lec16_ Optimal Control

8/12/2019 lec16_ Optimal Control

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MIT OpenCourseWarehttp://ocw.mit.edu

16.323 Principles of Optimal Control

Spring 2008

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16.323Lecture16ModelPredictiveControl

Allgower,F.,andA.Zheng,NonlinearModelPredictiveControl,Springer-Verlag,2000.

Camacho,E.,andC.Bordons,ModelPredictiveControl,Springer-Verlag,1999. Kouvaritakis, B., and M. Cannon, Non-Linear Predictive Control: Theory &

Practice,IEEPublishing,2001. Maciejowski, J.,PredictiveControlwithConstraints,PearsonEducationPOD,

2002. Rossiter, J. A., Model-Based Predictive Control: A Practical Approach, CRC

Press,2003.


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Spr2008 16.323 162 Note that the control algorithm is based on numerically solving an

optimizationproblemateachstepTypicallyaconstrainedoptimization

MainadvantageofMPC:Explicitlyaccountsforsystemconstraints.Doesntjust design a controller to keep the system away from

them.Caneasilyhandlenonlinearandtime-varyingplantdynamics,since

thecontrollerisexplicitlyafunctionofthemodelthatcanbemodifiedinreal-time(andplantime)

Manycommercialapplicationsthatdatebacktotheearly1970s,seehttp://www.che.utexas.edu/

~qin/cpcv/cpcv14.html

Muchofthisworkwasinprocesscontrol- verynonlineardynamics,butnotparticularlyfast.

Ascomputerspeedhas increased,therehasbeenrenewed interest inapplying thisapproach toapplicationswith faster time-scale: trajectorydesignforaerospacesystems.

June18,2008

Ref

PlantP

Trajectory

Generation

Noise

uRef

Output

PlantP

Trajectory

Generation

Noiseuud

xd du

Output

Feedback

Compensation

Implementation architectures for MPC (from Mark Milam)

Figure by MIT OpenCourseWare.
http://www.che.utexas.edu/~qin/cpcv/cpcv14.htmlhttp://www.che.utexas.edu/~qin/cpcv/cpcv14.htmlhttp://www.che.utexas.edu/~qin/cpcv/cpcv14.htmlhttp://www.che.utexas.edu/~qin/cpcv/cpcv14.html


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BasicFormulation

Spr2008 16.323 163 Givenasetofplantdynamics(assumelinearfornow)

x(k+ 1) = Ax(k) +Bu(k)z(k) = Cx(k)

andacostfunctionN

J = {z(k+j|k)Rzz +u(k+j|k)Ruu}+F(x(k+N|k))j=0

z(k+j|k)Rxx isjust a short hand for a weighted norm of thestate,andtobeconsistentwithearlierwork,wouldtake

z(k+j|k)Rzz =z(k+j|k)TRzzz(k+j|k)F(x(k+N|k))isaterminalcostfunction

NotethatifN ,andtherearenoadditionalconstraintsonzoru,thenthisisjustthediscreteLQRproblemsolvedonpage314.NotethattheoriginalLQR resultcouldhavebeenwrittenasjust

an inputcontrolsequence(feedforward),butwechoosetowriteitasalinearstatefeedback.

In the nominal case, there is no difference between these two implementationapproaches(feedforwardandfeedback)

Butwithmodelingerrorsanddisturbances,thestatefeedbackformismuchlesssensitive.

Thisisthemainreasonforusingfeedback.

Issue: Whenlimitsonxanduareadded,wecannolongerfindthegeneralsolutioninanalyticform mustsolveitnumerically.

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Spr2008 16.323 164 However,solvingforaverylonginputsequence:

Doesnotmakesenseifoneexpectsthatthemodeliswrongand/orthere are disturbances, because it is unlikely that the end of theplanwillbeimplemented(anewonewillbemadebythen)

Longerplanshavemoredegreesof freedomandtakemuch longertocompute.

