up readme

Author: baggepinnen

README.md

@@ -105,21 +105,20 @@ julia> P02_named.x
 
 ### Internals: Transformation of non-proper models to proper statespace form
 For some models, ModelingToolkit will fail to produce a proper statespace model (a non-proper model is differentiating the inputs, i.e., it has a numerator degree higher than the denominator degree if represented as a transfer function) when calling `linearize`. For such models, given on the form
-$$
-\dot x = Ax + Bu + \bar B \dot u
-$$
+$$\dot x = Ax + Bu + \bar B \dot u$$
 we create the following augmented descriptor model
-$$
+```math
 \begin{aligned}
 sX &= Ax + BU + s\bar B U \\
-[X_u = U]\\
-s(X - \bar B X_u) = AX + BU \leftrightarrow \\
-s \begin{bmatrix}I & -\bar B \\ 0 & 0 \end{bmatrix} = 
+[X_u &= U]\\
+s(X - \bar B X_u) &= AX + BU \\
+s \begin{bmatrix}I & -\bar B \\ 0 & 0 \end{bmatrix} &= 
 \begin{bmatrix} A & 0 \\ 0 & -I\end{bmatrix}
 \begin{bmatrix}X \\ X_u \end{bmatrix} + 
-\begin{bmatrix} B \\ I_u\end{bmatrix} U
+\begin{bmatrix} B \\ I_u\end{bmatrix} U \\
+sE &= A_e x_e + B_e u
 \end{aligned}
-$$
+```
 where $X_u$ is a new algebraic state variable and $I_u$ is a selector matrix that picks out the differentiated inputs appearing in $\dot u$ (if all inputs appear, $I_u = I$).
 
 This model may be converted to a proper statespace model (if the system is indeed proper) using `DescriptorSystems.dss2ss`.