LMIs in Control/pages/Schur Detectability

Schur Detectability

Schur detectability is a dual concept of Schur stabilizability and is defined as follows, the matrix pair is said to be Schur detectable if there exists a real matrix such that is Schur stable.

The SystemEdit

We consider the following system:

 

where the matrices  ,  ,  ,  ,  , and   are the state matrix, input matrix, state vector, and the input vector, respectively.

Moreover,   represents time in the discrete-time system and   is the next time step.

The state feedback control law is defined as follows:

 

where   is the controller gain. Thus, the closed-loop system is given by:

 

The DataEdit

  • The matrices   are system matrices of appropriate dimensions and are known.

The Optimization ProblemEdit

There exist a symmetric matrix   and a matrix W satisfying
 
There exists a symmetric matrix   satisfying
 
with   being the right orthogonal complement of  .
There exists a symmetric matrix P such that
 
 

The LMI:Edit

The LMI for Schur detecability can be written as minimization of the scalar,  , in the following constraints:

 
 
 
 

Conclusion:Edit

Thus by proving the above conditions we prove that the matrix pair   is Schur Detectable.

ImplementationEdit

A link to Matlab codes for this problem in the Github repository: Schur Detectability

Related LMIsEdit

LMI for Hurwitz stability
LMI for Schur stability
Hurwitz Detectability

External LinksEdit

  • [1] - LMI in Control Systems Analysis, Design and Applications

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