Schur Detectability

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

The System

We consider the following system:

${\displaystyle {\begin{aligned}x(k+1)=Ax(k)+Bu(k)\\y(k)=Cx(k)+Du(k)\\\end{aligned}}}$

where the matrices ${\displaystyle A\in \mathbb {R} ^{n\times n}}$, ${\displaystyle B\in \mathbb {R} ^{n\times r}}$, ${\displaystyle C\in \mathbb {R} ^{m\times n}}$,${\displaystyle D\in \mathbb {R} ^{m\times r}}$${\displaystyle x\in \mathbb {R} ^{n}}$,${\displaystyle y\in \mathbb {R} ^{m}}$ , and ${\displaystyle u\in \mathbb {R} ^{r}}$ are the state matrix, input matrix, state vector, and the input vector, respectively.

Moreover, ${\displaystyle k}$ represents time in the discrete-time system and ${\displaystyle k+1}$ is the next time step.

The state feedback control law is defined as follows:

${\displaystyle {\begin{aligned}u(k)=Kx(k)\end{aligned}}}$

where ${\displaystyle K\in \mathbb {R} ^{n\times r}}$ is the controller gain. Thus, the closed-loop system is given by:

${\displaystyle {\begin{aligned}x(k+1)=(A+BK)x(k)\end{aligned}}}$

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 matrix ${\displaystyle P}$ and a matrix W satisfying
${\displaystyle {\begin{aligned}{\begin{bmatrix}-P&A^{T}P+C^{T}W^{T}\\PA+WC&P\end{bmatrix}}<0\\\end{aligned}}}$
There exists a symmetric matrix ${\displaystyle P}$ satisfying
${\displaystyle {\begin{aligned}{\begin{bmatrix}-N_{c}^{T}PN_{c}&N_{c}^{T}A^{T}P\\PAN_{c}&-P\end{bmatrix}}<0\\\end{aligned}}}$
with ${\displaystyle N_{c}}$ being the right orthogonal complement of ${\displaystyle C}$.
There exists a symmetric matrix P such that
${\displaystyle {\begin{aligned}{\begin{bmatrix}-P&PA\\A^{T}P&-P-\gamma C^{T}C\end{bmatrix}}<0\\\end{aligned}}}$
${\displaystyle \gamma >1}$

The LMI:

The LMI for Schur detecability can be written as minimization of the scalar, ${\displaystyle \gamma }$, in the following constraints:

${\displaystyle {\begin{aligned}&{\text{min}}\quad \gamma \\&{\text{s.t.}}\end{aligned}}}$
${\displaystyle {\begin{aligned}{\begin{bmatrix}-P&A^{T}P+C^{T}W^{T}\\PA+WC&P\end{bmatrix}}<0\\\end{aligned}}}$
${\displaystyle {\begin{aligned}{\begin{bmatrix}-N_{c}^{T}PN_{c}&N_{c}^{T}A^{T}P\\PAN_{c}&-P\end{bmatrix}}<0\\\end{aligned}}}$
${\displaystyle {\begin{aligned}{\begin{bmatrix}-P&PA\\A^{T}P&-P-\gamma C^{T}C\end{bmatrix}}<0\\\end{aligned}}}$

Conclusion:

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

Implementation

A link to Matlab codes for this problem in the Github repository: Schur Detectability

Related LMIs

LMI for Hurwitz stability
LMI for Schur stability
Hurwitz Detectability

External Links

• - LMI in Control Systems Analysis, Design and Applications

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