From d93dd0f967bcfc4c549775df878b74e75bcd5f78 Mon Sep 17 00:00:00 2001
From: Christos Choutouridis <hoo2@hoo2.net>
Date: Fri, 4 Apr 2025 00:33:58 +0300
Subject: [PATCH] Work 1: Problem 3 matlab + report

---
 Work 1/report/Work1_report.pdf                | Bin 424112 -> 662761 bytes
 Work 1/report/Work1_report.tex                | 156 ++-
 Work 1/scripts/Prob1_responce_20s.png         | Bin 80319 -> 80319 bytes
 Work 1/scripts/Prob1_responce_90s.png         | Bin 107807 -> 107807 bytes
 Work 1/scripts/Prob3_Error_vs_Duration.png    | Bin 0 -> 26475 bytes
 Work 1/scripts/Prob3a_ComparisonPlot.png      | Bin 0 -> 69210 bytes
 Work 1/scripts/Prob3a_NoiseEffect.png         | Bin 0 -> 30694 bytes
 Work 1/scripts/Prob3a_NoiseStd0.0010.png      | Bin 0 -> 71976 bytes
 Work 1/scripts/Prob3a_NoiseStd0.0025.png      | Bin 0 -> 67939 bytes
 .../scripts/Prob3b_SamplingPeriodEffect.png   | Bin 0 -> 52734 bytes
 Work 1/scripts/Prob3c_AmplitudeEffect.png     | Bin 0 -> 33207 bytes
 Work 1/scripts/Problem3a.m                    | 114 +++
 Work 1/scripts/Problem3b.m                    |  80 ++
 Work 1/scripts/Problem3c.m                    |  78 ++
 Work 1/scripts/problem3_data_T10s.csv         | 102 ++
 Work 1/scripts/problem3_data_T20s.csv         | 202 ++++
 Work 1/scripts/problem3_data_T40s.csv         | 402 ++++++++
 Work 1/scripts/problem3_data_T60s.csv         | 602 ++++++++++++
 Work 1/scripts/problem3_data_T90s.csv         | 902 ++++++++++++++++++
 19 files changed, 2617 insertions(+), 21 deletions(-)
 create mode 100644 Work 1/scripts/Prob3_Error_vs_Duration.png
 create mode 100644 Work 1/scripts/Prob3a_ComparisonPlot.png
 create mode 100644 Work 1/scripts/Prob3a_NoiseEffect.png
 create mode 100644 Work 1/scripts/Prob3a_NoiseStd0.0010.png
 create mode 100644 Work 1/scripts/Prob3a_NoiseStd0.0025.png
 create mode 100644 Work 1/scripts/Prob3b_SamplingPeriodEffect.png
 create mode 100644 Work 1/scripts/Prob3c_AmplitudeEffect.png
 create mode 100644 Work 1/scripts/Problem3a.m
 create mode 100644 Work 1/scripts/Problem3b.m
 create mode 100644 Work 1/scripts/Problem3c.m
 create mode 100644 Work 1/scripts/problem3_data_T10s.csv
 create mode 100644 Work 1/scripts/problem3_data_T20s.csv
 create mode 100644 Work 1/scripts/problem3_data_T40s.csv
 create mode 100644 Work 1/scripts/problem3_data_T60s.csv
 create mode 100644 Work 1/scripts/problem3_data_T90s.csv

diff --git a/Work 1/report/Work1_report.pdf b/Work 1/report/Work1_report.pdf
index dd1d87edc3709c13aa98e1743d403a255899c992..275b13def4a51deefb2a89e774d630127a72796d 100644
GIT binary patch
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diff --git a/Work 1/report/Work1_report.tex b/Work 1/report/Work1_report.tex
index 81f1715..d16c234 100644
--- a/Work 1/report/Work1_report.tex	
+++ b/Work 1/report/Work1_report.tex	
@@ -76,7 +76,7 @@
 % Request a title page or header
 \InsertTitle
 
-\section{Εισαγωγή}
+\section*{Εισαγωγή}
 Η παρούσα εργασία έχει ως αντικείμενο τη μελέτη ενός απλοποιημένου δυναμικού συστήματος — συγκεκριμένα ενός γραμμικοποιημένου εκκρεμούς με ροπή εισόδου — με σκοπό την κατανόηση της συμπεριφοράς του και την εκτίμηση των φυσικών παραμέτρων του μέσω της μεθόδου των ελαχίστων τετραγώνων.
 Στο πρώτο μέρος της εργασίας εξετάζεται η δυναμική του συστήματος, η οποία περιγράφεται από μία διαφορική εξίσωση δευτέρας τάξης, και προσεγγίζεται τόσο θεωρητικά όσο και αριθμητικά μέσω προσομοίωσης στο περιβάλλον MATLAB.
 \par
@@ -84,7 +84,7 @@
 Στη συνέχεια, εφαρμόζεται ημιτονική διέγερση και εξετάζεται η απόκριση του συστήματος. Ιδιαίτερο ενδιαφέρον παρουσιάζει ο τρόπος με τον οποίο το σύστημα φτάνει στη μόνιμη περιοδική του συμπεριφορά, καθώς και ο ρυθμός με τον οποίο αποκλίνουν οι ταλαντώσεις από τη μεταβατική φάση.
 Η ανάλυση αυτή θέτει τη βάση για τα επόμενα μέρη της εργασίας, όπου επιχειρείται η εκτίμηση των παραμέτρων του μοντέλου με τη χρήση δεδομένων εξόδου του συστήματος.
 
