A graph-informed regret metric for optimal distributed control

TL;DR

This work proposes spatial regret, a graph-informed metric that quantifies the worst-case performance gap between distributed controllers and an oracle with enhanced sensing, defined as $\operatorname{SpReg}_q(\mathbf{Q},\hat{\mathbf{Q}}) = \max_{\left\lVert w \right\rVert_q \le 1} [ J_q(w,\mathbf{Q}) - J_q(w,\hat{\mathbf{Q}}) ]$. It develops a Youla-based synthesis framework that is exact and convex for $q=2$ (via an infinite-dimensional SDP with finite approximations) and provides a scalable LP/ADMM approach for $q=\infty$, enabling large-scale distributed control. The analysis proves well-posedness of the metric when the oracle's information graph contains the plant's and the oracle is optimal under $\mathcal{H}_2$, $\mathcal{H}_\infty$, or $\mathcal{L}_1$ criteria. Numerical experiments on power-grid models show that spatial-regret controllers outperform classical $\mathcal{H}_2$ and $\mathcal{H}_\infty$ controllers, particularly for localized multi-frequency disturbances, demonstrating the practical impact of graph-informed performance metrics on distributed control design.

Abstract

We consider the optimal control of large-scale systems using distributed controllers with a network topology that mirrors the coupling graph between subsystems. In this work, we introduce spatial regret, a graph-informed metric that measures the worst-case performance gap between a distributed controller and an oracle which is assumed to have access to additional sensor information. The oracle's graph is a user-specified augmentation of the available information graph, resulting in a benchmark policy that highlights disturbances for which additional sensor information would significantly improve performance. Minimizing spatial regret yields distributed controllers-respecting the nominal information graph-that emulate the oracle's response to disturbances that are characteristic of large-scale networks, such as localized perturbations. We show that minimizing spatial regret admits a convex reformulation as an infinite program with a finite-dimensional approximation. To scale to large networks, we derive a computable upper bound on the spatial regret metric whose minimization problem can be solved in a distributed way. Numerical experiments on power-system models demonstrate that the resulting controllers mitigate localized disturbances more effectively than controllers optimized using classical metrics.

Figures (6)

• Figure 1: Scheme of an example of a plant $P \in \mathop{\mathrm{\mathtt{NS}}}\limits(\mathop{\mathrm{\mathcal{G}}}\limits)$ controlled by $K \in \mathop{\mathrm{\mathtt{NS}}}\limits(\mathop{\mathrm{\mathcal{G}}}\limits)$, where time dependencies are omitted for clarity.
• Figure 2: Interaction graph of the 16-bus power system model. Blue lines show the undirected edges between buses. Red dashed arrows indicate oracle's additional connections. Green dashed lines highlight the 5-bus subsystem considered for the first example.
• Figure 3: Plot of the squared 2-norm of $\mathbf{F}_{\ell}^{[:,1]}(e^{j \omega})$ as a function of frequency.
• Figure 4: Plot of $\left\lVert z_t\right\rVert_2$ as a function of time for the three controllers with $w_t = \bar{w}_t$ and $w^{[1]}_t =\cos(0.1 t) + \cos (\frac{\pi}{5}t)$.
• Figure 5: Plot of the $\infty$-norm of $\mathbf{F}_{\ell}^{[:,1]}(e^{j \omega})$ as a function of frequency.

Theorems & Definitions (22)

• Definition 1
• Example 1
• Definition 2
• Example 2
• Remark 1
• Remark 2
• Theorem 1
• Lemma 1
• Theorem 2
• proof