modelling and optimization‐based control of hybrid

A.Bemporad MATHMOD2009‐February13,2009

ModellingandOptimization‐basedControlofHybridDynamicalSystems

AlbertoBemporadhttp://www.dii.unisi.it/~bemporad

UniversityofSiena

Department of InformationEngineering

/48A.Bemporad MATHMOD2009‐February13,2009

Talkoutline

• Modelsofhybridsystems

• Modelpredictivecontrol(MPC)ofhybridsystems

• Automotiveapplications

• Newresearchdirections

2


Hybriddynamicalsystems

3

Hybridsystems

ComputerScience

ControlTheory

Finitestate

machines

Continuousdynamicalsystems

systemu(t)

AB

C

C

A

B

B

C

y(t)1 2

35

4


Embeddedsystems

4

continuousdynamicalsystem

discreteinputs

symbolssymbols

continuousstates

continuousinputs

automaton/logic

interface

Examples:automobiles,industrial

processes,consumerelectronics,

homeappliances,...

Anelectronic(control)deviceis“embedded”

inaphysicalprocessandinteractswithit

Sensor-based image stabilization


“Intrinsicallyhybrid”systems

5

•Transmission

discretecommand(R,N,1,2,3,4,5)

•Four‐strokeengines

automaton,dependentoncrankshaftangle

continuousdynamicalvariables(velocities,torques)+


Keyrequirementsforhybridmodels

6

•Descriptiveenoughtocapturethebehaviorofthesystem

–continuousdynamics(physicallaws)

–logiccomponents(switches,automata,softwarecode)

–interconnectionbetweenlogicanddynamics

•Simpleenoughforsolvinganalysisandsynthesisproblems

“Makeeverythingassimpleaspossible,butnotsimpler.”—AlbertEinstein

linearhybridsystems

/48A.Bemporad MATHMOD2009‐February13,2009 7

Piecewiseaffine(PWA)systems

Canapproximatenonlinearand/ordiscontinuousdynamicsarbitrarilywell

(Sontag1981)

state+inputspace

x(k+1)

x(k)C1C2C3C4C5C6


DiscreteHybridAutomata(DHA)

8

(Torrisi,Bemporad,2004)

EventGenerator

FiniteStateMachine

ModeSelector

SwitchedAffineSystem

1

2

s

mode

timeoreventcounter

continuous

discrete


Switchedaffinesystem

Theaffinedynamicsdependonthecurrentmodei(k):

continuous

discrete


Eventgenerator

Eventvariablesaregeneratedbylinearthresholdconditionsovercontinuousstatesandinputs(timeeventscanbealsomodeled):

Example:[δ(k)=1]↔[xc(k)≤0]

continuous

discrete


Finitestatemachine

ThebinarystateofthefinitestatemachineevolvesaccordingtoaBooleanstateupdatefunction:

Example:

continuous

discrete


Theactivemodei(k)isselectedbyaBooleanfunctionofthecurrentbinarystates,binaryinputs,andeventvariables:

Modeselector

Example:0

1

0 1

thesystemhas3modes

continuous

discrete


Logicandlinearinequalities

13

(Glover1975,Williams1977,Hooker2000)

FiniteStateMachine

ModeSelector

SwitchedAffineSystem

1

2

s

EventGenerator

AnylogicformulainvolvingBooleanvariablesandlinearcombinationsofcontinuousvariablescanbetranslatedintoasetof(mixed‐)integerlinear(in)equalities

AlltheDHAblockscanbetranslatedintoasetof(mixed‐)integerlinearequalitiesandinequalities


Mixed‐logicaldynamicalsystems

14

(Bemporad,Morari1999)MixedLogicalDynamical(MLD)Systems

HYSDEL (Torrisi,Bemporad,2004)

DiscreteHybridAutomaton

Suitableforsolvingoptimizationproblems(mixed‐integerprogramming)

Continuousandbinaryvariables


Mixed‐integermodelsinoperationsresearch

15

Translationoflogicalrelationsintolinearinequalitiesisheavilyusedinoperationsresearch(OR)forsolvingcomplexdecisionproblemsbyusingmixed‐integer(linear)programming(MIP)

Example:Timetablegeneration(fordemandingprofessors…)

CPUtime:0.2s

Effort:10%mathematicalproblemsetup(mixed‐integerlinearmodel)30%database&webinterfaces60%dealwithprofessors’complaints,complaints,complaints…

