Exam practice

Model Identification and Data Analysis — Module 2 — Practice

Past-exam questions, filterable by chapter, year and difficulty. Answer for instant grading, reveal the worked solution, and watch your accuracy climb.

Seen 0 / 76 · 0 correct · 0% accuracy

Chapter

Year

Difficulty

2026-02-q1a2026Q01Pole-zero cancellation and hidden modesmedium5 pts
Consider the system whose input/output behaviour is described by: $$y(t)=\frac{\left(1-2z^{-1}\right)}{1-\tfrac52z^{-1}+z^{-2}}u(t-2)$$ a. Is the system asymptotically stable? Is it strictly proper? b. Identify the system matrices from the transfer function. c. Provide the analytical expression $\omega(t)$ as a function of time for the system.
2026-02-q1b2026Q01Hankel rank and system orderhard6 pts
Given the following impulse-response coefficients: $$\omega(0)=\tfrac43,\;\omega(1)=0,\;\omega(2)=0,\;\omega(3)=3,\;\omega(4)=-\tfrac32,\;\omega(5)=\tfrac34,\;\omega(6)=-\tfrac38,\;\omega(7)=\tfrac3{16}$$ a. Identify the system order. b. Identify the system matrices using the 4SID method.
2026-06-q1a2026Q01Controllable canonical formeasy5 pts
Given the system $$\begin{cases}x_1(t+1)=-x_1(t)-x_2(t)\\ x_2(t+1)=2x_1(t)+3x_2(t)+2u(t)\\ y(t)=-2x_1(t)-x_2(t)+u(t)\end{cases}$$ a. Compute the transfer function of the system. b. Write a state-space realization using the controllable canonical form.
2026-06-q1b2026Q014SIDmedium5 pts
Given the (noiseless) impulse-response samples $$\omega(0)=-7,\quad \omega(1)=2,\quad \omega(2)=-1,\quad \omega(3)=-4,\quad \omega(4)=2,\quad \omega(5)=8$$ a. Identify the system matrices using the 4SID method. b. Compute the observability and reachability matrices of the identified system.
2026-06-q1c2026Q01MATLAB for identificationeasy2 pts
In MATLAB, define the matrices $F=\begin{bmatrix}-1&1\\2&1\end{bmatrix}$, $G=\begin{bmatrix}3\\1\end{bmatrix}$, $H=[0\;\;1]$, $D=0$ and build their corresponding discrete-time state-space system with sampling time `Ts`. Then compute the observability matrix of the system.
```
F = [-1 1; 2 1];
G = [3; 1];
H = [0 1];
D = 0;
sys = ss(F, G, H, D, Ts);
O = obsv(sys);
```
2026-07-q1a2026Q01Realization non-uniquenessmedium6 pts
A system is subject to the input $$u(t)=\begin{cases}-\tfrac12 & \text{for } t=1\\ 0 & \text{otherwise}\end{cases}$$ The response of the system is $$y(0)=0,\;y(1)=\tfrac12,\;y(2)=-\tfrac14,\;y(3)=-1,\;y(4)=-\tfrac34,\;y(5)=\tfrac14,\;y(6)=1$$ a. Identify the order of the system. b. Identify the system matrices using the 4SID method. c. Compute a different state-space realization using the non-singular matrix $T=\begin{bmatrix}1&1\\0&1\end{bmatrix}$.
2026-07-q1b2026Q01Pole-zero cancellation and hidden modesmedium5 pts
Consider the system whose input/output behaviour is described by $$y(t)=\left[\frac{\tfrac13z^{-1}+z^{-2}}{(3+z)\left(1-\tfrac13z^{-2}\right)}\right]u(t)$$ a. Is the system asymptotically stable? Is it strictly proper? Motivate your answers. b. Provide the analytical expression of the impulse response $\omega(t)$ as a function of time. c. In MATLAB, you are given the transfer function `W` of the system. Compute its poles and zeros, and store its impulse-response values in the vector `omega`.
```
W_poles = pole(W);
W_zeros = zero(W);
[omega, t] = impulse(W);
```
2026-02-q2a2026Q02DRE and AREmedium6 pts
Consider the following system: $$\begin{cases}x(t+1)=2x(t)+v(t)\\ y(t)=x(t)+w(t)\end{cases}$$ with $v(t)\sim\mathrm{WN}\!\left(0,\tfrac34\right)$, $w(t)\sim\mathrm{WN}\!\left(0,\tfrac12\right)$ and $\mathbb E\!\left[w(t)^\top v(t)\right]=\tfrac12$. a. Can you apply the asymptotic theorems? b. Compute the DRE, the ARE solution(s) and graphically check the convergence to the ARE solution(s).
2026-02-q2b2026Q02Predictor vs filtermedium4 pts
Consider the system described by the following matrices: $$F=2,\quad H=-1,\quad V_1=1,\quad V_2=\tfrac25,\quad V_{12}=-\tfrac12$$ Knowing that the DRE converges to the only solution of the ARE $\bar P=\tfrac12$, answer the following questions: a. Compute the asymptotic Kalman gain and assess the filter stability. b. Compute the asymptotic state-error covariance for the **filter** (not predictor).
