udel: Uncertain delay

The class udel is defined by diagonally repeated delay uncertainties of the form:

  • Type 1: \Delta_{del1}=(e^{-s\tau}-1)I_{nr}
  • Type 2: \Delta_{del2}=\frac{1}{s}(e^{-s\tau}-1)I_{nr}


  • s is the Laplace operator
  • nr is the number of repetitions
  • \tau\in[0,\tau_\mathrm{max}] with \tau_\mathrm{max} being the maximum delay time.

The udel class can be defined by

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

Just specifying \Delta_\mathrm{del}=udel('name') defines an LTI diagonally repeated delay uncertainty of Type 1, which has a maximum delay of \tau_\mathrm{max}=1 second and which is repeated once.

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

DelayTypeSpecify the type of delay operator (i.e., Type 1 or Type 2) (Default = 1).
DelayTimeSpecify the maximum time delay \tau_\mathrm{max} for \tau (default = 1).
NumberOfRepetitions Specify the number of repetitions of the uncertainty (default = 1).
InputChannel/ OutputChannelSpecify which input and output channels of the uncertain plant are affected by \Delta_\mathrm{udel}. The channels should be specified as:

    \[\begin{array}{c} InputChannel=\left[ \!  \!  \! \begin{array}{ccc}C_{x}^{in}&\cdots&C_{y}^{in}\end{array} \!  \!  \! \right]\\OutputChannel=\left[ \!  \!  \! \begin{array}{ccc}C_{v}^{in}&\cdots&C_{w}^{in}\end{array} \!  \!  \! \right] \end{array}\]

Here the order of the channels is not relevant, while C_{m}^{in}, C_{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.
AddIQCSpecify whether to add an additional LMI constraint for obtaining potentially improved results at the cost of more computational complexity.

Note: The default value is ‘yes’.
MstrictlyPropSpecify whether the part of the plant that is seem by the uncertainty satisfies M(\infty)=0. If so, one of the LMI constraints on the IQC (see previous property) can be dropped, allowing for more freedom in the optimization problem.

Note 1: The default value is ‘no’.
Note 2: This test is automatically turned to ‘yes’ in the IQC analysis if indeed M(\infty)=0.
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 5.5 of [1] for the details on the mathematical derivation of the IQC-multiplier.

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