Page - 62 - in Adaptive and Intelligent Temperature Control of Microwave Heating Systems with Multiple Sources

3. ModelingMicrowaveHeating of both RLS and RKF is compared in real experiments, which will be presented inchapter 4. NonlinearRecursiveSystemIdentification Thetaskofnonlinearrecursivesystemidentification is toestimate the matrices [An(k−1)] and [Φn(k−1)] in equation 3.46. Compared with the linear case, the nonlinear system identification process is more complicated because of the nonlinear heating term Ψ. There is no way to combine the parameterAn(k−1) and the effective heating matrix [Φn(k−1)] inonematrix (as in thevectorθn(k)), andestimate them at the same time. Instead, in the nonlinear system identifica- tion, a two-step identification procedure has been developed, to es- timate them separately at different time using the extended Kalman filter (EKF)method. ThisestimationisalsoappliedontheMISO-form model,which issimilaras in the linearcase. • Step1: Estimate thefirstpart [A(k−1)] This step is done during the cooling part or whenever no power is injected from the microwave sources (vm(k) = 0 for all 1≤m≤ M). Since the input vector is zero, the nonlinear MIMO system model (equation 3.46) can be decomposed intoNMISO systems, suchas Yn(k) =An(k−1)Yn(k−1), 1≤n≤N. (3.62) TheparameterAn(k−1)canbeestimateddirectlyas An(k−1) =Yn(k)/Yn(k−1), 1≤n≤N. (3.63) • Step2: Estimate theeffectiveheatingmatrix [Φn(k−1)] WhentheinputvectorV isnotzero, thestateparameterAn(k−1) isassumedtobeconstantlikeAn(k−1) =Anct,whereAnct is thelast updated value ofAn. This assumption is valid in practice because the parameterAn(k−1) is mainly determined by the convection heattransfercoefficienth (seeequation 3.15andthecomparisonin equation 3.14) which is a constant. The varying part ofAn(k−1) 62