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 $(A,C),$ is said to be Schur detectable if there exists a real matrix $L$ such that $A+LC$ is Schur stable.

The System

We consider the following system:

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

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

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

The state feedback control law is defined as follows:

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

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

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

The Data

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

The Optimization Problem

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

The LMI:

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

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

Conclusion:

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

Implementation

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

External Links

[1] - LMI in Control Systems Analysis, Design and Applications

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