BP算法和RBF算法的网络逼近
-
- BP算法
- RBF
- 高斯径向基函数
- RBF仿真实例
- 参考资料
最近开始做自己的毕业设计,开始记录相关的智能控制学习心得和知识点.
BP算法
bp算法全称BackPropagation,也就是误差反向传播算法,它的基本思想是梯度下降法,采用梯度搜索技术,通过链式求导法则,最终使得网络输出和期望输出的误差方差最小.BP算法的图如下图所示:
三层网络:输入层,隐藏层和输出层.网络的前向传播和误差反向传播可参考这篇博客:大白话讲解BP算法.这里主要涉及到的是对于sigmod函数的求导.
f(x)=11+e−x f ( x ) = 1 1 + e − x
其求导结果如下:
f(x)′=f(x)(1−f(x)) f ( x ) ′ = f ( x ) ( 1 − f ( x ) )
对于链式求导法则,可以举一个简单的例子:
f(x)=x2,g(x)=2x+1,现在对y=f(g(x))求导 f ( x ) = x 2 , g ( x ) = 2 x + 1 , 现 在 对 y = f ( g ( x ) ) 求 导 则有:
dydx=dydg(x)∗dg(x)dx=2∗g(x)∗2=8x+4 d y d x = d y d g ( x ) ∗ d g ( x ) d x = 2 ∗ g ( x ) ∗ 2 = 8 x + 4
这里有一个BP算法逼近的示例(《智能控制》第4版 page131),系统的传递函数如下:
y(k)=u(k)3+y(k−1)1+y(k−1)2采样时间取1ms,输入信号为u(k)=0.5sin(6πt),神经网络为2−6−1结构,权值W1,W2的初始值取[−1,+1]之间的随机值,取学习速率η=0.5,动量因子α=0.1 y ( k ) = u ( k ) 3 + y ( k − 1 ) 1 + y ( k − 1 ) 2 采 样 时 间 取 1 m s , 输 入 信 号 为 u ( k ) = 0.5 s i n ( 6 π t ) , 神 经 网 络 为 2 − 6 − 1 结 构 , 权 值 W 1 , W 2 的 初 始 值 取 [ − 1 , + 1 ] 之 间 的 随 机 值 , 取 学 习 速 率 η = 0.5 , 动 量 因 子 α = 0.1
clear all
close all
xite = 0.5;%学习速率
alfa = 0.1;%动量因子,动量因子大一点可以减小振荡,加快收敛,太大了会发散
w2=rands(6,1);%隐藏层和输出层的连接(隐藏层6个神经元)
w2_1=w2;w2_2=w2_1;
w1=rands(2,6);%输入层和隐藏层的连接(输入层两个u(k)和y(k))
w1_1=w1;w1_2=w1;
dwl=0*w1;%2x6,初始化为0
x=[0,0]';%输入初始化
u_1 = 0;
y_1 = 0;
I=[0,0,0,0,0,0]';
Iout=[0,0,0,0,0,0]';
FI=[0,0,0,0,0,0]';
ts=0.001;%步长
for k = 1:1:1000
time(k) = k*ts;
u(k) = 0.5*sin(3*2*pi*k*ts);%输入u(k)
y(k)=u_1^3+y_1/(1+y_1^2);%实际输出y(k)
for j =1:1:6
I(j)=x'*w1(:,j);%I是一个三维的数组,输入层乘以权重到隐藏层
Iout(j)=1/(1+exp(-I(j)));%sigmod函数,即f(net)
end
yn(k)=w2'*Iout;%输出
e(k)=y(k)-yn(k)%误差
w2=w2_1+(xite*e(k))*Iout+alfa*(w2_1-w2_2);%隐藏层到输出层的梯度,此处求导不涉及sigmod 函数
for j=1:1:6
FI(j)=exp(-I(j))/(1+exp(-I(j)))^2;%sigmod函数的求导结果
end
for i=1:1:2
for j =1:1:6
dw1(i,j)=e(k)*xite*FI(j)*w2(j)*x(i);%隐藏层到输入层的误差梯度
end
end
w1=w1_1+dw1+alfa*(w1_1-w1_2);%更新输入层到隐藏层权重
yu=0;
for j=1:1:6
yu=yu+w2(j)*w1(1,j)*FI(j);
end
dyu(k)=yu;%Jacobian信息
x(1)=u(k);
x(2)=y(k);
w1_2=w1_1;w1_1=w1;
w2_2=w2_1;w2_1=w2;
u_1=u(k)
y_1=y(k)
end
figure(1);
plot(time,y,'r',time,yn,'b');
xlabel('time(s)');ylabel('y and yn');
figure(2);
plot(time,y-yn,'r')
xlabel('time(s)');ylabel('error');
figure(3);
plot(time,dyu);
xlabel('time(s)');ylabel('dy')
仿真结果如下:
BP算法只要有足够多的隐藏层和隐藏节点,理论上是可以逼近任意的非线性系统的,它属于全局逼近算法,具有较强的泛化能力.但是BP算法待优化参数多,收敛速度慢,难以确定隐藏层和隐藏节点数,很多时候要靠经验试凑.难以适应实时控制的要求.
