Digital RBF Neural Network Control

Abstract

This chapter introduces adaptive Runge–Kutta–Merson method for digital RBF neural network controller design. Two examples for mechanical controls are given, including digital adaptive control for a servo system and digital adaptive control for two-link manipulators.

Appendix

9.1.1 Programs for Sect. 9.1.2

Program: chap9_1.m

clear all;

close all;

ts=0.001;  %Sampling time

x_1=0.5;

for k=1:1:10000  %dx=10*sint*x

   t(k)=k*ts;

  k1=ts*10*sin(t(k))*x_1;

  k2=ts*10*sin(t(k))*(x_1+1/3*k1);

  k3=ts*10*sin(t(k))*(x_1+1/6*k1+1/6*k2);

  k4=ts*10*sin(t(k))*(x_1+1/8*k1+3/8*k3);

  k5=ts*10*sin(t(k))*(x_1+1/2*k1-3/2*k3+2*k4);

  x(k)=x_1+1/6*(k1+4*k4+k5);

  x_1=x(k);

end

figure(1);

plot(t,x,'r','linewidth',2);

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

9.1.2 Programs for Sect. 9.2.2

Main program: chap9_2.m

%Discrete RBF control for Motor

clear all;

close all;

ts=0.001; %Sampling time

node=5; %Number of neural nets in hidden layer

gama=100;

c=0.5*[-2 -1 0 1 2;

     -2 -1 0 1 2];

b=1.5*ones(5,1);

h=zeros(node,1);

alfa=3;

kp=alfa^2;

kv=2*alfa;

q_1=0;dq_1=0;tol_1=0;

xk=[0 0];

w_1=0.1*ones(node,1);

A=[0  1;

 -kp -kv];

B=[0;1];

Q=[50 0;

  0 50];

P=lyap(A',Q);

eig(P);

k1=0.001;

for k=1:1:10000

time(k)=k*ts;

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

dqd(k)=cos(k*ts);

ddqd(k)=-sin(k*ts);

tSpan=[0 ts];

para=tol_1;       %D/A

[t,xx]=ode45('chap9_2plant',tSpan,xk,[],para); %Plant

xk=xx(length(xx),:);  %A/D

q(k)=xk(1);

%dq(k)=xk(2);

dq(k)=(q(k)-q_1)/ts;

ddq(k)=(dq(k)-dq_1)/ts;

e(k)=q(k)-qd(k);

de(k)=dq(k)-dqd(k);

xi=[e(k);de(k)];

for i=1:1:node

  S=2;

  if S==1

  w(i,1)=w_1(i,1)+ts*(gama*h(i)*xi'*P*B+k1*gama* norm(xi)*w_1(i,1)); %Adaptive law

  elseif S==2

    k1=ts*(gama*h(i)*xi'*P*B+k1*gama*norm(xi)* w_1(i,1));

    k2=ts*(gama*h(i)*xi'*P*B+k1*gama*norm(xi)* (w_1(i,1)+1/3*k1));

    k3=ts*(gama*h(i)*xi'*P*B+k1*gama*norm(xi)* (w_1(i,1)+1/6*k1+1/6*k2));

    k4=ts*(gama*h(i)*xi'*P*B+k1*gama*norm(xi)* (w_1(i,1)+1/8*k1+3/8*k3));

    k5=ts*(gama*h(i)*xi'*P*B+k1*gama*norm(xi)* (w_1(i,1)+1/2*k1-3/2*k3+2*k4));

    w(i,1)=w_1(i,1)+1/6*(k1+4*k4+k5);

  end

end

h=zeros(5,1);

for j=1:1:5

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

end

fn(k)=w'*h;

M=10;

tol1(k)=M*(ddqd(k)-kv*de(k)-kp*e(k));

d(k)=-15*dq(k)-30*sign(dq(k));

f(k)=d(k);

F=3;

if F==1      %No compensation

  fn(k)=0;

  tol(k)=tol1(k);

elseif F==2 %Modified computed torque controller

  fn(k)=0;

  tol2(k)=-f(k);

  tol(k)=tol1(k)+tol2(k);

elseif F==3 %RBF compensated controller

  tol2(k)=-fn(k);

  tol(k)=tol1(k)+1*tol2(k);

end

q_1=q(k);

dq_1=dq(k);

w_1=w;

tol_1=tol(k);

end

figure(1);

subplot(211);

plot(time,qd,'r',time,q,'k:','linewidth',2);

xlabel('time(s)');ylabel('Position tracking');

legend('ideal position','position tracking');

subplot(212);

plot(time,dqd,'r',time,dq,'k:','linewidth',2);

xlabel('time(s)');ylabel('Speed tracking');

legend('ideal speed','speed tracking');

figure(2);

