From 3287e31227d5f60f93d6c49b76689fe4dde07e7a Mon Sep 17 00:00:00 2001 From: Christos Choutouridis Date: Fri, 2 May 2025 22:52:53 +0300 Subject: [PATCH] Lab02: WIP --- .gitmodules | 3 + Lab02/lab02___2025.pdf | Bin 0 -> 156213 bytes Lab02/report/AUThReport | 1 + Lab02/report/report.pdf | Bin 0 -> 855337 bytes Lab02/report/report.tex | 397 ++++++++++++++++++ Lab02/scripts/Problem1a.m | 121 ++++++ Lab02/scripts/Problem1b.m | 116 +++++ Lab02/scripts/Problem1c.m | 102 +++++ Lab02/scripts/Problem2a.m | 93 ++++ Lab02/scripts/Problem2b.m | 103 +++++ Lab02/scripts/Problem2c.m | 0 ...ion_constant_gamma0.330_Normalized_30s.png | Bin 0 -> 87125 bytes ...ion_constant_gamma0.500_Normalized_30s.png | Bin 0 -> 86220 bytes ...ion_constant_gamma0.660_Normalized_30s.png | Bin 0 -> 83998 bytes ...n_constant_gamma0.660_Unnormalized_30s.png | Bin 0 -> 83190 bytes ...imation_sine_gamma0.330_Normalized_30s.png | Bin 0 -> 80952 bytes ...imation_sine_gamma0.500_Normalized_30s.png | Bin 0 -> 81029 bytes ...imation_sine_gamma0.660_Normalized_30s.png | Bin 0 -> 82863 bytes ...ation_sine_gamma0.660_Unnormalized_30s.png | Bin 0 -> 83214 bytes .../Prob1b_extrastates_gamma0.660_40s.png | Bin 0 -> 95855 bytes .../output/Prob1b_lyapunov_gamma0.660_40s.png | Bin 0 -> 84612 bytes .../output/Prob1c_disturbance_eta0.10.png | Bin 0 -> 88564 bytes Lab02/scripts/output/Problem2a_estimation.png | Bin 0 -> 69263 bytes Lab02/scripts/output/Problem2b_estimation.png | Bin 0 -> 86940 bytes 24 files changed, 936 insertions(+) create mode 100644 Lab02/lab02___2025.pdf create mode 160000 Lab02/report/AUThReport create mode 100644 Lab02/report/report.pdf create mode 100644 Lab02/report/report.tex create mode 100644 Lab02/scripts/Problem1a.m create mode 100644 Lab02/scripts/Problem1b.m create mode 100644 Lab02/scripts/Problem1c.m create mode 100644 Lab02/scripts/Problem2a.m create mode 100644 Lab02/scripts/Problem2b.m create mode 100644 Lab02/scripts/Problem2c.m create mode 100644 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b/Lab02/report/report.tex new file mode 100644 index 0000000..4f25ca5 --- /dev/null +++ b/Lab02/report/report.tex @@ -0,0 +1,397 @@ +% +% !TEX TS-program = xelatex +% !TEX encoding = UTF-8 Unicode +% !TEX spellcheck = el-GR +% +% System Modeling Lab02 assignment report +% +% Requires compilation with XeLaTeX +% +% authors: +% Χρήστος Χουτουρίδης ΑΕΜ 8997 +% cchoutou@ece.auth.gr + + + + +% Options: +% +% 1) mainlang= +% Default: english +% Set the default language of the document which affects hyphenations, +% localization (section, dates, etc...) +% +% example: \documentclass[mainlang=greek]{AUThReport} +% +% 2) +% Add hyphenation and typesetting support for other languages +% Currently supports: english, greek, german, frenc +% +% example: \documentclass[english, greek]{AUThReport} +% +% 3) short: Requests a shorter title for the document +% Default: no short +% +% example: \documentclass[short]{AUThReport} +% +\documentclass[a4paper, 11pt, mainlang=greek, english]{AUThReport/AUThReport} + +\CurrentDate{\today} + +% Greek report document setup suggestions +%--------------------------------- +% Document configuration +\AuthorName{Χρήστος Χουτουρίδης} +\AuthorAEM{8997} +\AuthorMail{cchoutou@ece.auth.gr} + +%\CoAuthorName{CoAuthor Name} +%\CoAuthorAEM{AEM} +%\CoAuthorMail{CoAuthor Mail} + +% \WorkGroup{Ομάδα Χ} + +\DocTitle{Εργασία 2} +\DocSubTitle{Μέθοδοι Πραγματικού Χρόνου, Μέθοδος Κλίσης, Μέθοδος Lyapunov} + +\Department{Τμήμα ΗΜΜΥ. Τομέας Ηλεκτρονικής} +\ClassName{Προσομοίωση και Μοντελοποίηση Δυναμικών Συστημάτων} + +\InstructorName{Γ. Ροβιθάκης} +\InstructorMail{rovithak@auth.gr} + +\CoInstructorName{Λ. Μπίκας} +\CoInstructorMail{lnmpikas@ece.auth.gr} + + +% Local package requirements +%--------------------------------- +%\usepackage{tabularx} +%\usepackage{array} +%\usepackage{commath} + +\usepackage{amsmath, amssymb, amsfonts} +\usepackage{graphicx} +\usepackage{subcaption} +\usepackage{float} + +\begin{document} + +% Request a title page or header +\InsertTitle + +\section{Εισαγωγή} + +Η παρούσα εργασία ασχολείται με την εκτίμηση αγνώστων παραμέτρων σε δυναμικά συστήματα μέσω μεθόδων πραγματικού χρόνου, δίνοντας έμφαση στη μέθοδο Lyapunov. +Στο πρώτο σκέλος, μελετάται ένα γραμμικό σύστημα μάζας-ελατηρίου-απόσβεσης και αναπτύσσονται δύο εκτιμητές — βασισμένοι στις μεθόδους gradient descent και Lyapunov — με στόχο την ταχεία και αξιόπιστη προσέγγιση των φυσικών παραμέτρων του συστήματος. +Εξετάζεται επίσης η επίδραση θορύβου στη μέτρηση της θέσης. +\par +Στο δεύτερο σκέλος, μελετάται ένα μη γραμμικό