removed warnings, formatted matrices

src/sections/pole_placement.tex

@@ -111,4 +111,4 @@ \subsubsection{MIMO}
 \begin{itemize}
     \item If the system is controllable from any one of the inputs, pole placement works even though we might not use the full potential of all the inputs
     \item In addition to the poles, the closed-loop eigenvectors (modal shapes) can be placed
-\end{itemize}
+\end{itemize}

src/sections/stab_and_perf_robustness.tex

@@ -25,62 +25,64 @@ \subsubsection{The Small Gain Theorem (SGT)}
     \item and $\left\|\Delta\right\|_{\mathcal{H}_\infty}<1$
 \end{itemize}
 the FB interconnection of $\mathbf{G}$ and $\Delta$ is \textbf{stable iff}
+
 \begin{equation*}
-    \left\|\mathbf{G}\right\|_{\mathcal{H}_\infty}<1
+    {\left\|\mathbf{G}\right\|}_{\mathcal{H}_{\infty}}<1
 \end{equation*}
-Otherways, one can find a $\Delta$ so that $(\mathbf{I}-\mathbf{G\Delta})$ singular.\\
+Otherwise, one can find a $\Delta$ so that $(\mathbf{I}-\mathbf{G}\Delta)$ singular.
 
 \subsubsection{Modelling Uncertainty}
 \begin{center}
     \includegraphics[width = 0.5\linewidth]{uncertainty.png}
 \end{center}
 To take uncertainty into account, one first defines an \textbf{uncertainty model}, consisting of
 \begin{enumerate}
-    \item A nominal model $P$
+    \item A nominal model $\mathbf{P}$
     \item A set of models that is guaranteed to contain the system uncertainty, and is easier to handle
 \end{enumerate}
-and then designs a control system that meets the stability and performance specifications not only for $P$, but also for all other possible models in the uncertainty model.
+and then designs a control system that meets the stability and performance specifications for all possible models in the \textit{uncertainty model}.
 
+\newpar{}
 \ptitle{Assumptions on Uncertainty}
 
 In the following uncertainty models one assumes that $\Delta$
 \begin{itemize}
     \item is minimum-phase
-    \item does not cancle unstable poles of the nominal system
+    \item does not cancel unstable poles of the nominal system
     \item $|\Delta(j\omega)|<1,\quad\forall\omega$
 \end{itemize}
 and that there is a high-pass transfer function $W$ called \textbf{frequency weight} so that the total uncertainty is given by
 \begin{equation*}
-    W(s)\Delta(s)
+    \mathbf{W}(s)\Delta(s)
 \end{equation*}
 
 \paragraph{Common Uncertainty Models}
 \ptitle{Additive Uncertainty}
 \begin{equation*}
-    \tilde{P}(s)=P(s)+W(s)\Delta(s)
+    \tilde{\mathbf{P}}(s)=\mathbf{P}(s)+\mathbf{W}(s)\Delta(s)
 \end{equation*}
 \begin{center}
     \includegraphics[width = 0.75\linewidth]{additive_uncertainty.png}
 \end{center}
 
 \ptitle{Multiplicative Uncertainty}
 \begin{equation*}
-    \tilde{P}(s)=P(s)(1+W(s)\Delta(s))
+    \tilde{\mathbf{P}}(s)=\mathbf{P}(s)(1+\mathbf{W}(s)\Delta(s))
 \end{equation*}
 \begin{center}
     \includegraphics[width = 0.75\linewidth]{multiplicative_uncertainty.png}
 \end{center}
 \ptitle{Feedback Uncertainty}
 \begin{equation*}
-    \tilde{P}(s)=(I-P(s)W(s)\Delta(s))^{-1}P(s)
+    \tilde{\mathbf{P}}(s)={(\mathbf{I}-\mathbf{P}(s)\mathbf{W}(s)\Delta(s))}^{-1}\mathbf{P}(s)
 \end{equation*}
 \begin{center}
     \includegraphics[width = 0.75\linewidth]{fb_uncertainty.png}
 \end{center}
 
 \subsubsection{Robust Stability}
 \begin{itemize}
-    \item Equivalent to SISO bode plot obstacles (now obstacles are given by singular values)
+    \item Equivalent to SISO bode plot obstacles (now obstacles are given by singular values and in time domain)
     \item Now we impose constraints on the $\mathcal{H}_{\infty}$ norm
     \item \textbf{Caution}: depending on the convention the block diagrams vary and some matrices in a specific $\mathbf{G}$ will be permuted to the forms shown below.
 \end{itemize}
@@ -140,9 +142,9 @@ \subsubsection{Robust Performance}
     \item Therefore we want to limit the gain from $\xi$ to $y$
 \end{itemize}
 