•Hybridmodels:design,simulation,verification

•Controldesignforlinearsystemsw/constraintsandhybridsystems(on‐lineoptimizationviaQP/MILP/MIQP)

•ExplicitMPCcontrol(viamulti‐parametricprogramming)

•C‐codegeneration

•Simulinklibrary


HybridToolboxforMATLAB

16

Features: (Bemporad,2003‐2009)

Support:

http://www.dii.unisi.it/hybrid/toolbox

2200+downloadrequestssinceOctober2004


Easilymodeledasdiscrete‐timelinearhybridsystems

bouncingball

magneticallyactuatedfuelinjector

Examples:systemswithimpacts


T1 T2

Tamb

•#1turnstheheater(A/C)onwheneverheiscold(hot)

•If#2iscoldheturnstheheateron,unless#1ishot

•If#2ishotheturnsA/Con,unless#2iscold

•Otherwise,heaterandA/Careoff

Heater

ACsystem

uhotucold

Hybriddynamics

Example:roomtemperature


HYSDELmodel

http://www.dii.unisi.it/hybrid/toolbox

HybridToolboxforMatlab

>>S=mld('heatcoolmodel',Ts)

>>[XX,TT]=sim(S,x0,U);

gettheMLDmodelinMatlab

simulatetheMLDmodel


HybridMLDmodel•MLDmodel

•2continuousstates:

•1continuousinput:

•6auxiliarybinaryvars:

•2auxiliarycontinuousvars:

(roomtemperatureTamb)

(temperaturesT1,T2)

(powerflowsuhot,ucold)

(4thresholds+2forORcondition)

•20mixed‐integerinequalities

Possiblecombinationofintegervariables:26=64


HybridPWAmodel•PWAmodel

•2continuousstates:

•1continuousinput:(roomtemperatureTamb)

(temperaturesT1,T2)

•5polyhedralregions(partitiondoesnotdependoninput)

>>P=pwa(S);

bothoff

heateron

A/Con


1

2

3

e,p12e

e,p13e∼e(Bemporad,DiCairano,IEEETAC,submitted)

StochasticFiniteStateMachine(sFSM)

Discrete‐timeHybridStochasticAutomaton(DHSA)

Event‐basedContinuous‐timeHybridAutomaton(icHA)

Switchedintegraldynamics

(Bemporad,DiCairano,Julvez,2009)

k =eventcounter

AsynchronousFSM

k =discrete‐timecounter

Allcontrol/verificationtechniquesdevelopedforDHAcanbeextendedtoDHSAandicHA!

DHAextensions

ucconstantbetweeneventskandk+1


Talkoutline

✓ Modelsofhybridsystems

• Modelpredictivecontrol(MPC)ofhybridsystems



23


ModelPredictiveControl(MPC)

24

Amodeloftheprocessisusedtopredictthefutureevolutionoftheprocesstodecidethecontrolsignal

binaryinputs

continuousinputs

binarystates

continuousstates

modelpredictivecontroller

desiredbehavior

constraints

hybridprocess


Recedinghorizonphilosophy

25

•Onlyapplythefirstoptimalmove

Predictedoutputs

ManipulatedInputs

t t+1 t+N

ut+k

r(t)

t+1 t+2 t+N+1

•Attimet+1:Getnewmeasurements,repeattheoptimization.Andsoon…

Advantageofrepeatedon‐lineoptimization:FEEDBACK!

yt+k•Attime t:solveanoptimalcontrolproblemoverafinitefuturehorizonof N steps:


Recedinghorizonexample

26

‐predictionmodel

‐costfunction

‐constraints

‐recedinghorizonmechanism

howvehiclemovesonthemap

minimumtime,minimumdistance,etc.

driveonroads,respectone‐wayroads,etc.

event‐based(optimalroutere‐plannedwhenpathislost)

‐disturbances mainlydriver’sinattention!

