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+++ b/mt2.tex @@ -1,7 +1,7 @@ \documentclass{beamer} \usepackage{xparse} -\usepackage{xfrac} +%\usepackage{xfrac} \usepackage[siunitx, american]{circuitikz} \usepackage{hyperref} \usepackage{cancel} @@ -662,7 +662,9 @@ \section{Linearization} \begin{frame}{Break} - \Large{GIVE US FEEDBACK!\\\vspace{2em}\\\url{hkn.mu/feedback}\\\url{https://github.com/hkntutoring/ee16b-review/issues} } + GIVE US FEEDBACK \\ + \url{hkn.mu/feedback} \\ + \url{github.com/hkntutoring/ee16b-mt2-review/issues} \end{frame} \section[SVD]{Singular Value Decomposition} From dcce55cdac80bcb855b0f508caa78eb527d53068 Mon Sep 17 00:00:00 2001 From: geohh Date: Sat, 9 Nov 2019 19:53:00 -0800 Subject: [PATCH 2/6] Changes from sprint --- images/PCAexample.png | Bin 0 -> 242015 bytes mt2.pdf | Bin 310800 -> 468511 bytes mt2.tex | 1235 +++++++++++++++++++++++------------------ 3 files changed, 692 insertions(+), 543 deletions(-) create mode 100644 images/PCAexample.png diff --git a/images/PCAexample.png 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\frac{#3^{#2} #4}{#3 #5^{#2}} - } - } + \IfBooleanTF {#1}{% + \IfNoValueTF{#2}{% + \sfrac{#5 #4}{#5 #3} + }{% + \sfrac{#3^{#2} #4}{#3 #5^{#2}} + } + }{% + \IfNoValueTF{#2}{% + \frac{#3 #4}{#3 #5} + }{% + \frac{#3^{#2} #4}{#3 #5^{#2}} + } + } } \usetheme{hutch-pdflatex} \title{EE16B --- Midterm 2 Review} -\author{George Higgins Hutchinson, Parth Nobel, Patrick Wang, Matteo Ciccozzi, Jaymo Kang} -\institute{Presented by: Vishnu Iyer, George Higgins Hutchinson, Rehan Durrani} -\date{\today} +\author{Presented by: George Higgins Hutchinson, Naomi Sagan, Maran Palaniappan} +\institute{Authors: George Higgins Hutchinson, Parth Nobel, Patrick Wang, Matteo Ciccozzi, Jaymo Kang, Brianna Zhang, Rehan Durrani, Sarah Sun} +\date{November 10, 2019} \begin{document} \begin{frame} - \titlepage - \end{frame} + \titlepage +\end{frame} + +\begin{frame}{Disclaimer} +This is an unofficial review session and HKN is not affiliated with this course. All of the topics we're reviewing will reflect the material you have covered, our experiences in EE16B, and past exams. We make no promise that what we cover will necessarily reflect the content of this midterm. While some course staff members may be among the presenters, this review session is still not official. +\vspace{1em} - \begin{frame}{Disclaimer} - This is an unofficial review session and HKN is not affiliated with this course. All of the topics we're reviewing will reflect the material you have covered, our experiences in EE16B, and past exams. We make no promise that what we cover will necessarily reflect the content of this midterm. While some course staff members may be among the presenters, this review session is still not official. - \vspace{1em} - - This is licensed under the Creative Commons CC BY-SA: feel free to share and edit, as long as you credit us and keep the license. For more information, visit \\ \small{\url{https://creativecommons.org/licenses/by-sa/4.0/deed.en_US}} - - \end{frame} +This is licensed under the Creative Commons CC BY-SA: feel free to share and edit, as long as you credit us and keep the license. For more information, visit \\ \small{\url{https://creativecommons.org/licenses/by-sa/4.0/deed.en_US}} + +\end{frame} \section[State-Space]{State-Space Representations} @@ -91,20 +91,20 @@ \begin{frame} \frametitle{State Space Modeling:} \begin{tikzpicture} - \node at (-2, 2) {original space:}; - \node at (-2, 0) {new space:}; - - \node[draw,circle] (x) at (0, 2) {\(\vec x(t)\)}; - \node[draw,circle] (dx) at (3, 2) {\(\ddt{}\vec x(t)\)}; - \node[draw,circle] (xt) at (0, 0) {\(\tilde{\vec{x}}(t)\)}; - \node[draw,circle] (dxt) at (3, 0) {\(\ddt{}\tilde{\vec{x}}(t)\)}; - - \path[->] (x) edge node [above] {\(A\)} (dx); - \path[->] (xt) edge node [above] {\(\tilde A\)} (dxt); - \path[->] (x) edge [bend right] node [left] {\(T^{-1}\)} (xt); - \path[->] (xt) edge [bend right] node [right] {\(T\)} (x); - \path[->] (dx) edge [bend right] node [left] {\(T^{-1}\)} (dxt); - \path[->] (dxt) edge [bend right] node [right] {\(T\)} (dx); +\node at (-2, 2) {original space:}; +\node at (-2, 0) {new space:}; + +\node[draw,circle] (x) at (0, 2) {\(\vec x(t)\)}; +\node[draw,circle] (dx) at (3, 2) {\(\ddt{}\vec x(t)\)}; +\node[draw,circle] (xt) at (0, 0) {\(\tilde{\vec{x}}(t)\)}; +\node[draw,circle] (dxt) at (3, 0) {\(\ddt{}\tilde{\vec{x}}(t)\)}; + +\path[->] (x) edge node [above] {\(A\)} (dx); +\path[->] (xt) edge node [above] {\(\tilde A\)} (dxt); +\path[->] (x) edge [bend right] node [left] {\(T^{-1}\)} (xt); +\path[->] (xt) edge [bend right] node [right] {\(T\)} (x); +\path[->] (dx) edge [bend right] node [left] {\(T^{-1}\)} (dxt); +\path[->] (dxt) edge [bend right] node [right] {\(T\)} (dx); \end{tikzpicture} Always apply operations to right side first @@ -115,37 +115,38 @@ \begin{frame} \frametitle{State Space Modeling Procedure:} \begin{enumerate} - \item Set up differential equation of the form: $\ddt{}\vec{x}(t) = A\vec{x}(t) + \vec{b}u(t)$ \pause -\item Find $\lambda_i$ of \(A\); let $\tilde{A} = \begin{bmatrix} -\lambda_{1} & 0 & \cdots & 0 \\ -0 & \lambda_{2} & 0 & \vdots \\ -\vdots & 0 & \ddots & 0 \\ -0 & \cdots & 0 & \lambda_{n} \\ -\end{bmatrix}$ \pause +\item Set up differential equation of the form: $\ddt{}\vec{x}(t) = A\vec{x}(t) + \vec{b}u(t)$ \pause \item Find eigenvectors $\vec{v}_i$ of \(A\); let $T = \begin{bmatrix} \vec{v_{1}} & \vec{v_{2}} & \cdots & \vec{v_{n}} \end{bmatrix}$ \pause \item Convert $\vec{x}(t)$ to $\tilde{\vec{x}}(t)$ using: $\tilde{\vec{x}}(t) = T^{-1}\vec{x}(t)$ \pause +\item Find $\lambda_i$ of \(A\); let $\tilde{A} = \begin{bmatrix} +\lambda_{1} & 0 & \cdots & 0 \\ +0 & \lambda_{2} & 0 & \vdots \\ +\vdots & 0 & \ddots & 0 \\ +0 & \cdots & 0 & \lambda_{n} \\ +\end{bmatrix}$ \pause \item Solve $\frac{d}{dt}\tilde{\vec{x}}(t) = \tilde{A}\tilde{\vec{x}}(t) + T \vec{b} u(t)$ \pause \item Convert solution back to $\vec{x}(t)$ \end{enumerate} \end{frame} -\begin{frame} -\frametitle{State Space Modeling:} +%\begin{frame} +%\frametitle{State Space Modeling:} -Continuous time solution: -\[ - \tilde{x}(t) = e^{\lambda t} \tilde x(0) + \frac{e^{\lambda t} - 1}{\lambda } u(t) + w(t) -\] -Discrete time solution: -\[ - x_{d}(i+1) = e^{\lambda \Delta} x_{d}(i) + \frac{e^{\lambda \Delta} - 1}{\lambda } u(i) + w(i) -\] -\end{frame} +%Continuous time solution: +%\[ +% \tilde{x}(t) = e^{\lambda t} \tilde x(0) + \frac{e^{\lambda t} - 1}{\lambda } u(t) + w(t) +%\] + +%Discrete time solution: +%\[ +% x_{d}(i+1) = e^{\lambda \Delta} x_{d}(i) + \frac{e^{\lambda \Delta} - 1}{\lambda } u(i) + w(i) +%\] +%\end{frame} \begin{frame} \frametitle{State Space Modeling Example:} @@ -153,22 +154,22 @@ Given the following circuit: \begin{tikzpicture}[scale=2] - \draw (2, 0.75) to [R, *-, l_={\(R_2\)}] (1, 0.75) to [R, *-, l_={\(R_1\)}] (0, 0.75) to [voltage source, v_>={\(V_{in}\)}] (0, 0); - \draw (1, 0.75) node[label={\(V_1\)}] {} to [C, l={\(C_1\)}] (1, 0) node[ground] {}; - \draw (2, 0.75) node[label={\(V_2\)}] {} to [C, l={\(C_2\)}] (2, 0) node[ground] {}; - %\draw (0, 0) to (1.5, 0) to [C, l=2<\micro\farad>] (1.5, 1) to (0, 1) to [L, l=2<\henry>] (0, 0); - %\draw (0, 0) to (0.75, 0) to (0.75, 0.0) node[ground] {} to (0.75, 0) to (1.5, 0) to [C, l=2<\micro\farad>, v_<=\(v\)] (1.5, 1) to (0, 1) to[I=\(i\)] [L, l=2<\henry>] (0, 0); - \node [ground] (0, 0) {}; +\draw (2, 0.75) to [R, *-, l_={\(R_2\)}] (1, 0.75) to [R, *-, l_={\(R_1\)}] (0, 0.75) to [voltage source, v_>={\(V_{in}\)}] (0, 0); +\draw (1, 0.75) node[label={\(V_1\)}] {} to [C, l={\(C_1\)}] (1, 0) node[ground] {}; +\draw (2, 0.75) node[label={\(V_2\)}] {} to [C, l={\(C_2\)}] (2, 0) node[ground] {}; +%\draw (0, 0) to (1.5, 0) to [C, l=2<\micro\farad>] (1.5, 1) to (0, 1) to [L, l=2<\henry>] (0, 0); +%\draw (0, 0) to (0.75, 0) to (0.75, 0.0) node[ground] {} to (0.75, 0) to (1.5, 0) to [C, l=2<\micro\farad>, v_<=\(v\)] (1.5, 1) to (0, 1) to[I=\(i\)] [L, l=2<\henry>] (0, 0); +\node [ground] (0, 0) {}; \end{tikzpicture} in which $R_{1} = \SI{2}{\ohm}, R_{2} = \frac{8}{3} \si{\ohm}, C_{1} = \SI{1}{\coulomb}, C_{2} = \frac{3}{2} \si{\coulomb} $ \\ solve equations for $V_{1}$ and $V_{2}$ \end{frame} -\section[S,O,\&C]{Stability, Observability, and Controlability} +\section[S\&C]{Stability and Controllability} \begin{frame} -\frametitle{Stability, Observability, Controllability:} +\frametitle{Stability and Controllability:} given: \\ $\vec{x}(i+1) = A\vec{x}(i) + Bu(i)$ \\ @@ -184,7 +185,11 @@ Discrete time model: \\ if $|\lambda_{i}| < 1$ for all $\lambda_{i}$ of A, system is stable\\ -intuition: if any $|\lambda_{i}| >= 1$, state vector is increasing each time step will be infinitely magnified over time \\ +Intuition: +\begin{enumerate} +\item If any $|\lambda_{i}| > 1$, state vector is increasing each time step so will be infinitely magnified over time \\ +\item If any $|\lambda_{i}| = 1$, state vector is constant each time step so if we were to integrate this, it would eventually blow up to infinity \\ +\end{enumerate} \end{frame} \begin{frame} @@ -192,7 +197,10 @@ Continuous time model: \\ if the real parts of all eigenvalues of A are strictly negative, system is stable\\ -intuition: if real part of eigenvalue is positive, state vector is increasing over time and will be infinitely magnified over time \\ +Intuition: +\begin{enumerate} +\item If real part of eigenvalue is positive, state vector is increasing over time and will be infinitely magnified over time \\ +\end{enumerate} \end{frame} \begin{frame} @@ -202,35 +210,99 @@ $\begin{bmatrix} B && AB && ... && A^{n-1}B \end{bmatrix}$ -spans $R^{n}$, then system is controllable +spans $R^{n}$, then system is controllable \\ +Intuition: If you can go to any location in an $n$-dimensional space in infinite steps, that means you can go to any location in that space in $n$ steps as well, implying that the matrix is full rank. \end{frame} \begin{frame} \frametitle{Feedback:} -if system is controllable, we can set: +\begin{enumerate} +\item If system is controllable, we can set: $u(t) = K\vec{x}(t)$ \\ -plugging in, we get: +Plugging in, we get: $\vec{x}(t+1) = (A + BK)\vec{x}(t)$ \\ -we can find the eigenvalues of (A + BK) to check for stability +We can find the eigenvalues of (A + BK) to check for stability +\item If this system has noise, we can modify the equation to be: +$\vec{x}(t+1) = (A + BK)\vec{x}(t) + \vec{\omega}(t)$ \\ +Where $\vec{\omega}$ is noise.\\ +\end{enumerate} + + \end{frame} +\begin{frame}{Feedback Flow Diagram:} +\begin{tikzpicture}[scale=1.5] +\node[isosceles triangle, isosceles triangle apex angle=44,draw, +inner sep=0pt,anchor=lower side,rotate=0,draw=black, +line width=1.0pt, minimum height=1.2cm, fill=gray!2] (triangle) at (1.2,-0.05) {A}; +\draw[line width=1.0pt, fill=gray!2] (0, 0)node [below] {x(t)} -- (1.2, 0) ; +\draw[line width=1.0pt, fill=gray!2] (2.0, -0.05) -- (3.4, -0.05); +\draw[line width=1.0pt, fill=gray!2] (3.7,-0.05) circle (.3cm) node[center]{\LARGE +}; -\begin{frame} -\frametitle{Observability:} +\node[isosceles triangle, isosceles triangle apex angle=44,draw, +inner sep=0pt,anchor=lower side,rotate=0,draw=black, +line width=1.0pt, minimum height=1.2cm, fill=gray!2] (triangle) at (1.2,-1.55) {B}; +\draw[line width=1.0pt, fill=gray!2] (0, -1.5)node [below] {u(t)} -- (1.2, -1.5) ; +\draw[line width=1.0pt, fill=gray!2] (2.0, -1.55) -- (3.7, -1.55); +\draw[line width=1.0pt, fill=gray!2] (3.7, -1.55) -- (3.7, -.35); +\draw[line width=1.0pt] (4.0, -0.05) -- (4.8, -0.05) node[below] {x(t+1)} -- (4.8, 0.75) -- (0, .75) -- (0, 0); +\end{tikzpicture} +\end{frame} +\begin{frame}{Feedback Flow Diagram:} +\begin{tikzpicture}[scale=1.5] +\node[isosceles triangle, isosceles triangle apex angle=44,draw, +inner sep=0pt,anchor=lower side,rotate=0,draw=black, +line width=1.0pt, minimum height=1.2cm, fill=gray!2] (triangle) at (1.2,-0.05) {A}; +\draw[line width=1.0pt, fill=gray!2] (0, 0)node [left] {x(t)} -- (1.2, 0) ; +\draw[line width=1.0pt, fill=gray!2] (2.0, -0.05) -- (3.4, -0.05); +\draw[line width=1.0pt, fill=gray!2] (3.7,-0.05) circle (.3cm) node[center]{\LARGE +}; + +\node[isosceles triangle, isosceles triangle apex angle=44,draw, +inner sep=0pt,anchor=lower side,rotate=0,draw=black, +line width=1.0pt, minimum height=1.2cm, fill=gray!2] (triangle) at (1.2,-1.55) {B}; +\draw[line width=1.0pt, fill=gray!2] (0, -1.5)node [below] {u(t)} -- (1.2, -1.5) ; +\draw[line width=1.0pt, fill=gray!2] (2.0, -1.55) -- (3.7, -1.55); +\draw[line width=1.0pt, fill=gray!2] (3.7, -1.55) -- (3.7, -.35); +\draw[line width=1.0pt] (4.0, -0.05) -- (4.8, -0.05) node[below] {x(t+1)} -- (4.8, 0.75) -- (0, .75) -- (0, 0); +\node[isosceles triangle, isosceles triangle apex angle=44,draw, +inner sep=0pt,anchor=lower side,rotate=-90,draw=black, +line width=1.0pt, minimum height=1.2cm, fill=gray!2] (triangle) at (0,-.45) {K}; +\draw[line width=1.0pt] (0, 0) -- (0, -0.45); +\draw[line width=1.0pt] (0, -1.25) -- (0, -1.5); +\end{tikzpicture} +\end{frame} +\begin{frame}{Feedback Flow Diagram:} +\begin{tikzpicture}[scale=2] +\node[isosceles triangle, isosceles triangle apex angle=44,draw, +inner sep=0pt,anchor=lower side,rotate=0,draw=black, +line width=1.0pt, minimum height=2cm, fill=gray!2] (triangle) at (1.2,-0.05) {A+BK}; +\draw[line width=1.0pt, fill=gray!2] (0, 0)node [below] {x(t)} -- (1.2, 0) ; +\draw[line width=1.0pt] (2.2, -0.05) -- (3.4, -0.05) node[below] {x(t+1)} -- (3.4, 0.95) -- (0, 0.95) -- (0, 0); -if -$\begin{bmatrix} -C \\ -CA \\ -\vdots \\ -CA^{n-1} \\ -\end{bmatrix}$ -spans $R^{n}$, system is observable \\ -intuition: if observability matrix is full rank, it is invertible, and we can retrieve all the past states without loss of information \\ +\end{tikzpicture} \end{frame} +% \begin{frame}{Feedback:} +% \begin{tikzpicture}[right iso/.style={isosceles triangle,scale=0.5,sharp corners, anchor=center, xshift=-4mm}] +% draw |[right iso]| ; +% \end{tikzpicture} +% \end{frame} +% \begin{frame} +% \frametitle{Observability:} + +% if +% $\begin{bmatrix} +% C \\ +% CA \\ +% \vdots \\ +% CA^{n-1} \\ +% \end{bmatrix}$ +% spans $R^{n}$, system is observable \\ +% intuition: if observability matrix is full rank, it is invertible, and we can retrieve all the past states without loss of information \\ +% \end{frame} \begin{frame} -\frametitle{Stability, Controllability, Observability Example:} +\frametitle{Stability and Controllability Example:} +%\frametitle{Stability, Controllability, Observability Example:} given the following system: \\ $\vec{x}[t+1] = @@ -244,17 +316,17 @@ -1 \end{bmatrix} u[t]$\\ -$ -\vec{y}[t] = -\begin{bmatrix} -1 & 1 -\end{bmatrix} -\vec{x}[t]$ +%$ +%\vec{y}[t] = +%\begin{bmatrix} +%1 & 1 +%\end{bmatrix} +%\vec{x}[t]$ \end{frame} \begin{frame} \frametitle{Stability Check:} - +Remember: $\vec{x}[t+1] = A\vec{x}[t] + \vec{\omega}[t]= U \Lambda U^T \vec{x}[t] + \vec{\omega}[t]$ \\ \\ $\lambda = 6, -5$\\ \pause System is unstable \end{frame} @@ -270,29 +342,31 @@ 8 & -1 \end{bmatrix}$ which spans $R^{n}$ \\ \pause -System is controllable +System is controllable\\ +* Use results from previous slides to figure this out. + \end{frame} -\begin{frame} -\frametitle{Observability Check:} +% \begin{frame} +% \frametitle{Observability Check:} -$\begin{bmatrix} -C \\ -CA -\end{bmatrix}$ = -$\begin{bmatrix} -1 & 1 \\ -2 & 6 -\end{bmatrix}$ -which spans $R^{n}$ \\ \pause -System is observable -\end{frame} +% $\begin{bmatrix} +% C \\ +% CA +% \end{bmatrix}$ = +% $\begin{bmatrix} +% 1 & 1 \\ +% 2 & 6 +% \end{bmatrix}$ +% which spans $R^{n}$ \\ \pause +% System is observable +% \end{frame} - \section{Eigenvalue Placement} +\section{Eigenvalue Placement} - \begin{frame} - \frametitle{Eigenvalue Placement} - \centering\includegraphics[width = 10cm]{eigenvalue.jpg} +\begin{frame} +\frametitle{Eigenvalue Placement} +\centering\includegraphics[width = 10cm]{eigenvalue.jpg} \end{frame} \begin{frame} @@ -345,496 +419,571 @@ \end{itemize} \end{frame} -\begin{frame}{Controllable Canonical Form} -\begin{itemize} -\item Controllable Canonical Form (CCF) for any controllable system is a special form that allows us to simplify the process of eigenvalue placement. It takes on the following form: -\begin{columns} +%\begin{frame}{Controllable Canonical Form} +%\begin{itemize} +%\item Controllable Canonical Form (CCF) for any controllable system is a special form that allows us to simplify the process of eigenvalue placement. It takes on the following form: +%\begin{columns} + +%\begin{column}{0.25\textwidth} +%\begin{align*} +%A^* = +%\begin{bmatrix} +%0&1&0 &\hdots &0\\ +%0& 0 & 1& \hdots&0\\ +%\vdots& \hdots & \hdots& \hdots&0\\ +%0 & \hdots &\hdots & \hdots &1\\ +%\alpha_{0} & \alpha_{1}& \hdots & \hdots & \alpha_{n-1} +%\end{bmatrix} +%\end{align*} +%\end{column} +%\begin{column}{0.25\textwidth} +%\begin{align*} +%B^* = +%\begin{bmatrix} +%0\\ +%0\\ +%\vdots\\ +%0\\ +%1 +%\end{bmatrix} +%\end{align*} +%\end{column} +%\end{columns}\pause +%\item The characteristic polynomial of $A^*$ is $\lambda_n - \sum\limits_{i = 0}^{n-1} \alpha_i\lambda^i$ = 0. +%\item So how does it help with eigenvalue placement? \pause The last row of this matrix determines the eigenvalues of $A^*$ so modifying the last row will allow us to (easily) modify the eigenvalues. +%\end{itemize} +%\end{frame} + +%\begin{frame}{How to convert to CCF} +%\begin{itemize} +%\item Let $A,B$ be the matrices in standard form and let $A^*,B^*$ be the matrices in CCF. \pause +%\item Recall the matrix we used to check controllability?