Page - 51 - in Adaptive and Intelligent Temperature Control of Microwave Heating Systems with Multiple Sources

3.2. Grey-boxModeling make sure they can be directly used in the following controller de- sign. For each locationn, the original heat transfer equation 3.12 is converted into two different forms, regarding different approxima- tions. For the case of scalar superposition, the corresponding linear heat transferequation isshownas ρcp dTn dt · l3= −h˜n(Tn)(Tn−Ta) · l2+PnmwU, (3.29) where Tn is the measured temperature at the n-th location, and −h˜n(Tn) is the power dissipation coefficient. Defining the new vari- ableYn=Tn−Ta, theaboveequation is rewrittenas ρcp dYn dt · l3= −hˆn(Yn) ·Yn · l2+PnmwU, (3.30) with hˆn(Yn) =h+%σ′ [ (Yn+Ta) 2+T2a ] ·(Yn+2Ta) (3.31) In microwave heating applications, the temperatures of the load are always changing along time. All temperature dependent variables in equation 3.30, as well as the heating power vector Pnmw and the con- trol input vector U, can be written as time varying variables. There- foreequation 3.30canbeformulatedintoacontinuous-timeequation, suchas dYn(t) dt = −hˆn(t) ρcp l ·Yn(t)+ 1 ρcpl3 ·Pnmw(t)U(t). (3.32) From another point of view, in the above equation all these time- varying parameters, e.g.Yn(t), represent not functions depending on time,but rathervaluesofvectorsormatricesat the time t. Defining Anc(t) := −hˆn(t) ρcp l , Bnc(t) := 1 ρcpl3 ·Pnmw(t) = [ Bn,1c (t), B n,2 c (t), .. . ,B n,M c (t) ] , (3.33) 51