I am trying to use the small-gain theorem to prove the robust stability condition for additive uncertainty. I have been stuck on it for a while now.

Question: How can I use the small-gain theorem to prove the following condition. For a noice model $ \tilde G(s) = G(s) + \Delta(s) W(s) $, the condition for robust stability is $||W(s)C(s)S(s)||_{\infty} < 1$, where:

  • $C(s)$ is the controller transfer function
  • $S(s)$ is the sensitivity transfer function $S(s) = \frac{1}{1 + G(s)C(s)} $
  • $W(s)$ is a weighting transfer function in the uncertainty

Does anyone have a link to a complete proof using this method? I find notes on the internet that simply leave it as an exercise for the reader, but I am getting stuck along the way.

Any help is greatly appreciated.

  • $\begingroup$ Over my head, but some papers (e.g. on "disk margin") point to the reference text: A Course In Robust Control Theory (Dullerud & Paganini 2000) for these things. (it is more mathematical than most engineers, including myself, are used to). If you search you might find the whole thing online $\endgroup$
    – Pete W
    May 28 '21 at 14:58
  • $\begingroup$ Thanks @PeteW! I'll take a look at that. Much appreciated! $\endgroup$ May 28 '21 at 16:19
  • $\begingroup$ Is there a block diagram to go along with the equations? $\endgroup$
    – AJN
    May 30 '21 at 16:01

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