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:

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

where the matrices $A\in {ℝ}^{n\times n}$, $B\in {ℝ}^{n\times r}$, $C\in {ℝ}^{m\times n}$,$D\in {ℝ}^{m\times r}$$x\in {ℝ}^n$,$y\in {ℝ}^m$ , and $u\in {ℝ}^r$ 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:

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

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

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

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
$\begin{array}{c}\left[\begin{array}{cc}-P& A^{T}P+C^T{W}^T\\ PA+WC& P\end{array}\right]<0\\ \end{array}$
There exists a symmetric matrix $P$ satisfying
$\begin{array}{c}\left[\begin{array}{cc}-N_c^{T}P{N}_c& N_c^T{A}^{T}P\\ PA{N}_c& -P\end{array}\right]<0\\ \end{array}$
with $N_c$ being the right orthogonal complement of $C$.
There exists a symmetric matrix P such that
$\begin{array}{c}\left[\begin{array}{cc}-P& PA\\ A^{T}P& -P-\gamma C^{T}C\end{array}\right]<0\\ \end{array}$
$\gamma >1$

The LMI:

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

$\begin{array}{cc}& \text{min}\gamma \\ & \text{s.t.}\end{array}$
$\begin{array}{c}\left[\begin{array}{cc}-P& A^{T}P+C^T{W}^T\\ PA+WC& P\end{array}\right]<0\\ \end{array}$
$\begin{array}{c}\left[\begin{array}{cc}-N_c^{T}P{N}_c& N_c^T{A}^{T}P\\ PA{N}_c& -P\end{array}\right]<0\\ \end{array}$
$\begin{array}{c}\left[\begin{array}{cc}-P& PA\\ A^{T}P& -P-\gamma C^{T}C\end{array}\right]<0\\ \end{array}$

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

Related LMIs

LMI for Hurwitz stability
LMI for Schur stability
Hurwitz Detectability

External Links

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

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