Piecewise-affine state feedback for piecewise-affine slab systems using convex optimization

19 Pages·2005·0.3 MB·English

Systems&ControlLetters54(2005)835–853
www.elsevier.com/locate/sysconle
Piecewise-affine state feedback for piecewise-affine slab systems
using convex optimization
Luis Rodriguesa,∗, Stephen Boydb
aDepartmentofMechanicalandIndustrialEngineering,ConcordiaUniversity,2160BBishopSt.,B-304,Montréal,QC,CanadaH3G2E9
bDepartmentofElectricalEngineering,StanfordUniversity,PackardBuilding264,Stanford,CA94305,USA
Received31July2003;receivedinrevisedform14December2004;accepted23January2005
Abstract
ThispapershowsthatLyapunov-basedstatefeedbackcontrollersynthesisforpiecewise-affine(PWA)slabsystemscanbe
cast as an optimization problem subject to a set of linear matrix inequalities (LMIs) analytically parameterized by a vector.
Furthermore,itisshownthatcontinuityofthecontrolinputsattheswitchingscanbeguaranteedbyaddingequalityconstraints
totheproblemwithoutaffectingitsparameterizationstructure.Finally,itisshownthatpiecewise-affinestatefeedbackcontroller
synthesisforpiecewise-affineslabsystemstomaximizethedecayrateofaquadraticcontrolLyapunovfunctioncanbecastas
asetofquasi-concaveoptimizationproblemsanalyticallyparameterizedbyavector.Beforecastingthesynthesisintheformat
presentedinthispaper,Lyapunov-basedpiecewise-affinestatefeedbackcontrollersynthesiscouldonlybeformulatedasabi-
convexoptimizationprogram,whichisveryexpensivetosolveglobally.Thus,thefundamentalimportanceofthecontributions
ofthepaperreliesonthefactthat,forthefirsttime,thepiecewise-affinestatefeedbacksynthesisproblemhasbeenformulated
asaconvexproblemwithaparameterizedsetofLMIsthatcanberelaxedtoafinitesetofLMIsandsolvedefficientlytoapoint
neartheglobaloptimumusingavailablesoftware.Furthermore,itisshownforthefirsttimethat,insomesituations,theglobal
canbeexactlyfoundbysolvingonlyoneconcaveproblem.
©2005ElsevierB.V.Allrightsreserved.
Keywords:Piecewise-affinesystems;Statefeedback;Convexoptimization;Ellipsoids;Lyapunovfunction
1. Introduction
Piecewise-affinesystemsaremulti-modelsystemsthatofferagoodmodelingframeworkforcomplexdynamical
systemsinvolvingnonlinearphenomena.Infact,manynonlinearitiesthatappearfrequentlyinengineeringsystems
areeitherpiecewise-affine(e.g.,asaturatedlinearactuatorcharacteristic)orcanbeapproximatedaspiecewise-affine
functions. Piecewise-affine systems are also a class of hybrid systems, i.e, systems with a continuous-time state
∗
Correspondingauthor.Tel.:[phone]x3135;fax:[phone].
E-mailaddresses:[email protected](L.Rodrigues),[email protected](S.Boyd).
0167-6911/$-seefrontmatter©2005ElsevierB.V.Allrightsreserved.
doi:10.1016/j.sysconle.2005.01.002
836 L.Rodrigues,S.Boyd/Systems&ControlLetters54(2005)835–853
andadiscrete-eventstate.Forpiecewise-affinesystemsthediscrete-eventstateisassociatedwithdiscretemodes
ofoperation.Thecontinuous-timestateisassociatedwiththeaffine(linearwithoffset)dynamicsvalidwithineach
discretemode.Piecewise-affinesystemsposechallengingproblemsbecauseofitsswitchedstructure.Infact,the
analysis and control of even some simple piecewise-affine systems have been shown to be either an NP hard
problemorundecidable[4].
Stateandoutputfeedbackcontrolofcontinuous-timepiecewise-affinesystemshavereceivedincreasinginterest
overthelastyears[10,13,15,21].Theinterestingapproachpresentedin[13,15]reliesoncomputingupperandlower
bounds to the optimal cost of the controller obtained as the solution to the Hamilton–Jacobi–Bellman equation.
