RBF Neural Network Design and Simulation

Abstract

This chapter introduces RBF neural network design method, gives RBF neural network approximation algorithm based on gradient descent, analyzes the effects of Gaussian function parameters on RBF approximation, and introduces RBF neural network modeling method based on off-line training. Several simulation examples are given.

Appendix

2.1.1 Programs for Sect. 2.1.2.1

Simulink main program: chap2_1sim.mdl

RBF function: chap2_1rbf.m

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); %Input Layer

%i=1

%j=1,2,3,4,5

%k=1

c=[-0.5 -0.25 0 0.25 0.5]; %cij

b=[0.2 0.2 0.2 0.2 0.2]'; %bj

W=ones(5,1); %Wj

h=zeros(5,1); %hj

for j=1:1:5

h(j)=exp(-norm(x-c(:,j))^2/(2*b(j)*b(j))); %Hidden Layer

end

y=W'*h; %Output Layer

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);

Plot program: chap2_1plot.m

close all;

% y=y(:,1);

% x=y(:,2);

% h1=y(:,3);

% h2=y(:,4);

% h3=y(:,5);

% h4=y(:,6);

% h5=y(:,7);

figure(1);

plot(t,y(:,1),'k','linewidth',2);

xlabel('time(s)');ylabel('y');

figure(2);

plot(y(:,2),y(:,3),'k','linewidth',2);

xlabel('x');ylabel('hj');

hold on;

plot(y(:,2),y(:,4),'k','linewidth',2);

hold on;

plot(y(:,2),y(:,5),'k','linewidth',2);

hold on;

plot(y(:,2),y(:,6),'k','linewidth',2);

hold on;

plot(y(:,2),y(:,7),'k','linewidth',2);

2.1.2 Programs for Sect. 2.1.2.2

Simulink main program: chap2_2sim.mdl

RBF function: chap2_2rbf.m

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 = 8;

sizes.NumInputs = 2;

sizes.DirFeedthrough = 1;

sizes.NumSampleTimes = 0;

sys = simsizes(sizes);

x0 = [];

str = [];

ts = [];

function sys=mdlOutputs(t,x,u)

x1=u(1); %Input Layer

x2=u(2);

x=[x1 x2]';

%i=2

%j=1,2,3,4,5

%k=1

c=[-0.5 -0.25 0 0.25 0.5;

-0.5 -0.25 0 0.25 0.5]; %cij

b=[0.2 0.2 0.2 0.2 0.2]'; %bj

W=ones(5,1); %Wj

h=zeros(5,1); %hj

for j=1:1:5

h(j)=exp(-norm(x-c(:,j))^2/(2*b(j)*b(j))); %Hidden Layer

end

yout=W'*h; %Output Layer

sys(1)=yout;

sys(2)=x1;

sys(3)=x2;

sys(4)=h(1);

sys(5)=h(2);

sys(6)=h(3);

sys(7)=h(4);

sys(8)=h(5);

Plot program: chap2_2plot.m

close all;

% y=y(:,1);

% x1=y(:,2);

% x2=y(:,3);

% h1=y(:,4);

% h2=y(:,5);

% h3=y(:,6);

% h4=y(:,7);

% h5=y(:,8);

figure(1);

plot(t,y(:,1),'k','linewidth',2);

xlabel('time(s)');ylabel('y');

figure(2);

plot(y(:,2),y(:,4),'k','linewidth',2);

xlabel('x1');ylabel('hj');

hold on;

plot(y(:,2),y(:,5),'k','linewidth',2);

hold on;

plot(y(:,2),y(:,6),'k','linewidth',2);

hold on;

plot(y(:,2),y(:,7),'k','linewidth',2);

hold on;

plot(y(:,2),y(:,8),'k','linewidth',2);

figure(3);

plot(y(:,3),y(:,4),'k','linewidth',2);

xlabel('x2');ylabel('hj');

hold on;

plot(y(:,3),y(:,5),'k','linewidth',2);

hold on;

plot(y(:,3),y(:,6),'k','linewidth',2);

hold on;

plot(y(:,3),y(:,7),'k','linewidth',2);

hold on;

plot(y(:,3),y(:,8),'k','linewidth',2);

2.1.3 Programs for Sect. 2.2.2.1

Simulink main program: chap2_3sim.mdl

S function for plant: chap2_3rbf.m

function [sys,x0,str,ts]=s_function(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 = 1;

sizes.NumInputs = 2;

sizes.DirFeedthrough = 1;

sizes.NumSampleTimes = 0;

sys=simsizes(sizes);

x0=[];

str=[];

ts=[];

function sys=mdlOutputs(t,x,u)

persistent w w_1 w_2 b ci

alfa=0.05;

xite=0.5;

if t==0

b=1.5;

ci=[-1 -0.5 0 0.5 1;

