Data-Driven Optimal Distributed Controller Synthesis via Spatial Regret

Technical Summary: Data-Driven Optimal Distributed Controller Synthesis via Spatial Regret

Problem Formulation
The paper addresses the challenge of designing optimal distributed controllers for large-scale networked linear time-invariant (LTI) systems under structural communication constraints. Traditional optimal control methods, such as $H_2$ and $H_\infty$ designs, are often "graph-agnostic," treating all disturbances uniformly regardless of their location or the network's communication topology. This approach fails to capture the specific performance degradation caused by limited information flow in distributed settings.

Furthermore, existing methods for minimizing "spatial regret"—a metric quantifying the performance gap between a constrained distributed controller and an "oracle" with enhanced communication capabilities—rely on accurate state-space models. For large-scale systems, obtaining such parametric models is often infeasible or unreliable. The core problem is to synthesize a distributed controller that minimizes spatial regret directly from experimentally obtained frequency-response data, without requiring intermediate system identification, while preserving closed-loop stability and adhering to fixed communication structures.

Methodology
The proposed framework operates entirely in the frequency domain using measured Frequency Response Functions (FRFs). The methodology proceeds as follows:

1. Data Acquisition: The plant's input-output behavior is captured via the FRF, specifically estimating the transfer function $G_{22}$ from persistently exciting input-output data. Other transfer functions ($G_{11}, G_{12}, G_{21}$) are defined by the user to encode performance specifications and disturbance models.
2. Oracle Definition: An "oracle" controller $\hat{K}$ is defined with an enhanced communication topology ($\hat{K} \supseteq K$). This oracle serves as a hypothetical benchmark that can access additional measurements or communication links. The spatial regret is defined as the worst-case performance gap between the constrained controller $K$ and this oracle $\hat{K}$ over bounded disturbances.
3. Frequency-Domain Reformulation: The spatial regret minimization problem is reformulated in the frequency domain. Using Lemma 1, the regret metric is shown to be equivalent to the supremum of the maximum eigenvalue of the difference between the closed-loop transfer function matrices of the constrained controller and the oracle.
4. Iterative Convex Synthesis:
   • The controller is parameterized via a left factorization $K = Y^{-1}X$.
   • The resulting optimization problem is non-convex due to quadratic terms and the requirement for closed-loop stability.
   • The authors propose an iterative algorithm (Theorem 1) that linearizes the non-convex constraints using a convex lower bound derived from a previous stabilizing controller $K_c$.
   • At each iteration, a convex optimization problem (a semi-infinite program solved via frequency gridding) is solved to update $X$ and $Y$.
   • Crucially, if initialized with a stabilizing controller, the algorithm guarantees that the synthesized controller remains stabilizing at every iteration.

Key Contributions
The paper makes three primary contributions:

1. Extension of Well-Posedness: The authors extend the well-posedness results of spatial regret (previously established for specific controller classes) to arbitrary controller classes. They prove that if the oracle is designed to minimize a classical norm ($H_2$ or $H_\infty$) over the enhanced constraint set, the spatial regret metric is guaranteed to be non-negative for any constrained controller, ensuring the benchmark is valid.
2. Data-Driven Synthesis Framework: The paper recasts spatial regret minimization within a data-driven frequency-domain framework. This removes the dependency on parametric system identification, allowing controller synthesis directly from frequency-response measurements.
3. Stability-Preserving Algorithm: A tractable iterative convex algorithm is developed that synthesizes fixed-structure distributed controllers. The method ensures closed-loop stability throughout the iterations, provided the initialization is stabilizing.

Results
The methodology was validated through numerical experiments on a 5-bus power grid model with swing dynamics.

• Setup: The distributed controller was restricted to nearest-neighbor communication with a one-step delay. The oracle was granted immediate access to measurements from a specific bus (Bus 1) prone to disturbances.
• Comparison: The spatial regret controller ($K_{SR}$) was compared against classical $H_2$ ($K_2$) and $H_\infty$ ($K_\infty$) distributed controllers.
• Performance:
  • In the frequency domain, $K_{SR}$ achieved a favorable trade-off between the average performance of $K_2$ and the robustness of $K_\infty$ near resonance peaks.
  • For disturbances localized at the specific node (Bus 1), $K_{SR}$ closely mimicked the oracle's performance, significantly outperforming both $K_2$ and $K_\infty$.
  • Time-domain simulations with multi-sine disturbances on Bus 1 showed that $K_{SR}$ reduced the average regulated output energy by 21.72% relative to $K_2$ and 48.00% relative to $K_\infty$.

Significance and Claims
The paper claims that spatial regret provides a more nuanced performance metric for distributed control by explicitly accounting for the cost of communication constraints and disturbance localization. The proposed data-driven framework is significant because it enables the synthesis of such optimal controllers without the need for accurate parametric models, which are often unavailable for large-scale systems.

The authors acknowledge limitations: the generalization to arbitrary controller structures necessitates an iterative solution rather than a single convex program, and the method does not guarantee completeness over the set of all stabilizing controllers (it explores a subset defined by the left factorization). Additionally, the computational complexity scales with network size, suggesting that distributed implementations of the optimization problem remain an area for future research. The work builds upon previous data-driven control frameworks (e.g., Schuchert et al., 2024; Gupta et al., 2026) and extends the spatial regret concept (Martinelli et al., 2025) to a model-free setting.