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:
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| body | \dot{x} = A\vec{x} + Bu(t) |
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|
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| body | y = C \vec{x} + Du(t) |
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Where
,
,
, and
are matrices.
Example:
Say we have a system that can be described with
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| body | \ddot{\theta} = k\dot{\theta} + n\theta + u(t) |
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|
and
To model this, let
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| body | x = \left[_{\dot{\theta}}^{\theta}\right] |
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.
We need to find matrices
and
such that
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| body | \left[_\ddot{\theta}^\dot{\theta}\right] = A\left[_\dot{\theta}^\theta\right] + Bu(t) |
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|
.
We find that
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| body | A = \begin{bmatrix} 0 & 1 \\ 0 & 0 \end{bmatrix} |
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.