-\subsection{Παραδοτέα}
+\subsection*{Παραδοτέα}
 Τα παραδοτέα της εργασίας αποτελούνται από:
 \begin{itemize}
     \item Την παρούσα αναφορά.
@@ -92,7 +92,7 @@
     \item Το \href{https://git.hoo2.net/hoo2/SystemModling}{σύνδεσμο} με το αποθετήριο που περιέχει όλο το project με τον κώδικα της MATLAB, της αναφοράς και τα παραδοτέα.
 \end{itemize}
 
-\section{Θέμα 1 – Μοντελοποίηση και Προσομοίωση Συστήματος Εκκρεμούς}
+\section*{Θέμα 1 – Μοντελοποίηση και Προσομοίωση Συστήματος Εκκρεμούς}
 
 Το σύστημα που μελετάται περιγράφεται από τη γραμμικοποιημένη διαφορική εξίσωση:
 \[
@@ -132,16 +132,14 @@
 Η προσομοίωση του Θέματος 1 χρησίμευσε όχι μόνο για την παρατήρηση της δυναμικής του συστήματος, αλλά και για την παραγωγή των δεδομένων που χρησιμοποιήθηκαν στο Θέμα 2 για την εκτίμηση των παραμέτρων. Η ακρίβεια των αποτελεσμάτων του Θέματος 2 επιβεβαιώνει τη σωστή ρύθμιση και υλοποίηση της προσομοίωσης.
 
 \subsection*{Παρατήρηση Συμπεριφοράς Συστήματος}
-
 Βλέπουμε ότι το σύστημα παρουσιάζει περιοδική απόκριση, η οποία όμως δεν σταθεροποιείται γρήγορα.
 Όπως φαίνεται στο Σχήμα~\ref{fig:20s}, το πλάτος των ταλαντώσεων συνεχίζει να μεταβάλλεται ακόμη και μετά από 20 δευτερόλεπτα, γεγονός που δείχνει ότι το σύστημα βρίσκεται ακόμη σε μεταβατική κατάσταση.
-
+\par
 Η καθυστερημένη σύγκλιση οφείλεται κυρίως:
 \begin{itemize}
     \item στην πολύ μικρή τιμή του συντελεστή απόσβεσης ($c = 0.15$), και
     \item στο γεγονός ότι η συχνότητα της εισόδου ($\omega = 2$ rad/s) είναι κοντά στη φυσική συχνότητα του εκκρεμούς ($\omega_n = \sqrt{g/L} \approx 2.8$ rad/s), προκαλώντας φαινόμενο ενίσχυσης (quasi-resonance).
 \end{itemize}
-
 Για τον λόγο αυτό, επεκτείναμε τη διάρκεια προσομοίωσης στα $90$ δευτερόλεπτα.
 Όπως φαίνεται στο Σχήμα~\ref{fig:90s}, το σύστημα σταθεροποιείται τελικά σε περιοδική μορφή περίπου μετά από $50$ δευτερόλεπτα.
 
@@ -152,7 +150,7 @@
     Απόκριση του συστήματος για $t \in [0, 90]$ sec. Το σύστημα σταθεροποιείται σε περιοδική συμπεριφορά μετά τα $50$ sec.
 }
 
-\section{Θέμα 2 – Εκτίμηση Παραμέτρων με τη Μέθοδο Ελαχίστων Τετραγώνων}
+\section*{Θέμα 2 – Εκτίμηση Παραμέτρων με τη Μέθοδο Ελαχίστων Τετραγώνων}
 
 Η εξίσωση που περιγράφει τη δυναμική του εκκρεμούς είναι της μορφής:
 \[
@@ -167,8 +165,7 @@
 
 Στόχος του παρόντος ερωτήματος είναι η εκτίμηση του διανύσματος $\theta$ με χρήση μετρήσεων από την έξοδο του συστήματος και της εισόδου $u(t)$, εφαρμόζοντας τη μέθοδο των ελαχίστων τετραγώνων.
 
