Loop Shaping - Control
Loop ShapingBo Bernhardsson and Karl Johan ÅströmDepartment of Automatic Control LTH,Lund UniversityBo Bernhardsson and Karl Johan ÅströmLoop Shaping
Loop Shaping1Introduction2Loop shaping design3Bode’s ideal loop transfer funtion4Minimum phase systems5Non-minimum phase systems6Fundamental Limitations7Performance Assessment8SummaryTheme: Shaping Nyquist and Bode PlotsBo Bernhardsson and Karl Johan ÅströmLoop Shaping
IntroductionA powerful classic design methodElectronic Amplifiers (Bode, Nyquist, Nichols, Horowitz)Command signal followingRobustness to gain variations, phase margin ϕmNotions of minimum and non-minimum phaseBode Network Analysis and Feedback Amplifier Design 1945Servomechanism theoryNichols chartJames Nichols Phillips Theory of Servomechanisms 1947Horowitz (see QFT Lecture)Robust design of SISO systems for specified process variations2DOF, cost of feedback, QFTHorowitz Quantitative Feedback Design Theory - QFT 1993H - Loopshaping (see H Lecture)Design of robust controllers with high robustnessMc Farlane Glover Robust Controller Design Using NormalizedCoprime Factor Plant Descriptions 1989Bo Bernhardsson and Karl Johan ÅströmLoop Shaping
Harry Nyquist 1889-1976From farm life in Nilsby Värmland to Bell LabsDreaming to be a teacherEmigrated 1907High school teacher 1912MS EE U North Dakota 1914PhD Physics Yale 1917Bell Labs 1917Key contributionsJohnson-Nyquist noiseThe Nyquist frequency 1932Nyquist’s stability theoremBo Bernhardsson and Karl Johan ÅströmLoop Shaping
Hendrik Bode 1905-1982Born Madison WisconsinChild protégé, father prof at UIUC,finished high school at 14Too young to enter UIUCOhio State BA 1924, MA 1926 (Math)Bell Labs 1929Network theoryMissile systemsInformation theoryPhD Physics Columbia 1936Gordon McKay Prof of Systems Engineering at Harvard 1967(Bryson and Brockett held this chair later)Bo Bernhardsson and Karl Johan ÅströmLoop Shaping
Bode on Process Control and Electronic AmplifiersThe two fields are radically different in character and emphasis. . Thefields also differ radically in their mathematical flavor. The typicalregulator system can frequently be described, in essentials, bydifferential equations by no more than perhaps the second, third orfourth order. On the other hand, the system is usually highly nonlinear,so that even at this level of complexity the difficulties of analysis maybe very great. . As a matter of idle, curiosity, I once counted to findout what the order of the set of equations in an amplifier I had justdesigned would have been, if I had worked with the differentialequations directly. It turned out to be 55Bode Feedback - The History of and Idea 1960Bo Bernhardsson and Karl Johan ÅströmLoop Shaping
Nathaniel Nichols 1914 - 1997B.S. in chemistry in 1936 from CentralMichigan University,M.S. in physics from the University ofMichigan in 1937Taylor Instruments 1937-1946MIT Radiation Laboratory Servo Groupleader 1942-46Taylor Instrument Company Director ofresearch 1946-50Aerospace Corporation, San Bernadino, Director of the sensingand information divisionhttp://ethw.org/Archives:Conversations with the Elders - Nathaniel NicholsStart part 1 at Taylor: 26 min, at MIT:36 minBo Bernhardsson and Karl Johan ÅströmLoop Shaping
Isaac Horowitz 1920 - 2005B.Sc. Physics and MathematicsUniversity of Manitoba 1944.B.Sc. Electrical Engineering MIT 1948Israel Defence Forces 1950-51M.E.E. and D.E.E. Brooklyn Poly1951-56 (PhD supervisor Truxal whowas supervised by Guillemin)Prof Brooklyn Poly 1956-58Hughes Research Lab 1958-1966EE City University of New York 1966-67University of Colorado 1967-1973Weizmann Institute 1969-1985EE UC Davis 1985-91Air Force Institute of Technology 1983-92Bo Bernhardsson and Karl Johan ÅströmLoop Shaping
