Reaction Cancellation Example
 Reaction Cancellation Example Examples Modeling In order to prevent spam, users must register before they can edit or create articles.

## 1 Introduction to the Reaction Cancellation Example

Figure 1: Reaction Cancellation double Mass

This is an example that builds on the Reaction Cancellation article. This example takes the state space model created for the reaction cancellation article and uses it to demonstrate the performance of a controller design.

## 2 State Space Model for Reaction Cancellation

The state space model from the reaction cancellation article is the following

Using the following abbreviations Mount = Mnt, Mirror = Mr, and ReactionMass = RM the full state space reaction cancellation model for a single axis is

 $LaTeX: \begin{bmatrix}\ddot{\theta}_{Mr} \\ \dot{\theta}_{Mr} \\ \ddot{\theta}_{Mnt} \\ \dot{\theta}_{Mnt} \\ \ddot{\theta}_{RM} \\ \dot{\theta}_{RM}\end{bmatrix}$$LaTeX: =\begin{bmatrix}-\frac{B_{Mr}}{I_{Mr}} & -\frac{k_{Mr}}{I_{Mr}} & 0 & 0 & 0 & 0 \\ 1 & 0 & 0 & 0 & 0 & 0 \\ \frac{B_{Mr}}{I_{Mnt}} & \frac{k_{Mr}}{I_{Mnt}} & 0 & 0 & \frac{B_{RM}}{I_{Mnt}} & \frac{k_{RM}}{I_{Mnt}} \\ 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & -\frac{B_{RM}}{I_{RM}} & -\frac{k_{RM}}{I_{RM}} \\ 0 & 0 & 0 & 0 & 1 & 0\end{bmatrix}$$LaTeX: \begin{bmatrix}\dot{\theta}_{Mr} \\ \theta_{Mr} \\ \dot{\theta}_{Mnt} \\ \theta_{Mnt} \\ \dot{\theta}_{RM} \\ \theta_{RM}\end{bmatrix}+$$LaTeX: \begin{bmatrix}\frac{1}{I_{Mr}} & 0 \\ 0 & 0 \\ 0 & 0 \\ 0 & 0 \\ 0 & \frac{1}{I_{RM}} \\ 0 & 0\end{bmatrix}$$LaTeX: \begin{bmatrix}\tau_{Mr} \\ \tau_{RM}\end{bmatrix}$

Simplifying for the reaction cancellation in the x-axis

 $LaTeX: \dot{x}_{x}=\begin{bmatrix}\ddot{\theta}_{Mr} \\ \dot{\theta}_{Mr} \\ \ddot{\theta}_{Mnt} \\ \dot{\theta}_{Mnt} \\ \ddot{\theta}_{RM} \\ \dot{\theta}_{RM}\end{bmatrix}$

 $LaTeX: A_{x}=\begin{bmatrix}-\frac{B_{Mr}}{I_{Mr}} & -\frac{k_{Mr}}{I_{Mr}} & 0 & 0 & 0 & 0 \\ 1 & 0 & 0 & 0 & 0 & 0 \\ \frac{B_{Mr}}{I_{Mnt}} & \frac{k_{Mr}}{I_{Mnt}} & 0 & 0 & \frac{B_{RM}}{I_{Mnt}} & \frac{k_{RM}}{I_{Mnt}} \\ 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & -\frac{B_{RM}}{I_{RM}} & -\frac{k_{RM}}{I_{RM}} \\ 0 & 0 & 0 & 0 & 1 & 0\end{bmatrix}$

 $LaTeX: B_{x}=\begin{bmatrix}\frac{1}{I_{Mr}} & 0 \\ 0 & 0 \\ 0 & 0 \\ 0 & 0 \\ 0 & \frac{1}{I_{RM}} \\ 0 & 0\end{bmatrix}$

 $LaTeX: u_{x}=\begin{bmatrix}\tau_{Mr} \\ \tau_{RM}\end{bmatrix}$

 $LaTeX: \dot{x}_{x}=A_{x}x_{x}+B_{x}u_{x}$

For 2 axes the reaction cancellation state space equations become

 $LaTeX: \begin{bmatrix}\dot{x}_{x} \\ \dot{x}_{y}\end{bmatrix}=\begin{bmatrix}A_{x} & 0 \\ 0 & A_{y}\end{bmatrix} \begin{bmatrix}x_{x} \\ x_{y}\end{bmatrix}+\begin{bmatrix}B_{x} \\ B_{y}\end{bmatrix} \begin{bmatrix}u_{x} \\ u_{y}\end{bmatrix}$

For a rigid body model the mounting plate imposed torque can be determined with the mounting plate inertia and angular acceleration therefore

 $LaTeX: C_{x}=\begin{bmatrix}0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0\end{bmatrix}$

This Cx provides the angular position of the mounting plate

 $LaTeX: y=\begin{bmatrix}C_{x} & 0 \\ 0 & C_{y}\end{bmatrix}\begin{bmatrix}x_{x} \\ x_{y}\end{bmatrix}+0\begin{bmatrix}\tau_{x} \\ \tau_{y}\end{bmatrix}$