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





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


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

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


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

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.



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


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



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);









See also

iqc_slope, iqc_monotonic, iqc_slope_odd, iqc_polytope_stvp


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.