iqc_ltiunmod

Purpose

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

$$ \hat{w}(j\omega)=\Delta(j\omega)\hat{v}(j\omega),$$

where \(\hat{v}(j\omega)\) and  \(\hat{w}(j\omega)\)  represent the harmonic spectrum of the signals v(t), w(t) at the frequency \(\omega.\ \Delta(s)\) is an unknown Linear Time-Invariant (LTI) system with bounded \(H_{\infty}\)-norm, i.e. \(| \Delta(j\omega) | < g\ \forall \omega\), for some constant g.

Synopsis

w==iqc_ltiunmod(v,a,n,g);

[w,X]=iqc_ltiunmod(v,a,n,g);

Description

The IQC given by this M-file has the form

$$ g^{2} \cdot \left\langle v,Xv \right\rangle -\left\langle w,Xw \right\rangle \geq 0,\quad X>0,$$

where

$$ X(s)=x_{0}+\frac{x_{1}}{s+a_{1}}+\cdots+\frac{x_{N}}{s+a_{N}}$$

and \(a_{i}\) are positive constants, \(x_{i}\) are arbitrary scalar variables.

Inputs/Outputs

Inputs:

v     Input signal.

a     Row vector. The negative value of a(i) gives a pole of the multiplier X(s). Default: a=1

n    The dimension of the output signal w. Default: n= size of v

g    Upper bound of the \(H_{\infty}\)-norm of \(\Delta(s)\). Default: gain= 1

Outputs:

w   Basic signal has dimension n × 1.

X    A rectangular matrix variable with size (m + 1) × 1, where m is the length of a. The elements of X, X(i), are the scalar variables \(x_{i-1}\) in the multiplier X(s).

Example

Figure 1: System with LTI unmodeled dynamics

Consider the system in Figure 1, where 

$$ G(s)=\frac{1}{1.5}\begin{bmatrix} \frac{s}{s+100} & 0\\ \frac{s}{s+80} & 0\\ \frac{10}{s+10} & \frac{8}{s+8}\end{bmatrix},\quad \Delta(s)=\begin{bmatrix} \Delta^{(1)} & 0\\0 & \Delta^{(2)}\end{bmatrix}$$

 \(\Delta^{(1)}\) and \(\Delta^{(2)}\) represent LTI unmodeled dynamics, which have size 1 × 1 and 2 × 1, respectively. Both  \(\Delta^{(1)}(s)\)  and  \(\Delta^{(2)}(s)\) have \(H_{\infty}\)-norm less than 1. We are interested in studying the stability of the interconnected system.  The following IQCβ script shows that the gain from f to v is finite. The estimated gain (using the multipliers which have the form \(x_{0}+\frac{x_{1}}{s+1}\))  is 2.83. Hence, the system is stable under the appearance of the unmodeled dynamics \(\Delta^{(1)}\) and \(\Delta^{(2)}\).

>> s=tf([1 0],1);

>> G=1/1.5*[s/(s+100),0;s/(s+80),0;10/(s+10),8/(s+8)];

>> abst_init_iqc;

>> f=signal(2);

>> w1=signal;

>> w2=signal;

>> a=4;

>> v=M*(f+[w1;w2]);

>> w1==iqc_ltiunmod(v(1),a,1,1);

>> w2==iqc_ltiunmod(v(2:3),a,1,1);

>> gain=iqc_gain_tbx(f,v)

See also

iqc_diag, iqc_ltigain, iqc_ltvnorm, iqc_tvscalariqc_slowtv