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)