systems Indefinite MPC and linear approximated economic MPC fornonlinear Journal of Process Control

ContentslistsavailableatScienceDirect

Journal

of

Process

Control

jo u r n al h om ep ag e :w w w . e l s e v i e r . c o m / l o c a t e / j p r o c o n t

Indefinite

linear

MPC

and

approximated

economic

MPC

for

nonlinear

systems

夽

Mario

Zanon

a,∗

_,

_Sébastien

_Gros

b

_,

_Moritz

_Diehl

a,c

a_{DepartmentESAT-STADIUS/OPTECKULeuvenUniversity,KasteelparkArenberg10,3001Leuven,Belgium}

b_{DepartmentofSignalsandSystems,ChalmersUniversityofTechnology,Horsalsvagen11,SE-41296Goteborg,Sweden} c_{DepartmentofMicrosystemsEngineeringIMTEK,UniversityofFreiburg,Georges-Koehler-Allee102,79110Freiburg,Germany}

a

r

t

i

c

l

e

i

n

f

o

Articlehistory:

Received16January2014

Receivedinrevisedform30April2014 Accepted30April2014

Availableonline28May2014 Keywords:

EconomicMPC LQR

StabilitytheoryforMPC

a

b

s

t

r

a

c

t

Thestabilityproofforeconomicmodelpredictivecontrol(MPC)reliesonstrictdissipativity,whichisin generalhardtocheck.Inthiscontribution,wewillfirstanalyzethelinearquadraticcasewithpossibly indefinitecost.Theconditionsforstabilitywillberecalledanditwillbeshownthateverystabilizing LQR/MPChasanequivalentpositivedefiniteLQR/MPCwhichyieldsthesameopenloopandclosedloop behavior.ThisanalysiswillthenbeusedtoformulateanapproximatednonlineareconomicMPCscheme, whichhasstabilityguarantees.Anexamplewillbeusedtoillustratetheproposedtechniqueandshow itspotentialintermsofperformance.

©2014ElsevierLtd.Allrightsreserved.

1. Introduction

Economicmodelpredictivecontrol(MPC)hasrecentlygained popularity,asitdirectlyoptimizesagivenperformanceindex,as opposedtotrackingMPC,whichminimizesthedeviationfroma givenreference.ThemainadvantageofeconomicMPCover track-ingMPCbecomesobviousintransients,whenthesystemissteered tosteadystatesoastominimizethegivenperformanceindex.

ThestabilitytheoryofeconomicMPCinitiallyconsidered lin-earsystemsand convexobjectives[1,2].For nonlinearsystems, ananalysisofaverageperformanceboundswasproposedin[3]

andaverageconstraintswereconsideredin[4].Lyapunovstability ofeconomicMPCwasfirstprovenin[5]underastrongduality assumptionand generalized in [6,7] under a strictdissipativity assumption.Thenecessityofstrictdissipativityforoptimal steady-stateoperationhasbeenanalyzedin[11,12].Astabilityproofin theabsenceofterminalconstraintsisprovidedin[8].The exten-sionofthestabilityresultstoperiodicsystemshasbeenconsidered

夽 ThisresearchwassupportedbyResearchCouncilKUL:PFV/10/002Optimization inEngineeringCenterOPTEC,GOA/10/09MaNetandGOA/10/11Globalreal-time optimalcontrolofautonomousrobotsandmechatronicsystems.Flemish Govern-ment:FWO:PhD/postdocgrants;IWT:PhDGrants,projects:EurostarsSMART; BelgianFederalSciencePolicyOffice:IUAPP7(DYSCO,Dynamicalsystems,control andoptimization,2012–2017);EU:FP7-SADCO(MCITN-264735),FP7-TEMPO(MC ITN-607957),ERCHIGHWIND(259166).

∗ Correspondingauthor.Tel.:+4915756296036.

E-mailaddress:mario.zanon@imtek.uni-freiburg.de(M.Zanon).

in[9,10].EconomicMPCschemesbasedonLyapunovtechniques havebeenproposedin[13–15].

Inthenonlinearcase,thestrictdissipativityconditioncanbe extremelyhardtocheck,thusmakingitdifficulttoensurestability. Thiscontributionfocusesonthelinearquadraticcasewithpossibly indefinitestagecosts.Theanalysisbecomessimplerinthiscaseand theconditionsthatensurestabilityfortheLQRcontrollerhavebeen studiedin[16,17].

Themaincontributionsofthispaperare(a)theanalysisof lin-earquadraticeconomicMPCand(b)theconstructionofatracking nonlinearMPC(NMPC)schemewhichcloselymimicstheeconomic NMPC behavior while providing stability guarantees. The main resultsforthelinearquadraticcasearegiveninTheorems1and 2,whichstatethatanypossiblyindefinitestabilizingLQRorlinear quadraticMPCcanbereformulatedasapositivedefiniteonethat deliversthesametrajectory.Thisresultisthenusedforeconomic NMPC,toconstructanapproximatedeconomicNMPCschemeof trackingtypewhichprovidesstabilityguarantees.

Thepaperisstructuredasfollows.Section2proposesa motiva-tionalexamplewhichshowshowtrackingNMPCcanyieldnearly thesameperformanceaseconomicNMPC.InSection3,the con-ditionsforhavingastabilizingLQRarerecalled.Section4shows that,givenanyindefinitestabilizingLQR,thereexistsapositive definite LQRwhichyieldsthesame closedloop trajectory. Sec-tion5istheanalogueofSection4forthecaseofafinitehorizon (MPC).InSection6alinearexampleisproposedtoillustratethe theory.AnonlinearexamplewellknownintheeconomicNMPC literatureisproposedinSection7toillustratehowtheproposed http://dx.doi.org/10.1016/j.jprocont.2014.04.023

0 5 10 15 20 25 30 35 40 45 50 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 c A t 0 5 10 15 20 25 30 35 40 45 50 1 2 3 4 5 6 7 q t

Fig.1. StateandinputprofilesfortheeconomicNMPCscheme(blackline),trackingscheme0(blueline)andtrackingscheme1(redline).(Forinterpretationofthereferences tocolorinthisfigurelegend,thereaderisreferredtothewebversionofthisarticle.)

frameworkcanbeusedforapproximatedeconomicNMPCwith stabilityguaranteesinthenonlinearcase.