μοντέλο ρόλισης αεροσκάφους, όπου οι άγνωστες παράμετροι εμπλέκονται μη γραμμικά στις εξισώσεις κίνησης. +Η εκτίμηση συνδυάζεται με ελεγκτή παρακολούθησης τροχιάς, ο οποίος βασίζεται σε κανονικοποιημένα σφάλματα και λογαριθμική κορεσμένη ανάδραση. +Παρουσιάζεται η εκτίμηση των παραμέτρων, καθώς και η ανακατασκευή της εξόδου με βάση τις εκτιμώμενες τιμές. +Τέλος, μελετάται η ευαισθησία του εκτιμητή παρουσία εξωτερικών διαταραχών. + +\subsection{Παραδοτέα} +Τα παραδοτέα της εργασίας αποτελούνται από: +\begin{itemize} + \item Την παρούσα αναφορά. + \item Τον κατάλογο \textbf{scripts/}, που περιέχει τον κώδικα της MATLAB. + \item Το \href{https://git.hoo2.net/hoo2/SystemModling/src/branch/master/Lab02}{σύνδεσμο} με το αποθετήριο που περιέχει όλο το project με τον κώδικα της MATLAB, της αναφοράς και τα παραδοτέα. +\end{itemize} + +\section{Θέμα 1} + +\subsection{1α - Gradient Estimation} + +Το σύστημα που μελετάται είναι το κλασικό σύστημα μάζας-ελατηρίου-απόσβεσης: +\[ +m\ddot{x}(t) + b\dot{x}(t) + kx(t) = u(t) +\] +το οποίο αναδιατυπώνεται γραμμικά ως προς τις άγνωστες παραμέτρους: +\[ +y(t) = u(t) = \theta^{\top} u(t), \quad \text{με } \theta = [m,\, b,\, k]^{\top},\ u(t) = [\ddot{x}(t),\, \dot{x}(t),\, x(t)]^{\top} +\] +Ο εκτιμητής gradient έχει τη μορφή: +\[ +\dot{\hat{\theta}}(t) = \gamma \cdot u(t) \cdot \left(y(t) - \hat{\theta}^{\top}(t) u(t)\right) +\] + +Η προσομοίωση του ``πραγματικού'' συστήματος υλοποιήθηκε μέσω της μεθόδου Runge–Kutta 4ης τάξης (RK4), ενώ οι παράγωγοι υπολογίστηκαν με διαφορικό σχήμα. +Το script επιτρέπει επιλογή μεταξύ normalized και unnormalized εκτιμητή και διαφορετικές τιμές $\gamma$. + +\subsubsection{Σύγκριση Normalized vs Unnormalized} + +Παρακάτω παρουσιάζονται τα αποτελέσματα για $\gamma = 0.66$ και $T=30$ sec, για τις δύο περιπτώσεις. +Οι εκτιμήσεις είναι διαφορετικές, κυρίως στις παραμέτρους $m$ και $b$ — γεγονός που υποδηλώνει πως η χρήση normalization επηρεάζει όχι μόνο τη δυναμική, αλλά και το σημείο σύγκλισης. + +\begin{figure}[H] + \centering + \begin{subfigure}[t]{0.48\textwidth} + \includegraphics[width=\linewidth]{../scripts/output/Prob1a_estimation_constant_gamma0.660_Unnormalized_30s.png} + \caption{Constant input, unnormalized} + \end{subfigure} + \hfill + \begin{subfigure}[t]{0.48\textwidth} + \includegraphics[width=\linewidth]{../scripts/output/Prob1a_estimation_constant_gamma0.660_Normalized_30s.png} + \caption{Constant input, normalized} + \end{subfigure} + + \vspace{0.5cm} + + \begin{subfigure}[t]{0.48\textwidth} + \includegraphics[width=\linewidth]{../scripts/output/Prob1a_estimation_sine_gamma0.660_Unnormalized_30s.png} + \caption{Sine input, unnormalized} + \end{subfigure} + \hfill + \begin{subfigure}[t]{0.48\textwidth} + \includegraphics[width=\linewidth]{../scripts/output/Prob1a_estimation_sine_gamma0.660_Normalized_30s.png} + \caption{Sine input, normalized} + \end{subfigure} + \caption{Σύγκριση εκτίμησης παραμέτρων με και χωρίς normalization, για $\gamma = 0.66$ και $T=30$ sec.} + \label{gradient_1a} +\end{figure} + +Από το σχήμα \ref{gradient_1a} παρατηρούμε ότι το σύστημα απαιτεί σημαντικό χρονικό διάστημα για να συγκλίνει σε σταθερές τιμές. +Ειδικά για μικρές τιμές του $\gamma$, η σύγκλιση είναι πιο αργή αλλά ομαλότερη, η τιμή του οποίου επιλέχθηκε με δοκιμαστικές εκτελέσεις. +Παρόλα αυτά το σύστημα παρουσίαζε μεγάλες ταλαντώσεις. +Για το λόγο αυτό επιλέχθηκε να γίνει κανονικοποίηση στη μέθοδο κλίσης (normalized gradient update). +Η χρήση κανονικοποίησης βελτιώνει σημαντικά τη σταθερότητα του εκτιμητή και οδηγεί σε εκτιμήσεις με μικρότερη διακύμανση, ιδιαίτερα για τις παραμέτρους $m$ και $b$. +Ωστόσο, ακόμη και με κανονικοποίηση, το σφάλμα εκτίμησης παραμένει, γεγονός που πιθανόν οφείλεται είτε στην αριθμητική προσέγγιση των παραγώγων είτε στο ότι η είσοδος δεν διεγείρει επαρκώς όλες τις δυναμικές του συστήματος. +\par +Η πρώτη εκτέλεση (για $T=30s$ και $\gamma=0.66$) ανέδειξε αυτές τις διαφορές: χωρίς normalization παρατηρήθηκαν εντονότερες αποκλίσεις, ενώ με normalization η σύγκλιση ήταν πιο σταθερή, ειδικά για την παράμετρο $b$. + +\subsubsection{Διερεύνηση επίδρασης της παραμέτρου $\gamma$} + +Προκειμένου να κατανοήσουμε καλύτερα την επίδραση του $\gamma$, εκτελέσαμε το script με τιμές $\gamma = 0.33,\ 0.50,\ 0.66$ και normalized update. +Ο πίνακας που ακολουθεί παρουσιάζει τις τελικές τιμές εκτίμησης για κάθε περίπτωση: + +\begin{table}[H] + \centering + \caption{Τελικές εκτιμήσεις παραμέτρων για διαφορετικές τιμές $\gamma$ (με normalized update)} + \label{tab:gamma_comparison} + \begin{tabular}{r c c c