Robustness to Additive Uncertainty Proof using small gain theorem

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.

• 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 May 28 '21 at 14:58
• Thanks @PeteW! I'll take a look at that. Much appreciated! May 28 '21 at 16:19
• Is there a block diagram to go along with the equations?
– AJN
May 30 '21 at 16:01