EECE 574 - Adaptive Control - Model-Reference Adaptive Control

73 Pages·2013·0.79 MB·English

by  Guy Dumont

EECE 574 - Adaptive Control
Model-ReferenceAdaptiveControl-AnOverview
GuyDumont
DepartmentofElectricalandComputerEngineering
UniversityofBritishColumbia
January2013
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Model-Reference Adaptive Systems
TheMRACorMRASisanimportantadaptivecontrolmethodology1
1seeChapter5oftheÅströmandWittenmarktextbook,orH.Butler,
”Model-ReferenceAdaptiveControl-FromTheorytoPractice”,Prentice-Hall,1992
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Model-Reference Adaptive Systems
TheMITrule
Lyapunovstabilitytheory
DesignofMRASbasedonLyapunovstabilitytheory
Hyperstabilityandpassivitytheory
Theerrormodel
Augmentederror
Amodel-followingMRAS
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MITRule TheBasics
The MIT Rule
OriginalapproachtoMRACdevelopedaround1960atMITfor
aerospaceapplications
Withe=y−y ,adjusttheparametersθ tominimize
m
1
J(θ)= e2
2
Itisreasonabletoadjusttheparametersinthedirectionofthenegative
gradientofJ:
dθ ∂J ∂e
=−γ =−γe
dt ∂θ ∂θ
∂e/∂θ iscalledthesensitivityderivativeofthesystemandisevaluated
undertheassumptionthatθ variesslowly
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MITRule TheBasics
The MIT Rule
ThederivativeofJ isthendescribedby
dJ ∂e (cid:18)∂e(cid:19)2
=e =−γe2
dt ∂t ∂θ
Alternatively,onemayconsiderJ(e)=|e|inwhichcase
dθ ∂J ∂e
=−γ =−γ sign(e)
dt ∂θ ∂θ
Thesign-signalgorithmusedintelecommunicationswheresimple
implementationandfastcomputationsarerequired,is
(cid:18) (cid:19)
dθ ∂J ∂e
=−γ =−γsign sign(e)
dt ∂θ ∂θ
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MITRule Examples
MIT Rule: Example 1
Process: y=kG(s)whereG(s)isknownbutkisunknown
Thedesiredresponseisy =k G(s)u
m 0 c
Controllerisu=θu
c
Thene=y−y =kG(p)θu −k G(p)u
m c 0 c
Sensitivityderivative
∂e k
=kG(p)u = y
c m
∂θ k
0
MITrule
dθ k
=γ(cid:48) y e=−γy e
m m
dt k
0
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MITRule Examples
MIT Rule: Example 1
Figure: MITruleforadjustmentoffeedforwardgain(fromtextbook).
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MITRule Examples
MIT Rule: Example 1
Figure: MITruleforadjustmentoffeedforwardgain: Simulationresults(from
textbook).
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MITRule Examples
MIT Rule: Example 2
Considerthefirst-ordersystem
dy
=−ay+bu
dt
Thedesiredclosed-loopsystemis
dy
m
=−a y +b u
m m m c
dt
Applyingmodel-followingdesign(seelecturenotesonpoleplacement)
degA ≥0 degS=degR=degT =0
0
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MITRule Examples
MIT Rule: Example 2
Thecontrolleristhen
u(t)=t u (t)−s y(t)
0 c 0
Forperfectmodel-following
dy
= −ay(t)+b[t u (t)−s y(t)]
0 c 0
dt
=−(a+bs )y(t)+bt u
0 0 c
=−a y (t)+b u
m m m c
Thisimplies
a −a
m
s =
0
b
b
m
t =
0
b
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