Page - 73 - in Adaptive and Intelligent Temperature Control of Microwave Heating Systems with Multiple Sources

3.3. Black-boxModeling 3.3.2. NeuralNetworkModelingofHEPHAISTOS In this section, the neural network modeling of HEPHAISTOS is ex- plained. In the beginning, the fundamental principles of supervised learning are introduced. Then the neural network structures used in HEPHAISTOS are presented. In the end, the learning (training) algo- rithmsusedin thisdissertationarediscussed. Fundamentals inSupervisedLearning Thetrainingdataset is representedby D = {( UNN(1),Yd(1) ) , ( UNN(2),Yd(2) ) , .. . , ( UNN(Q),Yd(Q) )} , UNN(q) = [ uNN,1(q), uNN,2(q), . . ., uNN,M(q) ]T , 1≤ q≤Q, Yd(q) = [ Yd,1(q), Yd,2(q), . . ., Yd,N(q) ]T , 1≤ q≤Q. The numberQ indicates the size of the training data set D. The vector UNN(q) is the input vector and Yd(q) is the corresponding desired (correct)outputvector. Thetargetofsupervisedlearningis tofindthe weightw∗ that makes the estimated output of the networkYNN(q) = f ( w∗,UNN(q) ) equivalent toYd(q) forall1≤ q≤Q. Inordertofindthebestweightvectorw∗,anobjectiveorcostfunction Jhastobedefined,whichmeasurestheerrorbetweentherealandde- siredoutputs. Thereareseveraldifferentwaystodefinethecost func- tion. For instance, a simple choice is to use theL1-norm [HDB+96], suchas JL1= Q∑ q=1 ‖YNN(q)−Yd(q)‖ . (3.71) This L1-norm formed cost function is intuitive, but not very com- monly used in practice because the derivative is not easy to calculate. Themostcommoncostfunctionusedinsupervisedlearningisthefol- lowingL2-norm[HDB+96] JL2= min 1 2Q Q∑ q=1 ( YNN(q)−Yd(q) )T( YNN(q)−Yd(q) ) , (3.72) 73