\\ %\pause +%\[C = \begin{bmatrix} +%B & AB & \hdots & A^{n-1}B +%\end{bmatrix}\] \\ +%\[ C^* = \begin{bmatrix} +%B^*& A^*B^* & \hdots & A^{*n-1}B^* +%\end{bmatrix}\] +%\item We then have $T := C^*C^{-1}$, such that $A^* = TAT^{-1}$ and $B^* = TB$. +% \item Remember, all controllable matrices with single input can be transformed into CCF! +% \end{itemize} +% \end{frame} + +% \begin{frame}{Example} +% Consider the following discrete time system: +% \begin{align*} +% \vec{x}[t+1] = \begin{bmatrix} +% -2&2&0\\ +% 2&-4&2\\ +% 0&1&-1 +% \end{bmatrix}\vec{x}[t] + \begin{bmatrix} 1 \\ 0 \\0 \end{bmatrix}u[t] +% \end{align*} + +% \begin{enumerate}[(a)] +% \item Is the system stable? Is it controllable? +% \item Using an appropriate transformation ($\vec{z}[t] = T\vec{x}[t]$), bring the system to controllable canonical form. +% \item Using the state feedback $u[t] =$ \[\begin{bmatrix} f_1 & f_2 & f_3 \end{bmatrix}\] $\vec{z}[t]$ bring the eigenvalues of the system to $0,0.75, -0.25$. +% \end{enumerate} +% \end{frame} + +% \begin{frame}{Solutions to Example} +% \begin{enumerate}[(a)] +% \item The characteristic polynomial is: $\lambda^3 + 7\lambda^2 + 8\lambda = \lambda(\lambda^2 +7\lambda +8) = 0$, therefore the eigenvalues of A are $\{0,-5.56, -1.44\}$. As we can see there are $|\lambda_i|>1$ therefore the system is not stable. \\ +% The controllability matrix $C =$ \[\begin{bmatrix} +% 1&-2&8\\ +% 0&2&-12\\ +% 0&0&2 +% \end{bmatrix}\] +% $C$ has full rank so the system is controllable + +% \item As we previously mentioned the coefficients of the characteristic polynomial are closely related to the last row of the $A^*$ matrix. Therefore, the CCF of the system is: +% \begin{align*} +% \vec{z}[t+1] = +% \begin{bmatrix} +% 0&1&0\\ +% 0&0&1\\ +% 0&-8&-7 +% \end{bmatrix}\vec{x}[t] + \begin{bmatrix} 0 \\ 0 \\1 \end{bmatrix}u[t] +% \end{align*} +% \end{enumerate} +% \end{frame} + +% \begin{frame}{Example Solutions Continued} +% \begin{enumerate}[(c)] +% \item Our system then becomes: +% \begin{align*} +% \vec{z}[t+1] = +% \begin{bmatrix} +% 0&1&0\\ +% 0&0&1\\ +% f_1&f_2-8&f_3-7 +% \end{bmatrix}\vec{x}[t] +% \end{align*} +% Which means its characteristic polynomial is : $\lambda^3 - (f_3 -7)\lambda^2 -(f_2 - 8)\lambda - f_1$ = 0. \\ +% Now, we know the characteristic polynomial should be $\lambda(\lambda - \frac{3}{4})(\lambda + \frac{1}{4})$, so we can equate the two and solve for the feedback vector $\vec{f}\hspace{0.1cm}^T = \begin{bmatrix} 0 & \frac{1}{2}& \frac{3}{16}\end{bmatrix}$. +% \end{enumerate} +% \end{frame} + +% \section{Linearization} + +% \begin{frame}{Linearization} +% \begin{itemize} + + +% \item Recall that if we have $\frac{dx}{dt} = \lambda x(t) + bu(t)$ we know a solution to this is: $$x(t) = x(0)e^{\lambda t} + \int_0^t \! e^{\lambda(t - \tau)}u(\tau)\ d\tau$$ +% \item What if we had $\frac{dx}{dt} = f(x(t)) + bu(t)$, where $f$ is nonlinear (e.g $sin$)? \pause \\ +% \item Big Picture: linearize $f$ around an operating point and then treat it as a linear function in a small neighborhood of that point.\\ +% \item Why linearization? \pause +% \\ It allows you to treat the system as a linear one, which is very helpful as linear ODE are (usually) much easier to solve! +% \end{itemize} +% \end{frame} +% \begin{frame}{Linearizing a Single-Variable Function} +% \begin{itemize} +% \item Suppose we have $f(x)$ that is a non linear function. We can use a first order Taylor polynomial to linearize the function, this is equivalent to finding the slope of the tangent line of $f(x)$ at a particular point. +% \item From calculus: $f(x) \approx f(x^*) + f'(x^*)(x - x^*)$. +% \item As long as we are within some (very small) $\delta$ neighborhood of $x^*$ the linearization is valid. +% \item Example: Linearize $f(x) = 3e^{x^2 + 2}$ around $x^*$ +% \item Solution: \\ +% $f(x^*) = 3e^{x^{*2} + 2}$\\ +% $f'(x) = 3e^{x^2+2}(2x) = 6xe^{x^2+2}$\\ +% $f'(x^*) = 6x^*e^{x^{*2}+2}$\\ +% Therefore : +% $f(x) \approx 3e^{x^{*2} + 2} + 6x^*e^{x^{*2}+2}(x-x^*) $ +% \end{itemize} + +% \end{frame} +% \begin{frame}{Linearizing Steps for $\frac{dx(t)}{dt} = f(x(t)) + bu(t)$} +% \begin{enumerate}[(i)] +% \item Choose, or you may be given, a DC input point. That is, a point $u^* \equiv u(t)$ that is constant with time.\pause \\ +% \item Find a DC operating point, $x^* \equiv x(t)$. That is, solve $\frac{dx^*}{dt} = f(x^*) + bu^*$. Notice that this boils down to finding an $x^*$ such that $f(x^*) + bu^* = 0 $. \pause \\ +% \item Define $x_l(t) = x(t) - x^*$ and $u_l(t) = u(t) - u^*$, and re-write the ODE in terms of $ x_l(t)$ and $ u_l(t)$. By plugging in you get: $\frac{dx_l(t)}{dt} = f(x_l(t) + x^*) + b(u_l(t)+ u^*)$ \pause \\ +% \item It is ok to assume at this point that $u_l(t)$ is small, that means that the $u(t)$ in step 1 does not deviate too much from $u^*$.\pause \\ +% \item Linearize the ODE: $f(x_l(t) + x^*) \approx f(x^*) + f'(x^*)x_l(t)$. Here we assume that $x_l(t)$ is also small. This is something that we will need to verify in the next step! +% \end{enumerate} +% \end{frame} +% \begin{frame}{Linearizing Steps for $\frac{dx(t)}{dt} = f(x(t)) + bu(t)$ (Continued)} +% \begin{enumerate}[(vi)] +% \item Plug (vi) back into (iii) and we obtain : $\frac{dx_l(t)}{dt} \approx f'(x^*)f(x_l(t)) + \cancel{f(x^*)} + bu_l(t)+ \cancel{bu^*} = f'(x^*)f(x_l(t))+ bu_l(t)$ +% \item Verify the linearization! \\ +% How do we know if the linearization is valid? \pause Well, if we have $\frac{dx_l(t)}{dt} = \lambda x_l(t) + bu(t)$ we know the solution doesn't blow up if $\lambda < 0$ as we will have a term $e^{\lambda t}$. +% \\ This means that we want $m = f'(x^*) <$ 0. +% \end{enumerate} \pause +% So what do we do if $m>0 $? \pause \\ +% We need to go back and change our DC operating point $x^*$ +% \end{frame} +% \begin{frame}{Practice Problem} +% Linearize $\frac{dx(t)}{dt} = \cos(x(t)) + bu(t)$ about $u^* = 0$. \pause \\ +% %Show this hint after a while if people are stuck +% \textit{Hint:} $\cos(x^*) = 0$ has multiple solutions, which means that we can find numerous DC operating points, can you guess which one would result in a stable system ? +% \end{frame} +% \begin{frame}{Practice Problem Solution} +% \begin{enumerate}[(i)] +% \item We were given the DC input, $u^* = 0$ \pause \\ +% \item $\cos(x^*)$ = 0, which means that $x^* = k\frac{\pi}{2}$ for $k \in \{...-2,-1,1,2,...\}$. We will choose $x^* = \frac{\pi}{2}$ \pause \\ +% \item Let $x_l(t) = x(t) - \frac{\pi}{2}$ and $u_l(t) = u(t) - 0$. +% By plugging in we get: $\frac{dx_l(t)}{dt} = \cos(x_l(t) + \frac{\pi}{2}) + u_l(t)$ \pause \\ +% \item We assume that $u_l(t)$ is small.\pause \\ +% \item Linearize the ODE: $\cos(x_l(t) + \frac{\pi}{2}) \approx \cos(\frac{\pi}{2}) -\sin(\frac{\pi}{2})x_l(t)$. \pause +% \item Plug (v) back into ODE: +% $\frac{dx_l(t)}{dt} = -x_l(t) + u_l(t)$ +% \end{enumerate} +% \pause +% We see that our assumption that $x_l(t)$ is small is indeed satisfied as we will have a $e^{-t}$ term in the solution which means that $x_l(t)$ will decay.\\ \pause +% What if we had chosen a different DC Operating point, say $-\frac{\pi}{2}$? When we linearize the system we see that the solution will "explode" around that particular DC operating point. +% \end{frame} + +% \begin{frame}{Linearization of Vector Functions} +% What if we had $\frac{d\vec{x}}{dt} = \vec{f}(\vec{x}, \vec{u})$ ? We will see that the process is very similar to the scalar case we just analyzed!