2026-06-q2a2026Q02Asymptotic KF theoremshard7 pts
Consider the system $$\begin{cases}x(t+1)=(\alpha-1)x(t)+(1-\beta)v(t)\\ y(t)=x(t)+\beta v(t)+w(t)\end{cases}$$ with $v(t)\sim\mathrm{WN}(0,1)$, $w(t)\sim\mathrm{WN}(0,2)$, $v(t)\perp w(t)$, $\alpha\in\mathbb R$, $\beta\in\mathbb R$. a. Find $F$, $H$, $V_1$, $V_2$ and $V_{12}$. b. For which values of $\alpha$ and $\beta$ can you apply the first and second asymptotic theorems, respectively? c. Set $\alpha=0$ and $\beta=0$. Analyze graphically the convergence of the DRE to the unique steady-state solution $\bar P=2$ (no need to check the stability of the filter).
2026-06-q2b2026Q02Predictor vs filtermedium4 pts
Consider the system described by the following matrices: $$F=2,\quad H=-1,\quad V_1=4,\quad V_2=2,\quad V_{12}=-1$$ Knowing that the DRE converges to the only solution of the ARE $\bar P=7$, answer the following questions: a. Compute the asymptotic Kalman gain and assess the filter stability. b. Compute the asymptotic value of $\mathrm{Var}\!\left(x(t)-\hat x(t|t)\right)$ (state **filter**, not predictor).
2026-07-q2a2026Q02Noise covariances V1, V2, V12hard6 pts
Consider the following system $$\begin{cases}x(t+1)=x(t)+\sqrt3\,v(t)-\tfrac{\sqrt3}{2}w(t)\\ y(t)=2x(t)+\tfrac1{\sqrt3}w(t)\end{cases}$$ with $v\sim\mathrm{WN}(0,1)$, $w\sim\mathrm{WN}(0,1)$ and $\mathbb E\!\left[v(t)w(t)^\top\right]=1$. a. Compute $V_1$, $V_2$ and $V_{12}$. b. Compute the DRE. Compute also the ARE solution(s). c. Graphically check the convergence to the ARE solution(s).
2026-07-q2b2026Q02Predictor vs filtermedium4 pts
Consider the system described by the following matrices: $$F=1,\quad H=-2,\quad V_1=\tfrac13,\quad V_2=\tfrac14,\quad V_{12}=-\tfrac14$$ Knowing that the DRE converges to the only solution of the ARE $\bar P=\tfrac18$, answer the following questions: a. Compute the asymptotic Kalman gain and assess the filter stability. b. Compute the transfer function from $y(t)$ to $\hat y(t+1|t)$.
2026-02-q3a2026Q03MVC assumptions and canonical formhard9 pts
Consider the following process: $$y(t)=\frac{1+\beta z^{-1}}{1+\tfrac\beta4 z^{-1}}u(t-1)+\frac{1+3z^{-1}}{1+\tfrac\beta4 z^{-1}}e(t)$$ with $e\sim\mathrm{WN}(0,1)$ and $\beta\in\mathbb R$. a. For which values of $\beta\in\mathbb R$ is the stochastic part of the system in canonical form? If necessary, manipulate the system to satisfy the hypotheses of the canonical form. Explicitly check all canonical-form hypotheses. b. For the values of $\beta$ that satisfy point (a), verify the remaining MVC hypotheses and find the values of $\beta$ that satisfy them all. Explicitly mention all MVC hypotheses. c. Find the Minimum Variance Controller when $\beta$ is chosen to satisfy the MVC assumptions. Keep $\beta$ as a generic parameter.
2026-06-q3a2026Q03All-pass repairmedium7 pts
Consider the ARMAX process $$y(t)=-\tfrac12 y(t-1)+u(t-1)+\tfrac13 u(t-2)+e(t)+3e(t-1)$$ with $e\sim\mathrm{WN}(0,1)$. a. Manipulate the system to satisfy all MVC assumptions. Explicitly mention all MVC hypotheses and check if they are satisfied. b. Find the Minimum Variance Controller.
2026-06-q3b2026Q03MVC closed-loop transfer functionsmedium5 pts
Consider the following process and its minimum variance controller: $$y(t)=\frac{1-\tfrac14z^{-1}}{1+\tfrac15z^{-1}}u(t-1)+\frac{1+\tfrac12z^{-1}}{1+\tfrac15z^{-1}}e(t)$$ $$u(t)=\frac{1}{1-\tfrac14z^{-1}}\left(\left(1+\tfrac12z^{-1}\right)y^\circ(t)-\tfrac3{10}y(t)\right)$$ a. Draw the block diagram of the closed-loop system. b. Compute the transfer function from $e(t)$ to $u(t-1)$.
2026-07-q3a2026Q03k-step predictor and MVCmedium8 pts
Consider the following process: $$y(t)=\tfrac16 y(t-1)+u(t-2)+e(t)-\tfrac14 e(t-1)$$ with $e\sim\mathrm{WN}(0,1)$. a. Manipulate the system to satisfy all MVC assumptions. Explicitly mention all MVC hypotheses and check if they are satisfied. b. Find the 1-step-ahead and the 2-step-ahead predictors for the system in canonical representation. c. Find the Minimum Variance Controller.
2026-02-q42026Q04Black-box SW-sensor architecturesmedium3 pts
Describe the architectures for Black-Box SW-sensing that combine a linear-dynamical part with a non-linear static part (architectures
2026-07-q52026Q05Coloured noise and model extensionmedium3 pts
Consider the problem of using a Kalman filter when the noise $v_1(t)$ on the state equation is not white; how can this problem be managed? Describe the solution with an example of a simple 1st-order system.
