Controllability

Controllability is an important property of a control system and plays a crucial role in many regulation problems, such as the stabilization of unstable systems using feedback, tracking problems, obtaining optimal control strategies, or, simply prescribing an input that has a desired effect on the state.

Controllability and observability are dual notions. Controllability pertains to regulating the state by a choice of a suitable input, while observability pertains to being able to know the state by observing the output (assuming that the input is also being observed).

Broadly speaking, the concept of controllability relates to the ability to steer a system around in its configuration space using only certain admissible manipulations. The exact definition varies depending on the framework or the type of models dealt with.

The following are examples of variants of notions of controllability that have been introduced in the systems and control literature:

• State controllability: the ability to steer the system between states

• Strong controllability: the ability to steer between states over any specified time window

• Collective controllability: the ability to simultaneously steer a collection of dynamical systems

• Trajectory controllability: the ability to steer along a predefined trajectory rather than just to a desired final state

• Output controllability: the ability to steer to specified values of the output

• Controllability in the behavioural framework: a compatibility condition between past and future input and output trajectories