2. Motivationalexample

Asamotivationtothisarticle,wewouldliketostartwithan examplethatshowshowatrackingNMPCformulationcanyielda goodperformancecomparedtoaneconomicNMPCformulation. We propose the following CSTRexample from [5,7]. It consid-ersasinglefirst-order,irreversiblechemicalreactionA→Bwith reactionratekrcA,wherekr=0.4l/(molmin)istherateconstant. QuantitiescAandcBdenotethemolarconcentrationsofAandB respectively.Theprocessdynamicsaregivenby

˙ cA= q

VR(cAf −cA)−krcA (1)

˙

cB=_VRq(cBf−cB)+krcA (2)

wherecAf=1mol/l,cBf=0mol/ldenotethefeedconcentrationsof Aand Brespectively and VR=10lis the volumeof thereactor. Here,theflowthroughthereactorqisacontrolvariable.Inorder toensurelocalcontrollabilityatsteadystate,letusassumethat cA+cB=1.Thestatesarethusdefinedasx=cAandthecontrolsas u=q.Moreover,theflowrateisupperboundedbyq_≤20l/min.

Theeconomicstagecostisgivenby

(x,u)=−(2q(1−cA)−0.5q)+0.1(q−4)2, (3) whichgivesthesteadystatexs_=cs

A=0.5mol/landqs=4l/min. Consideranalternativetrackingformulationwherethestage costispositivedefiniteanddefinedas

 tr(x,u)=



cA−cs A q−qs



T H



cA−cs A q−qs



, ∈{0,1} (4) H0₌



14.227 0.825 0.825 0.066



, H1₌



1 0 0 1



(5)

Asimulationhasbeenruninordertocomparetheeconomic performanceofthetrackingNMPCschemewith=0andthe eco-nomicNMPC scheme.Forcompleteness,alsotheperformancea morestandardtrackingschemewith=1hasbeencompared.All simulationsareruninPythonusingCasadi[18]todiscretizethe optimalcontrolproblem(OCP)andIpopt[19]tosolvethe nonlin-earprogrammingproblem(NLP).Thecontinuoustimedynamics havebeendiscretizedwithasamplingtime Ts=0.5minusinga fixedstepsizeRunge-Kuttaintegratoroforder4.Thechosen pre-dictionhorizonisN=100andtheendpointconstraintxN=xshas beenintroduced.

The proposed scenario lasts 50s and considers a system at steadystatethatisperturbedevery12.5swiththefollowing per-turbationd=[0.4,−0.4,0.2,−0.2]whichaltersthestateasfollows: cA(k12.5)=cA(k12.5)+d(k),k=1,2,3,4.ThemetricGhasbeen pro-posedin[20]tomeasurethegainobtainedbyusingeconomicMPC overtracking:

G=100Peco−P  tr

Ps , =0,1, (6)

wherePeco,Ptr,Psdenotetheaverageprofitobtainedbyeconomic NMPC,trackingNMPCandsteadystateoperationrespectively.

ThesimulationresultsyieldG0_=0.0009andG1_=1.139,which suggestthatscheme0hasanegligibleperformancelosscompared totheeconomicNMPCscheme,whilescheme1hasasmall per-formancelosswhichishoweversensiblyhigherthantheoneof scheme0.ThestateandinputtrajectoriesaredisplayedinFig.1, whereitcanbeseenthatscheme0deliversaninputtrajectory whichisveryclosetotheonedeliveredbyeconomicNMPC,while thestatetrajectoryisevencloser.Scheme1,ontheotherhand, deliverssignificantlydifferenttrajectories.

Proving stability for economic NMPC schemes can be very involved.Moreover,trackingNMPCproblemstypicallyhavealeast squaresstructurewhichcanbeexploitedinefficientalgorithms.For economicNMPC,onthecontrary,suchstructureisnotpresentand thecomplexityoftheproblemtendstobesignificantlyhigher.This motivatesourinterestintrackingNMPCformulationsthatmimic economicNMPCbehavior.

3. ExistenceofastabilizingLQR

Inthissectionwewillfocusonlinearquadraticdiscretetime infinite horizon OCPs. The linear quadratic regulator (LQR) is definedbythefollowingOCP

V(x0)=inf x,u ∞



k=0 l(xk,uk) (7a) s.t. x0=x0, (7b) xk+1=Axk+Buk, k≥0, (7c) lim k→∞xk=0, (7d)

withstatex=[x0,x1,...]andcontrolu=[u0,u1,...].Thestagecost isdefinedasl(xk,uk)=



xk uk



T H



xk uk



,withmatrixH=



Q ST S R



symmetricandpossiblyindefinite.Thisoptimizationproblemhas beenwidelystudiedintheliteratureandexcellentanalysescanbe foundin[17,21]fordiscretetimesystemsandin[16]forcontinuous timesystems.

Thecontributionsabovedefinea setofnecessaryconditions fortheboundednessofproblem(3)andestablishthelinktothe existenceof solutionstothealgebraic Riccatiequation.In [16], moreover,theconnectionismadebetweenthealgebraicRiccati equation,alinearmatrixinequality,afrequencyconditionanda dissipativityconditionandthepropertiesofasetofoptimalcontrol problems.Mostoftheresultsforcontinuoustimecanbedirectly translatedtodiscrete time.Inthissection,wewillbrieflyrecall theconditionsfortheexistenceofastabilizingLQRwithapossibly indefinitestagecost.

ItiswellknownthatthesolutiontotheLQRproblem(3)can beobtainedbycomputingthestabilizingsolutionofthediscrete algebraicRiccatiequation(DARE),definedas

D(A,B,Q,R,S):={(P,K)| Q+AT_PA−P−(ST_+AT_PB)K=0,

K=(R+BT_PB)−1_(S+BT_PA), (A−BK)<1},

(8)

wheredenotesthespectralradius,i.e.(A)=max{||| iseigen-valueofA}.Ingeneral,theDAREhasmanysolutionsbutonlyone, ifit exists,isstabilizingthesystem(A,B),yieldsthesolutionto theLQRproblemandthecost-to-goisthengivenbyV(x0)=xT

0Px0. Notethat,inthecaseofanindefinitecost,i.e.H / 0,alsothe cost-to-goPcanbeindefinite.

Onenecessarybutnotsufficientconditionfortheexistenceof asolutionto(8)istheinvertibilityofmatrix(R+BT_PB).Inthe fol-lowing,wewillrecalltheconditionsforwhichtheLQRproblem(3) hasastabilizingsolution,whichalsoguaranteethat(R+BT_PB)0 isinvertibleandtheDARE(8)doeshaveanecessarilyunique sta-bilizingsolution.

