A poset framework to model decentralized control
problems
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Citation
Shah, P., and P.A. Parrilo. “A poset framework to model
decentralized control problems.” Decision and Control, 2009 held
jointly with the 2009 28th Chinese Control Conference.
CDC/CCC 2009. Proceedings of the 48th IEEE Conference on.
2009. 972-977. © Copyright 2009 IEEE
As Published
http://dx.doi.org/10.1109/CDC.2009.5400464
Publisher
Institute of Electrical and Electronics Engineers
Version
Final published version
Accessed
Thu May 26 01:49:18 EDT 2016
Citable Link
http://hdl.handle.net/1721.1/60325
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Detailed Terms
Joint 48th IEEE Conference on Decision and Control and
28th Chinese Control Conference
Shanghai, P.R. China, December 16-18, 2009
WeAln5.7
A Poset Framewerk to Model Deceotralized C·ontrol ProbleJDS
Parikshit Shah and Pablo A. Parrilo
Li::Ib:lr,il1DrJ for Inf"ormi::ttlcJ1 ,and. Deci%ian :S:Y$~~m:;;
lItla.!3llChusctt5 lnstitute of Teehnclngy, Cambridge, MA 02139_
,tu oy
,.y:li tern, with time de,]a ~ n :t:uu:l been
pre.... ious purer [9]. that ,JU'bj e [;10 1:0 certain
o:md:iti.oJl~ on the de']ay:;. ~t~.rJ 'LJb,y,te~ (rlmmely
I:he' triangj ~ Ine qLJ al it}) c~ 1h.~ .r=.ul ting pm b l.e-m. w m:qLJ 2Ld.r.al: Ic i::I LlY lrrviU"i.i1IJ t (a.:n.d t.b L11 ~lG9b ~ to c c n~x opt.:imbtatic[lJ- ~ Ibcw th.i::It ~~ Ls a cl'JML1riJL
E'I c I4:t iU.SOC ~-d. wl th. .syh r.rI5 'W Lth, t:i me ~J~ 'wlth,
th Ls. .m'ba.dd JtLV:l trY pre r.erty, MI.d. th.at the c c mpu tatlonal
tnLctab :iLLt Y :i I ~ m E"I Ly an il8~'bra.:iI: ['D ~ nee c f th Ls.
LJ [IderLy Lns: ]'02 t.
:2) Wi! ~ ei~ ;!IC I'X~J'1Sion c f I:he pn!l C!!!ldi.ng 1:0 .sp;!~i.2ILL y
di.s1.r ibutd 5.YI bem I. I ~ W lUi S bcw[I 'by :B..,.m i.-ih I:haI: .sp.atia LLy j nvarian ~ I f I bI! InS 'W i a •"funneL !Ya usaL~ im.pul~
~I':I.R W~ .2I.D1erIii!I'b Le tc [D1'J'IIe:X op1imi'ZlL1ic c _ 'lIs ing
c LJ r pc set .i1pprOlLC~h.. ~e ~ .ab~ b:i g~ oer;a]bE.~ t.~
fiJ nnc Lc:3'lJaI1i1Y CD nditioo..
3) ~ ~ dYthe rclJI.tio:n.ML.ip l'c tw«c pc aeti II.[] d q1J.Bj[":3 ti c
ir.rv3ri..Brli:~'., Vtle sbow 1:luI.t qu.adnI.ti.c j n...an ~ e i:'--.B!l, be
cawr:a.L Ly j nterp:rcta:I :as a trmlsi ti\.i ty property. and
"'... h!ilf~ct- [n III Prew.i DlI'- pili per I:t 0 I, th 8JC H.1I the rs mc¥r I:CI
1.h.IIt po;ets p]"[]"l(lftc III L:lsGlI] DUJlliel.:lcJ!: mmR"ll.'Drk 1m- II rcusnnIIbL~· ]~ ~L~ Dr d~1niILz.crt ~cDtroL prnblems, Ic thLs plllper
"e sh 0,..: m Cre ~ 0 nnee11 0 JlS be t"."ftJl ]XI ~ ill Drt rie-c eCt r H.l.:1:rRt:l
enntrnl, tl[§ll~'I w!" dmw tbat pTHiDWiLy kDOIfl."J1. ren.lbi IILboLJ\
s)..~t~ms 1JI:11h tI~ d.cilllLy!i FoUcw JUIItllnU:!-· LJ5:lc~ the P[fii~t fJ'lLm~­
"0 T k, 'W·1f III br.c me DtI CD ~:ttnL·dc C!i 0 f tkl s tC l.:rdL I'll \!"- d LItl!"[] s:l1JJUII L
SplLtie-tt"mpOnU !iY~ M..-tJDdL}~.. ~ stLJd).. crmdltluns under
"bL.:h 1h!" p~:!-. k.DOIfl."J1. ~ qLJllf'iJ-IItL.: Lm~lIIrllllJ1n- unrl p~
strueture IIIr~ LIflLJh·IIIL.mL T:hroLJ~h 1h:l!i pllLpct. 1fI."If hnpe to
("[]ml.n£e Ihe reuder c£ the :lm]XIrtunt role thut pm~1:.s ~m
tc PW)· :lc much cF tb~ [LJrrcDt thllilJ1-· cf dKflllrU:ll.u~d [octrcl~
1) v..~
~·,n ina
in
It i..:Ii w ~] -know--n ,hmt 'in ~rm1 d~cec l:.aLliz:e.1 con\ro L i:li
l)i~kLi,. [3] ha\~, ~.JJ tba\
certllin j[l:litarloe, 0 r ~b pm b Je.cJn:li .are jn rIiK: l in\r.aI:\Ii!I'b~.
II,
hard problem. :S]C[lrje] JUld
On 'h~ D1her harll:t~ "ViJLJlSIi!l1'i$ [13]. [14] ~:lie:nle.-j ~
~Zl2.5 wb~ dt::Lt:rlt:t'aliud coc t.m L L.s I7ai: tab]~. In iJ ~.,.1.ou5
pa.pu [lO]~ th~ i]lJ.th.oni w~ ab~ to ~riJ.Ii%ll: ~ ~mL'15
:in an ~Lng FtaDlI:wcrx LJI:I[lS pilTtiiJlJy crdt:~d I4:tl. Th.:is
pa.pu Lrx:lLJ~ ~~~1J11n1'll of th.i::It ,~ tc othJ:r l~tU[lSI.
R c tko-wwLtz and. La!l [9] Mil."':; ~nbl: d iJ c.rl ~ ri c n krJO"lN,n i15
qu.adrat:i ~ i.mrad.iul~, tbat ~ b~ b!riu I i1 ~W S of pro'bl ~ m5
in ~~nlJ"2lJi.'Ud~IJI:.mL t.h.a~ h~ ~ property thilt pro'bl~m5
N ~ I;Qm1aX jlJ I:he YCLJla p~. Ou.r n!mLI3. ~
~,la~ 1:0 Ihis pm pmiy M'I.d we I hcw th~ ro ~ ~tiD IJ b:i l:JJe,ir
wo.rk jlJ CLJr ~. ~ i:9.l5o ~ how .so~ p~jsti.ng
~.ILJll:.5 C[l qu.adrat:i~ i.mrad.iul~, cf rI~N-oT~.d 5.yltem!i with,
"Ii me ckaaylil CB11. "bf~ in1.cIp rcted Bi mmJ. Lti :abc L1 t L1 [Ii:! crI.)dng
parti.a.lLy D~;rc d &ets~
In a prcvic 1111 pap: r [10]. W~ de'l--elc pro a fnut1.C1N"OTk f Dr
dt:~tm.lj;mj ccntrol ~b~DLi 11IIing PJ;s.cm... We a.rgued that
~m. pmvide the dgbt LangLlJtge :and t-er-hnicaJ IDcls to t.alk.
lib c L1t 121 trlO.t'e: get1B'1i11 JX:i t:i c n of ~a ~ ditJ~ (a:I, c referred \[]
Il:li 11'~~ ~clJtml
in the litermtuJ"e) 1lJl1.OrJ:g 'LJb:liy:libl:m,.
pc:w.-t (whi["h i, .Ii ['[] mb ica\[] ri 1211
o"b1 ~ l) ~ the nJJoon of Ill] '~i.-.1ec~ II1geb r1I. [IS L Ill] .al gebr.llic
O"b1~L Thi, al8~'bra:i~ ~tu.J:ee mUcwe.1 11:- tc con,·wf)"
'lhe: pro b~m. "tile: ~n lll.mt 'COle' i.rJ~ ~c 8 ex a.:rn.p1=. cf"
~ ~n tri!.l:l ud conI:Tc L t.b il t bad. b~ [I .sJtm.r.n t c 'be I:r4-i:t..ab ~
:in the Ii~U2lh.1.~ ~ :I[I file t I ~ I.fi[" :I[II tiu:I C"e I Cf t.b Ls PJ I ~
paca:=llpn. In thLs ~.pu, ~ ~xteru:l. thil ~t ~ 'I.e
o1hB" cas 19, fo r :i ns.I:i!J1£~; ~~ 'W Lth tim ~ ad i::I)"'s.
