LMIs in Control/Stability Analysis/Hurwitz Stability

LMIs in Control/Stability Analysis/Hurwitz Stability


This is a set of LMI conditions for determining Hurwitz Stability of continuous time systems.


The System edit

 

The Data edit

  • The matrices   are system matrices of appropriate dimensions.
  •  ,   and   are state vector, output vector and input vector respectively.

The Optimization Problem edit

Find a symmetric postive definite matrix  , where  . Thus   and   where  .

The LMI: The Lyapunov Inequality edit

Matrix pair  , is Hurwitz stabilizable if and only if there exist a symmetric positive definite matrix   and a matrix   satisfying
 

Proof : Matrix pair  , is Hurwitz stabilizable if and only if

 ,   and  
This is the definition of Hurwitz Stability. Now, using this definition we can prove the above LMI if we find matrix   and matrix   and thus by substituting   in the above LMI we get,
 , which brings us to the Lyapunov Stability Theory.

Conclusion: edit

Thus by proving the above conditions we prove that the matrix pair   is Hurwitz Stabilizable. At the same time, we also prove that the   i.e. it is full rank and the real part of   is  .

Implementation edit

Please find the MATLAB implementation at this link below
https://github.com/omiksave/LMI

Related LMIs edit

Links to other closely-related LMIs

  • Schur Stability
  • Quadratic Hurwitz Stability
  • Quadratic Schur Stability
  • Quadratic D-Stability

External Links edit

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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