2025-02-q1a2025Q014SIDmedium6 pts
Given the following impulse-response coefficients: $$\omega(0)=1,\quad \omega(1)=-\tfrac14,\quad \omega(2)=-\tfrac1{16},\quad \omega(3)=-\tfrac1{64},\quad \omega(4)=-\tfrac1{256}$$ a. Identify the system matrices using the 4SID method. What is the system order? b. Compute the transfer function from the identified matrices. Is the system strictly proper? Is it asymptotically stable? c. In MATLAB, you are given a generic (noisy) dataset containing the input `u_vec` and output `y_vec` timeseries signals of a system of the third order. The sampling frequency of the vectors is 10 Hz. Write the MATLAB code to identify the system using the 4SID method. Display the identified $F$, $G$, $H$, $D$ matrices.
```
data = iddata(y_vec, u_vec, 1/10);   % iddata(OUTPUT, INPUT, Ts)
sys  = n4sid(data, 3);               % 4SID, known order n = 3
disp(sys.A); disp(sys.B); disp(sys.C); disp(sys.D);
```
2025-02-q1b2025Q01Controllable canonical formeasy4 pts
Consider the system described by the following transfer function: $$W(z)=\frac{2}{1+\tfrac13z^{-2}}$$ a. Compute the state-space realization in controllable canonical form. b. Provide the analytical expression of the impulse response $\omega(t)$ for the system.
2025-07-q1a2025Q014SIDmedium5 pts
Given the following impulse response: $$\omega(t)=\begin{cases}0 & \text{if } t \text{ odd}\\ \left(\tfrac25\right)^{\frac t2-1} & \text{if } t \text{ even}\end{cases}$$ a. Compute the first 6 impulse-response coefficients. b. Identify the system matrices using the 4SID method. What is the system order? c. Write the system of (b) as a discrete-time state-space MATLAB system `sys`. Check the observability of the system.
```
F = [0 1; 2/5 0];
G = [0; 1];
H = [1 0];
D = 5/2;
sys = ss(F, G, H, D, -1);
n = 2;
rank(obsv(sys)) == n
```
2025-07-q1b2025Q01Pole-zero cancellation and hidden modeshard6 pts
Consider the system whose input/output behaviour is described by: $$y(t)=\left[\frac{z^{-2}-a\cdot z^{-3}}{\left(1-\tfrac12z^{-2}\right)(z-a)}\right]u(t)$$ a. Determine all the poles and zeros of the system. Discuss the stability of the system with respect to the parameter $a$. b. Identify the system matrices from the transfer function (keep $a$ as a parameter in your answer). c. Provide the analytical expression $\omega(t)$ as a function of time for the system.
2025-09-q1a2025Q01Pole-zero cancellation and hidden modesmedium6 pts
Given the system described by the following equations: $$\begin{cases}x_1(t+1)=-2x_1(t)-2x_2(t)+\tfrac23u(t)\\ x_2(t+1)=5x_1(t)+\tfrac92x_2(t)-\tfrac13u(t)\\ y(t)=2x_1(t)+x_2(t)+u(t)\end{cases}$$ a. Discuss reachability and observability of the system. b. Compute the transfer function from $u(t)$ to $y(t)$. c. Identify poles and zeros of the system. Assess the system's stability.
2025-09-q1b2025Q01Hankel rank and system orderhard6 pts
Given the following impulse-response coefficients: $$\omega(0)=1;\;\omega(1)=0;\;\omega(2)=0;\;\omega(3)=1;\;\omega(4)=\tfrac12;\;\omega(5)=\tfrac14;\;\omega(6)=\tfrac18;\;\omega(7)=\tfrac1{16}$$ Identify the system matrices using the 4SID method. What is the system order?
2025-09-q1c2025Q01MATLAB for identificationeasy3 pts
You are given a generic dataset containing the input and output timeseries signals of a system: `u_vec` and `y_vec` with time vector `t_vec`. The sampling frequency of the vectors is 1000 Hz. Write the MATLAB code to identify the system `sys` using the 4SID method, assuming that the system has order 5. Write the code to compute the response of the identified system `sys` to the input vector `u_vec`.
```
data  = iddata(y_vec, u_vec, 1/1000);   % iddata(OUTPUT, INPUT, Ts)
sys   = n4sid(data, 5);                 % 4SID, order 5
y_sim = lsim(sys, u_vec, t_vec);
```
2025-02-q2a2025Q02Subsystem decouplingmedium5 pts
Given the system described by the following equations: $$\begin{cases}x(t+1)=\begin{bmatrix}-\tfrac35&0\\0&2\end{bmatrix}x(t)+v_1(t)\\ y(t)=[0\;\;2]\,x(t)+v_2(t)\end{cases}$$ where $v_1(t)=\begin{bmatrix}v_{11}(t)\\v_{12}(t)\end{bmatrix}\sim\mathrm{WN}\!\left(0,\begin{bmatrix}1/4&0\\0&1/2\end{bmatrix}\right)$ is uncorrelated with the state, $v_2(t)\sim\mathrm{WN}\!\left(0,\tfrac35\right)$, $v_{11}(t)\perp v_2(t)$, and $\mathbb E\!\left[v_{12}(t)v_2(t)^\top\right]=\tfrac15$. a. Can you apply the asymptotic K.F. theorems to the given system? b. Can the system be decoupled into two independent subsystems? If so, explain how and state whether you can apply the theorems to each of the subsystems.