-Applying the SGT (assuming $\Delta=G_{y\xi}$) yields
+Applying the SGT (assuming $\boldsymbol{\Delta}=\mathbf{G}_{y\xi}$) yields
 \begin{equation*}
-    \|G_{y\xi}\|_{\mathcal{H}_\infty}=\|(I+PC)^{-1}W\|_{\mathcal{H}_\infty}<1
+    \|\mathbf{G}_{y\xi}\|_{\mathcal{H}_\infty}=\|{(\mathbf{I}+\mathbf{PC})}^{-1}\mathbf{W}\|_{\mathcal{H}_\infty}<1
 \end{equation*}
 
 \subsubsection{Combined Robustness}
@@ -161,37 +163,42 @@ \subsubsection{Combined Robustness}
         \end{bmatrix}}_{\boldsymbol{\Delta}}
     \begin{bmatrix}z_1 \\
         z_2
-    \end{bmatrix}
+    \end{bmatrix}\qquad
+    \begin{matrix}
+        \mathrm{stability} \\
+        \mathrm{robustness}
+    \end{matrix}
 \end{equation*}
 
-\paragraph{Applying the SGT}
+\paragraph{Robust Disturbance Rejection}
 
 The transfer function matrix for the SGT is given by
 \begin{align*}
     \mathbf{M} & =
     \begin{bmatrix}
         G_{{z_{1}w_{1}}} & G_{{z_{1}w_{2}}} \\
         G_{{z_{2}w_{1}}} & G_{{z_{2}w_{2}}}
-    \end{bmatrix}                     \\
+    \end{bmatrix}                                                                                                                                                                                                                  \\
                & =\begin{bmatrix}
-                      -W_1PK(I+P_0K)^{-1} & -W_1PK(I+P_0K)^{-1} \\
-                      W_2(I+P_0K)^{-1}    & W_2(I+P_0K)^{-1}
+                      -\mathbf{W}_1\mathbf{P}_0\mathbf{K}{(\mathbf{I}+\mathbf{P}_0\mathbf{K})}^{-1} & -\mathbf{W}_1 \mathbf{P}_0\mathbf{K}{(\mathbf{I}+\mathbf{P}_0\mathbf{K})}^{-1} \\
+                      \mathbf{W}_2{(\mathbf{I}+\mathbf{P}_0\mathbf{K})}^{-1}                        & \mathbf{W}_2{(\mathbf{I}+\mathbf{P}_0\mathbf{K})}^{-1}
                   \end{bmatrix}
 \end{align*}
-%%% TBD %%%
-% Why are the delta blocks of the "other" BSB parts not in the TF matrix?
-%%% TBD %%%
 Therefore, we impose
 \begin{equation*}
     \left\|\mathbf{M}\right\|_{\mathcal{H}_\infty}<1
 \end{equation*}
-However, this is a conservative assumption as the SGT could be applied to an arbitrary $\boldsymbol{\Delta}$. Therefore, we adjust the condition to more specific uncertainty matrices.
+to achieve robust \textbf{disturbance rejection}.
+
 
-\paragraph{Applying the Structured Singular Value (SSV)}
+\subsubsection{Structured Singular Value (SSV)}
+The condition from the \textit{unstructured SGT} is a conservative assumption as the SGT could be applied to an arbitrary $\boldsymbol{\Delta}$.
+If $\boldsymbol{\Delta}$ has \textbf{block-diagonal structure}, less conservative robustness conditions can be applied.
 
+\newpar{}
 \ptitle{Definition of SSV}
 
-The SSV is defined with respect to a \textbf{class of perturbations} $\mathrm{D}$ as
+The SSV is defined with respect to a \textbf{class of perturbations} $\mathbb{D}$ as
 \begin{equation*}
     \mu(\mathbf{M}):=\frac1{\inf\{\sigma_{\max}(\boldsymbol{\Delta}):\det(1-M\boldsymbol{\Delta})=0\}},\quad\boldsymbol{\Delta}\in\mathbb{D}
 \end{equation*}
@@ -205,16 +212,16 @@ \subsubsection{Combined Robustness}
     \item $\mu(\mathbf{M})\geq0$
     \item If $\mathbb{D}$ is arbitrary: $\mu(\mathbf{M})=\|\mathbf{M}\|_{\mathcal{H}_\infty}$ (unstructured case)
     \item If $\mathbb{D}=\{\lambda I:\lambda\in\mathbb{C}\}$: $\mu(\mathbf{M})=\rho(\mathbf{M})$ (spectral radius, largest eigenvalue)
-    \item If $\mathbb{D}$ diagonal (complex): $\mu(\mathbf{M})=\mu(\mathbf{D}^{-1}\mathbf{MD})$ for any invertible $\mathbf{D}$ and
-    \item \begin{equation*}
+    \item If $\mathbb{D}$ is the set of diagonal (complex) matrices: $\mu(\mathbf{M})=\mu(\mathbf{D}^{-1}\mathbf{MD})$ for any invertible $\mathbf{D}$ and
+          \begin{equation*}
               \rho(\mathbf{M})\leq\mu(\mathbf{M})\leq\inf_{\mathbf{D}}\sigma_{\max}(\mathbf{D}^{-1}\mathbf{MD})\leq\sigma_{\max}(\mathbf{M})
           \end{equation*}
 \end{itemize}
 