However, the continuous-time controller resulting from the approach in [15] is a patched LQR that cannot be
guaranteed to avoid sliding modes at the switching and, therefore, is not provably stabilizing. Previous work of
theauthorshasconcentratedonLyapunov-basedcontrollersynthesismethodsforcontinuous-timepiecewise-affine
(PWA)systems[10,21].In[21],Lyapunov-basedcontrollersynthesiswasformulatedasabi-convexoptimization
problem.Thebi-convexitystructurearisesbecauseofthenegativityconstraintonthederivativeofthepiecewise-
quadraticLyapunovfunctionovertime.Thisconstraintleadstoabilinearmatrixinequality(BMI)[8].Bi-convex
optimization problems are non-convex, NP hard and, therefore, extremely expensive to solve globally from a
computationalpointofview[8].Basedonthisfact,Ref.[21]hasadaptedthreealternativeiterativealgorithmsfor
solvingthenon-convexproblemtoasuboptimalsolution.Althoughthecontrollersynthesisproblemforpiecewise-
affine systems using piecewise-quadratic Lyapunov functions is non-convex, Hassibi and Boyd [10] have shown
thatfortheparticularcaseofpiecewise-linearstatefeedbackofslabpiecewise-linearsystems(withoutaffineterms),
globallyquadraticstabilizationcouldbecastasaconvexoptimizationproblem.Unfortunately,ifaffinetermsare
included in the controller, as stated in [10], “it does not seem that the condition for stabilizability can be cast
asanLMI”,whichapparentlydestroystheconvexstructureoftheproblem,makingithardtosolveglobally.The
currentpapershowsthatpiecewise-affinestatefeedbackforpiecewise-affineslabsystemsusingagloballyquadratic
LyapunovfunctioncanindeedbesolvedtoapointneartheglobaloptimuminanefficientwaybyasetofLMIs.
Building on the result of [10], this paper formulates piecewise-affine state feedback as an optimization problem
involving a set of LMIs analytically parameterized by a vector. Three different algorithms will be suggested to
solve relaxations of the optimization problem to a point near the global optimum. One is based on gridding of
the domain of the parameterizing vector and yields solutions that approach the global optimum as the density
of the grid is increased. The others are based on trace maximization to approximately solve an LMI subject to
rank constraints [17], a problem that appears frequently in reduced order controller design. Although yielding
solutions approaching the global optimum, the algorithm involving gridding increases the computational cost as
the grid becomes denser and can be prohibitive for large systems. However, the gridding approach has already
been used in other recent research on analysis [9], LPV control [23], gain-scheduling control [1,2] and some
techniques already exist to alleviate the computational cost due to the gridding phase [3,22].The algorithms for
tracemaximizationareinspiredbytheworkpresentedin[11,7,6].Oneofthesealgorithmsisiterativebuttypically
involvesonlyoneortwoiterations,thustypicallybeinglesscomputationallyexpensivethanthegriddingalgorithm.
Theotherproposedtracemaximizationalgorithmissimplyaconcaveprogram,whichisthereforeefficientfrom
a computational point of view. It is also shown in the paper that constraints for continuity of the control inputs
can be added to the PWA state feedback problem without affecting its parameterization structure. Finally, it is
shownthatpiecewise-affinestatefeedbackcontrollersynthesisforpiecewise-affineslabsystemstomaximizethe
decay rate of a globally quadratic control Lyapunov function can be cast as a set of quasi-concave optimization
problems analytically parameterized by a vector. This problem can also be solved numerically using efficient
algorithms.
In this paper, four controller synthesis problems are formulated, relaxed to a finite set of convex optimization
problems and solved. The paper starts by presenting the assumptions that are common to all controller design
problems,followedbythestatementsofthefourproblems.Section4formulatesthecontrollersynthesisproblems
asoptimizationprograms.Section5presentsseveralalgorithmstosolvetheformulatedproblems.Finally,aftertwo
numericalexamples,thepaperfinishesbypresentingtheconclusions
L.Rodrigues,S.Boyd/Systems&ControlLetters54(2005)835–853 837
2. Problemassumptions
It is assumed that a PWA system and a corresponding partition of the state space with polytopic cells Ri, i ∈
I={1,...,M}aregiven(see[20]forgeneratingsuchapartition).Following[14,18,10],eachcellisconstructed
astheintersectionofafinitenumber(pi)ofhalf-spaces
Ri ={x|HiTx−g˜i<0}, (1)
whereHi =[hi1 h(cid:1)i2...hipi],g˜i =[g˜i1 g˜i2...g˜ipi]T.Moreover,thesetsRi partitionasubsetofthestatespace
X⊂Rn suchthat Mi=1Ri =X,Ri ∩Rj =∅, i (cid:14)=j,whereRi denotestheclosureofRi.Withineachcellthe
dynamicsareaffineoftheform
x˙(t)=Aix(t)+b˜i +Biu(t), (2)
where x(t) ∈ Rn and u(t) ∈ Rm. For system (2), we adopt the following definition of trajectories or solutions
presentedin[13].
