Gaussian kernel extended state observer–based control for systems with uncertainties and noise

The extended state observer is widely used for states and disturbances estimation in uncertain systems. However, the traditional extended state observer typically assumes that disturbances eventually converge to constants, making it difficult to guarantee theoretical convergence for disturbances with non-zero derivatives. To address this issue, we model the lumped disturbance as a stationary Gaussian process via a Matérn covariance kernel, reformulated as a stochastic differential equation within an extended system. The Gaussian kernel extended state observer is implemented through a Kalman filter, enabling accurate disturbance estimation. Stability is rigorously proven using a Lyapunov function for Itô processes, establishing mean-square exponential practical stability under detectability and stabilizability conditions. Comparative numerical simulations on a permanent magnet synchronous motor model demonstrate that the Gaussian kernel extended state observer largely outperforms the conventional solution in disturbance estimation accuracy and exhibits strong robustness to non-Gaussian noise. It yields smaller observer errors across most states, thereby offering a theoretically rigorous and practical framework for disturbance estimation and compensation in control systems with stationary stochastic disturbances.