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Wikipedia / Observability Gramian

In control theory, we may need to find out whether or not a system such as

matrices.

One of the many ways one can achieve such goal is by the use of the Observability Gramian.

Observability in LTI Systems

are invariant with respect to time.

. Then, we can say that the following statements are equivalent:

is observable.

matrix

.

observability matrix

has rank n.

matrix

.

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:

is indeed the solution for the Lyapunov equation under analysis.

Properties

is a symmetric matrix.

is unique. In order to prove so, suppose we have two different solutions for

. Then we have:

by the right, would lead us to

:

:

has to be unique.

Also, we can see that

a positive definite matrix.

More properties of observable systems can be found in, is observable" presented in section Observability in LTI Systems.

Discrete Time Systems

For discrete time systems as

is observable" (the equivalences are much alike for the continuous time case).

is stable), then the unique solution of

is positive definite and given by

is observable. More properties and proofs can be found in.

Linear Time Variant Systems

Linear time variant (LTV) systems are those in the form:

is observable or not. This can be done in a very similar way of the preceding cases.

if and only if matrix also called the Observability Gramian is given by

is nonsingular.

Again, we have a similar method to determine if a system is or not an observable system.

\boldsymbol{W}_{o}(t_{0},t_{1})

have the following property:

and by the property of the state transition matrix that claims that:

More about the Observability Gramian can be found in.

See also

References