LMIs in Control/pages/LMI for the Observability Grammian

LMI for the Observability Grammian

Observability is a system property which says that the state of the system can be reconstructed using the input and output on an interval . This is necessary when knowledge of the full state is not available. If observable, estimators or observers can be created to reconstruct the full state. Observability and controllability are dual concepts. Thus in order to investigate the observability of a system we can study the controllability of the dual system. Although system observability can be determined with multiple methods, one is to compute the rank of the observability grammian.

The SystemEdit

 

where  ,  , at any  .

The DataEdit

The matrices necessary for this LMI are   and  .

The LMI:LMI to Determine the Observability GrammianEdit

  is observable if and only if   is the unique solution to

 ,

where   is the observability grammian.

Conclusion:Edit

The above LMI attempts to find the observability grammian  of the system  . If the problem is feasible and a unique   is found, then the system is also observable. The observability grammian can also be computed as:  . Due to the dual nature of observability and controllability this LMI can be determined by determining the controllability of the dual nature, which results in the above LMI. The Observability and Controllability matricies are written as   and   respectively. They are related as follows:

 
 

Hence   is observable if and only if   is controllable. Please refer to the section on controllability grammians.

ImplementationEdit

This implementation requires Yalmip and Sedumi.

https://github.com/eoskowro/LMI/blob/master/Observability_Gram_LMI.m

Related LMIsEdit

Stabilizability LMI

Hurwitz Stability LMI

Detectability LMI

Controllability Grammian LMI

External LinksEdit

A list of references documenting and validating the LMI.


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