User:Margav06/sandbox/Click here to continue/Observer synthesis/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:

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

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:

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

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

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

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 $$\begin{align} \begin{bmatrix}  -P &  A^TP + C^TW^T    \\ PA + WC  & P \end{bmatrix} < 0\\ \end{align}$$ There exists a symmetric matrix $$ P $$ satisfying $$\begin{align} \begin{bmatrix}  -N_c^TPN_c &  N_c^TA^TP    \\ PAN_c  & -P \end{bmatrix} < 0\\ \end{align}$$ with $$ N_c $$ being the right orthogonal complement of $$ C $$. There exists a symmetric matrix P such that $$\begin{align} \begin{bmatrix}  -P &  PA    \\ A^TP  & -P-\gamma C^TC \end{bmatrix} < 0\\ \end{align}$$ $$ \gamma > 1 $$

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

$$ \begin{align} & \text{min} \quad \gamma\\ & \text{s.t.} \end{align}$$ $$\begin{align} \begin{bmatrix}  -P &  A^TP + C^TW^T    \\ PA + WC  & P \end{bmatrix} < 0\\ \end{align}$$ $$\begin{align} \begin{bmatrix}  -N_c^TPN_c &  N_c^TA^TP    \\ PAN_c  & -P \end{bmatrix} < 0\\ \end{align}$$ $$\begin{align} \begin{bmatrix}  -P &  PA    \\ A^TP  & -P-\gamma C^TC \end{bmatrix} < 0\\ \end{align}$$

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

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