Observability Gramian

Observability Gramian
In control theory, we may need to find out whether or not a system such as
is observable, where and are, respectively , and matrices.
One of the many ways one can achieve such goal is by the use of the Observability Gramian.
Observability in LTI Systems
Linear Time Invariant (LTI) Systems are those systems in which the parameters and are invariant with respect to time.
One can determine if the LTI system is or is not observable simply by looking at the pair . Then, we can say that the following statements are equivalent:
1. The pair is observable.
2. The matrix
is nonsingular for any .
3. The observability matrix
has rank n.
4. The matrix
has full column rank at every eigenvalue of .
If, in addition, all eigenvalues of have negative real parts (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.
Observability Gramian
The Observability Gramian can be found as the solution of the Lyapunov equation given by
In fact, we can see that if we take
as a solution, we are going to find that:
Where we used the fact that at for stable (all its eigenvalues have negative real part). This shows us that is indeed the solution for the Lyapunov equation under analysis.
Properties
We can see that is a symmetric matrix, therefore, so is .
We can use again the fact that, if is stable (all its eigenvalues have negative real part) to show that is unique. In order to prove so, suppose we have two different solutions for
and they are given by and . Then we have:
Multiplying by by the left and by by the right, would lead us to
Integrating from to :
using the fact that as :
In other words, has to be unique.
Also, we can see that
is positive for any (assuming the non-degenerate case where is not identically zero), and that makes a positive definite matrix.
More properties of observable systems can be found in, as well as the proof for the other equivalent statements of "The pair is observable" presented in section Observability in LTI Systems.
Discrete Time Systems
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 observable" and all the eigenvalues of have magnitude less than (is stable), then the unique solution of
is positive definite and given by
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 is positive definite, and all eigenvalues of have magnitude less than , the system is observable. More properties and proofs can be found in.
Linear Time Variant Systems
Linear time variant (LTV) systems are those in the form:
That is, the matrices , and 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 is observable or not. This can be done in a very similar way of the preceding cases.
The system is observable at time if and only if there exists a finite such that the matrix also called the Observability Gramian is given by
where is the state transition matrix of is nonsingular.
Again, we have a similar method to determine if a system is or not an observable system.
Properties of
We have that the Observability Gramian have the following property:
that can easily be seen by the definition of and by the property of the state transition matrix that claims that:
More about the Observability Gramian can be found in.
See also
Observability
Controllability Gramian
Gramian matrix
Hankel singular value
References
External links
Mathematica function to compute the observability Gramian
Control theory