Tabs and spaces

2 files changed, 28 insertions, 28 deletions

diff --git a/tex/lti.tex b/tex/lti.tex
index b01a31f..316bef1 100644
--- a/tex/lti.tex
+++ b/tex/lti.tex
@@ -4,9 +4,9 @@
 Let \(\mathcal{S}\) denote a system.
 \begin{table}[H]
   \begin{tabularx}{\linewidth}{p{.3\linewidth} X}
-		\toprule
-		\bfseries Property & \bfseries Meaning \\
-		\midrule
+    \toprule
+    \bfseries Property & \bfseries Meaning \\
+    \midrule
     static \(\leftrightarrow\)\newline dynamic & Static means that it is memoryless (in the statistical sense), whereas dynamic has memory. Static systems depend only on the input \(u\), dynamic systems on \(du/dt\) or \(\int u\,dt\). \\
     causal \(\leftrightarrow\)\newline acausal & Causal systems use only informations from the past, i.e. \(h(t < 0) = 0\). Real systems are always causal. \\
     linear \(\leftrightarrow\)\newline nonlinear & The output of a linear system does not have new frequency that were not in the input. For linear system the superposition principle is valid: \(\mathcal{S}(\alpha_1 x_1 + \alpha_2 x_2) = \alpha_1 \mathcal{S} x_1 + \alpha_2 \mathcal{S} x_2\). \\
@@ -14,8 +14,8 @@ Let \(\mathcal{S}\) denote a system.
     \midrule
     SISO, MIMO & Single input single output, multiple input multiple output. \\
     BIBO & Bounded input bounded output, i.e. there are some \(A\), \(B\) such that \(|x| < A\) and \(|y| < B\) for all \(t\), equivalently \(\int_\mathbb{R} |h|\,dt < \infty\).\\
-		\bottomrule
-	\end{tabularx}
+    \bottomrule
+  \end{tabularx}
 \end{table}
 
 \subsection{Impulse response}
diff --git a/tex/state-space.tex b/tex/state-space.tex
index e0f7960..fadc6dd 100644
--- a/tex/state-space.tex
+++ b/tex/state-space.tex
@@ -1,36 +1,36 @@
 \section{State space representation}
 
 \begin{figure}
-	\centering
-	\resizebox{\linewidth}{!}{
-		\input{tex/tikz/mimo}
-	}
-	\caption{A LTI MIMO system.}
+  \centering
+  \resizebox{\linewidth}{!}{
+    \input{tex/tikz/mimo}
+  }
+  \caption{A LTI MIMO system.}
 \end{figure}
 
 A system described by a system of linear differential equations of \(n\)-th order, can be equivalently be described by \(n\) first order differential equations. Which can be compactly written in matrix form as
 \begin{align*}
-	\dot{\vec{x}} &= \mx{A}\vec{x} + \mx{B}\vec{u} \\
-	\vec{y} &= \mx{C}\vec{x} + \mx{D}\vec{u}.
+  \dot{\vec{x}} &= \mx{A}\vec{x} + \mx{B}\vec{u} \\
+  \vec{y} &= \mx{C}\vec{x} + \mx{D}\vec{u}.
 \end{align*}
 If the system is time \emph{variant} the matrices are functions of time.
 
 \begin{table}
-	\begin{tabular}{ >{\(}c<{\)} >{\(}c<{\)} l }
-		\toprule
-		\text{\bfseries Symbol} & \text{\bfseries Size} & \bfseries Name \\
+  \begin{tabular}{ >{\(}c<{\)} >{\(}c<{\)} l }
+    \toprule
+    \text{\bfseries Symbol} & \text{\bfseries Size} & \bfseries Name \\
     \midrule
     \vec{x} & n & State vector \\
     \vec{u} & m & Output vector \\
     \vec{y} & k & Output vector \\
-		\midrule
-		\mx{A} & n\times n & System matrix \\
-		\mx{B} & m\times n & Input matrix \\
-		\mx{C} & n\times k & Output matrix \\
-		\mx{D} & k\times m & Feed forward matrix \\
-		\bottomrule
-	\end{tabular}
-	\caption{Matrices for a state space representation}
+    \midrule
+    \mx{A} & n\times n & System matrix \\
+    \mx{B} & m\times n & Input matrix \\
+    \mx{C} & n\times k & Output matrix \\
+    \mx{D} & k\times m & Feed forward matrix \\
+    \bottomrule
+  \end{tabular}
+  \caption{Matrices for a state space representation}
 \end{table}
 
 \subsection{Canonical representations}
@@ -45,16 +45,16 @@ The Jordan form diagonalizes the \(\mx{A}\) matrix. Thus we need to solve the ei
 
 The transformation to the eigenbasis \(\mx{T}\), obtained by using the eigenvector as columns of a matrix \(\mx{T} = \begin{bmatrix} \vec{v}_1 & \cdots & \vec{v}_n \end{bmatrix}\), is then used to compute
 \begin{align*}
-	\mx{\hat{A}} & = \mx{T}\mx{A}\mx{T^{-1}} & 
-	\mx{\hat{B}} & = \mx{T}\mx{B} \\
-	\mx{\hat{C}} & = \mx{C}\mx{T^{-1}} & 
-	\mx{\hat{D}} & = \mx{D}.
+  \mx{\hat{A}} & = \mx{T}\mx{A}\mx{T^{-1}} & 
+  \mx{\hat{B}} & = \mx{T}\mx{B} \\
+  \mx{\hat{C}} & = \mx{C}\mx{T^{-1}} & 
+  \mx{\hat{D}} & = \mx{D}.
 \end{align*}
 
 In this form the system is described with \(n\) decoupled states \(\xi_i\) with the equations \(\dot{\vec{\xi}} = \mx{\hat{A}}\vec{\xi} + \mx{\hat{B}}\vec{u}\) and \(\vec{y} = \mx{\hat{C}}\vec{\xi} + \mx{\hat{D}} \vec{u}\).
 
 \subsection{Stability}
-If \emph{all} eigenvalues \(\lambda\) are not zero and have a positive real part the system is asymptotically \emph{stable}. If \emph{all} eigenvalues are not zero but \emph{at least one} has a negative real part the system is \emph{unstable}. If even one eigenvalue is zero, no conclusion can be drawn.
+If \emph{all} eigenvalues \(\lambda\) are not zero and have a positive real part the system is asymptotically \emph{stable}. If \emph{all} eigenvalues are not zero but \emph{at least one} has a negative real part the system is \emph{unstable}.
 
 \subsection{Controllability}
 The state controllability condition implies that it is possible --- by admissible inputs --- to steer the states from any initial value to any final value within some finite time window. A LTI state space model is controllable iff the matrix