Talk:H-infinity methods in control theory

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[edit] Discussion 2005

It's somewhat misleading to refer to this as a tool for finding optimal controllers. Technically, it is correct, since we are trying to minimize the error outputs z and hence we introduce some notion of optimality. However, especially in relation to robust control, \mathcal{H}_{\infty} controllers tend to be quite conservative and suboptimal in practical senses (tending to be sluggish and/or high-order controllers) - since the underlying problem is to find a stabilizing controller for a whole set of plants. I thought I should open this as discussion rather than edit the page since it's more of a semantics argument than a cut-and-dry technical change. M0nstr42, 05:46, 16 Jun 2005 (UTC)

Optimal controller design is the process of choosing a controller which minimises some cost function. In the case of \mathcal{H}_{\infty} control, the cost we are trying to minimise is the \mathcal{H}_{\infty} norm of the closed loop system. However, as with all engineering problems, achieving optimality according to one performance measure will inevitably lead to compromises in other performance measures e.g. speed of response or controller order. In my opinion, such compromises should not prevent us from calling our \mathcal{H}_{\infty} based design "optimal"AndyHazell 12:48, 16 August 2005 (UTC).

Meh. What you say is true. It still seems like odd semantics to me. A cost function could be defined somewhat arbitrarily, which would mean every controller is optimal in some sense (i.e. I define my cost function to be zero for my controller and 1 for every other controller). "Optimal control", to me, generally implies that you want to work with something less abstract than the Hilbert 2-norm. I guess it's the difference between saying \mathcal{H}_{\infty} falls under the heading of "optimal control" versus saying that the controller is optimal. *shrugs* M0nstr42 01:20, 8 September 2005 (UTC)

I added a paragraph to explain this point of view. --Orzetto 21:50, 8 December 2005 (UTC)
More to the point, surely \mathcal{H}_{\infty} is itself a particular Hardy Space? We probably all mean \mathcal{H}_{\infty} Control.

My instructor uses the phrase "optimization-based control", which is a little less ambiguous. Nfette (talk) 22:43, 9 March 2008 (UTC)