iqc_delay1

Purpose

Defines IQCs for the relation

$$w=e^{-sT}v,$$

where 0  ≤ T ≤ T0 is an uncertain time delay.

Synopsis

w==iqc_delay1(v,T0,a)

[w,X,x]=iqc_delay1(v,T0,a)

Description

The IQCs have the form

$$\left\langle Hv,XHv\right\rangle - \left\langle w-v,X(w-v)\right\rangle \geq 0$$

where

$$H(s)=2sT_{0}\frac{T_{0}s+\sqrt{12.5}}{(T_{0}s)^{2}+aT_{0}s+b},\quad b=\sqrt{50},\quad a=\sqrt{2b+6.5}$$

and

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

where \(a_{i}>0\), and \(x_{i}\) are arbitrary scalar variables.

Inputs/Outputs

Inputs:

v   Input to the delay function.

T0 Maximum time delay. Default T0=1.

a   Pole locations of the multiplier X, a=[a1,..,an]. Default a=[].

Outputs:

w  Output from the delay function.

X  The multiplier X (optional).

x  The coefficients of X (optional).

Example

Figure 1: Simple feedback interconnection of system with time delay.

We Consider the system for the above Figure 1, where \(e^{-sT}\) is an uncertain delay with T ∈ [0, T0], and

$$G=\frac{0.5}{s^{2}+0.25s+1}$$

We want to find a bound on T0. The sequence of commands below show that the gain from f to v is finite (gain= 185.409) when T0 = 0.5. The system is thus stable when T0 ≤ 0.5.

>>G=tf(0.5,[1 0.25 1]);

>>abst_init_iqc;

>>w=signal;

>>f=signal;

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

>>w==iqc_delay1(v,0.5);

>>gain=iqc_gain_tbx(f,w);

See also

iqc_window, iqc_cdelay, iqc_delay

Reference

A. Megretski and A. Rantzer. System analysis via integral quadratic constraints. IEEE Transactions on Automatic Control, 42(6):819–830, June 1997.