Circle Criterion 

The Lur'e problem investigates the stability of an important class of control systems whose forward path consists of a linear time-invariant system and whose feedback path consists of a memoryless nonlinearity.

For single-input, single-output systems, Lur'e's problem can be solved graphically using the circle criterion. It says that the number () of unstable poles of the closed-loop system in which satisfies the sector constraint is given by , where is the number of unstable poles of and is the number of clockwise encirclements by the Nyquist plot of around the disk corresponding to the feedback in the sector ().

A stable system ().

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X lsys = TransferFunctionModel[{{{1 + s}}, 1 + 3 s + s^2}, s];

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X Cases[TransferFunctionPoles[lsys], _?(Re[#] >= 0 &), {2}]

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For feedback in the sector (), there are no encirclements ().

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X NyquistPlot[lsys, FeedbackSector -> {1, 5}]

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Various nonlinearities within the feedback sector.

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X r = Sequence[4, 3.5, 3, 2, 1.5]; \[Phi] = Table[Sin[u] + \[Alpha] u, {\[Alpha], {r}}];

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X Plot[{5 u, \[Phi], u}, {u, -10, 10}, PlotLegends -> {"upper bound", r, "lower bound"}]

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Simulate the stable () closed-loop system.

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X \[ScriptCapitalR] = Table[OutputResponse[ SystemsModelFeedbackConnect[lsys, NonlinearStateSpaceModel[ {{}, {\[Phi]}}, {}, {u}, {Automatic}, Automatic , SamplingPeriod -> None]], UnitStep[t], {t, 0, 10}], {\[Phi], Reverse[\[Phi]]}];

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X Plot[\[ScriptCapitalR], {t, 0, 10}, PlotLegends -> Reverse[{r}]]

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