The Necessity and Sufficiency of Anytime Capacity for Stabilization of a Linear System Over a ...

27 Pages·2006·0.69 MB·English

IEEETRANSACTIONSONINFORMATIONTHEORY,VOL.52,NO.8,AUGUST2006 3369
The Necessity and Sufficiency of Anytime Capacity
for Stabilization of a Linear System Over a Noisy
Communication Link—Part I: Scalar Systems
AnantSahai, Member, IEEE, and Sanjoy Mitter,Fellow, IEEE
Abstract—In this paper, we review how Shannon’s classical concerned, all that matters is the channel capacity and the na-
notion of capacity is not enoughto characterize a noisy commu- tureofthesource.Givenenoughtoleranceforend-to-enddelay,
nicationchannelifthechannelisintendedtobeusedaspartofa
the source can be encoded into bits and those bits can be reli-
feedbacklooptostabilizeanunstablescalarlinearsystem.While
ablytransportedacrossthenoisychanneliftherateislessthan
classical capacity is not enough, another sense of capacity (pa-
rametrized by reliability) called “anytime capacity” is necessary the Shannon capacity. As long as the source, distortion, and
for the stabilization of an unstable process. The required rate channel are well-behaved [1], [2], there is asymptotically no
is given by the log of the unstable system gain and the required loss in separating the problems of source and channel coding.
reliability comes from the sense of stability desired. A conse-
Thisprovidesajustificationforthelayeredarchitecturethatlets
quenceofthisnecessityresultisasequentialgeneralizationofthe
engineers isolate the problem of reliable communication from
Schalkwijk–Kailathschemeforcommunicationovertheadditive
white Gaussian noise (AWGN) channel with feedback. In cases that of using the communicated information. Recent advances
ofsufficientlyrichinformationpatternsbetweentheencoderand incodingtheoryhavealsomadeitpossibletoapproachtheca-
decoder,adequateanytimecapacityisalsoshowntobesufficient pacityboundsverycloselyinpracticalsystems.
fortheretoexistastabilizingcontroller.Thesesufficiencyresults
In order to extend our understanding of communication to
are then generalized to cases with noisy observations, delayed
interactive settings, it is essential to have some model for in-
controlactions,andwithoutanyexplicitfeedbackbetweentheob-
serverandthecontroller.Bothnecessaryandsufficientconditions teraction. Schulman and others have studied interaction in the
are extended to continuous time systems as well. We close with contextofdistributedcomputation[3],[4].Theinteractionthere
commentsdiscussingahierarchyofdifficultyforcommunication isbetweencomputationalagentsthathaveaccesstosomepri-
problems and how these results establish where stabilization
vate data and wish to perform a global computation in a dis-
problemssitinthathierarchy.
tributedway.Thecomputationalagentscanonlycommunicate
Index Terms—Anytime decoding, control over noisy channels, witheachotherthroughnoisychannels.InSchulman’sformula-
errorexponents,feedback,real-timeinformationtheory,reliability
tion,capacityisnotaquestionofmajorinterestsinceconstant
functions,sequentialcoding.
factor slowdowns are considered acceptable.1 Fundamentally,
thisisaconsequenceofbeingabletodesignallthesystemdy-
namics.Therichfieldofautomaticcontrolprovidesaninterac-
I. INTRODUCTION
tivecontexttostudy capacityrequirements sincethe plantdy-
FORcommunicationtheorists,Shannon’sclassicalchannel
namicsaregiven,ratherthansomethingthatcanbedesigned.In
capacity theorems are not just beautiful mathematical re-
control, we consider interaction between an observer that gets
sults, they are useful in practice as well. They let us summa-
toseetheplantandacontrollerthatgetstocontrolit.Thesetwo
rize a diverse range of channels by a single figure of merit:
canbeconnectedbyanoisychannel.
thecapacity.Formostnoninteractivepoint-to-pointcommuni-
Shannonhimselfhadsuggestedlookingtocontrolproblems
cationapplications,theShannoncapacityofachannelprovides
formoreinsightintoreliablecommunication[5].
anupperboundonperformanceintermsofend-to-enddistor-
tionthroughthedistortion-ratefunction.Asfarasdistortionis “ can be pursued further and is related to a duality
between past and future2 and the notions of control and
knowledge.Thuswemayhaveknowledgeofthepastand
ManuscriptreceivedMay2,2005;revisedJanuary3,2006.TheworkofS.
cannot control it; we may control the future but have no
K.MitterwasprovidedbytheArmyResearchOfficeundertheMURIGrant:
DataFusioninLargeArraysofMicrosensorsDAAD19-00-1-0466andtheDe- knowledgeofit.”
partmentofDefenseMURIGrant:ComplexAdaptiveNetworksforCooper-
ativeControlSubaward#03-132andtheNationalScienceFoundationGrant Wearefarfromthefirsttoattempttobringtogetherinforma-
CCR-0325774.ThematerialinthispaperwaspresentedinpartattheIEEECon- tionandcontroltheory.In[7],Ho,Kastner,andWongdrewout
ferenceonDecisionandControl(CDC),ParadiseIsland,Bahamas,December,
a detailed diagram in which they summarized the then known
2004.
A.SahaiiswiththeDepartmentofElectricalEngineeringandComputerSci- relationships among team theory, signaling, and information
ence,UniversityofCalifornia,Berkeley,CA94720USA(e-mail:sahai@eecs. theory from the perspective of distributed control. Rather
berkeley.edu).
than taking such a broad perspective, we instead ask whether
S.MitteriswiththeDepartmentofElectricalEngineeringandComputerSci-
ence,theMassachusettsInstituteofTechnology,Cambridge,MA02139USA
(e-mail:[email protected]). 1Furthermore,suchconstantfactorslowdownsappeartobeunavoidablewhen
CommunicatedbyY.Steinberg,AssociateEditorforShannonTheory. facingtheverygeneralclassofinteractivecomputationalproblems.
DigitalObjectIdentifier10.1109/TIT.2006.878169 2Thedifferingrolesofthepastandfuturearemadeclearin[6].
