## iqc_window

Purpose

Defines IQCs for the relation

$$w=\frac{e^{-sT}-1}{s}v,$$

where 0 < T  < T0  is an uncertain time delay.  This corresponds to convolution in the time domain with a rectangular window of uncertain length.

Synopsis

w==iqc_window(v,T0,a)

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

Description

The IQCs have the form

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

where

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

and

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

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

Inputs/Outputs

Inputs:

v    Input to window function.

T0  Maximum time delay. Default T 0=1.

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

Outputs:

w   Output from window function.

X    The multiplier X (optional).

x    The coefﬁcients of X (optional).

Example

Figure 1: Simple feedback interconnection of uncertain window ﬁlter and linear system.

Consider the system in Figure 1. We assume that $$T \in [0,\ 0.5]$$, and

$$G(s)=-1.4\frac{s+1}{s^{2}+0.3s+100}$$

We want to ﬁnd an estimate of the induced L2-gain from f to v.   Use of the iqc_window command gives the estimate gain=627.60.

>>G=ss([-0.3 -100;1 0],[1.4;0],[1 1],0);

>>abst_init_iqc;

>>w=signal;

>>f=signal;

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

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

>>gain=iqc_gain_tbx(f,v)

iqc_cdelay

References

Derived as iqc cdelayin

A. Megretski and A. Rantzer.   System analysis via integral quadratic constraints. IEEE Transactions on Automatic Control, 42(6):819–830, June 1997.

It was useful for an application in

U. Jönsson and A. Megretski. The Zames Falb IQC for critically stable systems. To appear in the proceedings of the 1998 American Control Conference.