-10 -5 0 5 10];

w=rands(5,1);

w_1=w;w_2=w_1;

end

ut=u(1);

yout=u(2);

xi=[ut yout]';

for j=1:1:5

h(j)=exp(-norm(xi-ci(:,j))^2/(2*b^2));

end

ymout=w'*h';

d_w=0*w;

for j=1:1:5 %Only weight value update

d_w(j)=xite*(yout-ymout)*h(j);

end

w=w_1+d_w+alfa*(w_1-w_2);

w_2=w_1;w_1=w;

sys(1)=ymout;

S function for plant: chap2_3plant.m

function [sys,x0,str,ts]=s_function(t,x,u,flag)

switch flag,

case 0,

[sys,x0,str,ts]=mdlInitializeSizes;

case 1,

sys=mdlDerivatives(t,x,u);

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 = 2;

sizes.NumDiscStates = 0;

sizes.NumOutputs = 1;

sizes.NumInputs = 1;

sizes.DirFeedthrough = 0;

sizes.NumSampleTimes = 0;

sys=simsizes(sizes);

x0=[0,0];

str=[];

ts=[];

function sys=mdlDerivatives(t,x,u)

sys(1)=x(2);

sys(2)=-25*x(2)+133*u;

function sys=mdlOutputs(t,x,u)

sys(1)=x(1);

Plot program: chap2_3plot.m

close all;

figure(1);

plot(t,y(:,1),'r',t,y(:,2),'k:','linewidth',2);

xlabel('time(s)');ylabel('y and ym');

legend('ideal signal','signal approximation');

2.1.4 Programs for Sect. 2.2.2.2

Matlab program: chap2_4.m

%RBF identification

clear all;

close all;

alfa=0.05;

xite=0.15;

x=[0,1]';

b=3*ones(5,1);

c=[-1 -0.5 0 0.5 1;

-1 -0.5 0 0.5 1];

w=rands(5,1);

w_1=w;w_2=w_1;

c_1=c;c_2=c_1;

b_1=b;b_2=b_1;

d_w=0*w;

d_b=0*b;

y_1=0;

ts=0.001;

for k=1:1:10000

time(k)=k*ts;

u(k)=sin(k*ts);

y(k)=u(k)^3+y_1/(1+y_1^2);

x(1)=u(k);

x(2)=y_1;

for j=1:1:5

h(j)=exp(-norm(x-c(:,j))^2/(2*b(j)*b(j)));

end

ym(k)=w'*h';

em(k)=y(k)-ym(k);

M=2;

if M==1 %Only weight value update

d_w(j)=xite*em(k)*h(j);

elseif M==2 %Update w,b,c

for j=1:1:5

d_w(j)=xite*em(k)*h(j);

d_b(j)=xite*em(k)*w(j)*h(j)*(b(j)^-3)*norm (x-c(:,j))^2;

for i=1:1:2

d_c(i,j)=xite*em(k)*w(j)*h(j)*(x(i)-c(i,j))*(b(j)^-2);

end

end

b=b_1+d_b+alfa*(b_1-b_2);

c=c_1+d_c+alfa*(c_1-c_2);

end

w=w_1+d_w+alfa*(w_1-w_2);

y_1=y(k);

w_2=w_1;

w_1=w;

c_2=c_1;

c_1=c;

b_2=b_1;

b_1=b;

end

figure(1);

subplot(211);

plot(time,y,'r',time,ym,'k:','linewidth',2);

xlabel('time(s)');ylabel('y and ym');

legend('ideal signal','signal approximation');

subplot(212);

plot(time,y-ym,'k','linewidth',2);

xlabel('time(s)');ylabel('error');

2.1.5 Programs for Sect. 2.3

Program of Gaussian membership function design: chap2_5.m

%RBF function

clear all;

close all;

c=[-3 -1.5 0 1.5 3];

M=1;

if M==1

b=0.50*ones(5,1);

elseif M==2

b=1.50*ones(5,1);

end

h=[0,0,0,0,0]';

ts=0.001;

for k=1:1:2000

time(k)=k*ts;

%RBF function

x(1)=3*sin(2*pi*k*ts);

for j=1:1:5

h(j)=exp(-norm(x-c(:,j))^2/(2*b(j)*b(j)));

end

x1(k)=x(1);

%First Redial Basis Function

h1(k)=h(1);

%Second Redial Basis Function

h2(k)=h(2);

%Third Redial Basis Function

h3(k)=h(3);

%Fourth Redial Basis Function

h4(k)=h(4);

