## iqc_ltvnorm

Purpose

Deﬁne IQC for a pair of L2 signal (v(t), w(t)) which have the relation $$||w||<g\cdot ||v||$$, i.e.,

$$\int_{0}^{\infty}|w(t)|^{2}dt < g^{2} \cdot \int_{0}^{\infty} |v(t)|^{2} dt,$$

where $$|\cdot|$$ stands for the Euclidean norm.

Synopsis

w==iqc_ltvnorm(v,n,g);

[w,x]=iqc_ltvnorm(v,n,g);

Description

The IQC deﬁned by this M-ﬁle has the form

$$g^{2}\cdot \left\langle v,Xv \right\rangle - \left\langle w,Xw \right\rangle \geq 0,\quad X>0,$$

where $$X$$ is a real scalar variable.

Inputs/Outputs

Inputs:

v    Input signal.

n   The dimension of the output signal w. Default: n=1

maximal ampliﬁcation ratio. Default: g=1

Outputs:

w  Basic signal has dimension n × 1.

x   The multiplier X.

Example Figure 1: System with norm bounded uncertainty

Consider the uncertain system in Figure 1. The transfer function is

$$G(s)=\frac{0.8}{s^{2}+0.21s+1},$$

and $$||\Delta||<0.26$$ is known to us. We would like to check whether the system is stable. The commands below show that the gain from f to v is ﬁnite (the gain is computed to be gain=953.6498). Hence, the system is stable under the appearance of the uncertainty.

>> s=tf([1,0],1);

>> G=0.8/(s*s+0.21*s+1);

>> abst_init_iqc

>> f=signal;

>> w=signal;

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

>> w==iqc_ltvnorm(v,1,0.26);

>> gain=iqc_gain_tbx(f,v)