## iqc_d_slope

Purpose

Defines IQCs for the relation

$$\begin{bmatrix}w_{1}\\ \vdots\\w_{n}\end{bmatrix}=\Phi\left(\begin{bmatrix}v_{1}\\ \vdots\\ v_{n}\end{bmatrix}\right):=\begin{bmatrix}\phi(v_{1})\\ \vdots\\ \phi(v_{n})\end{bmatrix}$$

where $$\phi$$ : RR satisfies the slope restriction

$$\alpha < \frac{\phi(x)-\phi(y)}{x-y} \leq \alpha+\beta - \epsilon$$

for all $$x,y \in$$ R, where $$\alpha,\beta \in$$ R, $$\beta > 0$$ and $$\epsilon > 0$$.

Synopsis

w==iqc_d_slope(v,a,alpha,beta,ign)

[w,xa,xb,xc,xd,dd]=iqc_d_slope(v,a,alpha,beta,ign)

Description

The IQCs have the form

$$\left\langle q,Dp \right\rangle \geq \left\langle q,Hp\right\rangle$$

where

$$p=\left( 1+\frac{\alpha}{\beta}\right)v - \frac{1}{\beta}w,\quad q=-\alpha v+w$$

$$D$$ is diagonal,

$$H=H^{T}$$ is a convolution operator whose kernel components $$h_{ij}(t)$$ satisfy $$h_{ij}(t) \geq 0\quad \forall t \in$$ R, $$i,j=1,\cdots,n$$ and $$D_{ii} \geq \sum_{j=1}^{n}\int_{-\infty}^{\infty} h_{ij}(t)dt,\quad \forall i=1,\cdots,n.$$

Inputs/Outputs

Inputs:

v Input to the diagonal nonlinearity $$\Phi$$.

a Symmetric cell (of dimension n × n) of vectors $$a_{ij}$$ that determines the pole location of multipliers $$H_{ij}$$ according to the following table (a > 0, b > 0, c > 0):

Default a {i, j} =Inf i, j = 1, . . . , 0.

alpha Slope lower bound for $$\phi$$. Default alpha=0.

beta Slope upper bound for $$\phi$$ . Default beta=1.

ign Selects second order multipliers according to the table above. Default ign=0.

Outputs:

w Output signal from the iqc d slope function.

xa Coefﬁcient vector for multipliers in xa-column (optional).

xb Coefﬁcient vector for multipliers in xb-column (optional).

xc Coefﬁcient vector for multipliers in xc-column (optional).

xd Coefﬁcient vector for multipliers in xd-column (optional).

dd Vector of diagonal elements of D (optional).

Example

Consider the system in Figure 1. where $$\phi$$ is odd and satisﬁes the slope restriction

Figure 1: Feedback system with slope-restricted and odd diagonal nonlinearity.

$$0 < \frac{\phi(x)-\phi(y)}{x-y} < 1$$

The following commands can be used to obtain the upper bound 0.0099 for the L2 gain from f to z:

>>G1=tf(1,[1 1]);

>>G2=tf(1,[1 1.01]);

>>a={100 [1 Inf]; [1 Inf] 100};

>>abst_init_iqc;

>>w=signal(2);

>>f=signal;

>>v1=G1*f;

>>v2=G2*f;

>>z=w(1)-w(2);

>>w==iqc_d_slope([v1;v2],a);

>>gain=iqc_gain_tbx(f,z);