Loop Transfer Function

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Professor Walter W. OlsonDepartment of Mechanical, Industrial and Manufacturing EngineeringUniversity of ToledoLoop Transfer Function-1RealImaginaryPlane of the Open Loop Transfer Function B(0)B(iw)

-1 is called the critical pointStableUnstable-B(iw)1Outline of Todays LectureReviewPartial Fraction Expansionreal distinct rootsrepeated rootscomplex conjugate rootsOpen Loop SystemNyquist PlotSimple Nyquist TheoremNyquist Gain ScalingConditional StabilityFull Nyquist Theorem

Partial Fraction ExpansionWhen using Partial Fraction Expansion, our objective is to turn the Transfer Function

into a sum of fractions where the denominators are the factors of the denominator of the Transfer Function:

Then we use the linear property of Laplace Transforms and the relatively easy form to make the Inverse Transform.

Case 1: Real and Distinct Roots

Case 1: Real and Distinct RootsExample

Case 2: Complex Conjugate Roots

Case 3: Repeated Roots

Heaviside Expansion

Loop NomenclatureReferenceInputR(s)+-Outputy(s)ErrorsignalE(s)Open LoopSignalB(s)PlantG(s)SensorH(s)PrefilterF(s)ControllerC(s)+-Disturbance/NoiseThe plant is that which is to be controlled with transfer function G(s)The prefilter and the controller define the control laws of the system.The open loop signal is the signal that results from the actions of the prefilter, the controller, the plant and the sensor and has the transfer function F(s)C(s)G(s)H(s)The closed loop signal is the output of the system and has the transfer function

Closed Loop System++Outputy(s)ErrorsignalE(s)Open LoopSignalB(s)PlantP(s)ControllerC(s)Inputr(s)

-1Open Loop System++Outputy(s)ErrorsignalE(s)Open LoopSignalB(s)PlantP(s)ControllerC(s)Inputr(s)

Note: Your book uses L(s) rather than B(s)To avoid confusion with the Laplace transform, I will use B(s)Sensor-1

Open Loop SystemNyquist PlotErrorsignalE(s)++Outputy(s)Open LoopSignalB(s)PlantP(s)ControllerC(s)Inputr(s)Sensor-1

-1RealImaginaryPlane of the Open Loop Transfer Function B(0)B(iw)

-1 is called the critical pointB(-iw)Simple Nyquist TheoremErrorsignalE(s)++Outputy(s)Open LoopSignalB(s)PlantP(s)ControllerC(s)Inputr(s)Sensor-1

Simple Nyquist Theorem:For the loop transfer function, B(iw), if B(iw) has no poles in the right hand side, expect for simple poles on the imaginary axis, then the system is stable if there are no encirclements of the critical point -1.-1RealImaginaryPlane of the Open Loop Transfer Function B(0)B(iw)

-1 is called the critical pointStableUnstable-B(iw)ExamplePlot the Nyquist plot for

-1ImReStableExamplePlot the Nyquist plot for

-1ImReUnstableNyquist Gain ScalingThe form of the Nyquist plot is scaled by the system gain

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Conditional StabiltyWhile most system increase stability by decreasing gain, some can be stabilized by increasing gainShow with Sisotool

Full Nyquist Theorem Assume that the transfer function B(iw) with P poles has been plotted as a Nyquist plot. Let N be the number of clockwise encirclements of -1 by B(iw) minus the counterclockwise encirclements of -1 by B(iw)Then the closed loop system has Z=N+P poles in the right half plane.

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SummaryOpen Loop SystemNyquist PlotSimple Nyquist TheoremNyquist Gain ScalingConditional StabilityFull Nyquist TheoremNext Class: Stability Margins-1ImReUnstable