1mohhmmhhhhhl ImhomhohhiI mmmhhhmhhhl - DTIC · aa!i' )'7 asnhoos di r rifconiro of cont inu proc ijino iii as i f n in ivros i nma m fk a t l inu a 'l 1 ii jji ii iiaf sr tr83 0877

AA!i' )'7 ASNHOOS DI R RIfCONIRO OF CONT INU PROCIJINO III AS I F N IN IVROS I NMA M fK I t A l

INU A 'l II II 1 JJI IIAF SR TR83 0877 F4q620 82? 0080

1mohhmmhhhhhlmmmhhhmhhhlImhomhohhiI~I'll'.momo

I m mi _ _ _ _ _----L- - W

• 4

+ ii

*La Uf02

-..

SI ijLI

,. ~ ICROCOP:Y RESOLUTION TEST CHAT

t ar ma Ma Or STmcaaos n-A

" - .. .. - ,--L.. --a "'i ':I

0

" 6 - it ! - . .S

,. .. . .. .. .. . ....4 -

kit-

SECURITY CLASSIFICATION OF TONS PA4E (tm m.. COW~S. ______________

REPOR DOCMENTATIO PAGE __________________

1jr'. REOR M5.f- OTACUS REMPICUT'S CATALOG NMBER

4. ITLE (d 8.MIej SL TPX OF REPORT & PCmo. CONVURED

Asynchronous Discrete Control of Annual Report7/l/82-6/30/8 3Continuous Processes ~PROSGOG EOTNME

,. umor6. CORACT" DO GRCT NSCR~.

Martin E. Kaliski _ _ _ _ _ _

Timothy L. Johnson F4 9620-82 -C-00 80

SPER? lUGN ORGANIZATION NAME AND ADDRESS go. Pr 6AELEMENZ PROJCT

Northeastern UniversityAmRKUITNDESAS360 Huntington Avenue0AFBoston, MA 02115 &3,___________

11. CONTROLLING OPPICZ "AMC AND AOONEM L 9PRTDT

Directorate of Math. & Info. Sciences 7L 30/83DTEAir Force Office of Scientific Research 98. NUMBER OF PAGES

ILS 1A MOIOIN NCY NAE ADDRESS(56 I..a ban .ua Om..ie IL. SECURITV CLASS. (of Oa. ,.p.,wj

Unclassified

Ickrts g~~FC ATI ON/ 00114 RDIN _

W. DSTRIOUTION STATEMNICT tot IN@ AasjpH) dfrpui.z'le

17. DISTRIOUTION STATEMENT to# 0v auWe. mgomE in..*te 20, It l aem be. RePWeO

IL. 1111PL6491TART NOTES

coding, automata theory, discrete control, switching theory,-feedback control

IF "ARAC V..u == u=aea. S6 mm'ie ~ebb rn N

This research concerns the analysis and synthesis of asynchronouidiscrete-state or hybrid-state feedback compensators forcontinuous or hybrid-state processes. New realisation theoriesfor asynchronous discrete sys team, based on automata indsemigroup theory, have been derived.- These theories suggest newarchitectures for asynchronous system

-D liWime. Mlos 9FI 10 as msr -O

____ ___ ___ __M I=l

ASYNCHRONOUS DISCRETE CONTROL OF CONTINUOUSPROCESSES

N.E. Kaliski and T.L. Johnson

July 1983 Accession ForNTIS iDTIC TABUnannoundQJustification

Distribution/Prepared by: Availability Codes nI

Northeastern University jAvail and/or It360 Huntington Avenue Dist SpecialBoston, MA 02115

Prepared fo r:

Mathematical and Information Sciences DirectorateAttfl: J. BurnsAir Force office of Scientific ResearchBolling Air Force BaseWashingt on D.C. 20332

AIR p~neumn3OwTI7I mSK (A7SO)

aW??v Id f Or rS1M,'.' 9 J AW AYR 19--12.

3,etiuo is nlmied

,un 3.

TABLE OF CONTENTS

Page

1 INTRODUCTION AND STATEMENT OF WORK I

2. RESEARCH PROGRESS 3

2.1 Analysis of Qualitative Properties of Asynchronous 3Hybrid Systems

2.2 Acceptance and Control for Asynchronous Hybrid 5Systems

2.3 Linguistic Approaches to Asynchronous System 8Modelling

2.4 Application to real-time Multitasking Systems; 8Stochastic Analysis

3. PUBLICATIONS 11

4. INTERACTIONS 13

7 7

: ' . .. . .

1. INTRODUCTION AND STATKM3NT OF WORK

This report summarizes the research undertaken during the

period from July 1, 1982 through June 39, 1983, the first year of

a three-year research program entitled *Asynchronous Discrete

Control of Continuous Processes*.

The research during this first contract year centered about

developing sound theoretical models for asynchronous systems,

models that extend and generalize previously developed research

of the co-investigators on synchronous discrete control. The

following sections of this report delineate this year's work.

The original proposal for this research described four

tasks:

(1.1) Analysis of Qualitative Properties ofAsynchronous Hybrid Systems

(1.2) Acceptance and Control for Asynchronous Hybrid

Systems

(1.3) Linguistic Approach to Asynchronous Systems

(1.4) Application to Real-time Multi-tasking Systems;Stochastic Analysis

These tasks are to be pursued in parallel over a 3-year period,

with the first two tasks receiving greatest emphasis during the

first year. The research involves a collaboration between 4Prof.

K.3. Kaliski of Northeastern University and Dr. T.L. Johnson of

Bolt Beranek and Newman Inc. Dr. Johnson has taken

responsibility for Tasks 1.1 and 1.4, while Prof. Kallski has

overseen Tasks 1.2 and 1.3.

..... .7

2. RESEARCH PROGRESS

2.1 Analysis of Qualitative Properties of Asynchronous HybridSystems

In order to focus on key issues in the realization of

asynchronous hybrid systems# the class of dynamic systems with

finite input, output and state sets -- termed Osimple

asynchronous machines* (SAR),-was treated during this year. The

time-variable was assumed to be real-valued. interesting results

were obtained under the mild assumption that the input is a

piecewise continuous (i.e., piecewise-constant) function. Uinder

these conditions, the semigroup properties can be used to show

that such system can be realized in terms of a finite set of

constant-input state-response functions that play a role

analogous to the impulse response functions of linear system

theory. Preliminary results reported in Johnson and Kaliski

(1983) show that in the timo-invariant case, such a rdalization

am be synthesised from ideal digital components similar to those

which ate, cmintclally available -- but in an architectural

cnfiguretion which i9 hoew.

2he key concept in deriving this new _relization is that

each input transition triggers a OcaspadeE ot subsequent state

transitions which Is pcedetermined by the machine structures this

cascade may be Overwitten by *ascades, due to subsequent input

tawemsints ot.I Sm~put tb*in beats a sequence of

mi..fte~ ~I~u tzIi~o tian.1arou~arohiteotura1

3

configurations have been described for generating the fundamental

state transition sequences; the applicable architecture depends

on the specializing assumptions which are introduced.

This realization theory overcomes a number of fundamental

problem associated with alternative representations. First, it

places the theory of asynchronous discrete systems in a clearIrelationship with the theory of semIgroups, which is almost

universally used in describing a wide variety of other classes of

dynamic systems: this. link has been-uissing from previous

results on asynchronous machines. Secondly# it successfully

resolves a problem which occurs if transition rates are bounded

by assumption -- i.e., that it is unreasonable to impose

constraints on input transition times that depend on internal

state dynamics. The ideal, mathematical reproestation is

well-defined even for unbounded or infinite transition rates; the

transition rate bounds arise ,only when the differences between

real and ideal 42inthesis components. are oonsidoco4. This is a

key requirement in the, corwsiderat ion of, genera; f eedback systems

composed of simple asynchronous mab.na sinoe a£q ack system

should itself be representable as a simple asynchronous machine I

For insth~ep unstable eedwak ste are well-detined and are

rakell"e so tuak x meahiihgl theory of stailty can be

#A Onticipstod. h tbeq "upps", tbeatuoor ajg slons

to.a boo CC p!beu"-&b ~ be p. v V04 1A ubsequent

774 -7j7~z--

reerh (1 th..- cae fhbi sae nu ndotu es

det mcdi (3) feedback system properties such as controllability,

observability and stability can be defined f or simple

asynchronous machines; (4) equivalent realizations which

partition the timing and data flow functions of asynchronous

machines can be sought.

From a practical Standpoint, the mwst significant

implication of the realization results is the possibility that a

general automatic synthesis procedure for aynchronous state

machines can be developed., Tbe theory swlqests a general

architecture which is significantly different from traditional

Von Neumann and Data Flow architectures, which require

considerable use of ad boo design methods by experienced

designers. A general astosistic Synthesis -procedure could have

major implications for 0eqmutor'aided ctscuit design and

achievable pert ormee at VX1SIC ebipe as well as Suggesting new

concepts f or the design of asynchronous feedback Systems. A

N~iatat* effort a£10.9 tbese limes Is anticipated.