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

xlabel('time(s)'),ylabel('Control input of single link');

if F==2|F==3

  figure(3);

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

  xlabel('time(s)'),ylabel('f and fn');

  legend('Practical uncertainties','Estimation uncertainties');

end

Program for plant: chap9_2plant.m

function dx=Plant(t,x,flag,para)

dx=zeros(2,1);

tol=para;

M=10;

d=-15*x(2)-30*sign(x(2));

dx(1)=x(2);

dx(2)=1/M*(tol+d);

9.1.3 Programs for Sect. 9.3.2

Main program: chap9_3.m

%Discrete RBF control for two-link manipulators

clear all;

close all;

T=0.001; %Sampling time

xk=[0 0 0 0];

tol1_1=0;

tol2_1=0;

ei=0;

node=5;

c_M=[-1 -0.5 0 0.5 1;

-1 -0.5 0 0.5 1];

c_C=[-1 -0.5 0 0.5 1;

-1 -0.5 0 0.5 1;

-1 -0.5 0 0.5 1;

-1 -0.5 0 0.5 1];

c_G=[-1 -0.5 0 0.5 1;

-1 -0.5 0 0.5 1];

b=10;

W_M11_1=zeros(node,1);W_M12_1=zeros(node,1);

W_M21_1=zeros(node,1);W_M22_1=zeros(node,1);

W_C11_1=zeros(node,1);W_C12_1=zeros(node,1);

W_C21_1=zeros(node,1);W_C22_1=zeros(node,1);

W_G1_1=zeros(node,1);W_G2_1=zeros(node,1);

Hur=[5 0;0 5];

for k=1:1:5000

if mod(k,200)==1

 k

end

time(k) = k*T;

qd1(k)=sin(2*pi*k*T);

qd2(k)=sin(2*pi*k*T);

dqd1(k)=2*pi*cos(2*pi*k*T);

dqd2(k)=2*pi*cos(2*pi*k*T);

ddqd1(k)=-(2*pi)^2*sin(2*pi*k*T);

ddqd2(k)=-(2*pi)^2*sin(2*pi*k*T);

para=[tol1_1 tol2_1];    %D/A

tSpan=[0 T];

[t,xx]=ode45('chap9_3plant',tSpan,xk,[],para); %A/D speed

xk = xx(length(xx),:); %A/D position

q1(k)=xk(1);

dq1(k)=xk(2);

q2(k)=xk(3);

dq2(k)=xk(4);

q=[q1(k);q2(k)];

z=[q1(k);q2(k);dq1(k);dq2(k)];

e1(k)=qd1(k)-q1(k);

de1(k)=dqd1(k)-dq1(k);

e2(k)=qd2(k)-q2(k);

de2(k)=dqd2(k)-dq2(k);

e=[e1(k);e2(k)];

de=[de1(k);de2(k)];

r=de+Hur*e;

dqd=[dqd1(k);dqd2(k)];

dqr=dqd+Hur*e;

ddqd=[ddqd1(k);ddqd2(k)];

ddqr=ddqd+Hur*de;

for j=1:1:node

   h_M11(j)=exp(-norm(q-c_M(:,j))^2/(b^2));

   h_M21(j)=exp(-norm(q-c_M(:,j))^2/(b^2));

  h_M12(j)=exp(-norm(q-c_M(:,j))^2/(b^2));

  h_M22(j)=exp(-norm(q-c_M(:,j))^2/(b^2));

end

for j=1:1:node

  h_C11(j)=exp(-norm(z-c_C(:,j))^2/(b^2));

  h_C21(j)=exp(-norm(z-c_C(:,j))^2/(b^2));

  h_C12(j)=exp(-norm(z-c_C(:,j))^2/(b^2));

  h_C22(j)=exp(-norm(z-c_C(:,j))^2/(b^2));

end

for j=1:1:node

   h_G1(j)=exp(-norm(q-c_G(:,j))^2/(b^2));

  h_G2(j)=exp(-norm(q-c_G(:,j))^2/(b^2));

end

T_M11=5*eye(node);

T_M21=5*eye(node);

T_M12=5*eye(node);

T_M22=5*eye(node);

T_C11=10*eye(node);

T_C21=10*eye(node);

T_C12=10*eye(node);

T_C22=10*eye(node);

T_G1=5*eye(node);

T_G2=5*eye(node);

for i=1:1:node

  W_M11(i,1)=W_M11_1(i,1)+T*T_M11(i,i)*h_M11(i) *ddqr(1)*r(1);

  W_M21(i,1)=W_M21_1(i,1)+T*T_M21(i,i)*h_M21(i) *ddqr(1)*r(2);

