LMIs in Control/pages/Hurwitz detectability

LMIs in Control/pages/Hurwitz detectability


Hurwitz DetectabilityEdit

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

The SystemEdit

 

where  ,  ,  , at any  .

The DataEdit

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

The Optimization ProblemEdit

There exist a symmetric positive definite matrix   and a matrix   satisfying
 
There exists a symmetric positive definite matrix   satisfying
 
with   being the right orthogonal complement of  .
There exists a symmetric positive definite matrix   such that
 
for some scalar  

The LMI:Edit

Matrix pair  , is Hurwitz detectable if and only if following LMI holds

  •  
  •  
  •  

Conclusion:Edit

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

ImplementationEdit

Find the MATLAB implementation at this link below
Hurwitz detectability

Related LMIsEdit

Links to other closely-related LMIs
LMI for Hurwitz stability
LMI for Schur stability
Schur Detectability

External LinksEdit

A list of references documenting and validating the LMI.

  • LMI Methods in Optimal and Robust Control - A course on LMIs in Control by Matthew Peet.
  • LMI Properties and Applications in Systems, Stability, and Control Theory - A List of LMIs by Ryan Caverly and James Forbes.
  • LMIs in Systems and Control Theory - A downloadable book on LMIs by Stephen Boyd.

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