%Fifth Redial Basis Function

h5(k)=h(5);

end

figure(1);

plot(x1,h1,'b');

figure(2);

plot(x1,h2,'g');

figure(3);

plot(x1,h3,'r');

figure(4);

plot(x1,h4,'c');

figure(5);

plot(x1,h5,'m');

figure(6);

plot(x1,h1,'b');

hold on;plot(x1,h2,'g');

hold on;plot(x1,h3,'r');

hold on;plot(x1,h4,'c');

hold on;plot(x1,h5,'m');

xlabel('Input value of Redial Basis Function');ylabel('Membership function degree');

Program of RBF approximation to test the effect of b and c: chap2_6.m

%RBF approximation test

clear all;

close all;

alfa=0.05;

xite=0.5;

x=[0,0]';

%The parameters design of Guassian Function

%The input of RBF (u(k),y(k)) must be in the effect range of Guassian function overlay

%The value of b represents the widenth of Guassian function overlay

Mb=1;

if Mb==1 %The width of Guassian function is moderate

b=1.5*ones(5,1);

elseif Mb==2 %The width of Guassian function is too narrow, most overlap of the function is near to zero

b=0.0005*ones(5,1);

end

%The value of c represents the center position of Guassian function overlay

%the NN structure is 2-5-1: i=2; j=1,2,3,4,5; k=1

Mc=1;

if Mc==1 %The center position of Guassian function is moderate

c=[-1.5 -0.5 0 0.5 1.5;

-1.5 -0.5 0 0.5 1.5]; %cij

elseif Mc==2 %The center position of Guassian function is improper

c=0.1*[-1.5 -0.5 0 0.5 1.5;

-1.5 -0.5 0 0.5 1.5]; %cij

end

w=rands(5,1);

w_1=w;w_2=w_1;

y_1=0;

ts=0.001;

for k=1:1:2000

time(k)=k*ts;

u(k)=0.50*sin(1*2*pi*k*ts);

y(k)=u(k)^3+y_1/(1+y_1^2);

x(1)=u(k);

x(2)=y(k);

for j=1:1:5

h(j)=exp(-norm(x-c(:,j))^2/(2*b(j)*b(j)));

end

ym(k)=w'*h';

em(k)=y(k)-ym(k);

d_w=xite*em(k)*h';

w=w_1+ d_w+alfa*(w_1-w_2);

y_1=y(k);

w_2=w_1;w_1=w;

end

figure(1);

plot(time,y,'r',time,ym,'b:','linewidth',2);

xlabel('time(s)');ylabel('y and ym');

legend('Ideal value','Approximation value');

2.1.6 Programs for Sect. 2.4

Program of RBF approximation to test the effect of hidden nets number: chap2_7.m

%RBF approximation test

clear all;

close all;

alfa=0.05;

xite=0.3;

x=[0,0]';

%The parameters design of Guassian Function

%The input of RBF (u(k),y(k)) must be in the effect range of Guassian function overlay

%The value of b represents the widenth of Guassian function overlay

bj=1.5; %The width of Guassian function

%The value of c represents the center position of Guassian function overlay

%the NN structure is 2-m-1: i=2; j=1,2,…,m; k=1

M=3; %Different hidden nets number

if M==1 %only one hidden net

m=1;

c=0;

elseif M==2

m=3;

c=1/3*[-1 0 1;

-1 0 1];

elseif M==3

m=7;

c=1/9*[-3 -2 -1 0 1 2 3;

-3 -2 -1 0 1 2 3];

end

w=zeros(m,1);

w_1=w;w_2=w_1;

y_1=0;

ts=0.001;

for k=1:1:5000

time(k)=k*ts;

u(k)=sin(k*ts);

y(k)=u(k)^3+y_1/(1+y_1^2);

x(1)=u(k);

x(2)=y(k);

for j=1:1:m

h(j)=exp(-norm(x-c(:,j))^2/(2*bj^2));

end

ym(k)=w'*h';

em(k)=y(k)-ym(k);

d_w=xite*em(k)*h';

w=w_1+ d_w+alfa*(w_1-w_2);

y_1=y(k);

w_2=w_1;w_1=w;

x1(k)=x(1);

for j=1:1:m

H(j,k)=h(j);

end

if k==5000

figure(1);

for j=1:1:m

plot(x1,H(j,:),'linewidth',2);

hold on;

end

xlabel('Input value of Redial Basis Function');ylabel('Membership function degree');

end

end

figure(2);

subplot(211);

plot(time,y,'r',time,ym,'b:','linewidth',2);

xlabel('time(s)');ylabel('y and ym');

legend('Ideal value','Approximation value');

subplot(212);

plot(time,y-ym,'r','linewidth',2);

xlabel('time(s)');ylabel('Approximation error');