  W_M12(i,1)=W_M12_1(i,1)+T*T_M12(i,i)*h_M12(i) *ddqr(2)*r(1);

  W_M22(i,1)=W_M22_1(i,1)+T*T_M22(i,i)*h_M22(i) *ddqr(2)*r(2);

  W_C11(i,1)=W_C11_1(i,1)+T*T_C11(i,i)*h_C11(i) *dqr(1)*r(1);

  W_C21(i,1)=W_C21_1(i,1)+T*T_C21(i,i)*h_C21(i) *dqr(1)*r(2);

  W_C12(i,1)=W_C12_1(i,1)+T*T_C12(i,i)*h_C12(i)* ddqr(2)*r(1);

  W_C22(i,1)=W_C22_1(i,1)+T*T_C22(i,i)*h_C22(i)* ddqr(2)*r(2);

  W_G1=W_G1_1+T*T_G1(i,i)*h_G1(i)*r(1);

  W_G2=W_G2_1+T*T_G2(i,i)*h_G2(i)*r(2);

end

MSNN_g=[W_M11'*h_M11' W_M12'*h_M12';

   W_M21'*h_M21' W_M22'*h_M22'];

GSNN_g=[W_G1'*h_G1';

   W_G2'*h_G2'];

CDNN_g=[W_C11'*h_C11' W_C12'*h_C12';

   W_C21'*h_C21' W_C22'*h_C22'];

tol_m=MSNN_g*ddqr+CDNN_g*dqr+GSNN_g;

Kp=[20 0;0 20];

Ki=[20 0;0 20];

Kr=[1.5 0;0 1.5];

Kp=[100 0;0 100];

Ki=[100 0;0 100];

Kr=[0.1 0;0 0.1];

ei=ei+e*T;

tol=tol_m+Kp*r+Kr*sign(r)+Ki*ei;

tol1(k)=tol(1);

tol2(k)=tol(2);

W_M11_1=W_M11;

W_M21_1=W_M21;

W_M12_1=W_M12;

W_M22_1=W_M22;

W_C11_1=W_C11;

W_C21_1=W_C21;

W_C12_1=W_C12;

W_C22_1=W_C22;

W_G1_1=W_G1;

W_G2_1=W_G2;

tol1_1=tol1(k);

tol2_1=tol2(k);

end

figure(1);

subplot(211);

plot(time,qd1,'r',time,q1,'k:','linewidth',2);

xlabel('time(s)'),ylabel('Position tracking of link 1');

legend('ideal position','position tracking');

subplot(212);

plot(time,qd2,'r',time,q2,'k:','linewidth',2);

xlabel('time(s)'),ylabel('Position tracking of link 2');

legend('ideal position','position tracking');

figure(2);

subplot(211);

plot(time,dqd1,'r',time,dq1,'k:','linewidth',2);

xlabel('time(s)'),ylabel('Speed tracking of link 1');

legend('ideal speed','speed tracking');

subplot(212);

plot(time,dqd2,'r',time,dq2,'k:','linewidth',2);

xlabel('time(s)'),ylabel('Speed tracking of link 2');

legend('ideal speed','speed tracking');

figure(3);

subplot(211);

plot(time,tol1,'r','linewidth',2);

xlabel('time(s)'),ylabel('Control input of link 1');

subplot(212);

plot(time,tol2,'r','linewidth',2);

xlabel('time(s)'),ylabel('Control input of link 2');

Program for plant: chap9_3plant.m

function dx=Plant(t,x,flag,para)

%%%%%%%%%%%%%% x(1)=q1; x(2)=dq1; x(3)=q2; x(4)=dq2; %%%%%%%%%%%%

dx=zeros(4,1);

p=[2.9 0.76 0.87 3.04 0.87];

g=9.8;

M0=[p(1)+p(2)+2*p(3)*cos(x(3)) p(2)+p(3)*cos(x(3));

p(2)+p(3)*cos(x(3)) p(2)];

C0=[-p(3)*x(4)*sin(x(3)) -p(3)*(x(2)+x(4))*sin(x(3));

p(3)*x(2)*sin(x(3)) 0];

G0=[p(4)*g*cos(x(1))+p(5)*g*cos(x(1)+x(3));

p(5)*g*cos(x(1)+x(3))];

tol=para(1:2);

dq=[x(2);x(4)];

ddq=inv(M0)*(tol'-C0*dq-G0);

%%%%%%% dx(1)=dq1; dx(2)=ddq1; dx(3)=dq2; dx(4)=ddq2; %%%%%%%%%%%%%

dx(1)=x(2);

dx(2)=ddq(1);

dx(3)=x(4);

dx(4)=ddq(2);