0018-9448/$20.00©2006IEEE
3370 IEEETRANSACTIONSONINFORMATIONTHEORY,VOL.52,NO.8,AUGUST2006
Shannon’sclassicalcapacityistheappropriatecharacterization two qualitatively distinct types of information when it comes
for communication channels arising in distributed control to transport over a noisy channel. In addition to the classical
systems. Our interest is in understanding the fundamental Shannontypeofinformationfoundintraditionalrate-distortion
relationshipbetweenproblemsofstabilizationandproblemsof settings,6thereisanessentialcoreofinformationthatcaptures
communication. theunstablenatureofthesource.WhileclassicalShannonrelia-
Tatikonda’srecentworkonsequentialratedistortiontheory bilitysufficesfortheclassicalinformation,thisunstablecorere-
provides an information-theoretic lower-bound on the achiev- quiresanytimereliabilityfortransportacrossanoisychannel.7
able performanceofa controlsystem overachannel. Because As also discussed in this paper, anytime reliability is a sense
this bound is sometimes infinite, it also implies that there is a of reliable transmission that lies between Shannon’s classical
fundamentalrateofinformationproduction,namelythesumof -sense of reliable transmission and his zero-error reliability
thelogsoftheunstableeigenvaluesoftheplant,thatisinvari- [25]. In [23], we also review how the sense of anytime relia-
antly attached to an unstable linear discrete-time process [8], bility is linked to classical work on sequential tree codes with
[9].Thisparticularnotionofratewasjustifiedbyshowinghow boundeddelaydecoding.8
to stabilize the system over a noiseless feedback link with ca- Thenewfeatureincontrolsystemsistheiressentialinterac-
pacity greater than the intrinsic rate for the unstable process.3 tivity. The information to be communicated is not a message
Nair et al. extended this to cover the case of unbounded dis- knowninadvancethatisusedbysomecompletelyseparateen-
turbancesandobservationnoiseundersuitableconditions[10], tity. Rather, it evolves through time and is used to control the
[11].Inadditiontonoiselesschannels,theresultswereextended veryprocessbeingencoded.Thisintroducestwointerestingis-
foralmost-surestabilizationinthecontextofundisturbed4con- sues.First,causalityisstrictlyenforced.Theencoderandcon-
trol systems with bounded initial conditions being stabilized trollermustactinrealtimeandsotakingthelimitoflargedelays
overcertainnoisychannels[12]. must be interpreted very carefully. Second, it is unclear what
We had previously showed that it is possible to stabilize thestatusofthecontrolledprocessis.Ifthecontrollersucceeds
persistently disturbed controlled Gauss–Markov processes instabilizingtheprocess,itisnolongerunstable.Asexplored
oversuitablepower-constrainedadditivewhiteGaussiannoise inSectionII-D,apurelyexternalnoninteractiveobservercould
(AWGN) channels [13], [14] where it turns out that Shannon treatthequestionofencodingthecontrolledclosed-loopsystem
capacity is tight and linear observers and controllers are suf- stateusingclassicaltoolsfortheencodingandcommunication
ficient to achieve stabilization [15]. In contrast, we showed ofastationary ergodicprocess.Despitehaving toobserveand
thattheShannoncapacityofthebinaryerasurechannel(BEC) encodetheexactsameclosed-loopprocess,theobserverinternal
is not sufficient to check stabilizability and introduced the tothecontrolsystemrequiresachannelasgoodasthatrequired
anytimecapacityasacandidatefigureofmerit[16].Following tocommunicatetheunstableopen-loopprocess.Thisseemingly
uponourtreatmentoftheBECcase,Martinsetal.havestudied paradoxicalsituationillustrateswhatcanhappenwhentheen-
more general erasure-type models and have also incorporated codingofinformationanditsusearecoupledtogetherbyinter-
bounded model uncertainty in the plant [17]. There is also activity.
relatedworkbyEliathatusesideasfromrobustcontroltodeal In this paper (Part I), the basic equivalence between feed-
with communication uncertainty in a mixed continuous/dis- back stabilization and reliable communication is established.
crete context, but restricting to linear operations [18], [19]. Thescalarproblem(Fig.2)isformallyintroducedinSectionII
Basar and his students have also considered such problems where classical capacity concepts are also shown to be inade-
and have studied the impact of a noisy channels on both the quate. In Section III, it is shown that adequate feedback any-
observations and the controls [20]. The area of control with time capacity is necessary for there to exist an observer/con-
communications constraints continues to attract attention and trollerpairabletostabilizetheunstablesystemacrossthenoisy
the reader is directed to the recent September 2004 issue of channel.Thisconnectionisalsousedtogiveasequentialany-
IEEETRANSACTIONSONAUTOMATICCONTROLandthearticles timeversionoftheSchalkwijk–KailathschemefortheAWGN
thereinforamorecomprehensivesurvey. channelwithnoiselessfeedback.
Manyoftheissuesthatariseinthecontrolcontextalsoarise Section IV shows the sufficiency of feedback anytime ca-
for the conceptually simpler problem of merely estimating an pacityforsituationswheretheobserverhasnoiselessaccessto
unstable open-loop process,5 across a noisy channel. For this thechanneloutputs.InSectionV,thesesufficiencyresultsare
estimation problem in the limit of large, but finite, end-to-end generalized to the case where the observer only has noisy ac-
delays, we have proved a source coding theorem that shows cess to the plant state. Since the necessary and sufficient con-
that the distortion-rate bound is achievable. Furthermore, it is ditions are tight in many cases, these results show the asymp-
possible to characterize the information being produced by an totic equivalence between the problem of control with “noisy
unstableprocess[23].Itturnsoutthatsuchprocessesproduce
6In[23],weshowhowtheclassicalpartoftheinformationdeterminesthe
3Thesequentialrate-distortionboundisgenerallynotattainedevenathigher shapeoftherate-distortioncurve,whiletheunstablecoreisresponsiblefora
ratesexceptinthecaseofperfectlymatchedchannels. shiftofthiscurvealongtherateaxis.
4Inseminalwork[12],thereisnopersistentdisturbanceactingontheunstable 7Howtocommunicatesuchunstableprocessesovernoisychannelshadbeen
plant. an open problem since Berger had first developed a source-coding theorem
fortheWienerprocess[24].Bergerhadconjecturedthatitwasimpossibleto
5The unstable open-loop processes discussed here are first-order nonsta-
transportsuchprocessesovergenericnoisychannelswithasymptoticallyfinite
tionaryautoregressiveprocesses[21],ofwhichanimportantspecialcaseisthe
end-to-enddistortionusingtraditionalmeans.
WienerprocessconsideredbyBerger[22].