Definition2.1(Johansson[13]). Letx(t) ∈ Xbeanabsolutelycontinuousfunction.Thenx(t)isatrajectoryof
thesystem(2)on[t0,tf]if,foralmostallt ∈[t0,tf]andLebesguemeasurableu(t),theequationx˙(t)=Aix(t)+
b˜i +Biu(t)holdsforx(t)∈Ri.
Any two cells sharing a common facet will be called level-1 neighboring cells. Let Ni =
{level-1 neighboringcellsof Ri}. It is also assumed that vectors cij ∈ Rn and scalars dij exist such that the
facetboundarybetweencellsRi andRj iscontainedinthehyperplanedescribedby{x ∈Rn|ciTjx−dij =0},for
i=1,...,M,j ∈Ni.Aparametricdescriptionoftheboundariescanthenbeobtainedas[10]
Ri ∩Rj ⊆{x=l˜ij +Fijs|s ∈Rn−1} (3)
fori =1,...,M,j ∈ Ni,whereFij ∈ Rn×(n−1) (fullrank)isthematrixwhosecolumnsspanthenullspaceof
ciTj,andl˜ij ∈ Rn isgivenbyl˜ij =cij(ciTjcij)−1dij.Forsystemswhosepolytopiccellsareslabs,calledpiecewise-
affineslabsystems,eachRi canbeouterapproximatedbyadegenerateellipsoidεi.Thiscoveringwillbeusedto
describetheregionsinsteadofthepolytopicdescription.Theellipsoidaldescriptionofpiecewise-affinesystemsis
usefulbecauseitoftenrequiresfewerparametersthanthepolytopicdescriptionanditenablestocastthesynthesis
problemasanoptimizationprograminvolvingasetofLMIsanalyticallyparameterizedbyavector.Todescribethe
ellipsoidalcovering,itisassumedthatmatricesEi andf˜i existsuchthat
Ri ⊆εi, (4)
where
εi ={x|(cid:19)Eix+f˜i(cid:19)(cid:1)1}. (5)
This covering is especially useful in the case where Ri is a slab because in this case the matrices Ei and f˜i are
guaranteed to exist and the covering (having one degenerate ellipsoid εi) is exact, i.e., εi ⊆ Ri and Ri ⊆ εi.
Moreprecisely,ifRi ={x|d1<ciTx<d2},thenthedegenerateellipsoidisdescribedbyEi =2ciT/(d2−d1)and
f˜i=−(d2+d1)/(d2−d1).Finally,itisassumedthatthecontrolobjectiveistostabilizethesystemtoagivenpoint
x .Withthechangeofcoordinatesz=x−x theproblemistransformedtothestabilizationoftheorigin.Inthese
cl cl
coordinates,thesystemdynamics(2)are
z˙(t)=Aiz(t)+bi +Biu(t), (6)
838 L.Rodrigues,S.Boyd/Systems&ControlLetters54(2005)835–853
wherebi =b˜i +Aixcl.Theparametricdescriptionofboundaries(3)iswrittenas
Ri ∩Rj ⊆{z=lij +Fijs|s ∈Rn−1}, (7)
wherelij =l˜ij −xcl fori=1,...,M,j ∈Ni.Thedescriptionofthepolytopiccellsis
Ri ={z|HiTz−gi<0}, (8)
wheregi =g˜i −HiTxcl,andtheellipsoidalcoveringisdescribedby
εi ={z|(cid:19)Eiz+fi(cid:19)(cid:1)1}, (9)
wherefi =f˜i +Eixcl.
3. Problemstatement
There are four Lyapunov-based controller synthesis problems that will be solved in this paper. For the four
problems,thepiecewise-affinestatefeedbackinputsignalisparameterizedbyKi andmi intheform
u=Kiz+mi, z∈Ri (10)
with−l0(cid:1)mi(cid:1)l0wherel0isavectorofupperboundsfortheentriesofmi,i=1,...,M.Thegloballyquadratic
candidatecontrolLyapunovfunctionisparameterizedbyP =PT as
V(z)=zTPz. (11)
Thefourproblemsare:
1. Problem1.Findapiecewise-affinestatefeedbackcontrollerthatexponentiallystabilizestheoriginof(6)and
findagloballyquadraticLyapunovfunctionthatprovesit.
2. Problem2.Thesameasproblem1withcontinuousinputsignalsattheswitchingboundaries.
3. Problem3.Fromthecontrollersthatexponentiallystabilizetheorigin,findtheonethatmaximizesthedecay
rateofthegloballyquadraticcontrolLyapunovfunction.