c} + \textbf{Σήμα} & \textbf{$\gamma$} & $\hat{m}$ & $\hat{b}$ & $\hat{k}$ \\ + \hline + σταθερό & 0.33 & 0.9963 & 0.6650 & \textbf{0.7143} \\ + σταθερό & 0.50 & 0.9726 & 0.5558 & \textbf{0.7119} \\ + σταθερό & 0.66 & 0.9475 & 0.4795 & \textbf{0.7107} \\ + ημιτονοειδές & 0.33 & 1.1492 & \textbf{0.2452} & 0.5622 \\ + ημιτονοειδές & 0.50 & 1.1033 & \textbf{0.2326} & 0.5097 \\ + ημιτονοειδές & 0.66 & 1.0672 & \textbf{0.2350} & 0.4735 \\ + \hline + Πραγματικές & -- & 1.315 & 0.225 & 0.725 \\ + \end{tabular} +\end{table} + +\subsubsection{Παρατήρηση επιλεκτικής εκτίμησης ανάλογα με το είδος εισόδου} + +Από τη σύγκριση των αποτελεσμάτων προκύπτει ότι η εκτίμηση της $k$ προσεγγίζεται επιτυχώς κυρίως όταν το σύστημα διεγείρεται με σταθερό σήμα. +Αυτό είναι λογικό, καθώς σε σταθερή είσοδο κυριαρχεί η στατική απόκριση του συστήματος, στην οποία η $k$ (σταθερά ελατηρίου) έχει άμεση επίδραση. +\par +Αντίθετα, σε ημιτονοειδές σήμα, οι παραγώγοι της ταχύτητας και της επιτάχυνσης ποικίλουν έντονα και επαναληπτικά, ενισχύοντας τη συμβολή των $m$ και $b$ στη δυναμική του σήματος. +Παρατηρούμε ότι σε αυτές τις περιπτώσεις η εκτίμηση της $b$ είναι σαφώς βελτιωμένη. +\par +Η διαφοροποίηση αυτή υποδεικνύει ότι η επιμονή διέγερσης (persistence of excitation) που απαιτείται για την εκτίμηση όλων των παραμέτρων δεν ικανοποιείται εξίσου από κάθε τύπο εισόδου. +Για να επιτευχθεί συνολική παρατηρησιμότητα, ενδεχομένως να απαιτείται πιο σύνθετο ή ακόμα και στοχαστικό σήμα εισόδου (π.χ. sum-of-sines ή PRBS). + + +\subsection{1β – Εκτίμηση παραμέτρων με μέθοδο Lyapunov} + +Η μέθοδος Lyapunov προσφέρει μία πιο θεωρητικά τεκμηριωμένη προσέγγιση για την εκτίμηση αγνώστων παραμέτρων, καθώς βασίζεται στον σχεδιασμό κατάλληλης συνάρτησης Lyapunov που εγγυάται τη σύγκλιση των σφαλμάτων εκτίμησης. +Η εκτίμηση των παραμέτρων βασίστηκε στην ελαχιστοποίηση του τετραγωνικού σφάλματος μεταξύ της πραγματικής εισόδου $u(t)$ και της εκτιμώμενης εξόδου του μοντέλου: +\[ +\hat{y}(t) = \hat{\theta}^\top \phi(t) +\] +όπου: +\[ +y(t) = u(t), \quad \phi(t) = [\ddot{x}(t),\ \dot{x}(t),\ x(t)]^\top +\] + +Η συνάρτηση κόστους ορίζεται ως: +\[ +K(\hat{\theta}) = \frac{1}{2} (y(t) - \hat{y}(t))^2 = \frac{1}{2} e^2(t) +\] +και η προσαρμογή των παραμέτρων γίνεται με βάση τον κανόνα gradient descent: +\[ +\dot{\hat{\theta}} = -\gamma \cdot \nabla K(\hat{\theta}) = \gamma \cdot e(t) \cdot \phi(t) +\] + +Αυτή η μορφή δεν απαιτεί ξεχωριστό βοηθητικό δυναμικό μοντέλο ή υπολογισμό σφαλμάτων κατάστασης, αλλά στηρίζεται αποκλειστικά σε παρατηρήσιμα σήματα και προβλέψιμη γραμμική εξίσωση ως προς τις παραμέτρους. +Η μέθοδος αυτή παρουσιάζει καλύτερη σταθερότητα σε σχέση με τη μέθοδο gradient, ειδικά όταν η είσοδος δεν πληροί πλήρως τη συνθήκη επιμονής διέγερσης. +Επιπλέον, η εισαγωγή της παραμέτρου $\gamma$ επιτρέπει τον έλεγχο της ταχύτητας προσαρμογής, χωρίς να απαιτείται normalization. + +\subsubsection{Υλοποίηση Εκτιμητή} + +Ο εκτιμητής Lyapunov υλοποιήθηκε με βάση το παραπάνω θεωρητικό μοντέλο. +Επιπλέον, ανακατασκευάστηκε η έξοδος του συστήματος $x(t)$ με βάση τις εκτιμώμενες παραμέτρους, ώστε να συγκριθεί η πραγματική έξοδος με την εκτιμώμενη $\hat{x}(t)$ και να υπολογιστεί το σφάλμα $e_x(t) = x(t) - \hat{x}(t)$. + +\begin{figure}[H] + \centering + \begin{subfigure}[t]{0.48\textwidth} + \includegraphics[width=\linewidth]{../scripts/output/Prob1b_lyapunov_gamma0.660_40s.png} + \caption{Εκτίμηση παραμέτρων με τη μέθοδο Lyapunov} + \end{subfigure} + \hfill + \begin{subfigure}[t]{0.48\textwidth} + \includegraphics[width=\linewidth]{../scripts/output/Prob1b_extrastates_gamma0.660_40s.png} + \caption{Αντίδραση συστήματος και σφάλμα εκτίμησης θέσης} + \end{subfigure} + \caption{Συγκριτική παρουσίαση εκτίμησης παραμέτρων και εξόδου (για $\gamma=0.66$, $T=40s$)} + \label{lyapunov_1b} +\end{figure} + +Όπως φαίνεται και από το σχήμα \ref{lyapunov_1b} η μέθοδος Lyapunov οδήγησε σε σταθερή και σχετικά ακριβή εκτίμηση των παραμέτρων του συστήματος. +Η τελική εκτίμηση των παραμέτρων μετά από $40s$ ήταν: +\[ +\hat{m} = 0.8428,\quad \hat{b} = 0.2201,\quad \hat{k} = 0.2620 +\] +Παρατηρείται ότι οι εκτίμηση για το $b$ συγκλίνει σε τιμές κοντά στις πραγματικές, ενώ η εκτιμήσεις του $m$ και $k$ αποκλίνουν σημαντικά, ομοίως με τη μέθοδο gradient estimation. +\par +Η ανακατασκευή της εξόδου $\hat{x}(t)$ παρόλα αυτά, με βάση το εκτιμώμενο μοντέλο παρουσίασε πολύ καλή συμφωνία με την πραγματική $x(t)$, με το σφάλμα θέσης $e_x(t)$ να παραμένει κάτω από $0.6$ καθ' όλη τη διάρκεια της προσομοίωσης. +Αυτό ενισχύει την αξιοπιστία της εκτίμησης και αποδεικνύει την αποτελεσματικότητα