TypicallydesignusingasmallNshortplanthatdoesnotnecessarilyachieveallofthegoals.ClassicalhardquestionishowlargeshouldN be?Ifplandoesntreachthegoal,thenmustdevelopanestimateofthe

remainingcost-to-go

Typicalproblemstatement: forfiniteN (F = 0)N

minJ = k)Rzz +u(k+j k)Ruu}u {z(k+j| |j=0

s.t. x(k+j+ 1|k) = Ax(k+j|k) +Bu(k+j|k)x(k|k) x(k)

z(k+j|k) = Cx(k+j|k)and |u(k+j|k)| um

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Spr2008 16.323 165 Considerconvertingthisintoamorestandardoptimizationproblem.

z(k|k) = Cx(k|k)z(k+ 1|k) = Cx(k+ 1|k) =C(Ax(k|k) +Bu(k|k))

= CAx(k|k) +CBu(k|k)z(k+ 2|k) = Cx(k+ 2|k)

= C(Ax(k+ 1|k) +Bu(k+ 1|k))= CA(Ax(k|k) +Bu(k|k))+CBu(k+ 1|k)= CA2x(k|k) +CABu(k|k) +CBu(k+ 1|k)...

z(k+N|k) = CANx(k|k) +CAN1Bu(k|k) + +CBu(k+ (N1)|k)

Combinetheseequationsintothefollowing:

z(k k) C|z(k+ 1

=

x(k k)|||k)k) CACA2z(k+ 2.. ... .

CANz(k+N|k) 0 0 0 0CB 0 0 0

CAB CB 0 0u(k|k)

u(k+ 1

...|k)+ ...

CAN1B CAN2B CAN3B CB u(k+N1|k)

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Spr2008 16.323 166Nowdefine

z(k k) u(k k)Z(k) ...

| U(k) ...|

z(k+N|k) u(k+N1|k)then,withx(k|k) =x(k)

Z(k) =Gx(k) +HU(k)Notethat

Nz(k+j|k)TRzzz(k+j|k) =Z(k)TW1Z(k)

j=0withanobviousdefinitionoftheweightingmatrixW1Thus

Z(k)TW1Z(k) +U(k)TW2U(k)= (Gx(k) +HU(k))TW1(Gx(k) +HU(k))+U(k)TW2U(k)

1= x(k)TH1x(k) +H2TU(k) + U(k)TH3U(k)2

whereH1 =GTW1G, H2 =2(x(k)TGTW1H), H3 =2(HTW1H+W2)

ThentheMPCproblemcanbewrittenas:minJ = H2TU(k) +1U(k)TH3U(k)U(k) 2

INs.t. U(k)umIN

June18,2008


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ToolboxesSpr2008 16.323 167 Key point: the MPC problem is now in the form of a standard

quadraticprogramforwhichstandardandefficientcodesexist.QUADPROG

Quadratic

programming.

%

X=QUADPROG(H,f,A,b) attempts to solve the %quadratic programming problem:min 0.5*x*H*x + f*x subject to: A*x


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MPCObservations

Spr2008 16.323 168 Currentformassumesthatfullstateisavailable- canhookupwithan

estimator

CurrentformassumesthatwecansenseandapplycorrespondingcontrolimmediatelyWith most control systems, that is usually a reasonably safe as

sumptionGiventhatwemust re-run theoptimization, probablyneedtoac

count for thiscomputationaldelay - different formof thediscretemodel- seeF&P(chapter2)

Iftheconstraintsarenotactive,thenthesolutiontotheQPisthatU(K) =H1H23

whichcanbewrittenas:u(k|k) = 1 0 . . . 0 (HTW1H+W2)1HTW1Gx(k)

= Kx(k)whichisjustastatefeedbackcontroller.Canapplythisgaintothesystemandchecktheeigenvalues.