Ifstabilizabilityisassumed andanyofthefollowing equiva-lentconditionsissatisfied,thentheLQRhasauniquestabilizing solution.

• Thelinearmatrixinequality(LMI):

∃

Ps.t.



Q+AT_{PA−P S}T_+AT_PB S+BT_{PA R+B}T_PB



0; (9)

• Thequadraticmatrixinequality(QMI):

∃

Ps.t.

Q+AT_PA−P−(ST_+AT_PB)(R+BT_PB)−1_(S+BT_{PA)0, (10a)}

R+BT_{PB0; (10b)}

• Thefrequencydomaininequality(FDI):

∀

zs.t.|z|=1 (z):=R+S(Iz−A)−1B+BT_(Iz−1_−AT₎−1_ST

+BT_(Iz−1_−AT₎−1_Q(Iz−A)−1_{B0; (11)} • Thedissipationinequality(DIE):

∃

V:Rnx_{→R s.t.}

∀

_xk,uk

l(xk,uk)+V(Axk+Buk)≥V(xk), (12) withnxthestatedimension.

TheconnectionbetweentheFDIanddetectabilityof(A,Q1/2₎ hasbeenestablishedin[17].

Inordertoshowthattheseconditionsareindeednecessaryfor theexistenceofastabilizingLQR,letusadapttwotheoremsfrom

[16]tothecaseofdiscrete-timesystems.Thosetheoremsanalyze theboundednessofsimilarOCPsandrelateittotheconceptof dissipativity.ThisallowsonetoestablishboundednessoftheLQR OCP,thusprovingtheexistenceofastabilizingsolutiontotheDARE

(8).LetusdefinethefeasiblesetZnonthehorizonnas Zn(x0)={(x0,...,xn,u0,...,un−1)|x0=x0,

xk+1=Axk+Buk, k=0,...,n}. (13) Letusintroducethefollowingoptimalcontrolproblems(OCP) V_f+(x0)= inf (x,u)∈Z∞(x0) ∞



k=0 l(xk,uk), (14) V+_{(x0)= inf} (x,u)∈Z∞(x0) ∞



k=0 l(xk,uk), s.t. lim k→∞xk=0, (15) V−_{(x0)=− inf} (x,u)∈Z∞(x0) 0



k=−∞ l(xk,uk), s.t. lim k→−∞xk=0, (16) V+ N(x0)= inf (x,u)∈ZN(x0),N≥0 N



k=0 l(xk,uk). (17)

OptimalityimpliesthatV+

N ≤0andVN+≤Vf+≤V+.If stabilizabil-ity isassumed, thenV+

N, Vf+, V+<∞and ifcontrollabilityis assumedV−>−∞.

Theorems1and2from[16]establishtheexistenceofa stabi-lizingLQR,byprovingtheboundednessofV+_{.Itisinterestingto} notehowboundednessofV+_{canbeestablishedbylookingatthe} boundednessofproblemsthatcanbequitedifferentfromit.Both theoremscanbedirectlytranslatedtodiscretetimeasfollows.

Proposition1. Assumethatthesystemxk+1=Axk+Bukis control-lable,thenthefollowingconditionsareequivalent:

1



N_k=0l(xk,uk)≥0foreveryN≥0andevery(x,u)∈ZN(0); 2V−(x)≤0;

3V_f+(x)≤0; 4V+

N(x)≤0;

5ThereexistsafunctionV(x)≤0whichsatisfiestheDIE(12). Moreover,wheneveranyoftheseconditionsissatisfied,then−∞≤ V−_(x)≤V+

N(x)≤Vf+(x)≤V+(x)≤∞.Finally,anyfunctionV(x) satis-fyingtheDIE(12)satisfiesV−_{(x)≤V(x)≤V}+_(x)andV−_{(x)≤V(x)≤} V+

N(x)ifV(x)≤0.

Proposition2. Assumethatthesystemxk+1=Axk+Bukis control-lable,thenthefollowingconditionsareequivalent:

1



N_k=0l(xk,uk)_≥0 for every N_≥0 and every (_x,u)∈ZN(0) with xN=0;

2V+_{(x)≥−∞;} 3V−(x)≤∞;

4ThereexistsafunctionV(x)whichsatisfiestheDIE(12).

Moreover,wheneveranyoftheseconditionsissatisfied,then−∞≤ V−_{(x)≤V(x)≤V}+_(x)≤∞.

Proof. Theprooffollowsthesameargumentsasin[16]. 

Remark1. ThefirstconditionofPropositions1and2istrivially satisfiedwhenH0.Boundednessoftheinfimaishencealways guaranteedforpositivesemidefinitecostfunctions,provided con-trollabilityisassumed.

Remark 2. Note that Condition 1 in both Propositions 1 and 2impliesthat thesystemisoptimallyoperated atsteadystate, thoughourdefinitionofoptimaloperationatsteadystatediffers slightlyfromtheonegivenin[22,11].

TheimportanceoftheDIE(12)fortheexistenceofastabilizing LQRisestablishedinProposition2.ForquadraticfunctionsV(x)= xT_{Px,theDIE(12)isequivalenttotheLMI(9).Thisiseasilyseenby} replacingthedynamicequationsxk+1=Axk+BukintheDIE,which directlygivestheLMI.TheconditionV+_{>−∞isthensatisfiedifand} onlyifthereexistsarealsymmetricsolutionP=PT_totheLMI(see

[16],Theorem3).TheQMIistheSchurcomplementoftheLMI,thus itisequivalenttotheLMIifR+BT_{PB0holds.Notealsothatthe} FDIimpliestheLMIanditallowstoestablishtheboundednessof theinfimaapriori,giventhesystemmatricesandthestagecost only.

Providedthatthepair(A,B)isstabilizable,theexistenceofa necessarilyuniquesymmetricsolutionP=PT_{toD(A,B,Q,R,S)such} that|(A−BK)|<0isguaranteedif,forall|z|=1,theFDIissatisfied, i.e.(z)>0,see[17,21].

Remark3. Notethat,inordertochecktheLMIcondition,the matrixPthatsatisfiesitdoesnotneedtobeasolutiontotheDARE. ThisconditioncanhencebecheckedbysolvingtheLMIforP:if asolutionexists,thentheconditioncanbesatisfied,thesystem isdissipativeandastabilizingLQRexists.Onthecontrary,ifthe solutiondoesnotexist,theconditioncannotbesatisfiedandthe systemisnotdissipative,thusthesystemcannotbestabilizedusing theproposedcostfunction.Notethatthischeckcanbeformulated asaconvexproblemthatisefficientlysolvable.

