[Paper Review] A Fast Algorithm for Sparse Controller Design

Matt Wytock, J. Zico Kolter|arXiv (Cornell University)|Dec 17, 2013

Control Systems and Identification35 references18 citations

TL;DR

This paper proposes a fast proximal Newton method with coordinate descent and efficient Lyapunov solve techniques for sparse controller design in large-scale systems. By reducing per-iteration cost through O(n) Lyapunov solutions via numerical linear algebra tricks, the algorithm achieves convergence orders of magnitude faster than prior methods while maintaining high accuracy on systems with thousands of states, including real power networks.

ABSTRACT

We consider the task of designing sparse control laws for large-scale systems by directly minimizing an infinite horizon quadratic cost with an $\ell_1$ penalty on the feedback controller gains. Our focus is on an improved algorithm that allows us to scale to large systems (i.e. those where sparsity is most useful) with convergence times that are several orders of magnitude faster than existing algorithms. In particular, we develop an efficient proximal Newton method which minimizes per-iteration cost with a coordinate descent active set approach and fast numerical solutions to the Lyapunov equations. Experimentally we demonstrate the appeal of this approach on synthetic examples and real power networks significantly larger than those previously considered in the literature.

Motivation & Objective

• Address the lack of scalable algorithms for sparse optimal control in large-scale systems with thousands of states.
• Overcome the O(n³) per-iteration cost of Newton methods in sparse controller design by accelerating Lyapunov equation solves.
• Develop a practical, high-precision algorithm that enables sparse controller design for real-world systems like power grids.
• Achieve convergence speeds orders of magnitude faster than existing first-order or SDP-based methods.

Proposed method

• Uses a proximal Newton (Newton-Lasso) framework to minimize a smooth H2 cost plus ℓ1-norm penalty on controller gains.
• Employs coordinate descent on the active set to solve each regularized Newton step efficiently.
• Applies precomputations and transformations to reduce Lyapunov solve cost from O(n³) to O(n) per inner product.
• Leverages numerical methods including the fast multipole method and Autonne-Kakagami factorization for low-rank structure exploitation.
• Solves the Lyapunov equations arising in Hessian computation using structure-aware factorizations and matrix decompositions.
• Integrates these components into a globally convergent second-order method for ℓ1-regularized optimal control.

Experimental results

Research questions

• RQ1 Can a second-order method be made practical for large-scale sparse controller design despite O(n³) Lyapunov solves?
• RQ2 Can coordinate descent on the active set reduce the cost of solving inner Newton steps in ℓ1-regularized optimal control?
• RQ3 Can O(n) Lyapunov solves be achieved through matrix structure exploitation and numerical linear algebra techniques?
• RQ4 How does the proposed algorithm compare in speed and accuracy to first-order methods like ADMM and ISTA on real-world power systems?
• RQ5 To what extent can sparse controllers achieve near-optimal performance in large-scale systems like power networks?

Key findings

• The proposed Newton-CD algorithm converges to accuracy better than 10⁻⁸ in less than 173 seconds on the 500-state PST 50 machine system, while ADMM remains above 10³ error after over an hour.
• On the IEEE 145 Bus system with PSS, Newton-CD achieved convergence in under 173 seconds, whereas ADMM and ISTA showed significantly slower convergence and less stable sparsity patterns.
• For all tested power systems, including the NPCC 140 Bus and NE 39 Bus, the algorithm found stable controllers with performance within 0.1% of LQR while remaining highly sparse.
• The sparsest stable controllers found were often local or near-local, depending only on a few neighboring states, demonstrating practical decentralization.
• The method achieved orders of magnitude speedup over ADMM and ISTA across all benchmarks, with convergence differences of up to 10⁴× in runtime.
• The algorithm successfully handled systems with up to 500 states, significantly larger than previous studies, and maintained high numerical precision throughout.