Neural Network Sliding Mode Control

Abstract

This chapter introduces adaptive neural sliding mode control based on RBF neural network approximation, including a simple sliding mode controller and sliding mode control for second-order SISO nonlinear system, the chattering phenomena is eliminated. The closed-loop system stability can be achieved based on the Lyapunov stability.

Appendix

5.1.1 Programs for Sect. 5.2.3

Main Simulink program: chap5_1sim.mdl

Control law program: chap5_1ctrl.m

function [sys,x0,str,ts] = spacemodel(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

global cij bj c

sizes = simsizes;

sizes.NumContStates = 5;

sizes.NumDiscStates = 0;

sizes.NumOutputs = 2;

sizes.NumInputs = 2;

sizes.DirFeedthrough = 1;

sizes.NumSampleTimes = 0;

sys = simsizes(sizes);

x0 = 0*ones(1,5);

str = [];

ts = [];

cij=0.10*[-1 -0.5 0 0.5 1;

-1 -0.5 0 0.5 1];

bj=5.0;

c=15;

function sys=mdlDerivatives(t,x,u)

global cij bj c

e=u(1);

de=u(2);

s=c*e+de;

xi=[e;de];

h=zeros(5,1);

for j=1:1:5

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

end

gama=0.015;

W=[x(1) x(2) x(3) x(4) x(5)]';

for i=1:1:5

sys(i)=-1/gama*s*h(i);

end

function sys=mdlOutputs(t,x,u)

global cij bj c

e=u(1);

de=u(2);

thd=0.1*sin(t);

dthd=0.1*cos(t);

ddthd=-0.1*sin(t);

x1=thd-e;

s=c*e+de;

W=[x(1) x(2) x(3) x(4) x(5)]';

xi=[e;de];

h=zeros(5,1);

for j=1:1:5

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

end

fn=W'*h;

g=9.8;mc=1.0;m=0.1;l=0.5;

S=l*(4/3-m*(cos(x1))^2/(mc+m));

gx=cos(x1)/(mc+m);

gx=gx/S;

if t<=1.5

xite=1.0;

else

xite=0.10;

end

ut=1/gx*(-fn+ddthd+c*de+xite*sign(s));

sys(1)=ut;

sys(2)=fn;

Plant program: chap5_1plant.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 = 2;

sizes.NumInputs = 1;

sizes.DirFeedthrough = 0;

sizes.NumSampleTimes = 0;

sys=simsizes(sizes);

x0=[pi/60 0];

str=[];

ts=[];

function sys=mdlDerivatives(t,x,u)

g=9.8;mc=1.0;m=0.1;l=0.5;

S=l*(4/3-m*(cos(x(1)))^2/(mc+m));

fx=g*sin(x(1))-m*l*x(2)^2*cos(x(1))*sin(x(1))/(mc+m);

fx=fx/S;

gx=cos(x(1))/(mc+m);

gx=gx/S;

%%%%%%%%%

dt=0*10*sin(t);

%%%%%%%%%

sys(1)=x(2);

sys(2)=fx+gx*u+dt;

function sys=mdlOutputs(t,x,u)

g=9.8;

mc=1.0;

m=0.1;

l=0.5;

S=l*(4/3-m*(cos(x(1)))^2/(mc+m));

fx=g*sin(x(1))-m*l*x(2)^2*cos(x(1))*sin(x(1))/(mc+m);

fx=fx/S;

sys(1)=x(1);

sys(2)=fx;

Plot program: chap5_1plot.m

close all;

figure(1);

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

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

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

figure(2);

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

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

figure(3);

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

xlabel('time(s)');ylabel('fx and estiamted fx');

legend('fx','estiamted fx');