2.2 AoPternee a Cstrol got £synchroaous sybtid System

Wfozta deaJa th1 Lust '0teract" Your:toolased on

dsvolop$ag *eraeWm ~e n to"*. for 4baacttatining

4P.

asynchronous coders. These coders, to interfaces between the

plant to be controlled, and the finite-state, controller itself,

are intrinsically hybrid devices which, as asynchronous devices,

react not only to input changes bust to the times at which these

changes occur. Thus models of coders explicitly incorporating

transition times were sought.

A mathematical abstraction of the asynchronous coding

process can be made by viewing such coders as mappings from a

subset of (RxR)+, the set of non-null, finite length sequences of

points in RxR (R the real numbers) in-to the binary alphabet

fo,1). This abstraction,, detailed in the attached informal

memoranduma Otwards a Theory of Finitary Asynchronous Codersm,

ad summarised ini the pap~er NTOlards a Theory of Asynchronous,

Real-Tim Codess and.Their Application* to IDiscrete-Control of

Continuous Pr9sesses', 1903 American Control Conferentce, allows

us to view asynchronous coders as special forms of their

synchronous counterparts -- forms defined over a limited subset

of allowable Input atuings. TVhis sabset cotisists of strings

whose second componsnts:(namely.tI-me), always increases.

Having made this abstraction, research turned to finding

meaningful ways to extend such *partially specified" maps to all

of RIM. so aS to permit our previously, developed theory of

synchronous coders to come into play. The memorandum cited above

dov4Upe am*, an. =esusiono , %eort' "or th* case at f initary

iS t*. )**deft: &no- jot a.( .' a ke )rCstY o1f such

codes - t~a~t stuct*A fip aoprty s ancbtrlflal onewhose isplications, as Of this vriting, are, still, beingresearched.

IY 11 tihl ' ji~ sequnces that form the domain ofasynchronousg codera may,# #e1r ~A-j.OCwge tsttmpig

onth tewonep,~d it 4t 00l1 aitural that -at tentionturned to cha~arss ~ al fivet Eittibutivou of theObreak points!, p~,a fthe.f ts Oftl research-- the breakpoints, after allp represent eve 'switcbtfig 'time".

We desired to develop nots general characterizations Ofbreakpoint 4tb ip t aple sodcl Le 19eE n hcited fnmranium above.

-suck a- ro4ia 4 ~ ero- fa to, it ad som e wasGownd to'-c r~thej by

no Stna ttWTIooai Atu**,46(Y Slwankamt) to develop

41 as *rbital b aio'and

5 ~ rnz~~rAJv Aflf

MWA6"In ipf womlitar tgem*a

A wit4

A .g4

MY

4414 ~ 51i4

l~ 1

Afi

AM=A

other side. The study of such timing functions is currently

being done.)

2.3 Linguistic Approaches to Asynchronous System Modelling

The above theoretical models will be used in subsequent

research to characterize wider classes of asynchronous coders.

We will attempt to extend the linguistic models that were so

successful in characterizing synchronous coders to the

asynchronous case.

2.4 Application to real-time Multitasking Systems; StochasticAnalysis

Only preliminary investigations of this topic have been

completed this year. The key issue of interest in multitasking

systems is the development of a fundamental model for process

synchronization; i.e. two concurrent processes may be represented

as parallel asynchronous machines connected by a communication

channel which may impose requirements for synchronization at

certain stages of a computation. A dual view of this problem is

to determine when a single process can be split into two

concurrent processes with minimal communications requirements.

The representation of the (asynchronous) Communications channel

is being viewed (in the deterministic case) from the perspective

of the realization theory of asynchronous machines.. As a

preliminary example, the partial realization of a UAM (univet"sal

asynchronous reeiver-transsitter) has been worked out this year.

Stochastic problems have not been considered this year.

However, it should be noted that the semigroup theory ofstochastic systems is well-developed, and this provides a point

of departure for developing a theory of stochastic asynchronous

machines via the results cited in Section 2.1.

I-

I -.

p!

3. PUBLICATIONS

The following works have been published and/or submitted for

publication this year.

Kaliski, N.E., *Towards a Theory of Finitary Asynchronous

Coders', Northeastern University Memo~randum, January 1983.

Kaliski, N.E., Kwankam, S.Y., and Halpern, P., 8A Theory of

Orbital Behavior in a Class of Nonlinear Systems: 'Chaos' and a

Signature-based Approachn, submitted to Intl-.3 .SysAt2M 5Sj±ja&,

February 1983.

Kaliski, N.E. and Wimpey, D.G., *Towards a Theory of

Asynchronous, Real-time Coders and Their Applications to Discrete

Control of Continuous Processess, Procg. Amerian CotlQ1

Cofrne San Francisco, CA, June 1.983, pp. 731-733.

Wimpey, D.G. and Johnson, T.L., "Finite-State Control of

Continuous-State Processes: The Discrete Time Case", £WL- 2.lLt

LBU nf.. 2a Dec±iin and Control, Orlando, FL, December 1982,

pp. 1230-1232.

Johnson, T.L. and Kaliski, N.E., *Realization Of

Asynchronous Finite-State N4achines*, S~ubmitted to I= Trans-

£Auto-. Contr. and 2=a Z=Z Cn. 2a R~a±Qn and Cnnkr.B, San

Antonio, TX, December 1983.

4. INTERACTIONS

Prof. Kaliski and Dr. Johnson met regularly every 1-2 weeks

to coordinate research progress during the year. Prof. D.G

wimpey of Northeastern University has continued to participate as

an unfunded collaborator in this line of research.

A proposal by BBN, Inc. to develop control system design

algorithms based on our earlier results for the discrete-time

case was submitted to Wright-Patterson AFB but not funded this

year for lack of program funds. A similar effort has since been

funded by the Army R&D Command.

Dr. Johnson has been involved in related research on modular

hierarchical (hybrid-state) control systems and has given a

number of presentations and lectures during the year on this

topic.

'I

!1

FIZ-STATE CONTROL Of CON UW-#ZTR lSaCuuSES !EZ 0!SZ TME CAW1

David c. ViapeyDeprtent of Slectrical Engineering

TmtyL. Jolmiecs

Laboratory f or Infomaetian ad DecisionSytmRom. 35-2053

Visseachusetta basttte of TcnlgCamridge. Mass. 02139

An lgoith fo th deig offinte-tade omsign.

penstor fo nolinw dscrte inisytemdn or Csk'S~kg

e isdevlope. Te prble toformlatd asa rs-- with ehe breslsi eT9 oIwee9i mulster ~ ~ ~ ~ f problem andth oum.iiteis I- Piiv Intgr

catesecfcaition opf: al A? and quantizattheleel an to exac folte sne the p lant t- Th taeea fth lut inghoiorpe ilatorrned. Tfhresoluthen plantestae zIolve problem oilr excet tandw tolg reushrefioite-rsguated agregon of theplan tadte deigno 1 sard h

of acontolle forthe esuling andeermiist celoledacheT witn hoe pann otmW ha ~ ~ ~ ~ ~ Nek rarget sto tCR

8 fac that gihe timesste Initial0- 1. orte sa b

straint~~tat In tixee reuatr orul bepnatr ia sthan itulto for Ptm(umust ~ ~ ~ ~ ~ col be finte-tat realtraable. Thn genpenralo iap1 so (3

t 1u thestrod - u tion , Pre ,i) wlaithoanit It~ prh e 0 omlt sas otetretrobe eateiltyt prolatof fotos syot exit

st plan to be coanrle to: ftp * o'. ait sethe planing desrion, ofuther cotditieneaP(Sir Tsmares req]). I hspprecnie

& ictarget sys e £ an (In. qetzeio levels f.urtherh-X

steofiin-t sap te f: isx l n ods u ny eino a weak regulator forr amcws Ceiuu Infalinies arslte: ll and fruanth Cntl state as a vdSeeiiaiso h

ifguter isrobte hT I suchl that forlant stat x isore ifarolem fruisn ad iteisming that thesolvn t euled-ro sow Intialsaegtmet s sltaedlard Teprobem os defntion I ebeue

callte to a synchru proisis gie bys f0~lt- aroa agt ot TC~ i s g~iven tie facTh00- lwothe oo wher aP tropaidtriiti c finit

ths reeac was N permed at t th aI? fiiesoy The prbe omlto seless otetrefore Infeseatiln peroblem for seocwittissystemspwo-

sieb the RAir ermOf lo n: NZPStfi a eseara The d dlescto of afne-tte uertmaitrife

