## iqc_gain_tbx

Purpose

Gives best estimate of the L2 gain of a pair of signals in the system of IQCs deﬁned using the ”abst” IQC-environment.

Synopsis

gain=iqc_gain_tbx(f,y);

Description Figure 1: Gain estimation of an uncertain system

Consider an uncertain system as shown in Figure 1, in which G is deﬁned as

\begin{align*}&\dot{x}=Ax+B_{1}f+B_{2}w,\\ &y=C_{1}x+D_{11}f+D_{12}w,\\ &v=C_{2}x+D_{21}f+D_{22}w.\end{align*}

and $$\Delta$$ is an uncertain operator which satisﬁes IQCs deﬁned by $$\sigma_{\Delta}(\lambda)>0$$, where

$$\sigma_{\Delta}=\int_{0}^{\infty}\begin{bmatrix}v\\w\end{bmatrix}^{T}\Pi(\lambda)\begin{bmatrix}v\\w\end{bmatrix}dt.$$

Note that $$\Pi$$ are parametrized by $$\lambda$$. Let $$\sigma_{p}(\gamma)$$ be deﬁned as

$$\sigma_{p}(\gamma)=\int_{0}^{\infty}\begin{bmatrix}y\\f\end{bmatrix}^{T}\begin{bmatrix}I &0\\0 & -\gamma I\end{bmatrix}\begin{bmatrix}y\\f\end{bmatrix}dt.$$

During the execution of iqc_gain_tbx, a system of LMIs will be generated to solve the following problem

\begin{align*}&\textrm{inf }\gamma,\ (\textrm{over }\lambda,\gamma),\ \textrm{such that},\\ &\sigma_{p}(\gamma)+\sigma(\lambda)<0. \end{align*}

By the Kalman-Yakubovich-Popov Lemma, this optimization problem is equivalent to

\begin{align*}&\textrm{inf }\gamma,\ (\textrm{over }\lambda,\gamma,P),\ \textrm{such that},\\ &\begin{bmatrix}PA+A^{T}P & PB\\B^{T}P & 0 \end{bmatrix}+\Xi<0. \end{align*}

where P is a positive symmetric matrix, and

\begin{align*} &\Xi=\Xi_{1}+\Xi_{2},\\ &\Xi_{1}=\begin{bmatrix} C_{1} & D_{11} & D_{12}\\ 0 & I & 0 \end{bmatrix}^{T}\begin{bmatrix} I & 0\\0 & -\gamma I \end{bmatrix} \begin{bmatrix} C_{1} & D_{11} & D_{12}\\0 & I & 0\end{bmatrix},\\ & \Xi_{2}=\begin{bmatrix} C_{2} & D_{21} & D_{22}\\ 0 & 0 & I \end{bmatrix}^{T}\Pi(\lambda)\begin{bmatrix} C_{2} & D_{21} & D_{22}\\ 0 & 0 & I \end{bmatrix}.\end{align*}

LMI Control Toolbox wil be revoked by iqc_gain_tbx.m to solve minimization problem. If ABST.options(5)> 100, the resulting LMI script is exported in the format of two ﬁles: lmi exe.m and lmi_exe.mat (containing the script and the dataﬁle respectively), as explained in the previous chapter.

Example