Tb Jt'1.i].Ln ccntribL1tlmui DF UUI ~r ~ ~ follcwLnS:
that UTIdel:" cerLain n:a.1.1JJ"B.l settiJl.g!.. ~ sc=t B1.rUctures
and qUIInrll.ti ~ 'in v ariJm I:e .a.re ~XDd.~1 «J (I! fl.. til ~ ~'. \\-'"e'
Ln bD rlLJ I:"e the' riC tiDc c f A q~t. w hi clL 'i.:- .Ii ~ t
DD1u 1.0 IUl e.::tLJi...·.alerloe' ]"Ie'lati cc. ~~ ,hcw th.II.I: LJ riner ,iD'IiJar "hul ~.ha1 m.cl'e' ~oer:a.l cocr:lit:icn,,~
qLJadnLt:ic :in\!.ar:iIi1rX:~ 'i.:- equi,~.a1en\ \[] q~~.
:PcR:\s ~ v9J ~1J-.s1JJd:i~ o'bL1ec1s :in cc mbl.nal:cric:s.. and.
ha~ 'b-e~rJ LJI~ :lC ~[l8l.needl'J:(t and I:Ompu~ SC~.rJ~ (.see:
[ ] [I] Ml.d. th~~ r.er~~.OCB ~~Ln).,
The ~\ 0 r thil Pap!U :iI CrgarJ b:oen al fo LI.o-w5.. In Se I:t:i c rI I I
'~: In bodu ~ I:he o.rder;..1:hec r.e t k i!J1.d. C"On1.ro L~ c~ t:i I: ~ Llm.iJl.ades. th.i::It w Ll l b ~ U I.e d. tbrouS:hou t ~: ~r:. In Se I:CCn ITI
'~ shJ.d.y systeml wit.b
sIu d y an ~ xtenlio,n
t:i~ d~]a'yJi. Ic ~~i.o-n
of thee mmL 15 cf &ctIic n
inVi1J'i.2Ic~ IYlbeml. In ~t:ion V'I '~ Il:u.dy ~ ccnn~tio,n
'b-etw~ [I qu.a d.ra t:i[; i.mrad.iul.oe ;!IC d, ~ I:5/j[l [ddJ,J1Ce~ .al~M.2IJi_
Ic ~t:icn VI '~ co;n.wLu.d.e DUJ" p~r.
~~d to 'lhe: noti c [lor 1I.
II.
it.
OmrflF -~}lftJ porlic
:Pm.ID.JI.'IINA RlI3.
Pr:r ~ r-m FIUJ'~i rflS
~;tKm 1: A p.ac1iilly o~ ~~ (cr :PDS~i) F =
(p. ~~ ~cns:ill:..s of 1I. set P ab.n.g with il bjcM"f m]allcn ::5~
'Wbkh
L)
:2)
b
I ~
(0llDw ins pre pmdi.e,I:
~ ~ ~ (rd:le:tj.,.ity).
~ ~ b and ,~ :;5 ~ jmpL~1 a = ,~ (iU11is'ym~try~'I
3) 12 ~ IJ :mj ,b :5 c impliea l1 ~ c (tr.Br1.ajtivity)_
:Pc ~ may ~ finite c T infici I:e d~ pendin8 L1p c [I the' tm"d:inali ty Dr 1h~~ LJnd~.dying ~l P.
p
978-1-4244-3872-3/09/$25.00 ©2009 IEEE
IV'~
m b:i .sp;!~i.2ILLy
972
WeAln5.7
E.tamplt! /: An example of :1 poset Wml three eJe,./t1I::llbs.
P
b. c )) ....,ilh order relations u. ~ ,b and a ~ e:
is .shown in FiglJ~ L In '\h~ dill.SJlUn (koo.....o as :1 Hasse
d.bg./=), ;:10 up = 0 ..... indicates the relatinn ~.
= [....
(L~..
A standard l:'OmlL.uy of '\his tbeo:R!m is the following [12.
Theorem L2.3].
Coro'lary J: Suppose A
E
I" is invertible, Then K
I
elf"
B. Cafltro~.lht!GFt!lk PrrU~.flalU!.r;
[0 tbi" pmrer .....e aJ'1! in~ in (jecentr.aLi:z.e.::I. .structures
on Ii n eM tic:le -i nun an t
teens, V\'e wi lLnot JlIII1:ku]arly
ernphasdze t.hl:,cnntlnuous or dist"J'1!t.e tim.e:coses lIS. our results
~)' e9u:1Uy w.e]l to bo'lb the .settings- We will ccnslder
.sys.1!::m.5. l'I.'Lth ~ follo.....Lng de:scnr-tLon: [j € Jl"- is 'Ib~ control
input, y € Il"· I..s ~: [ll.01ct output, ~, € Il"· ls ~. u:ogeoolJs
input, z € Il": is the s.)'st..em output. ~'e wUJ be int..eJ'1!sted in
Rpr.es~ting. OUI s~'stecu Vi01 transfer functinn matrices ;:IS
"y"
lkjlfiitra,I:1: Let tpbe
po~t.
II
Let Q be
RI
nog. The
:set o( IIlL Junetions
f: Px'P~Q
P(w)
fe r, y) = 0 if ..t "1 ~'f is called !he
i,u:rdence a'gebra of IF over Q. It is denoted b}' 1.,.(Q). [[
the ring is clear from the context, we: ..... ilJ simply demte this
by /,:0 (we ..... ilJ usually work over the field of rational (XOpI:r
transfer functlons, or related extended SpB.CeS).
''''hen the po:set tp i" Jinite, tbe ~t o( Fum:tion" in t.he
im=ideot"~ IIlgebra JtlIIj' be thought of a:;. nlIItri.ces with .a
:;pecific ~t)' pau.ern gho.en by t.be order J'1!]atioo" of t.he
wi'\h !he propert)'
~fifl.tkNt
J: Lid 'Pbe 01 pnset, The functlon
[(~)
e
Ir{Q) defined by
~('P)(X , y) = {
' I]
]:
if:r oJ 1-'
othe~:se
is cwJed thc:. s:eta-jrll1Clron of 'P.,
the 'Zem-function of the ~t i" an eLec:lent o( !:.be
im::ideot"e 1al8ebrn,
Example- :l: The matrix Rj'J'e,;entlltionof the :zet.a (unction
for the pcsld FromEum[lLe 1 I..s lIS. (0110.....5:
Cleml.j'"
,.- ~ i ~ I
The inddeot"e :Ugeb~ is '\h~ srl of O1ll JmIb'i.ces. in Q1ll.J .....bkb
have '\h~ um~ spu.sii)' p;I~lem u iis Ut01 funl:'tion_
Given two functions J. g ~ I.,.C.Q). thdr sum J + g sod scabir
rnultipli(1ltion cJ are ddine:d as 1I.:SI.lal. The: product h = J. g
is d.c:fute:d :115 follows:
M.. . y) =
.L
f(x , ,)~t:;:. )') ,
(] )
, ~ f'
k =ntion~ ;:Ibo\'.e. we ..... iLL f.=:juentLy 'lbiol,;: of the functioos in lhe inddeot"e ~geb~ of :1 pcs~~ lIS. .squ=e JmIb'i=
(of 3pprcE=C~le dimensi.oru:.) inheriting .01 spwsity p;I~lem
dk~led by the p=e~. Th~ .i1bo~ dz::finiHon of function
mulHpLi~~ion i..s m.rle 50 ili01t i~ is coosis.~n~ .....illi sbnd=d
JmIb'ix ClUltiplic01tioo.