4. Problem4.Thesameasproblem3withcontinuousinputsignalsattheswitchingboundaries.
4. Problemformulation
Thissectionformulatesmathematicallythestabilizationproblems1and2asoptimizationprogramsinvolvinga
setofLMIsanalyticallyparameterizedbyavector.Thedecayratemaximizationproblems3and4areformulated
asasetofquasi-concaveoptimizationprogramsanalyticallyparameterizedbythesamevector.
4.1. Stabilization—problem1
ThecandidatecontrolLyapunovfunction(11)becomesaLyapunovfunctionifforfixed(cid:1)(cid:2)0,V >0andV˙ <−
(cid:1)V.Using(6)and(10),sufficientconditionsforexponentialstabilityarethusP =PT>0and
z∈R ⇒[(A +B K )z+(b +B m )]TPz
i i i i i i i
+zTP[(Ai +BiKi)z+(bi +Bimi)]+(cid:1)zTPz<0. (12)
L.Rodrigues,S.Boyd/Systems&ControlLetters54(2005)835–853 839
Thisexpressioncanberecastas
(cid:2) (cid:3) (cid:2) (cid:3)(cid:2) (cid:3)
z T A¯TP +PA¯ +(cid:1)P Pb¯ z
z∈Ri ⇒ 1 i (Pb¯i)Ti 0i 1 <0, (13)
whereA¯i=Ai+BiKi andb¯i=bi+Bimi.Iftheconditionz∈Ri in(13)isrelaxedtoz∈εi andifexpression(9)
isusedalongwiththeS-procedure[24,5]withmultiplier(cid:2)i<0yieldsthesufficientconditionsP =PT>0and
(cid:2) (cid:3) (cid:2) (cid:3)(cid:2) (cid:3) (cid:2) (cid:3) (cid:2) (cid:3)(cid:2) (cid:3)
z T A¯TP +PA¯ +(cid:1)P Pb¯ z z T E¯TE ETf z
1 i (Pb¯i)Ti 0i 1 <−(cid:2)i 1 fiiTEii fiTfii −i 1 1 , (14)
Rearrangingexpression(14)thefollowingsufficientconditionsforquadraticstabilizationarederived:
P =PT>0, (cid:2)i<0, i=1,...,M,
(cid:2) (cid:3)
A¯TP +PA¯ +(cid:1)P +(cid:2) ETE Pb¯ +(cid:2) ETf
i (Pb¯i +i (cid:2)iEiTfi)Ti i i −(cid:2)ii(1−ifiTifii) <0. (15)
UsingnewvariablesQ=P−1,(cid:3)i =(cid:2)−i 1thissetofconditionsareequivalentto
Q=QT>0, (cid:3)i<0, i=1,...,M,
(cid:2) (cid:3)
A¯TQ−1+Q−1A¯ +(cid:1)Q−1+(cid:3)−1ETE Q−1b¯ +(cid:3)−1ETf
i (Q−1b¯i +i (cid:3)−i 1EiTfi)Ti i i −(cid:3)−i 1i(1−ifiTfii)i <0. (16)
Thefollowinglemmawillpresentasetofequivalentconditionstoinequalities(16).
Lemma4.1. Conditions(16)areequivalenttoQ=QT>0,(cid:3)i<0and
1−fiTfi<0, (17)
A¯ Q+QA¯T+(cid:1)Q+(cid:3) b¯ b¯T
i i i i i
+((cid:3)ib¯ifiT+QETi)(cid:3)−i 1(I −fifiT)−1((cid:3)ib¯ifiT+QETi)T<0 (18)
fori=1,...,M.
