From 8b39a0de2e5ba235a204e96c4d2b8feb7f665b53 Mon Sep 17 00:00:00 2001 From: Cosimo Russo Date: Thu, 28 May 2020 03:09:44 +0200 Subject: [PATCH] Minor fixes --- lectures/2020-05-27.tex | 6 +++--- 1 file changed, 3 insertions(+), 3 deletions(-) diff --git a/lectures/2020-05-27.tex b/lectures/2020-05-27.tex index a019ca8..0b47b1f 100644 --- a/lectures/2020-05-27.tex +++ b/lectures/2020-05-27.tex @@ -21,7 +21,7 @@ \begin{itemize} \item compute the \emph{loop-function} $L(z) = F_1(z) F_2(z)$ \item Build the \emph{characteristic polynomial} $\chi(z) = L_N(z) + L_D(z)$ (sum of numerator and denominator) - \item Find the roots of $\chi(z)$, closed loop system is asymptotically stable iif all the roots of $\chi(z)$ are strictly inside the unit circle + \item Find the roots of $\chi(z)$, closed loop system is asymptotically stable iff all the roots of $\chi(z)$ are strictly inside the unit circle \end{itemize} \end{remark} @@ -47,7 +47,7 @@ Since the system is L.T.I. we can use the superposition principle: \[ - y(t) = F_{y^0y}(z)y^0(t) + F_{ey}(z)e(t) + y(t) = \underbrace{F_{y^0y}(z)y^0(t)}_\text{TF from $y^0$ to $y$} + \underbrace{F_{ey}(z)e(t)}_\text{TF from $e$ to $y$} \] Let's compute these transfer functions: @@ -149,7 +149,7 @@ \subsection{Stability check} \chi(z) = z^{-k}B(z)\tilde{R}(z) + B(z)E(z)A(z) = B(z) \left( z^{-k}\tilde{R}(z)+E(z)A(z) \right) = B(z)C(z) \end{align*} -The system is asymptotically stable iif: +The system is asymptotically stable iff: \begin{itemize} \item All roots of $B(z)$ are stable (minimum phase assumption) \item All roots of $C(z)$ are stable (canonical representation)