Th~1?m J: Lc:t P be a posrt. Under the: usual cle.f.icition
of aidition. sOO multiplication as deftned in (1) the iTX:idence
:algebr:a. is an asSOci8ti\'e algebra (Le. it is cLosc:d under
llrir'jition" "t'IIi.ar ClUltipLicatioo RID.,-t (unction muJli.plil:.IIJ:ion).
Proo}:" ::ieee [10].
•
P1;t(.W)]
Pn(w)
. P:;:I (..:v)
'\h;:l~
~t.
=[. p] I (..:v)
•
.....bere P(w) € C'".+"-i....:,., +":<J is the overall sY5tem transfer
flJodion. Through lh.e ~ of lliis p3'per. we :1bb.n:.vi01~
no~~ion ;:md define P':1.~ = G. Furthermnre, in .seve~ C01S~
WJ:., ..... iLL assume '\h;:l~ sysbern G b01S 01n ~ number of inputs
and nurpuls (Le, flu = "F)' In '\h~ cases, we wiLL think of
G being composed of several 5Ub.sys.~m5o, each mb.s}'~m
having one: input III1d one mupat. "'IDle dea.Jing with finite
dimensional LTI systems the: signal and. operatorspaces wiLL
be the standard ones, In some sections we: will be dealing
.....ith S}",st.ems. with time-delays, in these C8SClS the S}·,st.ems.
arc: 00 longer finite-dimensional, sod the relevant spaces wiLL
need to be: appropriately exteooed (see [g)). We denote by
~I' the .set o( rali.onlll peeope!r t:nm"FI::J" (unctio~ Giveo a
contro]]tt K ~ '!t;''''''-. the closed-loop systc:m has l:rB.nsfer
fucction:
f(P. 11::)
=Pli + PI~IL(/- G.Kr
l
P:;:I.·
W~ are:
ioleresled in o!Ximal l:'Onb'olLe~ynili~s problecu of
lhe fonc::
[]]LnimLu:
II f(P, IL) II
(2)
mbjeci to
K .st.1bLlizes P
KeS,
.....b.eJ'1! 5 is some 5U~t"e of the 5J'.01t"e of contro]]u5. [n this.
II ' II is .01tl)' oorm 00 'It',;:"P cho.seo to iJ]7pJtl[X'~tely
c3p'luJ'1! the perfo=t"e of '\h~ closed-loop s)'slem. 10 lliis.
p~ S .....iLL mp=n~ diffenm~ coosb'01iots 00 the conl:mlLer
K (for eumpLe .sp;srsi~}'. or dd:1}' 'b:mnd.s.). H JmI}' b~ no~
~~ for gener3l P ;:md .s !here is no known let:bni9u~ for
.solving problem (2.).
Problem (2) as p:r:cs.ente:d is. 8 non-<:on\'e;( problem in K. [f
the: 5Ub~ C'on.:s.lnti.nt K E S wen:: not prc:senL then ~ral
tc:ctmiques exist [or soh'ing the: problem.. One approach
towBrd.s a solution to the (XOble:m is to .....rite 8J] explicit
par:arnl:teri.zatioc of :all smbiiizing. coctroJLers. for the probLc:m. It is desirBbLc: to hsvc: the closed-loop tnms[er function
be an affine fuomoo in t.be patmmeter. that the problem
become" COm'ex. There ~ riiffecent IIJ1prcRllo"::hes to perform
the par.lUtletriuti.on, (or ell:lI.It1ple the "'.I:buill, par.ll.JIle1.riuti.on
[9] .and the "o--ealied R-pe.mm.ebUatioo 14].
Rotko'A'ia .and uU pee:sent~ RI .notion t:'II1leri qrtD.dmJic ip;ll'~,
973
L
,
"0
WeAln5.7
rTu/1Cf!l w h idJ giv M ~ .m 81 ~i en t ~nditic n fer r~rm!1:Il'Z.jng .21. d.e'~l!!lJl:ri1liZil!a tonboL prcblmn jn .21. 'WlI.1j tm.t i..s ~vso
jn I:lu! Youla ~bmr [9].
~fin;fFan 4: GivelJ a s1jste
G an:l ~ m'b~c ef
~elJsb'a:inls fer th~ ~entrolwS. it is said tn b~ qrll1JnE1;aJly
;PiI~-wiariit wjth R:cS~t tn G if fer all K e S. KGK -E S:
Similar 1:0 qmldmtl~ jnvMj~ s.ysb~1tUi with ~t s.1nJ..w1JJrec ~ ~.mena'bl.i!! 1:0 ~enve~ ~1Df,lxi~Hon I: LO]. In I:he~
resl ef this. p.apar. ~ will net I!mphas~ I:h:is ~~~t ef
re.~I:Ti.zatiec/r;enw;:tifi..watielJ.Thee main I:hrust of I:he~
~ pm"' is. 1.0 s.how how po.-ls. W Mil ~~Il 'wi th jns.j 8 b l;
1"a
em
gj\lil:~
ni [lJll
~]as5.iU 0
f
pr~=-'bl.i!! I1Ui
W
Jtj ~ are
i:I.JtmmI.b lI:~
tn ~V So
re.~I:Ti.zatielJ.
:rn.
S''!:".snJWS W [[",~
Tu..m, DIuA~s
As ml!lJt.i.o:ru!a earher, in ~ieus werk. [LO] ~ ilu1her5
I:hat D12lDf exam.pLu 0 f ~1ra.Li2d tDellbo L pm b-
sh~O"M!d
]~m5 sru.d:i~
ilJ th.-a lib!ral:uR; [JaIl b~ medd~ Viil pesds. In
pro v ide arD I:her t!,xampl ~ e f I:h ~ j .e, ~da:i n
m-u.c1:U..1:=ed ti ~ -d.c1Jl,ycj. :if Stc IJlB . It 'W':3B ~liJWn by' Ro I:kctwi tz
ct at [9] tlm.t S'}~ S tc IJlB in\lDL'...·ing Ii me-clela~ \Ij.~hi.ch 0 bey
this ~ t.i.ocll
W1!
thee I:ri..ar1gk i [llqlLEll i ty Jm:...-e a prnP==rt}~ 1.-Jl.JIwn as qpa.:lm'li c
in\·m::uxc. In tbLi ieCtion ~ :ibow 'that thi.& mmLt i:i a
direct ro~n~~ of ~r]yieg ~t struc1.lJre MI.d ia.
1I.!iD.Xiated inc id·e[1.i:~e :alg'ebrlL The cmpbBl3 i S beJ:e i:i on the
~elJ:liln.1ctie[l ef the LlerjerLying ~t mn.1 nel ~ reJLlLtin~
qUlln['lltit: LmrnriIUlDe'. ~ hope' 'lD ['[]n\~illCZ: the ~ throug::h.
~ 1Ul.r,f ether ~XIUDpJe,. ef' the fu.D::iamen'lm1 rciI~ 1h.m. pe:li~
~m lc play in Jt1.I.]IitJ ef' 1h.e ~llrren\ ~ DIJ ~'lnlI.ize-::l.
~elJtml.
In thL,. ~d:i.o[l w~ ['[]n,.irJer LTI :s.~ \lliitb time ~Im'f:li.
G'h·~ .rJ. CIe~[ltrIi1Iier:l pl.rJ.nl w:itb Q:iMmuni~n ~lay:s.
~t~ the' di:lf~t :s.llb:s.)I:titem:lir ,,-,""e' ~elJ:lii.der ~ t.B.:Iilr.:
of d~ig~~ ~e[ll:rcll.er~ fer ~' :s.ub,.)I:litem:ti wh~b ieter.act
~[lg to It :li'i.miLar CIe,ja~ :litl'Ut:I."ure'. It ha:Ii ~Il ~·,Il [9]
~t !Ii'1J.Ii:h cem.mLlek~en st:n:lctuJ'~]j ~ ~nable: to conV5.
re~l:rtz..i1tle[ld~ te ~L.r q~tic :imr;ui2lJ1.Of. ~ 'wLIJ
show thi:Jl PJs~13. ~ [labJralL;y :in th:is ~tu p .iU1n that ~;)I
~scr:i~ ~; cemmLllJk~enconsl::r.a.lct.s LIl an LlltuLti~ ~iY.
