I attended the 2019 Conference on Decision and Control in Nice, France. The photo above is a panoramic view of Nice taken from the top of Castle Hill. My research group had a solid showing at this year’s CDC with 3 papers. In no particular order, here are short summaries of each paper and a links to the slides for each talk.

• “Integral quadratic constraints: Exact convergence rates and worst-case trajectories” with my postdoc Bryan Van Scoy. This paper considers the problem of robust stability; in a feedback interconnection where one of the components is uncertain (but the uncertainty is bounded), when can we guarantee that the interconnection will be stable? The most general robust stability result available is the IQC theorem of Megretski and Rantzer [[LINK]], which roughly states that robust stability can be established by checking the feasiblity of a particular linear matrix inequality (LMI). It is known that the result goes both ways: if the LMI is not feasible, then (at least in theory), there should exist some admissible choice of the uncertainty that will lead to instability. Bryan’s paper provides an explicit way of constructing such worst-case realizations by leveraging a theorem of strong alternatives for LMIs. This result is useful because it allows us to go beyond “could this system fail?”, and address the question “how could this system fail?”. Here are links to the paper and the slides from Bryan’s talk.

• “Explicit agent-level optimal cooperative controllers for dynamically decoupled systems with output feedback” with my student Mruganka Kashyap. This paper considers a cooperative control problem where a group of agents (these could be robots, drones, etc.) attempt to solve a cooperative task under limited communication. In general, problems of this type are difficult to solve. However, solving simpler cases can provide useful intuition for what might work well in more general scenarios. In this work, we assumed the agents were “dynamically decoupled” and have linear dynamics. We also assume all disturbances are Gaussian and the global objective (which may couple the agents’ states and inputs) is quadratic. Previous work has shown that these assumptions ensure that the optimal strategy is a linear one and can be computed efficiently. Our contribution takes this a step further and elucidates the exact structure (down to the level of individual agents) that is used for optimal control. The optimal structure is reminiscent of the classical LQG structure: there is an “estimator”, a “compensator”, and a separation principle too! Here are links to the paper and the slides from Mruganka’s talk. Incidentally, this was Mruganka’s first talk at an international conference! See the photo on the right!
• “Unified necessary and sufficient conditions for the robust stability of interconnected sector-bounded systems” with my student Saman Cyrus. The Lur’e problem is perhaps one of the most fundamental and well-studied examples of a robust control problem: assessing the stability properties of a known linear time-invariant plant in feedback with an unknown sector-bounded nonlinearity. This problem is is fundamental because many electrical, mechanical, and chemical systems (read: engineering systems!) are close to being linear. The departures from linearity often involve phenomena such as model errors, saturations, friction/stiction, or other hysteresis effects, which can often be well-modeled as sector-bounded nonlinearities. In the 60+ years of literature on this topic, various results have emerged: passivity theory, the small-gain theorem, the circle criterion, and more. Our paper presents an effort to unify these results (and their various incarnations) by using a very general mathematical framework. The benefit of our approach is that we can write a single robustness theorem that makes minimal assumptions yet can be specialized to recover many existing results. The advantage of having a single general theorem is that it allows us to derive new specializations! In our paper, we present one such new result: a necessary and sufficient condition for robust exponentially-weighted stability of Lur’e systems. Here are links to the paper and the slides from the talk.

One final highlight: I ran into my former PhD advisor, Sanjay Lall! Here is a photo of a small subset of my academic family: From left-to-right: Sanjay (my advisor), me, Bryan (my postdoc), and Mruganka (my student).