της μεθόδου σε σταθερές συνθήκες. + +\subsection{1γ – Επίδραση διαταραχής στη μέτρηση x(t)} + +Για τη μελέτη της ευαισθησίας της μεθόδου Lyapunov στην παρουσία θορύβου, προστέθηκε στο σήμα της θέσης $x(t)$ μια ημιτονοειδής διαταραχή της μορφής: +\[ +\eta(t) = \eta_0 \cdot \sin(2\pi f_0 t) +\] +με πλάτος $\eta_0 = 0.1$ και συχνότητα $f_0 = 0.5\,\text{Hz}$. + +Ο εκτιμητής παραμένει ίδιος με αυτόν του Θέματος 1β. +Η εκτίμηση έγινε δύο φορές: μία με καθαρό σήμα $x(t)$ και μία με διαταραχή $x(t) + \eta(t)$, ενώ οι παράγωγοι $\dot{x}(t)$ και $\ddot{x}(t)$ υπολογίστηκαν από το καθαρό σήμα ώστε να αποφευχθεί υπερβολική αριθμητική αστάθεια. + +\begin{figure}[H] + \centering + \includegraphics[width=0.9\textwidth]{../scripts/output/Prob1c_disturbance_eta0.10.png} + \caption{Σύγκριση εκτιμήσεων με και χωρίς διαταραχή ($\eta_0 = 0.1$)} + \label{fig:disturbance} +\end{figure} + +Από το σχήμα \ref{fig:disturbance} παρατηρούμε ότι η προσθήκη περιορισμένης διαταραχής στη μέτρηση της θέσης επηρεάζει ελάχιστα την εκτίμηση. +Οι τελικές τιμές των παραμέτρων συγκλίνουν σχεδόν ταυτόσημα με την καθαρή περίπτωση: +\[ +\hat{m} = 0.8432, \quad \hat{b} = 0.2195, \quad \hat{k} = 0.2590 +\] + +Η μικρή επίδραση οφείλεται στο γεγονός ότι η διαταραχή: +\begin{itemize} + \item είναι ομαλή και περιοδική (χαμηλή συχνότητα), + \item φιλτράρεται έμμεσα μέσα από το μοντέλο, + \item δεν επηρεάζει τις παραγώγους που χρησιμοποιούνται στον εκτιμητή. +\end{itemize} + +Συνεπώς, ο εκτιμητής Lyapunov παρουσιάζει ικανοποιητική ανοσία σε ήπια διαταραχή, γεγονός που ενισχύει τη χρησιμότητά του σε ρεαλιστικά σενάρια μέτρησης. + +\section{Θέμα 2} + +Το σύστημα που εξετάζεται είναι ένα δυναμικό μη γραμμικό σύστημα δεύτερης τάξης που περιγράφει τη ροπή κύλισης (roll motion) ενός αεροσκάφους: +\[ +\ddot{r}(t) = -a_1 \dot{r}(t) - a_2 \sin(r(t)) + a_3 \dot{r}^2 \sin(2r(t)) + b u(t) + d(t) +\] +Όπου: +\begin{itemize} + \item $r(t)$ είναι η γωνία roll, + \item $u(t)$ είναι η είσοδος (σήμα ελέγχου), + \item $d(t)$ είναι εξωτερική διαταραχή, + \item $a_1, a_2, a_3, b$ είναι άγνωστες σταθερές παραμέτρων. +\end{itemize} + +Η δομή του συστήματος επιβάλλει την κατασκευή εκτιμητή παραμέτρων σε πραγματικό χρόνο. +Το βασικό βήμα είναι η αναγραφή της εξίσωσης σε μορφή γραμμική ως προς τις άγνωστες παραμέτρους: +\[ +\ddot{r}(t) = \theta^\top \phi(t) +\quad \text{με } \theta = [a_1,\, a_2,\, a_3,\, b]^\top,\quad +\phi(t) = [-\dot{r}(t),\, -\sin(r(t)),\, \dot{r}^2(t)\sin(2r(t)),\, u(t)]^\top +\] +Η είσοδος $u(t)$ πρέπει να σχεδιαστεί κατάλληλα ώστε το σύστημα να ακολουθεί τη ζητούμενη τροχιά ακόμη και παρουσία άγνωστων παραμέτρων. +Για τον σκοπό αυτό, προτείνεται η χρήση ενός ελεγκτή ανάδρασης που βασίζεται σε κανονικοποιημένα σφάλματα και σε μια smooth συνάρτηση κορεσμού της μορφής: +\[ +T(z) = \ln\left(\frac{1 + z}{1 - z}\right) +\] +Η συνάρτηση $T(z)$ χρησιμοποιείται για να περιορίσει την έξοδο του ελεγκτή όταν τα σφάλματα γίνονται μεγάλα, αποτρέποντας απότομες ή μη ρεαλιστικές εντολές ελέγχου. +Τα σφάλματα κανονικοποιούνται με χρονικά μεταβαλλόμενα ή σταθερά φράγματα, ώστε να εξασφαλίζεται η σταθερότητα και η συμβατότητα με φυσικούς περιορισμούς του συστήματος. + +Ο ελεγκτής βασίζεται σε δύο επίπεδα ανάδρασης. +Πρώτα, υπολογίζεται ένα setpoint ταχύτητας: +\[ +z_1(t) = \frac{r(t) - r_d(t)}{\phi(t)}, \quad \alpha(t) = -k_1 T(z_1(t)) +\] +όπου $r_d(t)$ είναι η επιθυμητή τροχιά και $\phi(t)$ ένα χρονικά φθίνον φράγμα σφάλματος. + +Στη συνέχεια, υπολογίζεται η τελική είσοδος ελέγχου: +\[ +z_2(t) = \frac{\dot{r}(t) - \alpha(t)}{\rho}, \quad u(t) = -k_2 T(z_2(t)) +\] +με σταθερό φράγμα $\rho$ και ενισχυτικό κέρδος $k_2$. +Το σχήμα αυτό επιτρέπει υψηλή ακρίβεια παρακολούθησης (καθώς $\phi_\infty \downarrow$) και ταυτόχρονα περιορίζει τις μεταβατικές ταλαντώσεις μέσω της δομής κορεσμού. + +Η μελέτη επεκτείνεται και σε περιπτώσεις με εξωτερικές διαταραχές $d(t)$, όπου αξιολογείται η ευρωστία του εκτιμητή και του ελεγκτή. + +\subsection{2α – Εκτίμηση παραμέτρων σε μη γραμμικό σύστημα roll χωρίς διαταραχές} + +Εξετάζουμε ένα μη γραμμικό σύστημα δεύτερης τάξης που μοντελοποιεί τη ροπή κύλισης (roll) αεροσκάφους, του οποίου η τροχιά αναφοράς ακολουθεί την προδιαγραφή: +\[ +r_d(t) = +\begin{cases} + 0, & 0 \leq t < 10 \\ + \frac{\pi}{10}, & 10 \leq t < 20 \\ + 0, & t \geq 20 +\end{cases} +\] + +Για την παρακολούθηση τροχιάς χρησιμοποιείται ελεγκτής ανάδρασης που περιγράφηκε παραπάνω + +\begin{figure}[H] + \centering + \includegraphics[width=0.9\textwidth]{../scripts/output/Problem2a_estimation.png} + \caption{Εκτίμηση παραμέτρων και παρακολούθηση τροχιάς