8Reference[26]raisedthepossibilityofsuchaconnectionearlyon.
SAHAIANDMITTER:ANYTIMECAPACITYFORSTABILIZATIONOVERANOISYCOMMUNICATIONLINK—PARTI 3371
Fig.1. The“equivalence”betweenstabilizationovernoisyfeedbackchannels
andreliablecommunicationovernoisychannelswithfeedbackisthemainresult
establishedinthispaper.
feedback” and the problem of reliable sequential communica-
tion with noiseless feedback (Fig. 1). In Section VI, these re-
sultsarefurtherextendedtothecontinuoustimesetting.Finally,
SectionVIIjustifieswhytheproblemofstabilizationofanun- Fig. 2. Control over a noisy communication channel. The unstable scalar
stablelinearcontrolsystemis“universal”inthesamesensethat systemispersistentlydisturbedbyW andmustbekeptstableinclosed-loop
throughtheactionsofO;C.
the Shannon formulation of reliable transmission of messages
over a noisy channel with (or without) feedback is universal.
Thisisdonebyintroducingahierarchyofcommunicationprob-
A. TheControlProblem
lemsinwhichproblemsatagivenlevelareequivalenttoeach
otherintermsofwhichchannelsaregoodenoughtosolvethem.
Problems high in the hierarchy are fundamentally more chal-
(1)
lengingthantheonesbelowthemintermsofwhattheyrequire
fromthenoisychannel.
where is a -valued state process. is a -valued
In Part II, the necessity and sufficiency results are general-
control process and is a bounded noise/disturbance
izedtothecaseofmultivariablecontrolsystemsonanunstable
process s.t. . This bound is assumed to hold with
eigenvaluebyeigenvaluebasis.Theroleofanytimecapacityis
certainty. For convenience, we also assume a known initial
playedbyarateregioncorrespondingtoavectorofanytimere-
condition .
liabilities.Ifthereisnoexplicitchanneloutputfeedback,thein-
Tomakethingsinteresting,consider sotheopen-loop
trinsicdelayofthecontrolsystem’sinput–outputbehaviorplays
systemisexponentiallyunstable. Thedistributednature ofthe
animportantrole.Itshowsthattwosystemswiththesameun-
problem (shown in Fig. 2) comes from having a noisy com-
stableeigenvaluescanstillhavepotentiallydifferentchannelre-
municationchannelinthefeedbackpath.Theobserver/encoder
quirements.Theseresultsestablishthatininteractivesettings,a
system observes andgenerates inputs tothechannel.
single“application”canfundamentallyrequiredifferentsenses
Itmayormaynothaveaccesstothecontrolsignals orpast
of reliability for its data streams. No single number can ade-
channeloutputs aswell.Thedecoder/controller9system
quatelysummarizethechannelandanylayeredcommunication
observeschannel outputs and generates control signals .
architecture should allow applications toadjust reliabilities on
Both are allowed to have unbounded memory and to be
bitstreams.
nonlinearingeneral.
There are many results in this paper. In order not to burden
Definition2.1: Aclosed-loopdynamicsystemwithstate
thereaderwithrepetitivedetailsandunnecessarilylengthenthis
is -stableif forall .
paper,wehaveadoptedadiscursivestyleinsomeoftheproofs.
Thisdefinitionrequirestheprobabilityofalargestatevalue
Thereadershouldnothaveanydifficultyinfillingintheomitted
tobeappropriatelybounded.Aloosersenseofstabilityisgiven
details.
bythefollowing.
Definition2.2: Aclosed-loopdynamicsystemwithstate
is -stableifthereexistsaconstant s.t. forall
II. PROBLEMDEFINITIONANDBASICCHALLENGES
.
In both definitions, the bound is required to hold for all
Section II-A formally introduces the control problem of
possible sequences of bounded disturbances that satisfy
stabilizing an unstable scalar linear system driven by both
the given bound . We do not assume any specific proba-
a control signal and a bounded disturbance. In Section II-B,
bilitymodelgoverningthedisturbances.Ratherthanhavingto
classicalnotionsofcapacityarereviewedalongwithhowtosta-
specify a specific target for the tail probability , holding the
bilizeanunstablesystemwithafiniteratenoiselesschannel.In
-moment within bounds is a way of keeping large deviations
SectionII-C,itisshownbyexamplethattheclassicalconcepts
rare. The larger is, the more strongly very large deviations
areinadequatewhenitcomestoevaluatinganoisychannelfor
control purposes. Shannon’s regular capacity is too optimistic are penalized. The advantage of -stability is that it allows
constant factors to be ignored while making sharp asymptotic
andzero-errorcapacityistoopessimistic.Finally,SectionII-D
showsthatthecoreissueofinteractivityisdifferentthanmerely
9Becausethedecoderandcontrollerarebothonthesamesideofthecommu-
requiringtheencodersanddecoderstobedelay-free. nicationchannel,theycanbelumpedtogetherintoasinglebox.
3372 IEEETRANSACTIONSONINFORMATIONTHEORY,VOL.52,NO.8,AUGUST2006
statements. Furthermore, Section III-C shows that for generic exactlyzerowithsufficientlylarge .Itdoesnothaveasimple
DMCs,nosensestrongerthan -stabilityisfeasible. single-lettercharacterization[25].
Thegoalinthispaperistofindnecessaryandsufficientcon- Example 2.1: Consider a system (1) with and
ditions on the noisy channel for there to exist an observer . Suppose that the memoryless communication channel is a
andcontroller sothattheclosedloopsystemshowninFig.2 noiseless one bit channel. So and
is stable in the sense of definitions 2.1 or 2.2. The problem is while
considered under different information patterns corresponding .Thischannelhas
todifferentassumptionsaboutwhatinformationisavailableat .
the observer . The controller is always assumed to just have Useamemorylessobserver
accesstotheentirepasthistory10ofchanneloutputs.