\\ \pause +% First, let's see how to linearize $\vec{f}(\vec{x})$ around a DC operating point $\vec{x}^*$. Where $\vec{f} \in \R^{n \times 1}$ is a vector of scalar functions. \\\pause +% The idea is to linearize individually each one of the $f_i$ around the DC operating point. \\\pause +% For example: $f_1 \approx f_1(\vec{x}^*) + \frac{\partial f_1}{\partial x_1}(\vec{x}^*)(x_1 - x_1^*) + ... + \frac{\partial f_n}{\partial x_1}(\vec{x}^*)(x_n - x_n^*)$\\\pause +% Repeating this for all $n$ functions in $\vec{f}$ we see we get a system of scalar, linearized, multivariate functions which makes you think, wouldn't it be nice to express it in a shorthand matrix notation? +% \end{frame} + +% \begin{frame}{Jacobian Matrix} +% We can use the Jacobian to express everything nicely and neatly. \\ +% The Jacobian is the name given to the matrix of partial derivatives of $\vec{f}$, and it is denoted by $J_{\vec{x}}$ or $\nabla_{\vec{x}}\vec{f}$. \\ \pause +% Continuing from the previous slide we have: \\ +% \[ +% \begin{bmatrix} f_1(\vec{x}) \\ \vdots \\f_n(\vec{x}) \end{bmatrix} \approx +% \begin{bmatrix} f_1(\vec{x}^*) \\ \vdots \\f_n(\vec{x}^*) \end{bmatrix} + +% \begin{bmatrix} \frac{\partial f_1}{\partial x_1}(\vec{x}^*)& \hdots& \frac{\partial f_1}{\partial x_n}(\vec{x}^*) \\ \vdots & \ddots &\vdots \\ \frac{\partial f_n}{\partial x_1}(\vec{x}^*)& \hdots & \frac{\partial f_n}{\partial x_n}(\vec{x}^*) \end{bmatrix} +% (\vec{x} - \vec{x}^*) +% \] +% \\ + +% \end{frame} + +% \begin{frame}{Linearization with Jacobians Example} +% \begin{align*} +% \text{Linearize} \hspace{0.2 cm} +% \vec{f}(\vec{x}(t)) = +% \begin{bmatrix} +% \sin(x_1(t)*x_2(t)) + 2x_1(t)x_3^2(t)\\ +% x_3(t)\cos(x_2(t)) + \frac{x_1(t)}{x_3(t)}\\ +% x_1(t) + 2x_3(t)x_2^3(t) +% \end{bmatrix} +% \text{about} \hspace{0.2 cm} \vec{x}^* = \begin{bmatrix} +% 0\\ 2 \pi \\ \frac{2 \pi}{3} +% \end{bmatrix} +% \end{align*} +% \end{frame} + +% \begin{frame}{Solutions} +% Find the Jacobian: +% \small +% \[ +% \begin{bmatrix} +% x_2(t)\cos(x_1(t)*x_2(t)) + 2x_3^2(t)& x_1(t)\cos(x_1(t)*x_2(t)) & 4x_1(t)x_3(t)\\ +% \frac{1}{x_3(t)}& -x_3(t)\sin(x_2(t)) &\cos(x_2(t)) - \frac{x_1(t)}{x_3^2(t)} \\ +% 1 & 6x_3(t)x_2^2(t)& 2 x_2^3(t) +% \end{bmatrix} +% \] +% \normalsize +% Evaluate the Jacobian about $\vec{x}^*$: +% \[ +% \begin{bmatrix} +% 5 \pi & 0 & 0\\ +% \frac{2 \pi }{3}& 0 & 1\\ +% 1 & 36 \pi^3 & 16 \pi^3 +% \end{bmatrix} +% \] +% \\ +% Linearize: +% \begin{equation*} +% \vec{f}(\vec{x}(t)) \approx +% \begin{bmatrix} +% 0 \\ \frac{3\pi }{4} \\ 24 \pi^4 +% \end{bmatrix} + +% \begin{bmatrix} +% 5 \pi & 0 & 0\\ +% \frac{2 \pi }{3}& 0 & 1\\ +% 1 & 36 \pi^3 & 16 \pi^3 +% \end{bmatrix} +% \begin{bmatrix} +% x_1(t) - 0 \\ x_2(t) - \frac{3\pi }{4} \\ x_3(t) - 24 \pi^4 +% \end{bmatrix} +% \end{equation*} + +% \end{frame} + +% \begin{frame}{Steps to Linearize Vector ODE Systems} +% To linearize $\frac{d\vec{x}}{dt} = \vec{f}(\vec{x}(t), \vec{u}(t))$ we use a similar procedure as we did for the scalar case. \pause +% \begin{enumerate}[(i)] +% % \item If you're not given a DC input $\vec{u}^*$, determine one. \pause \\ +% \item Solve $\vec{f}(\vec{x}^*, \vec{u}^*) = \vec{0}$ to determine the equilibrium point. \pause \\ +% \item Let $\vec{x}_l(t) = \vec{x}(t) - \vec{x}^*$ and $\vec{u}_l(t) = \vec{u}(t) - \vec{u}^*$ \pause \\ +% \item Linearize $\vec{f}(\vec{x}, \vec{u})$ about $(\vec{x}^*, \vec{u}^*)$. That is: $\vec{f}(\vec{x}(t), \vec{u}(t)) \approx \vec{f}(\vec{x}^*, \vec{u}^*) + J_{\vec{x}}\vec{x}_l(t) + J_{\vec{u}}\vec{u}_l(t)$ \pause \\ +% \item Plug (iv) back into the ODE: $\frac{d\vec{x}}{dt} \approx \cancel{\vec{f}(\vec{x}^*, \vec{u}^*)} + J_{\vec{x}}\vec{x}_l(t) + J_{\vec{u}}\vec{u}_l(t)$ +% \end{enumerate} + +% \end{frame} + +% \begin{frame}{Linearizing Vector ODE Systems Example} +% Given a DC input $u^* = 1$, linearize: +% \[ +% \frac{d\vec{x}(t)}{dt} = \begin{bmatrix} +% x_1^2(t) - x_2(t)u(t)\\ +% x_2^2(t)(1 + x_1(t)) + \sin(\pi x_1(t)u(t)) +% \end{bmatrix} +% \] + +% \end{frame} + +% \begin{frame}{Solutions} +% Again, we will do this in steps: +% \begin{enumerate}[(i)] +% \item We are given $u^* = 1$ \pause +% \item We need to find a DC operating point, this means solving the following system of equations: +% \begin{align} +% x_1^{*2}- x_2^*u^* = 0 \\ +% x_2^{*2}(x_1^* + 1) + \sin(\pi x_1^*u^*) = 0 +% \end{align} +% The solution is $x_1^* = -1$ and $x_2^* = 1$. \pause \\ +% \item Let $\vec{x}_l(t) = \vec{x}(t) - \vec{x}^*$ and $\vec{u}_l(t) = \vec{u}(t) - \vec{u}^*$ \pause \\ +% \item Linearize, +% \[ \vec{f}(\vec{x}(t), u(t)) \approx +% \vec{f}(\vec{x}^*, 1) + \begin{bmatrix} +% -2 & -1 \\ +% 1 - \pi & 0 +% \end{bmatrix} +% \vec{x}_l(t) + \begin{bmatrix} +% -1 \\ \pi +% \end{bmatrix} +% u_l(t) +% \] + + + +% \end{enumerate} +% \end{frame} + +% \begin{frame}{Solutions Continued} +% \begin{enumerate}[(v)] +% \item Substitute linear approximation back into the system, +% \[ \frac{d\vec{x}(t)}{dt} \approx +% \cancel{\vec{f}(\vec{x}^*, 1)} + \begin{bmatrix} +% -2 & -1 \\ +% 1 - \pi & 0 +% \end{bmatrix} +% \vec{x}_l(t) + \begin{bmatrix} +% -1 \\ \pi +% \end{bmatrix} +% u_l(t) +% \] +% \end{enumerate} +% \end{frame} -\begin{column}{0.25\textwidth} -\begin{align*} -A^* = -\begin{bmatrix} -0&1&0 &\hdots &0\\ -0& 0 & 1& \hdots&0\\ -\vdots& \hdots & \hdots& \hdots&0\\ -0 & \hdots &\hdots & \hdots &1\\ -\alpha_{0} & \alpha_{1}& \hdots & \hdots & \alpha_{n-1} -\end{bmatrix} -\end{align*} -\end{column} -\begin{column}{0.25\textwidth} -\begin{align*} -B^* = + +\begin{frame}{Break} +\Large{Please give us feedback @ \url{hkn.mu/feedback}!\\\vspace{2em}Let us know of any issues in the slides @ \url{https://github.com/hkntutoring/ee16b-review/issues} }! +\end{frame} + +\section[Triangularization]{Upper Triangularization} + +\begin{frame}{Schur Decomposition} +\begin{itemize}[<+->] +\item Start with $M_n = A$ +\item Find an eigenvalue/eigenvector pair $\lambda_i, v_i$ for $M_{n-i+1}$ +\item Find $R_{n-i}$ such that $R_{n-i}^T v_{i} = \Vec{0}$ and $R_{n-i}^T R_{n-i} = I$ by Gram Schmidt. +\item Find the next $M$ with $M_{n-i} = R_{n-i}^T M_{n-i+1} R_{n-i}$ +\item \[U = [v_1, R_{n-1} v_2, R_{n-1} R_{n-2} v_3 \ldots ] \] +\item \[A = UMU^T \implies M = U^T A U \] +\end{itemize} +\end{frame} + +\begin{frame}{Upper Triangularization} +By Schur Decomposition, with $A = UMU^T$, you end up with a matrix of the form: +\[ \begin{bmatrix} -0\\ -0\\ -\vdots\\ -0\\ -1 +\lambda_1 & * & * & \cdots \\ +0 & \lambda_2 & * & \cdots \\ +0 & 0 & \lambda_2 & \cdots \\ +\vdots & \vdots & \vdots & \ddots \end{bmatrix} -\end{align*} -\end{column} -\end{columns}\pause -\item The characteristic polynomial of $A^*$ is $\lambda_n - \sum\limits_{i = 0}^{n-1} \alpha_i\lambda^i$ = 0. -\item So how does it help with eigenvalue placement? \pause The last row of this matrix determines the eigenvalues of $A^*$ so modifying the last row will allow us to (easily) modify the eigenvalues. +\] +\end{frame} + +\begin{frame}{Spectral Theorem} +If $A = A^T$ is a symmetric matrix, +\begin{itemize} [<+->] +\item All eigenvalues are real +\item Real eigenvectors exist +\item Eigenvectors form an orthonormal basis for $\mathbb{R}^n$ +\end{itemize} +\end{frame} + +\begin{frame}{Practice Question} +\begin{itemize} [<+->] +\item We know how to solve +\[A \Vec{x} \approx \Vec{y} \] +by viewing it as +\[\displaystyle \min \limits_{\Vec{x}} ||\Vec{y} - A\Vec{x}||^2 = \min \limits_{\Vec{x}} (\Vec{y} - A \Vec{x})^T (\Vec{y} - A \Vec{x}) \] +\item What if we wanted to minimize: +\[||C (\Vec{y} - A \Vec{x})||^2 \] +\item Simply expand into +\[||C \Vec{y} - CA \vec{x}|| \] +and solve as before with $\Vec{y} = C \Vec{y}$ and $A = CA$. \end{itemize} \end{frame} -\begin{frame}{How to convert to CCF} +\begin{frame}{Practice Question} \begin{itemize} -\item Let $A,B$ be the matrices in standard form and let $A^*,B^*$ be the matrices in CCF. \pause -\item Recall the matrix we used to check controllability?\\ \pause -\[C = \begin{bmatrix} -B & AB & \hdots & A^{n-1}B -\end{bmatrix}\] \\ -\[ C^* = \begin{bmatrix} -B^*& A^*B^* & \hdots & A^{*n-1}B^* -\end{bmatrix}\] -\item We then have $T := C^*C^{-1}$, such that $A^* = TAT^{-1}$ and $B^* = TB$. -\item Remember, all controllable matrices with single input can be transformed into CCF! +\item We know how to minimize +\[||C (\Vec{y} - A \Vec{x})||^2 \] +\item What if we wanted to minimize: +\[(\Vec{y} - A \Vec{x})^T P (\Vec{y} - A \Vec{x})\] +where $P$ is symmetric. +\item Recall by the spectral theorem that $P$ can be decomposed into $U \Lambda U^T$. +\item Let $\Lambda_{1/2}$ denote $\Lambda$ with its diagonal square rooted. +\item We know that $\Lambda_{1/2} = \Lambda_{1/2}^T$ because it is symmetrical. +\item Simply solve with $C = \Lambda_{1/2}U$. \end{itemize} \end{frame} -\begin{frame}{Example} -Consider the following discrete time system: +\section[SVD]{Singular Value Decomposition} + +\begin{frame}{SVD Theorem} +Any matrix $A \in \mathbb{R}^{m \times n}$ can be decomposed into the product of three matrices \begin{align*} -\vec{x}[t+1] = \begin{bmatrix} --2&2&0\\ -2&-4&2\\ -0&1&-1 -\end{bmatrix}\vec{x}[t] + \begin{bmatrix} 1 \\ 0 \\0 \end{bmatrix}u[t] +A &= U \Sigma V^T \\ +U &: m \times m \\ +\Sigma &: m \times n \\ +V^T &: n \times n \end{align*} +Such that $U, V$ are unitary matrices and $\Sigma$ only has nonnegative values along its main