# User Contributed Dictionary

### Noun

- The quality of being controllable; controllableness.

### References

# Extensive Definition

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

## 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.

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.

Note that controllability does not mean that once
you reach a state that you will be able to keep it there, but
merely that you can reach that state.

## Continuous Linear Time-Invariant (LTI) Systems

- \dot(t) = \mathbf(t) + \mathbf(t)
- \mathbf(t) = \mathbf(t) + \mathbf(t)

where

- \mathbf is the "state vector",
- \mathbf is the "output vector",
- \mathbf is the "input (or control) vector",
- \mathbf is the "state matrix",
- \mathbf is the "input matrix",
- \mathbf is the "output matrix",
- \mathbf is the "feedthrough (or feedforward) matrix".

The controllability matrix is given by

- R = \beginB & AB & A^B & ...& A^B\end

The system is controllable if the controllability
matrix has a full rank.

## Discrete Linear Time-Invariant (LTI) Systems

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

- \textbf(k+1) = A\textbf(k) + B\textbf(k)

Where A is an n \times n matrix. The test for
controllability is that the matrix

- C = \beginB & AB & A^B & ...& A^B\end

has full rank
(i.e., rank(C) = n). The rationale for this test is that if n
columns of C are linearly
independent: in this case each of the n states is reachable
giving the system proper inputs through the variable u(k).

#### Example

For example, consider the case when n=2. If \beginB & AB\end has rank 2 (full rank). In this case B and AB are linearly independent and span the entire plane. If the rank is 1 then B and AB are collinear and cannot possibly span the plane.Assume that the initial state is zero.

At time k=0: x(1) = A\textbf(0) + B\textbf(0) =
B\textbf(0)

At time k=1: x(2) = A\textbf(1) + B\textbf(1) =
AB\textbf(0) + B\textbf(1)

At time k=0 all of the reachable states are on
the line formed by the vector B. At time k=1 all of the reachable
states are linear combinations of AB and B. If the system is
controllable then these two vectors can span the entire plane and
can be done so for time k=2. The assumption made that the initial
state is zero is merely for convenience. Clearly if all states can
be reached from the origin then any state can be reached from
another state (merely a shift in coordinates).

This example holds for all positive n, but the
case of n=2 is easier to visualize.

#### Analogy for example of n=2

Consider an analogy to the previous example system. You are sitting in your car on an infinite, flat plane and facing north. The goal is to reach any point in the plane by driving a distance in a straight line, come to a full stop, turn, and driving another distance, again, in a straight line. If your car has no steering then you can only drive straight, which means you can only drive on a line (in this case the north-south line since you started facing north). The lack of steering case would be analogous to when the rank of C is 1 (the two distances you drove are on the same line).Now, if your car did have steering then you could
easily drive to any point in the plane and this would be the
analogous case to when the rank of C is 2.

If you change this example to n=3 then the
analogy would be flying in space to reach any position in 3D space
(ignoring the orientation
of the aircraft). You
are allowed to:
Although
the 3-dimensional case is harder to visualize, the concept of
controllability is still analogous.

## Nonlinear Systems

Nonlinear systems in the control-affine
form

- \dot = \mathbf + \sum_^m \mathbf_i(\mathbf)u_i

is locally accessible about x_0 if the
accessibility distribution R spans n space, when n equals the rank
of x and R is given by:

- R = \begin \mathbf_1 & \cdots & \mathbf_m & [\mathrm^k_\mathbf] & \cdots & [\mathrm^k_\mathbf] \end

Here, [\mathrm^k_\mathbf] is the repeated
Lie
bracket operation defined by

- [\mathrm^k_\mathbf] = \begin \mathbf & \cdots & j & \cdots & \mathbf \end

The controllability matrix for linear systems in
the previous section can in fact be derived from this
equation.

## 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 shares the current state in such a way that the concatenation is
contained in the behaviour, i.e. is part of the admissible system
behaviour.

## Stabilizability

A slightly weaker notion than controllability is that of Stabilizability. A system is determined to be stabilizable when all uncontrollable states have stable dynamics. Thus, even though some of the state cannot be controlled (as determined by the controllability test above) all the states will still remain bounded during the system's behaviour.## External links

The system described above or (A,B) is said to be
stabilizable if there exists a feedback matrix F such that A+BF has
all its eigenvalues in the open left-half plane C^.

controllability in Arabic: قابلية التحكم

controllability in German: Steuerbarkeit

controllability in Spanish:
Controlabilidad

controllability in French: Commandabilité

controllability in Korean: 제어 가능성

controllability in Italian:
Controllabilità

controllability in Polish: Sterowalność

controllability in Russian: Управляемость
(теория управления)