Page - 59 - in Adaptive and Intelligent Temperature Control of Microwave Heating Systems with Multiple Sources

3.2. Grey-boxModeling • ExponentiallyWeightedRecursiveLeastSquares (RLS) The exponentially weighted recursive least squares (RLS) algo- rithm is one of the most used algorithms in adaptive filtering and system identification [Li08], for tracking time-varying parameters. Given all observations ( Yn(i),Πn(i) ) from the beginning (i= 1) to the current time (i= k), the cost functionJRLS(k) of the expo- nentiallyweightedRLSisdefinedas [Lju98] JRLS(k) = 1 k−1 k∑ i=2 λk−i ( Ynr (i)−θne(i−1)Πn(i−1) )2 , (3.55) whereλ isa forgettingfactorwith0<λ≤1. Acost function is the functiondefinedto beminimized. Themini- mizationof thecost functionleadstotheoptimalcoefficientvector θne , suchas θne(k−1) = argθminJRLS(k). (3.56) Theinvolvementofλ indicatesthattheabovecostfunctionassigns more credits to recent data than old data, endowing the exponen- tially weighted RLS the ability of tracking time-varying systems. Thesmallerλ is, the faster it forgetsolddata. The detailed derivation process of the estimation θne(k) can be found in [WP97]. The final update equations of the exponentially weightedRLSwithk≥2aregivenas [Pol03] Kn(k−1) = [Pne(k−2)]Π(k−1) [ λσ2 +ΠT(k−1)[Pne(k−2)]Π(k−1) ]−1 , θne(k−1) = θne(k−2)+Kn(k−1) [ Ynr (k)−Yne (k) ] , [Pne(k−1)] = 1 λ [ 1−Kn(k−1)Π(k−1)][Pne(k−2)] . (3.57) In practice, the vector θne(0) is randomly initialized. and the co- variance matrix [Pne ] is initialized as [Pne(0)] = r [I1+M], wherer is arealnumber[Lju98]. Thevalueofrdenotesthelevelof theinitial estimationerror. Forexample,alarger indicatesalargeestimation 59