Schur detectability is a dual concept of Schur stabilizability and is defined as follows, the matrix pair is said to be Schur
detectable if there exists a real matrix such that is Schur stable.
We consider the following system:
where the matrices , , ,, , and are the state matrix, input matrix, state vector, and the input vector, respectively.
Moreover, represents time in the discrete-time system and is the next time step.
The state feedback control law is defined as follows:
where is the controller gain. Thus, the closed-loop system is given by:
- The matrices are system matrices of appropriate dimensions and are known.
There exist a symmetric matrix and a matrix W satisfying
There exists a symmetric matrix satisfying
with being the right orthogonal complement of .
There exists a symmetric matrix P such that
The LMI for Schur detecability can be written as minimization of the scalar, , in the following constraints:
Thus by proving the above conditions we prove that the matrix pair is Schur Detectable.
A link to Matlab codes for this problem in the Github repository:
LMI for Hurwitz stability
LMI for Schur stability
-  - LMI in Control Systems Analysis, Design and Applications
LMIs in Control/Tools