ultid: Uncertain LTI dynamics

The class ultid is defined by LTI diagonally repeated dynamic scaler or full-block uncertainties of the form:

$\Delta_\mathrm{ultid}=I_{nr}\otimes\Delta=\left(\begin{array}{cccc}\Delta&0&\cdots & 0\\0 & \ddots & \ddots & \vdots\\ \vdots & \ddots & \ddots & 0\\0 & \cdots & 0 & \Delta\end{array}\right).$

Here

$\Delta\in RH_\infty^{n_{out}\times n_{in}}$ is a stable LTI MIMO transfer matrix of dimension $n_{out}\times n_{in}$
$nr$ is the number of repetitions
$\|\Delta\|_\infty\leq\alpha$

The ultid class can be defined by

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

Just specifying $\Delta_\mathrm{ultid}=ultid('name')$ defines an LTI dynamic uncertainty $\Delta$ of dimension $1\times1$ , which is repeated once and which satisfies $\|\Delta\|_\infty\leq1$ .

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

[n_{out}, n_{in}] — 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). See link for further details.
PoleLocation	Specify the pole location of the basis function (default = -1). See link for further details.
SampleTime	Specify the sample time (default = 0).
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 5.1 of [1] for the details on the mathematical derivation of the IQC-multiplier.

Property	Description
Dimensions	Specify the size of the uncertainty as $[n_{out}, n_{in}]$ . Default = $[1,1]$ .
NormBounds	Specify the $H_\infty$ -norm of the uncertainty by means of the constant $\alpha\geq0$ (default = 1).
NumberOf Repetitions	Specify the number of repetitions of the uncertainty (default = 1).
InputChannel/ OutputChannel	Specify which input/output-channels of the uncertain plant are affected by the uncertainty. For each repetition of the block one should specify a row in the following matrix: $\begin{array}{c}InputChannel=\left[\! \! \!\begin{array}{ccc}C_{r_1,x}^{in}&\cdots& C_{r_1,y}^{in} \\\vdots& \vdots& \vdots\\C_{r_N,x}^{in}&\cdots& C_{r_N,y}^{in} \\ \end{array} \! \! \! \right]\\OutputChannel=\left[ \! \! \! \begin{array}{ccc}C_{r_1,v}^{out}&\cdots& C_{r_1,w}^{ out } \\\vdots& \vdots& \vdots\\C_{r_N,v}^{ out }&\cdots& C_{r_N,w}^{ out } \\ \end{array} \! \! \! \right]\end{array}$ Here the order of the channels is not relevant, while $C_{r_j,m}^{in}$ and $C_{r_j,n}^{out}$ respectively denote the $m^{th}$ and $n^{th}$ in- and output channel of the uncertain plant $M$ .