Measure-Valued Automatic Differentiation for Hybrid and Non-Smooth Systems

Muhammad Haris Khan

Keywords: Automatic Differentiation, Hybrid dynamical systems, Finite Radon measure

TL;DR: This paper introduces a mathematical method to precisely compute changes and sensitivities in systems with sudden jumps, improving analysis and control of dynamic events such as bouncing balls and robotic motions.

Abstract: Differentiating hybrid dynamical systems and optimisation layers with jumps poses fundamental challenges for classical automatic differentiation. When the output trajectory has parameter-dependent discontinuities in time or state, the derivative is not an ordinary function. We propose \emph{measure-valued automatic differentiation} (MV-AD), which treats the parameter derivative of the trajectory as a finite Radon measure consisting of an absolutely continuous density on smooth segments and Dirac atoms at event times. MV-AD generalises saltation-style jump sensitivity and obtains accurate gradients without requiring global differentiability. Experiments on bouncing-ball dynamics, a transversality study, a parametric quadratic program, and a queueing model show that MV-AD matches finite differences and analytic gradients (up to $10^{-11}$ versus analytic; $10^{-3}$ relative error versus FD) while scaling linearly with the number of events.

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Submission Number: 18