Defines the Zames-Falb IQCs for a integrator encapsulated in a deadzone nonlinearity with slope in [0, kdzn], see Figure.

The encapsulation defines the operator

$$ \begin{align*}&\dot{z}=\varphi(v-z),\quad z(0)=0\\&w=\varphi(v-z),\end{align*}$$

where \(\varphi\) denotes the deadzone nonlinearity. The IQCs holds for any encapsulated nonlinearity \(\varphi\) with slope in [0, kdzn]. However, the IQCs are particularly interesting for nonlinearities of deadzone type, i.e., when the slope is zero for small values of the input to the nonlinearity.

Figure 1: Illustration of the encapsulation operator.





The IQCs are of the form

$$\left\langle w,(h_{0}-H)(v-\frac{1}{k_{dzn}}w)\right\rangle + \left\langle w, Fw \right\rangle \geq 0,$$

where \(F=(H(s)-H(0))/s\), and


Here \(a\) is any nonzeros real number and \(N \geq 0\).

The following constraints need to be satisfied

(i) \(h(t)=\sum_{k=0}^{N}\textrm{sign}(a)x_{k}t^{k}e^{-at}/k! \geq \left\{\begin{array}{c}0,\ \forall t \geq 0,\ if\ a>0\\0,\ \forall t \leq 0,\ if\ a<0,\end{array}\right.\)

(ii) \(H(0)=\sum_{k=0}^{N}\frac{x_{k}}{a^{k+1}} \leq h_{0}\)

We refer to the manual for information on the implementation of constraint (i) and (ii).



v      Input to the encapsulated nonlinearity.

a      Pole location. Default a=1.

N      Defines number of terms in expansion for H. Default N=0.

kdzn Slope bound for the encapsulated nonlinearity. Default kdz=1.


w     Output from the encapsulated nonlinearity.

h_0  Parameter h0 (optional).

H      Multiplier H (optional).

F      Multiplier F = (H(s) − H(0))/s (optional).

x      The parameters defining H, x=[x0,..,xN] (optional).

Example 1

In order to obtain more than one pole in H(s) call the iqc_slope command 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

Consider the system in Figure 2, where

Figure 2: A system with integrator in feedback interconnection with a deadzone nonlinearity with slope 1.

Figure 3: Transformation of the systemn in Figure 2.  The integrator mode is now encapsulated with the deadzone nonlinearity.


The nominal dynamics \(G(s)/s\) is unbounded due to the integrator mode. After partial fractions expansion we get \(G(s) = k(G_{0}(s) − 1/s)\), where \(k = −G(0) = 0.9\), and


We can now transform the system as in Figure 3. Stability of the transformed system also implies stability of the original system in Figure 2. The following IQCβ code computes gain=4.6955, which shows that the system is stable.


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








See also

iqc_dzn_e_odd, iqc_sector, iqc_monotonic, iqc_slope_oddiqc_polytope_stvp


U. Jönsson and A. Megretski. The Zames Falb IQC for critically stable systems. To appear in the proceedings of the 1998 American Control Conference.

U Jönsson and A. Megretski. The Zames Falb IQC for critically stable systems. Technical Report LIDS-P-2405, Laboratory for Information and Decision Systems, Massachusetts Institute of Technology, 1997.