Proof. TheconditionsQ=QT>0,(cid:3)i<0,i=1,...,M arethesameasin(16).Toderiveinequalities(17)and
(18),usingSchurcomplement,theLMIin(16)isequivalentto1−fiTfi<0and
Q−1A¯ +A¯TQ−1+(cid:1)Q−1+(cid:3)−1ETE
i i i i i
+(Q−1b¯i +(cid:3)−i 1EiTfi)(cid:3)i(1−fiTfi)−1(Q−1b¯i +(cid:3)−i 1EiTfi)T<0. (19)
Leftmultiplyinginequality(19)byQandrightmultiplyingitbyQT=Qyieldstheequivalentcondition
A¯ Q+QA¯T+(cid:1)Q+(cid:3)−1QETE QT
i i i i i
+(b¯i +(cid:3)−i 1QETifi)(cid:3)i(1−fiTfi)−1(b¯i +(cid:3)−i 1QETifi)T<0. (20)
TheMatrixInversionLemma[16]statesthat(1−fiTfi)−1=1+fiT(I−fifiT)−1fi.Usingthisresult,expression
(20)canberewrittenas
A¯ Q+QA¯T+(cid:1)Q+(cid:3)−1QETE QT
i i i i i
+(b¯i +(cid:3)−i 1QETifi)(cid:3)i[1+fiT(I −fifiT)−1fi](b¯i +(cid:3)−i 1QETifi)T<0. (21)
840 L.Rodrigues,S.Boyd/Systems&ControlLetters54(2005)835–853
Expression(21)canbeexpandedas
A¯ Q+QA¯T+(cid:1)Q+(cid:3)−1QETE QT+(cid:3) b¯ b¯T+(cid:3)−1(QETf )(QETf )T+b¯ fTE Q
i i i i i i i i i i i i i i i i
+QETifib¯iT+((cid:3)ib¯ifiT+QETififiT)(cid:3)−i 1(I −fifiT)−1((cid:3)ib¯ifiT+QETififiT)T<0. (22)
Expression(22)canberewrittenas
A¯ Q+QA¯T+(cid:1)Q+(cid:3) b¯ b¯T+(cid:3)−1QET(I +f fT)(QET)T+b¯ fT(QET)T
i i i i i i i i i i i i i
+QET(b¯ fT)T+((cid:3) b¯ fT+QET
i i i i i i i
−QETi(I −fifiT))(cid:3)−i 1(I −fifiT)−1((cid:3)ib¯ifiT+QETi −QETi(I −fifiT))T<0. (23)
Inequality(23)canberearrangedintheform
A¯ Q+QA¯T+(cid:1)Q+(cid:3) b¯ b¯T+((cid:3) b¯ fT+QET)(cid:3)−1(I −f fT)−1((cid:3) b¯ fT+QET)T
i i i i i i i i i i i i i i i i
+(cid:3)−1QET(I +f fT)(QET)T+b¯ fT(QET)T+QET(b¯ fT)T+(cid:3)−1QET(I −f fT)(QET)T
i i i i i i i i i i i i i i i i
−((cid:3)ib¯ifiT+QETi)(cid:3)−i 1(QETi)T−(cid:3)−i 1QETi((cid:3)ib¯ifiT+QETi)T<0. (24)
Finally,thelasttworowsofexpression(24)cancelandweareleftwiththefollowingconditions:
1−fiTfi<0, (25)
A¯iQ+QA¯Ti +(cid:1)Q+(cid:3)ib¯ib¯iT+((cid:3)ib¯ifiT+QETi)(cid:3)−i 1(I −fifiT)−1((cid:3)ib¯ifiT+QETi)T<0. (cid:3) (26)
Thefollowingresultcannowbederived.
Corollary4.1. Forpiecewise-affineslabsystemsconditions(16)areequivalentto
Q=QT>0, (cid:3)i<0, i=1,...,M,
(cid:2) (cid:3)
A¯ Q+QA¯T+(cid:1)Q+(cid:3) b¯ b¯T (cid:3) b¯ fT+QET
i ((cid:3)ib¯ifiiT+QETi)Ti i i −i(cid:3)ii(1i −fifiTi) <0. (27)
Proof. Itfollowstriviallyfrom(25)to(26)applyingSchurcomplementandusingthefactthat1−fiTfi<0and
I −fifiT<0areequivalentwhenfi isascalar,whichisthecaseforpiecewise-affineslabsystems. (cid:3)
Remark1. ForanalternativepaththatenablestoderivesimilarresultsinamoregeneralsettingpleaseseeDuality
Lemma4in[12].
Fortheremainderofthepaperwewillconcentrateonpiecewise-affineslabsystems.Performingthesubstitution
A¯i =Ai +BiKi andintroducingnewvariablesYi =KiQin(27)yields
Q=QT>0, (cid:3)i<0, i=1,...,M,
(cid:2) (cid:3)
A Q+QAT+B Y +YTBT+(cid:1)Q+(cid:3) b¯ b¯T (cid:3) b¯ fT+QET
i i ((cid:3)ib¯iifiiT+Qi EiTi)T i i i −i(cid:3)ii(1i −fifiTi) <0, (28)
whereb¯i =bi +Bimi.Thepiecewise-affinestatefeedbackstabilizationproblemisnowformallydefined.
