Explicit Model Predictive Control - IMT School for

/54©2009byA.Bemporad ControllodiProcessoedeiSistemidiProduzione‐A.a.2008/09 1

ExplicitModelPredictiveControl

/54©2009byA.Bemporad ControllodiProcessoedeiSistemidiProduzione‐A.a.2008/09

KKTConditionsforOptimality

Whenf,giareconvexfunctionsandhjarelinear,theconditionisalsosufficient2


KKT‐GeometricInterpretation

rg1(U1)‐rf(U1)

rg2(U1)

U1

U2

‐rf(U2)rg2(U2)

rg3(U2)

g 3(U)·0

g 1(U)·0

g2 (U)·0

g(U)·0

3


KKTConditionsforQP

HaroldW.Kuhn(1925‐)

WilliamKarush(1917‐1997)

AlbertW.Tucker(1905‐1995)

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http://images.google.com/imgres?imgurl=www.sie.arizona.edu/SPEED-CS/raptormore/more/images/kuhn.jpg&imgrefurl=http://www.sie.arizona.edu/SPEED-CS/raptormore/more/hall_fame.html&h=185&w=144&prev=/images%253Fq%253Dkuhn%252Bharold%2526svnum%253D10%2526hl%253Den%2526lr%253Dlang_en%2526ie%253DUTF-8%2526safe%253Doff%2526sa%253DG





QPAlgorithms

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MPCComputations•Theon‐lineoptimizationproblemisaQuadraticProgram(QP)

(orLinearProgram,LP)

•Noneedtoreachglobaloptimum(seetheoremproof)

•Algorithms:

‐ Activesetmethods(small/mediumsize)

‐ Interiorpointmethods(largesize)

•Benchmarksoncommercial/publicdomainQP/LPsolvers:

•Note:usingLinearProgramming(LP)for1‐or∞‐norms,controlactionmaybelesssmooththanwithQP

http://plato.la.asu.edu/bench.html

LP optimizer always on a vertex

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QP optimizer may lie anywhere

feasible set


MatlabOptimizationToolbox‐LP

MuchmoreefficientLPsolversareavailable!

(GLPK,CDD,CPLEX,NAG,andmanyothers...)

HybridToolbox

LPSOL

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MatlabOptimizationToolboxQP

MuchmoreefficientQPsolversareavailable!(CPLEX,NAG,DANTZGMP,andmanyothers...)

HybridToolbox

QPSOL

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QPComplexity

•Non‐optimizedDantzig’sroutine(MPCToolboxforMatlab)•TestsinMatlab5.3onPentiumII300MHz

CPUtime

numberofvars1 10 1001ms

10ms

100ms

1s

GeorgeDantzig(1914‐2005)

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ExplicitFormofModelPredictiveControl

viaMultiparametricProgramming

10


MPC:ProsandCons

•PRO:systematicdesignapproach–Multivariablesystems– Stateandinputconstraints– Stabilityguarantees–Referencepreview,delays,…

•CON:computationcomplexity!– Largesamplingtime/fasthardware– Softwarereliability

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Maindrawbacksofon‐lineimplementation

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•Computationtimemaybetoolong:okforlargesamplingtimes(>10ms)butnotforfast‐samplingapplications(<1ms).Worst‐caseCPUtimeoftenhardtoestimate

•Requiresrelativelyexpensivehardware(notsuitableoninexpensive8‐bitµ‐controllerswithfewkBRAM)

•Softwarecomplexity:controlprofileu(x)hardtounderstand,solvercodedifficulttocertify(badinsafetycriticalapps)

AnywaytouseMPCwithouton‐linesolvers?

•ExcellentLP/QP/MIP/NLPsolversexisttoday(“LPisatechnology”–S.Boyd)

but...


On‐Linevs.Off‐LineOptimization

•On‐lineoptimization:givenx(t)solvetheproblemateachtimestept(thecontrollawu=u(x)isimplicitlydefinedbytheQPsolver)

multi‐parametricQuadraticProgram(mp‐QP)

QuadraticProgram(QP)

•Off‐lineoptimization:solvetheQPforallx(t)tofindthecontrollawu=u(x)explicitly

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Multiparametricprogrammingproblem

Giventheoptimizationproblem

determine:

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MultiparametricQuadraticProgramming

•Objective:solvetheQPofflineforallx2XtofindtheMPCcontrollawu=u(x)explicitly

(Bemporadetal.,2002)

•Assumptions: alwayssatisfiedifmpQPcomesfromanoptimalcontrolproblem!

alwayssatisfiedifweightmatrixR>0

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LinearityoftheSolution

From(1):

Insomeneighborhoodofx0,λandUareexplicitaffinefunctionsofx!

