IQC-invariance analysis

Next to performing the more “conventional” IQC-based analyses with the function iqcanalysis, it is also possible to perform invariance analyses with dynamic IQCs as was recently pointed out in, for example, [21], [22], among others.

Along the lines of [21], IQClab now includes various tests that can be performed by means of the new function iqcinvariance.

Option	Description
$e2x$	As a first application, we consider one of the most common examples for regional analysis, namely the computation of invariant sets in the state-space under the assumption that the external disturbances $w$ are bounded in energy. Given the uncertain plant $q=M_{11}p+M_{12}w,\ \ p=\Delta(q),$ with realization $\begin{array}{l}\dot{x}=Ax+B_1p+B_2w,\ \ \ \ x(0)=0\\q = C_1x+D_{11}p+D_{12}w\end{array}$ compute the smallest hyper ellipsoidal region $x(t)\in\left\{x\in\mathbb{R}^n: x^TH^{-1}x\leq\alpha^2\right\},$ by minimizing the trace of $H$ for $\\|w\\|\leq\alpha$ . This option can be selected by specifying the performance metric option ‘e2x’ (energy to state) to the performance block.
$e2z$	Similarly, we might wish to compute invariant sets for the output $z$ , again under the assumption that the external disturbances $w$ are bounded in energy. Given the uncertain plant $\begin{array}{l}q=M_{11}p+M_{12}w,\ \ p=\Delta(q),\\z=M_{21}p+M_{22}w\end{array}$ with realization $\begin{array}{l}\dot{x}=Ax+B_1p+B_2w,\ \ \ \ x(0)=0\\q = C_1x+D_{11}p+D_{12}w\\z = C_2x\end{array}$ and $M_{21}(\infty)=M_{22}(\infty)=0$ is strictly proper, compute the smallest hyper ellipsoidal region $z(t)\in\left\{z\in\mathbb{R}^k: z^TH^{-1}z\leq\alpha^2\right\},$ by minimizing the trace of $H$ for $\\|w\\|\leq\alpha$ . This option can be selected by specifying the performance metric option ‘e2z’ (energy to output) to the performance block.
$e2p$	In practical applications, one might also be interested in bounds on the individual components of $z_j$ , $j\in\{1,\cdots,k\}$ , which can be interpreted as providing guaranteed bounds on the energy to peak gain for the performance channels $w\rightarrow z_j$ , $j\in\{1,\cdots,k\}$ . This leads to the third test. Given the uncertain plant $\begin{array}{l}q=M_{11}p+M_{12}w,\ \ p=\Delta(q),\\z=M_{21}p+M_{22}w\end{array}$ with realization $\begin{array}{l}\dot{x}=Ax+B_1p+B_2w,\ \ \ \ x(0)=0\\q = C_1x+D_{11}p+D_{12}w\\z = C_2x\end{array}$ and $M_{21}(\infty)=M_{22}(\infty)=0$ is strictly proper, compute the peak gains $\gamma_j$ on the performance output channels $z_j$ , $j\in\{1,\cdots,k\}$ such that $\|z_j(t)\|\leq\sqrt{\gamma_j}\alpha\ \ \forall t\geq0,\ j\in\{1,\cdots,k\}$ by minimizing $\sum_{j=1}^k\gamma_j$ . This option can be selected by specifying the performance metric option ‘e2p’ (energy to peak) to the performance block.
$x2p$	Another useful test is the possibility to guarantee bounds on the peak gain of the performance output $z$ for a given non-zero initial condition $x_0\neq0$ . Given the uncertain plant $\begin{array}{l}q=M_{11}p,\ \ p=\Delta(q),\\z=M_{21}p\end{array}$ with realization $\begin{array}{l}\dot{x}=Ax+B_1p,\ \ \ \ x(0)=x_0\\q = C_1x+D_{11}p\\z = C_2x\end{array}$ and $M_{21}(\infty)=0$ is strictly proper, compute the peak gain $\gamma$ on the performance output channel $z$ such that $\\|z(t)\\|=\\|C_2x(t)\\|\leq\gamma\\|x_0\\|\ \ \forall t\geq0$ by minimizing $\gamma$ . This option can be selected by specifying the performance metric option ‘x2p’ (state to peak) to the performance block.

The application proceeds in exactly the same fashion as the function iqcanalysis. Once all uncertainty blocks have been defined and all IQC-multipliers have been assigned, one can proceed by performing the invariance analysis by means of the function iqcinvariance:

$prob=iqcinvariance(M,\Delta,varargin).$