Create a Spark Motivational Contest

Professor Walter W. OlsonDepartment of Mechanical, Industrial and Manufacturing EngineeringUniversity of ToledoStability Margins

1Outline of Todays LectureReviewOpen Loop SystemNyquist PlotSimple Nyquist TheoremNyquist Gain ScalingConditional StabilityFull Nyquist TheoremIs stability enough?Margins from Nyquist PlotsMargins from Bode PlotNon Minimum Phase Systems

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

Open 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

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)Nyquist Gain ScalingThe form of the Nyquist plot is scaled by the system gain

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

Definition of StableA system described the solution (the response) is stable if that systems response stay arbitrarily near some value, a, for all of time greater than some value, tf.

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.

Determination of Stabilityfrom EigenvaluesContinuous Time

Discrete Time

Unstable

Stable

Asymptotic Stability

Is Stability Enough?If not Why Not?MarginsMargins are the range from the current system design to the edge of instability. We will determine

Gain MarginHow much can gain be increased?Formally: the smallest multiple amount the gain can be increased before the closed loop response is unstable.

Phase MarginHow much further can the phase be shifted?Formally: the smallest amount the phase can be increased before the closed loop response is unstable.Stability MarginHow far is the the system from the critical point?

Gain and Phase Margin DefinitionNyquist Plot-1

Example

Using Matlab command

nyquist(gs)Example

Here the gain from the previous plot has beenmultiplied by 3.2359

The result is that stability is about to be lost

Example

Using Matlab command

nyquist(gs)

Gain and Phase Margin DefinitionBode PlotsPositive Gain MarginPhase Margin-1800Phase, degMagnitude, dBwwPhase Crossover FrequencyExample

Using Matlab command

bode(gs)

Example

Again, stability is about tobe lost.Example

Using Matlab command

bode(gs)

NoteThe book does not plot the Magnitude of the Bode Plot in decibels.

Therefore, you will get different results than the book where decibels are required.

Matlab uses decibels where needed.

Stability MarginIt is possible for a system to have relatively large gain and phase margins, yet be relatively unstable.

Stabilitymargin, sm

Non-Minimum Phase SystemsNon minimum phase systems are those systems which have poles on the right hand side of the plane: they have positive real parts. This terminology comes from a phase shift with sinusoidal inputsConsider the transfer functionsThe magnitude plots of a Bode diagram are exactly the same but the phase has a major difference:

Another Non Minimum Phase SystemA DelayDelays are modeled by the function which multiplies the T.F.

SummaryIs stability enough?Margins from Nyquist PlotsMargins from Bode PlotNon Minimum Phase Systems

Next Class: PID Controls