ultv: LTV parametric uncertainties

The class ultv is defined by LTV parametric uncertainties of the form:

    \[\Delta_\mathrm{ultv}(\delta)=\sum_{i=1}^{N}\delta_iT_i= \delta_1T_1+\cdots+ \delta_NT_N.\]


  • T_i, i\in\{1,\ldots,N\} are fixed matrices T_i\in\mathbb{R}^{n\times m} (having the same size as \Delta_\mathrm{ultv}).
  • \delta:[0,\infty]\rightarrow\Lambda is a piecewise continuous time-varying parameter vector that takes its values form the (compact) polytope

        \[\Lambda=\mathrm{co}\{\delta^1,\ldots,\delta^M\}=\left\{\sum_{a=1}^{M}b_a\delta^a: b_a\geq0,\ \sum_{a=1}^{M}b_a=1\right\}\]

    with \delta^a=(\delta_1^a,\ldots,\delta_N^a), a\in\{1,\ldots,M\} as generator points.
  • \Lambda is assumed to be star convex: [0,1]\Lambda\subset\Lambda

The ultis class can be defined by

  • \Delta_\mathrm{ultv}=ultv('name')
  • \Delta_\mathrm{ultv}=ultv('name',varargin)

Just specifying \Delta_\mathrm{ultv}=ultv('name') defines an LTV parametric uncertainty \delta on the interval [-1,1], which is repeated once (i.e., \Delta_\mathrm{ultv}(\delta)=\delta_1\in[-1,1] and T_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.

PolytopeWith the option Polytope on can specify the generator points of \Delta_\mathrm{ultv} as

    \[\left[\begin{array}{ccc}\delta_1^{p_1}&\cdots& \delta_N^{p_1}\\\vdots& \cdots&  \vdots\\ \delta_1^{p_M}&\cdots& \delta_N^{p_M}\end{array}\right]\]

Note: It is always assumed that the 0 is contained in the set.
UncertaintyMapSpecify the matrices T_1,\ldots,T_N as a cell array T=\{T_1,\ldots,T_N\}. This defines the uncertainty map \Delta_\mathrm{ultv}(\delta)=\delta_1T_1+\cdots+ \delta_NT_N.

Note: In case you wish to apply the DG-relaxation scheme, T_1,\ldots,T_N must be defined such that \Delta_\mathrm{ultv}(\delta)=\mathrm{diag}(\delta_1I_{\delta_1},\ldots, \delta_N <em>I_{\delta_N</em>}) with \delta_i\in[-1,1], i\in\{1,\ldots,N\} begin normalized.
InputChannel/ OutputChannelSpecify which input and output channels of the uncertain plant are affected by \Delta_\mathrm{ultv}. 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:

    \[ \begin{array}{c}InputChannel= \\ =\left\{\!\!\!\begin{array}{ccc}row_{in,1}\!\!\!&\cdots\!\!\!& row_{in,N} \end{array}\!\!\!\right\}\\OutputChannel =\\ =\left\{\!\!\!\begin{array}{ccc} row_{out,1}\!\!\!\!&\cdots\!\!\!\!&row_{ out ,N} \end{array}\!\!\!\right\}\end{array}  \]

Uncertainty characteristics

RelaxationTypeSpecify the relaxation type. Options are (default = ‘DG’):
– DG-scalings: ‘DG’
– Convex hull relaxation: ‘CH’
– Partial convexity: ‘PC’
– Zeroth order Polya relaxation: ‘ZP’
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 5.2 of [1] for the details on the mathematical derivation of the IQC-multiplier.

Previous page