diagonal. + +\end{frame} + +\begin{frame}{SVD: Compact Form} +We can also express the SVD as +\begin{align*} +A &= \mathcal{U} S \mathcal{V}^T \\ +\mathcal{U} &: m \times r \\ +S &: r \times r \\ +\mathcal{V}^T &: r \times n +\end{align*} +where $r$ is the rank of $A$. The compact form matrices maintain properties of the original matrices, but have entries removed whenever they correspond to zero singular values. -\begin{enumerate}[(a)] -\item Is the system stable? Is it controllable? -\item Using an appropriate transformation ($\vec{z}[t] = T\vec{x}[t]$), bring the system to controllable canonical form. -\item Using the state feedback $u[t] =$ \[\begin{bmatrix} f_1 & f_2 & f_3 \end{bmatrix}\] $\vec{z}[t]$ bring the eigenvalues of the system to $0,0.75, -0.25$. -\end{enumerate} \end{frame} -\begin{frame}{Solutions to Example} -\begin{enumerate}[(a)] -\item The characteristic polynomial is: $\lambda^3 + 7\lambda^2 + 8\lambda = \lambda(\lambda^2 +7\lambda +8) = 0$, therefore the eigenvalues of A are $\{0,-5.56, -1.44\}$. As we can see there are $|\lambda_i|>1$ therefore the system is not stable. \\ -The controllability matrix $C =$ \[\begin{bmatrix} -1&-2&8\\ -0&2&-12\\ -0&0&2 -\end{bmatrix}\] -$C$ has full rank so the system is controllable -\item As we previously mentioned the coefficients of the characteristic polynomial are closely related to the last row of the $A^*$ matrix. Therefore, the CCF of the system is: +\begin{frame}{SVD: Outer Product Form} +Lastly, we can express +\[ A = \sum_{i = 1}^r \sigma_i \vec{u}_i \vec{v}_i^T \] +where $\vec{u}_i, \vec{v}_i$ are the columns of $U, V$, respectively, and $\sigma_i$ are corresponding diagonal entry of the matrix $\Sigma$ +\end{frame} + +\begin{frame}{Computing SVD with $A^T A$} \begin{align*} -\vec{z}[t+1] = -\begin{bmatrix} -0&1&0\\ -0&0&1\\ -0&-8&-7 -\end{bmatrix}\vec{x}[t] + \begin{bmatrix} 0 \\ 0 \\1 \end{bmatrix}u[t] +A^T A &= U \Sigma V^T V \Sigma^T U^T \\ +&= U \Sigma^2 U^T \end{align*} -\end{enumerate} +This is an eigen decomposition since $\Sigma^2$ is diagonal and $U^{-1} = U^T$. Thus solving for the eigenvalues and eigenvectors of $A^T A$ give $\lambda_i = \sigma_i^2$ with eigenvectors which correspond to the right singular vectors. We need to sort by decreasing $\sigma_i$. \\ + +\alert{Side note:} $\Sigma^T \Sigma$ is not actually equal to $\Sigma^2$, but the former product yields a matrix with singular values squared on the diagonal entries, hence we call it $\Sigma^2$ \end{frame} -\begin{frame}{Example Solutions Continued} -\begin{enumerate}[(c)] -\item Our system then becomes: +\begin{frame}{Computing SVD with $A^T A$} +Given a right singular vector $\vec{v}_i$ which we found from the previous part, we can apply it \begin{align*} -\vec{z}[t+1] = -\begin{bmatrix} -0&1&0\\ -0&0&1\\ -f_1&f_2-8&f_3-7 -\end{bmatrix}\vec{x}[t] +A \vec{v}_i &= \left( \sum_{k = 1}^r \sigma_k \vec{u}_k \vec{v}_k^T \right) \vec{v}_i \\ +&= \sum_{k = 1}^r \sigma_k \vec{u}_k \vec{v}_k^T \vec{i} \\ +&= \sigma_i \vec{u}_i \\ +\vec{u}_i &= \frac{1}{\sigma_i} A \vec{v}_i \end{align*} -Which means its characteristic polynomial is : $\lambda^3 - (f_3 -7)\lambda^2 -(f_2 - 8)\lambda - f_1$ = 0. \\ -Now, we know the characteristic polynomial should be $\lambda(\lambda - \frac{3}{4})(\lambda + \frac{1}{4})$, so we can equate the two and solve for the feedback vector $\vec{f}\hspace{0.1cm}^T = \begin{bmatrix} 0 & \frac{1}{2}& \frac{3}{16}\end{bmatrix}$. -\end{enumerate} \end{frame} -\section{Linearization} +\begin{frame}{Computing SVD with $A A^T$} +Similar calculations yield $\sigma_i = \sqrt{\lambda_i}$ of $A A^T$ with eigenvectors as left singular vectors, and $\vec{v}_i = \frac{1}{\sigma_i} A^T \vec{u}_i$ +\end{frame} -\begin{frame}{Linearization} +\begin{frame}{Intepretation of SVD} \begin{itemize} +\item Unitary matrices act as rotation in a given space. A diagonal matrix stretches in a given coordinate space. - -\item Recall that if we have $\frac{dx}{dt} = \lambda x(t) + bu(t)$ we know a solution to this is: $$x(t) = x(0)e^{\lambda t} + \int_0^t \! e^{\lambda(t - \tau)}u(\tau)\ d\tau$$ -\item What if we had $\frac{dx}{dt} = f(x(t)) + bu(t)$, where $f$ is nonlinear (e.g $sin$)? \pause \\ -\item Big Picture: linearize $f$ around an operating point and then treat it as a linear function in a small neighborhood of that point.\\ -\item Why linearization? \pause -\\ It allows you to treat the system as a linear one, which is very helpful as linear ODE are (usually) much easier to solve! +\item \href{https://en.wikipedia.org/wiki/File:Singular_value_decomposition.gif}{SVD visualization (open in browser)} \end{itemize} \end{frame} -\begin{frame}{Linearizing a Single-Variable Function} -\begin{itemize} -\item Suppose we have $f(x)$ that is a non linear function. We can use a first order Taylor polynomial to linearize the function, this is equivalent to finding the slope of the tangent line of $f(x)$ at a particular point. -\item From calculus: $f(x) \approx f(x^*) + f'(x^*)(x - x^*)$. -\item As long as we are within some (very small) $\delta$ neighborhood of $x^*$ the linearization is valid. -\item Example: Linearize $f(x) = 3e^{x^2 + 2}$ around $x^*$ -\item Solution: \\ -$f(x^*) = 3e^{x^{*2} + 2}$\\ -$f'(x) = 3e^{x^2+2}(2x) = 6xe^{x^2+2}$\\ -$f'(x^*) = 6x^*e^{x^{*2}+2}$\\ -Therefore : -$f(x) \approx 3e^{x^{*2} + 2} + 6x^*e^{x^{*2}+2}(x-x^*) $ + +\begin{frame}{Intepretation of SVD} +For a product $A \vec{x}$, we can decompose every vector $\vec{x}$ into a linear combination of right singular vectors +\[ \vec x = \sum_{i = 1}^n \alpha_i \vec{v}_i \] +Thus, we can see exactly which parts of $\vec{x}$ affect the output. +\end{frame} + + +\begin{frame}{Compression of Low-Rank Matrices} +\begin{itemize}[<+->] +\item Suppose I had a matrix $A \in \mathbb{R}^{m \times n}$ with $m, n >> rank(A)$. How could I more efficiently store $A$ and compute products like $A \vec{x}$? +\vspace{2em} +\item With the SVD, we only have to save $r$ set of two vectors and a scalar, which saves us a lot of space if the rank is small with respect to the matrix. Also, less computation is carried out if we represent the matrix as the outer product form. \end{itemize} +\end{frame} +\section[PCA]{Principle Component Analysis} + +\begin{frame}{PCA} +PCA is a linear dimensionality reduction tool. Given data $\vec{x}_i \in \mathbb{R}^d$, we can create a mapping $T : \mathbb{R}^d \rightarrow \mathbb{R}^{d'}, d' < d$ such that the variance in the dataset is still captured + +PCA captures the direction along which the data varies the most. (Direction of Maximal Variance) \end{frame} -\begin{frame}{Linearizing Steps for $\frac{dx(t)}{dt} = f(x(t)) + bu(t)$} -\begin{enumerate}[(i)] -\item Choose, or you may be given, a DC input point. That is, a point $u^* \equiv u(t)$ that is constant with time.\pause \\ -\item Find a DC operating point, $x^* \equiv x(t)$. That is, solve $\frac{dx^*}{dt} = f(x^*) + bu^*$. Notice that this boils down to finding an $x^*$ such that $f(x^*) + bu^* = 0 $. \pause \\ -\item Define $x_l(t) = x(t) - x^*$ and $u_l(t) = u(t) - u^*$, and re-write the ODE in terms of $ x_l(t)$ and $ u_l(t)$. By plugging in you get: $\frac{dx_l(t)}{dt} = f(x_l(t) + x^*) + b(u_l(t)+ u^*)$ \pause \\ -\item It is ok to assume at this point that $u_l(t)$ is small, that means that the $u(t)$ in step 1 does not deviate too much from $u^*$.\pause \\ -\item Linearize the ODE: $f(x_l(t) + x^*) \approx f(x^*) + f'(x^*)x_l(t)$. Here we assume that $x_l(t)$ is also small. This is something that we will need to verify in the next step! -\end{enumerate} +% Image From https://blog.bioturing.com/2018/06/14/principal-component-analysis-explained-simply/ +\begin{frame}{PCA Illustrated} +\includegraphics[scale=0.4]{images/PCAexample.png} \end{frame} -\begin{frame}{Linearizing Steps for $\frac{dx(t)}{dt} = f(x(t)) + bu(t)$ (Continued)} -\begin{enumerate}[(vi)] -\item Plug (vi) back into (iii) and we obtain : $\frac{dx_l(t)}{dt} \approx f'(x^*)f(x_l(t)) + \cancel{f(x^*)} + bu_l(t)+ \cancel{bu^*} = f'(x^*)f(x_l(t))+ bu_l(t)$ -\item Verify the linearization! \\ -How do we know if the linearization is valid? \pause Well, if we have $\frac{dx_l(t)}{dt} = \lambda x_l(t) + bu(t)$ we know the solution doesn't blow up if $\lambda < 0$ as we will have a term $e^{\lambda t}$. -\\ This means that we want $m = f'(x^*) <$ 0. -\end{enumerate} \pause -So what do we do if $m>0 $? \pause \\ -We need to go back and change our DC operating point $x^*$ -\end{frame} -\begin{frame}{Practice Problem} -Linearize $\frac{dx(t)}{dt} = \cos(x(t)) + bu(t)$ about $u^* = 0$. \pause \\ -%Show this hint after a while if people are stuck -\textit{Hint:} $\cos(x^*) = 0$ has multiple solutions, which means that we can find numerous DC operating points, can you guess which one would result in a stable system ? -\end{frame} -\begin{frame}{Practice Problem Solution} -\begin{enumerate}[(i)] -\item We were given the DC input, $u^* = 0$ \pause \\ -\item $\cos(x^*)$ = 0, which means that $x^* = k\frac{\pi}{2}$ for $k \in \{...