Stability of MIMO PID With Backward Differences Under Fast Sampling: An Exact Spectral Criterion

What the math gives to ML

The paper identifies a concrete failure mode in sampled feedback systems: replacing a derivative by a backward difference adds the previous output as a controller state, producing fast modes that are not predicted by the ideal continuous-time loop. The transferable asset is the separation between slow task dynamics and fast discretization-induced dynamics, together with a matrix spectral condition involving the plant input/output Jacobians and derivative gain. This can be used to build neural feedback controllers with finite-difference derivative branches whose gains are explicitly constrained during training, rather than relying on smaller sampling periods or empirical stability checks.