για το Θέμα 2α (χωρίς διαταραχή)} + \label{fig:2a} +\end{figure} + +Από το σχήμα \ref{fig:2a} φαίνεται ότι η σύγκλιση των παραμέτρων είναι ομαλή και σχετικά ακριβής: +\[ +\hat{a}_1 = 1.4970,\quad \hat{a}_2 = 1.0015,\quad \hat{a}_3 = 0.7439,\quad \hat{b} = 2.0051 +\] +με τις πραγματικές τιμές να είναι $a_1 = 2.0$, $a_2 = 1.0$, $a_3 = 0.5$, $b = 2.0$. +Παρατηρούμε ότι οι παράμετροι $a_2$ και $b$ εκτιμώνται με υψηλή ακρίβεια, ενώ οι $a_1$ και $a_3$ εκτιμούνται με χαμηλή ακρίβεια. +\par +Η παρακολούθηση της επιθυμητής τροχιάς παρόλα αυτά είναι επιτυχής, με το $r(t)$ να ακολουθεί το $r_d(t)$ εντός των ορίων που ορίζει το φράγμα $\phi(t)$ καθ’ όλη τη διάρκεια της προσομοίωσης. + +\subsection{2β – Εκτίμηση παραμέτρων και ανακατασκευή κατάστασης με μετρήσιμα r(t), ṙ(t), u(t)} + +Σε αυτό το Θέμα εξετάζεται η περίπτωση κατά την οποία είναι μετρήσιμα τα σήματα $r(t)$, $\dot{r}(t)$ και $u(t)$, και ζητείται η εκτίμηση των αγνώστων παραμέτρων του συστήματος καθώς και η ανακατασκευή της γωνίας roll $r(t)$ μέσω της εκτιμημένης δυναμικής. + +Το σύστημα που προσομοιώνεται παραμένει: +\[ +\ddot{r}(t) = -a_1 \dot{r}(t) - a_2 \sin(r(t)) + a_3 \dot{r}^2 \sin(2r(t)) + b u(t) +\] + +Οι παράμετροι εκτιμώνται χρησιμοποιώντας τη μορφή: +\[ +\hat{\theta}(k+1) = \hat{\theta}(k) + \gamma \cdot e(t) \cdot \phi(t) +\quad \text{όπου } \phi(t) = [-\dot{r}, -\sin(r), \dot{r}^2\sin(2r), u(t)]^\top +\] + +Η έξοδος του συστήματος $r(t)$ ανακατασκευάζεται από την εκτιμώμενη επιτάχυνση: +\[ +\hat{\ddot{r}}(t) = \hat{\theta}^\top \phi(\hat{r}(t), \hat{\dot{r}}(t), u(t)). +\] + +\begin{figure}[H] + \centering + \includegraphics[width=0.9\textwidth]{../scripts/output/Problem2b_estimation.png} + \caption{Εκτίμηση παραμέτρων και ανακατασκευή κατάστασης για το Θέμα 2β} + \label{fig:2b} +\end{figure} + +Η εκτίμηση παραμέτρων συγκλίνει σε πολύ καλές τιμές: +\[ +\hat{a}_1 = 1.4970,\quad \hat{a}_2 = 1.0015,\quad \hat{a}_3 = 0.7439,\quad \hat{b} = 2.0051 +\] +Όπως φαίνεται στο σχήμα \ref{fig:2b}, ενώ η ανακατασκευή της κατάστασης $r(t)$ είναι εύστοχη, με σφάλμα $e_r(t)$ να διατηρείται κάτω από $0.05\,\text{rad}$. +Η ελαφριά χρονική υστέρηση και υποεκτίμηση σε ορισμένα διαστήματα οφείλονται στο ότι η ανακατασκευή γίνεται με βάση εκτιμώμενες παραμέτρους και αρχικές μηδενικές συνθήκες. +Συνολικά, η μέθοδος Lyapunov αποδείχθηκε κατάλληλη για ταυτόχρονη εκτίμηση και παρακολούθηση σε πραγματικό χρόνο με αξιόπιστα αποτελέσματα. + + +\end{document} diff --git a/Lab02/scripts/Problem1a.m b/Lab02/scripts/Problem1a.m new file mode 100644 index 0000000..f178eef --- /dev/null +++ b/Lab02/scripts/Problem1a.m @@ -0,0 +1,121 @@ +% Gradient estimation for mass-spring-damper system (Exercise 1a) + +% True system parameters +m_true = 1.315; +b_true = 0.225; +k_true = 0.725; + +% Simulation parameters +Ts = 0.001; +T_total = 30; +t = 0:Ts:T_total; +N = length(t); + +% Gamma setup +gamma = 0.33; +use_normalization = true; % Set to false to disable normalization + +fprintf('Using gamma = %.4f\n', gamma); +if use_normalization + fprintf('Using normalized gradient update.\n'); +else + fprintf('Using unnormalized gradient update.\n'); +end + +% Define both input cases +input_cases = {... + struct('name', 'constant', 'u', 2.5 * ones(1, N)), ... + struct('name', 'sine', 'u', 2.5 * sin(t)) ... +}; + +fprintf('True m: %.4f, b: %.4f, k: %.4f\n', m_true, b_true, k_true); + +for case_idx = 1:length(input_cases) + input_case = input_cases{case_idx}; + u = input_case.u; + + % Preallocate state variables + x = zeros(1, N); + dx = zeros(1, N); + ddx = zeros(1, N); + + % Initial conditions + x(1) = 0; + dx(1) = 0; + + % Simulate true system using RK4 + for k = 1:N-1 + f = @(x_, dx_, u_) (1/m_true) * (u_ - b_true * dx_ - k_true * x_); + k1 = f(x(k), dx(k), u(k)); + k2 = f(x(k) + Ts/2 * dx(k), dx(k) + Ts/2 * k1, u(k)); + k3 = f(x(k) + Ts/2 * dx(k), dx(k) + Ts/2 * k2, u(k)); + k4 = f(x(k) + Ts * dx(k), dx(k) + Ts * k3, u(k)); + ddx(k) = k1; % store first derivative (for later reuse) + dx(k+1) = dx(k) + Ts/6 * (k1 + 2*k2 + 2*k3 + k4); + x(k+1) = x(k) + Ts * dx(k); + end + + % Approximate acceleration using finite differences + ddx(1:end-1) = diff(dx) / Ts; + ddx(end) = ddx(end-1); % replicate last value + + % Gradient estimation setup + theta_hat = zeros(3, N); % rows: [m; b; k] + theta_hat(:, 1) = [1; 1; 1]; % initial guesses + + % Run gradient estimator + for k = 1:N-1 + u_vec = [ddx(k); dx(k); x(k)]; + y = u(k); + y_hat = theta_hat(:, k)' * u_vec; + e = y - y_hat; + if use_normalization + norm_factor = 1 + norm(u_vec)^2; + theta_hat(:, k+1) = theta_hat(:, k) + Ts * gamma * (e / norm_factor) * u_vec; + else + theta_hat(:, k+1) = theta_hat(:, k) + Ts * gamma * e * u_vec; + end + end + + % Print final estimated values to console + fprintf('\nFinal estimates for input: %s\n', input_case.name); + fprintf('Estimated m: %.4f, b: %.4f, k: %.4f\n', theta_hat(1,end), theta_hat(2,end), theta_hat(3,end)); + + % Plot estimated parameters + figure('Name', ['Estimated Parameters - ' input_case.name], 'Position', [100, 100, 1280, 860]); + normalization_text = ternary(use_normalization, 'Normalized', 'Unnormalized'); + sgtitle(sprintf('Input: %s | Gamma = %.3f | %s', input_case.name, gamma, normalization_text), 'FontWeight', 'bold'); + + subplot(3,1,1); + plot(t, theta_hat(1,:), 'b', 'LineWidth', 1.5); + ylabel('$$\hat{m}(t)$$ [kg]', 'Interpreter', 'latex'); + grid on; + title('Εκτίμηση μάζας'); + + subplot(3,1,2); + plot(t, theta_hat(2,:), 'r', 'LineWidth', 1.5); + ylabel('$$\hat{b}(t)$$ [Ns/m]', 'Interpreter', 'latex'); + grid on; + title('Εκτίμηση απόσβεσης'); + + subplot(3,1,3); + plot(t, theta_hat(3,:), 'k', 'LineWidth', 1.5); + ylabel('$$\hat{k}(t)$$ [N/m]', 'Interpreter', 'latex'); + xlabel('t [sec]'); + grid on; + title('Εκτίμηση ελαστικότητας'); + + % Save figure + if ~exist('output', 'dir') + mkdir('output'); + end + saveas(gcf, sprintf('output/Prob1a_estimation_%s_gamma%.3f_%s_%ds.png', input_case.name, gamma, normalization_text, T_total)); +end + +function out = ternary(cond, val_true, val_false) + if cond + out = val_true; + else + out = val_false; + end +end diff --git a/Lab02/scripts/Problem1b.m b/Lab02/scripts/Problem1b.m new file mode 100644 index 0000000..6156181 --- /dev/null +++ b/Lab02/scripts/Problem1b.m @@ -0,0 +1,116 @@ +% +% Problem 1b: Lyapunov-based Parameter Estimation +% + + +% True system parameters +m_true = 1.315; +b_true = 0.225; +k_true = 0.725; + +% Simulation parameters +Ts = 0.001; +T_total = 40; +t = 0:Ts:T_total; +N = length(t); + +% Gamma setup +gamma = 0.66; +fprintf('Using gamma = %.4f (Lyapunov Based Estimation)\n', gamma); + +% Define sine input only (as per problem statement) +u = 2.5 * sin(t); + +% Simulate the true system +x = zeros(1, N); +dx = zeros(1, N); +ddx = zeros(1, N); +x(1) = 0; dx(1) = 0; +for k = 1:N-1 + f = @(x_, dx_, u_) (1/m_true) * (u_ - b_true * dx_ - k_true * x_); + k1 = f(x(k), dx(k), u(k)); + k2 = f(x(k) + Ts/2 * dx(k), dx(k) + Ts/2 * k1, u(k)); + k3 = f(x(k) + Ts/2 * dx(k), dx(k) + Ts/2 * k2, u(k)); + k4 = f(x(k) + Ts * dx(k), dx(k) + Ts * k3, u(k)); + ddx(k) = k1; + dx(k+1) = dx(k) + Ts/6 * (k1 + 2*k2 + 2*k3 + k4); + x(k+1) = x(k) + Ts * dx(k); +end +ddx(1:end-1) = diff(dx) / Ts; +ddx(end) = ddx(end-1); + +% Estimation using Lyapunov structure +phi_all = [ddx; dx; x]; % shape: [3 x N] +theta_hat = zeros(3, N); +theta_hat(:, 1) = [1; 1; 1]; + +for k = 1:N-1 + phi = phi_all(:,k); + y = u(k); + y_hat = theta_hat(:,k)' * phi; + e = y - y_hat; + theta_hat(:,k+1) = theta_hat(:,k) + Ts * gamma * e * phi; +end + +% Final estimates +fprintf('\nFinal estimates:\n'); +fprintf('Estimated m: %.4f, b: %.4f, k: %.4f\n', theta_hat(1,end), theta_hat(2,end), theta_hat(3,end)); + +% Plot results +figure('Name', 'Lyapunov Estimation (notes form)', 'Position', [100, 100, 1280, 860]); +sgtitle(sprintf('Input: sine | Gamma = %.3f | Lyapunov', gamma), 'FontWeight', 'bold'); + +subplot(3,1,1); +plot(t, theta_hat(1,:), 'b', 'LineWidth', 1.5); +ylabel('$$\hat{m}(t)$$ [kg]', 'Interpreter', 'latex'); +grid on; title('Εκτίμηση μάζας'); + +subplot(3,1,2); +plot(t, theta_hat(2,:), 'r', 'LineWidth', 1.5); +ylabel('$$\hat{b}(t)$$ [Ns/m]', 'Interpreter', 'latex'); +grid on; title('Εκτίμηση απόσβεσης'); + +subplot(3,1,3); +plot(t, theta_hat(3,:), 'k', 'LineWidth', 1.5); +ylabel('$$\hat{k}(t)$$ [N/m]', 'Interpreter', 'latex'); +xlabel('t [sec]'); +grid on; title('Εκτίμηση ελαστικότητας'); + +if ~exist('output', 'dir') + mkdir('output'); +end +saveas(gcf, sprintf('output/Prob1b_lyapunov_gamma%.3f_%ds.png', gamma, T_total)); + +% Reconstruct estimated output x_hat(t) +x_hat = zeros(1, N); +dx_hat = zeros(1, N); +dx_hat(1) = 0; +for k = 1:N-1 + m_hat = theta_hat(1,k); +b_hat = theta_hat(2,k); +k_hat = theta_hat(3,k); +ddx_hat = (u(k) - b_hat * dx_hat(k) - k_hat * x_hat(k)) / m_hat; + dx_hat(k+1) = dx_hat(k) + Ts * ddx_hat; + x_hat(k+1) = x_hat(k) + Ts * dx_hat(k); +end +e_x = x - x_hat; + +% Plot extra figure with x, x_hat and e_x +figure('Name', 'System Response vs Estimation', 'Position', [100, 100, 1280, 860]); +sgtitle(sprintf('System Response and Error | Gamma = %.3f', gamma), 'FontWeight', 'bold'); + +subplot(2,1,1); +plot(t, x, 