Definition4.1. Thepiecewise-affinestatefeedbacksynthesisproblem(problem1)is:forfixed(cid:1)(cid:2)0
find Q,Yi,mi,(cid:3)i
s.t. Q=QT>0, (cid:3)i<0, (28),
−l1 ≺Yi ≺l1,−l0 ≺mi ≺l0, i=1,...,M,
where(cid:23),≺meancomponent-wiseinequalitiesandl ,l aregivenvectorbounds.
0 1
L.Rodrigues,S.Boyd/Systems&ControlLetters54(2005)835–853 841
Section 5 presents three algorithms to (approximately) solve this problem. The feasibility problem 1 can be
transformedintoanoptimizationproblemiftheQwithminimumconditionnumberissoughtasfollows:
Definition4.2. Theminimumconditionnumberpiecewise-affinestatefeedbackproblemis:forfixed(cid:1)(cid:2)0,(cid:4)>0
min (cid:5)
s.t. (cid:5)>0, (cid:4)I<Q<(cid:5)(cid:4)I,
Q=QT>0, (cid:3)i<0, (28),
−l1 ≺Yi ≺l1,−l0 ≺mi ≺l0, i=1,...,M,
where(cid:23),≺meancomponent-wiseinequalitiesandl ,l aregivenvectorbounds.
0 1
Usually(cid:4)isselectedtobeunitary.
4.2. Stabilization—problem2
To formulate problem 2, similarly to what was done in [21], the boundary description (7) is used to yield the
followingconstraintsforcontinuityofthecontrolsignals:
(Ki −Kj)Fij =0, (29)
(Ki −Kj)lij +(mi −mj)=0, ∀j ∈Ni. (30)
These constraints for continuity cannot be directly used in the problem from Definition 4.1 because Ki, i =1,
...,M, are not variables in that problem since the change of variables Yi =KiQ has been used. To be able to
expressconstraints(29)–(30)onthevariablesYi,i=1,...,M,definethematrix
Xij =[Fij lij]. (31)
NotethatXij isinvertiblebecauseFij isfullrankandlij doesnotbelongtothecolumnspaceofFij byconstruction.
Using(31),(29)–(30)canberewrittenas
(Ki −Kj)Xij =[0m×(n−1) mj −mi], ∀j ∈Ni. (32)
Then,using(32),thechangeofvariablesYi =KiQandinvertingXij,wecanwritetheconstraintsonYi,Yj as
Yi =Yj +[0m×(n−1) mj −mi]Xi−j1Q, ∀j ∈Ni. (33)
Thestabilizationproblem2isnowformallydefined.
Definition4.3. Thestabilizationproblem2is:forfixed(cid:1)(cid:2)0
find Q,Yi,mi,(cid:3)i
s.t. Q=QT>0, (cid:3)i<0, (28),(33),
−l1 ≺Yi ≺l1,−l0 ≺mi ≺l0, i=1,...,M,
where(cid:23),≺meancomponent-wiseinequalitiesandl ,l arevectorbounds.
0 1
Insummary,constraints(33)mustbeincludedintheoptimizationproblemtoguaranteethatthecontrolsignals
arecontinuousattheswitchingboundaries.
842 L.Rodrigues,S.Boyd/Systems&ControlLetters54(2005)835–853
4.3. Decayratemaximization—problems3and4
In the problems of Sections 4.1 and 4.2 the parameter (cid:1) was fixed. Let now (cid:1), the desired decay rate for the
globallyquadraticcontrolLyapunovfunction,beavariable.Then,wedefinetheperformancecriterion
J=(cid:1). (34)
Thecontrollerdesignproblemisnowtofindfromtheclassofcontrolsignalsparameterizedintheformu=Kiz+mi
ineachregionRi,theonethatmaximizestheperformanceJ.Thisisformallydefinednext.
Definition4.4. Thedecayrateoptimizationproblem3is:
max (cid:1)
s.t. Q=QT>0, (cid:3)i<0, (cid:1)>0, (28),
−l1 ≺Yi ≺l1,−l0 ≺mi ≺l0, i=1,...,M,
where(cid:23),≺meancomponent-wiseinequalitiesandl ,l arevectorbounds.
0 1
Finally, to formulate problem 4, it suffices to include the continuity constraints (33) in the optimization 4.4
yieldingthefollowingnewoptimizationproblem.
Definition4.5. Thedecayrateoptimizationproblem4is
max (cid:1)
s.t. Q=QT>0, (cid:3)i<0, (cid:1)>0, (28),(33),
−l1 ≺Yi ≺l1,−l0 ≺mi ≺l0, i=1,...,M,
where(cid:23),≺meancomponent-wiseinequalitiesandl ,l arevectorbounds.
