A First Course in FUZZY and NEURAL CONTROL

5.2. IMPLEMENTING NEURAL NETWORKS 171 ìintersectionîB, and the set notation for the logical function OR isA∪B,also calledAìunion ...

172 CHAPTER 5. NEURAL NETWORKS FOR CONTROL only one neuronC. For all neurons, the ìactivation functionîfis taken to be f(x)= Ω 1 ...

5.3. LEARNING CAPABILITY 173 Then comes the basic question. How do wefind an appropriate neural net- work to represent a given r ...

174 CHAPTER 5. NEURAL NETWORKS FOR CONTROL requirements simultaneously. However, as with all tools, neural networks have limitat ...

5.4. THE DELTA RULE 175 defined by GF(k)=max{|FA|:A⊂Rn,|A|=k} Note that for allk∈N, GF(k)≤ 2 k TheVapnik-Chervonenkis dimension( ...

176 CHAPTER 5. NEURAL NETWORKS FOR CONTROL Figure 5.5. Network with weightswij The goal is to minimizeEwith respect to the weigh ...

5.4. THE DELTA RULE 177 that a step activation functionmodels neurons that eitherfire or do notfire, while continuous activation ...

178 CHAPTER 5. NEURAL NETWORKS FOR CONTROL Therefore, 4 wjk = XN q=1 4 qwjk = −η XN q=1 ∂Eq ∂wjk = XN q=1 −ηδqjxqk The delta rul ...

5.5. THE BACKPROPAGATION ALGORITHM 179 Explicitly, (1) ∂E ∂w 0 = w 1 +w 2 − 2 w 0 +1=0 (2) ∂E ∂w 1 =2w 1 +w 2 − 2 w 0 =0 (3) ∂E ...

180 CHAPTER 5. NEURAL NETWORKS FOR CONTROL The following popular learning algorithm, referred to as thebackpropaga- tion algorit ...

5.5. THE BACKPROPAGATION ALGORITHM 181 We will write down the updating formulas for weights in a two-layer neural network. The g ...

182 CHAPTER 5. NEURAL NETWORKS FOR CONTROL wherejis in the output layer. The∂E q ∂oqj are known from previous calculations using ...

5.6. EXAMPLE 1: TRAINING A NEURAL NETWORK 183 Propagatexqforward from the input layer to the output layer using oi=fi √ p X k= ...

184 CHAPTER 5. NEURAL NETWORKS FOR CONTROL Figure 5.9. Two-layer feedforward perceptron analyze a single feedforward and backpro ...

5.7. EXAMPLE 2: TRAINING A NEURAL NETWORK 185 Computeallthenineweightupdates: 1.∆w 10 =−η(δh 1 )(x 0 )=−(0.1)(− 5. 098 ◊ 10 −^6 ...

186 CHAPTER 5. NEURAL NETWORKS FOR CONTROL perform trash identification. We explain here how a backpropagation neural net- work ...

5.7. EXAMPLE 2: TRAINING A NEURAL NETWORK 187 Figure 5.12. Leaf sample and segmented image Figure 5.13. Pepper sample and segmen ...

188 CHAPTER 5. NEURAL NETWORKS FOR CONTROL Shape Factor: This feature is defined asPerimeter 2 4 π∑Area. Convex Area: The comput ...

5.7. EXAMPLE 2: TRAINING A NEURAL NETWORK 189 0 50 100 150 200 250 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Bark Stick Leaf Pepper Fi ...

190 CHAPTER 5. NEURAL NETWORKS FOR CONTROL 0 50 100 150 200 250 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Bark Stick Leaf Pepper Figur ...