Exact Pole Placement for a ZOH-Discretized Extended State Observer: Noise Analysis and a JONSWAP Case Study

Extended state observers (ESOs) are usually tuned in continuous time and then transferred to sampled implementations, where the poles that the running code actually places, and the resulting noise sensitivity, can differ from the values the design intended. This paper examines a third-order ESO that combines exact zero-order-hold state prediction with a bandwidth-parameterized innovation gain vector. Its characteristic polynomial differs from the intended triple-pole polynomial by a defect proportional to the third gain, and the resulting spectral radius returns toward 1 as the requested bandwidth increases, although the observer remains Schur stable at every positive bandwidth. Matching the characteristic polynomial term by term gives a corrected gain vector that places all three poles at the intended location for the same update timing. When the disturbance varies from sample to sample, its increment enters the extended state through a specific input channel, and the resulting estimation lag depends on bandwidth. We then separate three noise measures that are often used interchangeably: the white-noise variance factor, the unit-impulse peak, and the induced gain for bounded input. A closed-loop simulation confirms that the spectral-radius rebound is a genuine disturbance rejection penalty: at high bandwidth the examined estimate alternates in sign from sample to sample, whereas the corrected design stays monotone. Finally, a JONSWAP (Joint North Sea Wave Project) case study uses the physically consistent spectral moment and log-uniform importance weights, and the corrected estimator is unbiased for the band-limited spectral moment that carries almost all of the physical spectral mass in the band used here. All numerical results are produced by a documented Python workflow that is released as an open repository.