Controllability Decomposition 

The controllability decomposition reveals the inner structure and the reachable subspace of the system. The subspace can be visualized for second- and third-order systems and in general is a manifold.

An affine system. »

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Its reachable subspace is a sphere.

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The transformation reveals that it is confined to a specific sphere because .

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The system starts and remains on the same sphere for all inputs.

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