# LMIs in Control/Matrix and LMI Properties and Tools/Negative Imaginary System DC Constraint

## Introduction

These systems are often related to systems involving energy dissipation. But the standard Positive real theory will not be helpful in establishing closed-loop stability. However, transfer functions of systems with a degree more than one can be satisfied with the negative imaginary conditions for all frequency values and such systems are called systems with negative imaginary frequency response.

## The System

Consider a square continuous time Linear Time invariant system, with the state space realization ${\displaystyle (A,B,C,D)}$

{\displaystyle {\begin{aligned}{\dot {x}}(t)=Ax(t)+Bu(t)\\y=Cx(t)+Du(t)\end{aligned}}}

## The Data

${\displaystyle A\in \mathbb {R} ^{n\times n},B\in \mathbb {R} ^{n\times m},C\in \mathbb {R} ^{m\times n},D\in \mathbb {S} ^{m}}$

## The LMI

Consider an NI transfer matrix ${\displaystyle {\mathbf {G}}_{1}(s)}$  and an NI transfer matrix ${\displaystyle {\mathbf {G}}_{2}(s)={\mathbf {C}}_{2}(s{\mathbf {1}}-{\mathbf {A}}_{2})^{-1}{\mathbf {B}}_{2}+{\mathbf {D}}_{2}}$ . The condition λ̅ ${\displaystyle ({\mathbf {G}}_{1}(0){\mathbf {G}}_{2}(0)<1}$  is satisfied if and only if

${\displaystyle {\mathbf {S}}^{T}(-{\mathbf {C}}_{2}{\mathbf {A}}_{2}^{-1}{\mathbf {B}}_{2}+{\mathbf {D}}_{2}){\mathbf {S}}<{\mathbf {1}}}$ ,

## Conclusion

The above equation holds true if and only if ${\displaystyle {\mathbf {S}}{\mathbf {S}}^{T}={\mathbf {G}}_{1}(0)}$ .

## Implementation

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