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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