LMIs in Control/Click here to continue/Observer synthesis/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.

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