Merge pull request #2 from CosimoRusso/fixes-25-05-2020

Fix typos, add some additional information

lectures/2020-05-25.tex

@@ -1,5 +1,5 @@
 \chapter{Minimum variance control}
-\newlecture{Sergio Savaresi}{19/05/2020}
+\newlecture{Sergio Savaresi}{25/05/2020}
 
 Design and analysis of feedback systems.
 It's not system identification nor software sensing.
@@ -56,7 +56,7 @@ \chapter{Minimum variance control}
     Intuitively it's very difficult to control non-minimum phase systems.
     You can take the wrong decision if you react immediately.
 
-    Also for human it's difficult, for example \emph{steer to roll} dynamics in a bicycle.
+    Also for human it's difficult, for example \emph{steer to roll} dynamics in a bicycle: if you want to steer left, you must first steer a little to the right and then turn left.
 
     Design of controller for non-minimum phase is difficult and requires special design techniques (no MVC but generalized MVC).
 \end{remark}
@@ -85,7 +85,7 @@ \chapter{Minimum variance control}
 \[
     J = E\left[ (y(t) - y^0(t))^2 \right]
 \]
-It's the variance of the tracking error, because of the it's called Minimum Variance Control.
+It's the variance of the tracking error, that's why it's called Minimum Variance Control.
 
 Some additional (small) technical assumptions:
 \begin{itemize}
@@ -233,7 +233,7 @@ \chapter{Minimum variance control}
 \[
     S: y(t) = \frac{b_1+b_1z^{-1}}{1-az^{-1}}u(t-1) + \frac{1}{1-az^{-1}}e(t)
 \]
-Note that this is an $ARMAX(1,0,1+1)$.
+Note that this is an $ARMAX(1,0,1+1)=ARX(1,2)$.
 \[
     k=1 \qquad B(z) = b_0+b_1z^{-1} \qquad A(z)=1-az^{-1} \qquad C(z) = 1
 \]
@@ -249,7 +249,7 @@ \chapter{Minimum variance control}
 
 Now we can impose that $\hat{y}(t|t-1)=y^0(t)$
 \[
-    b_0u(t) + b_1u(t-1) + ay(t) = y^0(t) \qquad u(t) = \left( y^0(t+1) - ay(t) \right)\frac{1}{b_0+b_1z^{-1}}
+    b_0u(t) + b_1u(t-1) + ay(t) = y^0(t+1) \qquad u(t) = \left( y^0(t+1) - ay(t) \right)\frac{1}{b_0+b_1z^{-1}}
 \]
 But we don't have $y^0(t+1)$, so we use $y^0(t)$.
 \[
@@ -260,7 +260,7 @@ \chapter{Minimum variance control}
     \begin{tikzpicture}[node distance=2cm,auto,>=latex']
         \node[sum] at (0,0) (sum) {};
         \node[int] at (1.5,0) (b1) {$\frac{1}{b_0+b_1z^{-1}}$};
-        \node[int] at (5,0) (b2) {$z^{-1}\frac{b_0+b_1z^{-1}}{1-az^[-1]}$};
+        \node[int] at (5,0) (b2) {$z^{-1}\frac{b_0+b_1z^{-1}}{1-az^{-1}}$};
         \node[int] at (3,-1.5) (b3) {$a$};
         \node[int] at (7,1.5) (b4) {$\frac{1}{1-az^{-1}}$};
         \node[sum] at (7,0) (sum2) {};