Page - 221 - in Adaptive and Intelligent Temperature Control of Microwave Heating Systems with Multiple Sources

A. Appendix A.1. DerivationofExtendedKalmanFilter When the input vectorV(k)is not zero, the state parameterAn(k−1) is assumed to be constant likeAn(k−1) =Anct, whereAnct is the last updated value ofAn. Then then-th MISO systems can be written as suchas Yn(k) =Anct ·Yn(k−1)+Ψn(k−1), Ψn(k−1) =VT(k−1)[Φn(k−1)]V(k−1). (A.1) Atanytimek, thevalueofΨn(k−1)canbecalculatedby Ψn(k−1) =Yn(k)−Anct ·Yn(k−1), (A.2) and then this value is used to estimate [Φn(k−1)] in equations A.1. In order to make the derivation process more intuitive to understand, theoriginalvariablesV(k−1),Ψn(k−1)and [Φn(k−1)]arereplaced bynewnotationsas Γ(k) =V(k−1), ∆Yn(k) = Ψn(k−1), [Λn(k)] = [Φn(k−1)]. (A.3) Thenthesecondequation in A.1canberewrittenas ∆Yn(k) =ΓT(k)[Λn(k)]Γ(k). (A.4) AsinthelinearRKF(equation 3.59), inEKFit isalsoassumedthatthe systemfulfills the followingequations ∆Ynr (k) =Γ T(k)[Λn(k)]Γ(k)+ ς(k), [Λn(k)] = [Λn(k−1)]+[ε(k−1)], (A.5) 221