usbsr: Sector bounded and slope restricted nonlinearities

The class usbsr is defined by diagonally repeated sector-bounded and slope-restricted nonlinearities of the form

    \[p=\Delta(q)=\left(\begin{array}{c}\delta(q_1)\\ \vdots\\ \delta(q_N)\end{array}\right),\ \ \ \  q=\left(\begin{array}{c}q_1\\ \vdots\\ q_N\end{array}\right)\]


  • The sector constraint (\delta(x)-ax)(bx-\delta(x))\geq0 for all x and fixed constants a\leq0\leq b
  • and optionally, the incremental sector constraint (slope restriction)

        \[c\leq\dfrac{\delta(x_1)-\delta(x_2)}{x_1-x_2}\leq d\]

    for all x_1, x_2, x_1\neq x_2 and with fixed constants c\leq0\leq d.


  • \delta\in slope(c,d)\Rightarrow\delta\in sector(a=c, b=d). Hence, if desired, one can consider tighter sector bounds with c\leq a and b\leq d.
  • For mathematically more precise formulations of the constraints see [1] and respectively [2] and [23] for the continuous- and discrete time version respectively.

The usbsr class can be defined by

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

Just specifying \Delta_\mathrm{sbsr}=usbsr('name') defines a sector bounded nonlinearity, which is repeated once, and which satisfies the constraint [a,b]=[0,1].

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

NumberOfRepetitions Specify the number of repetitions of the nonlinearity (default = 1).
SectorBoundsSpecify the sector constraints sector\in[a,b], a\leq0\leq b (default = [0,1]).
SlopeBounds Specify the slope bounds slope\in[c,d], c\leq0\leq d (default = [0,0]).
OddSpecify whether the nonlinearity is Odd (i.e., f(-x)=-f(x)):
no (default)
InputChannel/ OutputChannelSpecify which input and output channels of the uncertain plant are affected by \Delta_\mathrm{usbsr}. 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.
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

Previous page