2025-02-q2c2025Q02Kalman gain and filter stabilityeasy4 pts
Consider the system described by the following matrices: $$F=\tfrac12,\quad H=\tfrac14,\quad V_1=1,\quad V_2=\tfrac12,\quad V_{12}=\tfrac14$$ Knowing that the DRE converges to the only solution of the ARE $\bar P=1$, answer the following questions: a. Compute the asymptotic Kalman gain and assess the filter stability. b. Compute the asymptotic output-error covariance for the predictor.
2025-07-q2a2025Q02Asymptotic KF theoremsmedium6 pts
Consider the following system: $$\begin{cases}x(t+1)=2x(t)+v_1(t)\\ y(t)=h\cdot x(t)+v_2(t)\end{cases}$$ where $v_1(t)\sim\mathrm{WN}(0,q)$, $v_2(t)\sim\mathrm{WN}\!\left(0,\tfrac34\right)$ and $v_1(t)\perp v_2(t)$. a. For which values of $h$ and $q$ can you apply the asymptotic theorems? b. Set $h=\tfrac12$ and $q=5$. Compute the DRE and the ARE asymptotic solution(s). c. Check the convergence to the asymptotic solution(s) of point (b) using the graphical method.
2025-07-q2b2025Q02Predictor vs filtermedium4 pts
Consider the system described by the following matrices: $$F=3,\quad H=-4,\quad V_1=2,\quad V_2=5,\quad V_{12}=-3$$ Knowing that the DRE converges to the only solution of the ARE $\bar P=\tfrac14$, answer the following questions: a. Compute the asymptotic Kalman gain and assess the filter stability. b. Compute the asymptotic state-error covariance for the **filter**.
2025-09-q2a2025Q02Noise covariances V1, V2, V12hard6 pts
Consider the following system: $$\begin{cases}x(t+1)=3x(t)+w(t)\\ y(t)=-2x(t)-\tfrac38 w(t)+\sqrt{\tfrac7{32}}\,v(t)\end{cases}$$ where $w(t)\sim\mathrm{WN}(0,2)$, $v(t)\sim\mathrm{WN}(0,1)$, $w(t)\perp v(t)$, and the state is uncorrelated with the noises. a. Compute $V_1$, $V_2$ and $V_{12}$. Are the asymptotic Kalman theorems applicable? b. Compute the ARE solution(s) and discuss their existence. c. Graphically check the convergence of the DRE to the ARE solution(s).
2025-09-q2b2025Q02Predictor vs filtermedium4 pts
Given a system described by the following matrices: $$F=3,\quad H=6,\quad V_1=\tfrac14,\quad V_2=\tfrac14,\quad V_{12}=\tfrac14$$ The solution of the ARE is $\bar P=\tfrac1{18}$ and the asymptotic Kalman gain is $\bar K=\tfrac59$. Assume the DRE converges to the ARE solution. Assess the filter stability and compute the transfer function from $y(t)$ to $\hat x(t|t)$ (**filter**, not predictor).
2025-02-q3a2025Q03All-pass repairmedium7 pts
Consider the following process: $$y(t)=\frac{z+\tfrac13}{\left(z-\tfrac12\right)z^2}u(t)+\frac{z-4}{z^2-\tfrac34z+\tfrac18}e(t)$$ with $e\sim\mathrm{WN}(0,1)$. a. Manipulate the system to satisfy all MVC assumptions. Explicitly mention all MVC hypotheses and check if satisfied. b. Find the MVC controller.
2025-02-q3b2025Q03MVC closed-loop stabilityhard6 pts
Consider the following process and its minimum variance controller: $$y(t)=\frac{1-6z^{-1}}{1+\tfrac12z^{-1}}u(t-1)+\frac{1-\tfrac12z^{-1}}{1+\tfrac12z^{-1}}e(t)$$ $$u(t)=\frac{1}{1-6z^{-1}}\left(\left(1-\tfrac12z^{-1}\right)y^\circ(t)+y(t)\right)$$ a. Discuss the stability of the closed loop. b. Draw the block diagram of the closed-loop system. c. Compute the transfer function between $u(t)$ and the output $y(t)$.
2025-07-q3a2025Q03De-biasing non-zero-mean noisehard9 pts
Consider the following process: $$y(t)=\tfrac12 y(t-1)+u(t-1)+\tfrac13 u(t-2)+e(t)+4e(t-1)$$ with $e\sim\mathrm{WN}(1,1)$. **Note that the process has a non-zero mean.** a. Manipulate the system to satisfy all MVC assumptions. Explicitly mention all MVC hypotheses and check if they are satisfied. b. Find the 1-step-ahead predictor. c. Find the Minimum Variance Controller.
2025-09-q3a2025Q03MVC assumptions and canonical formhard9 pts
Consider the following process: $$y(t)=\frac{2z^{-2}-\tfrac12z^{-3}}{1+\beta z^{-1}}u(t)+\frac{1-\tfrac13z^{-1}}{1+\beta z^{-1}}e(t)$$ with $e\sim\mathrm{WN}(0,1)$ and $\beta\in\mathbb R$. a. For which values of $\beta\in\mathbb R$ is the stochastic part of the system in canonical form? Explicitly check all canonical-form hypotheses. b. For the values of $\beta$ that satisfy point (a), manipulate the system to satisfy all MVC assumptions. Explicitly mention all MVC hypotheses and check if they are satisfied. c. Find the Minimum Variance Controller when $\beta$ is chosen to satisfy the MVC assumptions.