Horowitz on FeedbackHorowitz IEEE CSM 4 (1984) 22-23It is amazing how many are unaware that the primary reason forfeedback in control is uncertainty. .And why bother with listing all the states if only one could actually bemeasured and used for feedback? If indeed there were severalavailable, their importance in feedback was their ability to drasticallyreduce the effect of sensor noise, which was very transpared in theinput-output frequency response formulation and terribly obscure inthe state-variable form. For these reasons, I stayed with theinput-output description.Bo Bernhardsson and Karl Johan ÅströmLoop Shaping
Important Ideas and TheoryConceptsArchitecture with two degrees of freedomEffect and cost of feedbackFeedforward and system inversionThe Gangs of Four and SevenNyquist, Hall, Bode and Nichols plotsNotions of minimum and non-minimum phaseTheoryBode’s relationsBode’s phase area formulaFundamental limitationsCrossover frequency inequalityToolsBode and Nichols charts, lead, lag and notch filtersBo Bernhardsson and Karl Johan ÅströmLoop Shaping
The Nyquist PlotIm L(iω)Strongly intuitiveStability and RobustnessStability margins ϕm , gm ,sm 1/MsFrequencies ωms , ωgc , ωpc 1 1/gmsmϕmRe L(iω)Disturbance attenuationCircles around 1, ωscProcess variationsEasy to represent in the Nyquist plotParameters sweep and level curves of T (iω) Measurement noise not easily visibleCommand signal responseLevel curves of complementary sensitivity functionBode plot similar but easier to use for design because its widerfrequency rangeBo Bernhardsson and Karl Johan ÅströmLoop Shaping
Impact of the Nyquist Theorem at ASEAFree translation from seminar by Erik Persson ABB in Lund 1970.We had designed controllers by making simplified models, applyingintuition and analyzing stability by solving the characteristic equation.(At that time, around 1950, solving the characteristic equation with amechanical calculator was itself an ordeal.) If the system was unstablewe were at a loss, we did not know how to modify the controller tomake the system stable. The Nyquist theorem was a revolution for us.By drawing the Nyquist curve we got a very effective way to design thesystem because we know the frequency range which was critical andwe got a good feel for how the controller should be modified to makethe system stable. We could either add a compensator or we coulduse an extra sensor.Why did it take 18 years? Nyquist’s paper was published 1932!Bo Bernhardsson and Karl Johan ÅströmLoop Shaping
Example: ASEA Depth Control of SubmarineToolchain: measure frequency response design by loopshapingFearless experimentationGeneration of sine waves and measurementSpeed dependenceBo Bernhardsson and Karl Johan ÅströmLoop Shaping
Example: ASEA Multivariable DesignBo Bernhardsson and Karl Johan ÅströmLoop Shaping
Control System Design - Loop Shaping1Introduction2Loop shaping design3Bode’s ideal loop transfer funtion4Minimum phase systems5Non-minimum phase systems6Fundamental Limitations7Performance Assessment8SummaryTheme: Shaping Nyquist and Bode PlotsBo Bernhardsson and Karl Johan ÅströmLoop Shaping
Loop Shaping DesignDetermine transfer function from experiments or physicsTranslate specifications to requirements on the loop transferfunction L P CImportant parametersGain crossover frequency ωgc and slope ngc at crossoverLow frequency slope of loop transfer function nHigh frequency roll offWatch out for fundamental limitationsThe controller is given by C Ldesired /PDesign can also be done recursivelyLag compensationLead compensationNotch filtersBo Bernhardsson and Karl Johan ÅströmLoop Shaping
RequirementsStablity and robustnessGain margin gm , phase margin ϕm , maximum sensitivity Ms P 1 P C Stability for large process variations: , P P C Load disturbance attenuation1Ycl (s) Yol (s)1 PCCan be visualized in Hall and Nichols chartsMeasurement NoiseCU (s) N (s)1 PCCommand signal following (system with error feedback)PCT can be visualized in Hall and Nichols charts1 PCFix shape with feedforward FHow are these quantities represented in loop shaping when wetypically explore Bode, Nyquist or Nichols plots?Bo Bernhardsson and Karl Johan ÅströmLoop Shaping