-\subsection{Υποερώτημα (α)}
-
+\subsection*{Υποερώτημα 2α}
 Στο πρώτο υποερώτημα θεωρείται ότι είναι διαθέσιμες μετρήσεις όλων των μεταβλητών κατάστασης, δηλαδή $q(t)$, $\dot{q}(t)$ και $\ddot{q}(t)$, καθώς και της εισόδου $u(t)$.
 Δημιουργείται λοιπόν το πρόβλημα παλινδρόμησης:
 \[
@@ -180,7 +177,6 @@
 \[
     \hat{\theta} = (X^T X)^{-1} X^T u
 \]
-
 Αφού εκτιμηθούν οι παράμετροι, χρησιμοποιούνται για να υπολογιστεί η επιτάχυνση $\ddot{q}_{\text{est}}(t)$ και στη συνέχεια η ανακατασκευή της απόκρισης $q_{\text{est}}(t)$ με αριθμητική ολοκλήρωση.
 Η διαδικασία υλοποιείται με ένα απλό βρόχο επανάληψης (for loop), όπου για κάθε χρονικό βήμα $k$ υπολογίζονται διαδοχικά:
 \[
@@ -189,21 +185,16 @@
     \[
     q_{\text{est}}(k) = q_{\text{est}}(k-1) + T_s \cdot \dot{q}_{\text{est}}(k-1)
 \]
-
 Στο Σχήμα~\ref{fig:prob2a} παρουσιάζονται η πραγματική και η εκτιμώμενη γωνία, καθώς και το σφάλμα $e_q(t)$ μεταξύ τους.
 
 \InsertFigure{!ht}{1}{fig:prob2a}{../scripts/Prob2_20s_Ts0.1.png}{
     Αποτελέσματα εκτίμησης παραμέτρων με χρήση όλων των μεταβλητών κατάστασης.
 }
-
 \paragraph*{Συμπεράσματα:}
 Η εκτίμηση παρουσιάζει υψηλή ακρίβεια, με σχετικό σφάλμα μικρότερο του 6\% για όλες τις παραμέτρους.
 Ο αλγόριθμος κατάφερε να ανακατασκευάσει την απόκριση με μικρό σφάλμα, επιβεβαιώνοντας τη θεωρητική εγκυρότητα της μεθόδου.
 
-\vspace{1em}
-
-\subsection{Υποερώτημα (β)}
-
+\subsection*{Υποερώτημα 2β}
 Στη δεύτερη περίπτωση θεωρούμε ότι διαθέσιμες είναι μόνο οι μετρήσεις της γωνίας $q(t)$ και της εισόδου $u(t)$.
 Οι παράγωγοι $\dot{q}(t)$ και $\ddot{q}(t)$ υπολογίζονται αριθμητικά από το σήμα $q(t)$ με χρήση διαφορικών τελεστών 2ης τάξης ακρίβειας (κεντρικών διαφορών):
 \[
@@ -211,7 +202,6 @@
     \quad
     \ddot{q}(t_k) \approx \frac{q_{k+1} - 2q_k + q_{k-1}}{T_s^2}
 \]
-
 Ακολουθείται η ίδια διαδικασία παλινδρόμησης με την περίπτωση (α), όπως και η ανακατασκευή της απόκρισης. Το Σχήμα~\ref{fig:prob2b} δείχνει τα αντίστοιχα αποτελέσματα.
 
 \InsertFigure{!ht}{1}{fig:prob2b}{../scripts/Prob2b_20s_Ts0.1.png}{
@@ -221,10 +211,7 @@
 \paragraph*{Συμπεράσματα:}
 Παρά τον περιορισμό στη διαθέσιμη πληροφορία, η εκτίμηση παρέμεινε εξαιρετικά ακριβής, με μικρές αποκλίσεις από την πραγματική τιμή. Το γεγονός αυτό δείχνει την ισχυρή ταυτοποιησιμότητα του συστήματος και τη δυνατότητα εφαρμογής της μεθόδου ακόμη και με ελλιπή δεδομένα.
 
-\vspace{1em}
-
-\subsection{Σύγκριση των δύο περιπτώσεων}
-
+\subsection*{Σύγκριση των δύο περιπτώσεων}
 Ο παρακάτω πίνακας συγκρίνει τις πραγματικές και εκτιμώμενες τιμές για τα δύο υποερωτήματα:
 
 \begin{center}
@@ -239,6 +226,133 @@
 
 Παρατηρούμε ότι και στις δύο περιπτώσεις οι εκτιμήσεις είναι ακριβείς και σταθερές, με τις αποκλίσεις να κυμαίνονται σε χαμηλά επίπεδα. Μάλιστα, το γεγονός ότι τα σφάλματα είναι αντίστοιχα ακόμα και όταν χρησιμοποιείται λιγότερη πληροφορία (περίπτωση 2β), αποτελεί ένδειξη για τη σταθερότητα και αξιοπιστία της μεθόδου. Ταυτόχρονα, αναδεικνύει την ιδιαίτερη σημασία της επιλογής σήματος διέγερσης και της ποιότητας των δεδομένων.
 
