The Laplace transform is a tool to facilitate solving for ODEs No need to do actually do the transform Lookup tables System Gs is stable if Its response is bounded and finite poles must have ID: 1003815
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1. To recapControl theory builds on differential equationsThe Laplace transform is a tool to facilitate solving for ODEs.No need to do actually do the transformLookup tablesSystem G(s) is stable ifIts response is bounded and finitepoles must have negative real partsStill, how does the cruise control example work?Getting there…Matone: An Overview of Control Theory and Digital Signal Processing (2)1LIGO-G1100863
2. RecallWhat is the transfer function of a system whose input and output are related by the following differential equation? Matone: An Overview of Control Theory and Digital Signal Processing (2)2LIGO-G1100863
3. RecallGiven the system’s transfer function determine the system’s differential equation to input . Matone: An Overview of Control Theory and Digital Signal Processing (2)3LIGO-G1100863
4. RecallDetermine which of the following transfer functions represent stable systems and which represent unstable systems. Use MATLAB’s step to verify your answer. Matone: An Overview of Control Theory and Digital Signal Processing (2)4LIGO-G1100863
5. Another exampleA simple mechanical accelerometer is shown below. The position y is with respect of the case, the case’s position is x. What is the transfer function between the input acceleration and the output Y? Matone: An Overview of Control Theory and Digital Signal Processing (2)5MykBx LIGO-G1100863
6. Control Theory 2 The response of a stable system G(s) is characterized by its Amplitude and Phase shiftto a sinusoidal excitationMatone: An Overview of Control Theory and Digital Signal Processing (2)6LIGO-G1100863
7. Frequency responseIt can be shown that ifthenwhere is the amplitude response and is the phase shift X This is how we measure transfer functionsMatone: An Overview of Control Theory and Digital Signal Processing (2)7LIGO-G1100863
8. Frequency responseThe dynamic behavior of a physical system can be determined by measuring its transfer function with a sinusoidal excitationMagnitude and phase response are a function of frequency (Frequency-response helps to understand the stability criteria Matone: An Overview of Control Theory and Digital Signal Processing (2)8LIGO-G1100863
9. Graphical analysis tool: Bode PlotCommon graphical representation of transfer function is complexplot of magnitude and phase ConventionLog-log scale for magnitude vs. frequency (Hz)Semi-log scale for phase (deg) vs. frequency (Hz)Other conventionsMagnitude in dB () vs. angular frequency (rad/s) Matone: An Overview of Control Theory and Digital Signal Processing (2)9LIGO-G1100863
10. Bode plot: →→Matone: An Overview of Control Theory and Digital Signal Processing (2)10LIGO-G1100863
11. bodeexamples.m Matone: An Overview of Control Theory and Digital Signal Processing (2)11LIGO-G1100863
12. Bode plot: →→Matone: An Overview of Control Theory and Digital Signal Processing (2)12LIGO-G1100863
13. bodeexamples.m Matone: An Overview of Control Theory and Digital Signal Processing (2)13LIGO-G1100863
14. Bode plot: →→Matone: An Overview of Control Theory and Digital Signal Processing (2)14LIGO-G1100863
15. bodeexamples.m Matone: An Overview of Control Theory and Digital Signal Processing (2)15LIGO-G1100863
16. Bode plot: →→Matone: An Overview of Control Theory and Digital Signal Processing (2)16LIGO-G1100863
17. bodeexamples.m Matone: An Overview of Control Theory and Digital Signal Processing (2)17LIGO-G1100863
18. Bode plot: Matone: An Overview of Control Theory and Digital Signal Processing (2)18LIGO-G1100863
19. bodeexamples.m Matone: An Overview of Control Theory and Digital Signal Processing (2)19LIGO-G1100863
