神经网络控制01 (英文)

1、Neural Network & Fuzzy Control SystemsNotes #1: Neural Network笔记#1: (神经网络：反向增值学习的算法 )笔记(英文)整理： 陈恳BACK PROPAGATION LEARNING ALGORITHMNEURALNETWORKSN(x)xS(y)de-+N(x)=S(y)=( S1(y1), S2(y2), , Sp(yp) ); S(): non-linear function.x and y are (1n) and (1p) vectors.d is the desired output, (1p) vector.e is

2、the error signal, (1p) vector.At iteration k, ek=dk N(xk) = dk S(yk) = (d1k S(y1k), , (dpk S(ypk) Instantaneous summed squared error:The error is observed at iteration k.Total errornT : total error of data pair, (x1,d1; ; x nT,d nT).Back propagation learning algorithm minimizes at each iteration.Doe

3、s this mean it also minimizes E?If each term of E is minimized, we expect that E is also minimized.Example: n=2, p=2, i.e., two inputs, two outputs.S2(x2)x1Input 1Input 2Sm11m22Sm12m21SSS1(x1)S1(y1)S2(y2)x2y2y1d1d2+e1e2desired output 1desired output 2input layeroutput layery1=m11S1(x1) + m21S2(x2)y2

4、=m12S1(x1) + m22S2(x2)Under the assumption, Si(x)= S (x), nonlinearity are the same.y1 y2 = S(x1) S(x2) =S(x1) S(x2) Actual network output:S(y1) S(y2)=S(y)=N(x) Error: e1=d1- S(y1) , e2=d2- S(y2)Since = are the only variables, then we have to minimize Ek with respect to these variables. This minimiz

5、ation is also known as the training of neural network. GRADIENT DESCENT ALGORITHM or we will consider two different networks:XHYconnection weightsinput layerhidden layeroutput layerXFGconnection weights3 hidden layerHYA)B)output layerinput layerLets look at the jth neuron at the output layer:mijjth

6、neuron other neurons We simplify SyjkS(yjk)S(yjk)by (See figure later page for a better view)At kth iteration, from the learning algorithm=; but ; nq is the number of neurons in hidden layer. For convenience, we take nq=p. = = =; but =Now consider qth neuron at the hidden layer,miqqth neuron other n

7、eurons AssumeSi(xi)=ximqjjth neuron from the learning algorithm=; but ; ni is the number of neurons in the input layer. = = =; = （注意：，p个输出） = again =Here we used the chain rule of differentiation, i.e., iff=f(x1, x2,xn ) thenThe way we used this relation in our derivation is through thenThe reason w

8、e do this is the fact that it is difficult to evaluate because it is not easy to see how much Ek will change if we change .B)Lets now consider a neuron that belongs to layer G,XFGHYmsrmrqneuron rneuron jneuron qOn this figure, I also showed neurons that we considered before, namely neurons j and q.H

9、ow do we train for the weight msr? from the learning algorithm=; but ; ns is the number of neurons in the layer F. = = =; = = = = =The weight msr can now be updated as See next pages for the detailed weight connecttions. Sxikith neuron xikSi(xik)miqSqth neuron hqkSq(hqk)mqjSjth neuron yjkSj(yjk)djkX

10、HYejk3-layer neural networkTHE NON-LINEAR FUNCTION SxSigmoid function0.5S(x)= Note: always positive.The advantage of Sigmoid function lies with the easy evaluation of its derivative.If the network has to handle negative as well as positive numbers, the Sigmoid function can be shifted as Sigmoid func

11、tion1.0S(x)x-1.0=Note that we dont just need to use the Sigmoid function only, what we need is a monotone non-decreasing differentiable function to represent the non-linearity of neuron.Sigmoid functionhS(x)xecShifted Sigmoid function g: gain Recommended for most networks.cheStep functionS(x)=e if x

12、cUsed in earlier networks such as perception Hopfield, etc. not differentiable.DISCUSSION ON GRADIENT DESCENT ALGORITHMGiven the Taylor expansion above, let us say we want to find the min or max of f(x), thenThis givesreplace x0=x(k) x=x(k+1)thenwhich corresponds to the training algorithmSIMPLE EXAM

13、PLE ON BACK PROPAGATION LEARNING.Input: x=-3.0 2.0Desired output: d=0.4 0.8Let us see this on the diagraminputoutputindex kTraining dataNow, we will create a neural network that will learn this input/output behavior.Here is the network:S(y)error e1Neuron 1SSm1m2SxNeuron 2Neuron 3ydNeuron 2 receives

14、a constant input, so called bias. The relation may have non-zero output for zero input. This is why we use bias.We want to minimize d1=0.4 , d2=0.8Since we have only two parameters (m1, m2), it is easy to construct the error surface graph as below.+m1m2误差 ExStarting from an initial condition (marked

15、 by +), the graph demonstrates how the minimum is achieved by the gradient descent technique. Think of a marble with no inertia sliding down to the one of the lowest point the error surface graph. m1 m2m2m10MOMENTUM ALGORITHM FOR BACKPROPAGATIONWe demonstrated that weight adjustments in back propaga

16、tion algorithm arewheremij()=follows a first order difference equation.More general updating can be accomplished by This third term is called the momentum term. The idea here is not to “forget” the previous gradient term, i.e., so if there are sudden random changes in , we will not be immediately ef

17、fected by it. To see how it works, consider the momentum update equation given above, i.e.,or a slight change of notation here.Applying z-transform, Low pass filter, |bk|1Current gradientWe see that the weight changes are not immediately effected by the current gradient but it is effected by a low p

18、ass filtered current gradient. EXAMPLE: IDENTIFICATION BY NEURAL NETWORKnoisey(t-1)y(t)m3m4m55m6m1m2Processy(t)=a1y(t-1)+ a2y(t-2)+ b1u(t-1)+ b2u(t-2)+v(t)u(t)m6x1x2h2h1S(h2)S(h1)S(y1)e(t) u(t-1) Initial conditions y(0), y(-1), u(0), u(-1)xi: ith neuron of input layer Xhi: ith neuron of hidden layer

19、 Hyi: ith neuron of output layer Yerror at iteration k: = S(y1)A) Neuron at the output layer Y,=; =For the second weight, the learning equation becomes=; = then,B) Neuron in hidden layer,=Similarly,=, are left to you as an exercise.Note also that S(x1)= x1=y(k-1) if S()=1S(x2)= x2=u(k-1)EXAMPLEeyx-NEURAL N