2.1.7 Programs for Sect. 2.5.2.1

Program of RBF training: chap2_8a.m

%RBF Training for MIMO

clear all;

close all;

xite=0.10;

alfa=0.05;

W=rands(5,2);

W_1=W;

W_2=W_1;

h=[0,0,0,0,0]';

c=2*[-0.5 -0.25 0 0.25 0.5;

-0.5 -0.25 0 0.25 0.5;

-0.5 -0.25 0 0.25 0.5]; %cij

b=10; %bj

xs=[1,0,0];%Ideal Input

ys=[1,0]; %Ideal Output

OUT=2;

NS=1;

k=0;

E=1.0;

while E>=1e-020

%for k=1:1:1000

k=k+1;

times(k)=k;

for s=1:1:NS %MIMO Samples

x=xs(s,:);

for j=1:1:5

h(j)=exp(-norm(x'-c(:,j))^2/(2*b^2)); %Hidden Layer

end

yl=W'*h; %Output Layer

el=0;

y=ys(s,:);

for l=1:1:OUT

el=el+0.5*(y(l)-yl(l))^2; %Output error

end

es(s)=el;

E=0;

if s==NS

for s=1:1:NS

E=E+es(s);

end

end

error=y-yl';

dW=xite*h*error;

W=W_1+dW+alfa*(W_1-W_2);

W_2=W_1;W_1=W;

end %End of for

Ek(k)=E;

end %End of while

figure(1);

plot(times,Ek,'r','linewidth',2);

xlabel('k');ylabel('Error index change');

save wfile b c W;

Program of RBF test: chap2_8b.m

%Test RBF

clear all;

load wfile b c W;

%N Samples

x=[0.970,0.001,0.001;

1.000,0.000,0.000];

NS=2;

h=zeros(5,1); %hj

for i=1:1:NS

for j=1:1:5

h(j)=exp(-norm(x(i,:)'-c(:,j))^2/(2*b^2)); %Hidden Layer

end

yl(i,:)=W'*h; %Output Layer

end

yl

2.1.8 Programs for Sect. 2.5.2.2

Program of RBF training: chap2_9a.m

%RBF Training for a Plant

clear all;

close all;

ts=0.001;

xite=0.50;

alfa=0.05;

u_1=0;y_1=0;

fx_1=0;

W=0.1*ones(1,7);

W_1=W;

W_2=W_1;

h=zeros(7,1);

c1=[-3 -2 -1 0 1 2 3];

c2=[-3 -2 -1 0 1 2 3];

c=[c1;c2];

b=1.5; %bj

NS=3000;

for s=1:1:NS %Samples

u(s)=sin(s*ts);

fx(s)=0.5*y_1*(1-y_1)/(1+exp(-0.25*y_1));

y(s)=fx_1+u_1;

u_1=u(s);

y_1=y(s);

fx_1=fx(s);

end

k=0;

for k=1:1:500

k=k+1;

times(k)=k;

for s=1:1:NS %Samples

x=[u(s),y(s)];

for j=1:1:7

h(j)=exp(-norm(x'-c(:,j))^2/(2*b^2)); %Hidden Layer

end

yl(s)=W*h; %Output Layer

el=0.5*(y(s)-yl(s))^2; %Output error

es(s)=el;

E=0;

if s==NS

for s=1:1:NS

E=E+es(s);

end

end

error=y(s)-yl(s);

dW=xite*h'*error;

W=W_1+dW+alfa*(W_1-W_2);

W_2=W_1;W_1=W;

end %End of for

Ek(k)=E;

end %End of while

figure(1);

plot(times,Ek,'r','linewidth',2);

xlabel('k');ylabel('Error index change');

save wfile b c W NS;

Program of RBF test: chap2_9b.m

%Online RBF Etimation for Plant

clear all;

load wfile b c W NS;

ts=0.001;

u_1=0;y_1=0;

fx_1=0;

h=zeros(7,1);

for k=1:1:NS

times(k)=k;

u(k)=sin(k*ts);

fx(k)=0.5*y_1*(1-y_1)/(1+exp(-0.25*y_1));

y(k)=fx_1+u_1;

x=[u(k),y(k)];

for j=1:1:7

h(j)=exp(-norm(x'-c(:,j))^2/(2*b^2)); %Hidden Layer

end

yp(k)=W*h; %Output Layer

u_1=u(k);y_1=y(k);

fx_1=fx(k);

end

figure(1);

plot(times,y,'r',times,yp,'b-.','linewidth',2);

xlabel('times');ylabel('y and yp');