Remark4. IftheLMIconditionisnotsatisfied,onemightbe inter-estedindefininganewcostforwhichtheconditionholds.Inorder tomodifytheinitialcostaslittleaspossible,onecansolvethe followingconvexproblem

min P,TT 2 ₍₁₈₎ s.t.



Q+AT_{PA−P S}T_+AT_PB S+BT_{PA R+B}T_PB



+T0, (19) whereT=



TQ TT S TS TR



and·canbeanymatrixnorm.Thenew costisthendefinedby ˜Q=Q+TQ, ˜R=R+TR, ˜S=S+TS.

Inthefollowing section,wewillassume thatthe aforemen-tionedconditionsaresatisfied,thusthatastabilizingLQRexists.

4. EquivalentLQRformulations

InSection3,theconditionsfortheexistenceofastabilizingLQR havebeengiven.In general,thestagecostdoesnot needtobe positivedefinite,i.e.H

Ɑ

0andthesolutionPtotheDAREcanalso beindefinite.InthissectionwewillshowthatanystabilizingLQR canbereformulatedasapositivedefiniteone,i.e.P0andH0, whichyieldsthesametrajectory.

Let us introduce the following notation. The DARE

D(A,B,Q,R,S) delivers a solution (P, K), where P=PT _{is a real} symmetric matrix and K is a feedback gain. Let us define the LQR trajectory as (x0):=P(A,B,Q,R,S), i.e. the LQR prob-lem P(A,B,Q,R,S) has associated DARE (P,K)=D(A,B,Q,R,S) and delivers the closed loop trajectory (x0)=(xk)k≥0 with xk+1=(A−BK)xk. In the following, we will consider two problems equivalent if they deliver the same trajectory, i.e.

P(A1,B1,Q1,R1,S1)=P(A2,B2,Q2,R2,S2).

Definition 1. Problem P(A2,B2,Q2,R2,S2) is a positive def-inite reformulation of the possibly indefinite LQR problem

P(A1,B1,Q1,R1,S1) if



Q2 ST 2 S2 R2



0 and P(A2,B2,Q2,R2,S2)= P(A1,B1,Q1,R1,S1).

Theorem1. AnystabilizingLQRproblemcanbereformulatedasa positivedefiniteLQRproblem.

Beforeprovingthetheorem,twolemmasareintroducedthatwill helpinestablishingthedesiredresult.Thefollowinglemma con-sidersthecaseofasystem(A,B)whichispre-stabilizedusinga linearfeedbackK.Itprovesthat,bysuitablymodifyingthecost,an LQRcanbedefinedforthepre-stabilizedsystem(A_K,B),whichis equivalenttotheoriginalLQRdefinedfor(A,B).

Lemma1. LetusassumethatastabilizingLQRexistsforsystem(A, B),withweightingmatricesQ,R,S.TheequivalenceP(A,B,Q,R,S)=

P(A_K,B,Q_K,R,S_K)holdsforanyarbitraryfeedbackmatrixK,when choosing A_K=A−BK, Q_K=Q−ST_K−KT_S+KT_{RK, S}

K=S−RK. Moreover, the real symmetric matrix P_K, computed as (P_K,K)=

D(A_K,B,Q_K,R,S_K),satisfiesP_K=Pwith(P,K)=D(A,B,Q,R,S).The feedbackgainK_KisgivenbyK_K=K−K.

Proof. TheprooffollowsbysubstitutionofA_K,Q_KandS_Kinto(8)

andsimplification[17].ThedetailsareprovidedinA.Notethatthe reformulationproposedinLemma1isequivalenttochoosingfor thecontrolinputuk=−Kxk+

v

_k,i.e.

xk+1=(A−BK)xk+Bvk=AKxk+Bvk. (20) 

Remark5. Notethat,fromuk=−Kxk+

v

kand

v

k=−KKxk,one obtains

−Kxk=uk=−Kxk+

v

_k=−Kxk−K_Kxk, whichalsoentailsK_K=K−K.

Corollary 1. By choosing K=K, the DARE reduces to QK+ AT

KPAK−P=0. Thisentails that, forany arbitrarily chosen matrix RK such that RK+BTPB0, the solution remains unchanged, i.e.

P(AK,B,QK,R,SK)=P(AK,B,QK,RK,SK).Oneobtainsthusa prob-lemwhosesolutionisKK=0and

v

_k=0.

Proof:TheDAREcorrespondingto_{P(AK,B,QK,RK,SK})isgivenby QK+ATKPAK−P−(SKT+AKPB)(RK+BTPB)−1(SK+BTPAK)=0.

Thelasttermisalwayszero,ascanbeseenbyreplacingthe expres-sionforK: SK+BT_{PAK= S−RK+B}T_PA−BT_PBK = S+BT_PA−(R+BT_PB)K = S+BT_PA−(R+BT_PB)(R+BT_PB)−1_(S+BT_PA) = 0_.

RegardlessofthechoiceofmatrixRK,theDAREsimplifiesandPis definedby

QK+AT

KPAK−P=0.

Remark6. TheconditionRK+BTPB0isrequiredfortheLQRto exist,seee.g.(3).Thisconditioncanalternativelybederivedfrom dynamicprogrammingarguments.

ThefollowinglemmastatesthattheLQRcostcanbemodified suchthatthetrajectory(x0)remainsunchangedwhilethe cost-to-gomatrixPcanbeanyarbitrarysymmetricmatrix.

Lemma2. LetusassumethatastabilizingLQRexistsforsystem(A,B), withweightingmatricesQ,R,S.GivenanyrealsymmetricmatrixP,the equivalenceP(A,B,Q,R,S)=P(A,B,Q_P,R_P,S_P)holdswhenchoosing Q_P=Q+AT_PA−P,S

P=S+ATPBandRP=R+BTPB.Thesolution P_PtoD(A,B,Q_P,R_P,S_P)isgivenbyP_P=P−Pandtheresulting feed-backgainisK_P=K.

Proof. BywritingtheDARED(A,B,Q_P,R_P,S_P),andreplacingthe expressionsforQ_P,R_P,S_P,oneimmediatelyobtainsD(A,B,Q,R,S), whichissolvedbymatrixPandfeedbackgainK.

