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 is said to be Schur detectable if there exists a real matrix such that is Schur stable. The System 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 Data The matrices are system matrices of appropriate dimensions and are known. The Optimization Problem 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: The LMI for Schur detecability can be written as minimization of the scalar, , in the following constraints: Conclusion: Thus by proving the above conditions we prove that the matrix pair 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 - LMI in Control Systems Analysis, Design and Applications Return to Main Page LMIs in Control/Tools