0 1
5. Solutionalgorithms
Previous work [10] stated that piecewise-affine state feedback controller synthesis using a quadratic control
Lyapunovfunctiondoesnotseemtobeconvex.Infact,itisclearfrom(28)thatthissynthesisproblemcannotbe
formulatedasoneconvexprogrambecause(28)isnotanLMIiftheparametersmi,i=1,...,M areunknownbut
ratheraninfinitesetofparameterizedLMIs.However,thissectionshowshowthepiecewise-affinestatefeedback
synthesisproblemforpiecewise-affineslabsystemsusingagloballyquadraticLyapunovfunctioncanberelaxed
andsolvedtoapointneartheglobaloptimuminanefficientwaybyafinitesetofLMIs.Threedifferentalgorithms
willbepresentednext.
5.1. SolutionAlgorithm#1:samplingmethod
The interesting observation here is that for fixed mi, i =1,...,M, expression (28) is indeed an LMI and the
problemisconvex.Therefore,althoughtheproblemformulatedin(28)cannotbecastasoneconvexprogram,it
is an infinite set of convex problems involving an LMI or, equivalently, an infinite number of LMIs analytically
parameterizedbythevector(cid:6)=[mT1mT2 ...mTM]T.Sinceeachelementmi,i=1,...,M hasboundedcomponents,
(cid:6) belongs to an hypercube. Effective meshing techniques can then be used to sample the hypercube and solve a
finitenumberofLMIs.Thefollowingalgorithmissuggestedtosolvethestatefeedbackproblems1and2:
Algorithm#1:samplingmethod.
1. Defineagridforthedomainofthevector(cid:6)tosampleitatNpoints,
L.Rodrigues,S.Boyd/Systems&ControlLetters54(2005)835–853 843
2. Forfixed(cid:1)(cid:2)0,solvethecorrespondingfeasibilityDefinition4.1foreachofthepointsinthegriduntilafeasible
pointisfound.
3. Ifstep2issuccessfulorifthemaximumnumberofiterationswasreached,stop.Otherwise,increasethegrid
densityandgobacktoStep2.
Remark2. Algorithm#1canbechangedtosolvetheproblemstatedinDefinition4.2andstoreforallgridpoints
theonethatyieldstheminimumvalueof(cid:5).Thealgorithmcanbefurtherimprovedifthederivativeofthecostwith
respectto(cid:6)iscomputedateachpoint.Then,foreachselectedsamplepoint,thenextsamplepointshouldbechosen
inthedirectionoppositetothevectorderivative.Thisreducesthenumberofpointsfromthegridthatneedtobe
used,thusreducingthecomputationalburdenofthealgorithm.
Algorithm#1increasesthecomputationalcostasthegridbecomesdenserandcanbeprohibitiveforlargesystems.
However,thegriddingapproachhasalreadybeenusedinotherrecentresearchonanalysis[9],LPVcontrol[23],
gain-schedulingcontrol[23,1,2]andsometechniquesalreadyexisttoalleviatethecomputationalcostduetothe
griddingphase[3,22].
5.2. SolutionAlgorithms#2and#3:tracemaximizationmethods
AnalternativealgorithmcanbedevelopedtosolvethestatefeedbackproblemstatedinDefinition4.1whenthe
ellipsoidalcoverforeachregionisformedbyonlyoneellipsoid,whichisthecaseforpiecewise-affineslabsystems.
Todevelopalternativealgorithmsforsuchcase,letusreturntoinequality(28)andperformthechangeofvariables
Zi =(cid:3)imi, (35)
Wi =(cid:3)imimTi =(cid:3)−i 1ZiZiT. (36)
Then,inequality(28)canberewrittenas
(cid:2) (cid:3)
A Q+QAT+B Y +YTBT+(cid:1)Q+(cid:3) b bT+b ZTBT+B Z bT+B W BT (·)
i i i i i i i i i i i i i i i i i i <0. (37)
(((cid:3)ibi+BiZi)fiT+QETi)T −(cid:3)i(1−fifiT)
This inequality is an LMI. Note that (35) is just a change of variables because mi is not involved in (37).After
knowing(cid:3)iandZi,micanbeobtainedasmi=(cid:3)−i 1Zi.Therefore,constraint(35)canbehandledafterthesolutionto
thecontrollerdesignoptimizationproblemisobtained.However,constraint(36)mustbeincludedinthecontroller
designoptimizationproblembecauseitinvolvesvariablesthatappearin(37).Therefore,thefollowingproblemcan
bedefined,whichisequivalenttoproblem1:
Definition5.1. Thepiecewise-affinestatefeedbacksynthesisproblem(problem1b)is:forfixed(cid:1)(cid:2)0
find Q,Wi,Zi,Yi,mi,(cid:3)i
s.t. Q=QT>0, (cid:3)i<0, (36),(37),
−l1 ≺Yi ≺l1,−l0 ≺mi ≺l0, i=1,...,M,
where(cid:23),≺meancomponent-wiseinequalitiesandl ,l aregivenvectorbounds.