+\section*{Θέμα 3 – Ανάλυση Ευαισθησίας της Εκτίμησης Παραμέτρων}
 
+\subsection*{Υποερώτημα 3α: Επίδραση Λευκού Θορύβου στις Μετρήσεις}
+Στο υποερώτημα αυτό προστέθηκε λευκός γκαουσιανός θόρυβος στα δεδομένα $q(t)$ και επαναλήφθηκε η διαδικασία εκτίμησης παραμέτρων από το Θέμα~2β.
+Χρησιμοποιήθηκαν δύο διαφορετικά επίπεδα θορύβου, με τυπική απόκλιση $\sigma = 0.001$ και $\sigma = 0.0025$, διατηρώντας σταθερή την είσοδο και την περίοδο δειγματοληψίας ($T_s = 0.1$ sec).
+Οι εκτιμήσεις συγκρίθηκαν με την αντίστοιχη χωρίς θόρυβο, τόσο ως προς το σφάλμα στις παραμέτρους όσο και στην ανακατασκευή της απόκρισης $q(t)$.
+
+\begin{figure}[H]
+%    \centering
+    \hspace{-2em}
+    \begin{minipage}{0.49\textwidth}
+        \centering
+        \includegraphics[width=1.15\linewidth]{../scripts/Prob3a_NoiseStd0.0010.png}
+        \caption{Εκτίμηση με $\sigma = 0.001$}
+    \end{minipage}
+    \hspace{0.5em}
+    \begin{minipage}{0.49\textwidth}
+        \centering
+        \includegraphics[width=1.15\linewidth]{../scripts/Prob3a_NoiseStd0.0025.png}
+        \caption{Εκτίμηση με $\sigma = 0.0025$}
+    \end{minipage}
+\end{figure}
+
+\paragraph*{Παρατηρήσεις:}
+Για μικρό θόρυβο ($\sigma = 0.001$) η ακρίβεια παραμένει υψηλή (σφάλμα περίπου $10\%$).
+Όταν ο θόρυβος αυξάνεται ($\sigma = 0.0025$), οι εκτιμήσεις επιδεινώνονται σημαντικά, κυρίως για το $mL^2$.
+Το γεγονός αυτό οφείλεται στην αριθμητική παραγώγιση, η οποία είναι ιδιαίτερα ευαίσθητη στον θόρυβο.
+
+\subsection*{Υποερώτημα 3β: Επίδραση της Περιόδου Δειγματοληψίας $T_s$}
+Στο υποερώτημα αυτό μελετάται η ακρίβεια των εκτιμήσεων σε συνάρτηση με την περίοδο δειγματοληψίας $T_s$.
+Η προσομοίωση έγινε για διάρκεια $T = 20$ sec, ενώ τα δεδομένα δειγματοληττήθηκαν ξανά για διαφορετικές τιμές:
+\[
+    T_s \in \{0.01, 0.05, 0.1, 0.2, 0.5\} \text{ sec}
+\]
+
+Η εκτίμηση έγινε με τον ίδιο τρόπο όπως στο Θέμα~2β.
+Παρακάτω παρουσιάζονται τα σφάλματα εκτίμησης για κάθε παράμετρο:
+\begin{table}[H]
+    \centering
+    \begin{tabular}{c|cc|cc|cc}
+        $T_s$ [sec]
+        & $mL^2$ & Σφάλμα (\%)
+        & $c$ & Σφάλμα (\%)
+        & $mgL$ & Σφάλμα (\%) \\
+        \hline
+        0.010 & 1.1708 & 0.09 & 0.1502 & 0.17 & 9.1928 & 0.04 \\
+        0.050 & 1.1737 & 0.16 & 0.1505 & 0.34 & 9.2009 & 0.04 \\
+        0.100 & 1.1826 & 0.92 & 0.1513 & 0.89 & 9.2249 & 0.31 \\
+        0.200 & 1.2188 & 4.00 & 0.1547 & 3.16 & 9.3210 & 1.35 \\
+        0.500 & 1.5186 & 29.59 & 0.1818 & 21.20 & 10.0959 & 9.78 \\
+    \end{tabular}
+    \caption{Εκτιμήσεις και ποσοστά σφάλματος παραμέτρων για διαφορετικές περιόδους $T_s$}
+\end{table}
+
+\InsertFigure{!ht}{1}{fig:3b}{../scripts/Prob3b_SamplingPeriodEffect.png}{
+    Σφάλμα εκτίμησης ως προς την περίοδο δειγματοληψίας $T_s$.
+}
+
+\paragraph*{Παρατηρήσεις:}
+Για $T_s \leq 0.1$ sec οι εκτιμήσεις είναι εξαιρετικά ακριβείς.
+Από το $T_s = 0.2$ sec παρατηρείται σταδιακή επιδείνωση, ενώ στο $T_s = 0.5$ sec οι αποκλίσεις είναι σημαντικές.
+Το φαινόμενο αυτό οφείλεται στην απώλεια πληροφορίας κατά τη δειγματοληψία, και υπογραμμίζει την ανάγκη για επιλογή κατάλληλης συχνότητας δειγματοληψίας ανάλογα με τη δυναμική του συστήματος.
+
+\subsection{Υποερώτημα 3γ: Επίδραση του Πλάτους Εισόδου $A_0$}
+
+Σε αυτό το υποερώτημα μελετάται η επίδραση του πλάτους της ημιτονικής εισόδου $A_0$ στην ακρίβεια της εκτίμησης παραμέτρων.
+Η διάρκεια προσομοίωσης διατηρήθηκε σταθερή στα $T = 20$ sec και η περίοδος δειγματοληψίας στα $T_s = 0.1$ sec.
+Το πλάτος $A_0$ έλαβε τις τιμές:
+\[