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Spr2008 16.323 169 WhatcanwesayaboutthestabilityofMPCwhentheconstraintsare

active? 33Dependsalotontheterminalcostandtheterminalconstraints.34

Classic result:35 Consider a MPC algorithm for a linear system withconstraints. Assumethatthereareterminalconstraints: x(k+N|k) = 0forpredictedstatex u(k+N|k) = 0forcomputedfuturecontroluTheniftheoptimizationproblemisfeasibleattimek,x= 0isstable.Proof: CanusetheperformanceindexJ asaLyapunovfunction.AssumethereexistsafeasiblesolutionattimekandcostJkCanusethatsolutiontodevelopafeasiblecandidateattimek+ 1,

bysimply

adding

u(k

+

N

+ 1)

=

0and

x(k

+

N

+ 1)

=

0.

Keypoint: canestimatethecandidatecontrollerperformance

Jk+1 = Jk{z(k|k)Rzz +u(k|k)Ruu} Jk{z(k|k)Rzz}

Thiscandidate issuboptimal fortheMPCalgorithm,henceJ decreasesevenfasterJk+1 Jk+1

WhichsaysthatJ decreases ifthestatecost isnon-zero(observabilityassumptions) butJ islowerboundedbyzero.

Mayneetal. [2000] provides excellent review of other strategies forprovingstabilitydifferentterminalcostandconstraintsets

33

Tutorial: modelpredictivecontroltechnology,Rawlings,J.B.AmericanControlConference,1999. pp. 662-67634Mayne,D.Q.,J.B.Rawlings,C.V.RaoandP.O.M.Scokaert,ConstrainedModelPredictiveControl: StabilityandOptimality,

Automatica,36,789-814(2000).35A.Bemporad,L.Chisci,E.Mosca: OnthestabilizingpropertyofSIORHC,Automatica,vol. 30,n. 12,pp. 2013-2015,1994.

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Example: HelicopterSpr2008 16.323 1610 ConsiderasystemsimilartotheQuansarhelicopter36

Thereare2controlinputsvoltagetoeachfanVf,Vb Asimpledynamicsmodelisthat:

e = K1(Vf +Vb)Tg/Jer = K2sin(p)p = K3(Vf Vb)

andtherearephysicallimitsontheelevationandpitch:0.5e 0.6 1p 1

ModelcanbelinearizedandthendiscretizedTs = 0.2sec.

0 5 10 15 20 250

2

4

6

8

10

12

N

State

Control

Time

Figure16.3:ResponseSummary36ISSN02805316ISRNLUTFD2/TFRT- -7613- -SEMPCtools1.0 ReferenceManualJohanAkessonDepartmentofAutomatic

ControlLundInstituteofTechnologyJanuary2006

June18,2008

p

e

r

Figure by MIT OpenCourseWare.


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Spr2008

0 10 20 300.1

0

0.1

0.2

0.3

0.4

Elevation[rad]

0 10 20 30

1

0.5

0

0.5

1

t [s]

Pitch[rad]

0 10 20 301

0

1

2

3

4

Rotation[rad]

0 10 20 302

1

0

1

2

3

4

Vf,

Vb[

V]

t [s]

16.323 1611

Figure16.4:ResponsewithN

= 3

0 10 20 300.1

0

0.1

0.2

0.3

0.4

Elevation[rad]

0 10 20 30

1

0.5

0

0.5

1

t [s]

Pitch[rad]

0 10 20 301

0

1

2

3

4

Rotation[rad]

0 10 20 302

1

0

1

2

3

4

Vf,

Vb[

V]

t [s]

Figure16.5:ResponsewithN =10

0 10 20 300.1

0

0.1

0.2

0.3

0.4

Elevation

[rad]

0 10 20 30

1

0.5

0

0.5

1

t [s]

Pitch[rad]

0 10 20 301

0

1

2

3

4

Rotation

[rad]

0 10 20 302

1

0

1

2

3

4

Vf,

Vb[

V]

t [s]

Figure16.6:ResponsewithN =25June 18 2008

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lec16_ Optimal Control