CelJ:liii1er It :Ii''y:Ii~D'] 'with ~t :s.llb:s.)I:titem:li (le'L N = ~ 1•... ~ Pill).
1h~ SfS~D'] b~ ~r:ib~ by ~ 1riimsf~ functIDIJ m.a'irix
G wh~; Gu{w) ~bc. lll~ r~q~f ~ ~1.~n
:inPJt ef' system i and Dlltput ef s~m. iu All ~qll:i..-al.erJJ: 'W1J.)I
'I.e ~s['["L~ tN: pl.act I..s to s~Lff tN: bnpul~ r~;spcnsB 8u{/).
l:'J.e:line tN: ~li1y b~tvre~1l ~ su'bs1~1'l15 i iiUJd j (den.c~d
'by D.;) iiUi folJews
ut
Dij = :illp [T" : BI~i(t) = 0 fDr alL t ~ oT] .
ND~
1h.at
Ji~: illL rs.ystems ~ 2l5Sll~~d. to b~ l:ii9L1sil, ~:
~L.ays D,ij ~ ~8a1J~.,
\-Ve rjefice
.:!: on ,N X R .a,i feLLo-w,...
:5IIy that (]. t] J ~ (i~ i~J if
.rJ. ,roejl2ltien
De.frnit~1
5:
~
~1 -
'1,2. D,ij.
SL~~ the :r.Y~te~ w~ lU'e rJemJi~ 'with
Ill"e' time Lmmi..arlt,.
",.ha1 tb~ ~etu:liti.on ~an:s. irttuitiv.el)l i:li tba\ (j.. 1IJ .:!: (i. ~:2.) if'
))I~ i lit ti~ ~0:i: i:li ie th~ ~et'l.e: ef' irtfLuenDe' ef I!J1. im.puj~
system j ~ tl~ ~",I. WI! shew J]ec:tt that if lbe~ ~~]a'yJi
a Ixiangll! ilJUl~lit1j ~Il lb~ rt!W.i.on ~n ~~'b.ecl in
Ddinitlen:S i..s a pmt~l nrder R:JatielJ.
.appl~,d~t
~ti.s.fy
PmprJafJirJfJ. /: SUPl='CEC DJJ = [I (LCc. effcct of inpot on
output withie S:3JL1.C aubsystems is wil:.OC-ut ckaay). D IJ > 0
(there i.!i nonzero delay between distinct aubsystems) and the
DI~i !HI.ti.B f f
fer .alL i~ j~ J:. distinct. Then :=S :is il l'a~tit:J. order relarlon,
Pro~f· 8iIJI:J! DJJ, = Dti 'by ~lJil::ien (F. 11,) ~n
t J ) . If
(;,
~
(j.
~,) ~.ea
(1. 12 )
(F. 11,) ) thmJ t J -/2 ~ 0 ~ea 12 - tJ ~ 0
'by cU!fi.n:it.Lo,1l ~"J = ~"2.
8ilJrlJ! D J) ~ 0 fer f ,-:I:- j it must 'bee ~ ~ I:h.at ~. = j giving
lI.[11i-srMrm!I:Iy. If (F. 11,) ~ (j ~"2) and. (j, t2 ) =f (i... 1]). '~~ bav~
11, :S ~, :S hI" :Mu-t:her. ~"2 - t J ~ D ~ and 13 - I~ ~ Dt j . :Hy (3)~
1] - ~",I ~ D~ + DJ:) ~ D~ and he,n~c (f. 11,) ~ ~i. f~). v.uifrlng
1:r2LrI.si t.ivi t,Y.
•
N DU=~ thB1 this trian gle inequali t f stru euire DD tbe delJI'f S i1i.
l'JlaC1ly tbf~ ooOOiti IJn that appeara i e [9]. Wb at is i eten=s t.ing
here is that ~ d.cLRy& a.ctuaLLy give rise to a nJll:UrB..1 ~t
sl.rU.ctLl.rc. :BB we bsv'C ju.&t ~in ted DLlt (~ PJiet ia dcterminm
pu.rcl~{ bf the dela}~B. ~ BCtuaL fLle~DeR.1 flJ~ Dr the
im.ptJ1:Ge re:li pee:Ge de ~ nm m a u.er). F Llrth~. th.e' ~t e f
(f. fJ)
~
(slrx:~ ~a)".s ~ eelJ~Hv~), UlUS
im.ptJl:Ge ~~:ti 8'~,;(~) whiclJ :lil2lti:lir~ thL,i d~lII'f c~tu~
fo.rm:li JUl I2Ilg~ b['II, 0 r fll.11C tLon,i u:n.der ~ e cvciIllOOIl. m:s.
~ ~~t pmpe:tiition ~.-'i.
~""':;:"'i~km 15: ~t \' = {D..
J
~ i3 8l~11 Rt ef
~--'.p
~ 1 ~i..J~n
.IlI:tIJJ 121 tI~
dela:y~:Lct
II,.
~tc
the c£Ct of (ma'lri:t) i.rn.pllse
~~
flUe) with t.be pmpcrt}~ that ~'~J(i) = 0 if (1. 0} l U. t).
ID~ili..-elf g~J(~) = 0 IJJe8l1B that the effect of MI, impulse
~
=
.at
time
Dr
.JUb:s.~.m. F M t~ ~'.
(J IIJ:
j ha:Ii ,not re~be(l the' e u t,rnL t
Thu:s. 1 ~ i:s. p~i~f ~ ~t cef
:s.llb:ti)l:tit.em
~ ~ 'lA·bkb o'bef:li the CIe'lay :li1:ructL1~ ~~~r:ib~ b)l T.
Civen III ~t []fimpu~ re,;P[]~ F = [f<i;(~)t IIDrJ G = {.e(I{~)}
CIe':Iine' F • G to ~ I::h.e mIIm ~ of bt1.ptJj~ ~ ~~J 'wi \h
(F .. G:~j(t) =
,4= i.l += 8J:/~)·
1
P~po~io~ 1: (]:i~e
i3
set of ~b.'fs 1q~ ,'wbkh ~tisf'~
~ ~em:lJtlnns ef PmfXJsLtien I. If F = tfij(l)] ~ G = {81~/~) J
.sucb 'ihilt F. G, -E ,1rr fer 1 :=:: i, j S It, lll.en F .. G e llip~
Proof· SuppeR~ (1. 0) '/:.' (F. 1). It suffLCM 1.0 s.how that
c
(jIi
-t
~j(~)
=
o.
Now~
(Ii' oil G)JJ(i) =
:t [
.1:.1,01"
/11:(1 - T)gli:j{T).h.
lit.,
If (F t' G1)(/) -:I:- 0 ~e ~ Jt1.I.]st b~ Eirm! i. 'T such ~hat
S.l:J{T) =I:. 0 .md fllt(1 - -r) =I:. O. This III I:ULtl means th.... t -r ~ ~.l:j
lI.[1n 1 - T ~ D.,... Thus ~j, 0) ~n (t.1") lI.[1n (t. 1") ~ (f.l). J3"y
l:r2LrI.sitivit,Y. ~j, D) =f (;. I). ce["l~ te our assump1iee.
•
Sie~
t.be i.rn.pllse ~~PJ[l~ ferm a oonvcLuti.onRl. aLgebraw
I:hc I:ran.!.fer fLlecti 0 e ma.tri ~ F( W J a:rx:l G(w) f erma mwti pl i.e ali..-e Rl gebra MI.d are thUB qLl aimti i2l.l Y in\lllJi.BD t. Th i.&
all.ou~cS LlS t.D coeclLldc the [oLLowing prnpositiiJ[l~
Prol' a $i1i0Pil 1: Cb:r1$ d~ .Il :se'L e f d~ La)l ~Il ~ III'
~b thm.1h~~:y :Ii~ff tb~ 1:Ji..ar18le '~qua:lit.Y (J). G,i\~ .rJ.pll2l.l1t
1 ~ with :uuoe del.rJ.)I ~I1$lT.IliIJQi. the ~ of ~entrdl:IeT~
x' f 1 ~ 'L,. qu.!ldr.llt~a:ljf ie...m:ian'L \lliith ~llc G.
