# 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 System**[edit | edit source]

Discrete-Time LTI System with state space realization

**The Data**[edit | edit source]

The matrices: System .

**The Optimization Problem**[edit | edit source]

The following feasibility problem should be optimized:

Maximize P while obeying the LMI constraints.

Then L is found.

**The LMI:**[edit | edit source]

Discrete-Time Detectability

The LMI formulation

**Conclusion:**[edit | edit source]

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

**Implementation**[edit | edit source]

A link to CodeOcean or other online implementation of the LMI

MATLAB Code

**Related LMIs**[edit | edit source]

[1] - Continuous time Detectability

**External Links**[edit | edit source]

A list of references documenting and validating the LMI.

- LMI Methods in Optimal and Robust Control - A course on LMIs in Control by Matthew Peet.
- LMI Properties and Applications in Systems, Stability, and Control Theory - A List of LMIs by Ryan Caverly and James Forbes.
- LMIs in Systems and Control Theory - A downloadable book on LMIs by Stephen Boyd.