-2,-1,1,2,...\}$. We will choose $x^* = \frac{\pi}{2}$ \pause \\ -\item Let $x_l(t) = x(t) - \frac{\pi}{2}$ and $u_l(t) = u(t) - 0$. -By plugging in we get: $\frac{dx_l(t)}{dt} = \cos(x_l(t) + \frac{\pi}{2}) + u_l(t)$ \pause \\ -\item We assume that $u_l(t)$ is small.\pause \\ -\item Linearize the ODE: $\cos(x_l(t) + \frac{\pi}{2}) \approx \cos(\frac{\pi}{2}) -\sin(\frac{\pi}{2})x_l(t)$. \pause -\item Plug (v) back into ODE: -$\frac{dx_l(t)}{dt} = -x_l(t) + u_l(t)$ + +\begin{frame}{PCA --- Computation} +\begin{enumerate}[<+->] +\item Store data row-major in $A \in \mathbb{R}^{n \times d}$ +\item De-mean $A$ \hfill \textit{Step is optional, only used for specific scenarios} +\item Take SVD: $A = U \Sigma V^T$ +\item Create $V_{d'} \in \mathbb{R}^{n \times d'}$ from vectors of $V$ corresponding to $d'$ greatest signular values +\item To project data into the representative subspace: $T(x) := V_{d'}^T x$ \end{enumerate} -\pause -We see that our assumption that $x_l(t)$ is small is indeed satisfied as we will have a $e^{-t}$ term in the solution which means that $x_l(t)$ will decay.\\ \pause -What if we had chosen a different DC Operating point, say $-\frac{\pi}{2}$? When we linearize the system we see that the solution will "explode" around that particular DC operating point. \end{frame} -\begin{frame}{Linearization of Vector Functions} -What if we had $\frac{d\vec{x}}{dt} = \vec{f}(\vec{x}, \vec{u})$ ? We will see that the process is very similar to the scalar case we just analyzed!\\ \pause -First, let's see how to linearize $\vec{f}(\vec{x})$ around a DC operating point $\vec{x}^*$. Where $\vec{f} \in \R^{n \times 1}$ is a vector of scalar functions. \\\pause -The idea is to linearize individually each one of the $f_i$ around the DC operating point. \\\pause -For example: $f_1 \approx f_1(\vec{x}^*) + \frac{\partial f_1}{\partial x_1}(\vec{x}^*)(x_1 - x_1^*) + ... + \frac{\partial f_n}{\partial x_1}(\vec{x}^*)(x_n - x_n^*)$\\\pause -Repeating this for all $n$ functions in $\vec{f}$ we see we get a system of scalar, linearized, multivariate functions which makes you think, wouldn't it be nice to express it in a shorthand matrix notation? +\begin{frame}{PCA: computation} +The mapping $T$ can then be expressed as +\[ T(\vec{x}) = V_k^T \vec{x} \] +If we apply this transformation onto the entire dataset (which has row vectors), we can say +\[ T(A) = B = A V_k \] +where $B \in \mathbb{R}^{n \times k}$ +\end{frame} +\begin{frame}{PCA: computation} +If we were to show the projected vectors in the original space, we can multiply back with the projection vectors +\[ A' = B V_k^T \] \end{frame} -\begin{frame}{Jacobian Matrix} -We can use the Jacobian to express everything nicely and neatly. \\ -The Jacobian is the name given to the matrix of partial derivatives of $\vec{f}$, and it is denoted by $J_{\vec{x}}$ or $\nabla_{\vec{x}}\vec{f}$. \\ \pause -Continuing from the previous slide we have: \\ -\[ -\begin{bmatrix} f_1(\vec{x}) \\ \vdots \\f_n(\vec{x}) \end{bmatrix} \approx -\begin{bmatrix} f_1(\vec{x}^*) \\ \vdots \\f_n(\vec{x}^*) \end{bmatrix} + -\begin{bmatrix} \frac{\partial f_1}{\partial x_1}(\vec{x}^*)& \hdots& \frac{\partial f_1}{\partial x_n}(\vec{x}^*) \\ \vdots & \ddots &\vdots \\ \frac{\partial f_n}{\partial x_1}(\vec{x}^*)& \hdots & \frac{\partial f_n}{\partial x_n}(\vec{x}^*) \end{bmatrix} -(\vec{x} - \vec{x}^*) -\] -\\ +\section{Discretization} +\begin{frame}{Discretization: Q1} +Note: this section follows hw8 q1 almost exactly. Suppose we have a scalar system +\[ \frac{d}{dt} x(t) = \alpha x + \vec{\beta}^T \vec u(t) \] +and we apply a constant input $\vec{u}_n$ for times $t \in [nT, (n + 1)T)$ for some $T > 0$. Given $x(nT)$ solve the differential equation \end{frame} -\begin{frame}{Linearization with Jacobians Example} +\begin{frame}{Discretization: Q1 Sol} +From $t = nT$ to $t = (n + 1)T$, $\vec{\beta}^T \vec{u}$ is a constant scalar. Thus, we can solve this like a normal differential equation. Let $x = x' - \frac{\vec \beta^T \vec u}{\alpha}$. Then \begin{align*} -\text{Linearize} \hspace{0.2 cm} -\vec{f}(\vec{x}(t)) = -\begin{bmatrix} -\sin(x_1(t)*x_2(t)) + 2x_1(t)x_3^2(t)\\ -x_3(t)\cos(x_2(t)) + \frac{x_1(t)}{x_3(t)}\\ -x_1(t) + 2x_3(t)x_2^3(t) -\end{bmatrix} -\text{about} \hspace{0.2 cm} \vec{x}^* = \begin{bmatrix} -0\\ 2 \pi \\ \frac{2 \pi}{3} -\end{bmatrix} +\frac{d}{dt} x(t) &= \alpha \left(x' - \frac{\vec \beta^T \vec u}{\alpha}\right) + \vec{\beta}^T \vec u(t) \\ +\differential{t}x = \differential{t}x' &= \alpha x' \\ +x' &= A e^{\alpha (t - nT)} \text{, for some integration constant \em{A}} \\ +x + \frac{\vec \beta^T \vec u}{\alpha} &= A e^{\alpha (t - nT)} \\ +x &= A e^{\alpha (t - nT)} - \frac{\vec \beta^T \vec u}{\alpha} \end{align*} \end{frame} - -\begin{frame}{Solutions} -Find the Jacobian: -\small -\[ -\begin{bmatrix} -x_2(t)\cos(x_1(t)*x_2(t)) + 2x_3^2(t)& x_1(t)\cos(x_1(t)*x_2(t)) & 4x_1(t)x_3(t)\\ -\frac{1}{x_3(t)}& -x_3(t)\sin(x_2(t)) &\cos(x_2(t)) - \frac{x_1(t)}{x_3^2(t)} \\ -1 & 6x_3(t)x_2^2(t)& 2 x_2^3(t) -\end{bmatrix} -\] -\normalsize -Evaluate the Jacobian about $\vec{x}^*$: -\[ -\begin{bmatrix} -5 \pi & 0 & 0\\ -\frac{2 \pi }{3}& 0 & 1\\ -1 & 36 \pi^3 & 16 \pi^3 -\end{bmatrix} -\] -\\ -Linearize: -\begin{equation*} -\vec{f}(\vec{x}(t)) \approx -\begin{bmatrix} -0 \\ \frac{3\pi }{4} \\ 24 \pi^4 -\end{bmatrix} + -\begin{bmatrix} -5 \pi & 0 & 0\\ -\frac{2 \pi }{3}& 0 & 1\\ -1 & 36 \pi^3 & 16 \pi^3 -\end{bmatrix} -\begin{bmatrix} -x_1(t) - 0 \\ x_2(t) - \frac{3\pi }{4} \\ x_3(t) - 24 \pi^4 -\end{bmatrix} -\end{equation*} - +\begin{frame}{Discretization: Q1 Sol} +At which point we can use our initial condition to get +\begin{align*} +x(nT) &= A - \frac{\vec \beta^T \vec u}{\alpha} \\ +A &= x(nT) + \frac{\vec \beta^T \vec u}{\alpha} \\ +x &= \left( x(nT) + \frac{\vec \beta^T \vec u}{\alpha} \right) e^{\alpha (t - nT)} - \frac{\vec \beta^T \vec u}{\alpha} +\end{align*} \end{frame} -\begin{frame}{Steps to Linearize Vector ODE Systems} -To linearize $\frac{d\vec{x}}{dt} = \vec{f}(\vec{x}(t), \vec{u}(t))$ we use a similar procedure as we did for the scalar case. \pause -\begin{enumerate}[(i)] -% \item If you're not given a DC input $\vec{u}^*$, determine one. \pause \\ -\item Solve $\vec{f}(\vec{x}^*, \vec{u}^*) = \vec{0}$ to determine the equilibrium point. \pause \\ -\item Let $\vec{x}_l(t) = \vec{x}(t) - \vec{x}^*$ and $\vec{u}_l(t) = \vec{u}(t) - \vec{u}^*$ \pause \\ -\item Linearize $\vec{f}(\vec{x}, \vec{u})$ about $(\vec{x}^*, \vec{u}^*)$. That is: $\vec{f}(\vec{x}(t), \vec{u}(t)) \approx \vec{f}(\vec{x}^*, \vec{u}^*) + J_{\vec{x}}\vec{x}_l(t) + J_{\vec{u}}\vec{u}_l(t)$ \pause \\ -\item Plug (iv) back into the ODE: $\frac{d\vec{x}}{dt} \approx \cancel{\vec{f}(\vec{x}^*, \vec{u}^*)} + J_{\vec{x}}\vec{x}_l(t) + J_{\vec{u}}\vec{u}_l(t)$ -\end{enumerate} - +\begin{frame}{Discretization: Q2} +Using the differential equation derived from question 1, create a discrete-time system to model the continuous time. In other words, if $x[n] = x(nT), \vec{u}[n] = \vec{u}(nT)$, find a relation such that +\[ x[n + 1] = A_d x[n] + B_d \vec{u}[n] \] \end{frame} -\begin{frame}{Linearizing Vector ODE Systems Example} -Given a DC input $u^* = 1$, linearize: -\[ -\frac{d\vec{x}(t)}{dt} = \begin{bmatrix} -x_1^2(t) - x_2(t)u(t)\\ -x_2^2(t)(1 + x_1(t)) + \sin(\pi x_1(t)u(t)) -\end{bmatrix} -\] - +\begin{frame}{Discretization: Q2 Sol} +We can solve the previous solution for $x((n + 1)T)$ +\begin{align*} +x((n + 1)T) &= \left( x(nT) + \frac{\vec \beta^T \vec u(nT)}{\alpha} \right) e^{\alpha ((n + 1)T - nT)} - \frac{\vec \beta^T \vec u(nT)}{\alpha} \\ +x[n + 1] &= e^{\alpha T} x[n] + \frac{e^{\alpha T} - 1}{\alpha} \vec{\beta}^T \vec{u}[n] \\ +\end{align*} +We see that $A_d = e^{\alpha T}, B_d = ((e^{\alpha T} - 1) / \alpha) \vec{\beta}^T$ \end{frame} -\begin{frame}{Solutions} -Again, we will do this in steps: -\begin{enumerate}[(i)] -\item We are given $u^* = 1$ \pause -\item We need to find a DC operating point, this means solving the following system of equations: -\begin{align} -x_1^{*2}- x_2^*u^* = 0 \\ -x_2^{*2}(x_1^* + 1) + \sin(\pi x_1^*u^*) = 0 -\end{align} -The solution