Automatic differentiation of Sylvester, Lyapunov, and algebraic Riccati equations
This work provides essential automatic differentiation capabilities for control theorists and engineers working with optimal control and observer design, addressing a current gap in numerical computing frameworks.
This paper addresses the absence of Sylvester, Lyapunov, and algebraic Riccati equation solvers in automatic differentiation libraries by deriving their forward and reverse-mode derivatives. The authors demonstrate the application of these derivatives on an inverse control problem.
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.