For discrete-time linear systems, the intrinsic rate of infor-
if
mationproduction(inunitsofbitspertime)equalsthesumof
if
the logarithms (base ) of the unstable eigenvalues [9]. In the
scalarcasestudiedhere,thisisjust .Thismeansthatitis andmemorylesscontroller
generically11 impossible to stabilize the system in any reason-
ablesenseifthefeedbackchannel’sShannonclassicalcapacity
if
.
if
B. ClassicalNotionsofChannelsandCapacity Assumethattheclosedloopsystemstateiswithintheinterval
Definition 2.3: A discrete time channel is a probabilistic .Ifitispositive,thenitisintheinterval .Atthe
system with an input. At every time step , it takes an input nexttime, wouldbeintheinterval .Theap-
and produces an output with probability12 plied control of shifts the state back to within the interval
where the notation is shorthand for the se- . The same argument holds by symmetry on the neg-
quence .Ingeneral,thecurrentchanneloutputis ative side. Since it starts at 0, by induction it will stay within
allowedtodependonallinputssofaraswellasonpastoutputs. forever. As a consequence, the second moment will
Thechannelismemorylessifconditionedon , isinde- staylessthan4foralltime,andalltheothermomentswillbe
pendentofanyotherrandomvariableinthesystemthatoccurs similarlybounded.
attime orearlier.Allthatneedstobespecifiedis . In addition to the Shannon and zero-error senses of relia-
The maximum rate achievable for a given sense of reliable bility,informationtheoryhasvariousreliabilityfunctions.Such
communication is called the associated capacity. Shannon’s reliability functions (or error exponents) are traditionally con-
classicalreliabilityrequiresthatafterasuitablylargeend-to-end sideredaninternalmatterforchannelcodingandwereviewed
delay13 that the average probability of error on each bit is asmathematicallytractableproxiesfortheissueofimplemen-
belowa specified .Shannonclassicalcapacity canalsobe tation complexity [1]. Reliability functions study how fast the
calculated in the case of memoryless channels by solving an probabilityoferrorgoestozeroastherelevantsystemparam-
optimizationproblem eterisincreased.Thus,thereliabilityfunctionsforblock-codes
aregiveninterms oftheblock length,reliability functionsfor
convolutionalcodesintermsoftheconstraintlength[27],and
reliability functions for variable-length codes in terms of the
expected block length [28]. With the rise of sparse code con-
where the maximization is over the input probability distribu- structionsanditerativedecoding,theprominenceoferrorexpo-
tionand representsthemutualinformationthroughthe nentsinchannelcodinghasdiminishedsincethecomputational
channel[1].Thisisreferredtoasasinglelettercharacterization burdenisnotsuperlinearintheblock-length.
ofchannelcapacityformemorylesschannels.Similarformulae For memoryless channels, the presence or absence of feed-
exist using limits in cases of channels with memory. There is back does not alter the classical Shannon capacity [1]. More
anothersenseofreliabilityanditsassociatedcapacity called surprisingly,forsymmetricDMC’s,thefixedblockcodingreli-
zero-errorcapacitywhichrequirestheprobabilityoferrortobe abilityfunctionsalsodonotchangewithfeedback,atleastinthe
highrateregime[29].Fromacontrolperspective,thisisthefirst
10InSectionIII.C.3,itisshownthatanythinglessthanthatcannotworkin indicationthatneitherShannon’scapacitynorblock-codingre-
general.
liabilityfunctionsaretheperfectfitforcontrolapplications.
11Therearepathologicalcaseswhereitispossibletostabilizeasystemwith
lessrate.Theseoccurwhenthedrivingdisturbanceisparticularlystructured
insteadofjustbeingunknownbutbounded.Anexampleiswhenthedisturbance C. Counterexample Showing Classical Concepts are
onlytakesonvalues(cid:6)1while(cid:21) = 4.Clearlyonlyonebitperunittimeis Inadequate
requiredeventhoughlog (cid:21)=2.
We use erasure channels to construct a counterexample
12ThisisaprobabilitymassfunctioninthecaseofdiscretealphabetsB,but
ismoregenerallyanappropriateprobabilitymeasureovertheoutputalphabet showing the inadequacy of the Shannon classical capacity in
B. characterizing channels for control. While both erasure and
13Traditionally,thecommunityhasusedblock-lengthforablockcodeasthe AWGNchannelsareeasytodealwith,itturnsoutthatAWGN
fundamentalquantityratherthandelay.Itiseasytoseethatdoingencodingand
decodinginblocksofsizencorrespondstoadelayofbetweennand2nonthe channelscannotbeusedforacounterexamplesincetheycanbe
individualbitsbeingcommunicated. treatedintheclassicalLQGframework[15].Thedeeperreason
SAHAIANDMITTER:ANYTIMECAPACITYFORSTABILIZATIONOVERANOISYCOMMUNICATIONLINK—PARTI 3373
for why AWGN channels do not provide a counterexample is
giveninSectionIII-C4.
1) Erasure Channels: The packet erasure channel models
situationswhereerrorscanbereliablydetectedatthereceiver.
In the model, sometimes the packet being sent does not make
itthroughwithprobability ,butotherwiseitmakesitthrough
correctly.Explicitly,Definition2.4.
Definition2.4: The -bitpacketerasurechannelisamem-
oryless channel with and
while .
It is well known that the Shannon capacity of the packet
erasure channel is bits per channel use regardless
of whether the encoder has feedback or not [1]. Furthermore,
because a long string of erasures is always possible, the zero-
errorcapacity ofthischannelis .Therearealsovariable-
Fig.3. Thecontrolsystemwithanadditionalpassivejointsource–channelen-
lengthpacketerasurechannelswherethepacket-lengthissome-
coderE watchingtheclosedloopstateX andcommunicatingittoapassive
thingtheencodercanchoose.See[30]foradiscussionofsuch estimatorD .ThecontrollerCimplicitlyneedsagoodcausalestimateforX
channels. andthepassiveestimatorD explicitlyneedsthesamething.Whichrequires
thebetterchannel?
To construct a simple counterexample, consider a further
abstraction.