20. bodeexamples.m Matone: An Overview of Control Theory and Digital Signal Processing (2)20LIGO-G1100863
21. bodeexamples.m Matone: An Overview of Control Theory and Digital Signal Processing (2)21LIGO-G1100863
22. Bode plot: SHO Matone: An Overview of Control Theory and Digital Signal Processing (2)22LIGO-G1100863
23. bodeexamples.m Matone: An Overview of Control Theory and Digital Signal Processing (2)23LIGO-G1100863
24. bodeexamples.m Matone: An Overview of Control Theory and Digital Signal Processing (2)24LIGO-G1100863
25. bodeexamples.m Matone: An Overview of Control Theory and Digital Signal Processing (2)25LIGO-G1100863
26. Bode plots for more complicated TFs? Break it into simpler partsIn general a transfer function can be re-written in terms of simpler ones (of 2nd order at most)then Matone: An Overview of Control Theory and Digital Signal Processing (2)26LIGO-G1100863
27. ExampleLet’s draw the bode plot forwhere Matone: An Overview of Control Theory and Digital Signal Processing (2)27LIGO-G1100863
28. Example Matone: An Overview of Control Theory and Digital Signal Processing (2)28LIGO-G1100863
29. 0.1Hz1Hz10HzLog(f)Log(f) Matone: An Overview of Control Theory and Digital Signal Processing (2)29LIGO-G1100863
30. 0.1Hz1Hz10HzLog(f)Log(f) Matone: An Overview of Control Theory and Digital Signal Processing (2)30LIGO-G1100863
31. bodeexamples.mMatone: An Overview of Control Theory and Digital Signal Processing (2)31LIGO-G1100863
32. >> z=-2*pi*0.1;>> p=-2*pi*[1 10];>> k=500;>> G=zpk(z,p,k);>> [m,p]=bode(G, w);bodeexamples.mMatone: An Overview of Control Theory and Digital Signal Processing (2)32LIGO-G1100863
33. ExerciseSketch bode plot for the following TFWhat is the DC gain (gain for )? What is the gain for ? Confirm results with MATLAB Matone: An Overview of Control Theory and Digital Signal Processing (2)33LIGO-G1100863
34. ExerciseSketch bode plot for the following TFWhat is the DC gain (gain for )? What is the gain for ? Confirm results with MATLAB Matone: An Overview of Control Theory and Digital Signal Processing (2)34LIGO-G1100863
35. ExerciseSketch bode plot for the following TFWhat is the DC gain (gain for )? What is the gain for ? Confirm results with MATLAB Matone: An Overview of Control Theory and Digital Signal Processing (2)35LIGO-G1100863
36. So far…A system’s TF is a complex functionCan be represented in terms of its magnitude and phaseBode plotsHelp visualize the TFPlot of magnitude vs. frequency and phase vs. frequency.Different conventionsWe have explored Bode plots of basic TFsBode plot of more complex TFs can be expressed in terms of simpler terms Matone: An Overview of Control Theory and Digital Signal Processing (2)36LIGO-G1100863
37. General Stability CriterionThe feedback control system is stable if and only if all the poles of the closed loop transfer function have a negative real part. Otherwise the system is unstable. Matone: An Overview of Control Theory and Digital Signal Processing (2)37LIGO-G1100863
38. In generalcrG(s)e-+H(s) Open loop gain Closed loop gain Stability: the poles’ real part of must be negative Matone: An Overview of Control Theory and Digital Signal Processing (2)38LIGO-G1100863
39. Loop stability and designIf the system is unstable, We can’t change butWe can design a different controller so as to make the system stableBut how should we change H? Let’s look closely at the root of the problem crG(s)e-+H(s)Matone: An Overview of Control Theory and Digital Signal Processing (2)39LIGO-G1100863
40. The problemcrG(s)e-+H(s) If ever becomes then system is unstable Matone: An Overview of Control Theory and Digital Signal Processing (2)40LIGO-G1100863
41. The general shape of crG(s)e-+H(s) has a limited bandwidth. Within bandwidth: Outside bandwidth: Matone: An Overview of Control Theory and Digital Signal Processing (2)41LIGO-G1100863