KKToptimalityconditions:

From(2):

(Zafiriou,1990)

Fixx02X

x‐space

x0•

X

solveQPtofind

identifyactiveconstraintsat

formmatricesbycollectingactiveconstraints:

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MultiparametricQPalgorithm•Imposeprimalanddualfeasibility:

•Removeredundantconstraints(thisrequiressolvingLP’s):

criticalregionCR0

CR0

x‐space

x0•

•CR0isthesetofallandonlyparametersxforwhichistheoptimalcombinationofactiveconstraintsattheoptimizer

linearinequalitiesinx!

X

from(4)

from(5)

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CR0

x‐space

x0• R1

R2

RN

R3

X

R4

Method#1:Splitandproceediteratively

Recursionsterminatebecausecombinationsofactiveconstraintsarefinite

(Bemporad,Morari,Dua,Pistikopoulos,2002)

(Tøndel,Johansen,Bemporad,2003)

Method#2:add/withdrawconstraintsfromactiveset

⇒add

⇒withdraw

Method#3:exploitthefacet‐to‐facetproperty

Stepout²outsideeachfacet,solveQP,getnewregion,iterate. (Baotic,2002)

CR1

(Spjøtvold,Kerrigan,Jones,Tøndel,Johansen,2006)

CR0CR0

x‐space

X

x‐space

(Spjøtvold,2008)

MultiparametricQPalgorithm


Propertiesofmultiparametric‐QP

continuous,piecewiseaffine

convex,continuous,piecewisequadratic,C1(ifnodegeneracy)

Corollary:ThelinearMPCcontrollerisacontinuouspiecewiseaffinefunctionofthestate

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SetofFeasibleParametersX*

WhyisthesetX*ofparametersforwhichtheQPproblemissolvableaconvexpolyhedralset?

X*istheprojectionofapolyhedronontotheparameterspace.ThereforeX*isapolyhedron.

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MultiparametricConvexProgramming

(Mangasarian,Rosen,1964)

V*andX*maynotbeeasytoexpressanalytically

(Bemporad,Filippi,2003)(approximatesolutionspossible)

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ComplexityReduction

Regionswherethefirstcomponentofthesolutionisthesamecanbejoined(whentheirunionisconvex).(Bemporad,Fukuda,Torrisi,Computational Geometry,2001)

‐2.5 ‐2 ‐1.5 ‐1 ‐0.5 0 0.5 1 1.5 2 2.5‐2.5

‐2

‐1.5

‐1

‐0.5

0

0.5

1

1.5

2

2.5CR1CR2CR3CR4CR5CR6CR7CR8CR9

‐2.5 ‐2 ‐1.5 ‐1 ‐0.5 0 0.5 1 1.5 2 2.5‐2.5

‐2

‐1.5

‐1

‐0.5

0

0.5

1

1.5

2

2.5CR1CR2CR3CR4CR5CR6CR7

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DoubleIntegratorExample

•System:

•Constraints:

•Controlobjective:minimize

•Optimizationproblem:forNu=2

sampling+ZOHTs=1s

(costfunctionisnormalizedbymaxsvd(H))

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mp‐QPsolution

Nu=2

gotodemo/demos/linear/doubleintexp.m (Hyb‐Tbx)

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Complexity

Nu=3 Nu=4 Nu=5

Nu=6

0 5 10 15 200

20

40CP

Utime

0 5 10 15 200

100

200

300

#Re

gion

s

#freemoves

(isthenumberofregionsfiniteforNu!1?)25


Complexity•Worst‐casecomplexityanalysis:

combinationsofactiveconstraints

•Strongestdependenceonthenumberqofconstraints

•StrongdependenceonthenumberNuoffreemoves

•Weakdependenceonthenumbernofparametersx

•Example:

•Usuallythenumberofregionsismuchsmaller,asmanycombinationsofactiveconstraintsareneverfeasibleandoptimalatanyparametervectorx

Dataaveragedover20randomlygeneratedsingle‐inputsingle‐outputsystemssubjecttoinputsaturation(q=2N)

