Missing figures of 03/06/2020

lectures/2020-06-03.tex

@@ -55,14 +55,34 @@ \section{Least square}
 \end{align*}
 
 This procedure has a drawback:
-\missingfigure{Form1}
+\begin{figure}[H]
+    \centering
+    \begin{tikzpicture}[node distance=2cm,auto,>=latex']
+        \node at (0,0.5) {$ \left\{ u(1), \ldots, u(N) \right\} $};
+        \node at (0,0)   {$ \left\{ y(1), \ldots, y(N) \right\} $};
+        \node at (6,0.5) {$ \left\{ u(1), \ldots, u(N), u(N+1) \right\} $};
+        \node at (6,0)   {$ \left\{ y(1), \ldots, y(N), y(N+1) \right\} $};
+        \node at (0,-0.8) {$\hat{\theta}_N$};
+        \node at (6,-0.8) {$\hat{\theta}_{N+1}$};
+
+        \draw[->] (1.8,0.25) -- (3.5,0.25) node[pos=0.5] {\footnotesize new data avail.};
+    \end{tikzpicture}
+\end{figure}
 
 $\hat{\theta}_N$ is not used to compute $\hat{\theta}_{N+1}$, it is necessary to repeat all the computation.
 The solution is \emph{Recursive Least Square}.
 
 \section{Recursive Least Square}
 
-\missingfigure{Fig1}
+\begin{figure}[H]
+    \centering
+    \begin{tikzpicture}[node distance=2cm,auto,>=latex']
+        \node[int, minimum height=1.5cm, minimum width=1.5cm] at (0,0) (RLS) {RLS};
+        \draw[<-,transform canvas={yshift=0.4cm}] (RLS) -- (-1.5,0) node[left] {$\hat{\theta}_{N-1}$};
+        \draw[<-,transform canvas={yshift=-0.4cm}] (RLS) -- (-1.5,0) node[left] {$\phi(N)$};
+        \draw[->] (RLS) -- (1.5,0) node[right] {$\hat{\theta}_{N}$};
+    \end{tikzpicture}
+\end{figure}
 
 \textbf{Advantages}
 \begin{itemize}
@@ -158,7 +178,18 @@ \subsection{Third form}
 \begin{remark}
     RLS is a rigorous version of LS (not an approximation), provided a correct initialization.
 
-    \missingfigure{Fig2}
+    \begin{figure}[H]
+        \centering
+        \begin{tikzpicture}[node distance=2cm,auto,>=latex']
+            \draw (0,0.2) -- (2.45,0.2) -- (2.45,0.7) -- (0,0.7) -- (0,0.2);
+            \node at (1.225, 0.45) {\footnotesize Initialization};
+            \draw (4.95, 0.2) -- (2.55, 0.2) -- (2.55, 0.7) -- (4.95, 0.7);
+            \node at (3.75, 0.45) {\footnotesize RLS};
+
+            \draw[->] (0,0) -- (5,0);
+            \draw (2.5,0.1) -- (2.5,-0.1) node[below] {$t_0$};
+        \end{tikzpicture}
+    \end{figure}
 
     \begin{itemize}
         \item Collect a first set of data, till $t_0$
@@ -172,7 +203,29 @@ \subsection{Third form}
 \section{Recursive Least Square with Forgetting Factor}
 
 Consider a time varying parameter
-\missingfigure{Fig3}
+\begin{figure}[H]
+    \centering
+    \begin{minipage}[t]{0.48\textwidth}
+        \centering
+        \begin{tikzpicture}[node distance=2cm,auto,>=latex']
+            \draw[->] (0,0) -- (5,0) node[below] {$t$};
+            \draw[->] (0,0) -- (0,3) node[left] {$\alpha(t)$};
+            \draw[domain=0:4.5,smooth,variable=\x] plot ({\x},{sin(\x*180/3.14*0.8+45)*0.8+1.2});
+            \draw[dashed, fill] (0,0.45) -- (4.5,0.45) circle (1pt);
+            \node[left] at (0,0.45) {$\hat{\alpha}_0$};
+        \end{tikzpicture}
+    \end{minipage}
+    \begin{minipage}[t]{0.48\textwidth}
+        \centering
+        \begin{tikzpicture}[node distance=2cm,auto,>=latex']
+            \draw[->] (0,0) -- (5,0) node[below] {$t$};
+            \draw[->] (0,0) -- (0,3) node[left] {$\alpha(t)$};
+            \draw[domain=0:4.5,smooth,variable=\x] plot ({\x},{sin(\x*180/3.14*0.8+45)*0.8+1.2});
+            \draw[dashed, fill] (0,1.05) -- (4.5,1.05);
+            \node[left] at (0,1.05) {$\hat{\alpha}_{LS}$};
+        \end{tikzpicture}
+    \end{minipage}
+\end{figure}
 
 $\hat{\alpha}_0$ is the correct estimation at time $N$, but it does not minimizes the objective function $J_N(\alpha)= \frac{1}{N} \sum_{t=1}^N \left( y(t) - \hat{y}(t|t-1, \alpha) \right)^2$ because it considers the entire time history of the system.
 
@@ -192,7 +245,26 @@ \section{Recursive Least Square with Forgetting Factor}
 \end{align*}
 
 \begin{remark}[Choice of $\rho$]
-    \missingfigure{Fig4}
+    \begin{figure}[H]
+        \centering
+        \begin{tikzpicture}[
+            node distance=2cm,auto,>=latex',
+            declare function={
+                f1(\x) = atan(\x*1.5-2)/180*3+1.5;
+                f2(\x) = f1(\x) + rand/4;
+                f3(\x) = f1(\x-0.6) + rand/30;
+            }]
+            \draw[->] (0,0) -- (5,0) node[below] {$t$};
+            \draw[->] (0,0) -- (0,3) node[left] {$\alpha(t)$};
+
+            \node[right, red] at (5,2.5) {$\rho \ll 1$};
+            \node[right, blue] at (5,2) {$\rho \approx 1$};
+
+            \draw[domain=0:4.5,smooth,variable=\x] plot ({\x},{f1(\x)});
+            \draw[domain=0:4.5,smooth,variable=\x,red] plot ({\x},{f2(\x)});
+            \draw[domain=0:4.5,smooth,variable=\x,blue] plot ({\x},{f3(\x)});
+        \end{tikzpicture}
+    \end{figure}
 
     If $\rho \ll 1$ there's high tracking speed but low precision. With $\rho \approx 1$ there's low tracking speed but greater precision.
 \end{remark}