Definition 2.5: The real packet erasure channel has
Noticethattherootoftheproblemisthat .In-
and while .
tuitively,thesystemisexplodingfasterthanthenoisychannel
ThismodelhasalsobeenexploredinthecontextofKalman
isabletogivereliability.Thiscausesthesecondmomenttodi-
filteringwithlossyobservations[31],[32].Ithasinfiniteclas- verge.Incontrast,thefirstmoment isboundedforall
sical capacity since a single real number can carry arbitrarily since .
manybitswithinitsbinaryexpansion,whilethezero-errorca- The adequacy of the channel depends on which moment is
pacityremains . required to be bounded. Thus no single-number characteriza-
2) The Inadequacy of Shannon Capacity: Consider the tionlikeclassicalcapacitycangivethefigure-of-meritneeded
problem from Example 2.1, except over the real erasure toevaluateachannelforcontrolapplications.
channel instead of the one bit noiseless channel. The goal is
for the secondmoment tobe bounded and recall that D. NoninteractiveObservationofaClosed-LoopProcess
. Let so that there is a 50% chance of any real ConsiderthesystemshowninFig.3.Inthis,thereisanad-
numberbeingerased.Assumetheboundeddisturbance ,as- ditionalpassivejointsource–channelencoder watchingthe
sumethatitiszero-meanandiidwithvariance .Byassuming closedloopstate andcommunicatingittoapassiveestimator
an explicit probability model for the disturbance, the problem throughasecondindependentnoisychannel.Boththepas-
is only made easier as compared to the arbitrarily-varying but siveand internalobservershaveaccesstothe same plantstate
boundedmodelintroducedearlier. andwecanalsorequirethepassiveencoderanddecodertobe
Inthiscase,theoptimalcontrolisobvious—set as causal—noend-to-enddelayispermitted.Atfirstglance,itcer-
thechannelinputanduse asthecontrol.Withevery tainlyappearsthatthecommunicationsituationsaresymmetric.
successfulreception,thesystemstateisresettotheinitialcon- Ifanything,theinternalobserverisbetteroffsinceitalsohasac-
cesstothecontrolsignalswhilethepassiveobserverisdenied
ditionofzero.Foranarbitrarytime ,thetimesinceitwaslast
accesstothem.
reset is distributed like a geometric- random variable. Thus,
Suppose that the closed-loop process (1) had already been
thesecondmomentis
stabilizedbytheobserverandcontrollersystemof2.1,sothat
thesecondmoment forall .Supposethatthenoisy
channelfacingthepassiveencoderisthereal -erasurechannel
oftheprevioussection.Itisinterestingtoconsiderhowwellthe
passiveobserverdoesatestimatingthisprocess.
The optimal encoding rule is clear, set . It is
certainly feasible to use itself as the estimator for
the process. This passive observation system clearly achieves
since the probability of nonera-
sure is . The causal decoding rule is able to achieve a finite
end-to-end squared error distortion over this noisy channel in
acausalandmemorylessway.
Thisexamplemakesitclearthatthechallengehereisarising
frominteractivity,notsimplybeingforcedtobedelay-free.The
passive external encoder and decoder do not have to face the
Thisdivergesas since . unstable nature of the source while the internal observer and
3374 IEEETRANSACTIONSONINFORMATIONTHEORY,VOL.52,NO.8,AUGUST2006
controllerdo.Anerrormadewhileestimating bythepassive
decoder has no consequence for the next state while a
similarerrorbythecontrollerdoes.
III. ANYTIMECAPACITYANDITSNECESSITY
Anytimereliabilityisintroducedandrelatedtoclassicalno-
tions ofreliability in[23]. Here,the focusis onthe maximum
rate achievable for a given sense of reliability rather than the
maximum reliability possible at a given rate. The two are of
courserelatedsincefundamentallythereisanunderlyingregion
offeasiblerate/reliabilitypairs.
Sincetheopen-loopsystemstatehasthepotentialtogrowex-
ponentially,thecontroller’sknowledgeofthepastmustbecome
certainatafastrateinordertopreventabaddecisionmadein
thepastfromcontinuingtocorruptthefuture.Whenviewedin
thecontextofreliablycommunicatingbitsfromanencodertoa
Fig.4. Theproblemofcommunicatingmessagesinananytimefashion.Both
decoder,thissuggeststhattheestimatesofthebitsatthedecoder
theencoderEanddecoderDarecausalmapsandthedecoderinprinciplepro-
mustbecomeincreasinglyreliablewithtime.Thesenseofany-
videsupdatedestimatesforallpastmessages.Theseestimatesmustconverge
time reliability is made precise in Section III-A. Section III-B tothetruemessagevaluesappropriatelyrapidlywithincreasingdelay.
thenestablishesthekeyresultofthispaperrelatingtheproblem
of stabilization to the reliable communication of messages in Arate sequentialcommunicationsystemachievesanytime
theanytimesense.Finally,someconsequencesofthisconnec- reliability ifthereexistsaconstant suchthat
tionarestudiedinSectionIII-C.Amongtheseconsequencesisa
sequentialgeneralizationoftheSchalkwijk–Kailathschemefor
communicationoveranAWGNchannelthatachievesadoubly (2)
exponential convergence to zero of the probability of bit error
universallyoveralldelayssimultaneously. holds for every . The probability is taken over the channel
noise,the bitmessages ,andallofthecommonrandom-
nessavailableinthesystem.
A. AnytimeReliabilityandCapacity If(2)holdsforeverypossiblerealizationofthemessages ,
thenthesystemissaidtoachieveuniformanytimereliability .
Theentiremessageisnotassumedtobeknownaheadoftime.
Communicationsystemsthatachieveanytimereliabilityare
Rather,itismadeavailablegraduallyastimeevolves.Forsim-
calledanytimecodesandsimilarlyforuniformanytimecodes.
plicityofnotation,let bethe bitmessagethatthechannel
Wecouldalternativelyhaveboundedtheprobabilityoferror
encodergetsattime .Atthechanneldecoder,notargetdelayis
by andinterpreted astheminimumdelay
assumed—i.e.,thechanneldecoderdoesnotnecessarilyknow
imposedbythecommunicationsystem.
when the message will be needed by the application. A past
Definition3.2: The -anytimecapacity ofachannel
message may even be needed more than once by the appli-
is the least upper bound of the rates (in bits) at which the
cation. Consequently, the anytime decoder produces estimates
channelcanbeusedtoconstructarate communicationsystem
whicharethebestestimatesformessage attime based
thatachievesuniformanytimereliability .
on allthe channel outputs receivedso far. If the application is
Feedback anytime capacity is used to refer to the anytime
usingthepastmessageswithadelay ,therelevantprobability
capacitywhentheencoderhasaccesstonoiselessfeedbackof
of error is . This corresponds to an un-
thechanneloutputswithunitdelay.
corrected error anywhere in the distant past (i.e., on messages
The requirement for exponential decay in the probability of
)beyond channelusesago.
error with delay is reminiscent of the block-coding reliability
Definition3.1: AsillustratedinFig.4,arate communica-
functions ofachannelgivenin[1].Thereisonecrucial
tionsystemoveranoisychannelisanencoder anddecoder
difference.Withstandarderrorexponents,boththeencoderand
pairsuchthatasfollows.
