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System Responses, Stability, and PolesThe Design of the LQR Controller

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.

In[10]:=

Out[10]//NumberForm=

The system is controllable with the tolerance .

In[11]:=

Out[11]=

The system, however, is not controllable if a greater tolerance must be attained.

In[12]:=

Out[12]=

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 .

System Responses, Stability, and PolesThe Design of the LQR Controller