iqc_white

Purpose

IQCs for deterministic white signals over the frequency interval [-b, b], where b is a user defined bandwidth.

Synopsis

f=iqc_white(n,b,a)

[f,Y,Z]=iqc_white(n,b,a)

Description

The IQCs are of the form

$$ \int_{-\infty}^{\infty} \hat{f}(j\omega)^{*}Y(j\omega)f(j\omega) \geq 0$$

where

$$Y = X_{0} + \sum_{k=1}^{N}\frac{1}{2}\left(\frac{Z_{k}}{s+a^{k}}+\frac{\bar{Z}_{k}}{s+\bar{a}_{k}}\right),\quad Z_{k}=X_{k}+iY_{k},\quad X_{k},Y_{k} \in \textbf{R},$$

satisfies

$$ \int_{-\infty}^{\infty} \textrm{tr}(Y(j\omega))d\omega \geq 0$$

The constraint (4.7) can be implemented as

$$ \textrm{tr}\left( bX_{0} + \sum_{k=1}^{N} \textrm{Re}(Z_{k}\textrm{arctan}(b/a_{k}))\right) \geq 0.$$

Input/Outputs

Inputs:

Size of signal. Default n=1.

Bandwidth of white signal. Default b=10.

Pole locations of Y , a=[a1,..,an]. Default a = 1.

Outputs:

f   The white signal.

The multiplier Y (optional).

The coefficients of Y structured as (optional)
\(Z=\begin{bmatrix} X_{1} & Y_{1}\\ \vdots & \vdots \\ X_{N} & Y_{N}\\ X_{0} & 0 \end{bmatrix}\)

Example

Consider the system in Figure 1. We assume that f is white over the bandwidth b and that we use the coloring filter \(H=\sqrt{bn/\pi}\), where n is the size of f. It can be shown that the L2-gain from f to z is

Figure 1: The weighting filter H colours the white noise signal w.

$$\int_{-b}^{b} \textrm{tr}(G^{*}G)d\omega \rightarrow ||G||_{2},\ \textrm{as} \ b \rightarrow \infty$$

Let us consider the case when G(s) = 1/(s + 1). The following sequence of commands gives the solution gain=0.705, which is close to the theoretical optimum 0.70485.  The choice of pole locations, in our case we use the single pole a = 1, is in general critical.

>>G=ss(-1,1,1,0);

>>b=100;

>>H=sqrt(b/pi);

>>a=[1];

>>abst_init_iqc;

>>f=iqc_white(1,b,a);

>>z=G*H*f;

>>gain=iqc_gain_tbx(f,z);

See also

iqc_window

References

U. J¨onsson and A. Megretski. Some new IQCs and their application. Under Preparation.