# Controllability

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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:

• State controllability
• Output controllability
• Controllability in the behavioural framework

## Contents

### State controllability

The state of a system, which is a collection of system's variables values, completely describes the system at any given time. In particular, no information on the past of a system will help in predicting the future, if the states at the present time are known.

Complete state controllability (or simply controllability if no other context is given) describes the ability of an external input to move the internal state of a system from any initial state to any other final state in a finite time interval.[1]:737

### Continuous Linear Time-Invariant (LTI) Systems

Consider the continuous linear time-invariant system

where

The $n \times nr$ controllability matrix is given by

The system is controllable if the controllability matrix has full rank (i.e. $\operatorname{rank}(R)=n$).

### Discrete Linear Time-Invariant (LTI) Systems

For a discrete-time linear state-space system (i.e. time variable $k\in\mathbb{Z}$) the state equation is