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.

中文翻译：

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Sylvester，Lyapunov和代数Riccati方程的自动微分

Sylvester，Lyapunov和代数Riccati方程是控制理论家的头等大事。它们用于计算无限水平的Gramian，解决连续或离散时间内的最佳控制问题以及设计观察器。尽管流行的数值计算框架（例如scipy）为这些方程式提供了有效的求解器，但大多数自动微分库中仍然缺少这些求解器。在这里，我们推导了所有这三种类型方程的解的正向和反向模式导数，并展示了它们在逆控制问题上的应用。