The Bode PlotStability and RobustnessGain and phase margins gm , ϕm , delay marginsFrequencies ωgc , ωpcDisturbance attenuationSensitivity function S 11 PCP/(1 P C) 1/C for low frequenciesProcess variationsCan be represented by parameter sweepMeasurement noiseVisible if process transfer function is also plottedUseful to complement with gain curves of GoFCommand signal responseLevel curves of T in Nichols plotWide frequency rangeBo Bernhardsson and Karl Johan ÅströmLoop Shaping
Physical Interpretations of the Bode PlotLogarithmic scales gives an overview of the behavior over widefrequency and amplitude rangesPiece-wise linear approximations admit good physicalinterpretations, observe units and 00Frequency ω10110Low frequencies GxF (s) 1/k , the spring line, system behaveslike a spring for low frequency excitationHigh frequencies GxF (s) 1/(ms2 ), the mass line,, systembehaves like a mass for high frequency excitationBo Bernhardsson and Karl Johan ÅströmLoop Shaping
Bode Plot of Loop Transfer FunctionA Bode plot of the loop transfer function P (s)C(s) gives a broadcharacterization of the feedback system1Performancelog L(iω)10Robustness and Performance ωgc 010Robustnss and noise attenuation-110-110011010log ω L(iω)-90-135Robustness-180-110010110log ωNotice relations between the frequencie ωgc ωsc ωbwRequirements above ωgcBo Bernhardsson and Karl Johan ÅströmLoop Shaping
Some Interesting FrequenciesIm Gl (iω)nωpcωbwωmsIm Gl (iω)ωpcnωmsωbwωscωgcRe Gl (iω)ωgcωscRe Gl (iω)The frequencies ωgc and ωsc are closeTheir relative order depends on the phase margin (borderlinecase ϕm 60 )Bo Bernhardsson and Karl Johan ÅströmLoop Shaping
Hall and Nichols Chart32log L(iω) Im L(iω)4201 20 4 50Re L(iω)5 1 4 3 2 1arg L(iω) [rad]Hall is a Nyquist plot with level curves of gain and phase forthe complementary sensitivity function T . Nichols log Hall.Both make is possible to judge T from a plot of P CConformality of gain and phase curves depend on scalesThe Nichols chart covers a wide frequency rangeThe Robustness Valley Re L(iω) 1/2 dashedBo Bernhardsson and Karl Johan ÅströmLoop Shaping0
Finding a Suitable Loop Transfer FunctionProcess uncertaintyAdd process uncertainty to the process transfer functionPerform the design for the worst case (more in QFT)Disturbance attenuationInvestigate requirements pick ωgc and slope that satisfies therequirementsRobustnessShape the loop transfer function around ωgc to give sufficientphase marginAdd high frequency roll-offMeasurement noiseNot visible in L but can be estimated if we also plot PBo Bernhardsson and Karl Johan ÅströmLoop Shaping
An ExampleTranslate requirements on tracking error and robustness to demandson the Bode plot for the radial servo of a CD player.From Nakajima et al Compact Disc Technology, Ohmsha 1992, page 134Major disturbance caused by eccentricity about 70µm, trackingrequirements 0.1µm, requires gain of 700, the RPM varies because ofconstant velocity read out (1.2-1.4 m/s) around 10 Hz.Bo Bernhardsson and Karl Johan ÅströmLoop Shaping
Bode on LoopshapingBode Network Analysis and Feedback Amplifier Design p 454The essential feature is that the gain around the feedback loop be reducedfrom the large value which it has in the useful frequency band to zero dB orless at some higher frequency without producing an accompanying phaseshift larger than some prescribed amount. .If it were not for the phase restriction it would be desirable on engineeringgrounds to reduce the gain very rapidly. The more rapidly the feedbackvanishes for example, the narrower we need make the region in which activedesign attention is required to prevent singing. .Moreover it is evidently desirable to secure a loop cut-off as soon as possibleto avoid the difficulties and uncertainties of design which parasitic elementsin the circuit introduce at high frequencies.But the analysis in Chapter XIV (Bode’s relations) shows that the phase shiftis broadly proportional to the rate at which the gain changes. . A phasemargin of 30 correspond to a slope of -5/3.Bo Bernhardsson and Karl Johan ÅströmLoop Shaping
