[Paper Review] Efficient Learning of Distributed Linear-Quadratic Controllers

Salar Fattahi, Nikolai Matni+1 more|arXiv (Cornell University)|Sep 21, 2019

Advanced Control Systems Optimization76 references8 citations

TL;DR

This paper proposes a robust, scalable method for learning distributed linear-quadratic controllers in unknown but sparse linear time-invariant systems. By combining structured system identification via Lasso with the System Level Synthesis framework, it achieves sub-linear sample complexity and near-linear time complexity, guaranteeing stability and near-optimal performance with minimal data—demonstrated to match oracle performance using only ~150 samples for a 40-node system.

ABSTRACT

In this work, we propose a robust approach to design distributed controllers for unknown-but-sparse linear and time-invariant systems. By leveraging modern techniques in distributed controller synthesis and structured linear inverse problems as applied to system identification, we show that near-optimal distributed controllers can be learned with sub-linear sample complexity and computed with near-linear time complexity, both measured with respect to the dimension of the system. In particular, we provide sharp end-to-end guarantees on the stability and the performance of the designed distributed controller and prove that for sparse systems, the number of samples needed to guarantee robust and near optimal performance of the designed controller can be significantly smaller than the dimension of the system. Finally, we show that the proposed optimization problem can be solved to global optimality with near-linear time complexity by iteratively solving a series of small quadratic programs.

Motivation & Objective

• To design distributed controllers for large-scale, unknown, sparse linear time-invariant systems with guaranteed stability and performance.
• To reduce the sample complexity required for learning optimal controllers below the system dimension, especially in sparse systems.
• To ensure computational efficiency by achieving near-linear time complexity in system size.
• To provide end-to-end theoretical guarantees on stability, robustness, and performance gap under model uncertainty.
• To enable practical deployment in safety-critical infrastructure such as smart grids and transportation networks.

Proposed method

• Leverages the System Level Synthesis (SLS) framework to parameterize distributed controllers with localized structure.
• Uses a Lasso-based regularized estimator to identify sparse system matrices $A_\star$ and $B_\star$ from a single trajectory of system data.
• Applies a bootstrap-based method (Algorithm 2) to compute tight, data-driven upper bounds on estimation error.
• Designs a robust distributed controller via Algorithm 1 that accounts for model uncertainty and guarantees stability.
• Employs a finite impulse response (FIR) structure for controller implementation to ensure practicality and localizability.
• Solves the resulting optimization problem by iteratively solving small quadratic programs, enabling near-linear time complexity.

Experimental results

Research questions

• RQ1 Can distributed controllers be learned efficiently in unknown, sparse linear systems with provable stability and performance guarantees?
• RQ2 What is the minimum number of samples required to learn a distributed controller with near-optimal performance?
• RQ3 Can the computational complexity of distributed controller synthesis be made near-linear in system dimension?
• RQ4 How does the choice of FIR length affect the performance and robustness of the learned controller?
• RQ5 Can bootstrapped estimation error bounds reliably bound the true model error in practice?

Key findings

• The proposed method achieves sub-linear sample complexity: only ~150 samples were sufficient to match the oracle controller cost in a 40-node system, approximately equal to the number of non-zero elements in the system matrices.
• The end-to-end controller performance closely matches the oracle cost, even with limited data, demonstrating near-optimality.
• The runtime scales empirically as $\mathcal{O}(n^{1.004})$, confirming near-linear time complexity, and the algorithm is highly parallelizable.
• Increasing FIR length from 4 to 8 improves performance significantly, but further increases (8 to 12) yield only marginal gains, indicating $L=8$ is a practical choice.
• The bootstrapped estimation error reliably bounds the true estimation error, validating the uncertainty quantification method.
• The method guarantees stability and near-optimal performance for the true system, even when the system matrices are unknown and only estimated from data.