RLS_Pure : Recursive Least Square Identification with Noise

This program objective is to identified parameters of third order transfer function. In real-life application usually there's some noise add in recorded signal. In this example we use the same transfer funtion, and still the same input step, only this time we add some noise. Author : Otniel Adrian Rachmat Contact information : Department of Electrical Engineering, Atma Jaya Catholic University Jendral Sudirman 51, Jakarta 12930 , Indonesia . email :[e-mail] email : [e-mail] Date : 09-06-2009 Version : 1.0 Reference : (1) System Modeling and Identification, Prentice Hall, Rolf Johansson,1993 (2) Neuro-Fuzzy and Soft Computing, J.-S.R.Jang, C.-T.Sun, E.Mizutani,1997 (3) FIR_RLS.m-file, Tamer Abdelazim Mellik, version 1.1.0, Note : The author doesn't take any responsibility for any harm caused by the use of this file

Contents

• Generate Transfer Function
• Simulate Input-Output Data
• Noise addition
• RLS Algorithm
• Comparison between real and identified transfer function
• Conclusion

Generate Transfer Function

In this example we try to identified some parameters in third order transfer function. First we generate some transfer function, using an identified parameters manually. Hence we use butter to generate third order lowpass digital Butterworth filter with determine cutoff frequency.

[num,den] = butter(3,0.75)
r_param = [num,den(2:end)];
Gz = tf(num,den,0.01)
ltiview('step',Gz);

num =

    0.4459    1.3377    1.3377    0.4459


den =

    1.0000    1.4590    0.9104    0.1978

 
Transfer function:
0.4459 z^3 + 1.338 z^2 + 1.338 z + 0.4459
-----------------------------------------
   z^3 + 1.459 z^2 + 0.9104 z + 0.1978
 
Sampling time: 0.01

Simulate Input-Output Data

Now that we got the transfer function, we must specified some input data to simulate identification process. Normally in identification we record a sequence of input-output data, in this term of case we use step input.

n_input = 10;       % Number of input data each step
n_step = 10;        % Number of step signal
sample = 4*n_step*n_input;
zero = zeros(n_input,1);
one = ones(n_input,1);
onen = one.*-1;
data = [zero;one;zero;onen];
in_data = [];
for n=1:n_step
    in_data = [in_data;data];
end
out_data = lsim(Gz,in_data);

figure
hold on;
plot(in_data,'b');
plot(out_data,'r');
title('Input-Output Data Identification') ;
xlabel('Samples')
ylabel('Input-Output Signal')

Noise addition

To get a better understanding of the real system we add some noise in the output signal we got.

out_sample = lsim(Gz,in_data);
noise = randn(sample,1);
noise = noise * std(out_sample)/(15*std(noise));
out_data = out_sample+noise;

figure
hold on;
plot(out_sample,'b');
plot(out_data,'r');
title('Output Pure-Noise Data Identification') ;
xlabel('Samples')
ylabel('Pure-Noise Signal')

RLS Algorithm

After we obtain input-output data pairs, we try to identified parameters of the original transfer function. This can be done by applied Recursive Least-Square with input-output data pairs. Vk as Input data at k-times. Wk as Output data at k-times. W(z) b0*Z^3 + b1*Z^2 + b2*Z + b3 ---- = ------------------------------ V(z) a0*Z^3 + a1*Z^2 + a2*Z + a3 Base on the function above, we must identified the parameters such as b0, b1, b2, b3, a1, a2, a3.

in_order = 4;
out_order = 3;
sys_order = in_order + out_order;

delta = 1e2;
Po = delta*eye(sys_order);
Qo = zeros(sys_order,1);
lamda = 0.999;

for n=4:sample
    for i=1:3
        W(i) = out_data(n-i);
    end
    for i=1:4
        V(i) = in_data(n-i+1);
    end
    reg = [V';W'];
    sample_out = out_data(n);
    est_out(n)= reg' * Qo;
    error(n) = out_data(n) - est_out(n);
    Po = (Po - ( (Po*reg*reg'*Po)/( lamda+(reg'*Po*reg) ) ))/lamda;
    Qo = Qo + (Po*reg* (sample_out - (reg'*Qo) ) );
    Recordedw(1:sys_order,n) = Qo;
end
f_param = [Qo(1:4);-Qo(5:7)];
numz = [Qo(1:4,1)];
denz = [1;-Qo(5:7,1)];
Fz = tf(numz',denz',0.01);

figure
hold on
plot(r_param,'b*');
plot(f_param,'R+');
title('Parameters system') ;
xlabel('Order')
ylabel('True and estimated output')

figure
plot(Recordedw(1:sys_order,sys_order:sample)');
title('Estimated parameters convergence') ;
xlabel('Samples');
ylabel('Parameter value');
axis([1 (sample)-sys_order min(min(Recordedw(1:sys_order,sys_order:(sample))')) max(max(Recordedw(1:sys_order,sys_order:(sample))')) ]);

figure
semilogy((abs(error))) ;
title('Error curve') ;
xlabel('Samples');
ylabel('Error value');

figure
hold on
plot(out_data);
plot(est_out,'r');
title('System output') ;
xlabel('Samples')
ylabel('True and estimated output')

Comparison between real and identified transfer function

In this comparison we found the identified parameters has a distinct value to the real one.

Gz
Fz
ltiview('step',Gz,Fz);

 
Transfer function:
0.4459 z^3 + 1.338 z^2 + 1.338 z + 0.4459
-----------------------------------------
   z^3 + 1.459 z^2 + 0.9104 z + 0.1978
 
Sampling time: 0.01
 
Transfer function:
0.4569 z^3 + 0.7619 z^2 + 0.007032 z - 0.3231
---------------------------------------------
    z^3 + 0.1872 z^2 - 0.09284 z - 0.194
 
Sampling time: 0.01

Conclusion

Using RLS we may identified parameters of transfer function by means of its input-output data. However, when there's noise in the same signal, we won't be able to get the exact desire value. But at least we got the closest system. Then, is there any way to got the desire ones?

Published with MATLAB® 7.1