LMIs in Control/Stability Analysis/Hurwitz Stability

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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 | edit source]

The Data[edit | edit source]

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

The Optimization Problem[edit | edit source]

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

The LMI: The Lyapunov Inequality[edit | edit source]

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 | edit source]

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 | edit source]

Please find the MATLAB implementation at this link below

Related LMIs[edit | edit source]

Links to other closely-related LMIs

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

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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