42. The general shape of Log(f)Log(f) “Bandwidth”Unity gainUnity Gain Frequency (UGF): Matone: An Overview of Control Theory and Digital Signal Processing (2)42DC gainLIGO-G1100863
43. Stability CriteriaA closed loop system is stable if the unity gain frequency is lower than the crossing. Log(f)Log(f) UGF crossing StableMatone: An Overview of Control Theory and Digital Signal Processing (2)43LIGO-G1100863
44. Stability Criteria:Rule of ThumbThe system is (almost always) stable if at the unity gain frequency. Log(f)Log(f) Slope at UGF: Matone: An Overview of Control Theory and Digital Signal Processing (2)44StableLIGO-G1100863
45. Nyquist stability criterionThe closed loop system is stable if the polar plot of the open loop transfer function does not encircle the point. >> G=10*tf(10,[1 10];>> nyquist(G)Matone: An Overview of Control Theory and Digital Signal Processing (2)45nyquist_example1.mLIGO-G1100863
46. Nyquist stability criterionThe closed loop system is stable if the polar plot of the open loop transfer function does not encircle the point. Checking the step response of 1/(1+Gol)Matone: An Overview of Control Theory and Digital Signal Processing (2)46nyquist_example1.mLIGO-G1100863
47. Back to cruise controlLet’s inspect the system’s loop stability. RecallMass m = 1000 kgCoefficient for air friction b = 50 kg/s GvfvrHKθ+-e-+c Matone: An Overview of Control Theory and Digital Signal Processing (2)47LIGO-G1100863
48. Cruise control: Bode plot of UGF @160 mHzcruise_freqdomain.mMatone: An Overview of Control Theory and Digital Signal Processing (2)48LIGO-G1100863
49. Cruise control: Nyquist plot of cruise_freqdomain.mMatone: An Overview of Control Theory and Digital Signal Processing (2)49LIGO-G1100863
50. With a little algebra: Matone: An Overview of Control Theory and Digital Signal Processing (2)50LIGO-G1100863
51. Let’s check step response of cruise_freqdomain.m Matone: An Overview of Control Theory and Digital Signal Processing (2)51LIGO-G1100863
52. Is the system with open loop transfer function stable? Matone: An Overview of Control Theory and Digital Signal Processing (2)52ExampleLIGO-G1100863
53. Two poles at 0 HzZero at 10 HzPole at 100 HzPole at 500 Hzfeedback_example4.mMatone: An Overview of Control Theory and Digital Signal Processing (2)53 LIGO-G1100863
54. Stableby how much?feedback_example4.mMatone: An Overview of Control Theory and Digital Signal Processing (2)54LIGO-G1100863
55. ProblemIf a system has an open loop transfer functionwhat values of make it stable? Use MATLAB to confirm this. Matone: An Overview of Control Theory and Digital Signal Processing (2)55LIGO-G1100863
56. Relative stabilityGain and phase marginMeasure of “relative” stabilityThe larger they are → the “safer we are”Gain marginBy how much can the gain increase until the system becomes unstable?Defined as Phase marginBy how much can the system tolerate a phase change at UGF?Defined as Rule of thumb: keep the phase margin above Matone: An Overview of Control Theory and Digital Signal Processing (2)56LIGO-G1100863
57. UGF @30 HzPhase margin: 500Gain margin: 20feedback_example4.mMatone: An Overview of Control Theory and Digital Signal Processing (2)57LIGO-G1100863
58. Step response of feedback_example4.mMatone: An Overview of Control Theory and Digital Signal Processing (2)58LIGO-G1100863
59. Pole-zero map of feedback_example4.mMatone: An Overview of Control Theory and Digital Signal Processing (2)59LIGO-G1100863
60. Performance to noise input d: with no feedbackcrG1d++e-+G2H Noise contribution to signal cMatone: An Overview of Control Theory and Digital Signal Processing (2)60LIGO-G1100863
61. Performance to noise input d: with feedbackcrG1d++e-+G2H Controlled signalSuppression factorNoise contribution to signal cMatone: An Overview of Control Theory and Digital Signal Processing (2)61LIGO-G1100863
