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 dualities 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

Table of contents

1 State Controllability 2 Output Controllability 3 Controllability in the behavioural framework

State Controllability

The states of a system is a collection of variables that at any given time completely describes the system. 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.

Thus state controllability is usually taken to mean that it is possible - by admissible inputs - to steer the states from any initial value to any final value within some time window.

A linear controllable system may be defined as a system which can be steered to any state from the zero initial state.

Output Controllability

Output Controllability means the ability to manipulate the outputs of a system by admissible inputs. For a system with several outputs, it might not be possible to manipulate these outputs independently by the admissible inputs, in which case the system is not output controllable.

Controllability in the behavioural framework

In the so-called behavioural system theoretic approach , due to Willems (see people in systems and control) the models considered do not directly define an input-output structure. In this framework systems are described by admissible trajectories of a collection of variables, some of which might be interpreted as inputs or outputs.

A system is then defined to be controllable in this setting, if any past part of a behaviour (state trajectory) can be concatenated with any future part of a behaviour with which it share the current state in such a way that the concatenation is contained in the behaviour, i.e. is part of the admissible system behaviour.