iqc_polytope

Purpose

Defines the IQCs for a polytopic uncertainty, i.e., \(w(t) = \Delta(t)v(t)\), where \(\Delta(t)\) takes values in the polytope \(C  =  \bar{co} \{ \Delta_{1},\cdots,\Delta_{N}\}\). We assume \(0 \in C\). The vertices are assumed to be \(\Delta_{i} \in \textbf{R}^{n\times m}\), where the m corresponds to the input dimension and n corresponds to the output dimension.

Synopsis

w==iqc_polytope(v,C);

[w,X,Y,Z]=iqc_polytope(v,C);

Description

The IQCs have the form

$$ \left\langle \begin{bmatrix} v\\w \end{bmatrix}, \begin{bmatrix} Z & Y\\ Y^{T} & -X \end{bmatrix} \begin{bmatrix} v\\w \end{bmatrix} \right\rangle \geq 0,$$

where \( X=X^{T} \geq 0,\ Z=Z^{T} \), and

$$ \begin{bmatrix} I\\ \Delta_{i} \end{bmatrix}^{T} \begin{bmatrix} Z & Y\\ Y^{T} & -X \end{bmatrix} \begin{bmatrix} I\\ \Delta_{i}\end{bmatrix} \geq 0$$

for \( i=1, \cdots, N\).

Inputs/Outputs

Inputs:

v   Input to the uncertainty.

C   The input C is a cell array containing the vertices of the polytope.

Outputs:

w  The output of the uncertainty.

X   The parameter X (optional).

Y   The parameter Y (optional).

Z   The parameter Z (optional).

Example

The system in Figure 1 consists of the linear system

$$ G(s)=\begin{bmatrix} \frac{0.15}{s^{2}+0.3s+1} & -\frac{0.72}{s+1}\end{bmatrix}$$

and the polytopic uncertainty

$$ \Delta(t) \in \bar{co} = \left\{ \begin{bmatrix} 1\\1 \end{bmatrix},\begin{bmatrix} -1\\1\end{bmatrix}, \begin{bmatrix} 1\\-1 \end{bmatrix}, \begin{bmatrix} -1\\-1 \end{bmatrix}\right\}.$$

Figure 1: Simple feedback interconnection of polytopic uncertainty and linear system.

The following sequence of code estimates the gain from f to v to be gain=234.2 by using iqc_polytope.

>>G1=ss([-0.3 -1;1 0],[1;0],[0 0.15],0);

>>G2=ss(-1,-0.72,1,0);

>>G=[G1 G2]

>>C{1}=[1;1];

>>C{2}=[1;-1];

>>C{3}=[-1;1];

>>C{4}=[-1;-1];

>>abst_init_iqc;

>>w=signal(2);

>>f=signal(2);

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

>>w==iqc_polytope(v,C);

>>gain=iqc_gain_tbx(f,v)

See also

iqc_polytope_stvpiqc_tvscalar

Reference

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