62. Setting the parameterscrG1d++e-+G2H zeros at 1, 10Hz; poles at 0, 100 Hz, k = 300 Matone: An Overview of Control Theory and Digital Signal Processing (2)62LIGO-G1100863
63. feedback_example6.mMatone: An Overview of Control Theory and Digital Signal Processing (2)63LIGO-G1100863
64. feedback_example6.mController has a low pass filter at 1kHz Matone: An Overview of Control Theory and Digital Signal Processing (2)64LIGO-G1100863
65. Suppression factor feedback_example6.mNoise term suppressionNo suppressionIf the DC gain is increased, so is the suppressionBandwidthMatone: An Overview of Control Theory and Digital Signal Processing (2)65LIGO-G1100863
66. feedback_example6.mMatone: An Overview of Control Theory and Digital Signal Processing (2)66Control signal, at the output of H.LIGO-G1100863
67. How can we correct an unstable loop? Typical compensatorsIntegral controllerThe output is proportional to the time integral of the inputDerivative controllerThe output is proportional to the time derivative of the input Matone: An Overview of Control Theory and Digital Signal Processing (2)67LIGO-G1100863
68. Phase-lag compensatorcompensator_phaselag.m Matone: An Overview of Control Theory and Digital Signal Processing (2)68LIGO-G1100863
69. Phase-lead compensatorcompensator_phaselead.m Matone: An Overview of Control Theory and Digital Signal Processing (2)69LIGO-G1100863
70. “Boost”compensator_boost.m Matone: An Overview of Control Theory and Digital Signal Processing (2)70LIGO-G1100863
71. ProblemIf a system has an open loop transfer functiondesign a compensator that would make the system stable with an UGF at 100 Hz. Use MATLAB to confirm this. Matone: An Overview of Control Theory and Digital Signal Processing (2)71LIGO-G1100863
72. Example: locking one LIGO arm ++e-+ CHSReference signal : the cavity “locks” to the laser frequency. For simplicity, units of length. Noise term : Mirror motion due to seismic noise, units of length Matone: An Overview of Control Theory and Digital Signal Processing (2)72LIGO-G1100863
73. Example: locking one LIGO arm ++e-+ CHSLigo arm cavity C: a cavity length change is converted to electronic signal eController H: designed in the control room, processes signal e and sends commands to the suspension SSuspension S: receives the signal from H and drives the mirror, units of lengthNoise term : Noise from coil drivers Matone: An Overview of Control Theory and Digital Signal Processing (2)73LIGO-G1100863
74. Example: no lock ++e-+ CHS Matone: An Overview of Control Theory and Digital Signal Processing (2)74LIGO-G1100863
75. Example: locking one LIGO arm ++e-+ CHS Suppression factor : Open Loop gain: Note on noise term : Noise term appears only when mirrors are driven → limited bandwidth desired Matone: An Overview of Control Theory and Digital Signal Processing (2)75LIGO-G1100863
76. Example: locking one LIGO armNeed to design H so as to haveEnough suppression of noise termsStable“Small” bandwidthMatone: An Overview of Control Theory and Digital Signal Processing (2)76LIGO-G1100863
77. Example: locking one LIGO armCavity transfer function C:Pole at 100 HzSuspension transfer function SSimple harmonic oscillator (SHO) with and quality factor Shown is What controller H can we use? cavityfeedback_exampleA.mMatone: An Overview of Control Theory and Digital Signal Processing (2)77LIGO-G1100863
78. Example: locking one LIGO armSet UGF at 100 HzNeed H with a zero after 1 Hz and before 100 HzTry phase leadwith and Bode plot of OLUGF at 100 HzPM ~40 degStable cavityfeedback_exampleA.mMatone: An Overview of Control Theory and Digital Signal Processing (2)78LIGO-G1100863
79. Example: locking one LIGO armDouble check stabilityStep response plot ofStep is driven to zero as it should (it is a suppression factor)In about 10 ms (~1/UGF) response is close to zeroTwo oscillation cycles – little ringing cavityfeedback_exampleA.mMatone: An Overview of Control Theory and Digital Signal Processing (2)79LIGO-G1100863