Bode’s Relations between Gain and PhaseWhile no unique relation between attenuation and phase can be statedfor a general circuit, a unique relation does exist between any givenloss characteristic and the minimum phase shift which must beassociated with it.Z log G(iω) log G(iω0 ) 2ω0dωπ 0ω 2 ω02Z1 d log G(iω) ω ω0π d log G(iω) dω logπ 0d log ωω ω02d log ωarg G(iω0 ) 2ω 2log G(iω) 0log G(iω0 ) π 2ω02πZ 0Z0 ω 1 arg G(iω) ω0 1 arg G(iω0 )dωω 2 ω02 d ω 1 arg G(iω)ω ω0dωlogdωω ω0Proven by contour integrationBo Bernhardsson and Karl Johan ÅströmLoop Shaping
The Weighting Functionf ω ω0 ω ω0 2log2π ω ω0 6f (ω/ω0 )543210-310-210-110Bo Bernhardsson and Karl Johan Åström0110ω/ω0Loop Shaping10210310
Do Nonlinearities Help?Can you beat Bode’s relations by nonlinear compensatorsFind a compensator that gives phase advance with less gain thangiven by Bode’s relationsThe Clegg integrator has the describing function4 i w1 . The gain is 1.62/ω and the phaselag is onlyN (iω) πω38 . Compare with integrator (J. C. Clegg A nonlinear Integratorfor Servomechanisms. Trans. AIEE, part II, 77(1958)41-42)2u, y10-1-20246Bo Bernhardsson and Karl Johan Åström8t10Loop Shaping121416
Control System Design - Loop Shaping1Introduction2Loop shaping design3Bode’s ideal loop transfer funtion4Minimum phase systems5Non-minimum phase systems6Fundamental Limitations7Performance Assessment8SummaryTheme: Shaping Nyquist and Bode PlotsBo Bernhardsson and Karl Johan ÅströmLoop Shaping
Loop Shaping for Gain VariationsThe repeater problemLarge gain variations in vacuum tubeamplifiers give distorsionThe loop transfer functionL(s) ωmax s nωgcωmingives a phase margin that is invariant to gainvariations.The slope n 1.5 gives the phase margin ϕm 45 .Horowitz extended Bode’s ideas to deal with arbitrary plant variationsnot just gain variations in the QFT method.Bo Bernhardsson and Karl Johan ÅströmLoop Shaping
1502 L(iω) ωBlue curve slope n 5/3 phase margin ϕm 30 Red curve slope n 4/3 phase margin ϕm 60 Making the curve steeper reduces the frequency range wherecompensation is required but the phase margin is smallerBo Bernhardsson and Karl Johan ÅströmLoop Shaping
A Fractional PID controller - A Current FadConsider the processP (s) 1s(s 1)Find a controller that gives L(s) s 1.5 , henceC(s) L(s)s(s 1) 1 s P (s)s ssA controller with fractional transfer function. To implement it weapproximate by a rational transfer functionĈ(s) k(s 1/16)(s 1/4)(s 1)(s 4)(s 16)(s 1/32)(s 1/8)(s 1/2)(s 2)(s 8)(s 32)High controller order gives robustnessBo Bernhardsson and Karl Johan ÅströmLoop Shaping
A Fractional Transfer Function2101 L(iω) 10010-110-210-110010110-128arg L(iω)-130-132-134-136-138-140-142-110010ω110The phase margin changes only by 5 when the process gain varies inthe range 0.03-30! Horowitz QFT is a generalization.Bo Bernhardsson and Karl Johan ÅströmLoop Shaping
Time ResponsesP (s) k,s(s 1)kL(s) s sC 1s ,s sk 1, 5, 25,y10.50012345t678910Notice signal shape independent in spite of 25 to 1 gain variationsBo Bernhardsson and Karl Johan ÅströmLoop Shaping
Fractional System Gain Curves GOF T (iω) 01010-1-11010-210-2-110010110211010 CS(iω) 10-110010210 S(iω) 001101010-110-110 P S(iω) 0-2-110P 010110k, k 1,s(s 1)21010k 5,Bo Bernhardsson and Karl Johan Åström-110k 25,Loop Shaping011010C 1s s210
Control System Design - Loop Shaping1Introduction2Loop shaping design3Bode’s ideal loop transfer funtion4Minimum phase systems5Non-minimum phase systems6Fundamental Limitations7Performance Assessment8SummaryTheme: Shaping Nyquist and Bode PlotsBo Bernhardsson and Karl Johan ÅströmLoop Shaping
