New preprint: A general absolute stability theorem

We have just posted a preprint titled: “Unified Necessary and Sufficient Conditions for the Robust Stability of Interconnected Sector-Bounded Systems”. It is available here and on arXiv. This is joint work with my student Saman Cyrus and myself. Our work provides a new absolute stability result that generalizes several existing works and enables new analyses, specifically tight certification of exponential stability.

The problem of “absolute stability” is a classical robust control problem that has been studied for over 75 years. The basic setup is that a known system is interconnected with another system that is nonlinear, uncertain, difficult to model, or otherwise troublesome. The goal is to certify “robust” stability. That is: can we guarantee that the interconnected system will be stable for all possible instances of the troublesome system?

The most classical version of this problem is when the troublesome system is “sector-bounded”. This happens for example if our system contains a saturation nonlinearity. Most results for this setting take the form: “if the known system satisfies a certain property, then the interconnected system will be robustly stable”. In other words, they are “sufficient” conditions for stability. Classical examples in the literature include passivity theory, the small-gain theorem, and the circle criterion. There have also been many efforts to generalize or unify these results. For example, researchers have observed that if we use a particular loop transformation, the three aforementioned results are one and the same!

Our contribution in this paper was to formulate a general version of the classical robust stability theorem that has the following properties:

  • The theorem is both necessary and sufficient. So if the condition fails, there must exist a nonlinearity that yields an unstable interconnection. This parallels existing necessary conditions from the literature.
  • It is formulated in a semi-inner product space. This distills the result to its essential features. In this general space, there need not be any notion of “time”. So we can do away with the typical requirements of stability, causality, and boundedness.
  • Existing results can be recovered by simply choosing the correct vector space and associated semi-inner product. This means we can recover passivity, small-gain, circle, discrete vs continuous time, and more.
  • We also provide a computationally verifiable linear matrix inequality (LMI) that allows one to efficiently check the conditions of the theorem in the special cases of ordinary stability (a known result) and exponential stability (a new result).
Finally, and perhaps most importantly, we can apply our general result to new semi-inner product spaces to obtain new and previously unknown results! In the paper, we showcase one such example, where we formulate a new result that gives necessary and sufficient conditions for exponential stability with a prescribed rate of convergence.