+    A_0 \in \{1, 2, 4, 6, 8, 16, 32\}
+\]
+
+Για κάθε τιμή, πραγματοποιήθηκε εκ νέου προσομοίωση, δειγματοληψία και εκτίμηση παραμέτρων με τη μέθοδο ελαχίστων τετραγώνων, όπως περιγράφεται στο Θέμα~2β.
+Παρακάτω παρουσιάζονται τα σφάλματα εκτίμησης για διαφορετικά πλάτη:
+\begin{table}[H]
+    \centering
+    \begin{tabular}{c|cc|cc|cc}
+        $A_0$
+        & $mL^2$ & Σφάλμα (\%)
+        & $c$ & Σφάλμα (\%)
+        & $mgL$ & Σφάλμα (\%) \\
+        \hline
+        1  & 1.1826 & 0.91 & 0.1514 & 0.90 & 9.2247 & 0.30 \\
+        2  & 1.1827 & 0.92 & 0.1513 & 0.90 & 9.2250 & 0.31 \\
+        4  & 1.1826 & 0.92 & 0.1513 & 0.89 & 9.2249 & 0.31 \\
+        6  & 1.1826 & 0.92 & 0.1513 & 0.90 & 9.2249 & 0.30 \\
+        8  & 1.1826 & 0.92 & 0.1513 & 0.90 & 9.2248 & 0.30 \\
+        16 & 1.1826 & 0.92 & 0.1513 & 0.90 & 9.2248 & 0.30 \\
+        32 & 1.1826 & 0.92 & 0.1513 & 0.89 & 9.2248 & 0.30 \\
+    \end{tabular}
+    \caption{Εκτιμήσεις και ποσοστά σφάλματος παραμέτρων για διαφορετικά πλάτη $A_0$}
+\end{table}
+
+\InsertFigure{!ht}{1}{fig:3c}{../scripts/Prob3c_AmplitudeEffect.png}{
+    Σφάλμα εκτίμησης παραμέτρων συναρτήσει του πλάτους εισόδου $A_0$.
+}
+
+\paragraph*{Παρατηρήσεις:}
+Τα αποτελέσματα δείχνουν ότι η ακρίβεια των εκτιμήσεων παραμένει σχεδόν σταθερή ανεξαρτήτως του πλάτους $A_0$.
+Το μέγιστο σχετικό σφάλμα κυμαίνεται εντός του $1\%$ για όλες τις παραμέτρους.
+Το εύρημα αυτό είναι αναμενόμενο, καθώς το υπό μελέτη σύστημα είναι γραμμικό ως προς τη ροπή εισόδου $u(t)$.
+Η εξίσωση:
+\[
+    mL^2 \ddot{q}(t) + c \dot{q}(t) + mgL q(t) = A_0 \sin(\omega t)
+\]
+δείχνει ότι το $A_0$ αποτελεί απλώς έναν γραμμικό πολλαπλασιαστικό συντελεστή της εισόδου, χωρίς να επηρεάζει τη δομή του συστήματος.
+
+Συνεπώς, οι πραγματικές παράμετροι του συστήματος δεν εξαρτώνται από το μέγεθος του $A_0$.
+Το μόνο που μπορεί να επηρεάζεται είναι η αριθμητική ακρίβεια των παραγώγων (και κατ’ επέκταση της εκτίμησης), μέσω του λόγου σήματος προς αριθμητικό σφάλμα κατά την παραγώγιση (numerical SNR).
+Ωστόσο, για τις τιμές $A_0$ που εξετάστηκαν, η ακρίβεια παρέμεινε υψηλή.
+
+\section*{Συμπεράσματα}
+
+Στην παρούσα εργασία μελετήθηκε η εκτίμηση παραμέτρων ενός γραμμικοποιημένου μοντέλου εκκρεμούς, με εφαρμογή της μεθόδου των ελαχίστων τετραγώνων (Least Squares).
+Αρχικά πραγματοποιήθηκε αριθμητική προσομοίωση του συστήματος, εξετάζοντας τη δυναμική του και τον ρυθμό σταθεροποίησης της απόκρισής του.
+\par
+Στη συνέχεια, χρησιμοποιήθηκαν τα δεδομένα της προσομοίωσης για την εκτίμηση των παραμέτρων $mL^2$, $c$ και $mgL$, με βάση μόνο τις μετρήσεις της γωνίας $q(t)$ και της εισόδου $u(t)$.
+Τα αποτελέσματα έδειξαν ότι, υπό ιδανικές συνθήκες, η μέθοδος οδηγεί σε πολύ ακριβείς εκτιμήσεις.
+\par
+Η ανάλυση των Υποερωτημάτων του Θέματος~3 ανέδειξε τη συμπεριφορά της μεθόδου σε ρεαλιστικότερα σενάρια.
+Διαπιστώθηκε ότι:
+\begin{itemize}
+    \item Η προσθήκη λευκού θορύβου στο $q(t)$ μειώνει σημαντικά την ακρίβεια, λόγω της ευαισθησίας της αριθμητικής παραγώγισης στον θόρυβο.
+    \item Η περίοδος δειγματοληψίας επηρεάζει έντονα την ποιότητα εκτίμησης: πολύ αραιή δειγματοληψία οδηγεί σε μεγάλη απώλεια πληροφορίας.
+    \item Το πλάτος της εισόδου $A_0$ δεν επηρεάζει ουσιαστικά την ακρίβεια, καθώς η δομή του συστήματος είναι γραμμική ως προς το $A_0$.
+\end{itemize}
+\par
+Συνολικά, η μελέτη ανέδειξε τις δυνατότητες αλλά και τους περιορισμούς της μεθόδου LS σε διακριτοχρονικές συνθήκες, παρέχοντας πολύτιμα συμπεράσματα για τη χρήση της σε εφαρμογές δυναμικής μοντελοποίησης.
 