'b', t, x_hat, '--r', 'LineWidth', 1.5); +legend('x(t)', 'x_{hat}(t)', 'Location', 'Best'); +ylabel('Θέση [m]'); +grid on; title('Αντίδραση Συστήματος και Εκτίμηση'); + +subplot(2,1,2); +plot(t, e_x, 'k', 'LineWidth', 1.5); +ylabel('e_x(t)'); +grid on; title('Σφάλμα θέσης: x(t) - x_{hat}(t)'); + + + +saveas(gcf, sprintf('output/Prob1b_extrastates_gamma%.3f_%ds.png', gamma, T_total)); + diff --git a/Lab02/scripts/Problem1c.m b/Lab02/scripts/Problem1c.m new file mode 100644 index 0000000..b68bdf2 --- /dev/null +++ b/Lab02/scripts/Problem1c.m @@ -0,0 +1,102 @@ +% +% Problem 1c: Effect of bounded sinusoidal disturbance on measurement x(t) +% + +% True system parameters +m_true = 1.315; +b_true = 0.225; +k_true = 0.725; + +% Simulation parameters +Ts = 0.001; +T_total = 40; +t_full = 0:Ts:T_total; + +% Generate full input signal +u_full = 2.5 * sin(t_full); + +% Simulate the true system +x = zeros(1, length(t_full)); +dx = zeros(1, length(t_full)); +ddx = zeros(1, length(t_full)); +x(1) = 0; dx(1) = 0; +for k = 1:length(t_full)-1 + f = @(x_, dx_, u_) (1/m_true) * (u_ - b_true * dx_ - k_true * x_); + k1 = f(x(k), dx(k), u_full(k)); + k2 = f(x(k) + Ts/2 * dx(k), dx(k) + Ts/2 * k1, u_full(k)); + k3 = f(x(k) + Ts/2 * dx(k), dx(k) + Ts/2 * k2, u_full(k)); + k4 = f(x(k) + Ts * dx(k), dx(k) + Ts * k3, u_full(k)); + ddx(k) = k1; + dx(k+1) = dx(k) + Ts/6 * (k1 + 2*k2 + 2*k3 + k4); + x(k+1) = x(k) + Ts * dx(k); +end +ddx(1:end-1) = diff(dx) / Ts; +ddx(end) = ddx(end-1); + +% Initial estimation (clean) using Lyapunov +T_total = 40; +index_limit = round(T_total / Ts); +t = t_full(1:index_limit); +N = length(t); +u = u_full(1:index_limit); +x = x(1:index_limit); +dx = dx(1:index_limit); +ddx = ddx(1:index_limit); + +phi_all = [ddx; dx; x]; +theta_hat = zeros(3, N); +theta_hat(:, 1) = [1; 1; 1]; +gamma = 0.66; +for k = 1:N-1 + phi = phi_all(:,k); + y = u(k); + y_hat = theta_hat(:,k)' * phi; + e = y - y_hat; + theta_hat(:,k+1) = theta_hat(:,k) + Ts * gamma * e * phi; +end + +% Disturbance settings +eta0 = 0.1; +f0 = 0.5; +eta = eta0 * sin(2 * pi * f0 * t); +x_noisy = x + eta; + +% Use clean derivatives, noisy position +phi_all_noise = [ddx; dx; x_noisy]; +theta_hat_noise = zeros(3, N); +theta_hat_noise(:, 1) = [1; 1; 1]; + +for k = 1:N-1 + phi = phi_all_noise(:,k); + y = u(k); + y_hat = theta_hat_noise(:,k)' * phi; + e = y - y_hat; + theta_hat_noise(:,k+1) = theta_hat_noise(:,k) + Ts * gamma * e * phi; +end + +fprintf('\n1c: Final estimates with disturbance:\n'); +fprintf('Estimated m: %.4f, b: %.4f, k: %.4f\n', ... + theta_hat_noise(1,end), theta_hat_noise(2,end), theta_hat_noise(3,end)); + +figure('Name', '1c - Parameter Estimation with Disturbance', 'Position', [100, 100, 1280, 860]); +sgtitle(sprintf('Lyapunov Estimation with Disturbance | η_0 = %.2f', eta0)); + +subplot(3,1,1); +plot(t, theta_hat(1,:), 'b', t, theta_hat_noise(1,:), '--b', 'LineWidth', 1.2); +ylabel('m(t)'); grid on; title('Μάζα'); +legend('Clear', 'With noise'); + +subplot(3,1,2); +plot(t, theta_hat(2,:), 'r', t, theta_hat_noise(2,:), '--r', 'LineWidth', 1.2); +ylabel('b(t)'); grid on; title('Απόσβεση'); +legend('Clear', 'With noise'); + +subplot(3,1,3); +plot(t, theta_hat(3,:), 'k', t, theta_hat_noise(3,:), '--k', 'LineWidth', 1.2); +ylabel('k(t)'); xlabel('t [s]'); grid on; title('Ελαστικότητα'); +legend('Clear', 'With noise'); + +if ~exist('output', 'dir') + mkdir('output'); +end +saveas(gcf, sprintf('output/Prob1c_disturbance_eta%.2f.png', eta0)); diff --git a/Lab02/scripts/Problem2a.m b/Lab02/scripts/Problem2a.m new file mode 100644 index 0000000..61288ee --- /dev/null +++ b/Lab02/scripts/Problem2a.m @@ -0,0 +1,93 @@ +% +% Problem 2a: Nonlinear system roll model parameter estimation without disturbance +% + +% True system parameters +a1 = 2.0; +a2 = 1.0; +a3 = 0.5; +b = 2.0; + +% Simulation setup +Ts = 0.001; +T_total = 30; +t = 0:Ts:T_total; +N = length(t); + +% Reference trajectory: step profile +r_d = zeros(1, N); +r_d(t >= 10 & t < 20) = pi/10; + +% Smooth bound phi(t) +phi0 = 1.5; +phi_inf = 0.05; +lambda = 0.5; +phi = (phi0 - phi_inf) * exp(-lambda * t) + phi_inf; + +% Control parameters +k1 = 1.0; +k2 = 1.0; +rho = 1.0; + +% Initial conditions +r = zeros(1, N); +dr = zeros(1, N); +ddr = zeros(1, N); + +% Parameter estimation setup +theta_hat = zeros(4, N); +theta_hat(:,1) = [1; 1; 1; 1]; +gamma = 0.66; + +% Output storage for control input and errors +alpha = zeros(1, N); +u = zeros(1, N); + +for k = 1:N-1 + % Compute normalized errors + z1 = (r(k) - r_d(k)) / phi(k); + z1 = max(min(z1, 0.999), -0.999); + alpha(k) = -k1 * log((1 + z1) / (1 - z1)); + + z2 = (dr(k) - alpha(k)) / rho; + z2 = max(min(z2, 0.999), -0.999); + u(k) = -k2 * log((1 + z2) / (1 - z2)); + + % True system dynamics + phi_true = [-dr(k); -sin(r(k)); dr(k)^2 * sin(2*r(k)); u(k)]; + ddr(k) = a1 * phi_true(1) + a2 * phi_true(2) + a3 * phi_true(3) + b * phi_true(4); + + % Integrate dynamics + dr(k+1) = dr(k) + Ts * ddr(k); + r(k+1) = r(k) + Ts * dr(k); + + % Estimation + phi_est = phi_true; % same form + y = ddr(k); + y_hat = theta_hat(:,k)' * phi_est; + e = y - y_hat; + theta_hat(:,k+1) = theta_hat(:,k) + Ts * gamma * e * phi_est; +end + +% Final estimates +fprintf('\n2a: Final estimated parameters:\n'); +fprintf('a1: %.4f, a2: %.4f, a3: %.4f, b: %.4f\n', theta_hat(1,end), theta_hat(2,end), theta_hat(3,end), theta_hat(4,end)); + +% Plot parameter estimates +figure('Name', 'Problem 2a - Parameter Estimation', 'Position', [100, 100, 1280, 860]); +sgtitle('Nonlinear Roll System - Parameter Estimation'); + +subplot(2,1,1); +plot(t, theta_hat', 'LineWidth', 1.4); +legend('a_1', 'a_2', 'a_3', 'b'); +ylabel('\theta estimates'); grid on; title('Εκτιμήσεις παραμέτρων'); + +subplot(2,1,2); +plot(t, r, 'b', t, r_d, '--r', 'LineWidth', 1.4); +legend('r(t)', 'r_d(t)'); +ylabel('Roll angle [rad]'); xlabel('Time [s]'); grid on; title('Παρακολούθηση τροχιάς'); + +if ~exist('output', 'dir') + mkdir('output'); +end +saveas(gcf, 'output/Problem2a_estimation.png'); diff --git a/Lab02/scripts/Problem2b.m b/Lab02/scripts/Problem2b.m new file mode 100644 index 0000000..4c60304 --- /dev/null +++ b/Lab02/scripts/Problem2b.m @@ -0,0 +1,103 @@ +% +% Problem 2b: Estimation of unknown parameters using Lyapunov (r, dr, u measurable) +% + +% True system parameters +a1 = 2.0; +a2 = 1.0; +a3 = 0.5; +b = 2.0; + +% Simulation setup +Ts = 0.001; +T_total = 30; +t = 0:Ts:T_total; +N = length(t); + +% Reference trajectory +r_d = zeros(1, N); +r_d(t >= 10 & t < 20) = pi/10; + +% Smooth bound phi(t) +phi0 = 1.5; +phi_inf = 0.05; +lambda = 0.5; +phi = (phi0 - phi_inf) * exp(-lambda * t) + phi_inf; + +% Control parameters +k1 = 1.0; +k2 = 1.0; +rho = 1.0; + +% Initial conditions +r = zeros(1, N); +dr = zeros(1, N); +ddr = zeros(1, N); + +% Estimated trajectory reconstruction +r_hat = zeros(1, N); +dr_hat = zeros(1, N); +dr_hat(1) = 0; r_hat(1) = 0; + +% Parameter estimation setup +theta_hat = zeros(4, N); +theta_hat(:,1) = [1; 1; 1; 1]; +gamma = 0.66; + +alpha = zeros(1, N); +u = zeros(1, N); + +for k = 1:N-1 + % Control law + z1 = (r(k) - r_d(k)) / phi(k); + z1 = max(min(z1, 0.999), -0.999); + alpha(k) = -k1 * log((1 + z1) / (1 - z1)); + + z2 = (dr(k) - alpha(k)) / rho; + z2 = max(min(z2, 0.999), -0.999); + u(k) = -k2 * log((1 + z2) / (1 - z2)); + + % System dynamics + phi_vec = [-dr(k); -sin(r(k)); dr(k)^2 * sin(2*r(k)); u(k)]; + ddr(k) = a1 * phi_vec(1) + a2 * phi_vec(2) + a3 * phi_vec(3) + b * phi_vec(4); + dr(k+1) = dr(k) + Ts * ddr(k); + r(k+1) = r(k) + Ts * dr(k); + + % Parameter estimation + y = ddr(k); + y_hat = theta_hat(:,k)' * phi_vec; + e = y - y_hat; + theta_hat(:,k+1) = theta_hat(:,k) + Ts * gamma * e * phi_vec; + + % Reconstruct r_hat from estimated theta + phi_hat = [-dr_hat(k); -sin(r_hat(k)); dr_hat(k)^2 * sin(2 * r_hat(k)); u(k)]; + dd_r_hat = theta_hat(:,k)' * phi_hat; + dr_hat(k+1) = dr_hat(k) + Ts * dd_r_hat; + r_hat(k+1) = r_hat(k) + Ts * dr_hat(k); +end + +fprintf('\n2b: Final estimated parameters:\n'); +fprintf('a1: %.4f, a2: %.4f, a3: %.4f, b: %.4f\n', theta_hat(1,end), theta_hat(2,end), theta_hat(3,end), theta_hat(4,end)); + +% Plot +figure('Name', 'Problem 2b - Estimation with State Comparison', 'Position', [100, 100, 1280, 860]); +sgtitle('Problem 2b - Parameter Estimation and State Reconstruction'); + +subplot(3,1,1); +plot(t, theta_hat', 'LineWidth', 1.4); +legend('a_1', 'a_2', 'a_3', 'b'); +ylabel('\theta estimates'); grid on; title('Εκτιμήσεις παραμέτρων'); + +subplot(3,1,2); +plot(t, r, 'b', t, r_hat, '--r', 'LineWidth', 1.4); +legend('r(t)', 'r_{hat}(t)'); +ylabel('Roll angle [rad]'); title('Πραγματικό vs εκτιμώμενο r(t)'); grid on; + +subplot(3,1,3); +plot(t, r - r_hat, 'k', 'LineWidth', 1.4); +ylabel('e_r(t)'); xlabel('Time [s]'); title('Σφάλμα θέσης: r(t) - r̂(t)'); grid on; + +if ~exist('output', 'dir') + mkdir('output'); +end +saveas(gcf, 'output/Problem2b_estimation.png'); diff --git a/Lab02/scripts/Problem2c.m b/Lab02/scripts/Problem2c.m new file mode 100644 index 0000000..e69de29 diff --git a/Lab02/scripts/output/Prob1a_estimation_constant_gamma0.330_Normalized_30s.png 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