the question number 9 and 5 is answered. Thank you. – Daniel Mårtensson Jul 31 '17 at 20:40
• 9. was answered by the simple example in the wikipedia article. That's why I'm asking you to make clear what you couldn't answer in your own research! We can't hammer the knowledge into your brain, so it's up to you to more closely define what you need helpf with. – Marcus Müller Jul 31 '17 at 20:47

1. They are system norms, a metric that you can compare two different systems in terms of their generalized gain and spread. You can look these up no need for attaching physical motivation.
2. You don't. They are for assesing and using in the optimization programs.
3. Yes, but only if you know what you are doing.
4. Can be. But again, they are for assessment and optimization. If you hit the minimum with a PID controller then it won't make any difference. But hitting the minimum is pretty much unlikely.
5. Unfortunately no.
6. see 5 (you can pull some tricks by bounding the nonlinearity with a linear system etc. or for very limited nonlinear operators you can apply them)
7. Not applicable. They are different things.
8. They are different norms. But both of them are norms. Read about them.
9. See the documentation of hinfsyn of matlab
10. Yes
• Thank you for the answers. They are just methods to find the robust control law $L$ ? – Daniel Mårtensson Aug 1 '17 at 0:20
• @DanielMårtensson No they are most of the cases used for coming up with a dynamic controller $K(s)$ – percusse Aug 1 '17 at 0:26
• Quite inaccurate. – percusse Aug 1 '17 at 1:08
• Starting from the robustness issue to Gaussian assumptions in the industrial processes that is certainly not true. You are asking H2 norm and claiming that LQG is the best. Had you had the full picture, you would be dealing with H2 norms all over the place because LQG is the special case of H2 norm control. That gives away your education level about LQG theory. Also these are model based solution that's why nobody bothers with them. Go to any industry and try to convince to model something accurately manage to make it work then we talk again. Nobody will give state space matrices to you – percusse Aug 1 '17 at 11:10
• I have now read aboot H infinity and you are totally right. H inf controller is totally robust and better that LQG. I found that H infinity control was quite easy. H infinity have also an observer too. Robust observer. – Daniel Mårtensson Aug 2 '17 at 14:29