0 1
The main obstacle here is that constraint (36) is not convex.Therefore the approach to be followed next is to
replacethisconstraintbyasetofconvexconstraintsandformulateaconvexoptimizationproblemwhosesolution
will also be a solution to problem 1b whenever the optimal value of the convex problem is zero. To do this,
observethatbecause(cid:3)i<0,ifconstraint(36)isverifiedthenrank(Wi)=1andbyaSchurcomplementargument
844 L.Rodrigues,S.Boyd/Systems&ControlLetters54(2005)835–853
thefollowingconstraintsareverified:
(cid:2) (cid:3)
W Z
W˜i = ZTi (cid:3)i (cid:1)0, i=1,...,M. (38)
i i
Letusnowdefinethefunctional
(cid:4)
Js=(cid:26) [trace(Wi)−(cid:3)−i 1ZiTZi]. (39)
i
Ifconstraint(38)isverified,weobservethatJs(cid:1)0andJs=0ifandonlyifWi=(cid:3)−i 1ZiZiT,i=1,...,M,which
agreeswith(36).Thisdiscussionmotivatesthedefinitionofthefollowingmaximizationproblem:
Definition5.2. Thetracemaximizationproblem(problem1c)is:forfixed(cid:1)(cid:2)0
max Js
s.t. Q=QT>0, (cid:3)i<0, (37),(38),
−l1 ≺Yi ≺l1,−l˜0 ≺Zi ≺l˜0, i=1,...,M,
where(cid:23),≺meancomponent-wiseinequalities,l˜0,l1aregivenvectorboundsandQ,Yi,Zi,Wi,(cid:3)i,i=1,...,M
aretheoptimizationvariables.
Fromthepreviousdiscussionitisclearthatiftheoptimalvalueofproblem1cinDefinition5.2iszero,thenthe
solutiontoproblem1cautomaticallyyieldsastabilizingpiecewise-affinestatefeedbackcontrollerwithguaranteed
Lyapunovdecayrate(cid:1),i.e,ityieldsasolutiontoproblem1bandtotheoriginalproblem1inDefinition4.1.Since
thefunctionalJsisnotconvex,followingtheideasexpressedin[11,7],anapproximationtothesolutionofproblem
1cisobtainedbythealgorithmthatispresentednext.LetnominalvaluesofthematricesZi ateachiterationk(cid:2)1
begivenbyZi,k−1,i=1,...,M.LetalsonominalvaluesfortheS-proceduremultiplierparametersbegivenas
(cid:3)i,0 andlet(cid:1)(cid:3)i =(cid:3)i −(cid:3)i,0,i=1,...,M.Thedescriptionofthealgorithmnowfollows.
Algorithm#2:Tracemaximization.
1.
find Q,Yi,Zi,Wi,(cid:3)i
s.t. Q=QT>0, (cid:3)i<0, (37),(38),
−l1 ≺Yi ≺l1,−l˜0 ≺Zi ≺l˜0, i=1,...,M.
Iftheproblemisnotfeasiblestop.Otherwise,set(cid:3)i,0=(cid:3)i,Zi,0=Zi,i=1,...,M,k=1andselectatolerance
parameter(cid:7)>0.
2. Solve
(cid:5)
max [trace(Wi,k)−2(cid:3)−i,01ZiT,k−1Zi,k +(cid:3)−i,01ZiT,k−1Zi,k−1]
i
s.t. Q=QT>0, (cid:3)i<0, (cid:1)(cid:3)i(cid:2)0, (37),(38),
−l1 ≺Yi ≺l1,−l˜0 ≺Zi ≺l˜0, i=1,...,M.
3. If the absolute value of Js obtained for the solution parameters of step 2 is less than (cid:7), or if the maximum
numberofiterationshasbeenreached,stop.Otherwise,setZi,k−1=Zi,k,i=1,...,M,k=k+1,andgoto
step2.
(cid:5)
Remark3. Notethatanequivalentfunctionaltobemaximizedis i[trace(Wi,k)−2(cid:3)−i,01ZiT,k−1Zi,k]becausethe
term(cid:3)−i,01ZiT,k−1Zi,k−1isconstantatiterationk.