Simultaneous stabalization - basic questions

Hello, I am wondering about the essential question of Simultaneous stabalization. I know it is different from robust stabalization (which stabalizes an infinite number of plants) in that the controller must stabalize a finite number of plants that are not interrelated.

My basic questions :

1) Do the plants being stabalized HAVE to include non-stable plants? Is it meaningful to prove a controller exists which stabalizes a finite number of stable plants?

2) Strong stabalization means the controller has to be stable too. Is this a necessary condition for the Simultaneous stabalization solution? I mean if one can find a controller which stabalizes a finite number of plants, does the controller HAVE to be stable? Or is it ok to just find any controller which stabalizes the plants?

I would appreciate any advice, references, etc... Thanks. Best regards, Jon

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Obiously yes, what would you stabilize here if everything were stable?

No. Because such a controller is ALWAYS GUARANTEED TO EXIST. Just consider a controller described by C(s) = 0 ...

You are done with the proof.

No. If you task is to find a controller, that stabilizes (simultaneously) several plants, then you are obviously happy if the controller just stabilizes (simultaneously) several plants...

If you need a STABLE controller that stabilizes (simultaneously) several plants, then you are obviously happy if the controller stabilizes (simultaneously) several plants and is itself stable...

Quite simple... Don't let definitions and formulas prevent you from simple thinking.


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Zdenek Hurak

Hi Zdenek,

Ok thanks for your reply. Yes, i agree if the family of plants considered are all stable, then the solution is trivial. In fact K doesnt have to be zero, but as long as it satisfies the small gain theorem for the 'worst' plant. But of course it still trivially exists.

I wonder if it is still trivial if some degree of performance is guarenteed as well as stability (e.g. zero offset)?

Initially i found the problem of Simultaneous stabalization confusing because i didnt see the difference between robust stabalization and Simultaneous stabalization. But it is clearer now. Jon

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