Reaction Cancellation Example

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Reaction Cancellation Example
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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}



2.1 Rearranging the Reaction Cancellation State Space Inputs and Outputs

3 Reaction Cancellation Simulink Model

3.1 Parameter Definition

3.1.1 Perfectly Matched Mirror and Reaction Mass

3.1.2 Reality - the Mirror and Reaction mass are never perfectly matched

3.2 Simulink Model Performance

3.2.1 Perfectly Matched

3.2.2 Reality

4 MATLAB files

5 See Also