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1 mrac for inverted pendulum

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1 mrac for inverted pendulum

  1. 1. Model ReferenceModel Reference Adaptive ControlAdaptive Control Survey of Control Systems (MEM 800)Survey of Control Systems (MEM 800) Presented byPresented by Keith SevcikKeith Sevcik
  2. 2. ConceptConcept  Design controller to drive plant response to mimic idealDesign controller to drive plant response to mimic ideal response (error = yresponse (error = yplantplant-y-ymodelmodel => 0)=> 0)  Designer chooses: reference model, controller structure,Designer chooses: reference model, controller structure, and tuning gains for adjustment mechanismand tuning gains for adjustment mechanism Controller Model Adjustment Mechanism Plant Controller Parameters ymodel u yplant uc
  3. 3. MIT RuleMIT Rule  Tracking error:Tracking error:  Form cost function:Form cost function:  Update rule:Update rule: – Change in is proportional to negative gradient ofChange in is proportional to negative gradient of modelplant yye −= )( 2 1 )( 2 θθ eJ = δθ δ γ δθ δ γ θ e e J dt d −=−= θ J sensitivity derivative
  4. 4. MIT RuleMIT Rule  Can chose different cost functionsCan chose different cost functions  EX:EX:  From cost function and MIT rule, control law can beFrom cost function and MIT rule, control law can be formedformed      <− = > = −= = 0,1 0,0 0,1 )(where )( )()( e e e esign esign e dt d eJ δθ δ γ θ θθ
  5. 5. MIT RuleMIT Rule  EX: Adaptation of feedforward gainEX: Adaptation of feedforward gain Adjustment Mechanism ymodel u yplantuc Π Π θ Reference Model Plant s γ− )()( sGksG om = )()( sGksGp = - +
  6. 6. MIT RuleMIT Rule  For system where is unknownFor system where is unknown  Goal: Make it look likeGoal: Make it look like using plant (note, plant model isusing plant (note, plant model is scalar multiplied by plant)scalar multiplied by plant) )( )( )( skG sU sY = k )( )( )( sGk sU sY o c = )()( sGksG om =
  7. 7. MIT RuleMIT Rule  Choose cost function:Choose cost function:  Write equation for error:Write equation for error:  Calculate sensitivity derivative:Calculate sensitivity derivative:  Apply MIT rule:Apply MIT rule: coccmm UGkUkGUGkGUyye −=−=−= θ δθ δ γ θ θθ e e dt d eJ −=→= )( 2 1 )( 2 m o c y k k kGU e == δθ δ eyey k k dt d mm o γγ θ −=−= '
  8. 8. MIT RuleMIT Rule  Gives block diagram:Gives block diagram:  considered tuning parameterconsidered tuning parameter Adjustment Mechanism ymodel u yplantuc Π Π θ Reference Model Plant s γ− )()( sGksG om = )()( sGksGp = - + γ
  9. 9. MIT RuleMIT Rule  NOTE: MIT rule does not guarantee errorNOTE: MIT rule does not guarantee error convergence or stabilityconvergence or stability  usually kept smallusually kept small  Tuning crucial to adaptation rate andTuning crucial to adaptation rate and stability.stability. γ γ
  10. 10.  SystemSystem MRAC of PendulumMRAC of Pendulum ( ) TdmgdcJ c 1sin =++ θθθ  cmgdcsJs d sT s ++ = 2 1 )( )(θd2 d1dc T 77.100389.0 89.1 )( )( 2 ++ = sssT sθ
  11. 11. MRAC of PendulumMRAC of Pendulum  Controller will take form:Controller will take form: Controller Model Adjustment Mechanism Controller Parameters ymodel u yplant uc 77.100389.0 89.1 2 ++ ss
  12. 12. MRAC of PendulumMRAC of Pendulum  Following process as before, writeFollowing process as before, write equation for error, cost function, andequation for error, cost function, and update rule:update rule: modelplant yye −= )( 2 1 )( 2 θθ eJ = δθ δ γ δθ δ γ θ e e J dt d −=−= sensitivity derivative
