Differentially Private Probabilistic Active Disturbance Rejection Control with Uncertainty-Calibrated Extended State Observers

Active disturbance rejection control (ADRC) is attractive because it estimates and compensates a lumped “total disturbance” with limited plant information, but privacy-sensitive networked deployment, measurement-noise amplification, and actuator saturation remain insufficiently addressed together. This paper proposes a Differentially Private Probabilistic ADRC (DP-PADRC) framework for nonlinear SISO systems under saturation. In contrast to adaptive ADRC schemes that schedule gains from raw residuals, and unlike model-based differentially private filters that rely on explicit stochastic plant models, the proposed method combines a linear ESO with a lightweight uncertainty surrogate computed from clipped and privatized innovations. The resulting controller is not Bayesian; rather, it is probabilistic in the sense that second-moment information from the released innovation stream is explicitly used to calibrate observer bandwidth and disturbance compensation. We further incorporate a saturation-aware gate so that scheduling remains well behaved when the commanded and applied inputs differ. An ISS-type mean-square bound is derived for the closed loop, making the dependence on the disturbance derivative, measurement-noise variance, clipping level, and privacy parameters (ε,δ) explicit. We also discuss the composition of privacy loss across repeated tuning windows and quantify the privacy-induced perturbation of the scheduling signal. Simulation-based nonlinear servo benchmarks show improved tracking/noise robustness over fixed-gain LADRC and a nonlinear ADRC baseline, while clarifying the privacy–performance trade-off and the scope of the method.