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: L_2e --> L_2e with state-space realization (A_d, B_d, C_d, D_d) 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.

Related LMIs

• Negative Imaginary Lemma

External Links

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