 
 \ptitle{Stability Condition}
 
-The $\mathbf{M}-\boldsymbol{\Delta}$ FB system is stable for all $\boldsymbol{\Delta}\in D$, $\left\|\boldsymbol{\Delta}\right\|_{\mathcal{H}_\infty}<1$ iff
+The $\mathbf{M}-\boldsymbol{\Delta}$ FB system is stable for all $\boldsymbol{\Delta}\in \mathbb{D}$, $\left\|\boldsymbol{\Delta}\right\|_{\mathcal{H}_\infty}<1$ iff
 \begin{equation*}
     \mu(\mathbf{M}(j\omega))\leq1,\quad\forall\omega\in\mathbb{R}
 \end{equation*}
@@ -223,8 +230,8 @@ \subsubsection{Combined Robustness}
 
 \begin{equation*}
     \mathbf{M}=\begin{bmatrix}
-        -\frac{W_1P_0K}{1+P_0K} \\
-        \frac{W_2}{1+P_0K}
+        -\frac{\mathbf{W}_1\mathbf{P}_0\mathbf{K}}{1+\mathbf{P}_0\mathbf{K}} \\
+        \frac{\mathbf{W}_2}{1+\mathbf{P}_0\mathbf{K}}
     \end{bmatrix}
     \begin{bmatrix}
         1 & 1
@@ -233,32 +240,32 @@ \subsubsection{Combined Robustness}
 yields
 
 \begin{align*}
-    \mu(\mathbf{M}(j\omega)) & =\left|\frac{W_1PK}{1+P_0K}(j\omega)\right|+\left|\frac{W_2}{1+P_0K}(j\omega)\right|\leq1 \\
-                             & =|W_1L(j\omega)|+|W_2(j\omega)|\leq|1+L(j\omega)|\quad\forall\omega\in\mathrm{R}
+    \mu(\mathbf{M}(j\omega)) & =\left|\frac{\mathbf{W}_1\mathbf{P}_0\mathbf{K}}{1+\mathbf{P}_0\mathbf{K}}(j\omega)\right|+\left|\frac{\mathbf{W}_2}{1+\mathbf{P}_0\mathbf{K}}(j\omega)\right|\leq1 \\
+                             & =|\mathbf{W}_1 \mathbf{L}(j\omega)|+|\mathbf{W}_2(j\omega)|\leq|1+\mathbf{L}(j\omega)|\quad\forall\omega\in\mathrm{R}
 \end{align*}
 where
 \begin{itemize}
-    \item $L=P_0K$
-    \item $W_1$ is a high-pass
-    \item $W_2$ is a low-pass
-    \item $|L|$ is a low-pass (low-pass assumption for physical systems)
+    \item $\mathbf{L}=\mathbf{P}_0 \mathbf{K}$
+    \item $\mathbf{W}_1$ is a high-pass
+    \item $\mathbf{W}_2$ is a low-pass
+    \item $|\mathbf{L}|$ is a low-pass (low-pass assumption for physical systems)
 \end{itemize}
 Therefore, one has for low frequencies
 \begin{equation*}
-    |W_1(j\omega)|+\frac{|W_2(j\omega)|}{|L(j\omega)|}\leq1
+    |\mathbf{W}_1(j\omega)|+\frac{|\mathbf{W}_2(j\omega)|}{|\mathbf{L}(j\omega)|}\leq1
 \end{equation*}
 \begin{equation*}
-    |L(j\omega)|\geq\frac{|W_2(j\omega)|}{1-|W_1(j\omega)|}\approx|W_2(j\omega)|
+    |\mathbf{L}(j\omega)|\geq\frac{|\mathbf{W}_2(j\omega)|}{1-|\mathbf{W}_1(j\omega)|}\approx|\mathbf{W}_2(j\omega)|
 \end{equation*}
 and for high frequencies
 \begin{equation*}
-    |W_1(j\omega)||L(j\omega)|+|W_2(j\omega)|\leq1
+    |\mathbf{W}_1(j\omega)||\mathbf{L}(j\omega)|+|\mathbf{W}_2(j\omega)|\leq1
 \end{equation*}
 \begin{equation*}
-    |L(j\omega)|\leq\frac{1-|W_2(j\omega)|}{|W_1(j\omega)|}\approx\frac1{|W_1(j\omega)|}
+    |\mathbf{L}(j\omega)|\leq\frac{1-|\mathbf{W}_2(j\omega)|}{|\mathbf{W}_1(j\omega)|}\approx\frac1{|\mathbf{W}_1(j\omega)|}
 \end{equation*}
 One can look at the condition with an advanced Nyquist condition.
 \begin{center}
-    \includegraphics[width = 0.5\linewidth]{chad_nyquist.png}
+    \includegraphics[width = 0.7\linewidth]{chad_nyquist.png}
 \end{center}