G [IE
974
WeAln5.7
po.ss.ibJe. bo e.n.d.ow X xT with a natural p::>.sel structure, DeJ.ice
In this section WI:: pm\.!.de ODe mnre C!ll.lIItl[1h! llf pcsd.
slruchL!;e arising in det"C!lll:raliud control peoblerns, namely
in Ute! ennbext llf sp.:lii..ilJLy disbibubed s)'sbems. It is po.z:ibJe.
1.c nahi rall y C!Xbelld the. results llf lbe ~t"C! dill g SU b sedi II n
1.c a d.1ISS llf illfi.n..ite dimem;,illllil sy.sbems th~l ~ SF'llbUy
distributed [L l). The:.se resulbs w&e prtl(XISL'd in [11] by these
autbers, Similar l"C!suL1.s WC!re: irrlepe:rrlet1Hy and sirnultaneou.sJy de''I5]Ilp:d by Rotkowil:z. et .al. ill [8]. These results
S'ecer.aliu:d in muLlipJe. directinns thl:' previous results llf
Bamieh and \'oulgaris (2).. \\'t: briefly rairn' OUT results in
this subsection, sirxe the)· [lied}' complement the preceding
results.
W-e consider systems that evolve along spatial coordinates.
(x ~ ,\j as. well as temporal coordinates (1 ~ T). Much
Iike '\em:ror.aJinVII:ruu:u;e. we .s.II.y thll.t a .sy:;.tr:m i:r; :r; PlI.tilltr:m:roraJ]y it1\'.llriBnt if the I!:EfC!d llf IlIlJ impui:r;e II.t :r;pll.tia.!
ecnrdinate rl II.t time II II.t lI.['Io1hl!:X ]llca1i.on "'':2 at ti.m.l!: Iz
de['l!"~ llt1]y on ...':2 - r] .ar1l'1 t.;!. - ~J' ,Such :r;y:r;tem:r; 11UI)' be
::peciti.erj by their spaliO-k~poral if1fJ'MI~ rrspofUf: laC.. ·• (,.
Thi:;. fUllrullll de:Krib~ the re:r;pcm:r;e llf '\he :r;y:r;tem lit ]lll:'lltilln
(.t, t) ullder the LnBuC!1'JC1:: llf ~ ill'1'J"uLse at (0, 0). (ji\"et1
a ~y.stem h(x,~) one defines the ];tfPpor~ jiUfditm J(....:) ;IS
foLlows.:
I(.'() =
.sup~" :
l(t. r) = 0 fllr .alL t S ·r~ .
11 rib'lillt'1 ~ nn
X
X
T
;:,:s,
flllLow.s. L:l (XI. 11) ~ (.t2. ~) if:
(5)
Nlltke I:h.4! 5oimLLarit,y between ice'lu.al.Lt~ (S) 1lt1d Defmltinn S..
Bath lnequalitles ~y that ~::: - t] shouldbe gl"C!ilter th~ thl::
deb.~ bl::tween the .subs.ysu:rn.s, thus mWllB (S) the .1'llIhJr.:sL
C!:tbetlSillll llfthl:' Pllse1. d4!Eitljtion 1:0 lhe sp;:lli~ invariance C::I5.C!.
Propo1Oi1io~
J: [] 1. PJ'OpllsLullll ]] SU[1[111R. the support
=
functiun ~ such thilt j(O)
0, f(r) :> 0 for x '" 0 and
.suk:lddiHve. Then ~ is ;3. partial ll~ re.];3.'\illll.
Ullder tbese asSUlJ'1jJtilltlS nn I:h.4! SU[1[111rt fucctlon, the LmpWR.
corrvnlutional ilg.l:'b=, lht! usual Yllul;3.
par3l:1'Jetrizollions. 3m empJllye4 ;:md ron,,~fic~lilltl foUllws.
R.5poJlS.l:'.s fonn ~
Our resul ts hoJ d fll r OlUL'Ii dimensi II nal sp.:liiil en II rdi mrt.l:'.S
(fOr example, nnrms are examples of sub;uidiHVI:' s u pport
in. higher climensicns). More inbere:.s1illgLy. subaddirive support functions. Me: a stricti}' larger class of furxtions
that contain C'O'OCEJVe functions as special C1ISeS [l l ], Pigure
fUlld.illtlS
2 shows an example of 3 subadditive: sul=PJrt furxtion that
is. nof C'O'OCsve. For further details, we: encourage the :rea.iI:r
to read [ll J.
l
(4)
Noticc! th.at thi:r; r1e.fillitilln llEltur.aJLy extellrl:r; '\hI!: notillt1 llf
de1.AybetW_ll .sub.sy:;.1.I!:m:oin~e:o::'Iill .semllll In "ill. (3).
ThC! .su:rrort fUl'JCtion =LUllterj lit ...· t~li:r; one the ddaJ'
it1\ 'Olve rj in the I!:ffect of llIlJimpul:r;e lit the origin to RD:b
...... Fllr C!:t~mpJI::, if the sy.stem utlder cllllsLderatLon were 11
.L·a .L
FIg. 2. SlIppa:t 1:( . Impulse nspalR.
WOlve. 1hoen I:h.4! SUPPllrt fUtlCtill[l wlluLd be e:Jt3ct.ly thc! LLgld
Cllce cl::ntere:d :d I:h.4! orlgitl.
B:unieh 1lt1d '\bulBilrh [2] cllt1Sidered sr:oti~U~ itlvm;:mt
s.)Istern.s wL'ih l.mp.JlR R5oplln25 WhllR .support functlon.s
WC!l"C! tlllt1tloegativC! 1lt1d CORctrI·~ (lbe)' c.alLC!d 50Uch funcHon50
"futlneL cauw'} Tbe.y .shc<wc!d ll1..al such impulse reSplltlR-s'
WC!l"C! cGll.l'OildioRan.'t· do.s~d j ..e>... if h(.t.l) alld S(x. I) llJ1I: tWll
!f.ysi:e.rn.s with .supparl fUncHon f(.x.), lhea. Ute!ir cocvoJution
(/I .. ,g)(rJ) =
f r /l(x JkJk.
f, t - r)g(f. r)dTdf
is al~ SUl=PJr\.ed on j'(.\'). As 3 oonseque-oce of this doS1.l.J'C
property. the: set of spatialL}' in't11dant controllers with the
:spo;:ifie:d support function j'i.\·) can b-e n:panunet:ri.ud 85 a
coD't"c:x set in the You1.3. domai[l.
ThC!Lr r~u]t r.ll!:J'C!llrj:r; lln N"O ~ .B:5:sumrtillt1:r;:
• "The spatial COCJrdinaws
ha\'C to b-e one-dimemioml1,
• "!be :surrert, fUl'JCtion 0lU.st 'be Cllt1l:..II.'>'I!_
Usi[lg, our (Xls-et framework we are abLe: 1.0 gene:ralize the:se
n:sults. n. turns out that the ke}' property is 5ubm!d;/Mry
(LI!:. j(x] + ....':2) :::;: j(rJ) + j(~)) of the .surrort fUllru llll.
(Nllte '\hat in the I'R'"illU:r; :r;C!Ctilltl" we ~ ~ th.at the Qe]II.Y:O
:-ati:r;fy thc! triangle i lJ.e1:!.UIIlity . SubJU;lr.litivity llf the :r;uppert
fullrullll:r; i:r; the IlIl.WmL gl!:nemLi.%llIillll til '\hi:r; ~.) If the
:wpport function.s im·Il]\·.eQ. .sliti:;.[y :;'Ub-Ili:\ditivit)· then it i:;.
I1 I I
V_ CONI'fEC11IJN BE'nII'EE1'4 Q1.\lDJl.JLltC [Mo'AJU~ AN)1'lUI.TlAl.
OJU:E:R !iJ1I.1.UtIi: E
[ll thi:r; :r;C!Ctilltl WI!: ' ....mt to :r;turly the clltltll!:Ction bl!:N'ee'n
qu.adratk Ln"oui.omt"C! ;md [111setS. We! ho:w4! .seen th~t p=t
slru.cture: LmplLes that the. problem ~ quadr.:stic.alLy im'm1lt1t.. WI::
nil..... illtel"C!sted itl urrlers/:omdLng thl:: com·l:::rse .
