Page - 65 - in Adaptive and Intelligent Temperature Control of Microwave Heating Systems with Multiple Sources

3.3. Black-boxModeling Definition Interpretation [Φne(k−1)] = E [ [Φn(k−1)]∣∣Ψnr(k−1)] Estimationof [Φn(k−1)] basedonthecurrent measuredtemperature Ψnr(k−1) [ Φnp(k−1) ] = E [ [Φn(k−1)]∣∣Ψnr(k−2)] Predictionof [Φn(k−1)] basedonthe former measuredtemperature Ψnr(k−2) [ene(k−1)] = [Φn(k−1)]− [Φne(k−1)] Estimationerrorbetween [Φn(k−1)]and [Φne(k−1)][ enp(k−1) ] = [Φn(k−1)]−[Φnp(k−1)] Predictionerrorbetween[Φn(k−1)]and [Φnp(k−1)] [Pne(k−1)] = E [ [ene(k−1)][ene(k−1)]T ] Covariancematrixof the estimationerror [ene(k−1)][ Pnp(k−1) ] = E [[ enp(k−1) ][ enp(k−1) ]T] Covariancematrixof thepredictionerror[enp(k−1)] Table3.2. ParametersandvectorsusedinEKF is from the input or observed space to an intermediate regressor vec- tor [Bri12], and the second layer mapping is from the regressor vector to theoutputspace. Fromamathematicalpointofview, thetwo-layer mapping can be also considered as a basis function expansion, which extract theinformationfromtheoriginalsystemandrepresentitusing differentcombinationsofbasis functions. Theblack-boxmodelingmethodusedhereistheneuralnetwork(NN) approach [JMM96]. In the temperature control system of HEPHAIS- TOS, two different neural networks are applied, functioning as the system estimator and the controller respectively. The NN system es- 65