Typo

lectures/2020-06-03.tex

@@ -33,7 +33,7 @@ \chapter{Recursive Identification}
 \section{Least square}
 
 \[
-    \hat{\theta_N} = \argmin_\theta \left\{ J_N(\theta) = \frac{1}{N} \sum_{t=1}^N \left( y(t) - \hat{y}(t|t-1, \theta) \right)^2 \right\}
+    \hat{\theta}_N = \argmin_\theta \left\{ J_N(\theta) = \frac{1}{N} \sum_{t=1}^N \left( y(t) - \hat{y}(t|t-1, \theta) \right)^2 \right\}
 \]
 
 We need to find the model predictor $\hat{y}(t|t-1, \theta)$.
@@ -100,7 +100,7 @@ \subsection{First form}
     \hat{\theta}_{N-1} &= S(N-1)^{-1} \sum_{t=1}^{N-1} \phi(t)y(t) \\
     \sum_{t=1}^{N-1} \phi(t)y(t) &= S(N-1)\hat{\theta}_{N-1} \\
     \sum_{t=1}^{N} \phi(t)y(t) &= \sum_{t=1}^{N-1} \phi(t)y(t) + \phi(N)y(N) \\
-    \sum_{t=1}^{N} \phi(t)y(t) &= S(N-1)\hat{\theta}_{N-1} + \phi(N)y(N) = \text{ equation } (1.2) \\
+    \sum_{t=1}^{N} \phi(t)y(t) &= S(N-1)\hat{\theta}_{N-1} + \phi(N)y(N) = \text{ equation } (7.2) \\
     S(N) \hat{\theta}_N &= S(N-1) \hat{\theta}_{N-1} + \phi(N)y(N)
 \end{align}
 
@@ -229,7 +229,7 @@ \section{Recursive Least Square with Forgetting Factor}
 
 $\hat{\alpha}_0$ 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.
 
-In order to identify a time varying parameter the RLS must be force to forget old data.
+In order to identify a time varying parameter the RLS must be forced to forget old data.
 The solution is provided by the minimization of $J_N$:
 \[
     J_N(\theta) = \frac{1}{N} \sum_{t=1}^N \rho^{N-t}\left( y(t) - \hat{y}(t|t-1,\theta) \right)^2

lectures/2020-06-04.tex

@@ -203,8 +203,8 @@ \subsection*{D to A converter}
 Another simple discretization technique frequently used is the discretization of time-derivative $\dot{x}$.
 
 \begin{align*}
-    \text{\textbf{eulero backward}} &\qquad \dot{x} \approx \frac{x(t)-x(t-1)}{\Delta T} = \frac{x(t)-z^{-1}x(t)}{\Delta T} = \frac{z-1}{z\Delta T} x(t) \\
-    \text{\textbf{eulero forward}} &\qquad \dot{x} \approx \frac{x(t+1)-x(t)}{\Delta T} = \frac{zx(t)-x(t)}{\Delta T} = \frac{z-1}{\Delta T} x(t)
+    \text{\textbf{Eulero backward}} &\qquad \dot{x} \approx \frac{x(t)-x(t-1)}{\Delta T} = \frac{x(t)-z^{-1}x(t)}{\Delta T} = \frac{z-1}{z\Delta T} x(t) \\
+    \text{\textbf{Eulero forward}} &\qquad \dot{x} \approx \frac{x(t+1)-x(t)}{\Delta T} = \frac{zx(t)-x(t)}{\Delta T} = \frac{z-1}{\Delta T} x(t)
 \end{align*}
 
 General formula
@@ -344,9 +344,9 @@ \subsection*{D to A converter}
 
         \draw[dotted] (1.5,1.875) -- (1.5,0) node[below] {};
         \node[below] at (0.75,0) {\footnotesize B.W. of interest};
-        \draw (4,0.1) -- (4,0) node[below] {$\scriptstyle f_S$};
+        \draw (4,0.1) -- (4,0) node[below] {$\scriptstyle \omega_S$};
         \draw (1.5,0) edge[bend left=20,->] (4,0);
-        \node at (2.75,0.45) {\footnotesize x10};
+        \node at (2.75,0.45) {$\scriptstyle \times 10$};
     \end{tikzpicture}
 \end{figure}