Utesetorate~~~ of shJtca an"ov~i. lses ias soe aftat fte atre~ copnstr Is-( a arge met T04C-0and as Intlawsaes i thi muig mall iD oband De fquazat leels. a1 erpaptere do nat atee bil8reachen the aor mo e lo 5.1fa.14lsisfud it tomnI ra

oover-~ belationite artitionole Is deine by hevd euwLe1Il-f% ~k (r))Teps t l sli n borsm

relatIon I Tenm the agregata model *tata-ttretutionU" 6p: P z In pis dfned as 4 .7 - Q a

SPPoU) u (P C P- f(7'i. U)fpj 0I 01 for With U.: Wa ) a UE4 Pr.U)t p C ;AG(w)].

u 4 tP Fommao.11 1: The solution of

Thus the nondterejaistic finitestate system (see E61) Xkl "(k k8e aC?. 0. 8p) silat~ae Pin the following somae. If.7 i

the Initial stae. of 6 ia the equivalence class 1-.12 2 k~l N(k, "k' q(S(xk)))

i.e. the block containing %0.and the input sequence w witX Zl resultsin Z k e 3' o Al W-.

is applied to both 8 and P , then the aet of possible Vt htteqetsto o eed nyastates reached by 8 contais the equivalence class.O htte4fitzttf a eed nyo(sodulo 1) of the state reached by P. 6 end P. The mop; chowe explicitly the decompoeition

of eekFlate-tat Coposteof a. *Aleo, In arder to comput and q we require11I. EIwicenc ofWa iieSaaCmeitows thP fI~*h

Clary a necessary cndtio for the existtee taf d I be wle ake foninuo-us. eglto

0of a weak finite-state regulator for P sta th ftl eiu JeI~ijMi 2.0 if twea euite o rea to ma

tis conttSion In berm cotollagrae t del 8 we Ks 2~ x W -0 0L with the property that solutios of themycompute the attainability seti 3:sse

T A i p 9 P: 3u CRm i.e. 8p(p~u)CT 1 1) zk s-f% '(k' k q(g(Wk))))

k a 1. 2, .. kl- ~';( k' (g( k ),q'( k )

where To a, A9( c P: pCT)Is athe aggregate target for any Za C0 X w6LPresult isnrk£T, Otty.

set. Then if X. is covered by Tk for isow k, i.e.- enX cVjlp: p g T AJ), there exsts a sata-fedback law H than ha U n

for 8 (and hae P) which solves the finite horims(hy (h. I~h, q(y)), q(y))regulator problem. Since the etteof P die not di-rectly miurable to general, the @electios of the con- n~h~y) - 19(h. q(y)).trol Inputs to he a*"lied at emy Uim stop mt hebased on an estimoa of the state of P. To obtain a I.slto fteIfnt oio rbefinite-itate state-eatimetor we will flee agoregateIV ouoafgetlfeRrinPobmieate-estimntea i.e. estimtes of the state of 8. In practice, construction of the successor teen

In principle the, the design of H my he carried for ; from the root down Is highly Inefficieet. Aout by the construction of a tree of aggregate state- "bottom-up" procedure to desirable, but this requiresaicinotci of P. The root 7f the tree is the Aggregate knowledge of a cot of etate-easntes in Tifor uhichinita state set In A (Ex3s a P: x e X0 and the all successive itate-sntent will remain innodes are elementa of 2 . If Pk~is a successor node Esentially. thia involves the comptation of a steady-to node Pk. With the connecting are labelled (u.7), crate reduced target tube E33 which we denotethen in' the -a aggregate itate etimate of P t dto2:Dfngiven that the Plant output vae 7 sad then input a Wa 4ZaU I DeieT to beth ae sbtapplied. Of comm.e the area would hae to he labelled of 2 with the proeet lo ':T are st.i~eby subsets, Instead of elements. of 0 x 279, sin1e aX)f' n I P , there exists ucRe et. ',luav)sthere are only finitely smy distinct state-eatimages. 4VThe design Is completed by selecting a feedbck lAw An aloih for the computation of 4: is gieK: 27 z a - I3 which results in all poesible #adoi of in [3. TAUn asoih actlly competes a minimalthe tree having scecessive ades ohich ea eveudoly ro hci ' hmal&Iweys subes at oe of It-Pa, smlyfi thu I fPe ha a

The deeip of en (open-loop) fitoz-esate state- Set.- subsets of 1' are not Included iseatluster for T psoeed s Mumsa . tofiva em sqwiV- Ow phac beow composed. the tree for ';my bealmso relationSe & a e II& YS 2 46 (EXIS cps I(X U1) constructed an follow.,

(C1 p S(a) - 1) - oa y1 *72ifft eat oeeaggregate states that P may he in gives output yl ios+ Pidentical to the a"c gie 7 . am, 41WIM ai Wls lP, for OeN& waW at. GMu6i1' #*,49 IV - gP/W (A I) end a, V + 2: 1a Ctxjl-S iaSw 0161""exsts Wei gse. ;Mq's)e xThen the statsaesimsen to a (9.0I X3 20,49, 11'as 2P a If: it R 2P 7ndetedi29Ma f p A wea fiaift-stau regeLator H suatg

for 1, T, aud ifthr e SiAM a fliie partitionP

of such that XaC I' for some I' i ' and some non- (51 Wimpey, D.G.. "'Tite-State Cntrol of Discrete-

n i Tim Continuous Processe: An Automata Motivatednelat~e L~teler J.Approach-, XIt..T., Ph.D. Thmeis, Dept. of Else.

The ar e esentially attainability sets can- ZnJ. end Com. 3d., January 1962.

structed from T,. A detaLled algorithm for the coupu- [61 mesna", F.C.. Fine-tate Models or Loeical

tation of the slnay be found in [51. The tree speci- ! , .

fying ;. wb represeted as a grah. to the state- t71 Tou, J.T., Darr .ip of PI Control

transition graph for a system with Input set Re x V £sZ!_ = Academic Fress. N York, 1963.

i.e. each we is labelled as (V, w) for son VCI, [81 Wgs, P.K.C., "A Method of Approximating DynamicalProcesses by Finite-State System". 1st. 3. Control

w a V. Thus. corresponding to each P' a 1 + and each Voc.8,eo by pinit8-2t , 19a8.

wsuch that a(*)flP, # s. we hem a traitiosat V ol. 8, No. 3, pp. 285-29%, 1948.

Ithen if utV. (F*', u, v)uI . since there are only [91 Sorkar, V. and Varaiya, P., "Fnite Chain Approx.finitely wany sets V emy select one representative for a Continuous Stochastic Cnerl Problem",

ftm each and define I as K(]P'. v) , I.e. K has . AC-24, No. 2, Arl L11I.

finite range YC . The structure of H t shoam in [101 Sorneoshenmk, 1.&., "'Finite Automaton Approim-Figure 1. tice of Behaviour of Linear Stationary Continuous

Plants", Lut. and aem. Control. Io. 7, pp. 1092-Clearly, the existence of the regulator H (if it fl02, 1977.

exists at &UtI) depends eD the choice of P. Tatialchoices for P should be coarse to simplify the con-

troller. and refinements should be mde if H cannot befound. The design process is therefore iterative. Figure 1. Structure of Reaulaor H.Techniques for chooeing the Initial partition and forupdating or refining P are under investigation. Theseare complex problems which are problem dependent. Tosimplify the design, a heirarchical controller struc-ture has beon suggested [51.

V. Discussio. Conclusions q

We have described a n w approach to the design ofdynamic digital compensetors. These compensators arefinite-state and are therefore directly realizableusing digital logic circuitry. Our procedure speci-fies all quantization levels and takes all quantize-tion effects Into account.

Previous attempts to design finite-state control-lets for coatinuous-state systems Mro concerned withmemoryless switching controllers. These problems wore

often formuated as optimal control problems and thedesign often employed approximations which resulted inclosed-loop systems with performances that war diffi-cult (if not Impossible) to predict [4, 71.

The problem of quantitation (or finite-stateaggregation) of cotinuous-state dynamic systems isnot new (8, 91. Our approach follow mot closely theIdeas of Kornousbenko [101 and is related to the prob-le- of obtaining structure-preserving covers forfinite automata (6].

Reforences

[(1 Gatto, M., and Guardabasi, G., "The RegulatorTheory for Finite Automata", taf. and Control,Vol. 31, pp. 1-16, 1976.

[21 Sontag, S.D., "Nonlinear Regulation: The Plece-

wise Linear Approach", U Trm. AC-26, No. 2,April 1981.

(31 -emires, S., "Set Theoretic Control of LargeSeale Uncertain Systems", M.Z.T., Ph.D. Thesis,Dept. of glee. ISn. and Coup. Set., May 1977.

