## iqc_cdelay

Purpose

Deﬁnes IQCs for the relation

$$w=(e^{-sT}-1)v,$$

where 0 ≤ T ≤ T0 is an uncertain time delay.

Synopsis

w==iqc_cdelay(v,T0,a)

[w,X,x]=iqc_cdelay(v,T0,a)

Description

The IQCs have the form

$$\left\langle Hv,XHv\right\rangle - \left\langle w,Xw\right\rangle \geq 0$$

where

$$H(s)=2sT_{0}\frac{T_{0}s+\sqrt{12.5}}{(T_{0}s)^{2}+aT_{0}s+b},\quad b=\sqrt{50},\quad a=\sqrt{2b+6.5}$$

and

$$X(s)=x_{0}+\frac{x_{1}}{s+a_{1}}+\cdots+\frac{x_{N}}{s+a_{N}}\geq 0$$

where $$a_{i}>0$$, and $$x_{i}$$ are arbitrary scalar variables.

Inputs/Outputs

Inputs:

v   Input to the cdelay function.

T0 Maximum time delay. Default T0=1.

a   Pole locations of the multiplier X, a=[a1,..,an]. Default a=[].

Outputs:

w  Output from the cdelay function.

X  The multiplier X (optional).

x  The coefﬁcients of X (optional).

Example

Figure 1: Feedback system with uncertain time delay.

Consider the system for the above Figure 1. The transfer function is

$$G(s)=\frac{0.5}{s^{2}+0.25s+1}$$

and we want to ﬁnd a bound on the maximal time delay T0. The commands below show that the gain from f to v is ﬁnite when T0  = 0.5 (the gain is computed to be gain=158.5407). Hence, T0 = 0.5 is a valid bound on the time delay.

>>G=tf(0.5,[1 0.25 1]);

>>abst_init_iqc;

>>v=signal;

>>w=signal;

>>f=signal;

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

>>w==iqc_cdelay(v,0.5);

>>gain=iqc_gain_tbx(f,v);