iqc_dzn_e_odd

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_odd(v,a,N,kdzn);

[w,h_0,H,F,xp,xm]=iqc_dzn_e_odd(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 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.

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_odd(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.