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Adaptive Control
Assignment 1
System Identification
姓 名: ****
学 号: *************
班 级: ***********
Answers：
1. a) Obtain the system model equation and write it in linear regression form.
The system model equation:
It’s auto regressive form:
b) Simulate the system by generating 1000 data points. Plot u(t) and y(t).
c) Obtain the least squares estimator for this system.
The least squares estimator for the parameter vector is:
The estimated value of system parameters are:
2.
a) Generate any input and get the response. Plot u(t) and y(t). Ignore the system noise
The ARX models：
It’s auto regressive form:
When input is a step function, the output is:
When input is a sin wave, the output is:
b) Write a recursive least squares program to identify this model and test your program.
The least squares estimate can be obtained from：
The estimated value of system parameters are:
Test my recursive least squares program ：
Clearly, the response with the least squares estimate is almost as same as the original system response.
c) Test the response and the recursive least squares program if a white noise is added.
Obviously, the response with the least squares estimate is almost as same as the original system response. So I think it is predicting the correct system parameters.
d) Comment on how different types of inputs, initial LN, and length of data affect the final estimation.
Case one: recursive least squares:
recursive least squares value A Step function signal error llEll2 A sin wave signal error llEll2 The
Estimated
value
of system
parameters
LN=1e+6*I
W=
-1.5363
0.8607
0.0416
0.0395
0.00148
0.00454 The
Estimated
value
of system
parameters
LN=1e+5*I W=
-1.5363
0.8607
0.0416
0.0395
0.00155 0.044297 The
Estimated
value
of system
parameters
LN=1e+4*I W=
-1.5363
0.8607
0.0416
0.0395
0.00478 0.356367 The
Estimated
value
of system
parameters
LN=1e+3*I W=
-1.5363
0.8607
0.0416
0.0395
0.04446 1.211363
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