Page - 55 - in Adaptive and Intelligent Temperature Control of Microwave Heating Systems with Multiple Sources

3.2. Grey-boxModeling and the system can be controlled accordingly. In order to get a good controlperformancefor the temperaturecontrol systemofHEPHAIS- TOS, thesystemidentificationalgorithmusedinourcaseneedstoful- fill the followingrequirements: • Implementation inarecursiveoronline form. • Abilityofdealingwithhigh-dimensional systems. • Abilityof trackingtime-varyingbehaviors. • Guaranteedconvergenceandfast convergingspeed. According to these four requirements, the exponentially weighted recursive least squares (RLS) [Lju98], recursive Kalman filter (RKF) [Lju98] and extended Kalman filter (EKF) [Lju98] [WVDM00] algo- rithms were selected as the system identification algorithms, with re- spect to the twodifferentmodels (equations 3.41and 3.46). LinearRecursiveSystemIdentification The taskof the linear recursivesystem identification is to estimate the valueofmatrices [A(k)]and [B(k)] inaonline(recursive) form. Inthe linear model (equation 3.41), both matrices are assumed to be time- varying. Since there is no prior knowledge indicating the varying trend of [A(k)] or [B(k)], the best choice is to assume both matrices fulfill the followingequations [ZL03] [A(k)] = [A(k−1)]+[ε1(k−1)] , [B(k)] = [B(k−1)]+[ε2(k−1)] . (3.47) Both [ε1(k−1)] and [ε2(k−1)] are zero-mean white noise added at time k− 1, and their covariance matrices are donated as [Ω1] and [Ω2] respectively. At the current time k, the matrices [A(k−1)] and [B(k−1)]canbeestimatedusingthecurrenttemperaturevectorY(k) andhistoricaldatasets { U(k−1),Y(k−1),U(k−2),Y(k−2), .. .}. Then the future matrices [A(k)] and [B(k)] can be predicted using theseestimations. The system identification process applied here is done in a multiple- input single output (MISO) way. In the beginning, the linear MIMO 55