Page - 103 - in Adaptive and Intelligent Temperature Control of Microwave Heating Systems with Multiple Sources

4.1. AdaptiveControl andtheneuralnetworkcontroller is representedby U(k) =G(Yt(k),Y(k),w(k)) , w(k) = [ w1(k), w2(k), . . ., wNw(k) ]T (4.21) whereF is the function that describes the dynamics of the real plant, G is thefunctionoftheNNcontroller. andw(k) is theweightvectorof the NN controller (Nw×1). Equation 4.21 means the control action is directly calculated based on the target and current measured temper- atures as well as the weights of the NN. With the same cost function inequation 4.2, theobjectiveofunsupervised learning in thisNNCis defined to find the optimal weight vector w∗ that minimizes the cost function,which is w∗= argwminJ(k) ⇒ ∂J(k) ∂w∗ = 0. (4.22) The weight update in the NN controller still uses the standard gradi- ent descent principle, following the direction that minimizes the cost function, suchas w(k+1) =w(k)−α(k) · ∂J(k) ∂w ∣∣∣∣ k =w(k)−α(k) · ∂J(k) ∂Y · ∂Y ∂U · ∂U ∂w ∣∣∣∣ k , (4.23) whereα(k) is thestepsizeof theupdate. If the dynamics of the real plantF is perfectly known, the partial dif- ferentiation ∂Y/∂U can be calculated and the gradient term can be directly implemented. However, for most cases, including HEPHAIS- TOS, therealdynamicsof thecontrolledplant is incompletelyknown, andthepartialdifferential term∂Y/∂U isnotdirectlycomputable. In this case, a stochastic approximation (SA) algorithm [KY97] has to be applied[SC98] w(k+1) =w(k)−α(k) ·(approximatedgradient)k , (4.24) which replaces the true gradient by an approximated gradient to up- date theweightsof theNNcontroller. 103