Page - 54 - in Adaptive and Intelligent Temperature Control of Microwave Heating Systems with Multiple Sources

3. ModelingMicrowaveHeating where Y(k) and A(k) are the same as defined in the linear model (equation 3.41). The vector Ψ is a augmented column vector which isdefinedas Ψ(k) := [ Ψ1(k),Ψ2(k), .. . ,ΨN(k) ]T , Ψn(k) := ∆t ρcpl3 ·VT(k)[Φnc(k)]V(k), 1≤n≤N. (3.43) Inordertohaveamoreclearnotation, thediscrete-timeeffectiveheat- ingmatrix [Φn(k)] isdefinedas [Φn(k)] := ∆t ρcpl3 · [Φnc(k)], (3.44) which leads to Ψn(k) = VT(k)[Φn(k)]V(k). (3.45) Similar to equation 3.41, the current temperature vector (equation 3.42) is representedas Y(k) = [A(k−1)] Y(k−1)+Ψ(k−1), (3.46) whichwillbeusedfor thenonlinearsystemidentification. Thegrey-boxmodelingapproachstartsfromtheverybasicheattrans- fer model, combining processes represented by equation 3.12. It ne- glects minor influences (thermal conduction) and approximates com- plex heating terms with computable expressions. In the end, two discrete-time models (equations 3.41 and 3.46) are generated. If all unknown matrices in the model, e.g. the matrix [A(k)], are accurately estimated, the control input vector U(k) or V(k) can be calculated to control thesystemforachievingthedesiredtarget. 3.2.4. OnlineSystemIdentification All unknown parameters and matrices in a system model can be esti- mated using different system identification algorithms. With the esti- matedparameters, thefuturebehaviorsofthesystemcanbepredicted 54