BydefinitionP=P+P_Pandonecanwrite

0=Q+AT_PA−P−(ST_+AT_PB)(R+BT_PB)−1_(S+BT_PA) =Q+AT_PA−P+AT_P PA−PP−(ST+ATPB+ATPPB) ×(R+BT_PB+BT_P PB) −1 (S+BT_PA+BT_P PA) =Q_P+AT_P PA−PP−(STP+ATPPB)(RP+BTPPB) −1 (S_P+BT_P PA)  Notethatchoosing_P=P,oneobtains_P

P=0.

Corollary 2. By successively applying Lemma 1 using K=K andLemma2,oneobtainsthatP(AK,B,QK,RK,SK)=P(AK,B,Q_K,P, R_K,P,S_K,P), where Q_K,P=QK+AT

KPAK−P, SK,P=SK+ATKPB and R_K,P=RK+BT_{PB.Notethat,asRKcanbechosenarbitrarilylarge,also} R_K,Pcanbechosenarbitrarilylarge.

Usingtheselemmasandcorollaries,itiseasiertoproveTheorem 1.

Proof. (Theorem1)UsingtheequivalencegiveninCorollary2,it followsthatP(A,B,Q,R,S)=P(AK,B,Q_K,P,R_K,P,S_K,P)andK_K,P=0. As we are interested in possibly indefinite but stabilizing LQR formulations, by hypothesis, the feedback gain K is stabilizing, i.e.(AK)=(A−BK)<1,wheredenotesthespectralradius,i.e. (A)=max{|||iseigenvalueofA}.ByLyapunovstabilitytheory, for anysymmetric matrix QL0, there exists a matrix PL such thatAT

KPLAK−PL+QL=0.ChoosingP=P−PL,andrecallingthe resultofCorollary1,oneobtainsAT

KPAK−P=QL−QK.ThenQK,P= QK+AT

KPAK−P=QL0.Onelaststepisneededtomakematrix H_K,P=



Q_K,P ST K,P S_K,P R_K,P



positivedefinite.ByCorollary2,onecan arbi-trarilychoosematrix R_K,P arbitrarilylargewithoutchangingthe solution(x0), northecost-to-goP. Theresult H_K,P0is then

obtainedbyselectingR_K,PS_K,PQ−1

K,PSTK,P.Whilethisreformulated problemalreadyyieldsanLQRwithpositivedefinitecost,itapplies tothe system(AK, B).By applying Lemma1 using K=−K one obtainsanLQRwithpositivedefinitecost,whichappliestothe sys-tem(A,B),i.e.P(A,B, ˜Q, ˜R, ˜S)=P(A,B,Q,R,S),where ˜Q =Q_K,P+ ST

K,PK+KTSK,P+KTRK,PK, ˜S =SK,P+RK,PKand ˜R =RK,P.Thiscanbe shownbyapplyingtheSchurcomplementtocheckpositive defi-niteness.Matrix ˜R ispositivedefinite,soitremainstoprovethat

˜

Q  ˜ST_R˜−1_{˜S.Thisfollowsimmediatelyfromthedefinition:} ˜

Q=Q_K,P+ST

K,PK+KTSK,P+KTRK,PK,

˜ST_R˜−1_{˜S =S}T

K,PR−1K,PSK,P+SK,PT K+KTSK,P+KTRK,PK. Theresult ˜Q  ˜ST_R˜−1_{˜SdirectlyfollowsfromQ}

K,PSTK,PR−1K,PSK,P.

Remark7. Inordertocomputetheequivalentpositivedefinite LQRmatrices ˜Q, ˜R, ˜S onecansolvethefollowingSDP

min ˜ P, ˜Q,˜R,˜S, ˜H ˜P −I 2 + ˜H −I2 (21a) s.t. ˜H=



˜ Q ˜ST ˜S _R˜



(21b) ˜ H0 (21c) ˜ P0 (21d) ˜ Q +AT_PA˜ _{− ˜P−}(˜ST_+AT_PB)K˜ ₌0, (21e) ( ˜R +BT_PB)K˜ ₋(˜S+BT_PA)˜ _{=0. (21f)} Due toour choice of objective, theproblem is convexand, by

Theorem1itisalsofeasible,thusthesolutiontothisSDPalways existsandisunique.OneadvantageofsolvingtheproposedSDPto computematrices ˜Hand ˜Pisthatthenumericalpropertiesofthose matricesareoptimalinthesenseofthechosenmatrixnormand reference.

5. EquivalentMPCformulations

In this section, we will analyze the MPC problem. We will restrictourattentiontothecaseofnoactiveinequalityconstraint atsteadystate.Notethatthisdoesnotexcludethepossibilityof havingstateandinputconstraintsintheMPCformulation,aslong astheyarenotactiveatsteadystate.WhenconsideringanMPC problem,thetimehorizonbecomesfiniteandtheproblemtobe solvedis V(x0)=inf x,u N



k=0 l(xk,uk)+xTNPNxN (22a) s_{.t. x0=x0,} (22b) xk+1=Axk+Buk, k=0,...,N−1, (22c) with l(xk,uk)=



xk uk



T H



xk uk



, matrices H=



Q ST S R



and PN symmetricandpossiblyindefinite.Whennoinequalityconstraints are imposed, the solutionto this problem can be obtainedby

propagatingbackwardsthediscreteRiccatiequation(DRE)

RN(A,B,Q,R,S,PN):={(P0,P1,...,PN,K0,K1,...,KN−1) |Pk−1 =Q+ATPkA−(ST+ATPkB)Kk−1,

Kk−1 =(R+BT_PkB)−1_(S+BT_PkA),

k=N,...,1.}. (23a)

TheclosedloopMPCtrajectorycanthenbecomputedby simulat-ingforwardintimethesystem

xk+1=AK0xk=(A−BK0)xk,

whiletheMPCpredictedtrajectorycanbecomputedbysimulating forwardintime

xk+1=AK_kxk=(_A−BKk)_xk.

Remark8. Note that,iftheterminal costmatrix is chosenas PN=P,wherePisobtainedbysolvingtheLQR,thenthetheoryfrom Section4applies.ThissectionfocusesonthecasePN /= P.

Let us define the MPC predicted trajectory as (x0):_=PN(_A,B,Q,R,S,PN), i.e. the associated DRE is (P0,P1,...,PN,K0,K1,...,KN−1)=RN(A,B,Q,R,S,PN) and the closed loop trajectory is given by (x0)=(xk)0≤k≤N with xk+1=(A−BKk)xk.Inthefollowing,we willdefinetwo problems equivalent if they deliver the same predicted trajectory, i.e.

PN( ˜A, ˜B, ˜Q, ˜R, ˜S, ˜PN)=PN(A,B,Q,R,S,PN).