RequirementsLarge signal behaviorLevel and rate limitations in actuatorsSmall signal behaviorSensor noiseResolution of AD and DA convertersFrictionDynamicsMinimum phase dynamics do not give limitationsThe essential limitation on loopshaping for systems with minimumphase dynamics are due to actuation power, measurement noise andmodel uncertainty.Bo Bernhardsson and Karl Johan ÅströmLoop Shaping
Controllers for Minimum Phase SystemsThe controller transfer function is given byC(s) Ldesired (s),P (s) C(iωgc ) 1 P (iωgc ) Since P (iω) typically decays for large frequencies, large ωgcrequires high controller gain. The gain of C(s) may also increase afterωgc if phase advance is required. The achievable gain crossoverfrequency is limited byActuation power and limitationsSensor noiseProcess variations and uncertaintyOne way to capture this quantitatively is to determine the largest highfrequency gain of the controller as a function of the gain crossoverfrequency ωgc . High gain is a cost of feedback (phase advance).Bo Bernhardsson and Karl Johan ÅströmLoop Shaping
Gain of a Simple Lead Networks!nss a Gn (s) ,G (s) k s ans/ k a nk 11 , ϕ log k,Phase lead: ϕn n arctan2n22 k2ϕsG (s) e s aMaximumqlead ϕ: gain for a given phasekn 1 2 tan2ϕn 2 tanPhase lead90 180 225 n 234-ϕn1 tan2n 425115014000ϕnn 6247304800 n, k e2ϕn 8246303300n 235402600Same phase lead with significantly less gain if order is high!High order controllers can be usefulBo Bernhardsson and Karl Johan ÅströmLoop Shaping
Lead Networks of Order 2, 3 and 3 G(iω) 10210110010-210-110010110210140arg G(iω)120100806040200-210-110010ω110210Increasing the order reduces the gain significantlywithout reducing the width of the peak too muchBo Bernhardsson and Karl Johan ÅströmLoop Shaping
Bode’s Phase Area FormulaLet G(s) be a transfer function with no poles and zeros in the right halfplane. Assume that lims G(s) G . Thenlog2G G(0)πZ arg G(iω)02dω ωπZ arg G(iω)d log ω The gain K required to obtain a given phase lead ϕ is an exponentialfunction of the area under the phase curve in the Bode plotarg( )G(iω)k e4cϕ0 /π e2γϕ02cγ πϕoBo Bernhardsson and Karl Johan ÅströmccLoop Shapingc
ProofIntegrate the functionIm slog G(s)/G( )siRΓiraround the contour, arg G(iω)/ω even fcnγRe s0 0Z Z0Hence G(ω) G(ω) dωlog i arg G( ) G( ) ω ! G(ω) G(ω) dω G(0) log i arg iπ log G( ) G( ) ω G( ) 2 G(0) log G( ) πZBo Bernhardsson and Karl Johan Åström arg G(iω) d log ω0Loop Shaping
Estimating High Frequency Controller Gain 1Required phase lead at the crossover frequencyϕl max(0, π ϕm arg P (iωgc ))Bode’s phase area formula gives a gain increase of Kϕ e2γϕlCross-over condition: P (iωgc )C(iωgc ) 1log C loglogωgcpKϕpKϕlog ωpKϕmax(1, eγ( π ϕm arg P (iωgc )) )eγϕl Kc max C(iω) ω ωgc P (iωgc ) P (iωgc ) P (iωgc ) Bo Bernhardsson and Karl Johan ÅströmLoop Shaping
Estimating High Frequency Controller Gain 2C CS C1 PCThe largest high frequency gain of the controller is approximately givenby (γ 1)Kc max C(iω) ω ωgceγϕlmax(1, eγ( π ϕm arg P (iωgc )) ) P (iωgc ) P (iωgc ) Notice that Kc only depends on the processCompensation for process gain 1/ P (iωgc ) Compensation for phase lag: eγϕl eγ( π ϕm arg P (iωgc ))The largest allowable gain is determined by sensor noise andresolution and saturation levels of the actuator. Results also hold forNMP systems but there are other limitations for such systemsBo Bernhardsson and Karl Johan ÅströmLoop Shaping
Example - Two and Eight Lags P (s 1) nKc 12 n/2 γ(n arctan ωgc π ϕm )eγ( π ϕm arg P (iωgc )) 1 ωgce P (iωgc ) πγ 1, ϕm , n 2, n 84ωgcKcϕl arg P (iωgc )10181.533.6168ωgcKcϕl arg P (iωgc )0.59.4782122079639.31741.0812225360501002005.3 1032.2 1048.7 10442.717843.81791.23.7 Bo Bernhardsson and Karl Johan Åström1032664011.41.5 Loop Shaping10430043544.41791.52.7 104315450
