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 ...
and plays a crucial role in many regulation problems, such as the stabilization of
unstable systems using feedback, tracking problems, obtaining
optimal control strategies, or, simply prescribing an input that has a desired effect on the state.
Controllability and
observability are
dual notions. Controllability pertains to regulating the state by a choice of a suitable input, while observability pertains to being able to know the state by observing the output (assuming that the input is also being observed).
Broadly speaking, the concept of controllability relates to the ability to steer a system around in its configuration space using only certain admissible manipulations. The exact definition varies depending on the framework or the type of models dealt with.
The following are examples of variants of notions of controllability that have been introduced in the systems and control literature:
* State controllability: the ability to steer the system between states
* Strong controllability: the ability to steer between states over any specified time window
* Collective controllability: the ability to simultaneously steer a collection of dynamical systems
* Trajectory controllability: the ability to steer along a predefined trajectory rather than just to a desired final state
* Output controllability: the ability to steer to specified values of the output
* Controllability in the behavioural framework: a compatibility condition between past and future input and output trajectories
State controllability
The
state
State most commonly refers to:
* State (polity), a centralized political organization that regulates law and society within a territory
**Sovereign state, a sovereign polity in international law, commonly referred to as a country
**Nation state, a ...
of a
deterministic system, 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 any initial state
and any 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 a finite time interval. When the time interval can also be specified, the dynamical system is often referred to as being strongly controllable.
Continuous linear systems
Consider the
continuous linear
In mathematics, the term ''linear'' is used in two distinct senses for two different properties:
* linearity of a '' function'' (or '' mapping'');
* linearity of a '' polynomial''.
An example of a linear function is the function defined by f(x) ...
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 of
:
where
is the
state-transition matrix, and
is the
Controllability Gramian.
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
*
is
positive semidefinite for
*
satisfies the linear
matrix differential equation
::
*
satisfies the equation
::
Rank condition for controllability
The
Controllability Gramian involves integration of the
state-transition matrix of a system. A simpler condition for controllability is a
rank condition analogous to the Kalman rank condition for time-invariant systems.
Consider a continuous-time linear system
smoothly varying in an interval