 \end{document}
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diff --git a/Work 1/scripts/Problem3a.m b/Work 1/scripts/Problem3a.m
new file mode 100644
index 0000000..669769c
--- /dev/null
+++ b/Work 1/scripts/Problem3a.m	
@@ -0,0 +1,114 @@
+% === Problem 3a: Effect of Noise on Parameter Estimation ===
+
+clear; close all;
+
+% True system parameters
+m = 0.75;
+L = 1.25;
+c = 0.15;
+g = 9.81;
+
+mL2_true = m * L^2;
+mgL_true = m * g * L;
+theta_true = [mL2_true; c; mgL_true];
+
+% Load clean data
+data = readtable('problem1_data.csv');
+t = data.t;
+q_clean = data.q;
+u = data.u;
+Ts = t(2) - t(1);
+N = length(t);
+
+% Derivatives from clean q
+dq_clean = zeros(N,1);
+ddq_clean = zeros(N,1);
+for k = 2:N-1
+    dq_clean(k) = (q_clean(k+1) - q_clean(k-1)) / (2*Ts);
+    ddq_clean(k) = (q_clean(k+1) - 2*q_clean(k) + q_clean(k-1)) / Ts^2;
+end
+
+% LS estimation on clean data
+X_clean = [ddq_clean, dq_clean, q_clean];
+theta_hat_clean = (X_clean' * X_clean) \ (X_clean' * u);
+rel_error_clean = abs((theta_hat_clean - theta_true) ./ theta_true) * 100;
+
+% Reconstruct q̂_clean
+q_hat_clean = zeros(N, 1);
+dq_hat_clean = zeros(N, 1);
+q_hat_clean(1) = q_clean(1);
+dq_hat_clean(1) = dq_clean(1);
+for k = 2:N
+    ddq_hat_clean_k = (1/theta_hat_clean(1)) * ...
+        (u(k-1) - theta_hat_clean(2)*dq_clean(k-1) - theta_hat_clean(3)*q_clean(k-1));
+    dq_hat_clean(k) = dq_hat_clean(k-1) + Ts * ddq_hat_clean_k;
+    q_hat_clean(k) = q_hat_clean(k-1) + Ts * dq_hat_clean(k-1);
+end
+
+% === Loop over noise levels ===
+noise_levels = [0.001, 0.0025];
+
+for i = 1:length(noise_levels)
+    noise_std = noise_levels(i);
+    q_noisy = q_clean + noise_std * randn(size(q_clean));
+
+    % Derivatives from noisy q
+    dq_noisy = zeros(N,1);
+    ddq_noisy = zeros(N,1);
+    for k = 2:N-1
+        dq_noisy(k) = (q_noisy(k+1) - q_noisy(k-1)) / (2*Ts);
+        ddq_noisy(k) = (q_noisy(k+1) - 2*q_noisy(k) + q_noisy(k-1)) / Ts^2;
+    end
+
+    % LS estimation on noisy data
+    X_noisy = [ddq_noisy, dq_noisy, q_noisy];
+    theta_hat_noisy = (X_noisy' * X_noisy) \ (X_noisy' * u);
+    rel_error_noisy = abs((theta_hat_noisy - theta_true) ./ theta_true) * 100;
+
+    % Reconstruct q̂_noisy
+    q_hat_noisy = zeros(N,1);
+    dq_hat_noisy = zeros(N,1);
+    q_hat_noisy(1) = q_noisy(1);
+    dq_hat_noisy(1) = dq_noisy(1);
+    for k = 2:N
+        ddq_hat_noisy_k = (1/theta_hat_noisy(1)) * ...
+            (u(k-1) - theta_hat_noisy(2)*dq_noisy(k-1) - theta_hat_noisy(3)*q_noisy(k-1));
+        dq_hat_noisy(k) = dq_hat_noisy(k-1) + Ts * ddq_hat_noisy_k;
+        q_hat_noisy(k) = q_hat_noisy(k-1) + Ts * dq_hat_noisy(k-1);
+    end
+
+    % Print results
+    fprintf('\n--- Noise std = %.4f ---\n', noise_std);
+    fprintf('Clean:  mL2=%.4f (%.2f%%), c=%.4f (%.2f%%), mgL=%.4f (%.2f%%)\n', ...
+        theta_hat_clean(1), rel_error_clean(1), ...
+        theta_hat_clean(2), rel_error_clean(2), ...
+        theta_hat_clean(3), rel_error_clean(3));
+    fprintf('Noisy:  mL2=%.4f (%.2f%%), c=%.4f (%.2f%%), mgL=%.4f (%.2f%%)\n', ...
+        theta_hat_noisy(1), rel_error_noisy(1), ...
+        theta_hat_noisy(2), rel_error_noisy(2), ...
+        theta_hat_noisy(3), rel_error_noisy(3));
+
+    % === Combined plot ===