Summary of Non-minimum Phase SystemsNon-minimum phase systems are easy to control. High performancecan be achieved by using high controller gains. The main limitationsare given by actuation power, sensor noise and model uncertainty.PC T1 PCC TL P (1 T )PThe high frequency gain of the controller can be estimated by (γ 1)Kc max C(iω) ω ωgceγϕleγ( π ϕm arg P (iωgc )) P (iωgc ) P (iωgc ) Notice that Kc only depends on the process; two factors:Compensation for process gain 1/ P (iωgc ) Gain required for phase lead: eγ( π ϕm arg P (iωgc ))Bo Bernhardsson and Karl Johan ÅströmLoop Shaping
Control System Design - Loop Shaping1Introduction2Loop shaping design3Bode’s ideal loop transfer funtion4Minimum phase systems5Non-minimum phase systems6Fundamental Limitations7Performance Assessment8SummaryTheme: Shaping Nyquist and Bode PlotsBo Bernhardsson and Karl Johan ÅströmLoop Shaping
RequirementsLarge signal behaviorLevel and rate limitations in actuatorsSmall signal behaviorSensor noiseResolution of AD and DA convertersFrictionDynamicsNon-minimum phase dynamics limit the achievable bandwidthNon-minimum phase dynamics give severe limitationsRight half plane zerosRight half plane poles (instabilities)Time delaysBo Bernhardsson and Karl Johan ÅströmLoop Shaping
Non-minimum Phase SystemsDynamics pose severe limitations on achievable performance forsystems with poles and zeros in the right half planeRight half plane polesRight half plane zerosTime delaysBode introduced the concept non-minimum phase to capture this. Asystem is minimum phase system if all its poles and zeros are in theleft half plane.Theme: Capture limitations due to NMP dynamics quantitativelyBo Bernhardsson and Karl Johan ÅströmLoop Shaping
Bode’s Relations between Gain and PhaseThere is a unique relation between gain and phase for a transferfunction with no poles and zeros in the right half plane.Z 2ω0log G(iω) log G(iω0 ) dωπ 0ω 2 ω02Zπ d log G(iω) 1 d log G(iω) ω ω0 dω logπ 0d log ωω ω02d log ωarg G(iω0 ) 2ω 2log G(iω) 0log G(iω0 ) π 2ω02πZ 0Z0 ω 1 arg G(iω) ω0 1 arg G(iω0 )dωω 2 ω02 d ω 1 arg G(iω)ω ω0dωlogdωω ω0Transfer functions with poles and zeros in the right half plane havelarger phase lags for the same gain. Factor process transfer functionasG(s) Gmp (s)Gnmp (s), Gnmp (iω) 1,Bo Bernhardsson and Karl Johan ÅströmLoop Shaping Gnmp (iω) 0
Normalized NMP Factors 1Factor process transfer function as P (s) Pmp (s)Pnmp (s), Pnmp (iω) 1 and Pnmp (iω) negative phase.Right half plane zero z 1ωgc not too largePnmp (s) 1 s1 s0-90-180-2100102100Time delay L 2ωgc not too large-90Pnmp (s) e 2sRight half plane pole p 1ωgc must be largePnmp (s) s 1s 1Bo Bernhardsson and Karl Johan Åström-180-2100102100-90-180-210Loop Shaping010210
Normalized NMP Factors 2Factor process transfer function as P (s) Pmp (s)Pnmp (s), Pnmp (iω) 1 and Pnmp (iω) negative phase.RHP pole zero pair z pOK if you pick ωgc properlyPnmp (s) (5 s)(s 1/5)(5 s)(s 1/5)p10-90-180-210010210p2RHP pole-zero pair z pImpossible with stable CPnmp (s) (1/5 s)(s 5)(1/5 s)(s 5)RHP pole and time delayOK if you pick ωgc properlyPnmp (s) 1 s 0.2se1 sBo Bernhardsson and Karl Johan Åström-180-270-360-210010210p30-90-180-210Loop Shaping010210
Examples of PnmpFactor process transfer function as P (s) Pmp (s)Pnmp (s) such thateach non-minimum phase factor is all-pass and has negative phaseP (s) 11 s1 s ,(s 2)(s 3)(s 1)(s 2)(s 3) 1 sP (s) s 3s 3s 1 ,(s 1))(s 2)(s 1)(s 2) s 1P (s) P (s) 1 e s ,s 1Pnmp (s) 1 s1 ss 1s 1Pnmp (s) e ss 1s 11 ss 2 ,(s 2)(s 3)(s 2)(s 3) 1 s s 2Bo Bernhardsson and Karl Johan ÅströmPnmp (s) Loop ShapingPnmp 1 ss 21 ss 2
Bode Plots Should Look Like This0 P , Pmp 10-110-210-3100arg(P , Pnmp , Pmp )101210103100-90-180-270-360010110Bo Bernhardsson and Karl Johan Åström2ωLoop Shaping10310
The Phase-Crossover InequalityAssume that the controller C has no poles and zeros in the RHP,factor process transfer function as P (s) Pmp (s)Pnmp (s) such that Pnmp (iω) 1 and Pnmp has negative phase. Requiring a phasemargin ϕm we getarg L(iωgc ) arg Pnmp (iωgc ) arg Pmp (iωgc ) arg C(iωgc ) π ϕmApproximate arg (Pmp (iωgc )C(iωgc )) ngc π/2 givesarg Pnmp (iωgc ) ϕlagnmpϕlagnmp π ϕm ngcπ2This inequality is called, the phase crossover inequality. Equality holdsif Pmp C is Bode’s ideal loop transfer function, the expression is anapproximation for other designs if ngc is the slope of the gain curve atthe crossover frequency.Bo Bernhardsson and Karl Johan ÅströmLoop Shaping