decodervarywithblocklengthordelay .Here,theencodingis
• -bitmessage enters14theencoderatdiscretetime
requiredtobefixedandthedecoderinprinciplehastoworkat
• The encoder produces a channel input at integer times
alldelayssinceitmustproduceupdated estimatesofthemes-
based on all information that it has seen so far. For en-
sage atalltimes .
coderswithaccesstofeedbackwithdelay ,thisalso
Thisadditionalrequirementiswhyitiscalled“anytime”ca-
includesthepastchanneloutputs .
pacity. The decoding process can be queried for a given bit at
• Thedecoderproducesupdatedchannelestimates for
anytimeandtheanswerisrequiredtobeincreasinglyaccurate
all basedonallchanneloutputsobservedtilltime .
thelongerwewait.Theanytimereliability specifiestheexpo-
nential rate at which the qualityof the answers must improve.
14Inwhatfollows,messagesareconsideredtobecomposedofbitsforsim-
plicityofexposition.Theithbitarrivesattheencoderattime andthusM Theanytimesenseofreliabletransmissionliesbetweenthatrep-
iscomposedofthebitsS . resentedbyclassicalzero-errorcapacity (probabilityoferror
SAHAIANDMITTER:ANYTIMECAPACITYFORSTABILIZATIONOVERANOISYCOMMUNICATIONLINK—PARTI 3375
becomeszeroatalargebutfinitedelay)andclassicalcapacity
(probabilityoferrorbecomessomethingsmallatalargebut
finitedelay).Itisclearthat .
Byusingarandomcodingargumentoverinfinitetreecodes,
it is possible to show the existence of anytime codes without
using feedback between the encoder and decoder for all rates
lessthantheShannoncapacity.Thisshows:
where isGallager’srandomcodingerrorexponentcalcu-
latedinbase2and istherateinbits[23],[33].Sincefeedback
playsanessentialroleincontrol,itturnsoutthatweareinter-
estedintheanytimecapacitywithfeedback.Itisinterestingto
note that there are many channels for which the block-coding
errorexponentsarenotincreasedatallwithfeedbackwhilethe
Fig.5. Theconstructionofafeedbackanytimecodefromacontrolsystem.
anytimereliabilitiesareincreasedconsiderably[6]. Themessagesareus(cid:20)edtogeneratethefW ginputswhicharecausallycom-
binedtogeneratefX gwithintheencoder.Thechanneloutputsareusedto
generatecontrolsi~gnalsa(cid:20)tboththeencoderanddecoder.Sincethesimulated
plantisstable,(cid:0)X~andX areclosetoeachother.Thepastmessagebitsare
B. NecessityofAnytimeCapacity estimatedfromtheXatthedecoder.
Anytimereliabilityandcapacityaredefinedintermsofdig-
ital messages that must be reliably communicated from point Whileboththeobserverandcontrollercanbesimulatedatthe
topoint.Stabilityisanotioninvolvingtheanalogvalueofthe encoderthankstothenoiselesschanneloutputfeedback,atthe
state of a plant in interaction with a controller over a noisy decoder only the channel outputs are available. Consequently,
feedbackchannel.Atfirstglance,thesetwoproblemsappearto thesechanneloutputsareconnectedtoacopyoftheblack-box
havenothingincommonexceptthenoisychannel.Evenonthat controller ,therebygivingaccesstothecontrols atthede-
pointthereisadifference.Theobserver/encoder inthecon- coder.Toextractthemessagesfromthesecontrolsignals,they
trol system may have no explicitaccess to the noisy output of arefirstcausallypreprocessedthroughasimulatedcopyofthe
thechannel.Itcanappeartobeusingthenoisychannelwithout unstableplant,exceptwithnodisturbanceinput.Allpastmes-
feedback.Despitethis,itturnsoutthattherelevantdigitalcom- sagesarethenestimatedfromthecurrentstateofthissimulated
munication problem involves access to the noisy channel with plant.
noiseless channel feedback coming back to the message en- Thekeyistothinkofthesimulatedplantstateasthesumof
coder. thestatesoftwodifferentunstableLTIsystems.Thefirst,with
Theorem 3.3: For a given noisy channel and , if statedenoted ,isdrivenentirelybythecontrolsandstartsin
there exists an observer and controller for the unstable state .
scalarsystemthatachieves forallsequencesof
boundeddrivingnoise ,thenthechannel’sfeedback
anytimecapacity bitsperchanneluse.
(3)
The proof of this spans the next few sections. Assume that
thereisanobserver/controllerpair thatcan -stabilizean
is available at both the decoder and the encoder due to the
unstablesystemwithaparticular andarerobusttoallbounded
presenceofnoiselessfeedback.16Theother,withstatedenoted
disturbancesofsize .Thegoalistousethepairtoconstruct
,isdrivenentirelybyasimulateddrivingnoisethatisgener-
arate anytimeencoderanddecoderforthechannel
atedfromthedatastreamtobecommunicated
withnoiselessfeedback,therebyreducing15theproblemofany-
timecommunicationtoaproblemofstabilization.
The heart of the construction is illustrated in Fig. 5. The
“black-box”observerandcontrollerarewrappedaroundasim- (4)
ulated plantmimicking (1).Since the must be generated
by the black-box controller and the is prespecified, the Thesum behavesexactlylikeitwascoming
disturbances mustbeusedtocarrythemessage.So,the from (1) and is fed to the observer which uses it to generate
encoder must embed the messages into an appropriate inputsforthenoisychannel.
sequence ,takingcaretostaywithinthe sizelimit.
The fact that the original observer/controller pair stabilized
theoriginalsystemimpliesthat issmall
15Intraditionalrate-distortiontheory,this“necessity”directionisshownby
goingthroughthemutualinformationcharacterizationsofboththerate-distor- andhence stayscloseto .
tionfunctionandthechannelcapacityfunction.Inthecaseofstabilization,mu-
tualinformationisnotdiscriminatingenoughandsothereductionofanytime 16If the controller is randomized, then the randomness is required to be
reliablecommunicationtostabilizationmustbedonedirectly. commonandsharedbetweentheencoderanddecoder.
