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LMIs in Control/Click here to continue/Observer synthesis/Schur Detectability

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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

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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

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  • The matrices are system matrices of appropriate dimensions and are known.

The Optimization Problem

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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:

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The LMI for Schur detectability can be written as minimization of the scalar, , in the following constraints:





Conclusion:

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Thus by proving the above conditions we prove that the matrix pair is Schur Detectable.

Implementation

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A link to Matlab codes for this problem in the Github repository: Schur Detectability

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LMI for Hurwitz stability
LMI for Schur stability
Hurwitz Detectability

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

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