I’m excited to share our new preprint on robust stabilization, by Saman Cyrus and me. The title is “Generalized necessary and sufficient robust boundedness results for feedback systems”, and it is now available on arXiv. This work is the culmination of Saman’s PhD and formed the core of his thesis, which he recently successfully defended (!).

Robust stabilization is an old problem, dating back at least 70 years to the seminal work of Lur’e, Zames, and Willems. The basic idea is that a known system G is in feedback with an unknown nonlinearity H constrained to lie in some known set, and we seek sufficient conditions on G that ensure stability of the interconnection. In this work, we looked at the case where the nonlinearity is constrained to lie in a cone (also known as “sector-bounded”). Results of this type include: small-gain, circle, passivity, and extended conicity. It may be possible to extend our ideas to dynamic versions of robust stability (multiplier theory, dissipativity, or integral quadratic constraints), but this is an area for future work.

Robust stabilization results are typically formulated in the extended space L2e and can be expressed in a variety of ways, depending how which assumptions are made (causality, stability, linearity, time-invariance, etc.). Moreover, most of these results are only sufficient while some others are “necessary and sufficient”. Necessity can also mean two different things; when the conditions fail, can we construct worst-case signals only? Or can we also construct worst-case nonlinearities?

Our aim in this work was to disentangle the different formulations of these results and distill them into a single unified result that makes as few assumptions as possible. We worked with relations in an arbitrary semi-inner product space, which allowed us to avoid the notions of “causality”, “stability”, “time-invariance”, and “well-posedness”. In fact, we don’t even assume a notion of “time”! In this setting, we prove a very general necessary and sufficient condition for robust boundedness that requires essentially no assumptions. When our condition fails to hold, we show how to explicitly construct worst-case signals, and an associated worst-case nonlinearity. Moreover, our construction produces a nonlinearity that happens to be linear, which is nice.

Specializing our result to causal operators on L2e, this is what happens:

- When our condition fails, we can still construct worst-case signals. This works even when the plant G is nonlinear. Note: although the S-procedure can be used to prove this result, it would require assuming G is linear and would be non-constructive. We provide a direct construction and no linearity assumption is needed.
- The second step of our construction (constructing a worst-case nonlinearity) does not work in the L2e case because it typically produces a non-causal nonlinearity.
- By making additional assumptions, namely that G is LTI, we can construct worst-case nonlinearities that are causal. Our construction produces an LTI nonlinearity (in fact, a pure delay).

Our work unifies existing results from the literature in the following way:

- Existing necessary-and-sufficient results are special cases of our general result.
- Existing sufficient-only results follow directly from our general necessary-and-sufficient condition once we apply the appropriate relaxations. We give examples in the text of how this works.

This turned out to be a very educational research project for both Saman and myself, on a rich and interesting topic, and I hope were able to do it justice! I also posted a Twitter thread summarizing the paper.