Page - 227 - in Adaptive and Intelligent Temperature Control of Microwave Heating Systems with Multiple Sources

A.2. DerivationofBackpropagationAlgorithm Replacing the new defined variables by the original variables, the co- variancematricesoftheestimationandthepredictionerrorsare [Pne(k−1)] =E [( [Φn(k−1)]− [Φne(k−1)] )( [Φn(k−1)]− [Φne(k−1)] )T] [ Pnp(k−1) ] =E [( [Φn(k−1)]−[Φnp(k−1)])([Φn(k−1)]−[Φnp(k−1)])T] . (A.18) Then at each timek, the effective heating matrix [Φn] can be updated as following. Predictionpart: [ Φnp(k−1) ] = [Φne(k−2)] ,[ Pnp(k−1) ] = [Pne(k−2)]+[Ω]. Estimationpart: [Kn(k−1)] = [Pnp(k−1)][Jn(k−1)]T([Jn(k−1)][Pnp(k−1)] [Jn(k−1)]T+σ2 [IM] )−1 , [Φne(k−1)] = [ Φnp(k−1) ] +[Kn(k−1)] ( Ψnr(k−1) −VT(k−1)[Φnp(k−1)]V(k−1)), [Pne(k−1)] = ( [IM]− [Kn(k−1)][Jn(k−1)] )[ Pnp(k−1) ] with [Jn(k−1)] = V(k−1)VT(k−1). 227