Automatic differentiation of Sylvester, Lyapunov, and algebraic Riccati equations

TL;DR

This paper derives the forward and reverse-mode derivatives of solutions to Sylvester, Lyapunov, and algebraic Riccati equations, enabling their use in automatic differentiation frameworks for control applications.

Contribution

It provides the first comprehensive derivation of derivatives for these equations, integrating them into automatic differentiation tools for control theory.

Findings

Derived derivatives enable gradient-based optimization in control problems.

Applied derivatives to inverse control problem demonstrating practical utility.

Enhanced automatic differentiation support for control equations.

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.