## iqc_dzn_e

Purpose

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.

Synopsis

w==iqc_dzn_e(v,a,N,kdzn);

[w,h_0,H,F,x]=iqc_dzn_e(v,a,N,kdzn);

Description

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

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

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).

Inputs/Outputs

Inputs:

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.

Outputs:

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

>>w==iqc_dzn_e(v,a1,N1,kdzn);

>>w==iqc_dzn_e_odd(v,a2,N2,kdz);

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.

$$G(s)=-\frac{0.9}{s(s+1)^{2}}.$$

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

$$G_{0}(s)=\frac{s+2}{(s+1)^{2}}.$$

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.

>>kphi=0.9;

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

>>G0=(s+2)/(s*s+2*s+1);

>>abst_init_iqc;

>>w=signal;

>>f=signal;

>>v=G0*w+f;

>>w==iqc_dzn_e(v,1,0,kphi);

>>gain=iqc_gain_tbx(f,v)