Page - 223 - in Adaptive and Intelligent Temperature Control of Microwave Heating Systems with Multiple Sources

A.1. DerivationofExtendedKalmanFilter Thenthepredictionerrormatrix [ enp(k) ] canberewrittenas[ enp(k) ] = [Λn(k)]−[Λnp(k)] = [Λn(k)]− [Λne(k−1)] = [Λn(k−1)]+[ε(k−1)]− [Λne(k−1)] = [Λn(k−1)]− [Λne(k−1)]+[ε(k−1)] = [ene(k−1)]+[ε(k−1)] (A.7) Based on this result, the covariance matrix [ Pnp(k) ] can also be trans- ferred intoanother form,suchas[ Pnp(k) ] =E [[ enp(k) ][ enp(k) ]T] =E [( [ene(k−1)]+[ε(k−1)] )( [ene(k−1)]+[ε(k−1)] )T] =E [ [ene(k−1)][ene(k−1)]T+[ε(k−1)][ε(k−1)]T ] = [Pne(k−1)]+[Ω] (A.8) Asrepresentedbyequation 3.61, theupdateequationof [Λne(k)] inthe recursivesystemidentification is representedby [Λne(k)] = [ Λnp(k) ] +[Kn(k)] [ ∆Ynr (k)−ΓT(k) [ Λnp(k) ] Γ(k) ] (A.9) Substitutingtheaboveexpression into thedefinitionof [ene(k)], thees- timationerrormatrixcanbewrittenas [ene(k)] = [Λ n(k)]− [Λne(k)] = [Λn(k−1)]+[ε(k−1)]− [Λne(k)] = [Λn(k−1)]+[ε(k−1)]−[Λnp(k)] − [Kn(k)] [ ∆Ynr (k)−ΓT(k) [ Λnp(k) ] Γ(k) ] = [Λn(k−1)]+[ε(k−1)]− [Λne(k−1)] − [Kn(k)] [ ∆Ynr (k)−ΓT(k) [ Λnp(k) ] Γ(k) ] (A.10) 223