2025-09-q3b2025Q03MVC closed-loop transfer functionsmedium5 pts
Consider the following process and its minimum variance controller: $$y(t)=\frac{1+\tfrac18z^{-1}}{1+\tfrac15z^{-1}}u(t-1)+\frac{1-\tfrac13z^{-1}}{1+\tfrac15z^{-1}}e(t),\qquad e(t)\sim\mathrm{WN}(0,1)$$ $$u(t)=\frac{1}{1+\tfrac18z^{-1}}\left(\left(1-\tfrac13z^{-1}\right)y^\circ(t)+\tfrac8{15}y(t)\right)$$ a. Draw the block diagram of the closed-loop system. b. Discuss the stability of the closed loop. c. Compute the transfer function from $y^\circ(t)$ to $u(t)$.
2025-07-q52025Q05KF vs black-box SW-sensingmedium3 pts
Compare and briefly discuss the main characteristics (pros and cons) of a model-based SW-sensing method (K.F.) and a black-box SW-sensing method.
2024-01-08-q12024Q014SIDmedium8 pts
Given the following impulse response: $$\omega(t)=\begin{cases}0 & t<t_0\\ \tfrac13(-2)^{2-t} & t\ge t_0\end{cases}\qquad\text{where } t_0\ge0$$ a. Is the system asymptotically stable? For which values of $t_0$ is it strictly proper? From now on, set $t_0=0$. b. Compute the transfer function associated with the system. What is the order of the system? c. Identify the system matrices from the transfer function. d. Identify the system matrices using the 4SID method. What is the order of the retrieved system? Did you retrieve the same matrices as in question (c)?
2024-01-29-q12024Q014SIDmedium8 pts
Given the following impulse-response coefficients: $$\omega(0)=0,\quad \omega(1)=3,\quad \omega(2)=0,\quad \omega(3)=\tfrac13,\quad \omega(4)=0,\quad \omega(5)=\tfrac1{27},\quad \omega(6)=0$$ a. Identify the system matrices using the 4SID method. What is the system order? b. Check the observability and reachability of the model identified in (a). c. Compute the transfer function of the model identified in (a). d. Provide the analytical expression of the impulse response $\omega(t)$ for the identified system. e. What happens to the matrices $F$, $G$, $H$ and $D$ identified with 4SID when $\omega(0)=1$?
2024-06-q1a2024Q01State space to TF and impulse responseeasy3 pts
Given the following impulse response: $$\omega(t)=\begin{cases}0 & t<t_0\\ \tfrac13(-3)^{2+t} & t\ge t_0\end{cases}\qquad\text{where } t_0\ge0$$ a. Is the system asymptotically stable? For which values of $t_0$ is it strictly proper? b. Set $t_0=2$. Compute the transfer function associated with the system. What is the system order?
2024-06-q1b2024Q01MATLAB for identificationeasy2 pts
In MATLAB, you are given a generic extended observability matrix `O4` and a generic extended reachability matrix `R4` for a system of order $n=3$ with $\omega(0)=3$. Write the code to find the system matrices $F$, $G$, $H$ and $D$.
```
% O4 and R4 are provided
F = inv(O4(1:3, :)) * O4(2:4, :);
G = R4(:, 1);
H = O4(1, :);
D = 3;
```
2024-06-q1c2024Q014SIDmedium3 pts
Given the following impulse-response coefficients: $$\omega(0)=0,\quad \omega(1)=-\tfrac14,\quad \omega(2)=0,\quad \omega(3)=1,\quad \omega(4)=0,\quad \omega(5)=-4,\quad \omega(6)=0$$ a. Identify the system matrices using the 4SID method. What is the system order? b. Compute the eigenvalues of the identified system and assess the stability of the system.
2024-07-q1a2024Q014SIDmedium5 pts
A system is subject to the following input: $$u(t)=\begin{cases}3 & \text{if } t=2\\ 0 & \text{else}\end{cases}$$ The response of the system is $$y(0)=0,\; y(1)=0,\; y(2)=6,\; y(3)=0,\; y(4)=\tfrac38,\; y(5)=0,\; y(6)=\tfrac3{32},\; y(7)=0$$ a. Identify the system order. b. Identify the system matrices using the 4SID method. c. Provide the MATLAB code to compute the eigenvalues of the matrix $F$ found in question (b), then write the system of (b) as a discrete-time state-space MATLAB object.
```
F = [0 1; 1/4 0];
G = [0; 1/8];
H = [1 0];
D = 2;
eig(F)
sys = ss(F, G, H, D, -1);
```
2024-07-q1b2024Q01Pole-zero cancellation and hidden modeshard5 pts
Consider the system whose input/output behaviour is described by: $$y(t)=\frac{-3z^{-2}\left(1-3z^{-1}\right)}{1-\tfrac72z^{-1}+\tfrac32z^{-2}}u(t)+2\,u(t)$$ a. Is the system asymptotically stable? Is it strictly proper? b. Identify the system matrices from the transfer function.
2024-07-q1c2024Q01Observability and reachabilityeasy3 pts
Given the following matrices: $$F=\begin{bmatrix}9/2&-2&0\\1&0&0\\0&1&0\end{bmatrix},\quad H=\begin{bmatrix}0&-3/4&3\end{bmatrix},\quad G=\begin{bmatrix}2\\0\\0\end{bmatrix},\quad D=1$$ Check the observability of the system.
