Controllability is an important property of a
control system
A control system manages, commands, directs, or regulates the behavior of other devices or systems using control loops. It can range from a single home heating controller using a thermostat controlling a domestic boiler to large industrial c ...
, 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
Observability is a measure of how well internal states of a system can be inferred from knowledge of its external outputs.
In control theory, the observability and controllability of a linear system are mathematical duals.
The concept of observ ...
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
State may refer to:
Arts, entertainment, and media Literature
* ''State Magazine'', a monthly magazine published by the U.S. Department of State
* ''The State'' (newspaper), a daily newspaper in Columbia, South Carolina, United States
* ''Our S ...
of a
deterministic system
In mathematics, computer science and physics, a deterministic system is a system in which no randomness is involved in the development of future states of the system. A deterministic model will thus always produce the same output from a given star ...
, 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 final state in a finite time interval.
That is, we can informally define controllability as follows:
If for some initial state
and some final state
there exists an input sequence to transfer the system state from
to
in a finite time interval, then the system modeled by the
state-space representation is controllable. For the simplest example of a continuous, LTI system, the row dimension of the state space expression
determines the interval; each row contributes a vector in the state space of the system. If there are not enough such vectors to span the state space of
, then the system cannot achieve controllability. It may be necessary to modify
and
to better approximate the underlying differential relationships it estimates to achieve controllability.
Controllability does not mean that a reached state can be maintained, merely that any state can be reached.
Controllability does not mean that arbitrary paths can be made through state space, only that there exists a path within the prescribed finite time interval.
Continuous linear systems
Consider the
continuous
Continuity or continuous may refer to:
Mathematics
* Continuity (mathematics), the opposing concept to discreteness; common examples include
** Continuous probability distribution or random variable in probability and statistics
** Continuous ...
linear
Linearity is the property of a mathematical relationship (''function'') that can be graphically represented as a straight line. Linearity is closely related to '' proportionality''. Examples in physics include rectilinear motion, the linear r ...
system
[A linear time-invariant system behaves the same but with the coefficients being constant in time.]
:
:
There exists a control
from state
at time
to state
at time
if and only if
is in the
column space
In linear algebra, the column space (also called the range or image) of a matrix ''A'' is the span (set of all possible linear combinations) of its column vectors. The column space of a matrix is the image or range of the corresponding mat ...
of
:
where
is the
state-transition matrix
In control theory, the state-transition matrix is a matrix whose product with the state vector x at an initial time t_0 gives x at a later time t. The state-transition matrix can be used to obtain the general solution of linear dynamical systems ...
, and
is the
Controllability Gramian In control theory, we may need to find out whether or not a system such as
\begin
\dot(t)\boldsymbol(t)+\boldsymbol(t)\\
\boldsymbol(t)=\boldsymbol(t)+\boldsymbol(t)
\end
is controllable, where \boldsymbol, \boldsymbol, \boldsymbol and \boldsymb ...
.
In fact, if
is a solution to
then a control given by
would make the desired transfer.
Note that the matrix
defined as above has the following properties:
*
is
symmetric
Symmetry (from grc, συμμετρία "agreement in dimensions, due proportion, arrangement") in everyday language refers to a sense of harmonious and beautiful proportion and balance. In mathematics, "symmetry" has a more precise definiti ...
*
is
positive semidefinite for
*
satisfies the linear
matrix differential equation
A differential equation is a mathematical equation for an unknown function of one or several variables that relates the values of the function itself and its derivatives of various orders. A matrix differential equation contains more than one funct ...
::
*
satisfies the equation
::
Rank condition for controllability
The
Controllability Gramian In control theory, we may need to find out whether or not a system such as
\begin
\dot(t)\boldsymbol(t)+\boldsymbol(t)\\
\boldsymbol(t)=\boldsymbol(t)+\boldsymbol(t)
\end
is controllable, where \boldsymbol, \boldsymbol, \boldsymbol and \boldsymb ...
involves integration of the
state-transition matrix
In control theory, the state-transition matrix is a matrix whose product with the state vector x at an initial time t_0 gives x at a later time t. The state-transition matrix can be used to obtain the general solution of linear dynamical systems ...
of a system. A simpler condition for controllability is a
rank
Rank is the relative position, value, worth, complexity, power, importance, authority, level, etc. of a person or object within a ranking, such as:
Level or position in a hierarchical organization
* Academic rank
* Diplomatic rank
* Hierarchy
* ...
condition analogous to the Kalman rank condition for time-invariant systems.
Consider a continuous-time linear system
smoothly varying in an interval