LMIs in Control/pages/Hurwitz detectability

Hurwitz Detectability

Hurwitz detectability is a dual concept of Hurwitz stabilizability and is defined as the matrix pair ${\displaystyle (A,C)}$, is said to be Hurwitz detectable if there exists a real matrix ${\displaystyle L}$ such that ${\displaystyle A+LC}$ is Hurwitz stable.

The System

${\displaystyle {\begin{aligned}{\dot {x}}(t)&=Ax(t)+Bu(t),\\y(t)&=Cx(t)+Du(t)\\x(0)&=x_{0}\\\end{aligned}}}$

where ${\displaystyle x(t)\in \mathbb {R} ^{n}}$, ${\displaystyle y(t)\in \mathbb {R} ^{m}}$, ${\displaystyle u(t)\in \mathbb {R} ^{q}}$, at any ${\displaystyle t\in \mathbb {R} }$.

The Data

• The matrices ${\displaystyle A,B,C,D}$ are system matrices of appropriate dimensions and are known.

The Optimization Problem

There exist a symmetric positive definite matrix ${\displaystyle P}$ and a matrix ${\displaystyle W}$ satisfying
${\displaystyle A^{T}P+PA+W^{T}C+C^{T}W<0}$
There exists a symmetric positive definite matrix ${\displaystyle P}$ satisfying
${\displaystyle N_{c}^{T}(A^{T}P+PA)N_{c}<0}$
with ${\displaystyle N_{c}}$ being the right orthogonal complement of ${\displaystyle C}$.
There exists a symmetric positive definite matrix ${\displaystyle P}$ such that
${\displaystyle A^{T}P+PA<\gamma C^{T}C}$
for some scalar ${\displaystyle \gamma >0}$

The LMI:

Matrix pair ${\displaystyle (A,C)}$, is Hurwitz detectable if and only if following LMI holds

• ${\displaystyle A^{T}P+PA+W^{T}C+C^{T}W<0.}$
• ${\displaystyle N_{c}^{T}(A^{T}P+PA)N_{c}<0}$
• ${\displaystyle A^{T}P+PA-\gamma C^{T}C<0}$
  

Conclusion:

Thus by proving the above conditions we prove that the matrix pair ${\displaystyle (A,C)}$ is Hurwitz Detectable.

Implementation

Find the MATLAB implementation at this link below
Hurwitz detectability

Related LMIs

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

External Links

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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