3376 IEEETRANSACTIONSONINFORMATIONTHEORY,VOL.52,NO.8,AUGUST2006
Sosetting
(6)
Fig.6. ThedatabitsareusedtosequentiallyrefineapointonaCantorset.
Itsnaturaltreestructureallowsbitstobeencodedsequentially.TheCantorset
alsohasfinitegapsbetweenallpointscorrespondingtobitsequencesthatfirst
differinaparticularbitposition.Thesegapsallowtheuniformlyreliableex- givesthedesiredresult.Manipulate(6)toget
tractionofbitvaluesfromnoisyobservations.
1) Encoding Data Into the State: As long as the bound
issatisfied,theencoderisfreetochooseanydisturbance17for
thesimulatedplant.Thechoicewillbedeterminedbythedata
rate andthespecificmessagestobesent.Ratherthanworking
withgeneralmessages ,considerabitstream withbit be-
comingavailableattime .Everythinggeneralizesnaturallyto
nonbinaryalphabetsforthemessages,butthenotationiscleaner
inthebinarycasewith .
isthepartof drivenonlybythe
Tokeepthisbounded,choose
(7)
whichisstrictlypositiveif .Applyingthatsubstitu-
tiongives
This looks like the representation of a fractional number in
base which is then multiplied by . This is exploited in
the encoding (illustrated in Fig. 6) by choosing the bounded
disturbancesequencesothat18
(5)
where isthe thbit19ofdatathattheanytimeencoderhasto
Sobychoosing
sendand isjustthetotalnumberofbitsthatareavailable
bytime . areconstantstobespecified.
To see that (5) is always possible to achieve by appropriate
choice of , use induction. (5) clearly holds for . Now (8)
assumethatitholdsfortime andconsidertime
thesimulateddisturbanceisguaranteedtostaywithinthespec-
ifiedbounds.
2) Extracting Data Bits From the State Estimate: Given a
channel with access to noiseless feedback, for any rate
, it is possible to encode bits into the simulated scalar
plantsothattheuncontrolledprocessbehaveslike(5)byusing
17In[23],asimilarstrategyisfollowedassumingaspecificdensityfortheiid
disturbanceW .Inthatcontext,itisimportanttochooseasimulateddisturbance disturbances given in (6) and the formulas (7) and (8). At the
sequencethatbehavesstochasticallylikeW .Thisisaccomplishedbyusing outputendofthenoisychannel,itispossibletoextractestimates
commonrandomnesssharedbetweentheencoderanddecodertoditherthekind
forthe thbitsentforwhichtheerrorevent
ofdisturbancesproducedhereintooneswiththedesireddensity.
18Foraroughunderstanding,ignorethe(cid:15) andsupposethatthemessagewere
encodedinbinary.ItisintuitivethatanygoodestimateoftheX(cid:20) stateisgoing
toagreewithX(cid:20) inallthehighorderbits.Sincethesystemisunstable,allthe
encodedbitseventuallybecomehigh-orderbitsastimegoeson.Sonobiterror (9)
couldpersistfortoolongandstillkeeptheestimateclosetoX(cid:20) .The(cid:15) inthe
encodingisatechnicaldevicetomakethisreasoningholduniformlyforallbit and thus
strings,ratherthanmerely“typical”ones.Thisisimportantsinceweareaiming
forexponentiallysmallboundsandsocannotneglectrareevents.
19For the next section, it is convenient to have the disturbances balanced
aroundzeroandsowechoosetorepresentthebitS as+1or(cid:0)1ratherthan
theusual1or0. (10)
SAHAIANDMITTER:ANYTIMECAPACITYFORSTABILIZATIONOVERANOISYCOMMUNICATIONLINK—PARTI 3377
Proof: Here isusedtodenotemembersoftheunderlying 3) Probability of Error for Bounded Moment and Other
samplespace.20 Senses of Stability: Proof of Theorem 3.3: Using Markov’s
Thedecoderhas whichiscloseto since inequality:
issmall.Toseehowtoextractbitsfrom ,firstconsider
howtorecursivelyextractthosebitsfrom .
Startingwiththefirstbit,noticethatthesetofallpossible
thathave isseparatedfromthesetofallpossible
thathave byagapof
CombiningwithLemma3.1,gives
Since representsthedelaybetweenthetimethatbit was
readytobesentandthedecodingtime,thetheoremisproved.
Allthatwasneededfromtheboundedmomentsenseofsta-
bilitywassomeboundontheprobabilitythat tookonlarge
values.Thus,theproofaboveimmediatelygeneralizestoother
Noticethatthisworst-casegap21isapositivenumberthatis
sensesofstochasticstabilityifwesuitablygeneralizethesense
growingexponentiallyin .Ifthe first bits arethesame,
ofanytimecapacitytoallowforotherboundsontheprobability
thenbothsidescanbescaledby togetthesame
oferrorwithdelay.
expressions above and so by induction, it quickly follows that
Definition3.4: Arate communicationsystemachieves -
the minimum gap between the encoded state corresponding to
anytimereliabilitygivenbyafunction if
twosequencesofbitsthatfirstdifferinbitposition isgivenby
isassumedtobe forallnegativevaluesof .
The -anytimecapacity ofanoisychannelisthe
if
(11)
leastupperboundoftherates atwhichthechannelcanbeused
otherwise
to construct a sequential communication system that achieves
-anytimereliabilitygivenbythefunction .
Because the gaps are all positive, (11) shows that it is always
Noticethatfor -anytimecapacity, forsome
possibleto perfectlyextractthe databits from byusing an
.
iterative procedure.22 To extract bit information from an input
Theorem3.5: Foragivennoisychannelanddecreasingfunc-
tion ,ifthereexistsanobserver andcontroller forthe
1) Initializethreshold andcounter .
unstable scalar system that achieves
2) Compare input to . If , set . If
for all sequences of bounded driving noise , then
,set .
forthenoisychannelconsideredwiththe
3) Increment counter and update threshold
encoder having access to noiseless feedback and having
theform forsomeconstant .
4) Gotostep2)aslongas
Proof: Foranyrate
Sincethegapsgivenby(11)arealwayspositive,theproce-
dureworksperfectlyifappliedtoinput .Atthedecoder,
applytheprocedureto instead.
Withthis,(9)iseasytoverifybylookingatthecomplemen-
taryevent .Thebound(11)thusimplies
thatwearelessthanhalfwayacrosstheminimumgapforbit
attime .Consequently,thereisnoerrorinthestep2)compar- Sincethedelay ,thetheoremisproved.
isonoftheprocedureatiterations .
