Embedded Model Control and the Error Loop

Drag-Free Control of the GOCE Satellite: Noise and Observer Design

“…That is consistent with (10), (12), and (13), except for the noise in (12), which is simplified to a white noise, being significant only close to . The detail of matrices in (18) is as follows: where , , , is invertible, and and derive from the corresponding block matrices of .…”

“…Then, assuming statistical independence and factoring out the flat spectral density of , the spectral density of the total noise may be expressed in a compact form by (12) Assuming and in (12) implies that is minimal across the mission band . The bowl-shape resulting from (12) is typical of servo-accelerometers; the second derivative of the sensor noise shapes the high-frequency rim, referred to as noise; actuator noise and drift shape the low-frequency side.…”

“…where is the degree of robustness, is the upper bound to the spectral density of the residual CoM acceleration , and and have been defined in (10) and (12), respectively. Inequality (32) is a rather complex expression of the gain in (26), but it can be solved analytically under the following assumptions.…”

“…2) The highest peak of occurs at , implying that may be approximated by the highfrequency asymptote [5], [12], as follows:…”

“…To this end, disturbance spectral densities like (10), (12), and (13) are not converted to weighting functions [7], but to simple stochastic equations to be real-time updated by the estimated innovation [8]. Observable though unpredictable disturbances can thus be explicitly rejected in a timely manner by the control law.…”