## iqc_popov

Purpose

Deﬁnes the combined sector and Popov IQC for a memoryless nonlinearity. This gives the classic Popov criterion.

Synopsis

w==iqc_popov(v,alpha,beta)

[w,x,y]=iqc_popov(v,alpha,beta)

Description

Let $$\varphi$$ be a sector bounded nonlinearity with $$\alpha v^{2} \leq v \varphi(v) \leq \beta v^{2}$$, for all $$v \in \textbf{R}$$, and $$t \geq 0$$.

Then $$\varphi$$ satisfies the IQCs

$$x\langle w -\alpha v,\beta v - w\rangle + y \langle w,dv/dt\rangle \geq 0$$

where $$x \geq 0$$ and $$y \in \textbf{R}$$.

The iqc_popov command generates the IQCs above.

Input/Outputs

Inputs:

v         Scalar input to nonlinearity

alpha  Lower sector bound. Default alpha=0

beta   Upper sector bound (beta > alpha). Default beta=1

Outputs:

w      Scalar output from nonlinearity

x       The decision variable x(optional)

y       The popov variable y(optimal)

Example Figure 1: Simple feedback interconnection of nonlinearity and linear system.

Consider the system in Figure 1. We assume that $$\varphi \in \textrm{sector}[0, 1]$$ (i.e., $$\alpha = 0$$, and $$\beta = 1$$), and

$$G(s)=- \frac{s+1}{s^{2}+0.1s+1}$$

We want to ﬁnd an estimate of the induced L2-gain from f to v. Using the iqc_sector command is not enough. However, the following sequence of commands shows that with iqc_popov we get the estimate gain=14.14.

>>G=-ss([-0.1 -1;1 0],[1;0],[1 1],0);

>>abst_init_iqc;

>>w=signal;

>>f=signal;

>>v=G*(w+f);

>>w==iqc_popov(v,0,1);

>>gain=iqc_gain_tbx(f,v);