Page - 224 - in Adaptive and Intelligent Temperature Control of Microwave Heating Systems with Multiple Sources

A. Appendix The real measured/calculated ∆Ynr (k) can be approximated by its Taylorseriesat [Λn(k)] = [ Λnp(k) ] , suchas ∆Ynr (k) =Γ T(k) [ Λnp(k) ] Γ(k)+[Jn(k)] [ [Λn(k)]−[Λnp(k)]]+ ς(k) +H.O.T. ≈ΓT(k)[Λnp(k)]Γ(k)+[Jn(k)][[Λn(k)]−[Λnp(k)]]+ ς(k) ≈ΓT(k)[Λnp(k)]Γ(k)+[Jn(k)][enp(k)]+ ς(k) ≈ΓT(k)[Λnp(k)]Γ(k)+[Jn(k)][[ene(k−1)]+[ε(k−1)]] + ς(k), (A.11) whereH.O.T.standsforhigherorder termsand [Jn(k)] is theJacobian matrixdefinedby [Jn(k)] = ∂∆Ynr ∂ [Λn] ∣∣∣∣ [Λn(k)]=[Λnp(k)] = Γ(k)ΓT(k). (A.12) Substituting the above expression A.11 into equation A.10, the esti- mationerrormatrix is further transferred into [ene(k)] = [Λ n(k−1)]+[ε(k−1)]− [Λne(k−1)] − [Kn(k)] [ ∆Ynr (k)−ΓT(k) [ Λnp(k) ] Γ(k) ] = [Λn(k−1)]+[ε(k−1)]− [Λne(k−1)]− ς(k)[Kn(k)] − [Kn(k)][Jn(k)][ [ene(k−1)]+[ε(k−1)]] = [ene(k−1)]+[ε(k−1)]− ς(k)[Kn(k)] − [Kn(k)][Jn(k)][ [ene(k−1)]+[ε(k−1)]] = [ [IM]− [Kn(k)][Jn(k)] ][ [ene(k−1)]+[ε(k−1)] ] − ς(k)[Kn(k)] = [ [IM]− [Kn(k)][Jn(k)] ][ enp(k) ]− ς(k)[Kn(k)], (A.13) where [IM] is the identitymatrixwith thedimensionM×M. 224