5.1.2 Programs for Sect. 5.3.2

Main Simulink program: chap5_2sim.mdl

Control law program: chap5_2ctrl.m

function [sys,x0,str,ts] = spacemodel(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

global xite cij bj h c

sizes = simsizes;

sizes.NumContStates = 10;

sizes.NumDiscStates = 0;

sizes.NumOutputs = 3;

sizes.NumInputs = 5;

sizes.DirFeedthrough = 1;

sizes.NumSampleTimes = 0;

sys = simsizes(sizes);

x0 = 0.1*ones(10,1);

str = [];

ts = [];

cij=[-1 -0.5 0 0.5 1;

-1 -0.5 0 0.5 1];

bj=5;

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

c=5;

xite=0.01;

function sys=mdlDerivatives(t,x,u)

global xite cij bj h c

thd=u(1);

dthd=0.1*cos(t);

ddthd=-0.1*sin(t);

x1=u(2);

x2=u(3);

e=thd-x1;

de=dthd-x2;

s=c*e+de;

xi=[x1;x2];

for j=1:1:5

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

end

for i=1:1:5

wf(i,1)=x(i);

end

for i=1:1:5

wg(i,1)=x(i+5);

end

fxn=wf'*h';

gxn=wg'*h'+0.01;

ut=1/gxn*(-fxn+ddthd+xite*sign(s)+c*de);

gama1=10;gama2=1.0;

S1=-gama1*s*h;

S2=-gama2*s*h*ut;

for i=1:1:5

sys(i)=S1(i);

end

for j=6:1:10

sys(j)=S2(j-5);

end

function sys=mdlOutputs(t,x,u)

global xite cij bj h c

thd=u(1);

dthd=0.1*cos(t);

ddthd=-0.1*sin(t);

x1=u(2);

x2=u(3);

e=thd-x1;

de=dthd-x2;

s=c*e+de;

for i=1:1:5

wf(i,1)=x(i);

end

for i=1:1:5

wg(i,1)=x(i+5);

end

xi=[x1;x2];

for j=1:1:5

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

end

fxn=wf'*h';

gxn=wg'*h'+0.01;

ut=1/gxn*(-fxn+ddthd+xite*sign(s)+c*de);

sys(1)=ut;

sys(2)=fxn;

sys(3)=gxn;

Plant program: chap5_2plant.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 = 4;

sizes.NumInputs = 1;

sizes.DirFeedthrough = 0;

sizes.NumSampleTimes = 0;

sys=simsizes(sizes);

x0=[pi/60 0];

str=[];

ts=[];

function sys=mdlDerivatives(t,x,u)

g=9.8;mc=1.0;m=0.1;l=0.5;

S=l*(4/3-m*(cos(x(1)))^2/(mc+m));

fx=g*sin(x(1))-m*l*x(2)^2*cos(x(1))*sin(x(1))/(mc+m);

fx=fx/S;

gx=cos(x(1))/(mc+m);

gx=gx/S;

sys(1)=x(2);

sys(2)=fx+gx*u;

function sys=mdlOutputs(t,x,u)

g=9.8;mc=1.0;m=0.1;l=0.5;

S=l*(4/3-m*(cos(x(1)))^2/(mc+m));

fx=g*sin(x(1))-m*l*x(2)^2*cos(x(1))*sin(x(1))/(mc+m);

fx=fx/S;

gx=cos(x(1))/(mc+m);

gx=gx/S;

sys(1)=x(1);

sys(2)=x(2);

sys(3)=fx;

sys(4)=gx;

Plot program: chap5_2plot.m

close all;

figure(1);

subplot(211);

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

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

legend('Ideal position signal','Position signal tracking');

subplot(212);

plot(t,0.1*cos(t),'r',t,x(:,3),'k:','linewidth',2);

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

legend('Ideal speed signal','Speed signal tracking');

figure(2);

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

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

figure(3);

subplot(211);

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

xlabel('time(s)');ylabel('f and estiamted f');

legend('True f','Estimation f');

subplot(212);

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

xlabel('time(s)');ylabel('g and estimated g');

legend('True g','Estimation g');