Controllability Gramian
Encyclopedia
In control theory
, the controllability Gramian is a Gramian used to determine whether or not a linear system is controllable
.
For the time-invariant
linear system
the controllability Gramian is given by
If the matrix
is nonsingular, i.e. 
has full rank, for any 
, then the pair 
is controllable
.
Linear time-variant state space models of form
,
are controllable in an interval
if the rows of the matrix product
where 
is the state transition matrix are linearly independent.
The Gramian is used to prove the linear independency of
.
To have linear independency Gramian matrix
have to be nonsingular, i.e., invertible.
Control theory
Control theory is an interdisciplinary branch of engineering and mathematics that deals with the behavior of dynamical systems. The desired output of a system is called the reference...
, the controllability Gramian is a Gramian used to determine whether or not a linear system is controllable
Controllability
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....
.
For the time-invariant
Time-invariant system
A time-invariant system is one whose output does not depend explicitly on time.This property can be satisfied if the transfer function of the system is not a function of time except expressed by the input and output....
linear system
the controllability Gramian is given by
If the matrix
is nonsingular, i.e. has full rank, for any , then the pair is controllableControllability
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....
.
Linear time-variant state space models of form
,are controllable in an interval
if the rows of the matrix product where is the state transition matrix are linearly independent.The Gramian is used to prove the linear independency of
.To have linear independency Gramian matrix
have to be nonsingular, i.e., invertible.
