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(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 1} +\DocSubTitle{Εκτίμηση Άγνωστων Παραμέτρων - Μέθοδος Ελαχίστων Τετραγώνων} + +\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{float} + +\begin{document} + +% Request a title page or header +\InsertTitle + +\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 +Αρχικά, το σύστημα αναδιατυπώνεται σε μορφή κατάστασης ώστε να καταστεί κατάλληλο για ανάλυση και προσομοίωση. +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} +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}{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} + +Το σύστημα που μελετάται περιγράφεται από τη γραμμικοποιημένη διαφορική εξίσωση: +\[ +mL^2 \ddot{q}(t) + c \dot{q}(t) + mgL q(t) = u(t) +\] +όπου $q(t)$ είναι η γωνία του εκκρεμούς, $u(t)$ η ροπή εισόδου, και $m$, $L$, $c$, $g$ φυσικές σταθερές του συστήματος. + +Ορίζοντας ως διάνυσμα κατάστασης: +\[ +x(t) = \begin{bmatrix} q(t) \\ \dot{q}(t) \end{bmatrix} +\] +οι εξισώσεις κατάστασης γράφονται ως: +\[ +\dot{x}(t) = A x(t) + B u(t) +\] +όπου: +\[ +A = +\begin{bmatrix} + 0 & 1 \\ + -\frac{g}{L} & -\frac{c}{mL^2} +\end{bmatrix}, +\quad +B = \begin{bmatrix} 0 \\ \frac{1}{mL^2} \end{bmatrix} +\] + +Η συνάρτηση μεταφοράς του συστήματος προκύπτει από την αρχική εξίσωση ως: +\[ +G(s) = \frac{Q(s)}{U(s)} = \frac{1}{mL^2 s^2 + c s + mgL}. +\] + +Για την προσομοίωση χρησιμοποιήθηκε ημιτονική είσοδος $u(t) = A_0 \sin(\omega t)$, με $A_0 = 4$ και $\omega = 2$. +Οι υπόλοιπες παράμετροι ήταν $m = 0.75$, $L = 1.25$, $c = 0.15$, $g = 9.81$. +Ο αρχικός χρόνος προσομοίωσης ορίστηκε σε $20$ δευτερόλεπτα, όπως ζητείται στην εκφώνηση. + +\subsection*{Παρατήρηση Συμπεριφοράς Συστήματος} + +Το σύστημα παρουσίασε περιοδική απόκριση, η οποία όμως δεν σταθεροποιήθηκε 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. + +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. + +\InsertFigure{!ht}{1}{fig:20s}{../scripts/Prob1_responce_20s.png}{ + Απόκριση του συστήματος για $t \in [0, 20]$ sec. Η μεταβατική φάση παραμένει ενεργή. +} +\InsertFigure{!ht}{1}{fig:90s}{../scripts/Prob1_responce_90s.png}{ + Απόκριση του συστήματος για $t \in [0, 90]$ sec. Το σύστημα σταθεροποιείται σε περιοδική συμπεριφορά μετά τα $50$ sec. +} + + +\end{document} diff --git a/Work 1/scripts/Prob1_responce_20s.png b/Work 1/scripts/Prob1_responce_20s.png new file mode 100644 index 0000000000000000000000000000000000000000..be998bff463977f249c2bdd813ba52ec0e24632c GIT binary patch literal 80028 zcmeFZg;!Nw)INF;6$A;Dkd{WeTR;#HNhJj7M!GwtLpiIu=idy*PPFM=Ce+KoQyac3O))921Ap0{puYIhSUp#-4jB-1HMU0 zRd@y79$3Csv4O#GA4C5T43ql7RWKx5329NJHAIpp*km%hhP2=-3R_WSTX_ouTL*1x zeVFu11*w<9^v@KG^q)y<+Zw&Mur{%we=t*>qS(Zc+h!bdZG z7#jEy{Ggek!Ru*11cEc6cHW=&~Oybo`1;_Z!DUa`Q`Z1?BagpPOMhpjc zX`dimG<7!EW2H6v?Ejk88|9VnC|q~Y$2Z+ZPpi7ZtKDd(F7_K>u%F%CY)NNdJBCB=~0e--kLF#U#SNKQO}{IQ)0bL!kzu z|Nf8oznA~-%>1tr{6DsbWyR3#wy`V`yjqi;IsL=6xL}f7ZZO8$?hN+L?AwRF(BJ9X z7)mGmXP}qI`rz}zE%4oL;edwgo@ll%w7>E{Z{${Pj~3_-)hZ~N`Za;t?u*Vpd1ZF@eOLkHh;cVA4O)C3W5eIWA16n0%^ODf>K z3%;)t``1j7)4}paUafmd?F+m0K`;i^7cUrm97igri|sasRWx_EEIck!{D~h6=X@w$ zFG{+~aorcZIS};s_g{kMA-%M;bpI&nd@uu@ko%d>=|=7K27_veLZZih8#Odz9W5Ip 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