‐setpoint desiredlocationx=GPSpositionu=navigationcommands

Fastest routeShortest routeAvoid motorwaysWalking routeBicycle routeLimited speed


MPCofhybridsystems

27

MixedIntegerQuadraticProgram(MIQP)

(Bemporad,Morari,1999)

Alternative:Mixed‐integerlinear(MILP)formulations

(Bemporad,Morari1999)

Closed‐loopconvergenceresultsofhybridMPCavailable(Lazar,Heemels,Weiland,Bemporad,2006) (DiCairano,Lazar,Bemporad,Heemels,2008)


HybridMPC–Roomtemperatureexample

>>[XX,UU,DD,ZZ,TT]=sim(C,S,r,x0,Tstop);

>>C=hybcon(S,Q,N,limits,refs);

>>refs.x=2; % just weight state #2>>Q.x=1;>>Q.rho=Inf; % hard constraints>>Q.norm=2; % quadratic costs>>N=2; % optimization horizon>>limits.xmin=[25;-Inf];

>> C

Hybrid controller based on MLD model S <heatcoolmodel.hys>

2 state measurement(s) 0 output reference(s) 0 input reference(s) 1 state reference(s) 0 reference(s) on auxiliary continuous z-variables 20 optimization variable(s) (8 continuous, 12 binary) 46 mixed-integer linear inequalitiessampling time = 0.5, MILP solver = 'glpk' Type "struct(C)" for more details.>>

28


HybridMPC–Roomtemperatureexample

29


Maindrawbacksofon‐lineimplementation

30

•Computationtimemaybetoolong:okforlargesamplingtimes(>0.1s)butnotforfast‐samplingapplications(<1ms).Worst‐caseCPUtimehardtoestimate

•Requiresrelativelyexpensivehardware(notsuitableoninexpensive8‐bitµ‐controllerswithfewkBRAM)

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

AnywaytouseMPCwithouton‐linesolvers?

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

but...

minU

1

2U ′HU + x′(t)F ′U +

1

2x′(t)Y x(t)

subj. to GU ≤ W + Sx(t)


Explicitmodelpredictivecontrol

31

Idea:solvetheQPforallx(t)withinagivenrangeofRn off‐line

ThelinearMPCcontrollerisacontinuouspiecewiseaffinefunctionofthestatevector

multi‐parametricprogrammingproblem

(Bemporadetal.,2002)


•ThehybridMPCcontrollerispiecewiseaffineinx,r(controllawmaybediscontinuous)

Note:Withquadraticcosts,partitionmaynotbefullypolyhedral.Thenbetterkeepoverlappingpolyhedra

(Borrelli,Baotic,Bemporad,Morari,Automatica,2005)(Mayne,ECC2001) (Alessio,Bemporad,ADHS2006)

Explicithybridmodelpredictivecontrol


ExplicitHybridMPC–Roomtemperature

>>E=expcon(C,range,options);

>> E

Explicit controller (based on hybrid controller C) 3 parameter(s) 1 input(s) 12 partition(s)sampling time = 0.5 The controller is for hybrid systems (tracking)This is a state-feedback controller. Type "struct(E)" for more details.>>

Sectioninthe(T1,T2)‐spaceforTref=30

33


ExplicitHybridMPC‐Example

generatedC‐code

utils/expcon.h

34

ExplicitHybridMPC–Roomtemperature


Talkoutline


✓ Modelpredictivecontrol(MPC)ofhybridsystems



35


AutomotiveapplicationsofMPC

36

PhDstudents:Bernardini,Borrelli,DiCairano,Giorgetti,Ripaccioli,Trimboli(2001‐2009)&Hrovat,Kolmanovsky,Tseng(Ford)

tractioncontrol

Homogeneous Stratified

enginecontrol

semiactivesuspensions

tiredeflection

suspensiondeflection

idlespeedcontrol


AutomotiveapplicationsofMPC

37

magneticactuators

activesteeringair‐to‐fuelratio

hybridelectricvehicles

robotizedgearbox

(Borodani,Mannelli,CRF)

(PhDstudents:Bernardini,Borrelli,DiCairano,Giorgetti,Ripaccioli,Trimboli‐2001‐2009)&Hrovat,Kolmanovsky,Tseng(Ford)


EnergymanagementofHEVs

38

•risingfuelprices•tighteningemissionregulations•performanceimprovement

WhyHybridElectricVehicles(HEV)?

AdvancedPowerSystems•morecomponents•morecontrolledvariables•moreconstraints•severaloperatingmodes

UseMPCassystematicmodelingandmodel‐basedcontrolapproach


HybridMPCforEnergymanagementofHEVs

39

High‐fidelityindustrialmodelofanadvanced4x4hybridelectriccarwith:

turbochargeddieselengine

highvoltageNiMHbattery

twoelectricmotorsacting(oneperaxle)

Objectives:•manageIC,ERAD,andCISGpowerrequests•minimizefuelconsumption•keepbatterypreferablywithin30‐70%offullcharge•fulfillvariousmechanical&electricalconstraints

dx(t)

dt= Ax(t) + Bu(t)



