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);

See also

iqc_popov, iqc_popov_vect, iqc_monotonic, iqc_slopeiqc_slope_odd