(41 tarden, J.S., "Optimal Feedback Characteristicsfrom Stochastic Automalon Models", Use.AC-lA, go. 1, February 1969.

A

Rea ation aL haini ka- a Mchinna

T.L. Johnson** N.H. KaliskiBBN, Inc. Northeastern University,1 Moulton Street 405 Dana HallCambridge# NA 12238 360 Huntington Avenue

Boston, MA 12115

Ph. 617/497-3413 Ph. 617/437-3034

A

Asynchronous finite-state machines accurately represent the jaction of computer control systems on continuous and discrete

processes. Despite their importance, the lack of a general

analytic representation for asynchronous machines has prevented

the development of adequate techniques for analysis and design of

finite-state asynchronous control systems# while ad hoc real-timecontrol software has proliferated. In this paper, a general

realization method is presented and is applied to a portion of a

universal asynchronous receiver/transmitter (UART).

*Research supported by APO Matbematical and Information

Sciences Directorate under Contract ?4ff20-82-C-008.

Preferred address for correspondence and return of proofs.

-..... . _ .. . . ... .-.-

,L,.

A simple real-time control program (e.g., one which

repeatedly polls a number of input lines, performs certain

logical tests and computations, and then deposits results in

several output registers) operates asynchronously. While such

programs undoubtedly characterize most computer control

applications in operation today, very little theory for therigorous analysis (let alone designi) of such systems is

currently available. In fact, technical standards for defining

the behavior of such systems, short of multichannel timing

diagrams, do not exist. Some consequences of thie situation areas followss (1) systems are designed by trial and errori (2)control software is not generally transferable from oneapplication to the nezt; (3) prior performance estimates of

proposed systems cannot be obtainedp (4) system functions cannot

be efficiently documented, short of providing complete

hardware/software descriptions. Clearly, these consequences are

undesirable.

A. number of strategies for avoiding these problems have

evolved, but none are very adequate. The practitioner, if he is

even aware of the preceding problems, is generally quite contentwith the status quo. Trial-and-error debugging of softvare is

fast and relatively inexpensive compared to analytical design

procedures. For simple systems, this approach indeed works well;

but for large systems (e.g., the U.S. Space Shuttle [1]), thecomplexity becomes overwhelming, and even exhaustive simulationcan fail to reveal subtle timing errors. The response of thetraditional deedeatc eastrol designer, by coatrast, is to forcethe system to operate synabronously (or with mltirate sampling),and to recuse logLcal operations (as distinct from real-nuaber

"- " "1

1.:. n"....i I -

operations) to an absolute minimum. While this approach is

usually 30re amenable to analysis and documentation, it forcesthe designer to overlook certain attractive control strategiesand forces the processor to operate inefficiently; thus,performance may be significantly compromised [2].

The realization problem for asynchronous finite-statemachines is a first step in the development of analytical methods

for the design of asynchronous computer control systems. Thaprimary issues of in &rat are todefine a natural set of

primitives that can ba used Ito haracteri.. the hehaviior of

asynhronous discreto-State nyste. and to illustrate a aneral

synthesis graodutra whereby speCIfication of the primitives leadaI - to a realistic imlenaon in previous research of the

authors and a colleague [31, these objectives have beensubstantially achieved for discrete synhannum systems. Inanyncghn = systems, the actual timing of events may effect theevolution of states, whereas in synchronous systems,, only thechronological ordering of events is important (i.e.,*synchronous* does not necessarily imply a uniform samplingrate).

several simplifications are made here in order to isolatethe main Issues: (1) only the purely finite-state case isconsidered, although the technical framework admits an extensionto hybrid-state systems;g (2) feedback interconnection of systemsIs not considered# though with certaibn additional conditions, theissue of closodness under feedback could be addressed; (3)existence of invariantsp equivalent representationa, and

* miniality are not treated here. While the present results arethus Incomplete, they do foaus on pragmatic Issues of Interest.

2

II. Background and Notation

Conceptual models for asynchronous state machines have been

employed primarily in automata theory, switching and

pomnications theory, and digital circuit design. Kohavi [4

provides a reasonable overview of the available methods and

technical Ksues, while Unger [5] Miller [6] have given detailed

design exatples for such syste. Petri nets 171 have als been

used to represent the behavior of asynchronous machines, though

they are often impractical for large systems. Kalman, Falb and

Arbib [8] have developed a conceptual framework for mathematical

systems thery which includes both automata theory and dynamic

systems theory, but this is based primarily on formal analogiesbetween existing theories; they did not explore the specific

properties of hybrid systems, and in particular discrete-state,

continuous time systems. In 191, an axiomatic framework for therepresentation of hybrid-state systems was introduced, and the

special case of asynchronous machines was briefly discussed.

The key technical issue in developing a representation for

asynchronous machines, aside from the choice of state variables,

is essentially one of existence of solutions. Two aspects of

this problem are: (1) On multiple input lines, transitions can

occur at arbitrarily close tines, which may lead to indeterminacy

in the state transition function. (2) Even with constant Inputs,

the state may switch at a rate which increases so rapidly with

tims that its value in not continuable with respect to time.

Both of these phenomena suggest the need to limit the maxiaum

Iwitchitig rate, or to impose a minimum switching delay time oan

the system. The avoidance of hazards and races is also an

important practical design consideration for switching circuits

that is closely related to these matbematical difficulties. it

an asynchronous circuit is not'designed to avoid these problems,

its actual behavior may be indeterminate In the sense that

- _ _ _ _ _ _ l- --i i I I

successive trials with identical" inputs produce different

results, i.e., the behavior depends on details of design and

implementation which are impractical to model. Limit cycle

oscillations, one manifestation of the second problem above, may

also occur in actual circuits, and their detailed structure is

again implementation-dependent. These phenomena are almost

always undesirable in practice; however, they are inherent in the

discontinuous nature of the systems under study. In the next

section, a further discussion is given of the way in which these

phenomena are represented.

The general setting for these results is dynamic systems

theory [l]. Only a slight generalization of the usual

definition of a dynamic system is required for the current

problem, due to the fact that the input and output sets admit

only a discrete topology (8, pp. 163-1641. The general notation

is introduced here, and is specialized in the following section.

nafIn~ton 2.1: A continuous-time ftnamig myjELt, E , on an

open interval TCR, consists of

U an InLt x",u an LAht AVm of functions u:?-G,Y an mLti MIk,I an gUiak UmqW of functions ytT-Y,

X a AtAls t,a Ltati M-W js TzxTxUzX, and a xjadan" uprsT&lUxX-. The State transition map is assumed to satisfy the

following semigroup axioms

i) IdNAuUk. JCt,t,u(.),zo)=z o for all tT, u(.)eU, zoeX.

(ii) £suinQ A" For any elemente tegtel, toct, and any.zoCe,

if ul(T)ft2(T) , toii the & o

4

(III) !zanitxitx. For any t 2 ltllto, all elements of T,any U.)CU, and any xd X, Jd(t 21to1u(-),xo).

The dynamic system Em(U#U,YvV,X,r) is thought of as thepair of equations

x(t) - 0(t't0,'u(.),X(tofl

for t~t0, both in T.

in this section, the properties of asynchronous machines# asa subclass of dynamic systems, are examined, and correspondingideal circuit realization results are given. some preliminarynotions are the following: notation follows that of the previoussection.

D5LD±ifll3.,Lt: A gartIal fungtion of finite rgnga (PlPO?)Is a partial function zi T-, where T4kR andZum(zliz 2 ... ,zq),qk.l is a finite set. A fUnalao gkf finkto rangnMOMl is a PrOrR such that T.T (Overbar denotes closure withrespect to the set operations on It).

h~iJ~tM .41 A point te! is a point ag- Ai M duai ofa VOn xsa1Z# with qJ2 r if z takes, oa mJtle. values 4- s everyopen set in T containing t. (Notes Por qwsl, points ofdiscontinuity cant xst).ae

~LtDL M =tI A VOR $s142, is a mnraI& uEkma

pMA.SM f Wor *I'0Mr'~h'eli24i~ tV4 r

max.s9rA:j.i~,mt epc oth aoge*aaoo

3~ 4Las. A 3011 a :iw*Z with a'ooutitable Wamb~iv of Points Of&ts~nkpRJtyig evogy fluite ,*ubintrvol of I is mewrakbl*.

ibis fellows frcm prqorties -of lao u'eawSable #ts.

Io4R~n, 3.,A: A V(WARt' uitk, a fir4t&number of pintsOf 4iscoentinuaity in every finite subinterval of T in termed a

ammzev aon4inUMi -f nnetL-W' at& U&J"~ -XMM' PCVMU).

