State variable observer theory estimates

Do not be confused with the unknown inputs observers theory in which the goal is to estimate state variables of a system subjected to unmeasured inputs. Here, the goal is precisely to estimate these unknown inputs and not only certain state variables. 

Modem system theory [43] offers mathematical methods, which—on the basis of these variables together with thermodynamics, chemical kinetics and a reactor model—enable the estimation of additional state variables (e.g. Kalman-Bucy filter, Luenberger observer). The more accurate the process model which is assumed as a base, and the more numerous and meaningful the starting measured variables are, the greater is the number and accuracy of the additional, estimated state variables. Consequently, variables which are usually available (e.g. temperature, pressure) should, if possible, be supplemented by further variables which are more meaningful. 

Because the time scale of the anticipated atmospheric motion associated with the temperature fluctuations is only one day, the thermal wind approximation caimot be used. However, diurnal variations in the pressure and wind fields can be estimated from the observed temperature field using classical tidal theory. The basic concept of the formulation is sketched here a detailed treatment can be found in Chapman Lindzen (1970). The theory is based on a linearization of the primative equations. A motionless atmospheric reference state is assumed with temperature profile To z) and a corresponding geopotential surface 4>o(z). It is further assumed that the diabatic heating and all other quantities vary as exp[i(j < — cot)] where s is a longitudinal wavenumber and co is 2 7r/(solar day) or integer multiples thereof. The amplitudes of the time-varying, dependent variables are taken to be sufficiently small so that only terms of first order need be retained. With these assumptions. 

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