I attended the 2023 Conference on Decision and Control in Singapore. The photo above is a panoramic view from the top (57th floor) of the Marina Bay Sands hotel. This was my first time in Singapore, and probably the farthest I have ever traveled in my life! This year, I had one paper with my former student Mruganka Kashyap, who recently graduated, and I also co-organized a tutorial session with my colleague and former postdoc Bryan Van Scoy entitled “Analysis and Design of Optimization Algorithms Using Tools from Control Theory”. Here are short summaries of the works I was involved in and links to papers and slides.

*Guaranteed stability margins for decentralized linear quadratic regulators*by Mruganka Kashyap and me (slides).

This paper was also accepted as a dual submission to the IEEE Control Systems Letters.It is well known that classical linear quadratic regulators (LQR compensators) have guaranteed stability margins. Namely, any LQR controller has gain margin $(\tfrac{1}{2},\infty)$ and a phase margin of $60^\circ$. It is also well-known that the classical LQG regulator (with output feedback and a Kalman filter in the compensator) has no guaranteed stability margins.

Our paper answers the question of whether

*decentralized LQR*has guaranteed stability margins. Here, “decentralized LQR” refers to a special kind of LQR problem where the dynamics are distributed on a graph. In particular, there is an underlying partially ordered set (poset) that determines the block-sparsity pattern of the state-space matrices. Additionally, the controller is required to satisfy a similar information constraint; each sub-controller only has access to the states of its*ancestors*. The LQR cost function is allowed to arbitrarily couple all states and inputs of all subsystems. Although decentralized control problems are hard in general, this flavor of decentralized LQR problem is tractable (see here and here, for example). I found this to be a fascinating question because on the one hand, decentralized LQR is a state-feedback problem, so perhaps one would expect it to have guaranteed stability margins like its centralized counterpart. But, on the other hand, the optimal decentralized LQR compensator is*dynamic*, and just like with LQG, it contains a filter that each sub-controller must use to estimate the states it cannot directly observe. So perhaps there should be no guaranteed stability margins?It turns out the answer is “sometimes”. Under certain mild assumptions (the $B$ and $R$ matrices must be block-diagonal partitioned conformally to the different subsystems), we obtain the same stability margins as with centralized LQR, independently for each sub-controller. We also show that the assumptions on $B$ and $R$ are both necessary by way of a simple counter-example.

*Tutorial session: Analysis and Design of Optimization Algorithms Using Tools from Control Theory*(website).

Co-organized with Bryan Van Scoy.First-order methods provide robust and efficient solutions to large-scale optimization problems. Recent advances in the analysis and design of first-order methods have been fueled by tools from controls, including integral quadratic constraints and multipliers from robust control. Similar advances have been made in the optimization community through the (related) performance estimation framework. Together, these tools have transformed the way in which we analyze and design optimization methods. This tutorial session gave an introduction to these tools in the context of algorithm analysis and presented some recent advances and open problems.

The session contained six talks (one introductory talk, and five invited talks with accompanying tutorial papers). I prepared a website containing links to all the slides and papers from our tutorial session. Enjoy!