+    figure('Name', sprintf('Noise std = %.4f', noise_std), 'Position', [100, 100, 1000, 800]);
+
+    subplot(2,1,1);
+    plot(t, q_clean, 'b', ...
+         t, q_hat_clean, 'g--', ...
+         t, q_hat_noisy, 'r:');
+    legend('Actual q(t)', 'Estimated (clean)', 'Estimated (noisy)');
+    title(sprintf('Estimated Output (σ = %.4f)', noise_std));
+    ylabel('q(t) [rad]');
+    grid on;
+
+    subplot(2,1,2);
+    bar([rel_error_clean, rel_error_noisy]);
+    set(gca, 'XTickLabel', {'mL^2', 'c', 'mgL'});
+    legend({'No Noise', sprintf('With Noise (σ=%.4f)', noise_std)}, 'Location', 'northwest');
+    ylabel('Relative Error [%]');
+    title('Estimation Error');
+    grid on;
+
+    % Save figure
+    filename = sprintf('Prob3a_NoiseStd%.4f.png', noise_std);
+    saveas(gcf, filename);
+end
diff --git a/Work 1/scripts/Problem3b.m b/Work 1/scripts/Problem3b.m
new file mode 100644
index 0000000..8334db1
--- /dev/null
+++ b/Work 1/scripts/Problem3b.m	
@@ -0,0 +1,80 @@
+% === Problem 3b: Effect of Sampling Period Ts on Estimation Accuracy ===
+
+clear; close all;
+
+% True parameters
+m = 0.75;
+L = 1.25;
+c = 0.15;
+g = 9.81;
+
+mL2_true = m * L^2;
+mgL_true = m * g * L;
+theta_true = [mL2_true; c; mgL_true];
+
+% Parameters of input
+A0 = 4;
+omega = 2;
+
+% Time settings
+T_final = 20;
+dt = 1e-4;
+t_full = 0:dt:T_final;
+
+% Simulate system with very fine resolution
+odefun = @(t, x) [
+    x(2);
+    (1/(m*L^2)) * (A0*sin(omega*t) - c*x(2) - m*g*L*x(1))
+];
+x0 = [0; 0];
+[t_sim, x_sim] = ode45(odefun, t_full, x0);
+
+% Sampling periods to test
+Ts_list = [0.01, 0.05, 0.1, 0.2, 0.5];
+n_cases = length(Ts_list);
+rel_errors_all = zeros(3, n_cases);
+
+for i = 1:n_cases
+    Ts = Ts_list(i);
+
+    % Resample at this Ts
+    t_sampled = t_sim(1):Ts:t_sim(end);
+    q = interp1(t_sim, x_sim(:,1), t_sampled);
+    u = A0 * sin(omega * t_sampled);
+
+    N = length(t_sampled);
+
+    % Compute dq and ddq with central differences
+    dq = zeros(N,1);
+    ddq = zeros(N,1);
+    for k = 2:N-1
+        dq(k) = (q(k+1) - q(k-1)) / (2*Ts);
+        ddq(k) = (q(k+1) - 2*q(k) + q(k-1)) / Ts^2;
+    end
+
+    % LS Estimation
+    idx = 2:N-1;        % Truncate to valid range (2:N-1)
+    X = [ddq(idx), dq(idx), q(idx)'];
+    y = u(idx).';
+    theta_hat = (X' * X) \ (X' * y);
+
+    rel_error = abs((theta_hat - theta_true) ./ theta_true) * 100;
+    rel_errors_all(:, i) = rel_error;
+
+    % Print
+    fprintf('Ts = %.3f s → mL^2=%.4f (%.2f%%), c=%.4f (%.2f%%), mgL=%.4f (%.2f%%)\n', ...
+        Ts, theta_hat(1), rel_error(1), ...
+        theta_hat(2), rel_error(2), ...
+        theta_hat(3), rel_error(3));
+end
+
+% === Plot ===
+figure('Name', 'Problem 3b - Effect of Sampling Period', 'Position', [100, 100, 1000, 600]);
+plot(Ts_list, rel_errors_all', '-o', 'LineWidth', 2, 'MarkerSize', 4);
+legend({'mL^2', 'c', 'mgL'}, 'Location', 'northwest');
+xlabel('Sampling Period Ts [sec]');
+ylabel('Relative Error [%]');
+title('Effect of Ts on Parameter Estimation');
+grid on;
+
+saveas(gcf, 'Prob3b_SamplingPeriodEffect.png');
diff --git a/Work 1/scripts/Problem3c.m b/Work 1/scripts/Problem3c.m
new file mode 100644
index 0000000..4f5b13e
--- /dev/null
+++ b/Work 1/scripts/Problem3c.m	
@@ -0,0 +1,78 @@
+% === Problem 3c: Effect of Input Amplitude A0 on Estimation Accuracy ===
