In control theory, we may need to find out whether or not a system such as
One of the many ways one can achieve such goal is by the use of the Controllability Gramian.
are invariant with respect to time.
One can observe if the LTI system is or is not controllable simply by looking at the pair
Then, we can say that the following statements are equivalent: If, in addition, all eigenvalues of
is stable), and the unique solution of the Lyapunov equation
is positive definite, the system is controllable.
The solution is called the Controllability Gramian and can be expressed as
In the following section we are going to take a closer look at the Controllability Gramian.
The controllability Gramian can be found as the solution of the Lyapunov equation given by
(all its eigenvalues have negative real part).
is indeed the solution for the Lyapunov equation under analysis.
is stable (all its eigenvalues have negative real part) to show that
In order to prove so, suppose we have two different solutions for
is positive for any t (assuming the non-degenerate case where
More properties of controllable systems can be found in Chen (1999, p. 145), as well as the proof for the other equivalent statements of “The pair
For discrete time systems as
One can check that there are equivalences for the statement “The pair
is controllable” (the equivalences are much alike for the continuous time case).
We are interested in the equivalence that claims that, if “The pair
is stable), then the unique solution of
That is called the discrete Controllability Gramian.
We can easily see the correspondence between discrete time and the continuous time case, that is, if we can check that
More properties and proofs can be found in Chen (1999, p. 169).
Linear time variant (LTV) systems are those in the form:
Again, as well as in the continuous time case and in the discrete time case, one may be interested in discovering if the system given by the pair
This can be done in a very similar way of the preceding cases.
matrix, also called the Controllability Gramian, given by
and by the property of the state transition matrix that claims that:
More about the Controllability Gramian can be found in Chen (1999, p. 176).