# LMIs in Control/pages/Schur Detectability

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