40

HEVmodelforMPCcontrol:•3realinputs(torques)

•56aux.realvariables

•490mixedintegerineq.≈lineardynamics

MLDhybridmodel:

≈piecewiseaffinemaps

•1realoutput(fuel)

•9realstates(dynamics)

•7binarystates(gears)

•32binaryinputs(maps)

(Bemporadetal.,IEEETAC,2005)

minξ

J(ξ, x(t)) ! Qρρ2 +

N∑

k=1

(Γxxk − xref)TS(Γxxk − xref)+

+

N−1∑

k=0

(Γuuk − uref)TR(Γuuk − uref) + (yk − yref)

TQ(yk − yref)

subj. to

x0 = x(t)xk+1 = Axk + B1uk + B3zkyk = Cxk + D1uk + D3zkE3zk ≤ E1uk + E4xk + E50.3− ρ ≤ SoCk ≤ 0.7 + ρ

Q = qfuel, R =[

rτ,IC 0 00 rτ,CISG 00 0 rτ,ERAD

]S =

[sv,veh 0 0

0 sSoC 00 0 sv,int

], Qρ = 105

yref ! frate,ref

uref ! [τIC,req τERAD,req τCISG,req]′

xref ! [vveh,ref SoCref 0]′



41

• HybridoptimalcontrolproblemforMPC:

• Setpoints:

• Weights:

hybridMLDmodel

constraintonSoC


HybridMPCforEnergymanagementofHEVs• Simulations(MPC#1)

42

qfuel sSoC sv,int fuel cons (norm) max |vveh − vveh,ref | max |SoC − SoCref |0 ∗ ∗ ∗ 1 ∗ ∗1 1e-2 2e6 10 0.79 2.105 0.13642 1e1 1e6 1 0.76 2.789 0.2484

0 200 400 600 800 1000 1200−5

0

5

10

15

20

25

30

35Vehicle Speed

• Performanceanalysis

•CPUtime:average0.13spertimestep(worst:0.29s)onPC2GHz+CPLEX•Simulationsbasedonhigh‐fidelitynonlinearmodel+hybridMPC

conventionalvehicleMPC#1MPC#2

Note:drivingcyclenotknowninadvance!

vehiclespeed stateofcharge torques

(Ripaccioli,Bemporad,Assadian,Dextreit,DiCairano,Kolmanovsky,HSCC2009)


Talkoutline


✓ Modelpredictivecontrol(MPC)ofhybridsystems

✓ Automotiveapplications


43


MPCinWirelessAutomation

44

MPC

MPC

MPC

Real‐timeoptimization

Large‐scalesystem

PID

PID

PID

•Characterizedbyspatialdistribution

•Needfordecentralizedcontrolcomputations

•Needforflexibleinformationgatheringsystem

•Wirelesscommunicationsintroducenewproblemsincontroldesigntotakecareof(packetdrops,delays&jitter,batteryenergyconsumption,etc.)

x1 x2 x3 x4

x5 x6 x7 x8

u4

u3

u1 u2

4

2

3

1

DecentralizedMPC

Wirelesssensornetworks



45


Europeanproject“WIDE”

46

UNI

• ICT-FP7-Call 2 (2008-2011), total budget 2.7M€. Started Sept. 1, 2008

• Objective ICT-2007.3.7 “Networked Embedded and Control Systems”

DEcentralizedandWIrelessControlofLarge‐scaleSystems

http://ist-wide.dii.unisi.it

3rdWIDEPhDSchoolonNetworkedControlSystems,Siena,July7‐9,2009



47

Objective:trackpositionandtemperaturereferenceswhileenforcingsafetyconstraints

position

steady‐statetemperature

HybridMPCproblem:•2binaryinputs(lamps)•1continuousinput(speed)•PWLstatefunction(heating)•Outputs:temp,position•Sampling=4Hz

•Implementation:(1)on‐lineoptimization(QP,mixed‐integer),or(2)off‐linemulti‐parametricoptimizationandPWLcontrol

•Matlab/Simulinktoolsareavailabletoassistthewholedesignprocess

•Hybridsystemsasaframeworkfornewapplications,wherebothlogicandcontinuousdynamicsarerelevant

•Modelpredictivecontrol(MPC)isarathergeneralandsystematiccontroldesignmethodologyformulti‐variablesystemswithconstraints


Conclusions

http://www.dii.unisi.it/hybrid/toolbox48

modelling and optimization‐based control of hybrid

Documents