Presentation on theme: "Stability Margins Professor Walter W. Olson"— Presentation transcript:
1 Stability Margins Professor Walter W. Olson Department of Mechanical, Industrial and Manufacturing EngineeringUniversity of ToledoStability Margins
2 Outline of Today’s Lecture ReviewOpen Loop SystemNyquist PlotSimple Nyquist TheoremNyquist Gain ScalingConditional StabilityFull Nyquist TheoremIs stability enough?Margins from Nyquist PlotsMargins from Bode PlotNon Minimum Phase Systems
3 Loop Nomenclature Disturbance/Noise Reference Error Input signal +-Outputy(s)ErrorsignalE(s)Open LoopSignalB(s)PlantG(s)SensorH(s)PrefilterF(s)ControllerC(s)Disturbance/NoiseReferenceInputR(s)The 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 theprefilter, the controller, the plant and the sensor and has the transfer functionF(s)C(s)G(s)H(s)The closed loop signal is the output of the system and has the transfer function
4 Open Loop System Error signal Input Output E(s) r(s) y(s) Controller Note: Your book uses L(s) rather than B(s)To avoid confusion with the Laplace transform, I will use B(s)Open Loop SystemErrorsignalE(s)Inputr(s)Outputy(s)ControllerC(s)PlantP(s)+Open LoopSignalB(s)Sensor-1
5 Simple Nyquist Theorem ErrorsignalE(s)+Outputy(s)Open LoopSignalB(s)PlantP(s)ControllerC(s)Inputr(s)Sensor-1-1RealImaginaryPlane of the Open LoopTransfer FunctionB(0)B(iw)-1 is called thecritical pointStableUnstable-B(iw)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.
6 Nyquist Gain ScalingThe form of the Nyquist plot is scaled by the system gain
7 Conditional StabiltyWhlie most system increase stability by decreasing gain, some can be stabilized by increasing gainShow with Sisotool
8 Definition of StableA system described the solution (the response) is stable if that system’s response stay arbitrarily near some value, a, for all of time greater than some value, tf.
9 Full Nyquist TheoremAssume 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.
10 Determination of Stability from Eigenvalues Continuous TimeDiscrete TimeUnstableStableAsymptotic Stability
12 MarginsMargins are the range from the current system design to the edge of instability. We will determineGain 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?
13 Gain and Phase Margin Definition Nyquist Plot -1
21 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.
22 Stability MarginIt is possible for a system to have relatively large gain and phase margins, yet be relatively unstable.Stabilitymargin, sm
23 Non-Minimum Phase Systems Non 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:
24 Another Non Minimum Phase System A Delay Delays are modeled by the function which multiplies the T.F.
25 Summary Is stability enough? Margins from Nyquist Plots Margins from Bode PlotNon Minimum Phase SystemsNext Class: PID Controls