RBF
径向基函数(Radial Basis Function,RBF)神经网络是一种三层网络.其学习过程和BP算法类似,二者的主要区别在于:(1)隐藏层的激活函数不同,BP算法的激活函数是sigmod函数,RBF的激活函数是高斯基函数;(2)BP算法是全局的逼近,每次学习都要进行所有网络连接的权重迭代更新,RBF是一种局部逼近,其输入层到输层是非线性映射,但其隐藏层到输出层是线性映射的.且输入层到隐藏层直接求和,不需要参数迭代.
高斯径向基函数
首先给出高斯径向基函数的表达式:
f(x)=exp(−||x−c||22b)其中,c是函数的中心,b是函数的宽度 f ( x ) = e x p ( − | | x − c | | 2 2 b ) 其 中 , c 是 函 数 的 中 心 , b 是 函 数 的 宽 度
我们一定对正交基很熟悉,实际上函数基我们可以理解为一簇函数,是一个函数空间,对于RBF有一篇博客写得很好: RBF
我们从数据拟合的角度来理解,如下图所示,为了得到x1对应的值,我们在拟合图一的点时,每个点都是基函数的中心,则x1的值就是它所在的两个样本点间的求和,离哪个点近一点,哪个点给的权重就多一点,也就是中心距离设置大一点.
RBF仿真实例
我们同样是跟踪BP仿真的系统,模型结构如下:
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RBF函数:
function[sys,x0,str,ts]=spacemodel(t,x,u,flag)
switch flag,
case 0,
[sys,x0,str,ts]=mdlInitializeSizes;
case 3,
sys=mdlOutputs(t,x,u)
case {2,4,9}
sys=[];
otherwise
error(['Unhandled flag=',num2str(flag)]);
end
function[sys,x0,str,ts]=mdlInitializeSizes
sizes = simsizes;
sizes.NumContStates = 0;
sizes.NumDiscStates = 0;
sizes.NumOutputs = 7;
sizes.NumInputs = 1;
sizes.DirFeedthrough = 1;
sizes.NumSampleTimes = 0;
sys = simsizes(sizes);
x0=[];
str =[];
ts =[];
function sys = mdlOutputs(t,x,u)
x=u(1);
c=[-0.5 -0.25 0 0.25 0.5];
b=[0.2 0.2 0.2 0.2 0.2]';
W=ones(5,1)
h=zeros(5,1)
for j=1:1:5
h(j)=exp(-norm(x-c(:,j))^2/(2*b(j)*b(j)));
end
y=W'*h;
sys(1)=y;
sys(2)=x;
sys(3)=h(1);
sys(4)=h(2);
sys(5)=h(3);
sys(6)=h(4);
sys(7)=h(5);结果可视化:
close all
figure(1);
plot(t,y(:,1),'k','linewidth',2);
xlabel('time(s)');ylabel('y');
figure(2);
plot(y(:,2),y(:,3),'r','linewidth',2)
xlabel('x');ylabel('hj');
hold on;
plot(y(:,2),y(:,4),'k','linewidth',2);
hold on
plot(y(:,2),y(:,5),'g','linewidth',2);
hold on
plot(y(:,2),y(:,6),'b','linewidth',2);
hold on
plot(y(:,2),y(:,7),'y','linewidth',2);
基函数如下:
参考资料
- BP算法
- RBF
- 基函数和函数空间
- BP算法推导