## iqc_tvscalar

Purpose

Deﬁne IQC for the relation

$$w(t)=\delta(t)v(t),$$

where $$\delta(t)$$ is a scalar function which belongs to $$L_{\infty}$$ with the property $$| \delta(t) | ≤ D,\ \forall t$$.

Synopsis

w==iqc_tvscalar(v,D);

[w,X,Y]=iqc_tvscalar(v,D);

Description

The IQCs have the form

$$D^{2}\langle v,Xv \rangle - \langle w,Xw \rangle + \langle v,Yw \rangle + \langle w,Y^{T}v \rangle \geq 0,$$

where $$X=X^{T} \geq 0$$, and $$Y=-Y^{T}$$ are square matrices of real values.

Inputs/Outputs

Inputs:

v    Signal with dimentson n.

D   Positive scalar.  Upper bound of the $$L_{\infty}$$-norm of $$\delta(t)$$, i.e., $$| \delta(t) | <D,\ \forall t$$.  Default D=1.

Outputs:

w  Basic signal has the same dimension with v.

X  The multiplier X.

Y  The multiplier Y .

Example

Consider the system in Figure 1. The system G(s) is deﬁned according to

$$G(s) := D+C(sI-A)^{-1}B,$$

Figure 1: System with slowly time-varying coefﬁcient

where

$$A=\begin{bmatrix} -(a+bkT_{d}) & -bk\\1 & 0\end{bmatrix},\quad B = \begin{bmatrix} -2 & -2b\\0 & 0\end{bmatrix},\quad C=\begin{bmatrix} a+bkT_{d} & bk\\ kT_{d} & k\end{bmatrix},\quad D=\begin{bmatrix} 1 & 2b\\0 & 1\end{bmatrix}$$

and $$a = -0.3,\ b = 0.8,\ k = 2.5,\ T_{d} = 0.86,\ | \delta(t) | < 0.22,\ \forall t$$ is also known to us. We would like to check whether this system is stable. The commands below show that the gain from f to v is ﬁnite (the gain is computed to be gain=118.215). Hence, the system is stable under the appearance of the time-varying coefﬁcient.

>> a=-0.3;

>> b=0.8;

>> k=2.5;

>> Td=0.86;

>> A=[-(a+b*k*Td), -b*k; 1 0];

>> B=[-2, -2*b; 0, 0];

>> C=[a+b*k*Td, b*k; k*Td, k];

>> D=[1, 2*b;0, 1];

>> G=ss(A,B,C,D);

>> abst_init_iqc;

>> w=signal(2);

>> f=signal(2);

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

>> w==iqc_tvscalar(v,0.22);

>> gain=iqc_gain_tbx(f,v)