Especially for nonlinear systems.
It is known that for subactuated systems many times they have states that can not be controlled but only stabilized.
Dimel,
Your question is vague. What do you call subactuted systems? Let us assume that subactuated means uncontrollable, in the sense that the system is decomposable in a part that can be controlled and another that is not affected by control. If the uncontrolled part is stable, although it is impossible to change the dynamics from the control signal(s), the system is said to be stabilizable. I don't know if that is the "relation" you are looking for.
Thanks Luis,
My intention was to know if for a underactuated system , when calculating the controllability matrix I should expect (maybe always) to obtain a not full rank controllability matrix?
Underactuated (usually) means there are fewer independent control effectors than degrees of freedom (dof). A quadrotor, for example, is underactuated in that it has 4 independent control effectors but 6 dof. However, it is fully controllable (in the Kalman sense) in that the controlability matrix is full rank. Control of the non-actuated dof (horizontal position) is obtained via the coupling between the attitude and the lateral velocities.
Stabilizable generally refers to the system rather than to states. Take care in thinking of "states" as uncontrollable, it is better to think in terms of the system modes (at least for linear systems). Strictly speaking, the state of the system is the values of the state variable vector, x. The state variables are, of course, not unique, so the concept of controllablility of a state variable is unclear. Modes are unique.
Note that controlability for non-linear systems is a bit more involved than testing the controlability matrix rank. I am not very experienced with nonlinear systems.
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