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Extensions•Trackingofreferencer(t):

•Rejectionofmeasureddisturbancev(t):

•Softconstraints:

•Variableconstraints:

•Othermodels(hybrid,uncertain)andothernorms()

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ReferenceTracking,MIMOSystem

•System:

•Constraints:

•Controlobjective:

sampling+ZOH(Ts=1s)

gotodemolinear/mimo.m

0 50 100 150 2000

0.2

0.4

0.6

0.8outputy(t),referencer(t)

0 50 100 150 2000.5

0.6

0.7

0.8

0.9

1inputu(t)

N=20Nu=1

(Hyb‐Tbx)

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Polyhedralpartitionofthestate‐spaceforu=[00]’andr=[0.630.79]’

MPClaw

State‐spacepartition

ReferenceTracking,MIMOSystem

‐100 ‐80 ‐60 ‐40 ‐20 0 20 40 60 80 100‐100

‐80

‐60

‐40

‐20

0

20

40

60

80

100x‐space

16

23

5

7

89

4

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Complexity

averageon20randomSISOsystems(inputsaturation)

•Numberofregions·#combinationsofactiveconstraints

‐Largelydependson#constraints(usuallycombinatorially)

‐Alsodependson#freevariables

‐Weaklydependson#states

#constrai

nts

#states

#regions

ExplicitMPCtypicallylimitedto6‐8freecontrolmovesand8‐12states+references

•Example

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Averagetimeon100random3Dparameters(2Nconstraints)

Worst‐casetimeon100random3Dparameters(2Nconstraints)

(IntelCentrino1.4GHz)

Complexity‐QPvs.Explicit

Needtovisitregionsmoreefficientlythanlinearsearch

Needtoreducenumberofregions

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Example:AFTI‐16•Linearizedmodel:

gotodemolinear/afti16.m (Hyb‐Tbx)

Explicitcontroller:8parameters,51regions

‐10 ‐8 ‐6 ‐4 ‐2 0 2 4 6 8 10‐10

‐8

‐6

‐4

‐2

0

2

4

6

8

10Sectionwithx=[0000]’,u=[00]’

r1

r2

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MPCofaDCServomotor

V

R

JM

JL

θM

θLT

ρβM

βL

θs

T re

gotodemolinear/dcmotor.m (Hyb‐Tbx)

Explicitcontroller:7parameters,101regions

‐1.5 ‐1 ‐0.5 0 0.5 1 1.5‐3

‐2

‐1

0

1

2

3Sectionwithx1=x4=0,r=[00]’,u=0

x2

x3

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(Bemporad,Borrelli,Glielmo,Vasca,2001)

•ControlObjective

•smallfrictionlosses

•fastengagement

•drivercomfort

•Constraints:

•clutchforce

•clutchforcederivative

•minimumenginespeed

Slipphase:ωe >ωv

Clutchengaged:ωe=ωv=ω

DryClutchEngagement

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LinearMPCDesign•Linearmodelduringslip

+disturbancemodel

•Sampling

T=10ms

•Constraints:0· ¢u·80N,0·u·5000N,x1¸50rad/s,x2¸0

•Controlobjective:

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gotodemo/demos/dryclutch/dryclutch.m (Hyb‐Tbx)

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MPCTuning


Q=2,R=1

N=10,Nu=1

Choice:

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MPCsimulation


+LinearObserver

Alternative:explicithybridMPC

•Switchingmodel(slippingmode,engagedmode)•DesignanexplicitMPCcontrollerfortheswitchedsystem

ωerad/s

ωvrad

/s

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MPCController‐ExplicitSolution


• TuneanMPCcontrollerbysimulation,usingtheMPCSimulinkToolbox

• GettheexplicitsolutionoftheMPCcontroller.

• Validatethecontrolleronexperiments.

