## iqc_diag

Purpose

Define IQC for the relation

$$w(t)=D(t)v(t)$$

where the size of input signal v(t) is n, and D(t) is a diagonal matrix function of the form

$$D(t)=\begin{bmatrix}\delta_{1}(t) & 0 & \cdots & 0\\0 & \delta_{2}(t) & \cdots & 0\\ & \vdots & & \\0 & 0 & \cdots & \delta_{n}(t)\end{bmatrix}$$

whose elements $$\delta_{i}(t)$$ belong to $$L_{\infty}$$ with $$|\delta_{i}|\leq 1$$, for all $$t\in R$$.

Synopsis

w==iqc_diag(v);

[w,X]=iqc_diag(v);

Description

The IQC defined by this M-file has the form

$$\left\langle v,Xv\right\rangle - \left\langle w,Xw\right\rangle \geq 0,\quad X>0,$$

where X is a real diagonal matrix variable with size n × n.

Inputs/Outputs

Inputs:

v   Signal with dimension n.

Outputs:

w  Basic signal has the same dimension with v.

X  The multiplier X.

Example

Consider the system

$$G\ :\ \dot{x}=(A+B\Delta(t)C)x,\quad x(0)=x_{0},$$

where $$\Delta$$ has the form

$$\Delta(t)=\begin{bmatrix}\delta_{1}(t) & 0\\0 & \delta_{2}(t)\end{bmatrix}$$

All elements $$\delta_{i}(t)$$ have the properties $$|\delta_{i}(t)|\leq 1\ \forall t$$. A, B, C have the values

$$A=\begin{bmatrix} -8.9130 & 1.5647\\2.5647 & -3.1850\end{bmatrix},\quad B=\begin{bmatrix}1 & 0\\0 & 1\end{bmatrix},\quad C=\begin{bmatrix} 2 & 0\\0 & 2\end{bmatrix}.$$

Figure 1: System with multiple unknown coefﬁcients

We are interesting in whether this system is stable. The system can be represented as in the block diagram in above Figure 1, where f represents the contribution from the initial condition ($$f=Ce^{At}x_{0}$$, for t ≥ 0 and 0 otherwise). If the gain from f to v (or to w) is bounded then we also know that the original system is stable. The IQC Toolbox script below gives the result g=3.6434, which implies that the system is stable.

>> A=[-8.9130,1.5647; 2.5647,-3.1850];

>> B=eye(2);

>> C=2*eye(2);

>> G=ss(A,B,C,zeros(2));

>> abst_init_iqc

>> f=signal(2);

>> w=signal(2);

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

>> w==iqc_diag(v);

>> g=iqc_gain_tbx(f,v)