# LMIs in Control/pages/Positive Real Lemma

Positive Real Lemma

The Positive Real Lemma is a variation of the Kalman–Popov–Yakubovich (KYP) Lemma. The Positive Real Lemma can be used to determine if a system is passive (positive real).

## The System

{\displaystyle {\begin{aligned}{\dot {x}}(t)&=Ax(t)+Bu(t)\\y(t)&=Cx(t)+Du(t)\\x(0)&=x_{0}\end{aligned}}}

where ${\displaystyle x(t)\in \mathbb {R} ^{n}}$ , ${\displaystyle y(t)\in \mathbb {R} ^{m}}$ , ${\displaystyle u(t)\in \mathbb {R} ^{q}}$ , at any ${\displaystyle t\in \mathbb {R} }$ .

## The Data

The matrices ${\displaystyle A,B,C,D}$  are known.

## The LMI: The Positive Real Lemma

Suppose ${\displaystyle {\hat {G}}(s)(A,B,C,D)}$  is the system. Then the following are equivalent.

${\displaystyle 1)\quad G\;{\text{is passive, i.e.}}\;\left\langle u,Gu\right\rangle _{L_{2}}\geq 0\;({\hat {G}}(s)+{\hat {G}}(s)^{*}\geq 0)}$
${\displaystyle 2)\quad {\text{There exists a}}\;X>0\;{\text{such that}}}$
${\displaystyle {\begin{bmatrix}A^{T}X+XA&XB-C^{T}\\B^{T}X-C&-D^{T}-D\end{bmatrix}}\leq 0}$

## Conclusion:

The Positive Real Lemma can be used to determine if the system ${\displaystyle G}$  is passive. Note from the (1,1) block of the LMI we know that ${\displaystyle A}$  is Hurwitz.

## Implementation

This implementation requires Yalmip and Sedumi. https://github.com/eoskowro/LMI/blob/master/Positive_Real_Lemma.m