  13. 13. MRAC of PendulumMRAC of Pendulum  Assuming controller takes the form:Assuming controller takes the form: ( ) cplant plantcpplant cmpmodelplant plantc u ss y yu ss uGy uGuGyye yuu 2 2 1 212 21 89.177.100389.0 89.1 77.100389.0 89.1 θ θ θθ θθ +++ = −      ++ == −=−= −=
  14. 14. MRAC of PendulumMRAC of Pendulum ( ) plant c c cmc y ss u ss e u ss e uGu ss e 2 2 1 2 2 2 1 2 2 2 2 1 2 2 1 89.177.100389.0 89.1 89.177.100389.0 89.1 89.177.100389.0 89.1 89.177.100389.0 89.1 θ θ θ θ θ θθ θ θ +++ −= +++ −= ∂ ∂ +++ = ∂ ∂ − +++ =
  15. 15. MRAC of PendulumMRAC of Pendulum  If reference model is close to plant, canIf reference model is close to plant, can approximate:approximate: plant mm mm c mm mm mm y asas asae u asas asae asasss 01 2 01 2 01 2 01 1 01 2 2 2 89.177.100389.0 ++ + −= ∂ ∂ ++ + = ∂ ∂ ++≈+++ θ θ θ
  16. 16. MRAC of PendulumMRAC of Pendulum  From MIT rule, update rules are then:From MIT rule, update rules are then: ey asas asa e e dt d eu asas asa e e dt d plant mm mm c mm mm       ++ + = ∂ ∂ −=       ++ + −= ∂ ∂ −= 01 2 01 2 2 01 2 01 1 1 γ θ γ θ γ θ γ θ
  17. 17. MRAC of PendulumMRAC of Pendulum  Block DiagramBlock Diagram ymodel e yplantuc Π Π θ1 Reference Model Plant s γ− 77.100389.0 89.1 2 ++ ss Π + - mm mm asas asa 01 2 01 ++ + mm mm asas asa 01 2 01 ++ + mm m asas b 01 2 ++ s γ Π - + θ2
  18. 18. MRAC of PendulumMRAC of Pendulum  Simulation block diagram (NOTE:Simulation block diagram (NOTE: Modeled to reflect control of DC motor)Modeled to reflect control of DC motor) am s+am am s+am -gamma s gamma s Step Saturation omega^2 s+am Reference Model 180/pi Radians to Degrees 4.41 s +.039s+10.772 Plant 2/26 Degrees to Volts 35 Degrees y m Error Theta2 Theta1 y
  19. 19. MRAC of PendulumMRAC of Pendulum  Simulation with small gamma = UNSTABLE!Simulation with small gamma = UNSTABLE! 0 200 400 600 800 1000 1200 -100 -50 0 50 100 150 ym g=.0001
  20. 20. MRAC of PendulumMRAC of Pendulum  Solution: Add PD feedbackSolution: Add PD feedback am s+am am s+am -gamma s gamma s Step Saturation omega^2 s+am Reference Model 180/pi Radians to Degrees 4.41 s +.039s+10.772 Plant 1 P du/dt 2/26 Degrees to Volts 35 Degrees 1.5 D y m Error Theta2 Theta1 y
  21. 21. MRAC of PendulumMRAC of Pendulum  Simulation results with varying gammasSimulation results with varying gammas 0 500 1000 1500 2000 2500 0 5 10 15 20 25 30 35 40 45 ym g=.01 g=.001 g=.0001 707. sec3 :such thatDesigned 56.367.2 56.3 2 = = ++ = ζ s m T ss y
  22. 22. LabVIEW VI Front PanelLabVIEW VI Front Panel
  23. 23. LabVIEW VI Back PanelLabVIEW VI Back Panel
  24. 24. Experimental ResultsExperimental Results
  25. 25. Experimental ResultsExperimental Results  PD feedback necessary to stabilizePD feedback necessary to stabilize systemsystem  Deadzone necessary to prevent updatingDeadzone necessary to prevent updating when plant approached modelwhen plant approached model  Often went unstable (attributed to inherentOften went unstable (attributed to inherent instability in system i.e. little damping)instability in system i.e. little damping)  Much tuning to get acceptable responseMuch tuning to get acceptable response
  26. 26. ConclusionsConclusions  Given controller does not perform well enoughGiven controller does not perform well enough for practical usefor practical use  More advanced controllers could be formed fromMore advanced controllers could be formed from other methodsother methods – Modified (normalized) MITModified (normalized) MIT – Lyapunov direct and indirectLyapunov direct and indirect – Discrete modeling using Euler operatorDiscrete modeling using Euler operator  Modified MRAC methodsModified MRAC methods – Fuzzy-MRACFuzzy-MRAC – Variable Structure MRAC (VS-MRAC)Variable Structure MRAC (VS-MRAC)

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