LA:. "'dces. quadratk ill...m~ce Lmply C!:tL.stel'JCl:: llf [111RtIi I.L.I:: struc ture:?" Qu.ad.ratk LIlv.1I'i.anCC! is re:.al ~y 11 "!rom sLti\.l ty
pmpm-ty. A.s arg,uC!d C!MLiI:':r~ :J'C'5oC!bs. providl:' \:.be. rigbt J~I:'
lo de&crik crilt1sili,,1:' re:Lali=. In whal follll .....s, .w.e. ~ this.
=
clltltll:'Ction mlll"C! cllllcre.l:e. Clltltll:'Ction.s bl:'lWC!l:'n 'ldr.:slil:i
inn:rLallct! 1llld parr.rully ~.sIf!J .sInrcIure's 35 defined ill ;3.
beouc-th='lic 2l1.illg by Hll ~ Chu [:5] Ju\'15 been s.1.udk.:L
arid JX)inted out by Rotko ·itz 17]. The: team thron::tic probLe:m co ['IS iders. a s.oc:[lsrio lJ-en:, there are IJJULtipJe decision
IJUlkers .....bo must. c:ach make a decision in ~mc OIdI:r.. "The
paper oonsiders a .sc-enario whe:re the order in wbich decisions
are: Il12Ide satisfy certain p.m:edtmce Mlmion3. (Though this.
tc:nniooLog}' is not. us.:d in these: papers. these pR:CI:dc:nce
~]atilltl:r;
BI1:c in fllCt dll~Jy r~JEIte:o::'I to pllrti.lll orr,Il!:X re1.Atilltl:r;.) The- peper by He 15J ~'.s '\hat probLem:r; ......i'\h thi:;.
I'n!'-""-e<le II ~ :r;tructure (c a.!le<:l. p.ar1i aJ] y ne:r;terj prob lem:r;) BI1:c
.IlJOet1l1ble til Clltl\·.eJ: optimi'Z.ll.tillll. II.['Id mo=, '\hat llptimal
Clltltm]le:r.s .lire ]inea:r. Rotkow:itz ~-:o thIIt .eJ:i:r;tet1l:': llf the:r;e
975
WeAln5.7
pr~,dJ,JX:~ retal:LoJJJi is l5I:luivalmlt to quadratic lmmi..arl~.
Our results are similar in spirit:., in r~t PmpesHien :S is
!l!s.mlJt.~.LLy ~ecl:.aioed ilJ [1]. :He~;vM".. '~ pmvi.dJJ! .21. 6Ju!r
~bu~MriZ21tlen of qml~tl[; lrnrm:-iiU1~~ in berms of pcs~
:EI.IJ d quose rs,
Consider tbe prob letn 0 f Clesign ing an op Ii mal CD [l1:ID ller
,K c: S B.S described in I;XOble m 2. In tb ~ section we revis it
the mode L where (JeCCIJ tralizatlon oonstrain I:B arc viewe d 6
spar:;i ty constrain 1:; i [I the IXm tro Ll CT. Let ,K [IE fr" X fir- ,. Denee m ~b:n~:t ef ier:li~ ef eK viii, 9' !: 11..... ~ 1'iI,.] X [1 ~ ..... J'oII!l,.
Then the n'b.$pll~ 1:On.:*'lIBillt i:;. d~~o m; K~j' = I) for all
(F.. .}l IE
5· Qumdrmcc:
irl!..~~ ~r:lLl~:Ii to \h~ feLLo....."ing
[9. Tbenrem 215]:
\rBIJ:tiiti\~ property ill tbi.; ~j
'Th wnm ,;2: The subspace Sis quedratical. Ly i nvari8J] t with
respect to a apecified plant G if B11.d only' if fur all K. E S
:El.lJd all r.. j.1l.~ '"l
KucG;.:K.ull - ~) =
o.
(IS)
R.ema:rk
~ t LlS i [I terpr:e1 a:jLlati~ [I (I§) In ::m Intuitivc W'S}~.:Let
d.c"rDtc the cO[lstminl ,K"J -:j:. D b}~ i -4K j ('wbich (Jermt~
th.at th.cR: is 8 p.atb from i 1D j in the controller) a.:n.d G j: -:j:. 0
by j -4.:r 1 (Le. that there is a path, fmm j to Ir. i[l the pl::mt).,
Thee the: equation (6) :Ii~:Ii tlust:
lUii
•. """x ;~ j """G ~:t """t:. I :impL~s Ii' -ti,r d.
tiad. i::l,LJmdnnil: ie...-m:i1lJl.Ce re(jLlt"e:li to tnm:riti~, ["je:liLJre ef'
'lhi:li (i£Jen~ml) gmpb~ \Ve next ~. 'hmt ilJ t:h..i:li :K:~narie
qUIIrjrllti i:: inVl!rilm ~ ~e ~ penif:li to e'J:i:li tee ~ e f' pc ~ t :;'1nJ.ctLJre'.
oj Po~'s
Cb ["ISiCIer a p I.2IIJ t G lIJ]d a ~ ml tr.al i:z.ecl ~11 tre Lpm'b b!m wil.h
spar:;ity celJstmint.s K c: mJ.Cb thal fl ,) = 0 f'Or B.ll (ill J) E :T
fur ~rn.c i[ldel; cset :I.e 'Cen~deT the ~ case i.e. n, =
II...,. L ~ t N =
~
J'.rJ· \\~, :;.my t:h..at Jl Bi "'~["I r:I~ntmLize-::l.
.s
1] .....
~eet.n:lj pro'b~1J"J i:li plQII/-eOril~ml~· ~~d~?c if ~ gi\"oe'n
plmn t mJ:.o :;.mti:tifie:li !:he, :rpar~t:r ~Il t:s. e f' 1h.~ ~ e ct.n:lj ~r
a E S). Irl 'lhi:li ~t'L1pr eI1OC~' thlll i::tLJa1r'atil: i["l...·.ari~~
(i.~.
~qLl:l..-aLelJt to t.~: f.ad thai:r :Is triU15:itLv~Ly 1:1.o2d~ :I.~.
(~'I.. J) -E .T, (j: Il~ E
PropCI!Jrl;o~
~ e IJI:.m l
T . (k, f) e .T ~ (;, l) E T .
5: Ccn5ider
(8~
a pl.an~tontrolJM" SfD1.m!I!l:.ri~
pro bLem. Su ppo.m 1h~ fo IJ e wi IJg .as sLl mp I::i e n~ ~
tnJ.e;
of~l~~t:
1) (i. i) [~ .:J
2) Fer r:li.$licct F a.nd j
~ iecd~1lOe
ha\~, (i.
j)
~l~allf ilJ..-arian1..
~t ~ ever 1'iI,:r ~e~el:$ :liUdl that
.alge'bnL
.sr
ef~. (~caJ.L thar S i:li t.be $et ef
metric ~ that :Ii mC:li ~:Ii 'lh.e :rpar~t:r c nnstrai rlQi n f .:J.)
Pmof· S:lIJ~ 'both G and. K iiU'!: PiI, X PiI, matrices, :it I.s
erlJJLJS,h tel:OJl5tru.ct lL poset on .ftj" ~LelDl:I1ts and. .show- that
~ 5.pars Lty pi] rI!:m of SexiX:tly ~rxls to the :l1J dderx: I:
.al ge;bra 0 f th 1.5. pc sc:L J...!: t LJ S denCol:: n ur I:aJrlL:bte fer t.bI::
p.arl:i al n rdt: r ~ as f e LI.o-w5.: f ~ ; I.f ((, 11 ce: :J. Wr!, ,neI:d. t.e
VI! rify I:h.a t lb is is iIJ~
a pmi.i.al n rd.!!! r J:di]'[jn n,
SilJD!! (i. r) ., :1.. W~ ~I.urLy bi3Y~ ; ~~ ; th~ yuiIflng
~j...il:y. If r :!f j a.IJ[I, j :5 i I:l1.!I!IJ it must be, f:b:; [i~ that
(i. J) tI. :I lLrIn (J, f) f. :1. Hovrewr the ~ ttl nd lUi s LJ mp ti n n in
~ siatemml ef ~ propesitielJ e~~L~s th.e; ~biLity ef
.su& b ;, j bed IJg dis t~ l, th~ ; = j .2Ll1.d "Ne, haw, an li -.sy m.:rn.e; by.
a
Fi[lB.lly.. all~ae Wl; h8'YC i ~ j a.:n.d ] ::51 (i.~. (i~ JJ ~ .:1 and
(]. f) [~ 3). CbDO!ie ilJ~ t BllCb that t = j and Llse quaj.ralic
invllJi.BI::J~ to cDncL~ from Ci:jLlatiiJlJ (SJ I:haJ: (r.. f) ~ ::J. Th~
r ::51"l \-erifylng ~ti¥jty,.
