Temporal regularization of petal modes in SCAO systems: The Soft-Limiter approach

Extremely large telescopes (ELTs) rely on segmented mirrors and thick support structures that fragment the pupil. In single-conjugate AO (SCAO), this produces differential pistons, or petal modes. The Pyramid Wavefront Sensor (PWFS), baseline for ANDES, MICADO, HARMONI, and GMT-NGSAO, has poor sensitivity to these modes in the visible under realistic residual turbulence. We aim to prevent the "Island Effect", where poorly observed differential pistons accumulate and degrade the PSF, without compromising continuous-mode correction. We propose the Minimum Mean Square Error (MMSE) Soft-Limiter, a deterministic, lightweight controller combining a subspace MMSE estimator derived from von Kármán priors with a continuous temporal regularization. It acts as a leaky integrator applied only to the relative petal degrees of freedom. End-to-end SPECULA simulations of ELT/ANDES show that, while a classical integrator loses lock and wanders across multiple fringes at seeing $\ge 1.0''$, the MMSE Soft-Limiter confines the differential piston within a single fringe over the $0.6-1.2''$ range. It bounds the petal excursion and its science-band impact under severe turbulence, while keeping optimal continuous-mode correction at good seeing, without the Strehl penalty of spatial mitigation such as Edge Actuator Coupling. We derive and verify the operating window for the leakage gain, $g \lesssim g_{\mathrm{eff}}/π$: below it, the fast loop retains genuine physical steps up to $λ_{WFS}/2$ with a small droop. This maintains a stable modulo-$λ_{WFS}$ wavefront on the fast sensor, letting a slower phasing sensor resolve integer ambiguities. As the window narrows with turbulence, its boundaries cross near $1.0''$ seeing; beyond this, retaining physical discontinuities becomes the sole responsibility of the phasing channel, a key design driver for segmented ELTs.