Automatic differentiation of Sylvester, Lyapunov, and algebraic Riccati equations

2020/11/23 by Ta-Chu Kao, Kao, Ta-Chu, Guillaume Hennequin +1
Computer Science · Engineering · Physics and Astronomy · #Control Systems and Identification #FOS: Computer and information sciences #FOS: Mathematics #Machine Learning (cs.LG) #Mathematical Software (cs.MS) #Model Reduction and Neural Networks #Numerical Methods and Algorithms #Optimization and Control (math.OC)

paper · pdf · doi:10.48550/arxiv.2011.11430

openalex publication_date 2020/11/23 · openalex created_date 2020/12/07 · openalex updated_date 2026/07/28

Abstract

Sylvester, Lyapunov, and algebraic Riccati equations are the bread and butter of control theorists. They are used to compute infinite-horizon Gramians, solve optimal control problems in continuous or discrete time, and design observers. While popular numerical computing frameworks (e.g., scipy) provide efficient solvers for these equations, these solvers are still largely missing from most automatic differentiation libraries. Here, we derive the forward and reverse-mode derivatives of the solutions to all three types of equations, and showcase their application on an inverse control problem.

Citations

• On the limited memory BFGS method for large scale optimization
• Owl: A General-Purpose Numerical Library in OCaml
• PyTorch: An Imperative Style, High-Performance Deep Learning Library

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