## iqc_domharmonic

Purpose

IQCs for signals with dominant harmonics, i.e., the spectrum is concentrated to be in a certain frequency band. There are three alternatives, 1) bandpass, 2) lowpass, and 3) highpass characteristic:

1. supp $$\hat{f}(j\omega)=[-b,-a]\cup[a,b]$$ (supp means support)

2. supp $$\hat{f}(j\omega)=[-a,-a]$$, if b=[], a>0

3. supp $$\hat{f}(j\omega)=(-\infty,-|a|]\cup[|a|,\infty)$$, if b=[], and a<0

Synopsis

f=iqc_domharmonic(n,a,b,N1,N2)

[f,M,x,d]=iqc_domharmonic(n,a,b,N1,N2)

Description

The IQCs are of the form

$$\left\langle f,Mf\right\rangle \geq 0,$$

where $$M=xH^{*}H-d$$, and for the three cases we use

1. $$H$$ is $$N_{1}$$ order Butterworth bandpass filter with cut-off frequencies at $$\omega_{1}=a/N_{2}$$ and $$\omega_{2}=bN_{2}$$, and

• $$x\textrm{Re}H(ja)-d \geq 0$$

• $$x\textrm{Re}H(jb)-d \geq 0$$

• $$x,d \geq 0$$

2. $$H$$ is $$N_{1}$$ order Butterworth lowpass filter with cut-off frequency at $$\omega=aN_{2}$$, and

• $$x\textrm{Re}H(ja)-d \geq 0$$

• $$x,d \geq 0$$

3. $$H$$ is $$N_{1}$$ order Butterworth lowpass filter with cut-off frequency at $$\omega=|a|/N_{2}$$, and

• $$x\textrm{Re}H(j|a|)-d \geq 0$$

• $$x,d \leq 0$$

Figure 1 illustrates how “close to optimal” multipliers M look for the three different cases. It is important to note that the choice of parameter $$N_{2}$$ can be critical. The reason is that the Butterworth filters are maximaly flat with gain 1/2 at the cut-off frequency. It is therefore useful to use $$N_{2} > 1$$. The example in Figure 2 shows that when using $$N_{2} = 1$$ the multiplier $$M$$ will take large values in the passband while when we use $$N_{2} = 2$$ we have a multiplier of the type in Figure 1.

Figure 1: Multipliers for the three cases. An optimal ﬁlter has value zero in the passband and minus inﬁnity outside this frequency band.

Figure 2: Let us assume that we are interested in case 2 with a = 1. In solid line we show the multiplier $$M_{1} = H_{1}^{*}H_{1} - d_{1}$$, where $$H_{1}$$ is a fourth order Butterworth ﬁlter with cut-off frequency at $$\omega = 1$$ and $$d_{1}$$ is chosen such that $$M_{1}(j1) = 0$$.  In dashed line we show the ﬁlter $$M_{2} = H_{2}^{*}H_{2}-d_{2}$$, where $$H_{2}$$ is a fourth order Butterworth ﬁlter with cut-off frequency at $$\omega = 2$$ and $$d_{2}$$ is chosen such that $$M_{2}(j1) = 0$$. For the ﬁrst multiplier we used $$N_{2} = 1$$ while $$N_{2} = 2$$ for the second multiplier.

Inputs/Outputs

Inputs:

n    Size of signal. Default n=1.

a    Frequency bound (lower/upper). Default a=1.

b    Frequency bound (upper). Default b=[].

N1  Order of Butterworth filter.

N2  Determines cut-off frequency of filter.

Outputs:

f    The signal with dominant harmonics.

M  The multiplier M (optional).

x   The parameter x (optional).

d   The parameter d (optional).

Example

We want to compute the L2-gain from f to z for the system $$z=Gf$$ when

$$G=\frac{100}{s^{2}+0.1s+100},$$

and f has all its energy in the frequency interval [−5, 5]. The following sequency of commands computes the gain to be gain=1.3343, which is close to the theoretical value $$|G(j5)|=1.3333$$.

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

>>G=100/(sˆ2+0.1s+100);

>>abst_init_iqc;

>>f=iqc_domharmonic(1,5,[],9,2);

>>z=G*f;

>>gain=iqc_gain_tbx(f,z)

It is interesting to note that if we instead use N2 = 1, i.e., f=iqc_domharmonic(1,5,[],9,1) then we get the gain estimate gain=70.7897. This is expected from the discussion above. In fact, theoretically we obtain the gain $$||G||/\sqrt{2}$$ = 70.7116 when N2 = 1.