Kalman Filter 09/28

After all-hands we had a Kalman Filter meeting where we learned about state-space modelling and discussed preliminary goals for Kalman Filtering.

STATE-SPACE MODELLING

State-space modelling is a way of representing a physical system (often with high order diff eqs) by way of max first-order differential equations like so:

$//<![CDATA[ \begin{array}{l}\dot{x} = A\vec{x} + Bu(t)\end{array} //]]>$

$//<![CDATA[ \begin{array}{l}y = C \vec{x} + Du(t)\end{array} //]]>$

Where  $//<![CDATA[ \begin{array}{l}A\end{array} //]]>$,  $//<![CDATA[ \begin{array}{l}B\end{array} //]]>$,  $//<![CDATA[ \begin{array}{l}C\end{array} //]]>$, and  $//<![CDATA[ \begin{array}{l}D\end{array} //]]>$ are matrices.

Example:

Say we have a system that can be described with

$//<![CDATA[ \begin{array}{l}\ddot{\theta} = k\dot{\theta} + n\theta + u(t)\end{array} //]]>$

and

$//<![CDATA[ \begin{array}{l}y(t) = \theta\end{array} //]]>$

To model this, let 

$//<![CDATA[ \begin{array}{l}x = \left[_{\dot{\theta}}^{\theta}\right]\end{array} //]]>$.

We need to find matrices  $//<![CDATA[ \begin{array}{l}A\end{array} //]]>$ and  $//<![CDATA[ \begin{array}{l}B\end{array} //]]>$ such that

$//<![CDATA[ \begin{array}{l}\left[_\ddot{\theta}^\dot{\theta}\right] = A\left[_\dot{\theta}^\theta\right] + Bu(t)\end{array} //]]>$.

We find that 

$//<![CDATA[ \begin{array}{l}A = \begin{bmatrix} 0 & 1 \\ 0 & 0 \end{bmatrix}\end{array} //]]>$.