11r'k _=.: if 2. is a PCLFOFR~o It Is. measurable.. in suary,EUVW60f 'J CWV gff 4:;- A.P1t tebrait iilbluiiaon).

The special class of dynamic systems of tttt'her*eL*s tUeclass of simple asnchr6noum machinles.

M~iLLBR I: A ginL a~rau uaakW~ IBM) is a

u-{11 *.,~*,sW a finite integerus u1r*)t u isI Cight-60tipousj

to. atftr that tbeir z at* *"-*m got al1

toas %9.txA MYP~

a..i. *LAq1M~W

These formal definitions are of interest only to the extent

that the functions j6 and r can be simplified, which is next shown

to be the case. This is analogous to the expression of the

transition mapping in terms of the transition matrix, for linear

systems, and the expression of this in turn as a matrix of

impulse-response functions that can be written in terms of a

finite set of parameters for linear time-invariant systems.

A SAN can be thought of as a system

x(t) - O(t,toU('),Xo)EX ; t>t O (3.1)

y(t) - r(t,toU(t),x(t))EY; t,toeT (3.2)

Since uPCFOFR and [t,t O] is a finite interval, u has a finiteset of points of discontinuity, denoted tl,...,t3 , such that

to<tl<t 2 ...<tN<t andUoCU tol T < t (3.3)

u(T) du, U, tij T < ti+1 1 i{l,N}(by right continuity)

UN U, tN< T < t

and furthermore, ui+l#ui.* (The degenerate cases m-1 and/or N-1

are not written out explicitly here]. By the causality property,

0 in (3.1) depends only on u(.) in the interval [t,t o ] . By the

semigroup property of %,

x(t) - 0(ttNuM,¢(tNtluMl,0(.... ,¢(tiltooXo) ... )

y(t) - r(t, touN,x(t)) (3.4)

**he case of a single input line with multiple values isconsidered here. for multiple linea, it is necessary todistinguish which input line has changed.

7[

i

where (by slight abuse of notation), uo...uN represent constant

functions of values uo...u N , respectively, on the intervals in

question.

From (3.4), it is apparent that the fundamental solutionp(t, T,U,X): TxTxUX ; t>i~to0; t,Te T

is sufficient to define the transition mapping. This function isdefined for a finite t of possible input-state combinations,and hence admits a sum-of-products (mini sum) decomposition inthe form

0(t,T,u,x) = V [0.. (t,T) (uAui)A(xAxJ)] (3*5)i~j il, .. ,m;j-l,...n

where V is the *or" operator and A is the "andu operator.

Furthermore, the range of lj is finite for each i,j, and

hence this function may be written as k0ij(t,T) = V xk I(t,Ti (T)) (3.6)we k-1

where T j(T)-{CTaeTIjlj(a,:i-xk), and V is the characteristic

function defined as

1ttETJC0 { otherwise

Furthermore, by definition the sets Tij(T) have the completeness

and orthogonality properties that for each Trt o

nU Tk (T) ={aT,aeT1 - T-(tT)

Tk.(T)IT"j (T) - (the empty set)

The readout map may be similarly specifiedt

(3.7)r(t, ToUX0 = V [rij(t,T) (uAui)A(xAxJl) i-1...M,

- . .J -,... n.

8 l

P k k kkaTlrij(t,r) kr.(t,T) = V y *(tsi (T)) ; S.. =yT Y

k=l i.

(3.8)

Summarizing these results for time-varying SAM's, one obtains

Theorem 3.11: Any SAM is completely characterized bycomplete mutually orthogonal subsets TiilT) and Sijlr),

parameterized by T and defined on T-(toT) where to is theinfimal element of T. These sets have the followinginterpretation: T~j(T) is the set of times in T that the statetakes on value xk when the system starts in state x at timeT and the constant input u=ui is applied. Si (T) is the set oftimes in T that the output takes on value when the system

starts at time T and the final state and input have values xi and

u1.

In general, in the time-invariant case, it is well-inot*that 0(tT,u(.),x) depends only on the difference t-T, and thatr(t,T,u(t),x(t)) depends only on u(t) and x(t). The implicationsof the time-invariant assumption for the finite-state case are

clear from the preceding constructions.

oroll&u. ,i: Any time-invariant SAN is completelycharacterized by the complete mutually orthogonal subsets T j, of

T; and the indicator function 8 . These sets have the following

interpretation: Ttj is the set of times in T, relative to an

initial time tYCT that the state takes on value xk when the

system starts in state xJ at time to and the constant input usu1

Lis applied. SOj takes value 1 if the current output value

y occurs whenever the current state takes value 0i and the

current input takes, value u1, and Is zero otherwise.

At this juncture, it should be noted that the assumptions

___1---.

about the input space allowed 6 (and r) to be defined only on theset of all finite partitions of TxT (e.g., all possible Onetsu

that cover the plane); but this is all that is required.

Secondly, it should be noted that the definition of a dynamic

system does not specify any space of Ostate-functionsa. In fact,

the sets Tk in Corollary 3 may be almost arbitrarily ill-

behaved. This means, in turn, that the rate of switching between

different state values may actually be infinite. This situation

exists because no continuity conditions have been imposed on 0 or

r. However, the assumption that the output be piecewise

continuous does imply that certain compatibility conditions exist

between the state transition and readout mappings.

C 3.2: When a SAM is characterized as in Theorem

3.1 or Corollary 3.1, certain compatibility conditions among the

sets T~i and Skj must exist in order to assure that the input-

output mapping Wtoxo:UY defined by~r(trtorU(t) wx(t) ) a r(ttto'U(t) wA(t'ttoU(-)'Xo))

takes U into (piecewise continuous functions) y for all xoEX. A

necessary condition (in addition to completeness and

orthogonality) for satisfaction of these conditions is that the

sets Tj(T) and Stj(T) have a finite number of boundary points in

every finite subinterval of T-IteT] for each TET, in Theorem

3.1; or that the sets Tfj have a finite number of boundary points

in every finite subinterval of T. in Corollary 3.1.

The necessary conditions of the Corollary can be established

by noting that they imply that the state function x(t) isS pieeise continuous In time and that the readout sap preserves

piecewise-€ontinuity (in the tilme-varying case). These

conditions are far from sufficient, thought -the readout map may

Omamk* sequences of states which occur at Infinite rates in the

ifi

-9

general case, so that *irregularity" in the state-transition

function is compensated by *regularity* in the readout map.

For Corollary 3.1, it is apparent that the most interesting

realization results will be obtained in the time-invariant case,

and hence only time-invariant SANs will be considered in the

sequel. A useful insight in provided by (3.1)z each input

transition triggers a pattern of state transitions (depending on

what "state" the system is in when it occurs) which persists only

until the next input transition. The output thus may be viewed

as resulting from a sequence of truncated cascades, each

truncated cascade being drawn from a finite (but possibly large)

set of waveforms. The definition of the output space for a SAN

in such that the output can only make a transition whenever

either the *state* or input makes a transition.

In considering circuit realization and synthesis issues, it

is desirable to consider the subset of SANs which have piecewise-

continuous state functions.

DmtLLnII±LI:3. A SAN is termed xsgulu if for all torT,

u(.)CU and z0CX, x(t) as defined by (3.1) is a PCFOFR when viewed

as a function from T to X.

Necessary conditions for a SAM to be regular have been

stated in Corollary 3.2. The ideal circuit realization for

regular SANs requires that certain idealized elementary building-

blocks be defined.

naLaLot£laa L; A SWigiJa ,,,lpilaw-r (DNUX) is a

memoryless element witb addeses lines, al...aq , where input aican assume n1 discrete values (aij,"1,.e..,ni), nlqI . Ini digitalinput lines labelled va..a each taking no values, and one

output line, w, taking no valuls, defined such that

, 11 .

bq

--- -- --- -- ----- --

w (t) =Wa~~2t.. qt t1 a W.. qW(3.9)

Renark 3J.: An (ideal) digital multiplexor can be synthesizedfrom (ideal) digital switches.

D.8inition : A digital function generator (DFG) for thefunction f:T Z, where TC-R is an open interval and Z-{zl...z q} isa finite set, has a binary-valued input line, b, and a q-valuedoutput line, z, defined such that

z(t) = f(t-T)

where T is the time of the most recent *- l transition on b. }Remark 3.J: Formally, the set of all discontinuity points ofb(t) is partitioned into times {T9 I } and {TlMJ, and the supremumof { 3il} (--,t) is defined to be T.

Dalnition 3.,: An ideal trigger element (IT)it fox thetransition ij9. on the input line v taking values vl...v q has abinary output b(t) such that

'1 for all t such that v(t-) = V(T) - V and

b(t) v v(t + r v(T) = vZ

0 otherwise

An ideal transition &lement (IT) with input line v detects alldiscontinuity points of v (i.e., its output is the union of allideal trigger elements for that input line).

