# LMIs in Control/Matrix and LMI Properties and Tools/Discrete Time/Discrete Time Negative Imaginary Lemma

**The System**Edit

Given a square, discrete-time LTI system **G**: *L _{2e} --> L_{2e}* with state-space realization (A

_{d}, B

_{d}, C

_{d}, D

_{d}) where

, , , and .

In this system, and and .

**The Data**Edit

, , , and

**The LMI:**Edit

The system **G** posed above is considered to be negative imaginary under either of the sufficient and necessary conditions:

- There exists , where P > 0 such that

2. There exists , where Q > 0 such that

**Conclusion**Edit

By using the LMI described above, a discrete LTI system can be evaluated for the negative imaginary condition.

**Implementation**Edit

This LMI can be implemented in any LMI solver such as YALMIP, using an algorithmic solver like MOSEK.

**Related LMIs**Edit

## External LinksEdit

- LMI Properties and Applications in Systems, Stability, and Control Theory - A List of LMIs by Ryan Caverly and James Forbes.