ultis: LTI static diagonally repeated parametric uncertainties

The class ultis is defined by LTI diagonally repeated parametric uncertainties of the form:

$\Delta_\mathrm{ultis}=\left(\begin{array}{cccc}\delta_1I_{n_1}&0&\cdots & 0\\0 & \ddots & \ddots & \vdots\\ \vdots & \ddots & \ddots & 0\\0 & \cdots & 0 & \delta_NI_{n_N} \end{array}\right).$

Here

$\delta_i\in\mathbf{S}\subset\mathbb{R}$ , $i\in\{1,\ldots,N\}$ .
$\mathbf{S}$ is star convex (i.e., $[0,1]\mathbf{S}\subset\mathbf{S}$ )
$\|\Delta\|_\infty\leq\alpha$

Note: In case the length of the basis function (i.e. one of the properties to be specified for this class) is 1, the class generalizes to the class of LTV diagonally repeated parametric uncertainties that can vary arbitrarily fast.

The ultis class can be defined by

$\Delta_\mathrm{ultis}=ultis('name')$
$\Delta_\mathrm{ultis}=ultis('name',varargin)$

Just specifying $\Delta_\mathrm{ultis}=ultis('name')$ defines an LTI parametric uncertainty $\delta$ of dimension $1\times1$ , which is repeated once and which satisfies $\delta\in\mathbf{S}=[-1,1]$ .

Specifying and/or changing properties proceeds as summarized in the following two tables for properties related to the uncertainty and to IQC-multiplier respectively.

[nr_1,\ldots,nr_N] — Uncertainty characteristics

Property	Description
BasisFunctionType	Specify the type of basis function to be used in the multiplier (default = 1). See link for further details.
Length	Specify the length of the basis function (default = 1). Note: In case of multiple diagonally repeated uncertainties, one can specify one common length, or a different one for each $\delta_i$ respectively as $l$ or $[l_1,l_2,\ldots]$ . See link for further details.
PoleLocation	Specify the pole location of the basis function (default = -1). Note: In case of multiple diagonally repeated uncertainties, one can specify one common pole-location, or a different one for each $\delta_i$ respectively as $pl$ or $[pl_1,pl_2,\ldots]$ . See link for further details.
SampleTime	Specify the sample time (default = 0).
RelaxationType	Specify the relaxation type. Options are (default = ‘DG’): – DG-scalings: ‘DG’ – Convex hull relaxation: ‘CH’ – Partial convexity: ‘PC’ – Zeroth order Polya relaxation: ‘ZP’
RelaxationProp	Specify the relaxation constraint type. Options are (default = ‘S’) – Static relaxation constraints: ‘S’ – Dynamic relaxation constraints: ‘D’
PrimalDual	Specify whether the multiplier should be a primal/dual parametrization (default = ‘Primal’). – Primal multipliers: ‘Primal’ – Dual multipliers: ‘Dual’ Note: For a standard IQC-analysis, all multipliers must be primal ones.

Multiplier characteristics

Note: See Section 5.3 of [1] for the details on the mathematical derivation of the IQC-multiplier.

Property	Description
NumberOfRepetitions	Specify the number of repetitions of the uncertainty (default = 1). Note: In case of more than one uncertainty, one needs to specify the number of repetitions as $[nr_1,\ldots,nr_N]$ .
Bounds	Specify the domain on which the uncertainty is defined (default = $\left[-1;1\right]$ ). $\left[\begin{array}{ccc}-\delta_1&\cdots&-\delta_N\\\delta_1&\cdots&\delta_N\end{array}\right]$
Polytope	Alternatively, instead of using the option Bound one can specify the option Polytope: $\left[\begin{array}{ccc}\delta_1^{p_1}&\cdots& \delta_N^{p_1}\\\vdots& \cdots& \vdots\\ \delta_1^{p_M}&\cdots& \delta_N^{p_M}\end{array}\right]$ Note: It is always assumed that the 0 is contained in the set.
InputChannel/ OutputChannel	Specify which input and output channels of the uncertain plant are affected by $\Delta_\mathrm{ultis}$ . For each $\delta_i$ , the channels should be specified as: $\begin{array}{c} row_{in,i}=\left[\begin{array}{ccc}C_{x_i}^{in}&\cdots&C_{y_i}^{in}\end{array}\right]\\row_{out,i}=\left[\begin{array}{ccc}C_{v_i}^{in}&\cdots&C_{w_i}^{in}\end{array}\right] \end{array}$ Here the order of the channels is not relevant, while $C_{m_i}^{in}$ , $C_{n_i}^{out}$ respectively denote the $m^{th}$ and $n^{th}$ in- and output channel of the uncertain plant $M$ . Note here thatthe row length of $row_{in,i}$ and $row_{out,i}$ equals the number of repetitions of $\delta_i$ . The option InputChannel/ OutputChannel should then be specified as a cell: $\begin{array}{c}InputChannel= \\ =\left\{\!\!\!\begin{array}{ccc}row_{in,1}\!\!\!&\cdots\!\!\!& row_{in,N} \end{array}\!\!\!\right\}\\OutputChannel =\\ =\left\{\!\!\!\begin{array}{ccc} row_{out,1}\!\!\!\!&\cdots\!\!\!\!&row_{ out ,N} \end{array}\!\!\!\right\}\end{array}$