2024-09-q1a2024Q01Pole-zero cancellation and hidden modeshard5 pts
Consider the system whose input/output behaviour is described by: $$y(t)=\frac{z^{-2}\left(1-\tfrac32z^{-1}\right)}{1-2z^{-1}+\tfrac34z^{-2}}u(t)$$ a. Is the system asymptotically stable? Is it strictly proper? b. Identify the system matrices from the transfer function. c. Provide the analytical expression $\omega(t)$ as a function of time for the system.
2024-09-q1b2024Q014SIDmedium5 pts
Given the following impulse response: $$\omega(t)=\begin{cases}0 & \text{if } t \text{ odd}\\ \left(-\tfrac13\right)^{\frac t2+1} & \text{if } t \text{ even}\end{cases}$$ a. Compute the first 5 impulse-response coefficients. b. Identify the system matrices using the 4SID method. What is the system order? c. Write the system of (b) as a discrete-time state-space MATLAB system `sys`. Given a known input vector `u` and its corresponding time vector `t`, write the code to compute the response of `sys` to `u`.
```
F = [0 1; -1/3 0];
G = [0; 1/9];
H = [1 0];
D = -1/3;
sys = ss(F, G, H, D, -1);
y = lsim(sys, u, t);
```
2024-01-08-q22024Q02DRE and AREhard8 pts
Given the system described by the following equations: $$\begin{cases}x(t+1)=-2x(t)+v_1(t)\\ y(t)=\tfrac12 x(t)+v_2(t)\end{cases}$$ where $v_1(t)\sim\mathrm{WN}(0,3)$ is uncorrelated with the state, $v_2(t)\sim\mathrm{WN}\!\left(0,\tfrac12\right)$ and $\mathbb E\!\left[v_1(t)\,v_2^\top(t)\right]=-1$. a. Can you apply the asymptotic theorems? b. Analyze the convergence of the DRE to the ARE solution. Compute the asymptotic Kalman gain. Analyze the stability of the resulting asymptotic predictor. c. Compute the transfer function from $y(t)$ to $\hat x(t|t)$ (state **filter**, not predictor).
2024-01-29-q22024Q02Asymptotic KF theoremshard8 pts
Given the system described by the following equations: $$\begin{cases}x(t+1)=x(t)\\ y(t)=\tfrac12 x(t)+\alpha w(t)\end{cases}$$ where $w(t)\sim\mathrm{WN}(0,1)$ is uncorrelated with the state and $\alpha>0$. a. Can you apply the asymptotic theorems? b. Does the DRE converge to the ARE for all $\alpha>0$? *(Hint: select a feasible value of $\alpha$ to show the convergence of $P(t)$ if needed.)* c. Compute the Kalman gain and analyze the stability of the resulting asymptotic predictor as a function of $\alpha$. d. Write the equations of the asymptotic 1-step-ahead predictor. e. Compute $\mathrm{Var}\!\left(x(t)-\hat x(t|t)\right)$ (state **filter**, not predictor).
2024-06-q2a2024Q02DRE and AREhard5 pts
Consider the following system: $$\begin{cases}x(t+1)=\alpha x(t)+v(t)\\ y(t)=x(t)-\tfrac1\alpha v(t)\end{cases}\qquad v(t)\sim\mathrm{WN}(0,1),\;\alpha\ne0$$ a. Can you apply the asymptotic theorems? b. Compute the ARE solution(s) and discuss their existence as $\alpha$ varies (convergence of the solutions is not required).
2024-06-q2c2024Q02Predictor vs filtermedium3 pts
Consider the system described by the following matrices: $$F=-2,\qquad H=1,\qquad V_1=\tfrac34,\qquad V_2=\tfrac12,\qquad V_{12}=-\tfrac12$$ Knowing that the DRE converges to the only solution of the ARE $\bar P=\tfrac12$, and that the resulting Kalman gain is $\bar K=-\tfrac32$, answer the following questions: a. Assess the filter stability. b. Compute the transfer function from $y(t)$ to $\hat x(t|t)$ (**filter**, not predictor).
2024-07-q2a2024Q02DRE and AREmedium4 pts
Consider a system whose DRE is: $$P(t+1)=\frac{4P(t)}{P(t)+\tfrac14}$$ a. Compute the asymptotic solution(s) of the ARE. b. Check graphically the convergence of the DRE to the ARE solution(s).
2024-07-q2b2024Q02Subsystem decouplingmedium4 pts
Given the system described by the following equations: $$\begin{cases}x(t+1)=\begin{bmatrix}1&0\\0&\tfrac14\end{bmatrix}x(t)+v_1(t)\\ y(t)=[2\;\;0]\,x(t)+v_2(t)\end{cases}$$ where $v_1(t)=\begin{bmatrix}v_{11}(t)\\v_{12}(t)\end{bmatrix}\sim\mathrm{WN}\!\left(0,\begin{bmatrix}4&0\\0&1\end{bmatrix}\right)$ is uncorrelated with the state, $v_2(t)\sim\mathrm{WN}(0,1)$, $v_{12}(t)\perp v_2(t)$, and $\mathbb E\!\left[v_{11}(t)v_2(t)^\top\right]=\tfrac12$. a. Can you apply the asymptotic K.F. theorems to the given system? b. Can the system be decoupled into two independent subsystems? If so, write the equations of the two subsystems.