+
+clear; close all;
+
+% True system parameters
+m = 0.75;
+L = 1.25;
+c = 0.15;
+g = 9.81;
+
+mL2_true = m * L^2;
+mgL_true = m * g * L;
+theta_true = [mL2_true; c; mgL_true];
+
+% Simulation settings
+omega = 2;       % input frequency
+Ts = 0.1;        % sampling period
+dt = 1e-4;       % integration resolution
+T_final = 20;    % simulation time
+
+% Amplitudes to test
+A0_list = [1, 2, 4, 6, 8, 16, 32];
+n_cases = length(A0_list);
+rel_errors_all = zeros(3, n_cases);
+
+for i = 1:n_cases
+    A0 = A0_list(i);
+
+    % Simulate system
+    t_full = 0:dt:T_final;
+    odefun = @(t, x) [
+        x(2);
+        (1/(m*L^2)) * (A0*sin(omega*t) - c*x(2) - m*g*L*x(1))
+    ];
+    x0 = [0; 0];
+    [t_sim, x_sim] = ode45(odefun, t_full, x0);
+
+    % Resample at Ts
+    t_sampled = t_sim(1):Ts:t_sim(end);
+    q = interp1(t_sim, x_sim(:,1), t_sampled);
+    u = A0 * sin(omega * t_sampled);
+    N = length(t_sampled);
+
+    % Estimate derivatives
+    dq = zeros(N,1);
+    ddq = zeros(N,1);
+    for k = 2:N-1
+        dq(k) = (q(k+1) - q(k-1)) / (2*Ts);
+        ddq(k) = (q(k+1) - 2*q(k) + q(k-1)) / Ts^2;
+    end
+
+    % LS Estimation
+    idx = 2:N-1;
+    X = [ddq(idx), dq(idx), q(idx)'];
+    y = u(idx).';
+    theta_hat = (X' * X) \ (X' * y);
+
+    rel_error = abs((theta_hat - theta_true) ./ theta_true) * 100;
+    rel_errors_all(:, i) = rel_error;
+
+    % Print
+    fprintf('A0 = %d → mL^2=%.4f (%.2f%%), c=%.4f (%.2f%%), mgL=%.4f (%.2f%%)\n', ...
+        A0, theta_hat(1), rel_error(1), ...
+        theta_hat(2), rel_error(2), ...
+        theta_hat(3), rel_error(3));
+end
+
+% === Plot ===
+figure('Name', 'Problem 3c - Effect of A0', 'Position', [100, 100, 1000, 600]);
+plot(A0_list, rel_errors_all', '-o', 'LineWidth', 2, 'MarkerSize', 4);
+legend({'mL^2', 'c', 'mgL'}, 'Location', 'northeast');
+xlabel('Input Amplitude A_0');
+ylabel('Relative Error [%]');
+title('Effect of Input Amplitude on Parameter Estimation');
+grid on;
+ylim([0 1.1]);
+
+saveas(gcf, 'Prob3c_AmplitudeEffect.png');
diff --git a/Work 1/scripts/problem3_data_T10s.csv b/Work 1/scripts/problem3_data_T10s.csv
new file mode 100644
index 0000000..494a448
--- /dev/null
+++ b/Work 1/scripts/problem3_data_T10s.csv	
@@ -0,0 +1,102 @@
+t,q,dq,ddq,u
+0,0,0,0,0
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diff --git a/Work 1/scripts/problem3_data_T20s.csv b/Work 1/scripts/problem3_data_T20s.csv
new file mode 100644
index 0000000..2a5f2d2
--- /dev/null
+++ b/Work 1/scripts/problem3_data_T20s.csv	
@@ -0,0 +1,202 @@
+t,q,dq,ddq,u
+0,0,0,0,0
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diff --git a/Work 1/scripts/problem3_data_T40s.csv b/Work 1/scripts/problem3_data_T40s.csv
new file mode 100644
index 0000000..92676e7
--- /dev/null
+++ b/Work 1/scripts/problem3_data_T40s.csv	
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diff --git a/Work 1/scripts/problem3_data_T60s.csv b/Work 1/scripts/problem3_data_T60s.csv
new file mode 100644
index 0000000..41138a5
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diff --git a/Work 1/scripts/problem3_data_T90s.csv b/Work 1/scripts/problem3_data_T90s.csv
new file mode 100644
index 0000000..3dc0dff
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