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

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

The System


\begin{align} \dot x(t)&=Ax(t)+Bu(t),\\ y(t)&=Cx(t)+Du(t)\\ x(0)&=x_0\\ \end{align}$$

where $$x(t)\in \R^n$$, $$y(t)\in \R^m$$, $$u(t)\in \R^q$$, at any $$t\in \R$$.

The Data

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

The Optimization Problem
There exist a symmetric positive definite matrix $$ P $$ and a matrix $$ W $$ satisfying $$ A^TP + PA + W^TC + C^TW < 0 $$ There exists a symmetric positive definite matrix $$ P $$ satisfying $$ N_c^T ( A^TP + PA) N_c < 0 $$ with $$ N_c $$ being the right orthogonal complement of $$ C $$. There exists a symmetric positive definite matrix $$ P $$ such that $$ A^TP + PA < \gamma C^TC $$ for some scalar $$ \gamma > 0 $$

The LMI:
Matrix pair $$(A,C)$$, is Hurwitz detectable if and only if following LMI holds
 * $$ A^TP + PA + W^TC + C^TW < 0. $$
 * $$ N_c^T ( A^TP + PA) N_c < 0 $$
 * $$ A^TP + PA - \gamma C^TC < 0 $$

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

Implementation
Find the MATLAB implementation at this link below Hurwitz detectability

Related LMIs
Links to other closely-related LMIs LMI for Hurwitz stability LMI for Schur stability Schur Detectability

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 * https://en.wikibooks.org/wiki/LMIs_in_Control