## iqc_sector

Purpose

Deﬁnes IQCs for a sector bounded scalar nonlinearity.

Synopsis

w==iqc_sector(v,alpha,beta)

[w,x]=iqc_sector(v,alpha,beta)

Description

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

$$x \langle w - \alpha v, \beta v - w \rangle \geq 0,$$

where $$x > 0$$ and $$w = \varphi(v)$$.

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 (optional).

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

$$G(s)=-0.5\frac{s+1}{s^{2}+0.1s+1}$$

We want to ﬁnd an estimate of the induced L2-gain from f to v.  The following sequence of commands computes the estimate gain=48.8608.  Use of better IQCs for the nonlinearity will give better gain estimates.

>>G=-ss([-0.1 -1;1 0],[0.5;0],[1 1],0);

>>abst_init_iqc;

>>w=signal;

>>f=signal;

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

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

>>gain=iqc_gain_tbx(f,v);