Seite - 56 - in Adaptive and Intelligent Temperature Control of Microwave Heating Systems with Multiple Sources

3. ModelingMicrowaveHeating model (equation 3.41) has to be transformed intoN different MISO ARXmodels (AutoRegressivemodelwitheXogenousinputs) [BJR13], suchas Yn(k) =An(k−1)Yn(k−1)+Bn(k−1)U(k−1) =θn(k−1)Πn(k−1), (3.48) θn(k−1) := [ An(k−1),Bn(k−1) ] , = [ An(k−1), Bn,1(k−1), . . .,Bn,M(k−1) ] , (3.49) Πn(k−1) := [ Yn(k−1),UT(k−1) ]T , = [ Yn(k−1), u1(k−1), . . .,uM(k−1) ]T , (3.50) whereYn(k) is the true temperature value at timek,θn(k−1) is the augmentedcoefficientvectorfortheoutputYn(k)withthedimension 1×(1+M), andΠn(k−1) is the augmented data vector with the di- mension(1+M)×1. Accordingtotheassumptionsmadeinequations 3.47, the coefficient vectorθn(k) can also be represented as a varying variable, suchas θn(k) =θn(k−1)+ε(k−1), θn(k−1) =θn(k−2)+ε(k−2), (3.51) where ε is a zero-mean white noise with the covariance matrix [Ω]. The first equation indicates that the future augmented coefficient vec- torθn(k)canbepredictedbasedonthecurrentθn(k−1). Thesecond equationsmeansthat thecurrentθn(k−1)canbeestimatedusingthe former vectorθn(k−2). The definitions of estimation and prediction aswellasotherparametersusedinthelinearsystemidentificationcan befoundintable 3.1. In equation 3.48, bothYn(k)andΠn(k−1)are known at timek. The unknownaugmentedvectorθn(k−1)hastobeestimated, fromwhich bothAn(k−1)andB(k−1)canbeobtainedsimultaneously. Inprac- tice, therealmeasuredtemperaturevalueYnr (k) iswrittenas Ynr (k) = Y n(k)+ ς(k) =θn(k−1)Πn(k−1)+ ς(k), (3.52) 56