The inc~,[1~ ~bra Df thia ]XIsct ~ I:.be cset of clcmc[ltB
~b I::h..IIt ,K~ct = 0 if' i '$ ;. (i.~. (i~ i) € 3) whiclt i:li ~~~tly
s.
•
~ d~finjlinl1
er
R Ex:rsmlct
oj· QttoaeIS
~erE:m ~~Derll1'i~eT~
to what -f:l;t-en1 ~ th.e ~ ccding
It tLJ~ eLJl that ~ ~mn in r~t l'e,ja.x
-.01
~ ~erl.d .B.:Ii:IiUmpcen (anti-:;.ymmetry). It i:li ~ib~' \0 bllVoe
Jl
mere' g~oeT.al rlJJ IinI1 e f A
pMtial
o.nj.er in t.be .IIb:li ~
rmt1.e1:rJ. !rl, tlust :Ii e-!I.tic g eli:;.ti IJd. ~,~~ ~ m1 be
~i::l,LJ i...met1t end !:he, pmtillL ord~ i.; defined e ["I 'lhe: qu.o ti.erJ t
or
1m ti-:ry
r
r
2t modulo thl: ~qLJL~e~~ Th.l: R5IJltlng o'bj~t I..s ~mlliu­
to ~, peRl (~i1I.lJ1:d. iii qLJetl.ent peRi er qrJDH~~ seJOe,Umes Lt Is.
calld. a p~.~~~~. :in the: Litou.aJ:url:)~ The~ I..s a co~speedlng
.al p raJ [" e b~~~t,
a.l8 ~'bra, calld.
aJ8f!brn []].
~;tKm 7: A ql:lla2i!" Q = (Q,::5) is a s~ Q 'with a
'binary Toe;] aUe ["I ~ SlJ rn I:h aI: ~ i S R~fl.ec:t i VI! and I:raru:i H~.
Thus it is p:iS&ibl..f fer dist~t ~l!!!merl}s r.. j Lu satlsry ; ~ j
lI.t'l.d. J ~ r (mil! wilL [ilLL SLJ ~h ~cZ:rtm1] is. ~'qLJ iVii!.l.ee t and. am~
tlti a by r.::: ),. Thi s neti 0 [I 0 f a i:jLl 0 set CIqXLl.res I:.bf~ in1JJ i t.i.on
that if i and j ~_an 00 mmuni~coBk: tJJ cac.h 0 t:h.c r :BJ] d if they
ha..-e th.: s8JI.1.C ].evel of i [lfOIIIlElt.i.on richnes s then t:h.cy are
~~qLl i-..llI.el11., One defines t:h.c anaLog1lC of arl inc iDence algcbra
as fulJeM.
DlifiI'iI i ~ Fo.re s: ~t :F ~ II ring JUld Q = (Q. ~) b ~ II
qu.o~t. ut the 1/~TiJC/~H"I5I~ mm.ri.x aJ,ebra J\1 ~ the ~t ef
fLJel:tlierl.:Ii f : Q u~ Q ...., l? with !:he, property 1h.mt J~{i..11 = [j ir
F t. ; for' aU F.. j.
w~ ]I:lI. y.e: :I t ~.s an oe~s1 .e:toen::I!Iie 1:0 th.1: r~a:3er te YoU Lf,y th.at
~ .s1~ d[j rn ~
lIlli1l~ to ~ Ln.i:: Lde rI~
017:1 m!rix
c
eM Ls ar1 iU5IJ~la.tLvl: alg~hR. Flgu.rI: 3 ~ an oeJ:i1Impll: ef
a 't~t MI.d. 'lh~ sp.ar~ty pii9.tbUtl ef I:he a.UiXw.e~d siructLlraL
r.rG!Itd~ aLg~b~. The lIJ]aleg~ ef Prepesitien S to q~~ is.
I::Im fo lJe wi IJ~.
Pmp" ~J~"o~ 6: ·CDn.Si dJ,r 21 P l.2Ie t- ~el1 bo IJ ~ s y ml:l1etri ~
centrol pmbk:m., Su~ t:h.c feLLCtWing DSLllJl];ltiOJUii ~ I:n:Ie
of th.e ilJOe:1;
'\IIi"e'
A
It is natumJ to aali.:
arourn
(6) I.s
i.:s.
(7)
The ~ti~ ~~~ N[;O~ men! ~nt lJew. lIIbat
quadrat:it; ic..-ari~ ~ saying ~ thai: I::Im ovmalJ graph ef
th~ ~I.o.ma kop (wh~b js ~emFds~ ef ~ ~mebi.:n.2lti.on ef
w'bgJ:Eq:.tl.s of the pLant mld the ron1roLl cr) ia tr'B.n!.i t ivcLy
cLo~ d., Th.e oond itic [I Jnr..B.I] s th.at if I L~ net :aL I.ctw"c d 1c
cel]]muni.cal:e to i in the oontrollcr th.CTI there m~t ~ist [l~1J
~th foom. I to r
th-c clesed. leep (bc~.L1se aLlch B, path,
\t"OULd pmdllCC 8 WRy fer ~ to rommLllJicatc to r by going
once MOUrn the cLoEed ]eep).
\~berl, the grIIpb i rI, i.d-e !:he, pl..arll JUld I:h..e, ~e ntrcil1er i:li id~ IJ-
A. E.xiai"mc~
3) Tbe prcbzm is
Then \h~ eJ:i:li~
t.3 ::;. (;~ i) f 3·
1) (i~ i)
976
~t:
f. 3
WeAln5.7
I
l
~
:
•1
1
Fl!.
2
:'!>
.
..
0
0
0 0
•
•
•
0
-t
VI. CCf:IIC:1.USleN
~]
We; p=n~d a peset based fJ"iUrU:'WOJ"k to shidy decenlralcnntml pcoblerns. We were able; b::o .e.-.:1Jmd our previous
....m en. deI:e;ncraliu:d centro] using posets b::o other s.eltiegs.
We; .showed. a W3y to interpret snroe ~l:'XisHng results
on s.ysl:6JnS wiUJ l:im.e del3ys using pcslrts. We; were abl.e
b::o e;xte;rr.l thme results tn Ute case of s.p3~i3lJy di.dribut~d.
iz~
sy:s,1rnl.:5.. We also studied the connection bet....oeen qusdratic
invarianee and posets aOO sho .....ed that they were equivalent
in certain settings, Und.c:r .50m~ll..hat.more general conditions•
.....e s.hO'Wc:d that. quadratic lnvariarree is equivalent to the
existence of quosetstructure.
A q lIl:Do:tlnd 11K' 1("Ud:,Y ("II'IcnI C r .111 ........,., lad :ob:ucb.....1 m:oId:t
". ' l.rxI..l:.ob:- ~ nan= "I"mcnl::<
~
:oI~. E.L:tn:nll ..... 111".
2) The problem is
qUlldrsticall~.
invariant.
Thee there exist:;. III qU02t Q O'\'I::I" Iff elements such thJIt oS
is the structural matrix .al~bTa of Q.
Again we construct a candidate quoset and
Proof
verify the lI:uo;::ill1e-:t prorertie-',a. \\':e.my thet i ::S j if (i. j) (!
The mLthor.s ......o uld like to thank a.n IU:lCeyr::J1OUS ref~ fer
.IlI1 in:r;igbtful mod dl::tIIiI~ re ...·ie ....·_
a. The verificatinn ef the propertie:r; me- very :r;imiJm te tha.t
of Prnpe:r;itien :5..