3mak j: An ideal transition element is defined for allveFOFRI, while an ideal trigger element may be only defined forvcPCOI'R. A one-shot circuit approximates an ideal 1*1transition element with a binary Input line.

.12

3/nJiitin : A logical transformation (LT) for the

memoryless transformation G: V1Z where V-(v 1 ...vs ) andZ-zl... zq} are finite sets with input line v and output line z,

is defined such that

z(t) = Gv(t)

for (almost) all tET.

Symbols for these elements are shown in Figure 1. From thediscussion preceding Theorem 3.1 and the remarks prior to

Corollary 3.2, it is apparent that a SAM may be realized as shown

in Figure 2. Every input transition initializes the function

generators for Oij(.,*) as defined in (3.5), and the multiplexor

performs the logical operations (uAui)A(xAxJ) to select the

appropriate function. The transformation 8 in Figure 2 is

defined as in Corollary 3.1, based on the more general expression

y V V V y Si(uAu)A(xAx)i-1 Jul 1=1 (3.16)

The interconnection of elements in Figure 2 has the effect of

computing the transition mapping using the semigroup property

according to (3.1). These results are restated as follows.

Theorm 2: A time-invariant SAM satisfying the conditions

of Corollary 3.1 may be realized by one IT, an DFG's and one LT

as shown in Figure 2.*

The nature of the definitions of the ideal elements is such

that regularity is not required in Theorem 3.2. However, in

practice, actual circuit building blocks corresponding to these

*One inPut line is assumed) if there are maltIple input lines,one IT element for each line is required (see example).

13

1qDMUX FM 4z

ni

a, q

a) digital multiplexor b) digital functiongenerator

vT1 b v LF (G) z

c) ideal trigger element d) logical transformation

FIG. 1. IDEAL CIRCUIT ELEKENTS

U- LF

• .. U** _.-.----yOPO to,,)

FiG. 2. UALIZATI01 OF llU ABNCYEROUMO NMUAII (SA)

t -- -. i Ji4

ideal elements have built-in switching delays which limit their

performance. The nature of these limitations becomes more

apparent when the problem of synthesizing the DFG's in Figure 2

is considered. In addressing this problem, the relationship of

the present theory with traditional state-transition based

approaches will also become apparentl in fact, these approaches

constitute a rather special case of the foregoing theory. This

process requires a few more definitions (see Figure 3)1

De. nitln 3ii: An ideal resettable intearator (RI) has a

binary-valued input b and a real-valued output, h defined by

h(t) - t-tb

where tb is the time of the most recent 061 transition on b.

f inition 3.12: An ideal compurator (C) is a memoryless

element with two real inputs h1 and h2 and a binary output, b,

defined such that

b(t) - 1 whenever h 1 (t) > h 2 (t)

0 otherwise

Dh inition 2.: An ideal hybrid Stack (RS) with data

sequences {hk} and {Zk], k-,l,... has two binary inputs, b1 and

b2 , one real output h, and one discrete output zeZ, a finite set.

These are related as follows

h(t) = hkERz(t) - zkCZ

where k is the number of 0-4 transitions of b2 since the most

recent 9+l transition of bl, i.e., b1 resets the stack pointer

and b2 increments the stack pointer.

mMK 3A s ft6 Ideal hybrid stack can be apOrOzimated by atandem Interconnection of very long analog and digital shift

registers.

.. ..

b f h ,'

RI

a) resattable integrator b) comparatorb2

ih*KS0

c) hybrid stack element

FIG. 3. MORE IDEAL CIRCUIT ELEMENTS

(!13-I~hRAN CSI)

Corollary 3.3: If the conditions of Theorem 3.2 hold and

the SAM is regular, it may be represented as in Figure 2, with

the DFG's synthesized as in Figure 4.

Since the SAN is regular, the number of transitions of any

Oij(.,O} in Figure 2 is finite for any finite interval and

countable on [,0). Let fhk} denote the sequence of

intertransition times and let [zk) denote the sequence of output

values (elements of X) of Alj(.uI). These are stored in the HS

of Figure 4. The values zk are delivered at the appropriate

times by the clocking circuit consisting of the RI, C and IT,

while b1 (the input shown on Figure 2) correctly resets the

circuit whenever a new input value occurs.

The obvious limitation of Corollary 3.3 is that the stacks

(in general) have infinite memory requirements. However, there

is great variety of interesting special cases in which the stack

elements can be finitely generated. For instance, if the number

of transitions for each j is finite, each corresponding hybrid

stack can be terminated (formally) with an element hxmD, fKtas.

Upon reaching this element, the comparator output of Figure 4

will never change state, and hence the output z. will remain

unless and until b1 is tetriggered. In this case, the hybrid

stacks can be replaced with hybrid shift registers. A more

interesting, and quite general, special case is where the

elements of the stack may be recursively generated. This case

corresponds to the one considered in [9J and (111. Two final

definitions art required for this case.

n3.Utt An ideal real recursive function (Mli)

generator for the function ftR*R with kernel hdR has two binary

input lines bl and b2 and one real-valued output line h. These

are related as follows

L -.

h(t) .fk(ho)

where k is the number of 0+11 transitions of b2 since the most

recent On1 transition of b1 , and fk(.) denotes the k-th iterate

of f. Thus, bi serves as a reset line and b2 causes successive

iterates to be generated.

maxk 3.1: The properties of iterates of certain functions have

been studied by Klein and Kaliski [12]; they point out relations

to ergodic theory and coding theory.

/±flJ IL'±2/ jal: An ideal digital recursive function

generator (DRF) for the function g:X'*Z with kernelzoz,:m{zl...zgj, has two binary input lines b1 and b2 , and one

discrete output ztZ. These are related as followsz(t)-gk(zo)

where k is the number of I-l transitions on b2 since the most

recent I-l transition of b1 , and gk(.) denotes the k-th iterate

of g.

Bauazah IJI: In Definition 3.15, since Z is a finite set, gK.g,

for some finite K which is bounded by q2 . Thus, in the absence

of resets, the sequence of values assumed by z in a DRF is

necessarily periodic.

SCaioaa LA. There exist subsets of the regular SANs

which admit a finite parameterization. Onesuch subset is termed

the inaMnden y raeurmivlv Senaramted SAN's (IRQ8AN). These

are characterized by the existence of independent functional

recursions for the stack elements in Corollary 3.3 (Figure 4).

The realisation for these systms is sbown in Figure 6.

|1

RRF(Ifho) h DRFMg, zo) z

a) r4i -r§ursive function b) digital rencursive functiongenerator g0nerator

FIG. 5. COMPUTATIONAL ELEMNTSba/OR

C I

rZG. 6 . RMZXAt~c U3PI RUU IV3L GRIERATED

As an example, one mode of operation of the Signetics 2651

Programmable Communications Interface tm (PCI) is considered [13].

This is a universal synchronous/asynchronous data communications

controller chip which accepts programmed instructions from a

microprocessor and supports several serial communications

disciplines, both synchronous and asynchronous. It serializes

parallel data characters output from the microprocessor for

transmission and can simultaneously receive serial data and

convert it to parallel data for input to the microprocessor. It

is packaged in a 28-pin DIP. A block diagram of the PCI is shown

in Figure 7.

Since the PCI has 11 8-bit binary user-accessible registers,

a rough bound on the number of possible states is 288. In the

design of such devices, it is common to employ non-minimal(l)

realizations to simplify the internal logic. As a special case,

consider the lCZ 9onfigursd for as an asynchronous transmitter

for 5-bit haractets with no parity and two stop bits. In this

mod, the input lines can be taken as

?TC ezternal transmitter clock (pin 9)T = 8transmit enable (command register, bit 0)al chip enable (pin 11)

and the relevant outputs are

TxR0 1 transmitter ready (pin 15, and status registerbit I)

Tx=T transmitter empty (pin 18, and status register,bit 2)

TD transmitted data (pin 19)

In this mode, CR causes parallel data to be clocked into the

2L

a.w

oumo.

now7 WE I3KO IGUIB25 PORKALcOKWIAIOS I!3FCE(CI.KDIID RM 11

'moIt.wim31

"F-

transmit data holding register (THR). When TxEN is asserted by

programming the command register (and CTS is low), transmission

will commence and TIREN will be asserted (low). Normally, the

microprocessor will test this status bit, deposit new parallel

data and pulse CE, which resets TIRM (high). However, if new

data has not been received from the microprocessor by the timethe last data bit is being set, T;ERT is asserted (low) andmarking bits are transmitted at the output. If data is

subsequently received from the microprocessor, the end of the CE

pulse resets TjERT and TjO5; as soon as the data is clocked into

the THR, then TRDY is reset (low) and the new data istransmitted, etc.