Definition 2. Problem PN( ˜A, ˜B, ˜Q, ˜R, ˜S, ˜PN) is a posi-tive definite reformulation of the possibly indefinite MPC problem PN(A,B,Q,R,S,PN) if



˜ Q ˜ST ˜S _R˜



0, _PN˜ 0 and PN( ˜A, ˜B, ˜Q, ˜R, ˜S, ˜PN)=PN(A,B,Q,R,S,PN).

Theorem1 considered thecase of an infinite horizon linear quadraticOCP(LQRproblem).Thefollowingtheoremconsidersthe caseofafinitehorizonOCPwithaterminalcostwhichisdifferent fromtheLQRcosttogo(MPCproblem).

Theorem2. EverystabilizingpossiblyindefiniteMPCschemeofthe form(5)withtimeinvariantmatricesA,B,Hcanbeformulatedasan equivalentpositivedefiniteMPCscheme.

Beforeprovingthetheorem,letusfirstintroducethefollowing lemma.

Lemma3. TwoDREsRN(A,B,Q,R,S,PN)andRN(A,B, ˜Q, ˜R, ˜S, ˜PN) yieldthesamefeedbacksequenceK0,...,KN−1if

˜

PN− ˜P=PN−P, ˜R+BT_PB˜ _=R+BT_{PB, ˜S +B}T_{PA =}˜ _S+BT_{PA, (24)} withPand ˜PcomputedbysolvingtheassociatedDAREsD(A,B,Q,R,S) andD(A,B, ˜Q, ˜R, ˜S).

Proof. Let us first consider the two equivalent LQR prob-lems P(A,B,Q,R,S)=P(A,B, ˜Q, ˜R, ˜S) with associated DAREs (P,K)=D(A,B,Q,R,S) and ( ˜P,K)=D(A,B, ˜Q, ˜R, ˜S). Defining k=Pk−P and ˜k= ˜Pk− ˜P, the two DREs can berewritten as

RN(A,B,Q,R,S,P+N)andRN(A,B, ˜Q, ˜R, ˜S, ˜P+ ˜N),i.e. P+k−1=Q+ATPA+ATkA−(ST+ATPB+ATkB)Kk−1, (25a) Kk−1=(_R+BT_PB+BT_kB)−1(_S+BT_PA+BT_kA), (25b) and ˜ P+ ˜k−1= ˜Q +ATPA +˜ AT˜kA−(˜ST+ATPB +˜ AT˜kB)Kk−1, (26a) Kk−1=( ˜R+BTPB˜ +BT˜kB)−1(˜S +BTPA +˜ BT˜kA). (26b)

Theidentityk= ˜kisobtainedifcondition(24)issatisfied.This canbeseenbyreplacingthevaluesofPand ˜PfromtheDAREinto Eqs.(Proof)and(Proof),whichgives

k−1=ATkA−(ST+ATPB)K−(ST+ATPB+ATkB)Kk−1, (27a) Kk−1=(R+BT_PB+BT_kB)−1_(S+BT_PA+BT_{kA), (27b)} and ˜ k−1=AT_kA˜ _−(˜ST_+AT_PB)K˜ ₋(˜_ST_+AT_{PB +}˜ _AT_kB)Kk−1,˜ _(28a) Kk−1=( ˜R +BT_{PB +}˜ _BT_kB)˜ −1_{(˜S +B}T_{PA +}˜ _BT_kA).˜ _(28b) Byreplacingcondition(24)into(Proof)oneimmediatelyobtains

(Proof). 

Remark9. Lemma3providessufficientconditionsforthe equiv-alenceoftwo DREsbut doesnot provideanyguarantee onthe existenceoftwoDREswhich satisfythoseconditions.It willbe clearintheproofofTheorem2thatthisisindeedpossible.

Proof. (Theorem2)Theprooffollowsfromcombiningtheresults ofTheorem1andLemma3.Inparticular,conditions(24)canbe easilyenforcedinTheorem1.FromCorollary1wehavethatST

K+ BT_PAK=ST

K,P+BTPPAK=0,foranychoiceofmatrixP.Asmatrix R_K,P canbearbitrarilychosen,onecanchooseR_K,P=R+BT_PB− BT_P

PB.

AsinTheorem1,wecanchooseP=P−PL,toobtainP_P=PL.In ordertoguaranteepositivedefiniteness,letuschooseQLarbitrarily butpositivedefiniteandscaleitbyafactor >0.TheLyapunov equationbecomes AT

KPLAK− PL+ QL=0and,subsequently, ST

K,P =− BTPLAK, R_K,P =R+BT_{PB− B}T_PLB.

TheconditionforensuringpositivedefinitenessR_K,PST

K,PQK,P−1SK,P thusbecomes

R+BT_{PB B}T_{PLB+ (B}T_AT

KPLB)QL−1(BTPLAKB), (29) whichholdsfor sufficientlysmall.Thetransformationbackfrom system(AK,B)tosystem(A,B)canthenbedoneasinTheorem1.

Remark10. InthesamewayasforTheorem1,alsoforTheorem 2,theMPCreformulationcanbecomputedbysolvingthefollowing SDP min ˜ P, ˜Q,˜R,˜S, ˜H˜P−I 2 + ˜H −I2 (30a) s.t. ˜H=



˜ Q ˜ST ˜S _R˜



(30b) ˜ H 0 (30c) ˜ P0 (30d) ˜ Q +AT_PA˜ _{− ˜P−}(˜ST_+AT_PB)K˜ ₌0, (30e) ( ˜R +BT_PB)K˜ ₋(˜S +BT_PA)˜ ₌0, (30f) ˜ R+BT_PB˜ _=R+BT_{PB, (30g)} ˜S+BT_PA˜ _=S+BT_{PA. (30h)}

Thenewterminalcostisthengivenby ˜PN=PN−P+ ˜P.

Remark11. ThereformulationobtainedinTheorem2yieldsa convexifiedstagecostwhichisindependentofthechosenterminal cost.Itcanbeverifiedthattheresultisalsovalidforthecaseofan endpointconstraintintheMPCscheme.

