iqc_slope_odd

Purpose

Defines the Zames-Falb IQCs for an odd slope restricted nonlinearity, i.e., a nonlinearity satisfying \( \varphi(-x)=-\varphi(x)\), and a slope condition

$$\alpha(x_{1}-x_{2})^{2} \leq (\varphi(x_{1})-\varphi(x_{2}))(x_{1}-x_{2}) \leq \beta(x_{1}-x_{2})^{2},$$

where \( -\infty < \alpha \leq \beta < \infty\).

Synopsis

w==iqc_slope_odd(v,a,N,alpha,beta)

[w,h0,H,xp,xm]=iqc_slope_odd(v,a,N,alpha,beta)

Description

The IQCs are of the form

$$\langle -\alpha v+w,(h_{0}-H)(\beta v-w)\rangle \geq 0,$$

where \( h_{0} \geq 0\), and

$$H(s)=\sum_{k=0}^{N}\frac{x_{k}}{(s+a)^{k+1}},$$

where a is a nonzero real number. The parameters \(h_{0}\) and \(x_{k}\) are subject to the constraint

$$ ||h||_{1} = \int_{-\infty}^{\infty} |h(t)| dt \leq h_{0},$$

where

$$ h(t)=\sum_{k=0}^{N}\frac{\textrm{sign}(a)x_{k}}{k!}t^{k}e^{-at}\theta(\textrm{sign}(a)t),$$

where \(\theta(t)\) is the unit step function, i.e., \(\theta(t) = 1,\ t \geq 0\) and \(\theta(t) = 0\) otherwise. We refer to the manual regarding the implementation of the constraint in above equation.

Inputs/Outputs

Inputs:

v Input to nonlinearity.

a Pole location for the multiplier H. Default a=1.

N Length of the expansion that defines H. Default N=0.

alpha Lower bound on slope. Default alpha=0.

beta Upper bound on slope. Default beta=0

Outputs:

w Output from nonlinearity.

h0 The parameter h0 (optional).

H The multiplier H (optional).

x The decision variables (optional).

Example 1

In order to obtain more than one pole in H(s) call the iqc slope oddcommand twice as

w==iqc_slope_odd(v,a1,N1,alpha,beta);

w==iqc_slope_odd(v,a2,N2,alpha,beta);

The constraints (i) and (ii) will be conservative in general except for the case when a1 and a2 have different signs. See the manual for a more thorough discussion.

Example 2

It is sometimes convenient to combine Zames and Falbs IQC with the Popov IQC. Consider, for example, the system in Figure 1, where

$$ G(s)=\frac{(2s^{2}+s+2)(s+100)}{(s+10)^{2}(s^{2}+5s+20)},$$

Figure 1: Simple feedback interconnection of nonlinearity and linear system.

and where \(\varphi\) is odd and has its slope restricted to the intervall [0, 100].

The Popov criterion cannot be used to prove stability for this system but it turns out that by combining an IQC from iqc slope oddwith a Popov IQC we get a good enough characterization of \(\varphi\) to prove stability. The following sequence of commands was used to prove that the gain from f to v is gain= 2.8360.

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

>>G=((2*s*s+s+2)*(s+100))/((s+10)*(s+10)*(s*s+5*s+20));

>>abst_init_iqc;

>>w=signal;

>>f=signal;

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

>>w==iqc_slope_odd(v,-1,1,0,100);

>>w==iqc_popov_vect(v);

>>gain=iqc_gain_tbx(f,v);

See also

iqc_slope, iqc_monotonic, iqc_slope_odd, iqc_polytope_stvp

Reference

G. Zames and P.L. Falb. Stability conditions for systems with monotone and slope-restricted nonlinearities. SIAM Journal of Control, 6(1):89–108, 1968.