RBF Neural Network Control Based on Gradient Descent Algorithm

Abstract

This chapter introduces three kinds of RBF neural network control laws based on gradient descent rule, including supervisory control law, model reference adaptive control law, and self-adjust control law; the weight value learning algorithms are presented. Several simulation examples are given.

Appendix

3.1.1 Programs for Sect. 3.1.2

The program of RBF supervisory control: chap3_1.m

%RBF Supervisory Control

clear all;

close all;

ts=0.001;

sys=tf(1000,[1,50,2000]);

dsys=c2d(sys,ts,'z');

[num,den]=tfdata(dsys,'v');

y_1=0;y_2=0;

u_1=0;u_2=0;

e_1=0;

xi=0;

x=[0,0]';

b=0.5*ones(4,1);

c=[-2 -1 1 2];

w=rands(4,1);

w_1=w;

w_2=w_1;

xite=0.30;

alfa=0.05;

kp=25;

kd=0.3;

for k=1:1:1000

time(k)=k*ts;

S=1;

if S==1

yd(k)=0.5*sign(sin(2*2*pi*k*ts)); %Square Signal

elseif S==2

yd(k)=0.5*(sin(3*2*pi*k*ts)); %Square Signal

end

y(k)=-den(2)*y_1-den(3)*y_2+num(2)*u_1+num(3)*u_2;

e(k)=yd(k)-y(k);

xi=yd(k);

for j=1:1:4

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

end

un(k)=w'*h';

%PD Controller

up(k)=kp*x(1)+kd*x(2);

M=2;

if M==1 %Only Using PID Control

u(k)=up(k);

elseif M==2 %Total control output

u(k)=up(k)+un(k);

end

if u(k)>=10

u(k)=10;

end

if u(k)<=-10

u(k)=-10;

end

%Update NN Weight

d_w=-xite*(un(k)-u(k))*h';

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

w_2=w_1;

w_1=w;

u_2=u_1;

u_1=u(k);

y_2=y_1;

y_1=y(k);

x(1)=e(k); %Calculating P

x(2)=(e(k)-e_1)/ts; %Calculating D

e_1=e(k);

end

figure(1);

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

xlabel('Time(second)');ylabel('Position tracking');

legend('Ideal position signal','Tracking position signal');

figure(2);

subplot(311);

plot(time,un,'k','linewidth',2);

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

legend('Control input with RBF');

subplot(312);

plot(time,up,'k','linewidth',2);

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

legend('Control input with P');

subplot(313);

plot(time,u,'k','linewidth',2);

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

legend('Total control input');

3.1.2 Programs for Sect. 3.2.2

Programs: chap3_2.m

%Model Reference Aapative RBF Control

clear all;

close all;

u_1=0;

y_1=0;

ym_1=0;

x=[0,0,0]';

c=[-3 -2 -1 0 1 2 3;

-3 -2 -1 0 1 2 3;

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

b=2;

w=rands(1,7);

xite=0.35;

alfa=0.05;

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

c_1=c;c_2=c;

b_1=b;b_2=b;

w_1=w;w_2=w;

ts=0.001;

for k=1:1:3000

time(k)=k*ts;

yd(k)=0.50*sin(2*pi*k*ts);

ym(k)=0.6*ym_1+yd(k);

y(k)=(-0.1*y_1+u_1)/(1+y_1^2); %Nonlinear plant

for j=1:1:7

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

end

u(k)=w*h;

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

dyu(k)=sign((y(k)-y_1)/(u(k)-u_1));

d_w=0*w;

for j=1:1:7

d_w(j)=xite*ec(k)*h(j)*dyu(k);

end

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

%Return of parameters

u_1=u(k);

y_1=y(k);

ym_1=ym(k);

x(1)=yd(k);

x(2)=ec(k);

x(3)=y(k);

w_2=w_1;w_1=w;

end

figure(1);

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

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

legend('Ideal position signal','Tracking position signal');

figure(2);

plot(time,u,'r','linewidth',2);

xlabel('time(s)');ylabel('Control input');

3.1.3 Programs for Sect. 3.3.3

RBF NN Self-adjust Control:chap3_3.m

%Self-Correct control based RBF Identification

clear all;

close all;

xite1=0.15;

xite2=0.50;

alfa=0.05;

w=0.5*ones(6,1);

v=0.5*ones(6,1);

cij=0.50*ones(1,6);

bj=5*ones(6,1);

h=zeros(6,1);

w_1=w;w_2=w_1;

v_1=v;v_2=v_1;

u_1=0;y_1=0;

ts=0.02;

for k=1:1:5000

time(k)=k*ts;

yd(k)=1.0*sin(0.1*pi*k*ts);

%Practical Plant;

g(k)=0.8*sin(y_1);

f(k)=15;

y(k)=g(k)+f(k)*u_1;

for j=1:1:6

h(j)=exp(-norm(y(k)-cij(:,j))^2/(2*bj(j)*bj(j)));

end

Ng(k)=w'*h;

Nf(k)=v'*h;

ym(k)=Ng(k)+Nf(k)*u_1;

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

d_w=0*w;

for j=1:1:6

d_w(j)=xite1*e(k)*h(j);

end

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

d_v=0*v;

for j=1:1:6

d_v(j)=xite2*e(k)*h(j)*u_1;

end

v=v_1+d_v+alfa*(v_1-v_2);

u(k)=-Ng(k)/Nf(k)+yd(k)/Nf(k);

u_1=u(k);

y_1=y(k);

w_2=w_1;

w_1=w;

v_2=v_1;

v_1=v;

end

figure(1);

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

xlabel('Time(second)');ylabel('Position tracking');

legend('Ideal position signal','Tracking position signal');

figure(2);

plot(time,g,'r',time,Ng,'k:','linewidth',2);

xlabel('Time(second)');ylabel('g and Ng');

legend('Ideal g','Estimation of g');

figure(3);

plot(time,f,'r',time,Nf,'k:','linewidth',2);

xlabel('Time(second)');ylabel('f and Nf');

legend('Ideal f','Estimation of f');