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.
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.
[1]: 737 That is, we can informally define controllability as follows: If for any initial state
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.
to better approximate the underlying differential relationships it estimates to achieve controllability.
Consider the continuous linear system [note 1] There exists a control
defined as above has the following properties: The Controllability Gramian involves integration of the state-transition matrix of a system.
and define Consider the matrix of matrix-valued functions obtained by listing all the columns of the
[3] The above methods can still be complex to check, since it involves the computation of the state-transition matrix
For a discrete-time linear state-space system (i.e. time variable
states is reachable by giving the system proper inputs through the variable
at an initial time, arbitrarily denoted as k=0, the state equation gives
If the system is controllable then these two vectors can span the entire plane and can be done so for time
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.
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
You are allowed to: Although the 3-dimensional case is harder to visualize, the concept of controllability is still analogous.
Nonlinear systems in the control-affine form are locally accessible about
is the repeated Lie bracket operation defined by The controllability matrix for linear systems in the previous section can in fact be derived from this equation.
In particular: For a linear continuous-time system, like the example above, described by matrices
This phenomenon is caused by constraints on the input that could be inherent to the system (e.g. due to saturating actuator) or imposed on the system for other reasons (e.g. due to safety-related concerns).
The controllability of systems with input and state constraints is studied in the context of reachability[5] and viability theory.
[6] In the so-called behavioral system theoretic approach due to Willems (see people in systems and control), 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 said to be stabilizable when all uncontrollable state variables can be made to have stable dynamics.
, where xT→z denotes that there exists a state transition from x to z in time T. For autonomous systems the reachable set is given by : where R is the controllability matrix.
Proof We have the following equalities: Considering that the system is controllable, the columns of R should be linearly independent.