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

## The System

Given a square, discrete-time LTI system G: L2e --> L2e with state-space realization (Ad, Bd, Cd, Dd) where

${\displaystyle A_{d}\in \mathbb {R} ^{nxn}}$ , ${\displaystyle B_{d}\in \mathbb {R} ^{nxm}}$ , ${\displaystyle C_{d}\in \mathbb {R} ^{mxn}}$ , and ${\displaystyle D_{d}\in \mathbb {R} ^{mxm}}$ .

In this system, ${\displaystyle C_{d}(z1-A_{d})^{-1}B_{d}+D_{d}=B_{d}^{T}(z1-A_{d}^{T})^{-1}C_{d}^{T}+D_{d}^{T}}$  and ${\displaystyle det(1+A)\neq 0}$  and ${\displaystyle det(1-A)\neq 0}$ .

## The Data

${\displaystyle A_{d}\in \mathbb {R} ^{nxn}}$ , ${\displaystyle B_{d}\in \mathbb {R} ^{nxm}}$ , ${\displaystyle C_{d}\in \mathbb {R} ^{mxn}}$ , and ${\displaystyle D_{d}\in \mathbb {R} ^{mxm}}$

## The LMI:

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

1. There exists ${\displaystyle P\in \mathbb {S} ^{n}}$ , where P > 0 such that

${\displaystyle A_{d}^{T}PA_{d}-P\leq 0,}$

${\displaystyle C_{d}+B_{d}^{T}(A_{d}^{T}-1)^{-1}P(A_{d}+1)=0}$

2. There exists ${\displaystyle Q\in \mathbb {S} ^{n}}$ , where Q > 0 such that

${\displaystyle A_{d}QA_{d}^{T}-Q\leq 0,}$

${\displaystyle B_{d}+(A_{d}-1)^{-1}Q(A_{d}^{T}+1)C_{d}^{T}=0}$

## Conclusion

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

## Implementation

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