Sector-bounded and slope-restricted nonlinearities

In this section it is demonstrated how to create a sector-bounded and slope-restricted nonlinearity block. To do so, suppose that this nonlinearity has the following properties:

  • Name: ‘delta’
  • Operator type: p=\Delta(q)=\left(\begin{array}{c}\delta(q_1)\\\delta(q_2)\\ \delta(q_3)\end{array}\right), q=\left(\begin{array}{c}q_1\\q_2\\q_3\end{array}\right) is a three times repeated nonlinearity satisfying the constraints:
    • Sector constraint: \delta\in\mathrm{sector}(0,1)
    • Slope constraint: \delta\in\mathrm{slope}(0,1)
    • The nonlinearity is an odd function.
  • Input channels of M connecting the uncertainty block: 1:3
  • Outputs channels of M connecting the uncertainty block: 1:3

This uncertainty can be created as follows:

delta = iqcdelta('delta','LinNonlin','NL','InputChannel', 1:3,'OutputChannel',1:3,'SectorBounds',[0,1],'SlopeBounds',[0,1],'Odd','yes');

In a next step, you have to assign an IQC-multiplier. The appropriate class for this is called usbsr (for details see here).

Now suppose you want the basis function have the following properties:

  • Length: 3 (this corresponds to a McMillan degree of 2)
  • Pole location: -1

This can be specified as:

delta = iqcassign(delta,'usbsr','Length',3, 'PoleLocation',-1);

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