For convenience, the state set may be chosen from the timingdiagram (Figure 8), to consist of states A, SlS 2 ,S3,S4, S5 ,BC,and D, where

A - start bit transmissionS1 - Do transmission--- DI transmissionS3 = D2 transmissionS4~ D3 transmission,S D, transmissionB5 sop bit 1 transmissionC stop bit 2 transmissionD = mark transmission

Thus there are a total of (2)3 3 8 input values and 9 statevalues, for a possible maximum of 8 x 9 = 72 state transition

functions to be generated. However, most of these are trivial.

Two examples will be considered: (1) the type of transition

occurring between A and 81, and (2) a transition between D and A

after !IR! has been recently cleared by a data transmission.

CaMa IJI: This case is defined by an input transition to

22

-- - - --- -- tmmm m

Im

TxRDY, TxEMT

ITeD° --°-' 1 -""" ' a' -3 4""0 A "-2 a'

BY S-IGETICS 2651 PCI. (5-BI CHRCES NO PRT,2

o p ts FI, TD (9,) h c anbe a

PIG. S. TIMING DIAGRAM FOR SYNCRONOUS CHARACTER TRANSMISSIONBY SIGNETICS 2651 PCI (5-BIT CHARACTERS, No PARITY, 2STOP BITS). MODIFIED PROM 113]

(TjC,TxEN, E) (0,0,) when the state is A. In this case there

is an almost immedpate (650 nsec) transition to state S1 , and theoutputs are (TxfT, TrT, TXD) - (,ilDo) .This can be realizedby the one-state timing circuit of Figure 6 which switches frob Ato h at 95o nsec after the transition.

Cain tims This cae is defined by an Input transition from

to (0,1,1) in state D, as hown at the end of the timingdiagram. In this case, there is a significant delay while the

holding register accepts the data and transfers it to the .

transmitter shift register for serialization. Thus the state

transition function shifts from D to A after & time which may be

note than one clock cycle. The (current) output is (,,)

This time delay may be realized as in Case (1), however one

22

should also proceed to consider the situation after the next

clock cycle, as shown in the timing diagram, which illustrates

the difference between a single input line having 8 values and 3

input lines having 2 values. For a single input line having 8

values, every new clock transition would reset this function

generator, which is incorrect in the present case. with 3 input

lines, only a transition on the CK line will reset this function

generator, while transitions on the other input lines (in

particular, T; C) will leave it unaffected. (See footnote follow-ing (3.3).)

Most of the other state transitions are well-approximated as

instantaneous, as in Case (1), and admit simplified realizations

when this approximation is acceptable. The state transition

functions are simple in this mode of operation because there is

an external clock, i.e., generally, state transitions cannot

occur without a clock transition, which is by definition an input

transition. The state transition functions are defined for

nstagt inputs, which is equivalent to predicting what happens

if the clock sta following the input transitionl The PCI

admits another mode of operation, where the transmitter clock is

taken from an internal baud rate generator. In this case, the

state transition functions for constant inputs would switch

spontaneously as dictated by the internal clock, and the

realization would contain an internal clocking mechanism that was

still practical to implement as described in Corollary 3.4.

_ _ _ _ 24

The realization theory presented in Section 3 is based on

semigroup theory and thus, unlike many common asynchronous

machine representations, allows one to draw on a wealth ot

intuition about dynamic system theory. In fact, only the

simplest case has been presented here: generalizations to

hybrid-state and pulse modulated systems are clearly possible,

for example. The results have some favorable properties wnich

one would not necessarily expect: it is relatively easy to

identify candidates for state variables for representative

applications, and furthermore, the functions (ij) required for

the realization may often be readily determined from experiments

on model systems. Finally, there is a significant class of

systems which can be accurately realized by a finite number of

elements and which correspond to practical digital devices. The

theory provides a rigorous basis for the delineation of Ocontrol"

(sequencing, and/or timing elements), and computational or "data

flow" elements.

References

1. AILUsLon N*k A EAAR TachnoLOUs Nov. 91, 1981,"2Shttla Launch Delay Attributed to Oil. Boftwareftablemis, pp. 21-21.

2. young# R.D., and Ewatny, B.G., rormmlation and Dynamicbehavior of a Variable Btructure Bevomaecanisd"n, Pnac.L22 nnt.m A&IL Cantzl, Cwd., Paper 3P-3,. (June,

Srlotteaville, VA.)

3. Wiapey, D.G., Johnson, ?.L., and Kaliski, N.S.,*Real nation of A/D and D/A Coders", Izac.. =11 ainL°Aan" CALn. Coa rn Paper WA-sA. (June 1981,Charlottesville, VA.)

4. Kohavi, 3., klc±B a4hl~a£~a a ~ rzNcGraw-BEll Publishing Co., Ltd., New Delhi, 1978.

< 25.

-- -,-v,&,--~

5. Unger, S.., Aynrnous g aicngCircuits, J. Wiley & Sons, Inc., New York, 1969.

6. Miller, R.E., Switching Thgory (Vol. II), J. Wiley &Sons, Inc., New York, 1966.

7. Peterson, J.L., 'Petri Nets", = mputing M LY-= ,Vol. 9, No. 3, pp. 223-252, 1977.

8. Kalman, R.E., Falb, P.L. and Arbib, K.A., Xg ir& inHathaaica1System Theory, McGraw-Hill, New York,1969.

9. Johnson, T.L., 'Finite-State Control of ContinuousP, Prc ,th I= Cngress, Helsinki, Finland,

1978.

10. Willems, J.C., and Hitter, S.K., "Controllability,Observability, Pole Allocatoin and StateReconstruction', I=i raD. Auto , Conftrol, Vol AC-16,No. 6, pp. 582-596 (1971).

11. Johnson, T.L., 'Analytic Models of Multitask,Processes Proc. 2th I= Conf. an DAinn And

Control, pp. 738-740, December 1981, San Diego, CA.

12. Klein, Q., and Kaliski, M., OFunctinal Equivalence in aClass of Autonomous One-Dimensional Nonlinear DiscreteTime Systems', Inform k Contl, Vol. 42, No. 2, 1979.

13. Signetics, Inc., 'Programmable Communications Interface(PCI)", Application Notes 2651-1, July 1978.

...... 6.... . _26

TOWIARDS A TEtORY of ASYNCHR.ONOUS. * ULDM CODERS AND THEIRAPPLICATIONS TO DISCRETZ-CONTROL Of CONTINUOUS PROCESSES~ p 34

f byDr. Martin 9. WaiSkand Dr. David C. ViUMe

Department of Electrical and Computer EngineeringNortheastern University

Bostoni, IA 02115 USA*Suppor~ted Ln Part by the United Stares Air Fore* Officeof Scientific %esearch, AFSC, Contract F4962O-42-C-3O83

ABSTRACT ways of extending those real-time coders to all of 12Thispapr dvelps noel f area-tie cder in a manner that allow weil-behaved devices to be

-- a t ap r f eelzed a coneofrteal-hichmp cod- constructed. This paper will approach the problem of--a ypeof gnerlize A/ covertr wich apsa.- specifying the "don't-cares" of these maps so as to

quences of n-tupics of real numbers into the binary ahyint-at (nir)relztnso efud4alphabet (0,I). Coders form an essential Interface Several examples of real-time coders will be given.

In the discrete-control environment between the con-tinuous state plant to be controlled. and the discrete Notation: For X a given set, X+ denotes the setcontroller itself. Theae are partially-specified of all finite-length non-null sequences of elements ofInput/output maps, and finite-state realizations of X.them are considered. Examples of the developed theoryare given. II. Genralizing the Simple Coder Model

Standard coders do not explicitly Imcoqporate time1. Introduction into their structure. One way to introduce time Into

the coder model is by making It an explicit Input toA key element In the theory of discrete-control the coder-its value reflecting the "arrival" ties of

of continuous system is the coder-the (hybrid) the other coder inputs. Doing this transfor= a coderinterface between the continuous-valued part of the C: -(, inonewchmp 3)4-{ l.

