LMIs in Control/pages/Schur Detectability

From Wikibooks, open books for an open world
Jump to navigation Jump to search

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[edit | edit source]

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[edit | edit source]

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

The Optimization Problem[edit | edit source]

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:[edit | edit source]

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

Conclusion:[edit | edit source]

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

Implementation[edit | edit source]

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

Related LMIs[edit | edit source]

LMI for Hurwitz stability
LMI for Schur stability
Hurwitz Detectability

External Links[edit | edit source]

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

Return to Main Page[edit | edit source]

LMIs in Control/Tools