2024-07-q2c2024Q02Output prediction error varianceeasy3 pts
Given a system described by the following matrices: $$F=-2,\quad H=1,\quad V_1=\tfrac12,\quad V_2=\tfrac15,\quad V_{12}=-\tfrac14$$ The solution of the ARE is $\bar P=\tfrac14$ and the resulting asymptotic Kalman gain is $\bar K=-\tfrac53$. Assume the DRE converges to the ARE solution. Compute the variance of the output prediction error $\mathrm{Var}\!\left[y(t)-\hat y(t|t-1)\right]$.
2024-09-q2a2024Q02Noise covariances V1, V2, V12hard6 pts
Consider the following system: $$\begin{cases}x(t+1)=3x(t)+w(t)\\ y(t)=-2x(t)-\tfrac38 w(t)+\sqrt{\tfrac7{32}}\,v(t)\end{cases}$$ where $w(t)\sim\mathrm{WN}(0,2)$, $v(t)\sim\mathrm{WN}(0,2)$, $w(t)\perp v(t)$, and the state is uncorrelated with the noises. a. Compute $V_1$, $V_2$ and $V_{12}$. Are the asymptotic Kalman theorems applicable? b. Compute the ARE solution(s) and discuss their existence. c. Graphically check the convergence of the DRE to the ARE solution(s).
2024-09-q2b2024Q02Predictor vs filtermedium4 pts
Given a system described by the following matrices: $$F=2,\quad H=2,\quad V_1=1,\quad V_2=\tfrac13,\quad V_{12}=\tfrac13$$ The solution of the ARE is $\bar P=\tfrac23$ and the resulting asymptotic Kalman gain is $\bar K=1$. Assume the DRE converges to the ARE solution. a. Write the asymptotic Kalman predictor and filter equations. b. Compute the transfer function from $y(t)$ to $\hat x(t|t)$ (**filter**, not predictor).
2024-01-08-q32024Q03MVC assumptions and canonical formmedium8 pts
Consider the following process: $$y(t)=-\tfrac56 y(t-1)-\tfrac16 y(t-2)+u(t-2)+3e(t-1)-\tfrac12 e(t-2)$$ with $e\sim\mathrm{WN}(0,1)$. a. Are the assumptions for the design of a minimum variance controller satisfied? If they are not, and if it is possible, manipulate the model of the system so that the assumptions of minimum variance control are satisfied. b. Design the minimum variance controller for the system and write it using the time formulation. Discuss the stability of the closed-loop system. c. Write the time equation of $u(t)$ when $y^\circ(t)=6\;\forall t$.
2024-01-29-q32024Q03All-pass repairhard8 pts
Consider the following process $$y(t)=\tfrac34 y(t-1)+u(t-1)+e(t)+4e(t-1)$$ with $e\sim\mathrm{WN}(0,1)$. a. Are the assumptions for the design of a minimum variance controller satisfied? If they are not, and if it is possible, manipulate the model of the system so that the assumptions of minimum variance control are satisfied. b. Design the minimum variance controller for the system. Discuss the stability of the closed-loop system. c. Draw the block diagram of the closed-loop system. d. Compute the transfer function between the reference $y^\circ(t)$ and the input $u(t)$, using any approach you like.
2024-06-q3a2024Q03MVC assumptions and canonical formhard8 pts
Consider the following process: $$y(t)=-\tfrac13 y(t-1)+2u(t-2)+e(t-2)+\alpha\,e(t-3)\qquad \alpha\in\mathbb R,\; e(t)\sim\mathrm{WN}(0,1)$$ a. For which values of $\alpha$ are the MVC assumptions satisfied? b. Set $\alpha=-2$. Write the system in canonical representation and find its 2-step-ahead predictor. c. Design the Minimum Variance Controller.
2024-07-q3a2024Q03All-pass repairmedium6 pts
Consider the following process: $$y(t)=\frac{2z}{z^2-\tfrac34z+\tfrac18}u(t)+\frac{z-3}{z^2-\tfrac34z+\tfrac18}e(t)$$ with $e\sim\mathrm{WN}(0,1)$. a. Are the assumptions for the design of a minimum variance controller satisfied? If they are not, and if it is possible, manipulate the model of the system so that the assumptions of minimum variance control are satisfied. Explicitly mention all MVC hypotheses. b. Design the minimum variance controller for the specified system.
2024-07-q3b2024Q03MVC closed-loop stabilityhard6 pts
Consider the following process and its minimum variance controller: $$y(t)=\frac{1-2z^{-1}}{1-\tfrac13z^{-1}}u(t-1)+\frac{1}{1-\tfrac13z^{-1}}e(t)$$ $$u(t)=\frac{1}{1-2z^{-1}}\left(y^\circ(t)-\tfrac13 y(t)\right)$$ a. Discuss the stability of the closed loop. b. Draw the block diagram of the closed-loop system. c. Compute the transfer function between $u(t)$ and the output $y(t)$. Compare it to the result in (a) and comment on it.
2024-09-q3a2024Q03De-biasing non-zero-mean noisehard8 pts
Consider the following process: $$y(t)=\frac{4}{z+\tfrac12}u(t)+\frac{(z+3)z}{z^2+\tfrac56z+\tfrac16}e(t)$$ with $e\sim\mathrm{WN}(1,1)$. **Note that the process has a non-zero mean.** a. Manipulate the system to satisfy all MVC assumptions. Explicitly mention all MVC hypotheses. b. Find the 1-step-ahead predictor. c. Find the MVC controller.
