Seite - 104 - in Adaptive and Intelligent Temperature Control of Microwave Heating Systems with Multiple Sources

4. ControlSystemDesign SAisapowerfulmethodthatisusedforoptimizationproblemswhere traditional analytical methods could not solve. In this section, the so-called simultaneous perturbation stochastic approximation (SPSA) learning algorithm is introduced to demonstrate how the NN con- troller isdesignedandapplied inHEPHAISTOS.SPSAwasfirstlyde- veloped in [Spa87] as a general SA algorithm used in parameter esti- mation. Since the middle of the 1990s, it has been gradually extended to the field of neural network learning and control, and widely im- plemented in practical applications, such as in [SC94] [Spa98] [BK04] [SSSN08]. The procedures to implement SPSA in a NN controller are described in the following. At each timek, the weight vector of the NN controller is updated by theequation w(k+1) =w(k)−α(k) ·h(w(k)), (4.25) whereh(w(k)) is thesimultaneousperturbationapproximationofthe original gradient∂J(k)/∂w. The approximation term is calculated by theequation h(w(k)) = J(+)(k)−J(−)(k) 2c(k)∆(k) , (4.26) where ∆(k) = [∆1(k),∆2(k), .. . ,∆Nw(k)] T , w(±)(k) =w(k)±c(k)∆(k), U(±)(k) =G ( Yt(k),Y(k),w (±)(k) ) , Y(±)(k) =Y(k+1) =f ( Y(k),U(±)(k) ) , J(±)(k) =J ( Y(±)(k),U(±)(k) )∣∣∣ p=1 , (4.27) and c(k) is the tuning constant that fulfills certain regularity condi- tions. Severalpointsneedtobenoticedregardingthe implementation of the learningprocess: • The simultaneous perturbation vector ∆(k) is randomly gener- ated. All elements ∆i(k) are independent, bounded and symmet- rically distributed random variables [SC98]. In practice, these el- ements could be randomly generated around 0 with amplitudes 104