is $x_1^* = -1$ and $x_2^* = 1$. \pause \\ -\item Let $\vec{x}_l(t) = \vec{x}(t) - \vec{x}^*$ and $\vec{u}_l(t) = \vec{u}(t) - \vec{u}^*$ \pause \\ -\item Linearize, -\[ \vec{f}(\vec{x}(t), u(t)) \approx -\vec{f}(\vec{x}^*, 1) + \begin{bmatrix} --2 & -1 \\ -1 - \pi & 0 -\end{bmatrix} -\vec{x}_l(t) + \begin{bmatrix} --1 \\ \pi -\end{bmatrix} -u_l(t) -\] +\begin{frame}{Discretization: Q3} +Instead of a scalar, we instead have a diagonal matrix $A$ such that +\[ \frac{d}{dt} \vec{x} = A \vec{x} + B \vec{u} \] +Discretize this system in the same was as Q2. +\end{frame} +\begin{frame}{Discretiziation: Q3 Sol} +Expanding the original system out line-by-line gives +\[ \frac{d}{dt} x_i = a_i x_i + b_i \vec{u}_i \] +where $x_i$ is the $i$th variable of $\vec{x}$, $a_i$ is the diagonal entry of $A$, and $b_i$ is the row of $B$. +\end{frame} +\begin{frame}{Discretization: Generic Matrix} +Math not shown, but we can perform a change of basis from our original space to our diagonal space, and then apply the results of the previous part. -\end{enumerate} \end{frame} -\begin{frame}{Solutions Continued} -\begin{enumerate}[(v)] -\item Substitute linear approximation back into the system, -\[ \frac{d\vec{x}(t)}{dt} \approx -\cancel{\vec{f}(\vec{x}^*, 1)} + \begin{bmatrix} --2 & -1 \\ -1 - \pi & 0 -\end{bmatrix} -\vec{x}_l(t) + \begin{bmatrix} --1 \\ \pi -\end{bmatrix} -u_l(t) -\] +\section{Notes} +\begin{frame}{Notes} +These are some potential topics that would be good to look at +\begin{enumerate} +\item Proofs for all the theorems we mentioned +\item Outlier Rejection \end{enumerate} \end{frame} - - -\begin{frame}{Break} - GIVE US FEEDBACK \\ - \url{hkn.mu/feedback} \\ - \url{github.com/hkntutoring/ee16b-mt2-review/issues} +\begin{frame}{Thank You!} +Thank you everyone for coming and good luck on your midterm! \\ +In addition, HKN volunteers will be giving out oreos at the end of your midterm so please take one if you want! \end{frame} - \section[SVD]{Singular Value Decomposition} - - \begin{frame}{SVD Theorem} -Any matrix $A \in \mathbb{R}^{m \times n}$ can be decomposed into the product of three matrices - \begin{align*} - A &= U \Sigma V^T \\ - U &: m \times m \\ - \Sigma &: m \times n \\ - V^T &: n \times n - \end{align*} - Such that $U, V$ are unitary matrices and $\Sigma$ only has nonnegative values along its main diagonal. - - \end{frame} - - \begin{frame}{SVD: Compact Form} - We can also express the SVD as - \begin{align*} - A &= \mathcal{U} S \mathcal{V}^T \\ - \mathcal{U} &: m \times r \\ - S &: r \times r \\ - \mathcal{V}^T &: r \times n - \end{align*} - where $r$ is the rank of $A$. The compact form matrices maintain properties of the original matrices, but have entries removed whenever they correspond to zero singular values. - - \end{frame} - - - \begin{frame}{SVD: Outer Product Form} - Lastly, we can express - \[ A = \sum_{i = 1}^r \sigma_i \vec{u}_i \vec{v}_i^T \] - where $\vec{u}_i, \vec{v}_i$ are the columns of $U, V$, respectively, and $\sigma_i$ are corresponding diagonal entry of the matrix $\Sigma$ - \end{frame} - - \begin{frame}{Computing SVD with $A^T A$} - \begin{align*} - A^T A &= U \Sigma V^T V \Sigma^T U^T \\ - &= U \Sigma^2 U^T - \end{align*} - This is an eigen decomposition since $\Sigma^2$ is diagonal and $U^{-1} = U^T$. Thus solving for the eigenvalues and eigenvectors of $A^T A$ give $\lambda_i = \sigma_i^2$ with eigenvectors which correspond to the right singular vectors. We need to sort by decreasing $\sigma_i$. \\ - - \alert{Side note:} $\Sigma^T \Sigma$ is not actually equal to $\Sigma^2$, but the former product yields a matrix with singular values squared on the diagonal entries, hence we call it $\Sigma^2$ - \end{frame} - - \begin{frame}{Computing SVD with $A^T A$} -Given a right singular vector $\vec{v}_i$ which we found from the previous part, we can apply it - \begin{align*} - A \vec{v}_i &= \left( \sum_{k = 1}^r \sigma_k \vec{u}_k \vec{v}_k^T \right) \vec{v}_i \\ - &= \sum_{k = 1}^r \sigma_k \vec{u}_k \vec{v}_k^T \vec{i} \\ - &= \sigma_i \vec{u}_i \\ - \vec{u}_i &= \frac{1}{\sigma_i} A \vec{v}_i - \end{align*} - \end{frame} - - \begin{frame}{Computing SVD with $A A^T$} - Similar calculations yield $\sigma_i = \sqrt{\lambda_i}$ of $A A^T$ with eigenvectors as left singular vectors, and $\vec{v}_i = \frac{1}{\sigma_i} A^T \vec{u}_i$ - \end{frame} - - \begin{frame}{Intepretation of SVD} - \begin{itemize} - \item Unitary matrices act as rotation in a given space. A diagonal matrix stretches in a given coordinate space. - - \item \href{https://en.wikipedia.org/wiki/File:Singular_value_decomposition.gif}{SVD visualization (open in browser)} - \end{itemize} - - - - - \end{frame} - - \begin{frame}{Intepretation of SVD} - For a product $A \vec{x}$, we can decompose every vector $\vec{x}$ into a linear combination of right singular vectors - \[ \vec x = \sum_{i = 1}^n \alpha_i \vec{v}_i \] - Thus, we can see exactly which parts of $\vec{x}$ affect the output. - \end{frame} - - - \begin{frame}{Compression of Low-Rank Matrices} - \begin{itemize}[<+->] - \item Suppose I had a matrix $A \in \mathbb{R}^{m \times n}$ with $m, n >> rank(A)$. How could I more efficiently store $A$ and compute products like $A \vec{x}$? - \vspace{2em} - \item With the SVD, we only have to save $r$ set of two vectors and a scalar, which saves us a lot of space if the rank is small with respect to the matrix. Also, less computation is carried out if we represent the matrix as the outer product form. - \end{itemize} - \end{frame} - - \section[PCA]{Principle Component Analysis} - - \begin{frame}{PCA} - PCA is a linear dimensionality reduction tool. Given data $\vec{x}_i \in \mathbb{R}^d$, we can create a mapping $T : \mathbb{R}^d \rightarrow \mathbb{R}^{d'}, d' < d$ such that the variance in the dataset is still captured - \end{frame} - - \begin{frame}{PCA --- Computation} - \begin{enumerate}[<+->] - \item Store data row-major in $A \in \mathbb{R}^{n \times d}$ - \item De-mean $A$ - \item Take SVD: $A = U \Sigma V^T$ - \item Create $V_{d'} \in \mathbb{R}^{n \times d'}$ from vectors of $V$ corresponding to $d'$ greatest signular values - \item To project data into the representative subspace: $T(x) := V_{d'}^T x$ - \end{enumerate} - \end{frame} - - \begin{frame}{PCA: computation} - The mapping $T$ can then be expressed as - \[ T(\vec{x}) = V_k^T \vec{x} \] - If we apply this transformation onto the entire dataset (which has row vectors), we can say - \[ T(A) = B = A V_k \] - where $B \in \mathbb{R}^{n \times k}$ - \end{frame} - \begin{frame}{PCA: computation} - If we were to show the projected vectors in the original space, we can multiply back with the projection vectors - \[ A' = B V_k^T \] - \end{frame} - - \section{Discretization} - - \begin{frame}{Discretization: Q1} - Note: this section follows hw8 q1 almost exactly. Suppose we have a scalar system - \[ \frac{d}{dt} x(t) = \alpha x + \vec{\beta}^T \vec u(t) \] - and we apply a constant input $\vec{u}_n$ for times $t \in [nT, (n + 1)T)$ for some $T > 0$. Given $x(nT)$ solve the differential equation - \end{frame} - - \begin{frame}{Discretization: Q1 Sol} - From $t = nT$ to $t = (n + 1)T$, $\vec{\beta}^T \vec{u}$ is a constant scalar. Thus, we can solve this like a normal differential equation. Let $x = x' - \frac{\vec \beta^T \vec u}{\alpha}$. Then - \begin{align*} - \frac{d}{dt} x(t) &= \alpha \left(x' - \frac{\vec \beta^T \vec u}{\alpha}\right) + \vec{\beta}^T \vec u(t) \\ - \differential{t}x = \differential{t}x' &= \alpha x' \\ - x' &= A e^{\alpha (t - nT)} \text{, for some integration constant \em{A}} \\ - x + \frac{\vec \beta^T \vec u}{\alpha} &= A e^{\alpha (t - nT)} \\ - x &= A e^{\alpha (t - nT)} - \frac{\vec \beta^T \vec u}{\alpha} - \end{align*} -\end{frame} -\begin{frame}{Discretization: Q1 Sol} - At which point we can use our initial condition to get - \begin{align*} - x(nT) &= A - \frac{\vec \beta^T \vec u}{\alpha} \\ - A &= x(nT) + \frac{\vec \beta^T \vec u}{\alpha} \\ - x &= \left( x(nT) + \frac{\vec \beta^T \vec u}{\alpha} \right) e^{\alpha (t - nT)} - \frac{\vec \beta^T \vec u}{\alpha} - \end{align*} - \end{frame} - - \begin{frame}{Discretization: Q2} - Using the differential equation derived from question 1, create a discrete-time system to model the continuous time. In other words, if $x[n] = x(nT), \vec{u}[n] = \vec{u}(nT)$, find a relation such that - \[ x[n + 1] = A_d x[n] + B_d \vec{u}[n] \] - \end{frame} - - \begin{frame}{Discretization: Q2 Sol} - We can solve the previous solution for $x((n + 1)T)$ - \begin{align*} - x((n + 1)T) &= \left( x(nT) + \frac{\vec \beta^T \vec u(nT)}{\alpha} \right) e^{\alpha ((n + 1)T - nT)} - \frac{\vec \beta^T \vec u(nT)}{\alpha} \\ - x[n + 1] &= e^{\alpha T} x[n] + \frac{e^{\alpha T} - 1}{\alpha} \vec{\beta}^T \vec{u}[n] \\ - \end{align*} - We see that $A_d = e^{\alpha T}, B_d = ((e^{\alpha T} - 1) / \alpha) \vec{\beta}^T$ - \end{frame} - - \begin{frame}{Discretization: Q3} - Instead of a scalar, we instead have a diagonal matrix $A$ such that - \[ \frac{d}{dt} \vec{x} = A \vec{x} + B \vec{u} \] - Discretize this system in the same was as Q2. - \end{frame} - - \begin{frame}{Discretiziation: Q3 Sol} - Expanding the original system out line-by-line gives - \[ \frac{d}{dt} x_i = a_i x_i + b_i \vec{u}_i \] - where $x_i$ is the $i$th variable of $\vec{x}$, $a_i$ is the diagonal entry of $A$, and $b_i$ is the row of $B$. - \end{frame} - \begin{frame}{Discretization: Generic Matrix} - - Math not shown, but we can perform a change of basis from our original space to our diagonal space, and then apply the results of the previous part. - - \end{frame} - \end{document} From b1a84237b403bab39fbb8b782a2797032f002604 Mon Sep 17 00:00:00 2001 From: geohh Date: Sun, 10 Nov 2019 12:19:27 -0800 Subject: [PATCH 3/6] Fixed typo --- mt2.pdf | Bin 468511 -> 468642 bytes mt2.tex | 2 +- 2 files changed, 1 insertion(+), 1 deletion(-) diff --git a/mt2.pdf b/mt2.pdf index 7406902154195cd74f8a4c12d20f51ddb5f8e277..cca0a3a8741a088f28e770e930d949bd64b67236 100644 GIT binary patch delta 34490 zcmY(pQ*$l~)3uvqhBLNp+qP}nwsXg}Z96lzZQHi3y>>k(?{_eMKv#A3=)QX7P2m(y zWr?~Cctk9F zg)LiV$X(gN$#EDD{$(dPSP01JHalz&psQ{5bCxSv{IsLSyrRUrlFU=Nxe`2nMRt10 zMUGL}TR+NF5mP#bWsKA!96Nff8&;)MnNJl%U#xE;r9W9-_N^EUog22%`(b+z=*r6O 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z|Jw8ivt_zCzqBxUhIm?CO!Ho&c}tQ?;d#`n2!Zmh|94E eS7Ifu!Zo-S%dkQ`11DJ4{{Zv~yi%8TCI=}701yQL diff --git a/mt2.tex b/mt2.tex index bfe1a89..db512f1 100644 --- a/mt2.tex +++ b/mt2.tex @@ -836,10 +836,10 @@ \section{Eigenvalue Placement} \begin{frame}{Computing SVD with $A^T A$} \begin{align*} -A^T A &= U \Sigma V^T V \Sigma^T U^T \\ -&= U \Sigma^2 U^T +A^T A &= V \Sigma^T U^T U \Sigma UV^T \\ +&= V \Sigma^2 V^T \end{align*} -This is an eigen decomposition since $\Sigma^2$ is diagonal and $U^{-1} = U^T$. Thus solving for the eigenvalues and eigenvectors of $A^T A$ give $\lambda_i = \sigma_i^2$ with eigenvectors which correspond to the right singular vectors. We need to sort by decreasing $\sigma_i$. \\ +This is an eigen decomposition since $\Sigma^2$ is diagonal and $V^{-1} = V^T$. Thus solving for the eigenvalues and eigenvectors of $A^T A$ give $\lambda_i = \sigma_i^2$ with eigenvectors which correspond to the right singular vectors. We need to sort by decreasing $\sigma_i$. \\ \alert{Side note:} $\Sigma^T \Sigma$ is not actually equal to $\Sigma^2$, but the former product yields a matrix with singular values squared on the diagonal entries, hence we call it $\Sigma^2$ \end{frame} From 75c7003028bbda95d73d52ccfb857cc994b8bbf6 Mon Sep 17 00:00:00 2001 From: geohh Date: Sun, 10 Nov 2019 14:48:21 -0800 Subject: [PATCH 5/6] updated notation convention for sampling period to \Delta --- mt2.pdf | Bin 468639 -> 468746 bytes mt2.tex | 20 ++++++++++---------- 2 files changed, 10 insertions(+), 10 deletions(-) diff --git a/mt2.pdf b/mt2.pdf index 2b0e20840ff2c49c66e0eae8d98c2d1554aa8fc6..3b292cb6ee1040dba80155483c76fd06c9bb91aa 100644 GIT binary patch delta 41472 zcma%hLv$ug*kx?nwr$(CZQF0h9oy*Gw$rg~+qRSGZ{}aj?B?7}ol}ci-RI(|KT~*- z(|Er^fI~n+KtsSnz?&c{l>pL?2OMZUFPgV%@mRn2dYY^PJyL=A(UOKw%}XHrl=(o+ z*6J_m_#aze5|YiNQ!Rs$wGu{0S^pz7kHCC%njcM7g6Pq zS0K-j>pQ8H1p#E4dw?#OdOVm!XxqDwNVDwFSG2)ahV!qXY^PkDqi_W}4|9WB1f%%_ 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Given $x(nT)$ solve the differential equation +and we apply a constant input $\vec{u}_n$ for times $t \in [nT, (n + 1) \Delta)$ for some $T > 0$. Given $x(nT)$ solve the differential equation \end{frame} \begin{frame}{Discretization: Q1 Sol} -From $t = nT$ to $t = (n + 1)T$, $\vec{\beta}^T \vec{u}$ is a constant scalar. Thus, we can solve this like a normal differential equation. Let $x = x' - \frac{\vec \beta^T \vec u}{\alpha}$. Then +From $t = n \Delta$ to $t = (n + 1) \Delta$, $\vec{\beta}^T \vec{u}$ is a constant scalar. Thus, we can solve this like a normal differential equation. Let $x = x' - \frac{\vec \beta^T \vec u}{\alpha}$. Then \begin{align*} \frac{d}{dt} x(t) &= \alpha \left(x' - \frac{\vec \beta^T \vec u}{\alpha}\right) + \vec{\beta}^T \vec u(t) \\ \differential{t}x = \differential{t}x' &= \alpha x' \\ -x' &= A e^{\alpha (t - nT)} \text{, for some integration constant \em{A}} \\ -x + \frac{\vec \beta^T \vec u}{\alpha} &= A e^{\alpha (t - nT)} \\ -x &= A e^{\alpha (t - nT)} - \frac{\vec \beta^T \vec u}{\alpha} +x' &= A e^{\alpha (t - n \Delta)} \text{, for some integration constant \em{A}} \\ +x + \frac{\vec \beta^T \vec u}{\alpha} &= A e^{\alpha (t - n \Delta)} \\ +x &= A e^{\alpha (t - n \Delta)} - \frac{\vec \beta^T \vec u}{\alpha} \end{align*} \end{frame} \begin{frame}{Discretization: Q1 Sol} @@ -938,22 +938,22 @@ \section{Discretization} \begin{align*} x(nT) &= A - \frac{\vec \beta^T \vec u}{\alpha} \\ A &= x(nT) + \frac{\vec \beta^T \vec u}{\alpha} \\ -x &= \left( x(nT) + \frac{\vec \beta^T \vec u}{\alpha} \right) e^{\alpha (t - nT)} - \frac{\vec \beta^T \vec u}{\alpha} +x &= \left( x(n \Delta) + \frac{\vec \beta^T \vec u}{\alpha} \right) e^{\alpha (t - n \Delta)} - \frac{\vec \beta^T \vec u}{\alpha} \end{align*} \end{frame} \begin{frame}{Discretization: Q2} -Using the differential equation derived from question 1, create a discrete-time system to model the continuous time. In other words, if $x[n] = x(nT), \vec{u}[n] = \vec{u}(nT)$, find a relation such that +Using the differential equation derived from question 1, create a discrete-time system to model the continuous time. In other words, if $x[n] = x(n \Delta), \vec{u}[n] = \vec{u}(n \Delta)$, find a relation such that \[ x[n + 1] = A_d x[n] + B_d \vec{u}[n] \] \end{frame} \begin{frame}{Discretization: Q2 Sol} We can solve the previous solution for $x((n + 1)T)$ \begin{align*} -x((n + 1)T) &= \left( x(nT) + \frac{\vec \beta^T \vec u(nT)}{\alpha} \right) e^{\alpha ((n + 1)T - nT)} - \frac{\vec \beta^T \vec u(nT)}{\alpha} \\ -x[n + 1] &= e^{\alpha T} x[n] + \frac{e^{\alpha T} - 1}{\alpha} \vec{\beta}^T \vec{u}[n] \\ +x((n + 1) \Delta) &= \left( x(n \Delta) + \frac{\vec \beta^T \vec u(n \Delta)}{\alpha} \right) e^{\alpha ((n + 1)\Delta - n \Delta)} - \frac{\vec \beta^T \vec u(n \Delta)}{\alpha} \\ +x[n + 1] &= e^{\alpha \Delta} x[n] + \frac{e^{\alpha \Delta} - 1}{\alpha} \vec{\beta}^T \vec{u}[n] \\ \end{align*} -We see that $A_d = e^{\alpha T}, B_d = ((e^{\alpha T} - 1) / \alpha) \vec{\beta}^T$ +We see that $A_d = e^{\alpha \Delta}, B_d = ((e^{\alpha \Delta} - 1) / \alpha) \vec{\beta}^T$ \end{frame} \begin{frame}{Discretization: Q3} From b9d6b82b9118cde956ac9d3dbc1959e16e80ce20 Mon Sep 17 00:00:00 2001 From: geohh Date: Sun, 10 Nov 2019 14:51:37 -0800 Subject: [PATCH 6/6] updated notation convention in some places I missed last commit --- mt2.pdf | Bin 468746 -> 468734 bytes mt2.tex | 4 ++-- 2 files changed, 2 insertions(+), 2 deletions(-) diff --git a/mt2.pdf b/mt2.pdf index 3b292cb6ee1040dba80155483c76fd06c9bb91aa..bff406d157c924a210c67893d3995c3280452708 100644 GIT binary patch delta 31286 zcma&NQ*bW85-q%A?$}OtY}?tfZQC}!*wz=@wryj_wrwZ(oO@pG|Mu5-659{pWWNK&&>#<&~CTV}bj@A95eybuu zIpv_Q)g;Ix?N1OXY5d51CghMD{mZ`ol9unW^(7%0O;XM5UmtXp#NnttAday8PZr~+ zqY(shv?4vcX4S24><(S~9~PUbb$m;3S0z+K(a-Ia!%&^wPcFY!aACDQD!L}65WtJT zECq&pYd7kMG;g)L2RGe9#NGaGFbg~H9^_vgZbq$*<{aG9{f>uBXXBL~8qR#YI@5CwHf->b_6(K z^C;#Y?{A(rMu7lVghoC1D!|XR3E(_m*PyrbOT<;X=Eixt!3f)kEBQ@EBhy#wm}N%T z2llRd;_2H&j#ED7-k0LIvqEEsz~G9KvfA^lx|JO{G^o%?lHzS6WmnH)V80--YIzaS~ka6=3q66RuXo 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Given $x(nT)$ solve the differential equation +and we apply a constant input $\vec{u}_n$ for times $t \in [n \Delta, (n + 1) \Delta)$ for some $T > 0$. Given $x(n \Delta)$ solve the differential equation \end{frame} \begin{frame}{Discretization: Q1 Sol} @@ -937,7 +937,7 @@ \section{Discretization} At which point we can use our initial condition to get \begin{align*} x(nT) &= A - \frac{\vec \beta^T \vec u}{\alpha} \\ -A &= x(nT) + \frac{\vec \beta^T \vec u}{\alpha} \\ +A &= x(n \Delta) + \frac{\vec \beta^T \vec u}{\alpha} \\ x &= \left( x(n \Delta) + \frac{\vec \beta^T \vec u}{\alpha} \right) e^{\alpha (t - n \Delta)} - \frac{\vec \beta^T \vec u}{\alpha} \end{align*} \end{frame}

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