Kalman filtering to suppress spurious signals in Adaptive Optics control Page: 7 of 54
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incorporate the closed-loop DM correction.
Given (1) and (2), we simply specify the variables and values, and use the Algebraic
Riccati Equation and the state equation to solve for the controller. (See, for example,
Sections 2 and 3 of Poyneer and Vran .)
In the case of colored measurement noise, we begin with the regular AO model
state equation, which models common path aberrations, now with slightly different
xc[t + 1] = Acxc[t] + Bcwc[t]. (3)
Now the measurement equation, instead of having white noise v[t] has colored noise
y[t] = Ccxc[t] + Du[t] + z[t]. (4)
In order to use the Kalman equations on this model, we need to convert it to one
where the measurement noise is white. We do this by expressing the colored noise z[t]
with its own state space model.
n[t + 1] = An[t] + Br[t], (5)
z[t] = Cn[t] +v[t]. (6)
In this model v[t] remains the temporally white measurement noise of the original
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Poyneer, L & Veran, J P. Kalman filtering to suppress spurious signals in Adaptive Optics control, article, March 29, 2010; Livermore, California. (digital.library.unt.edu/ark:/67531/metadc835045/m1/7/: accessed January 17, 2019), University of North Texas Libraries, Digital Library, digital.library.unt.edu; crediting UNT Libraries Government Documents Department.