Page - 60 - in Adaptive and Intelligent Temperature Control of Microwave Heating Systems with Multiple Sources

3. ModelingMicrowaveHeating error and then the estimationθne will jump away from the current statequickly. In thisdissertationr= 100 isused. • RecursiveKalmanfilter (RKF) Kalman filter [Rib04] is another powerful algorithm that has been widely implemented in adaptive filtering, adaptive control and systemidentificationareas. InRKFthesystemisassumedthesame as in theRLSalgorithmas Ynr (k) = θ n(k−1)Πn(k−1)+ ς(k), (3.58) θn(k−1) = θn(k−2)+ε(k−2), (3.59) where ς(k) is themeasurementerroras inequation 3.52andε(k− 2) is the white noise as in equation 3.51. This assumption also providesRKFtheability to track time-varyingparameters. ThecompleteupdateprocessofRKFcanbedividedintotwoparts. The first part is the prediction of the unknown vectors at the cur- renttimek,basedonthedataobtainedattheformertimestepk−1, asgiven in [Rib04]withk≥2 θnp(k−1) =θne(k−2),[ Pnp(k−1) ] = [Pne(k−2)]+[Ω]. (3.60) The first equation in 3.60 indicates that the predicted coefficient vectorθp(k−1)based on the old data is the equivalent to the pre- vious estimation θe(k−2). The influence of the unknown noise variableε(k−1) is reflected in the second equation, where the co- variance matrix of prediction errorPp(k) equals to the covariance matrix of the previous estimation error Pe(k−1) plus the covari- ancematrixof thenoiseΩ. The second part is called estimation, which is to estimate the cur- rent variables using the currently measured temperatures and for- 60 merpredictions, suchasgiven in [Rib04]withk≥2