2024-09-q3b2024Q03MVC closed-loop transfer functionsmedium5 pts
Consider the following process and its minimum variance controller: $$y(t)=\frac{1+\tfrac14z^{-1}}{1+\tfrac13z^{-1}}u(t-3)+\frac{1+\tfrac12z^{-1}}{1+\tfrac13z^{-1}}e(t),\qquad e(t)\sim\mathrm{WN}(0,1)$$ $$u(t)=\frac{1}{\left(1+\tfrac14z^{-1}\right)\left(1+\tfrac16z^{-1}-\tfrac1{18}z^{-2}\right)}\left(\left(1+\tfrac12z^{-1}\right)y^\circ(t)-\tfrac1{54}y(t)\right)$$ a. Discuss the stability of the closed loop. b. What are the values of the following terms: $C(z)$, $A(z)$, $E(z)$ and $\tilde R(z)$? *(Recall that $C(z)=E(z)A(z)+\tilde R(z)z^{-k}$.)* c. Draw the block diagram of the closed-loop system.
2024-01-08-q42024Q04Sinusoid amplitude and phase estimationmedium3 pts
Describe the procedure to estimate a sinusoidal signal (estimation of its amplitude and phase) from a noisy one, assuming that the frequency $\Omega$ of the sinusoid is a-priori known.
2024-01-29-q42024Q04Frequency-domain experiment designmedium3 pts
Describe in detail the "step 1" (experiment design and pre-processing) of a frequency-domain system identification method.
2024-07-q42024Q044SIDmedium3 pts
Describe in detail and briefly discuss the 4SID identification procedure, starting from a NOISE-FREE impulse-response signal.
2024-01-08-q52024Q05MVC derivationhard3 pts
Find the general formula (proof is requested) of the optimal Minimum Variance controller for a generic ARMAX$(n,m,p+k)$ system.
2024-06-q52024Q05Realization non-uniquenessmedium3 pts
Explain and discuss the "realization" of a transfer function into a state-space representation.
2024-07-q52024Q05Gray-box identificationmedium3 pts
Describe what offline grey-box identification is and make two examples of input signals which can be used for the identification dataset.
2024-09-q52024Q05Discretization and samplingmedium3 pts
Describe the equations of the state-space method for the discretization of a continuous-time dynamical linear system. Discuss the transformation of the poles and zeros of the system from continuous to discrete time.
2023-07-q12023Q014SIDmedium8 pts
Given the following impulse-response coefficients: $$\omega(0)=1,\quad \omega(1)=0,\quad \omega(2)=-\tfrac12,\quad \omega(3)=0,\quad \omega(4)=\tfrac14,\quad \omega(5)=0,\quad \omega(6)=-\tfrac18,\quad \omega(7)=0$$ a. Identify the system matrices using the 4SID method. What is the system order? b. Compute the transfer function of the identified system. c. Identify the system matrices from the transfer function. Did you retrieve the same matrices as in (a)? d. Provide the analytical expression of the impulse response $\omega(t)$ for the identified system.
2023-07-q22023Q02Asymptotic KF theoremshard8 pts
Given the system described by the following equations: $$\begin{cases}x(t+1)=\begin{bmatrix}\tfrac12&0\\0&1\end{bmatrix}x(t)+v_1(t)\\ y(t)=[0\;\;1]\,x(t)+v_2(t)\end{cases}$$ where $v_1(t)=\begin{bmatrix}v_{11}(t)\\v_{12}(t)\end{bmatrix}\sim \mathrm{WN}\!\left(0,\begin{bmatrix}1&0\\0&1\end{bmatrix}\right)$ is uncorrelated with the state, $v_2(t)\sim\mathrm{WN}\!\left(0,\tfrac12\right)$, $v_{11}(t)\perp v_2(t)$, and $\mathbb E\!\left[v_{12}(t)v_2(t)^\top\right]=\tfrac12$. a. Can you apply the asymptotic K.F. theorems to the given system? b. Can the system be decoupled into two independent subsystems? If so, evaluate for each subsystem: (i) convergence of the DRE to the ARE solution, (ii) the asymptotic Kalman gain, (iii) stability of the asymptotic predictor.
2023-07-q32023Q03MVC assumptions and canonical formmedium8 pts
Consider the following process $$y(t)=-\alpha y(t-1)+u(t-1)+\tfrac13 u(t-2)+4e(t)+e(t-1),\qquad e\sim\mathrm{WN}(0,1)$$ a. For which values of $\alpha\in\mathbb R$ are the assumptions of minimum variance control satisfied? If necessary, write the system in canonical representation. b. If the assumptions at the previous point are satisfied, design the minimum variance controller for the system. Discuss the stability of the closed-loop system. c. Write the time equation of $u(t)$ when $y^\circ(t)=4\;\forall t$. d. Compute the values of $y(t)$, $t=0,1,2,3$, when the reference is $y^\circ(t)=0$ for $t<1$ and $y^\circ(t)=4$ for $t\ge1$, and the noise is null.
2023-07-q42023Q04State space to TF and impulse responseeasy3 pts
Consider the state-space (SS) representation of a discrete-time linear time-invariant system. Starting from this representation: a) Find the formula to obtain the Transfer-Function representation. b) Find the formula to obtain the Impulse-Response representation.
2023-07-q52023Q05Gray-box identificationmedium3 pts
Explain and discuss how the Kalman Filter can be used to solve grey-box system-identification problems.