# LMIs in Control/pages/Hurwitz detectability

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.