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Ferdowsi University of Mashhad School of Engineering

System Identification Exercises – First Series

Due date: 1390/8/20

In this series of exercises, we are going to analyze the different aspects of linear regression and examine the performance of the least squares methods, in parameter estimation and structure identification.

Matlab is considered as the preferable environment for accomplishing this series of exercises. Accompany your report with corresponding m-file(s). Thorough explanation on the code lines and several steps of each of the parts is expected.

– Let’s consider the following polynomial, as the descriptive equation of a system.

y = 1 + u2 + u3 + u5 + u7 . (1) Assume that all we know about the structure of y = f (x) is that it is a polynomial, with a degree of lower than 10. So, we are just aware of the system’s structure, which is as follows:

y = θ1x1 + θ2 x2 + θ3 x3 + · · · + θ8x8 + θ9x9 + θ10 x10 = xT θ. (2)

– To begin with, generate a vector of 1000 random numbers, uniquely distributed in the range of [0, 1], as the inputs of the system. Calculate the corresponding outputs of the system to each of the input samples and make them pair. Keep aside 300 of these pairs, to evaluate the constructed models with and put the rest in use to train the models.

1. Estimate the model’s parameters, utilizing the LS method. Using the evaluation data, (testing data) plot the system and model’s output as well as error for e ∈ [0, 1]. Calculate some of squared error (SSE). Where does this error originate from?

2. Now, make an effort to solve the problem of selecting the most significant regressors, using the OLS. Increase the number of regressors from one to ten. Show the variations of the output error, as the number of regressors vary. Force a suitable condition to separate the efficient regressors from the rest. Is the achieved structure match with anticipated according to the original system? How much error could be ignored, if we rule out the inefficient regressors?

3. Assuming the model structure to be as the system is, estimate the model’s parameters, employing the RLS. Index the 700 pairs of training samples (ui , yi ) randomly and take their order as the time indices. Observe the gradual convergence courses of θ(t) and P (t).

yˆ = θ1x1 + θ3x3 + θ4x4 + θ6x6 + θ8 x8.

4. This time, consider the 4th parameter, θ4 , to be time-variant, as it varies from 1 to 2, as the training time goes from 1 to 700. Now, try to identify the system using the RLS. To do so, you should generate the new (u, y) pairs in advance, as you have done before. Like the previous part, observe the gradual convergence courses of θ(t) and P (t).

5. Now, give it a try to improve the estimation of the 4th part, using the Forgetting Factor and the Kalman Filter. Fix the Forgetting Factor of λ and the co-variance matrix of Q appropriately, so that they could disclose their most effective results. Raise the pace of the θ4 variations, so that you could distinguish the difference between three methods, if you could have not already.

6. Contaminate the system with Gaussian noise.

y = 1 + u2 + u3 + u5 + u7 + n; n ∼ N (0, σ2)

Examine the effect of minor, average and major noise on the estimation results of the 2nd and 5th parts. Which method is more successful in modeling of the noise?

7. In part 6 let σ2 =1, use the Regularization method and choose the best α, to estimate the parameters. Study the effect of α variation on the bias and covariance of the parameter estimation. Plot the system and model’s output, error and error bars and compare the results with what you have obtained from the first part. Is there any chance to place a threshold on |θi |, so that you could mark the regressors with two labels of significant and insignificant?

8. Restrict the training data between 0 and 0.7 (e ∈ [0, 0.7]), while you keep the testing data as before, e ∈ [0, 1]. repeat the 4th part for the new experiment conditions and plot the model and system’s outputs, and the error bars. Observe the effect of this restriction on the variance of the error.

Optional part:

Consider a black box system identification problem where only input/output measurement data is available. Assume that all we know about the structure of y = f (x) is that it is a polynomial, with a degree of lower than 8. So, we are just aware of the system’s structure, which is as follows:

y = θ1x1 + θ2 x2 + θ3 x3 + · · · + θ8x8 = xT θ.

Use Matlab data file “measurement_data.mat” which contains 1000 input/output data pairs for identification. Keep aside 300 of these pairs, to evaluate the constructed model with and put the rest in use to train the model. In the new conditions, repeat the first and 2nd part to identify the new system.

❖ Accompany your report with corresponding m-file(s) and email them to: sralizadeh@

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