LMIs in Control/pages/Discrete-Time Detectability

Discrete-Time Detectability

A discrete time system operates on a discrete time signal input and produces a discrete time signal output. They are used in digital signal processing, such as digital filters for images or sound. The class of discrete time systems that are both linear and time invariant, known as discrete time LTI systems.

Discrete-Time LTI systems can be made detectable using observer gain L, which can be found using LMI optimization, such that the close loop system is detectable.

The SystemEdit

Discrete-Time LTI System with state space realization  
 

The DataEdit

The matrices: System  .

The Optimization ProblemEdit

The following feasibility problem should be optimized:

Maximize P while obeying the LMI constraints.
Then L is found.

The LMI:Edit

Discrete-Time Detectability

The LMI formulation

 

Conclusion:Edit

The system is detectabe iff there exits a  , such that  . The matrix   is Schur with  

ImplementationEdit

A link to CodeOcean or other online implementation of the LMI
MATLAB Code

Related LMIsEdit

[1] - Continuous time Detectability

External LinksEdit

A list of references documenting and validating the LMI.

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