C. Implications
20Ifthebitstobesentaredeterministic,thisisthesamplespacegivingchannel
Atthis point,itisinterestingtoconsiderafewimplications
noiserealizations.
ofTheorem3.5.
21Thetypicalgapislargerandsotheprobabilityoferrorisactuallylower
thanthisboundsaysitis. 1) Weaker Senses of Stability Than -Moment: There are
22ThisisaminortwistontheprocedurefollowedbyserialA/Dconverters. sensesofstabilityweakerthanspecifyingaspecific thmoment
3378 IEEETRANSACTIONSONINFORMATIONTHEORY,VOL.52,NO.8,AUGUST2006
oraspecifictaildecaytarget .Anexampleisgivenbythe mentaltothestabilizationproblemregardlessoftheobservers
requirement uniformly for all . orcontrollers.
Thiscanbeexploredbytakingthelimitof as . Thus for DMCs and a given , we are either limited to a
Wehaveshownelsewhere[23],[33]that: power-law tail for the controlled state because of an anytime
reliabilitythatisatmostsinglyexponentialindelayoritispos-
sibletoholdthestateinsideafiniteboxsincethereisadequate
feedback zero-error capacity. Nothing in between can happen
withaDMC.
where istheShannonclassicalcapacity.Thisholdsforalldis-
3) LimitingtheControllerEffortorMemory: Iftherewasa
cretememorylesschannelssincethe -anytimereliabilitygoes
hardlimit onactuator effort ( for some ), then
to zero at Shannon capacity but is for all lower rates even
the only way to maintain stability is to also have a hard limit
withoutfeedbackbeingavailableattheencoder.Thus,classical
onhowbigthestate canget.Theorem3.5immediatelygives
Shannoncapacityisthenaturalcandidatefortherelevantfigure
afundamentalrequirementforfeedbackzero-errorcapacity
of merit.
since forsufficientlylarge .
ToseewhyShannoncapacitycannotbebeaten,itisuseful
Similarly, consider limited-memory time-invariant con-
toconsideranevenmorelaxsenseofstability.Supposethere-
trollerswhichonlyhaveaccesstothepast channeloutputs.If
quirementwereonlythat
the channel has a finite output alphabet and no randomization
uniformly for all . This imposes the constraint that the prob-
is permitted at the controller, limited memory immediately
abilityofalargestatestaysbelow foralltime.Theorem
translates into only a finite number of possible control inputs.
3.5 would thus onlyrequires the probability of decoding error
Since there must be a largest one, it reduces to the case of
to be less than . However, Wolfowitz’ strong converse to
havingahardlimitonactuatoreffort.
thecodingtheorem[1]impliesthatsincetheblock-lengthinthis
We conjecture that even with randomization and time-vari-
caseiseffectivelygoingtoinfinity,theShannoncapacityofthe
ation,finitememoryat thecontrollerrequiresthatthe channel
noisychannelstillmustsatisfy .Addingafinitetol-
musthavefeedbackzero-errorcapacity .Intuitively,if
eranceforunboundedlylargestatesdoesnotgetaroundtheneed
thechannelhaszero-errorcapacity ,itcanmisbehave
tobeabletocommunicate bitsreliably.
forarbitrarilylongtimesandbuildupahuge“backlog”ofun-
2) Stronger Senses of Stability Than -Moment: Having
certainty that can not be resolved at the controller. With finite
decreaseonlyasapowerlawmightnotbesuitableforcertain
memory,thecontrollerhasnowayofknowingwhatuncertainty
applications.Unfortunately,thisis allthatcanbe hopedforin
it is actually facing and so is unable to properly interpret the
genericsituations.ConsideraDMCwithnozeroentriesinits
channeloutputstodevisethepropercontrolsignals.
transitionmatrix.Define .Forsuchachannel,
4) TheAWGNCaseWithanAverageInputPowerConstraint:
with or without feedback, the probability of error after time
Thetightrelationshipbetweencontroland communicationes-
stepsislowerboundedby sincethatlowerboundstheprob-
tablishedinTheorem3.5allowstheconstructionofsequential
abilityofallchanneloutputsequencesoflength .Thisimplies
codes for noisy channels with noiseless feedback if we know
that the probability of error can drop no more than exponen-
howtostabilizelinearplantsoversuchchannels.Considerthe
tially in for such DMCs. Tighter upper-bounds on anytime
problem of stabilizing an unstable plant driven by finite vari-
reliabilitywithfeedbackareavailablein[34]and[6].
ance driving noise over an AWGN channel. A linear observer
Theorem3.5thereforeimpliesthattheonly -sensesofsta-
and controller strategy achieve mean-square stability for such
bilitywhicharepossibleoversuchchannelsarethoseforwhich
systems since the problem fits into the standard LQG frame-
work[14].
By looking more closely at the actual tail probabilities
achieved by the linear observer/controller strategy, we obtain
a natural anytime generalization of Schalkwijk and Kailath’s
scheme [36], [37] for communicating over the power con-
whichisapowerlaw.Thisrulesoutthe “risksensitive”sense strained additive white Gaussian noise channel with noiseless
ofstabilityinwhich isrequiredtodecreaseexponentially.In feedback. Its properties are summarized in Fig. 7, but the
thecontextofTheorem3.3,thisalsoimpliesthatthereisan highlightisthatitachievesdoublyexponentialreliabilitywith
beyondwhichallmomentsmustbeinfinite! delay,universallyoverallsufficientlylongdelays.
Corollary 3.1: If any unstable process is controlled over a Theorem 3.6: It is possible to communicate bits reli-
discrete memoryless channel with no feedback zero-error ca- ably acrossa discrete-time average-powerconstrained AWGN
pacity, then the resulting state can have at best a power-law channelwithnoiselessfeedbackatanyrate
bound(Paretodistribution)onitstail. whileachievinga -anytimereliabilityofatleast
This is very much related to how sequential decoding must
havecomputationaleffortdistributionswithatbestaParetodis-
tribution [35]. In both cases, the result follows from the inter- (12)
action of two exponentials. The difference is that the compu-
tational search effort distributions assumed a particular struc- for some constant that depends only on the rate , power
tureonthedecodingalgorithmwhiletheboundhereisfunda- constraint ,andchannelnoisepower .