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 Observability Gramian.
are invariant with respect to time.
One can determine if the LTI system is or is not observable simply by looking at the pair
has full column rank at every eigenvalue
is stable) and the unique solution of
is positive definite, then the system is observable.
The solution is called the Observability Gramian and can be expressed as
In the following section we are going to take a closer look at the Observability Gramian.
The Observability 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
(assuming the non-degenerate case where
More properties of observable systems can be found in,[1] 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 observable" (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 Observability Gramian.
We can easily see the correspondence between discrete time and the continuous time case, that is, if we can check that
[2] Linear time variant (LTV) systems are those in the form:
have entries that varies with time.
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 Observability Gramian is given by
is the state transition matrix of
and by the property of the state transition matrix that claims that: