iqc_ltigain

Purpose

Deﬁne IQC for a pair of L2 signals (v(t), w(t)) which have the relation

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

where δ is an unknown constant which ranges between − 1 and 1.

Synopsis

w==iqc_ltigain(v,a);

[w,P,Q,X,Y,Z]=iqc_ltigain(v,a);

Description

The IQC given by this M-ﬁle has the form

$$\left\langle v,Pv\right\rangle -\left\langle w,Pw \right\rangle + 2\left\langle v, Qw \right\rangle \geq 0,\quad P>0$$

where

\begin{align*} &P(s)=X_{0}+\frac{1}{s+a_{1}}X_{1}+\cdots+\frac{1}{s+a_{N}}X_{N},\\ &Q(s)=Z_{0}+\frac{s}{s^{2}-a_{1}^{2}}Y_{1}+\frac{a_{1}}{s^{2}-a_{1}^{2}}Z_{1}+\cdots+\frac{s}{s^{2}-a_{N}^{2}}Y_{N}+\frac{a_{N}}{s^{2}-a_{N}^{2}}Z_{N},\end{align*}

$$X_{i}$$ are arbitrary square matrices, $$Y_{i}$$ are arbitrary symmetric matrices, and $$Z_{i}$$ are arbitrary skew symmetric matrices.

Inputs/Outputs

Inputs:

v Input signal.

a Row vector. The negative value of each element of vector a gives the a pole of the multiplier P(s), while both positive and negative values of a(i) give a pair of the poles of the multiplier Q(s). Default: a=1

Outputs:

w Basic signal which has the same dimension as v.

P Multiplier P (s).

Q Multiplier Q(s).

X A cell array with size m × 1, where m is the length of a. The elements of X, X { i } , are rectangular matrix variables that appears in the multiplier P (s).

Y A cell array with size m × 1. The elements of Y, Y{ i } , are symmetric matrix variables that appears in the multiplier Q(s).

Z A cell array with size m × 1. The elements of Z, Z{ i } , are skew symmetric matrix variables that appears in the multiplier Q(s).

Example

Figure 1: System with unknown constant

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

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

where

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

and a = −0.3, b = 0.8, k = 2.5, Td = 0.86. δ is an unknown constant with the properties $$|\delta| < 0.22$$. 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=11.1456). Hence, the system is stable under the appearance of the unknown bounded constant.

>> 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==0.22*iqc_ltigain(v);

>> gain=iqc_gain_tbx(f,v)