Controllability is an important property of a control system and 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 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 the prescribed finite time interval.
Continuous linear systems
Consider the continuous linear system
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 of :
The state-transition matrix is also smooth. Introduce the n x m matrix-valued function and define = .
Consider the matrix of matrix-valued functions obtained by listing all the columns of the , :
.
If there exists a and a nonnegative integer k such that , then is controllable.
If is also analytically varying in an interval , then is controllable on every nontrivial subinterval of if and only if there exists a and a nonnegative integer k such that .
The above methods can still be complex to check, since it involves the computation of the state-transition matrix . Another equivalent condition is defined as follow. Let , and for each , define
=
In this case, each is obtained directly from the data The system is controllable if there exists a and a nonnegative integer such that .
Example
Consider a system varying analytically in and matrices
, Then and since this matrix has rank 3, the system is controllable on every nontrivial interval of .
Continuous linear time-invariant (LTI) systems
Consider the continuous linear time-invariant system
where
is the "state vector", is the "output vector", is the "input (or control) vector", is the "state matrix", is the "input matrix", is the "output matrix", is the "feedthrough (or feedforward) matrix".