# 1 mrac for inverted pendulum

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Information about 1 mrac for inverted pendulum
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Published on March 20, 2014

Author: nazir1988

Source: slideshare.net

## Description

inverted pendulum

Model ReferenceModel Reference Adaptive ControlAdaptive Control Survey of Control Systems (MEM 800)Survey of Control Systems (MEM 800) Presented byPresented by Keith SevcikKeith Sevcik

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

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

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

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 = - +

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 =

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 −=−= '

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

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

 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

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

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

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 +++ = −      ++ == −=−= −=

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 +++ −= +++ −= ∂ ∂ +++ = ∂ ∂ − +++ =

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

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

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

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

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

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

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

LabVIEW VI Front PanelLabVIEW VI Front Panel

LabVIEW VI Back PanelLabVIEW VI Back Panel

Experimental ResultsExperimental Results

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

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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