WI:: Le~
R~I:'l:S
thi:;. rnutine :r;ter tn the n=-jer. •
Il) M. Aldwrr. G. P. Bartee an::! M. wfrI. S1IUcrur.aL rI:lliIlx .aL:!}!bnn
lI:I!. 1hdr .1 m1c15 or 1rrnrtlln1 SlIbs,paoes. Unu, A.f.ytortl .:-i U,
We have thus seen that. condition cn from Proposition 5
can be relaxed, aOO that. in the relaxed setting quadratic
.Af.~fNlkNu. ~'U:2:'i-)! . :nJ5.
III 8. Bamieh andP. \C:rn.Js.uis. A ooma chuoi:Il:riuI:iaJ af distri:c.J..b! d
im'mWX:1:: i:;. equh"lll.eet to e-xi:r;tenl:e ef que:r;et, :r;trul:ture- in
the peoblern, Wha.t. lIIIppe-n:-. whc!n. crmdition (1) is reillxe-:t
(i.e. allnw cnnstraints lei;
0 fer :r;e= if? ","'I:: .IlI1:r;wer thi:;.
in the next prcpe:s.:iti.o.n.
rontrol prcl:i.lems in ,;p.U::i;:IJlJ inv.uiu sJslerm with COOlIITUni.-.nian
ronslnlin1s. SY.Ilml" b corumJ" UJ't'J".1, 541:6):575-5&3. ms.
13] v. D. 8h:ndd m d. J. N. Tsi.tsii:J is. .~ mrn:y cl a:n::pi.t:JJ:ioiil
~i~ l!!.sults in systerm md . cnntml, ...III = 1 L""I2.. 3ril9): l1oW1214. Zf'.UJ.
14] S. P_ 8l:1)'d and C. H. &r.1tt.. Lmt'iU Cl1fll1rXJzr Dn~' Lmlll's of
=
a.[)
a.
Defmitw,1 9: m,,-c:c
'NC. C1IIJ 3" = .:J \ U
the
or.:r. This. is simply tbe operation of adding
the ~fiexi,,"(: relation to the set
which lJUIy oot a priori
satisfy re(jellivit}·.
1?fle.1.i~"e ClO5U1?
Pnf"rTD:JP>--' , J1..... tico: H.II. 1'}9] .
1:1] Y .:C . He ..",I X ...c. Chu
:r
16]
W.e will SO]y that the .se~ T p::o!I5= quo~I .strolU:rr.~ if
the; I:oJ1:.-:l:ien of relatioru (,I,J) e
utisfy the; axiems ef a
::r
19M.
1.7) M. Rctkawlu:
a. I n ~ :ttIu,,1un:::L "anu:"lty." .and I.b::aapllm:olll:J. ]n P~~ "J rlr.- 4-;No lEEE C""J....... ".... "" D«u,'""
un.:! C"n!'r<:'I. 200&.
1.1I) M.Ratlo::n>l.t" IUId R. C'c,!.JU. Cam"" ~Uh.:ol1: rE dI:olrlbuL-d t:t:IDIn:IlIcn Iar :tp:olic-b::mJ1t""l ~'Z'I<::m:t ... 1ib :tulu.ddltl.~ :oul'p::ut fun<:tlanl.
[n .f<:o~.5"U1r. ....nnlL.li A""",""""" C"nJ..;rJOt:.- "" C~uni~"n, C.::w-
qu.o.s.c:t. i-e=_
1) (r. r) E:r
2) (r. j) Eat mn.d (j):)E:r implie:r; (r.k) Ear.
propos-mOR 7: Su~ we haw. a pbnt-ceni:rcll«
s.yJnJDetrk coe~l preblem ....·i1h 'I ~ified. index .se~ (ef
spom;i~ r:onscraiels) :J. (The; s.p;ani~ coescraiels an:: Utus.
Kit 0 for J) e3.) The problem is qudraHcal])· jn"'Mbl1~
jf and. eely if
bas. a qu.o.se~ s~r::;hm:.
=
cr.
r:iy
first note that taking rr;B~xlv~ des.ure of a
'Ir.1esil:ivdy do.sed. .se~ does net 3ff.ed ;my ef ~be tl'il3Hons.
'behv.een di.s.tieI:t de;meels. D58ne;T
T . Define ~ .:!S j if
0. j) e .T. Suppue; WI:' 3dd. Ute r.eRexi..-e rd~ees .so that
I U (U.i.:N ~(;• .1)1). Cons..idm- the tr.msiti..-e desure ef r.
The only W'ly .new ml3l:iens. c;m 'be ~ is by cembining:
Prooj:·
~
=
r=
tnmsitive rcl8tions with. the: ne:....·l}" l"dkd :reflel;i,,-e :n:]atiocs.
Thus if ; ;:S j and j .:5 x.we kcow that for distinct i.
Vt-c:
:alm:rdy bave i : : ; x. If j =; or j =" we gc:l co cew :n:Jatiocs.
H~ncc: r is its own tmrlsitive c]osure.
Suppose the Teflexive closure: is .a quoset. lAA:. know that
in the do5UR: O]Xl'8tion. 00 ~w re:latiom bet....·ec:c disticct
j."
TIUDt1 rla:ilia1 ihocry ~inretmnian
in crt:imal rcnlr'lll rrebl.ml-pu1 I, lEEE Trurm:r<"li"""" "'1'1
~ C<;y1oln,( 1'7( I ):] :5-22, ]97;2
a .-C', lU:rto. On the rau~ cr =mbirul:ctial tl::-uy ]. Thll::ry cr
Miibiul Iuntticm, P~liJy Jr...oOV}'"",,' '~i...lji..l.#. 2(~):~oID-~6B .
IInI:t1lt'a
rmr.
un.:I C~~.»J&.
I.~ M .RaikDv.1to': an::! S. l...aI.L
A dllEllCl!!II:uI:Ian tf caR\U ,pmI:(.!!ms
In deceni:r'a.l!:zl!d cenirDl. IEEE
T~n"",,:r "n
A.JN.:.rNfk c""rr.,/.
:'i1(2):2.74-2Btl. 2DDtl.
IlD) f!. shlb .and P. A.PlI:t1lo. A panllII tetIe!" 1IJ1p.n::a:h ixJ deceni:r'a.l!:zl!d
ronlrul. In F=-.·....1l».Y "'irk.. 01]<A 'EEE Gmf..;rn.:.. "" n,~Wi:m .:-i
c""rr.,/. 2.OOB.
[ll) f!. shlb .and P. A.PlI:t1lo. A JlUilIII tetIe!" 1IJ1p.n::a:h ixJ deceni:r'a.l!:zl!d
rontrol l:l ,;plEiillJ .iImri:uT1
S)'ltertll.
In
fitm-~,'"
Annllll/ .../luI.."
CGlIfi.'.mIl.-oZ' .." c.."lIIII./oIJiroI'm. CMiIri. ::.wI clJIRf1lit~!- 2CI:G.
Ill] E. Sp~1 n:I C. 1. CJ Dca:aell. ~ .~. ~I CIekku.
11.m.
I l3]
I lot]
ele=t:r; .....-ere in'\roo::hl.ceo::l. heeD: 1nm:r;iti,,'ity is una.ff'el:ter!.
:8)' (B) the probl.erni:;. Cjua.drAI::kBliy im:mllll]t. Co~jy. if
thl:: problem i:r; qumr!rIItil:..lllly iOVllJ"ia.n1,. WI:: :know from (8) tha.t
Ii:;. tnuu:itively d~. Thu:r; if we !:Ilk the! reBexho.e de:r;ure.
by Prore:r;itiee 6 the! ~:r;ulting .set i:r; II q~t.
•
977
P.~.
CentroI of ee.strd ~
~ CommJ" cGlIfi.'.rzncoZ', 2DDD.
VlIul~lris.. A calMl: ~ri2tian of
r..
In of'nxl'l"a'm;;r of ,hE
r.l.zsses af
~ms
rcnlr'lll ... i!b ~cili", im.r.::tit:ol :anrl rammunit::ltit:ol :ttIu"turu .
Prr=udinS:r
af fir.-
~.~
C"',ald C"Pj"..;rp...-"
DJ] ,
in
£n