# Hand-in Problems

# FRTN35 2019

### Instructions:

FRTN35 System Identification course includes three mandatory hand-in problem sets, which should be solved individually. The problem sets are composed of exercises requiring analytic calculations as well as numerical calculations performed in Matlab or Julia. The purpose of the hand-in problem sets is to give You the opportunity to develop Your skills and get feedback from Your TA. The hand-in problem sets are also designed to help You prepare for the final exam and the course project.

Present Your results in a pdf report including well motivated answers to the problems stated in the problem text. **Notice that the interpretation of Your results should be an integral part of the solution.** Plots and hand-written calculations may be included as appendices and referred to in the solution text. Calculations should be clearly written and easy to follow. Number all equations for easy reference.

### Hand-in problem 1

#### 1.

Compute the transfer function Y(z)/U(z) and the noise spectrum of the system

*S*: y(k) = 1.5y(k-1) -0.9y(k-2) + 0.4u(k-1) + w(k) + 0.7w(k-1)

where {w(k)} is a zero-mean uncorrelated noise sequence with variance Var(w(k))=1. Include in your report a graphical spectral representation of the transfer function and the noise spectrum, also make sure you can differentiate between the concepts *spectrum* and *bode diagram*.

#### 2.

Consider the continuous time signal y = sin(3*t)+10*sin(10*t).

- In Matlab or Julia, create a vector representing a sampled version of the signal with sampling interval t=0.1s. Use the commands
`fft/pwelch/periodogram`to calculate the power spectrum of the sampled signal (in Julia, these are found in the package DSP.jl). Explain in a few words the relation between the function y and the characteristics of the spectrum. In particular, comment on location and magnitude of resonance peaks.

- Create a new vector representing the signal y sampled using the sampling interval t=0.5s. Again calculate and plot the power spectrum of the sampled signal using the command
`fft, pwelch`, or similar. Explain the result. Does the spectrum look the way you expected?

#### 3.

(*Optional*) Compute a one-step-ahead predictor for S. Compute the expected prediction error variance of your predictor.

To be handed in to <FRTN35@list.control.lth.se>, deadline on September 15.

### Hand-in problem 2

#### 1.

Consider exercise 6.1 in the textbook with the following modifications

*S*: y(k) + ay(k-1) = bu(k-1) + w(k) + cw(k-2)

Show that the parameter estimates of the parameter vector (a, b)^T will be asymptotically biased. Structure your answer and motivate each step accordingly.

#### 2.

(*Optional*)

Suppose that a true description of a certain system is given by

y(k) + a_1 y(k-1) + ... + a_n y(k-n) = b_1 u(k-1) + ... + b_m (k-m) + e(k)

where e(k) is white noise independent of the input. Let the regressor phi(k) be defined as usual as

phi(k) = [-y(k-1) ... -y(k-n) u(k-1) ...u(k-m)]^T .

Moreover let phi*(k) be given by

phi*(k) = [-y0(k-1) ...-y0(k-n) u(k-1) ...u(k-m)]^T,

where y0(k) + a_1 y0(k-1) + ... + a_n y0(k-n) = b_1 u(k-1) + ... + b_m u(k-m) .

Note that y0 is the noise-free response of the true system

Prove that for the vector of instrumental variables z(k) = [u(k-1) u(k-2)]^T, we have

E [z(k) phi(k)^T ] = E [z(k) phi*(k)^T ] .

To be handed in to <FRTN35@list.control.lth.se>, deadline on September 29.

### Hand-in problem 3

#### 1.

The revised problem can be found here.

Time series data (Unix) are given by data.mat. Use Matlab command

>> load data

Time series data (Unix/Windows): u.txt , y.txt. Use Matlab command

>> load u.txt -ascii

>> load y.txt -ascii

#### 2.

(*Optional*)

You can find the optional exercise here.

To be handed in to <FRTN35@list.control.lth.se>, deadline on October 6.