Controllability

Controllability is an important property of a control system, and the controllability property plays a crucial role in many control problems, such as stabilization of unstable systems by feedback, or optimal control.

Controllability and observability are dual aspects of the same problem.

Roughly, the concept of controllability denotes the ability to move a system around in its entire configuration space using only certain admissible manipulations. The exact definition varies slightly within the framework or the type of models applied.

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

The state of a deterministic system, which is the set of values of all the system's state variables (those variables characterized by dynamic equations), completely describes the system at any given time. In particular, no information on the past of a system is needed to help in predicting the future, if the states at the present time are known and all current and future values of the control variables (those whose values can be chosen) are known.

Complete state controllability (or simply controllability if no other context is given) describes the ability of an external input (the vector of control variables) to move the internal state of a system from any initial state to any other final state in a finite time interval.

Note that controllability does not mean that once a state is reached, that state can be maintained, but merely that that (or any) state can be reached.

Consider the continuous linear system

There exists a control ${\displaystyle u}$ from state ${\displaystyle x_{0}}$ at time ${\displaystyle t_{0}}$ to state ${\displaystyle x_{1}}$ at time ${\displaystyle t_{1}>t_{0}}$ if and only if ${\displaystyle x_{1}-\phi (t_{0},t_{1})x_{0}}$ is in the column space of

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