Closing the Gap to Quadratic Invariance: A Regret Minimization Approach to Optimal Distributed Control

In this work, we focus on the design of optimal controllers that must comply with an information structure. State-of-the-art approaches do so based on the
$\mathcal{H}_{2}$
or
$\mathcal{H}_{\infty}$
norm to minimize the expected or worst-case cost in the presence of stochastic or adversarial disturbances. Large-scale systems often experience a combination of stochastic and deterministic disruptions (e.g., sensor failures, environmental fluctuations) that spread across the system and are difficult to model precisely, leading to sub-optimal closed-loop behaviors. Hence, we propose improving performance for these scenarios by minimizing the regret with respect to an ideal policy that complies with less stringent sensor-information constraints. This endows our controller with the ability to approach the improved behavior of a more informed policy, which would detect and counteract heterogeneous and localized disturbances more promptly. Specifically, we derive convex relaxations of the resulting regret minimization problem that are compatible with any desired controller sparsity, while we reveal a renewed role of the Quadratic Invariance (QI) condition in designing in-formative benchmarks to measure regret. Last, we validate our proposed method through numerical simulations on controlling a multi-agent distributed system, comparing its performance with traditional
$\mathcal{H}_{2}$
and
$\mathcal{H}_{\infty}$
policies.