Task:

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MPCRegulationofaBallonaPlate


Ball&plate:experimentalsetup

Ball&PlateSystemHost

Target

Ball&PlateCode Data

u1,u2

a,b

x,y

TCP/IP

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• Specifications:Angle: ‐17deg…+17degPlate: ‐30cm…+30cm InputVoltage:‐10V…+10V Computer:PENTIUM166 SamplingTime:30ms

• Model:LTI14states Constraintsoninputsandstates

y

α

β

x

α'β'

δγ

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Ball&plate:modelandspecifications


MPCTuning

Samplingtime: Ts=30ms

N=50

Nu=2 Nu N

u

1(hard constraint)

5

y

Predictionhorizon:

Freecontrolmoves:

Outputconstrainthorizon:

Inputconstrainthorizon:

Weightonpositionerror:

Weightoninputvoltagechanges:

1(soft constraint)

1

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Implementation

•Solvemp‐QPandimplementexplicitMPC

E.g:Real‐TimeWorkshop+xPCToolbox

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ExplicitMPCSolution

x‐MPC:sectionsatαx=0,αx=0,ux=0,rx=18,rα=0

Region1:

Region6:

Region16:

Saturationat‐10

Saturationat+10

LQRController(nearequilibrium)

6

1 166

161

o

x:22Regions,y:23RegionsController:

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Ballandplateexperiment


BallandplateexperimentBallandplateexperimentinLEGO,usingexplicitMPCandHybridToolbox

•20Hzsamplingfrequency

•camerausedforpositionfeedback

•explicitMPCcodedusingintegernumbers

(byDanieleBenedettelli,Univ.ofSiena,July2008)


Example‐LaboratoryHelicopter

6states,2inputs

Upper/lowerconstraintsonboth

inputsandtwoofthestates.

Sizeofmp‐QPwithN=3:

dim(U)=6

dim(x)=6

24constraints

Off‐linecomputationtimes: On‐linecomputationtimes:

(Tøndel,2003)

Horizon Time(s) #polyhedra

1 1 33

2 19 395

3 163 2211

Horizon Implicitsolution Explicitsolution

1 19ms 0.3ms

2 20ms 9ms

3 25ms 52ms

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CommentsonExplicitMPC•MultiparametricQuadraticPrograms(mp‐QP)canbesolvedefficiently

•ExplicitsolutionofMPCcontrolleru=f(x)isPiecewiseAffine

Eliminateheavyon‐linecomputationforMPC

MakeMPCsuitableforfast/small/cheapprocesses

•ModelPredictiveControl(MPC)canbesolvedoff‐lineviamp‐QP

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AutomotiveapplicationsofexplicitMPC

Homogeneous Stratified

tractioncontrol

enginecontrol

semiactivesuspensions idlespeedcontrol magneticactuators

(Borrelli,Bemporad,Fodor,Hrovat,2001)

(N.Giorgetti,G.Ripaccioli,AB,I.Kolmanovsky,D.Hrovat,2006)

(DiCairano,Bemporad,Kolmanovsky,Hrovat,2006)

(DiCairano,Yanakiev,Bemporad,Kolmanovsky,Hrovat,2008)

(Giorgetti,Bemporad,Tseng,Hrovat,2005)

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ExplicitMPCforidlespeedcontrol

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•Fordpickuptruck,V84.6Lgasolineengine

•Process:

‐1 output(enginespeed)toregulate‐2 inputs(airflow,sparkadvance)‐inputdelays

‐regulateenginespeedatconstantrpm‐saturationlimitsonairflowandspark‐lowerboundonenginespeed(¸450rpm)

•Objectivesandspecs:

•Relatedtomostclassicalproblemincontrol:Watt’sgovernor(1787)

•ProblemsuitableforMPCdesign (Hrovat,1996)


ExplicitMPCDesignFlow

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NLSimulinkmodel

MPCsetup(on‐lineQP)

controlspecs

multi‐parametricsolver

Matlabtool:HybridToolbox

predictionmodel

identification

PWAcontrollaw(C‐code)

closed‐loopsimulation

experiments

reviseweights&observer

addintegralaction


ExplicitMPCforidlespeed‐Experiments

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baselinecontroller(linear)

explicitMPC

set‐point

Loadtorque(powersteering) Powersteering+airconditioning

peakreducedby50% convergence10sfaster



mpQPinportfoliooptimizationMarkowitzportfoliooptimization

pi=expectedreturnofasseti

§ij=covarianceofassetsi, j

x=expectedminimumreturnofportfolio

zi=moneyinvestedinasseti

Objective:minimizevariance(=risk)

Constraint:guaranteeaminimumexpectedreturn

(Bemporad,NMPCplenary,2008)

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mpQPinportfoliooptimizationMultiparametricQPsolution

(Bemporad,2008)

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Explicit Model Predictive Control - IMT School for