Reasonable Values of ϕnmplagAdmissible phase lag of non-minimum phase factor Pnmp as afunction of the phase margin ϕm and the slope ngc (roll-off) at the gaincrossover frequency100ϕlagnmp80ngc 0.560ngc 140ngc 1.520030ϕmϕmϕmϕm40 π6 , ngcπ4 , ngcπ3 , ngcπ4 , ngc5060ϕm7080 12 give ϕlagnmp 7π12 1.83 (105 )1π 2 give ϕlagnmp 2 (90 ) 1 give ϕlagnmp π6 0.52 (30 ) 1.5 give ϕlagnmp 0Bo Bernhardsson and Karl Johan ÅströmLoop Shaping90
Loop Shaping1Introduction2Loop shaping design3Bode’s ideal loop transfer funtion4Minimum phase systems5Non-minimum phase systems6Fundamental Limitations7Performance Assessment8SummaryTheme: Shaping Nyquist and Bode PlotsBo Bernhardsson and Karl Johan ÅströmLoop Shaping
System with RHP ZeroPnmp (s) z sz sCross over frequency inequalityπωgc π ϕm ngc ϕlagnmparg Pnmp (iωgc ) 2 arctanz2ϕlagnmpπωgcπ ϕm ngc ) tan tan( z2242Compare with inequality for ωsc in Requirements Lecture00ωsc /zωscMs 1 zMs10ωgc /z10-110-10204060ϕlagnmpBo Bernhardsson and Karl Johan Åström8010010Loop Shaping11.5Ms22.5
Water TurbineTransfer function from valve opening to power, (T time for water toflow through penstock)GP A P0 1 2u0 sTu0 1 u0 sTA first principles physics model is available in kjå Reglerteori 1968 sid 75-76Bo Bernhardsson and Karl Johan ÅströmLoop Shaping
Drum Level ControlSteam valveFeedwaterFFLDrumOilTurbineAirRaiserDown comerThe shrink and swell effect: steam valve opening to drum levelBo Bernhardsson and Karl Johan ÅströmLoop Shaping
System with Time DelayPnmp (s) e sT 1 sT /21 sT /2Cross over frequency inequalityωgc T π ϕm ngcπ ϕlagnmp2The simple rule of (ϕlagnmp π/4) gives ωgc T π 0.8. Pade41using the inequality for RHPapproximation gives the zero at z 2Tzero gives similar result. Comp inequality in Requirements lecture1.41.21.2ωgc T1.61.4ωgc TMs 1ωsc T 2Ms1.610.80.60.60.40.22010.80.44060ϕlagnmpBo Bernhardsson and Karl Johan Åström801000.2Loop Shaping11.5Ms22.5
System with RHP PolePnmp (s) Cross over frequency inequality 2 arctans ps pπp π ϕm ngc ϕlagnmpωgc2ωgc1 ptan ϕlagnmp /2Compare with inequality for ωtc in Requirements lecture11ωgc /pMtωtc pMt 110ωgc /p100100204060ϕlagnmpBo Bernhardsson and Karl Johan Åström8010010Loop Shaping11.5Mt22.5
System with complex RHP Zero(x i y s)(x i y s)(x i y s)(x i y s)y ωy ω 2 arctan 2 arctanxx2ω z ζ2ωx 2 arctan 2 2 arctan 2x y2 ω2 z ω 2Pnmp gc / z 0.60.70.80.91Damping ratio ζ 0.2 (dashed), 0.4, 0.6. 0.8 and 1.0, red dashedcurve single RHP zero. Small ζ easier to control.Bo Bernhardsson and Karl Johan ÅströmLoop Shaping
System with RHP Pole and Zero Pair(z s)(s p)z p,Ms (z s)(s p)z pCross over frequency inequality for z p p ωgcϕlagampωgcpp 2 arctan ϕlagamp , 1 tan 2 arctan zωgczωgcz2pThe smallest value of the left hand side is 2 p/z , which is achieved for ωgc pz , hence ϕlagnmp 2 arctan (2 pz/(z p))zPlot of ϕlagnmp for 2, 3, 5, 10, 20, 50 and Ms 3, 2, 1.5, 1.2, 1.1, 1.05pPnmp (s) ϕlagnmp18090010 -210 -1010 ωgc / pzBo Bernhardsson and Karl Johan ÅströmLoop Shaping10 110 2
An ExampleFrom Doyle, Francis Tannenbaum: Feedback Control Theory 1992.P (s) s 1,s2 0.5s 0.5Pnmp (1 s)(s 0.5)(1 s)(s 0.5)Keel and Bhattacharyya Robust, Fragile or Optimal AC-42(1997)1098-1105: In this paper we show by examples that optimum and robustcontrollers, designed by the H2 , H , L1 and µ formulations, can produceextremely fragile controllers, in the sense that vanishingly small perturbationsof the coefficients of the designed controller destabilize the closed loopsystem. The examples show that this fragility usually manifests itself asextremely poor gain and phase margins of the closed loop system.Pole at s 0.5, zero at s 1, ϕlagnmp 2.46 (141 ),Ms (z p)/(z p) 3,ϕm 2 arcsin(1/(2Ms )) 0.33 (19 )Hopeless to control robustlyYou don’t need any more calculationsBo Bernhardsson and Karl Johan ÅströmLoop Shaping
