matlab arx模型,ARARX模型的辨识算法.doc

ARARX模型的辨识算法

Homework

Recursive extended least squares identification

In this section, I focus on the following ARMAX model

where {} and {} are respectively input and output series, is white noise, and are some polynomials with respect to the backward operator :

, , and for t ≤ 0.

I use the residual based recursive extened least squares algorithm.

The procedure of MATLAB:

clc;clear all;

t=500;

for k=1:t

u(k)=1;

end

v=randn(1,t)/100;

y(1)=v(1);y(2)=1.5*y(1)+0.5*u(1)+v(2)-v(1);

y(3)=1.5*y(2)-0.7*y(1)+0.5*u(1)+u(2)+v(3)-v(2)+0.2*v(1);

for k=4:t

y(k)=1.5*y(k-1)-0.7*y(k-2)+u(k-1)+0.5*u(k-2)+v(k)-v(k-1)+0.2*v(k-2);

end

s=zeros(6,t);

q=zeros(6,t);

p=100000*eye(6);

v1=zeros(1,t);

v1(1)=y(1);

q(1:6,2)=[-y(1) 0 u(1) 0 v1(1) 0];

L=p*q(1:6,2)/(1+q(1:6,2)'*p*q(1:6,2));

s(1:6,2)=L*y(2);

v1(2)=y(2)-q(1:6,2)'*s(1:6,2);

for k=3:t

q(1:6,k)=[-y(k-1) -y(k-2) u(k-1) u(k-2) v1(k-1) v1(k-2)];

L=p*q(1:6,k)/(1+q(1:6,k)'*p*q(1:6,k));

s(1:6,k)=s(1:6,k-1)+L*(y(k)-q(1:6,k)'*s(1:6,k-1));

p=(eye(6)-L*q(1:6,k)')*p;

v1(k)=y(k)-q(1:6,k)'*s(1:6,k);

end

a1=-1.5*ones(1,t);a2=0.7*ones(1,t);b1=ones(1,t);

b2=0.5*ones(1,t);d1=-ones(1,t);d2=0.2*ones(1,t);

i=1:1:t;

figure(1)

subplot(311);

plot(i,a1,'b',i,s(1,1:t),'r');legend('blue--给定参数a1 red--估计参数a11');

subplot(312);

plot(i,a2,'b',i,s(2,1:t),'r');legend('blue--给定参数a2 red--估计参数a22');

subplot(313);

plot(i,b1,'b',i,s(3,1:t),'r');legend('blue--给定参数b1 red--估计参数b11');

figure(2)

subplot(311);

plot(i,b2,'b',i,s(4,1:t),'r');legend('blue--给定参数b2 red--估计参数b22');

subplot(312);

plot(i,d1,'b',i,s(5,1:t),'r');legend('blue--给定参数d1 red--估计参数d11');

subplot(313);

plot(i,d2,'b',i,s(6,1:t),'r');legend('blue--给定参数d2 red--估计参数d22');

e1=zeros(1,t);e2=zeros(1,t);e3=zeros(1,t);e4=zeros(1,t);e5=zeros(1,t);e6=zeros(1,t);

for k=1:t

e1(k)=norm(s(1,k)+1.5,2)/norm(-1.5,2);

e2(k)=norm(s(2,k)-0.7,2)/norm(0.7,2);

e3(k)=norm(s(3,k)-1,2)/norm(1,2);

e4(k)=norm(s(4,k)-0.5,2)/norm(0.5,2);

e5(k)=norm(s(5,k)+1,2)/norm(-1,2);

e6(k)=norm(s(6,