LMIs in Control/pages/Hurwitz detectability

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LMIs in Control/pages/Hurwitz detectability

Hurwitz Detectability[edit | edit source]

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 System[edit | edit source]

where , , , at any .

The Data[edit | edit source]

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

The Optimization Problem[edit | edit source]

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 | edit source]

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

Conclusion:[edit | edit source]

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

Implementation[edit | edit source]

Find the MATLAB implementation at this link below
Hurwitz detectability

Related LMIs[edit | edit source]

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

External Links[edit | edit source]

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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