80. Example: locking one LIGO armBode plot of the suppression factor Suppression of ~1500x at 100 mHzNo suppression above 100 HzNotice spike at 100 HzThis spike is responsible of the ringing in the step responsedecreased if phase margin is increased cavityfeedback_exampleA.mMatone: An Overview of Control Theory and Digital Signal Processing (2)80LIGO-G1100863
81. Example: locking one LIGO armLet’s increase the low frequency gain with a “boost”Try Hwith , and OL bode plotUGF at 100 HzPM ~40 degStable cavityfeedback_exampleB.mMatone: An Overview of Control Theory and Digital Signal Processing (2)81LIGO-G1100863
82. Example: locking one LIGO armDouble check stabilityStep response plot ofVery similar responseStep is driven to zero as it should (it is a suppression factor)In about 10 ms (~1/UGF) response is close to zeroTwo oscillation cycles – little ringing cavityfeedback_exampleB.mMatone: An Overview of Control Theory and Digital Signal Processing (2)82LIGO-G1100863
83. Example: locking one LIGO armBode plot of the suppression factor More suppression at low frequencies: at 100 mHzNo suppression above 100 HzNotice spike at 100 HzSimilar ringing cavityfeedback_exampleB.mMatone: An Overview of Control Theory and Digital Signal Processing (2)83LIGO-G1100863
84. Example: locking one LIGO armLet’s also “cut-off” the drive to the coils at high frequencyIntroduce a low pass LP 6th order Butterworth filter with cut-off frequency at 2 kHz.Try Hwith , and OL bode plotUGF at 100 HzPM ~30 degStable cavityfeedback_exampleC.mMatone: An Overview of Control Theory and Digital Signal Processing (2)84LIGO-G1100863
85. Example: locking one LIGO armDouble check stabilityStep response plot ofVery similar responseStep is driven to zero as it should (it is a suppression factor)In about 10 ms (~1/UGF) response is close to zero~Two oscillation cycles cavityfeedback_exampleC.mMatone: An Overview of Control Theory and Digital Signal Processing (2)85LIGO-G1100863
86. Example: locking one LIGO armBode plot of the suppression factor Same suppression: at 100 mHzNo suppression above 100 HzNotice spike at 100 HzA little higher than beforeSimilar ringing cavityfeedback_exampleC.mMatone: An Overview of Control Theory and Digital Signal Processing (2)86LIGO-G1100863
87. Example: locking one LIGO armIncreasing the gain by 2x: OL bode plotUGF at ~133 HzShould have gone to 200 Hz but the slope is not 1/f (because of the cavity pole at 100 Hz)PM ~20 degStable but with little phase margin left cavityfeedback_exampleD.mMatone: An Overview of Control Theory and Digital Signal Processing (2)87LIGO-G1100863
88. Example: locking one LIGO armStep response plot ofRinging has increased cavityfeedback_exampleD.mMatone: An Overview of Control Theory and Digital Signal Processing (2)88LIGO-G1100863
89. Example: locking one LIGO armBode plot of the suppression factor Suppression has increasedGain was increased by factor 2Notice spike at 100 Hz is more pronounced cavityfeedback_exampleD.mMatone: An Overview of Control Theory and Digital Signal Processing (2)89LIGO-G1100863
90. SummaryWe have explored the stability criteriaThe feedback control system is stable if and only if all the poles of the closed loop transfer function have a negative real part. Otherwise the system is unstable.Stability in terms of the open loop gainA closed loop system is stable if the unity gain frequency is lower than the crossing.Rule of thumb: the system is (almost always) stable if at the unity gain frequency Matone: An Overview of Control Theory and Digital Signal Processing (2)90LIGO-G1100863