Example - The X-29Advanced experimental aircraft. Many design efforts with manymethods and high cost.Requirements ϕm 45 could notbe met. Here is why! Process hasRHP pole p 6 and RHP zero z 26. Non-minimum phase factor oftransfer functionPnmp (s) (s 26)(6 s)(s 26)(6 s)The smallest phaselag ϕlagnmp 2.46(141 ) of Pnmp is too large.The zero pole ratio is z/p 4.33 gives Ms z pz p 1.61 ϕm 2 arcsin( 2Ms ) 0.64(36 ). Not possible to get a phase margin of 45 !Bo Bernhardsson and Karl Johan ÅströmLoop Shaping
Bicycle with Rear Wheel SteeringRichard Klein at UIUC has built several UnRidable Bicycles (URBs).There are versions in Lund and UCSB.Transfer functionV0amℓV0 s aP (s) mgℓbJs2 Jrmgℓ 3 rad/sJV0RHP zero at z aPole at p Pole independent of velocity but zero proportional to velocity. There isa velocity such that z p and the system is uncontrollable. Thesystem is difficult to control robustly if z/p is in the range of 0.25 to 4.Bo Bernhardsson and Karl Johan ÅströmLoop Shaping
RHP Pole and Time DelayNMP part of process transfer functionPnmp (s) s p sLe,s pMs epLpL 2p ωgc L ϕlagnmpωgcπϕlagnmp π ϕm ngc2Plot of ϕlagnmp for pL 0.01, 0.02, 0.05, 0.1, 0.2, 0.7arg Pnmp (iωgc )
Control System Design - Loop Shaping 1 Introduction 2 Loop shaping design 3 Bode's ideal loop transfer funtion 4 Minimum phase systems 5 Non-minimum phase systems 6 Fundamental Limitations 7 Performance Assessment 8 Summary Theme: Shaping Nyquist and Bode Plots Bo Bernhardsson and Karl Johan Åström Loop Shaping
robust loop-shaping POD controller design in large power systems. By applying the model reduction and modern robust loop-shaping control technique, the FACTS robust loop-shaping POD controller is realized. This controller exploits the advantages of both conventional loop-shaping and modern . H robust control technique.
power system: the automatic load frequency control loop (ALFC) and the automatic voltage control loop (AVR). There is little interaction from the ALFC loop to the AVR loop but the AVR loop does affect the ALFC loop to a certain extent. The AVR loop tends to be much faster than the ALFC loop with time constants of much less than one second.
the opinions of those with direct ties to the West Loop. I live in the West Loop I work in the West Loop I own property in the West Loop I own a business in the West Loop I am generally interested in the West Loop* Other (please specify) * 0 20 40 60 80 100 23.6% 59.7% 5.6% 4.9% 85.4% 32.6% 18 West Loop Design Guidelines Community Feedback 1 .
The Do-While and For loop structures are pretest loops. In a posttest loop, the condition is evaluated at the end of the repeating section of code. This is also called an exit-controlled loop. The Do/Loop Until construct is a posttest loop. Do While Loop The while loop allows you to di
while loop Format: while condition: loop body loop body 18 one or more instructions to be repeated condition loop body false true After the loop condition becomes false during the loop body, the loop body still runs to completion (before its check before the next turn) and exit the loop and go on with the next step.
2. The Loop Dashboard Use the Loop Dashboard to visualize the performance of any individual control loop in detail over a specified time period. This helps in assessing control loop performance and diagnosing potential causes of suboptimal performance. Consult the Loop Dashboard section of this documentation for details. 3. Loop KPI Analysis
In a pretest loop, the loop control expression is tested first. If it is true then the loop actions are executed; if it is false the loop is terminated. In a post-test loop, the loop actions are executed first. Then the loop control expression is tested. If it is true, a new iteration is started; otherwis
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