6. Numericalexample

Inordertoillustratetheresultspresentedintheprevious sec-tion,letusconsiderthelinearsystem

xk+1 =Axk+Buk, A =

⎡

⎣

−0.3319_−0.3393 00.7595.1250 10.5399.4245 −0.5090 0.9388 0.8864

⎤

⎦

_{, B=}

⎡

⎣

0_−1.3835.1060 −0.1496

⎤

⎦

_. Thestagecostischosenindefiniteanddefinedas

H=



Q ST S R



=

⎡

⎢

⎣

−1.0029 −0.0896 1.1050 −0.0420 −0.0896 1.6790 −0.5762 0.2112 1.1050 −0.5762 −0.4381 −0.2832 −0.0420 0.2112 −0.2832 0.6192

⎤

⎥

⎦

. (31)

ToobtainapositivedefiniteequivalentLQRonecaneitherapply theproceduredevelopedintheproofofTheorem1orsolveSDP(7), whichresultsinmatrices ˜Hand ˜Pwiththefollowingproperties

( ˜HTh1) =

⎡

⎢

⎢

⎣

615.9651 1.0000 0.9516 0.0017

⎤

⎥

⎥

⎦

, ( ˜HSDP) =

⎡

⎢

⎢

⎣

1.3444 0.6252 0.0010 0.0010

⎤

⎥

⎥

⎦

, (˜PTh1) =

⎡

⎣

3310.0529.4083 1.0004

⎤

⎦

, (˜PSDP) =

⎡

⎣

10.0140.2345 0.0383

⎤

⎦

.

Inparticular,itcanbeseenthatsolvingSDP(7)yieldsbetter con-ditionedmatrices,asaneffectofthechosencostfunction,which aimsatfindingmatrices ˜Hand ˜Pasclosetotheidentityas possi-ble.Inthisexample,theFrobeniusnormwasusedintheobjective function.

Tovisualizetheresults,asimulationhasbeenrunstartingfrom x0=[1,0,0].TheresultingtrajectoryandthevalueofthethreeLQR formulationsaredisplayedinFig.2.Itcanbeseenthatall formu-lationsstabilizethesystemandtheobtainedtrajectoriescoincide. However,onlythetworeformulatedLQRyieldapositivedefinite valuefunction,whichisalsoaLyapunovfunctionforthesystem.

7. ApproximatedeconomicNMPC

EconomicNMPCconsistsinoptimizingacostwhichisnot neces-sarilypositivedefinite,thusstandardstabilityproofsdonothold. AstabilityproofforeconomicNMPChasbeenproposedin[7].It reliesontheconceptofstrictdissipativity,whichisunfortunately extremelyhardtocheckinthegeneralcase.Thismotivatesasearch forNMPCschemesthatapproximateeconomicbehavior,butfor whichitisrathereasytoprovestability.

Localinformationaboutthesystemcanbeobtainedbylooking atthesecondorderexpansionatsteady-state.Theoptimalsteady stateiscomputedbysolvingthefollowingNLP

min

x,u (x,u) s.t. x=f(x,u), (32) wherexandudenotethestateandcontrolvectorsandfunctionf(x, u)istheconsidereddiscretetimesystem.UsingwT_=[xT_uT_],the LagrangianofProblem(32)isgivenbyL(w,)=(x,u)−T_(x− f(x,u)),whereisthevectorofLagrangemultipliersassociated withtheequalityconstraints.Thesecondorderexpansionatsteady stateisthenobtainedbyusingthelinearsystem(A,B)with

A =

∂

f(x,u)

∂

x





ws , B=

∂

f(x,u)

∂

u





ws ,

wherexsandusdenotethestateandcontrolvectorsatsteadystate. The cost function is given by 2(x,u)=1/2(w−ws)TH(w−ws), wherematrixHisgivenby

H=



_{Q S} ST _R



=

∂

2 L(w,)

∂

w2





ws .

NotethatmatrixHisingeneralindefinite,henceanNMPCscheme using2(·,·)asstagecostisingeneralnotoftrackingtypeand stabilityisguaranteedifthesystemisdissipative.Checkingifthe linearsystemisdissipativeisstraightforward:iftheLMI,QMIorFDI conditionissatisfied,thenthesystemisdissipative.Alternatively onecancheckifthecost,rotatedasin[5]isconvex.

Ifthenonlinearsystemislocallycontrollableatsteadystatebut notdissipative,steady-stateoperationisnotoptimal[12].If nev-erthelessoneisinterestedinstabilizingthesystematsteadystate, theobjectivecanberegularizedsoastomakethesystem dissi-pative,seee.g.[5,7].Addingregularizationcanmakethesystem dissipative,butthestagecostcanneverthelessstillbeindefinite.As thesystemisdissipative,theindefiniteLQRorlinearMPCscheme willbestabilizingforthelinearsystem.AccordingtoTheorem2, onecanconvexifytheindefinitestagecosttocomputeapositive definitestagecostandtheconvexifiedLQRandMPCproblemsare thenequivalenttotheoriginalones.Thisnewpositivedefinitestage costcanthenbeusedinanNMPCcontexttostabilizethenonlinear plant.Asthecostisquadraticpositivedefinite,theNMPCschemeis oftrackingtype.TheclassicalNMPCstabilityproofsarethen appli-cable[23].Moreover,thestagecostisasecondorderapproximation ofthenonlineareconomicstagecost.

Notethattheconvexificationprocedureproposedinthispaper reliesonasecond orderapproximationoftheeconomicNMPC, whichfailstoexistifthesystemisnotlocallystabilizableatsteady state,i.e.if(A,B)isnotstabilizable.Techniquestoovercomethis limitationarethesubjectofongoingresearch.

7.1. Numericalexample

Inordertoillustratetheseconcepts,letuslookattheCSTR exam-plefrom[5,7].AnalogouslyasinSection2,wewillassumethat cA+cB=1inordertohavelocalstabilizabilityatsteadystate.The realeconomicstagecostis

(x,u)=−(2q(1−cA)−0.5q), (33) andthequadraticterm (q−4)2_{with =0.1isaddedinorderto} makethesystemdissipative.WhileinSection2weconsidered eco-nomicNMPCwiththeregularizedstagecost,letusnowconsider theoriginaleconomicstagecost(33).

TheoptimalsteadystatecomputedbysolvingProblem(32)is cA=0.5mol/Landq=4L/min.Thesecondorderexpansionatsteady stategives A =0.6757, B=0.0203, H =



Q S ST _R



=



0 1.1832 1.1832 −0.0229



.

