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2.2.3 Testing the Controllability
The controllability of the system is investigated by means of the controller-Hessenberg form, which is constructed in a numerically robust way and provides information on how close a system is to an uncontrollable state (see Chapter 4).
This is the controller-Hessenberg form of the system.
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The system is controllable with the tolerance  .
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The system, however, is not controllable if a greater tolerance must be attained.
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In general, a system is uncontrollable with tolerance tol if there exists a perturbation with a norm less than or equal to tol that makes the perturbed system uncontrollable. Larger feedback gains are necessary to stabilize a controllable system that is close to an uncontrollable state. In the example, the system becomes uncontrollable for relatively small tolerances somewhere between  and  .
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