LMIs in Control/pages/Hurwitz detectability

LMIs in Control/pages/Hurwitz detectability

Hurwitz Detectability

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

The System

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

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

The Data

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

The Optimization Problem

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

The LMI:

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

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

Conclusion:

Thus by proving the above conditions we prove that the matrix pair $(A,C)$ 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.

Return to Main Page:

• https://en.wikibooks.org/wiki/LMIs_in_Control