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'''Controllability''' is a fundamental concept in the field of [[control theory]], which is a branch of [[engineering]] and [[mathematics]] that deals with the behavior of [[dynamical systems]]. A system is said to be controllable if it is possible to move the system from any initial state to any desired final state within a finite time interval, using an appropriate control input.

== Definition ==
In mathematical terms, consider a linear time-invariant (LTI) system described by the state-space equations:
\[ \dot{x}(t) = Ax(t) + Bu(t) \]
\[ y(t) = Cx(t) + Du(t) \]
where:
* \( x(t) \) is the state vector,
* \( u(t) \) is the control input,
* \( y(t) \) is the output vector,
* \( A \), \( B \), \( C \), and \( D \) are matrices of appropriate dimensions.

The system is said to be controllable if, for any initial state \( x(0) \) and any final state \( x_f \), there exists a control input \( u(t) \) that transfers the system from \( x(0) \) to \( x_f \) in a finite time.

== Controllability Matrix ==
The controllability of a system can be determined using the [[controllability matrix]] \( \mathcal{C} \), defined as:
\[ \mathcal{C} = [B \ AB \ A^2B \ \cdots \ A^{n-1}B] \]
where \( n \) is the number of states in the system. The system is controllable if and only if the controllability matrix \( \mathcal{C} \) has full rank, i.e., its rank is equal to \( n \).

== Types of Controllability ==
There are several types of controllability, including:
* [[State controllability]]: The ability to move the state vector to any desired value.
* [[Output controllability]]: The ability to move the output vector to any desired value.
* [[Complete controllability]]: The ability to move both the state and output vectors to any desired values.

== Applications ==
Controllability is a crucial property in the design and analysis of control systems. It ensures that the system can be driven to a desired state, which is essential for tasks such as [[stabilization]], [[tracking]], and [[regulation]]. Applications of controllability can be found in various fields, including [[aerospace engineering]], [[automotive engineering]], [[robotics]], and [[process control]].

== Related Concepts ==
* [[Observability]]
* [[Stability]]
* [[State-space representation]]
* [[Control theory]]
* [[Linear time-invariant system]]

== See Also ==
* [[Kalman rank condition]]
* [[Controllability Gramian]]
* [[Pole placement]]
* [[Lyapunov stability]]

== References ==
{{Reflist}}

== External Links ==
{{Commons category|Controllability}}

[[Category:Control theory]]
[[Category:Systems theory]]
[[Category:Engineering]]
[[Category:Mathematics]]

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Controllability - Revision history

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