Page - 61 - in Adaptive and Intelligent Temperature Control of Microwave Heating Systems with Multiple Sources

3.2. Grey-boxModeling Kn(k−1) = [Pnp(k−1)]Π(k−1)[σ2 +ΠT(k−1)[Pnp(k−1)]Π(k−1)]−1, θne(k−1) =θnp(k−1)+Kn(k−1) [ Ynr (k)−θnp(k−1)Π(k−1) ] , [Pne(k−1)] = [ 1−Kn(k−1)Π(k−1)][Pnp(k−1)]. (3.61) Thevectorsupdatedfromthepredictionpart (equations 3.60)can be directly used in the estimation part (equations 3.61). After the estimation, the estimated coefficient vector θne(k−1) can be em- ployed to predict future temperature valueYn(k+ 1) because of θnp(k) =θ n e(k−1). Both [Ω]andσ2 canbeeitherpredefinedbythe user or estimated online using other approaches [A˚JPJ08]. When the covariances of the noise signals are perfectly known, RKF is guaranteedtoprovidetheoptimalestimation[Rib04]. But inprac- tice, this condition is not always fulfilled. Normally the covari- ances [Ω] and σ2 are used as the tuning elements of RKF to ad- just the estimation properties. Detailed derivation process of RKF refers to [WVDM00]or [Rib04]. It should be noted that, ifλ is set to be 1 and the matrix [Ω] is set to be a zero matrix, the two results (equations 3.55 and 3.61) would be exactly the same. In other words, the RLS solution coincidis es with theRKFsolutionwhenthecoefficientvectorθn(k) isdeterministicand thecost functionJRLS(k) (equation 3.55) isnon-weighted. As mentioned in the beginning of this section, the solutions of RLS and RKF given by equations 3.55 and 3.61 are expressed in a MISO form. For a MIMO model that containsN different measured tem- peratures, the update process has to be repeated forN times per esti- mationperiod. Analternative to theMISO-formsystemidentification is to replace equation 3.48 by the MIMO equation 3.41, and directly applyRLSorRKFtotheMIMOARXmodel. Inthiscase,onlyoneup- date process is needed per estimation period. But the problem is that the state matrix [A(k−1)] resulted from the MIMO update approach is not diagonal, which is against equation 3.39. Correspondingly the estimation result of the MIMO-form system identification is signifi- cantly different from the result of the MISO-form. The performance 61