control//controllability and observability//controllability

Controllability is the structural property of a system with given actuators that says whether its inputs can drive the state from any starting point to any other in finite time, and it is checked before a controller is designed, when deciding how many actuators to install and where. If some direction of the state answers to no input, no controller will ever move it, however clever: the question is settled by the plant and its actuators, and no gain or algorithm reopens it.

Controllability is the structural property of a system with given actuators that says whether its inputs can drive the state from any starting point to any other in finite time, and it is checked before a controller is designed, when deciding how many actuators to install and where. If some direction of the state answers to no input, no controller will ever move it, however clever: the question is settled by the plant and its actuators, and no gain or algorithm reopens it.

The push can be very indirect. A planar drone has no motor that pushes it sideways and is controllable all the same: a difference of thrust tilts it, the tilt turns the thrust sideways, and that sideways force moves it (planar drone). The test that follows the push through the dynamics, step by step, is the rank of the controllability matrix. Controllability is the mirror of observability, which asks whether the sensors reveal every direction of the state: swapping AAA for ATA^{\mathsf T}AT and BBB for CTC^{\mathsf T}CT turns one test into the other (estimation-control duality).

A shared input is the classic way to lose it. Two identical carts pushed by the same force can have their mean position taken anywhere, and the distance between them never changes. A fleet that only receives a common radio command steers its centroid and never its formation, which is why each vehicle gets its own loop (fleet). A quadrotor that loses a motor can no longer control roll, pitch, yaw and thrust all at once, and survives by giving up yaw (control allocation).

Stabilizability is often all a design needs. If the uncontrollable part is stable by itself (with friction, the distance between the two carts eventually stops changing), the controller can leave it alone and still hold the system.

The rank answers yes or no, and engineering lives in how much. A controllability matrix that is nearly degenerate, with a tiny singular value (SVD), means some directions can only be moved with enormous efforts: controllable in theory, uncontrollable with your motors. That degree of controllability is what sizes actuators on ship stabilizers, satellites and structures fighting vibration.

What a controller then does with the inputs is state feedback; how controllability and observability together decide the smallest model of a system is in controllability and observability.