ultid: Uncertain LTI dynamics

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

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


  • \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 .

DimensionsSpecify the size of the uncertainty as [n_{out}, n_{in}]. Default = [1,1].
NormBoundsSpecify the H_\infty-norm of the uncertainty by means of the constant \alpha\geq0 (default = 1).
NumberOf RepetitionsSpecify the number of repetitions of the uncertainty (default = 1).
InputChannel/ OutputChannelSpecify 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.
Uncertainty characteristics

BasisFunctionTypeSpecify the type of basis function to be used in the multiplier (default = 1). See link for further details.
LengthSpecify the length of the basis function (default = 1). See link for further details.
PoleLocationSpecify the pole location of the basis function (default = -1). See link for further details.
SampleTimeSpecify the sample time (default = 0).
PrimalDualSpecify 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.

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