# 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

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

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

## The Data

$A_{d}\in \mathbb {R} ^{nxn}$ , $B_{d}\in \mathbb {R} ^{nxm}$ , $C_{d}\in \mathbb {R} ^{mxn}$ , and $D_{d}\in \mathbb {R} ^{mxm}$ ## The LMI:

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

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

$A_{d}^{T}PA_{d}-P\leq 0,$ $C_{d}+B_{d}^{T}(A_{d}^{T}-1)^{-1}P(A_{d}+1)=0$ 2. There exists $Q\in \mathbb {S} ^{n}$ , where Q > 0 such that

$A_{d}QA_{d}^{T}-Q\leq 0,$ $B_{d}+(A_{d}-1)^{-1}Q(A_{d}^{T}+1)C_{d}^{T}=0$ ## 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.