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);

See also

iqc_d_slope_odd, iqc_slope, iqc_slope_odd, iqc_monotonic

Reference

LIDS-TR (??) Integral Quadratic Constraints for . . . . ,1998.