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

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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

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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

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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,[1] 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

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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.[2]

Linear Time Variant Systems

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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  

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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.[3]

See also

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References

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  1. ^ Chen, Chi-Tsong (1999). Linear System Theory and Design Third Edition. New York, New York: Oxford University Press. p. 156. ISBN 0-19-511777-8.
  2. ^ Chen, Chi-Tsong (1999). Linear System Theory and Design Third Edition. New York, New York: Oxford University Press. p. 171. ISBN 0-19-511777-8.
  3. ^ Chen, Chi-Tsong (1999). Linear System Theory and Design Third Edition. New York, New York: Oxford University Press. p. 179. ISBN 0-19-511777-8.
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