As anticipated, this system is not dissipative, hence, the LMIcondition (9) cannot besatisfied. In [5], theregularization 0.505(cA+cB+q) is added to the cost. In [7], the regularization (q−4)2_{isaddedtothecostwith =0.1.Intheframework} pro-posedinthispaper,theregularizationcanbechosenastheminimal one which is needed in order to satisfy the LMI(9) using the approachproposedinRemark4.Choosingtorestrictthe regular-izationtotheform (q−4)2_{asproposedin[7]andsolvingSDP(7),} oneobtainsthattheneededregularizationforthelinearsystemis =0.0275.Notethatthisvalueisonlyvalidforthelinearsystem anddoesnotguaranteestabilityoftheeconomicNMPCschemea

0 10 20 30 40 50 60 −1 0 1 x1 0 10 20 30 40 50 60 −0.5 0 0.5 x2 0 10 20 30 40 50 60 −1 0 1 x3 t 0 10 20 30 40 50 60 −2 −1 0 VIndef 0 10 20 30 40 50 60 0 0.2 0.4 VSDP 0 10 20 30 40 50 60 0 5 10 VTh3 t

Fig.2.TrajectoryandvaluefunctionforthethreeLQRformulations.

priori.ItisinsteadguaranteedthatthetrackingNMPCusingthe reg-ularizedandconvexifiedcostisstabilizingforthenonlinearplant. ByconvexifyingthecostintheframeworkofTheorem2usingthe minimalregularization =0.0275,oneobtains

H2 ₌



25.0142 0.5528 0.5528 0.0132



.

Inthefollowing,wewillcallscheme2theNMPCschemewhich usesatrackingstagecostdefinedbymatrixH2_{andregularized} eco-nomicNMPCtheeconomicNMPCschemewhichusesthestagecost proposedin[7],i.e.cost(33)regularizedwith0.1(q−4)2_.Using =0.1,onewouldobtainmatrixH0_{usedinSection2.Thisexplains} whyscheme0yieldsstateandinputprofilesthatcloselymimicthe onesobtainedwiththeregularizedeconomicNMPC(cf.Section2

andFig.1).

ThetrackingNMPCscheme2hasbeencomparedtothe reg-ularized economic NMPC formulation in the same scenario as in Section2. In this second example, though, gain G is com-putedusing theaverageprofit ofeach schemecomputedusing the actual stage cost (33). For the tracking scheme that uses

H2_,oneobtainsG2_{=−8.0599,whichindicatesthattheeconomic} NMPC formulation, regularized with the term 0.1(q−4)2 per-forms worse than the tracking NMPC formulation which uses H2_{.ForthetrackingimplementationwhichusesH}1 _instead,one obtainsG1_{=2.1907,whichindicatesthattheregularizedeconomic} NMPCformulationperformsbetterthanthissecondtrackingNMPC formulation.

ThestateandinputprofilesaredisplayedinFig.3,whereitcan beseenthatthetrackingscheme2outputstrajectoriesthatare sig-nificantlydifferentfromtheregularizedeconomicNMPCscheme. Inparticular,trackingscheme2hasamoreaggressivebehaviorand itactivatestheinputboundsinseveraloccasions.

Providedthateithertheterminalcostandconstraintare cho-senadequately[23] orthehorizon ischosen longenough[24], thetrackingNMPCschemehastheadvantageofimmediately pro-vidingstabilityguarantees.Moreover,theunderlyingOCPcanbe efficientlysolvedbynumericalalgorithmsforfastNMPC,e.g.the realtimeiteration(RTI)scheme[25]withtheGauss-Newton Hes-sianapproximation.ForeconomicNMPCinstead,theunderlying OCPisingeneralmoredifficulttosolve.

0 5 10 15 20 25 30 35 40 45 50 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 cA t 0 5 10 15 20 25 30 35 40 45 50 0 2 4 6 8 10 12 14 16 18 20 q t

Fig.3.StateandinputtrajectoriesobtainedwiththeeconomicNMPCscheme(blackline),withthetrackingscheme2(blueline)andwiththetrackingscheme1(redline). (Forinterpretationofthereferencestocolorinthisfigurelegend,thereaderisreferredtothewebversionofthisarticle.)

8. Conclusions

In this paper we have analyzed economic MPC for the lin-earquadraticcase.Boththecasesoffiniteandinfiniteprediction horizonshavebeenconsideredandithasbeenshownthatevery stabilizingLQRorMPCschemecanbereformulatedsuchthatthe costis positive definite, whilepreserving thesameclosed loop behavior.

Approximatedeconomic NMPCwithstability guaranteeshas beenproposed,basedonthepreviousresults.Theapplicationto a simple well knownexample has shown thepotential of this approachalsointermsofeconomicperformance.

Futureworkwillfocusonextendingtheproposedframework tomoregeneralsettings, includingperiodicsystemsand active constraintsatsteadystate.

AppendixA. ProofofLemma1

By replacing the definitions of A_K, Q_K and S_K in

D(A_K,B,Q_K,R,S_K),oneobtains 0=Q_K+AT KPKAK−PK−(STK+ATKPKB)(R+BTPKB) −1 (S_K+BT_P KAK) =Q−ST_K−KT_S+KT_RK+(A−BK)T_P K(A−BK)−PK −(ST_−KT_R+(A−BK)T_P KB)(R+BTPKB) −1 ×(S−RK+BT_P K(A−BK))

Thelasttermofthesumcanberewrittenas (ST_−KT_R+(A−BK)T_P KB)(R+BTPKB) −1 (S−RK+BT_P K(A−BK)) =−KT(R+BT_P KB)(R+BTPKB) −1







I (S−RK+BT_P K(A−BK)) +(_ST_+AT_P KB)(R+BTPKB) −1 (_S−RK+BT_P K(A−BK)) =−KTS+KTRK−KTBT_P KA+K T BT_P KBK +(ST_+AT_P KB)(R+BTPKB) −1 (S+BT_P KA) −(ST_+AT_P KB)(R+BTPKB) −1 (R+BT_P KB)







I K =−KTS+KTRK−KTBTP_KA+KTBTP_KBK −ST_K−AT_P KBK+(ST+ATPKB)(R+BTPKB) −1 (S+BT_P KA). ReplacingthisexpressioninD(A_K,B,Q_K,R,S_K)andsimplifyingone obtains Q+AT_P KA−PK−(ST+ATPKB)(R+BTPKB) −1 (S+BT_P KA)=0, whichdirectlyentailsP_K=P.

TheresultK_K=K−Kisobtainedasfollows K_K =(R+BT_PB)−1_(S K+BTPAK) =(R+BT_PB)−1_(S−RK+BT_PA−BT_PBK) =(R+BT_PB)−1_(S+BT_PA)−(R+BT_PB)−1_(R+BT_PB)K =K−K. References

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