Page - 63 - in Adaptive and Intelligent Temperature Control of Microwave Heating Systems with Multiple Sources

3.2. Grey-boxModeling caused by different thermal radiation effects can be omitted, es- pecially when the temperature is in a stable range. Then then-th MISOsystemscanbewrittenas Yn(k) =Anct ·Yn(k−1)+Ψn(k−1), Ψn(k−1) =VT(k−1)[Φn(k−1)]V(k−1). (3.64) Atanytimek, thevalueofΨn(k−1)canbecalculatedby Ψn(k−1) =Yn(k)−Anct ·Yn(k−1), (3.65) and then the effective heating matrix [Φn(k−1)] can be estimated accordingto thevalueofΨn(k−1). As inthelinearRKF(equation 3.59), inEKFit isalsoassumedthat thesystemfulfills the followingequations Ψnr(k−1) =VT(k−1)[Φn(k−1)]V(k−1)+ ς(k−1), [Φn(k−1)] = [Φn(k−2)]+[ε(k−1)], (3.66) whereΨnr(k−1) is therealcalculatedvalueofΨn(k−1), ς(k−1) is azero-meanmeasurementorcalculationerrorwiththecovariance σ2 and [ε(k−1)] is a zero-mean white noise with the covariance matrix [Ω]. The definitions of parameters and vectors used in EKF isshownbytable 3.2. The detailed derivation process is enclosed in the appendix A.1. TheupdateruleofEKFisgivenas following. Predictionpartwithk≥2 :[ Φnp(k−1) ] = [Φne(k−2)] ,[ Pnp(k−1) ] = [Pne(k−2)]+[Ω]. (3.67) 63