91. SummaryNoise suppressionHow close to instability is a system? Gain and phase marginMeasure of “relative” stabilityThe larger they are → the “safer we are”Rule of thumb: keep the phase margin to more than Typical compensatorsPhase-lagPhase-lead“Boost”Cavity lock example Matone: An Overview of Control Theory and Digital Signal Processing (2)91LIGO-G1100863
92. Problem for the afternoonIdentify a (single-input-single-output) control system at LIGO – its plant TF along with its controller TF (LSC, ASC, SUS, MC, PSL, …)Sketch the block diagram and model the system with MATLAB. Generate the corresponding bode plot. Can you measure its OL TF? Where is the UGF and how does it compare with the model?For what range of frequencies can the UGF be placed at by simply adjusting the systems’ gain? What DC gain does it have, what suppression? LIGO-G1100863Matone: An Overview of Control Theory and Digital Signal Processing (2)92
93. Problem for the afternoonOptical levers are/were used to damp the fundamental mode of the suspensions. The controller has no DC gain (check this). Sketch the block diagram and model the system with MATLAB. Generate the corresponding bode plot. Can you measure its OL TF? Where is the UGF and how does it compare with the model?For what range of frequencies can the UGF be placed at by simply adjusting the systems’ gain? What DC gain does it have, what suppression? LIGO-G1100863Matone: An Overview of Control Theory and Digital Signal Processing (2)93opticallever.m
94. Solutions to problemsMatone: An Overview of Control Theory and Digital Signal Processing (2)94LIGO-G1100863
95. What is the transfer function of a system whose input and output are related by the following differential equation?Sol: Taking the Laplace transform of the equationWhich can be re-written as Matone: An Overview of Control Theory and Digital Signal Processing (2)95LIGO-G1100863
96. Given , determine the system’s differential equation to input .Sol:oror Matone: An Overview of Control Theory and Digital Signal Processing (2)96LIGO-G1100863
97. Determine which of the following transfer functions represent stable systems and which represent unstable systems. Use MATLAB’s step to verify your answer., unstable, stable, unstable, stable, unstable Matone: An Overview of Control Theory and Digital Signal Processing (2)97LIGO-G1100863
98. A simple mechanical accelerometer is shown below. The position y is with respect of the case, the case’s position is x. What is the transfer function between the input acceleration and the output Y? Matone: An Overview of Control Theory and Digital Signal Processing (2)98MykBx Sol: LIGO-G1100863
99. Sketch bode plot for the following TFWhat is the DC gain (gain for )? What is the gain for ? Confirm results with MATLAB Matone: An Overview of Control Theory and Digital Signal Processing (2)99bodeexcercise1.mLIGO-G1100863
100. Sketch bode plot for the following TFWhat is the DC gain (gain for )? What is the gain for ? Confirm results with MATLAB Matone: An Overview of Control Theory and Digital Signal Processing (2)100LIGO-G1100863
101. Sketch bode plot for the following TFWhat is the DC gain (gain for )? What is the gain for ? Confirm results with MATLAB Matone: An Overview of Control Theory and Digital Signal Processing (2)101LIGO-G1100863
102. SolutionIf a system has an open loop transfer functionwhat values of make it stable?Sol: UGF can be set after the pole at 10 and before pole at 100Set and . Find corresponding k.Set and 0. Find corresponding k Matone: An Overview of Control Theory and Digital Signal Processing (2)102LIGO-G1100863
103. SolutionIf a system has an open loop transfer functiondesign a compensator that would make the system stable with an UGF at 100 Hz. Use MATLAB to confirm this.Sol: two zeros at 10, decreasing the gain by 3xH=zpk([],[-10 -10 -10],1e3) * zpk([-10 -10],[],0.3) Matone: An Overview of Control Theory and Digital Signal Processing (2)103LIGO-G1100863