% Example ID3.2 Illustrate properties of least-square estimate
clc;echo on
% Copyright 1996, Babu Joseph, Washington University in St.Louis
%
% This program illustrates what happens when there is not sufficient 
% excitation in the input signal
%
% First create some data by exciting a process
% We will use a first order process 
% with gain 1 and time constant 5
num=[1];den=[5 1 ];
[numd,dend]=c2dm(num,den,1,'zoh');
% Define input as a random  binary sequence
t=0:1:10;u=0.1*ones(10,1);
y=dlsim(numd,dend,u) +randn(10,1);
% idplot([y u]);
% Now generate the matrices for least square estimation of parameters
%
A=[-y(1:9) u(1:9)];b=y(2:10);
%
% The estimated coefficients of the transfer function are
%
x=A\b
%
% The following are the actual coefficients
%
xactual=[ dend(2) numd(2)]'
%
% Computed coefficients can be compared to actual coefficients
%
e=b-A*x;norm(e)
mean(e)
cond(A)
% accuracy of the parameters are given by
var=inv(A'*A)*1.0
% where we have used the fact that the variance of the noise is 1
% the diagonal elements of var gives the variance in first and second parameter

% idplot([y u])