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

$\begin{array}{cc}\dot{x}(t)& =Ax(t)+Bu(t)\\ y(t)& =Cx(t)+Du(t)\\ x(0)& =x_0\end{array}$

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

The Data

The matrices $A,B,C,D$ are known.

The LMI: The Positive Real Lemma

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

$1)G\text{is passive, i.e.}{⟨u,Gu⟩}_{L_2}\ge 0(\hat{G}(s)+\hat{G}(s{)}^{*}\ge 0)$

$2)\text{There exists a}X>0\text{such that}$

$\left[\begin{array}{cc}{A}^{T}X+XA& XB-C^T\\ B^{T}X-C& -D^T-D\end{array}\right]\le 0$

Conclusion:

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

Implementation

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

Related LMIs

KYP Lemma (Bounded Real Lemma)

External Links

A list of references documenting and validating the LMI.

• LMI Methods in Optimal and Robust Control - A course on LMIs in Control by Matthew Peet.
• LMI Properties and Applications in Systems, Stability, and Control Theory - A List of LMIs by Ryan Caverly and James Forbes.
• LMIs in Systems and Control Theory - A downloadable book on LMIs by Stephen Boyd.