*system And the discrete-valued pert of the system However, this new coder is defined only ower a limited(typcaly amicrcomute). (inpy, 98Z~Kalaki subset of RX) + - naely those finite length strings

&dLemons, 1980) o h omTecoder io a kind of generalized analog-to- o h om

dgtlconverter, which In the discrete-tIme case,naps squseces of eloomts of an. for some n?-l, into (rl.,tl) ... (rk~tk)a finite set, which we may take as the binary alphabet wiht t2(.. kineiealasrvsf-(0,I). An extensive study of coder design. using boh wards. *(Due to practical restrictions Inherent inalgebraic methods (draw from standard "finite" auto- phsclIlentiosfcdrtistitymata theory) and linguistic methods (based upon a con- phsclile ntiosfcdrtistitymonotone sequence of time value* is fu~rther constrained:capt of languages defined over real alphabet&) has there exists some real value TI such that, for 2.,already been pursuad by the authors in the synchronous k, tj - t 1-l )- TI. This reflects the fact that inputcase, at cited above. This case, wherein both the changes cannot occur arbitrarily quickly.)transitions of the discrete-time plant, and of the turertohiclsofodse ra-im(finite-state) controller, occur In synchrony, under coders," which me will demote as ATC's.I the regulation of an external clock, is clearly aspea" cu, and many application contexts, whereactions are triggerd, by ,aents occuing asynchroncosy, III. An luesple: A lesl-TIme Discrete Integrator

* do sat fall under the umrella of this model. Thus,In this paper, a theory of coders which are intrinsi- Consider the following "imeegrator-Ilhe" *ssale.cally asceronous, and ich explicitly iWncoprate If the coder input to (C1 ,l) o then the weaput Is 0.the motion of time, Is Initiated. If the coder Input Is (rl~tl)... (rk.tk) with k'1, then

lie sdl asymchronous activity by vun Ideal the output io camputed as follow:$ We form the am:"samples and holds" to record the exect mocent In timeI at which aves occur. this allow us to define wheat rl~t2-tl) + r2Ct3-s) +4... r k-l (tk-t k-I)we 411l call "real-time coders." to their smstelemeataty fans they sode strings of real numbers and if the suin is greeter then 0 the output in 11are m8"s defined from a set S Into (0,l) where S eon- otherwise It to 0.sLats omly of fianie-lengt strins of pairs of the Gee my of Implementing sch a coer Is to extendI. fog (r,t), where "r" Is a real uber (produced by its dom&to a f (%a)+and to then employ pro-som process) and "t" is the time at which "r" Is viously developed techniques (Kaliski and Lemons. 190;"seen." Because time always Increases, the strings In VitpeM. 1I6). The mwet natural my of exteadirtr It IsS are alweys of the fore by just uslng the above rule for arlbitrary strings

CuL~vl) ... (ok,*k). regardless of the salurea of the(rl~t) .. (r~tk)sequence Vl'VI.... owh. That is, If the saieaene to of

where the t-walue fo=m a monoosm iaftasia Sequence length 1, the output is S1 If its length Is greaterand whets the differences t2-tl. 0-t0,..., are always then It we compute the ahewe saw an test Its poaltive-greater than soe apriari lever bound TI. mesa. If it is positive, them the output Is I g other-

capable *9 implmfLag sh ap is te view them 40 Now thia exeded I/0 amp caon be realised by the

22)eeiaload *mspeial retice -oao s t mw maei tucdsdw"Lo era

ecpossible sum of the form: f(AC)-f(BC).

n102-1) + u k-i (vk-v k-I) - ta vk A mucb more powerful "converse" to Lomes I is alsowhe. he uk inthse amLa iesam. he tat- true when 5 has the property that "once a aralaewher th "" In hes sus tothesam. Th strt- "S" it never "sets back in," i.e.. *If A to (01 i

Ing state of the coder is the congruence class of the mot In S, then AB is not In S for any 3 in (0,1).null string. which consists of strings whose sum, Call such S (for Lack of a better expression) "tightlyformed as above, Is 0 and whose WUk (lst term) La 0. structmuted."-4.e text state and output maps are as follows: (Noechs: they' are well-defined.) Linea 2: Let S be a tightly structured subset of

Next state map: (0.1). and P a partition (Pi,...,1PO ofS, for whichthe partition propet-7 holds for f:S -- %0,.1;. Then

S(lv)... (uk.vk) ]. (c~d) -- there exists an excansam Z of f to all of 0,i'+ whichis finite state realizable.

((uL~vi) ... (uk,vk) (c~d) I hs w ae

output cap: Theorem 1. (Fundametal Realizability Theorem

£(ul.vl) ... (uk~vk) 3. (c,d) >frPrilySei e / 'as

POS ( ul(v2-vl) + ... + ua k-1 (wit-v k-1) Lotr S he a tightly structured subset of (0,11..Let f be an 1/0 nap from S into (0,I). Then there

+ uk Cd-wit)) exists an extension S of f to all of (0.1)4 Which Lafinite-state realizable If end only if 5 obeys the

where P05(x Is 1 If and only If x is -0. "partition property" with respect to f:

A natural enbedding of the state set Into RzR "There exists a finite partition P..(fl,.,Pk)*Xiss- of S such that for any jal,...k, and any A

andBIn lj, if C InO.)+Is such that AC(ul~vi) ... (uk~vk) 3 'and BC are both in S, then AC and BC belong

to the same block ft of P as wrell (0 setCul(v2-vl) + ... + u k-i (yb-v k-l)-uk vk~uk) necessarily equal to J). Furthermore.

fCAC)f(Ic)."':slnl this embedding, a two-dimensional reas-

:to= of this coder can be specified as follows:V. Generalizing to 1/0 Maps an RxR

Next state map:Al2 of this Seneralizes to maps on gm, using te

(%.7). (c,d) -- > (x + (y-c)d, c) concept of a finita-autanaton defined over a realalphabet CgImpey. 1982). In particular It generaliess

Output nap* to Real-Tize Coders, where, by the very mature of thedomains of such coders (the set S), the tl*t structure

Cx,,'), (c.d) -> POS(% + yd) criterion La mew. (The Input to & RTC go" "bad" as

Startng satesciently far, oncea it becomes bad It remains bed).gobegin wt nitiiedfate fadtr

Colo)ninistic finite-state reel autematon (ISM. x is a753£ If there exist a finite number of states Qi, ... 6

IV The Kay Theoretical Problem: Structured Qit, and, "aisted with each state Qj * j'.l, ... .kv aExtensions of Partially Specified 1/0 Maps partition PJ of the real numbers. A single state of H

Is labelled as the starting stage of X. aid certainA~s :he above esseple so cleerly Indicates, in states. of M are classified as accepting states. Them

*r~ oefcieyraimra-iecdrad to XIs maual soitdwith a coder CKt at- (.)OMP17 te aread deeloed teor fo synhroous Clearly all of this generalize to Impute that are incoders. ~ ~ ~ ~ ~ ~ ~ xi voms Vago.ytoetn h T rm K-W alac fiaite-stage autoeoo a 2-Its onsraied oeaf toallof RW+.It btsam- dimensional 111£, and, for brevity demote se a device

tension can be wd na wa ht-rwre-teby the Symbols 20153£.intriasic structure of the tIC, then we have effected The motioe of a mom-datermislattc M33, alSOthe development ofage esg ol e etit smeralins In a straightfor'ward member. in pertsarcu-. attention to finite-state extenaions In this paper. If ome *aotiaqe with ach state Oj a moms C4 of a

Towds this @sal, let us begin our -diseussion (or ho thma one hias a am-detaimistie, neupleteLY'4wn the follwing leens dram froM the basnic Ideas specified "Ul. The pwoof that suck a dewie is amys2f automata thory. The pceefs of &em& I an 2 are equivelen:to 1i deteaimiatic MUA is viutaupatraighwforier d n ttd for lea f c" . Identical to th' e one used iS fimits ausemata eory.

L~al:Lea fU -2 0 1) ee gien I/ map we omit 2ite.tlycryeerf*reLemm It ot fs --. (o0 bea $I" 1/ maLImes I and 2Imdaeycm* a ~

Weined on a subset S of 0,1.Suppose& that thare tine coders. as their domains are tightly structured.Is a finite stt nehso X hs re"a&se me sn M gI Specifically, lot us formally defines a RM~ (This$ of f to to'*+-. Then there to a finite partition definition complemens the WaoM&s Noe give% InP1,. ..,Ph) of I osbat the fellawing property to settle& Xl ao"e.)

9 rue, for all strins A~ gmdAin i,.Dsfinuition: A Real-flag Coder (UQC S Wn aop

If A "d At*eImS e Sftblakof P. A C I& fm lass(0.0 bwer'is be fllow&* submit of

Ifs af Uedgsoty te Oe& 46at A 46d a we WI) ad I a (VwIO) -MACt~t) such OhatA2 -

uhere -AX is some fixed real constant. (Q2,0) Q~43 (412.1) -->Q1

Sots that S Is tightly Structured. There is (Q3,0) Q2 (Q3.1) -->Q3nothing in the proofs of !a= I and 2 that dependsupon the fact that the inu-s to the automata are 0 or1. Thus rhe Lams generalize to the reel-tine coder Vt. Conclusionscase and, thus. ve have:

The process of "cretively" selecting "don't care"Theorem 2: V?''adc:al Fialte-State Realiz- values so as to provide for meaningful extensions ofi.±: s~r f :r ?aa.-e dara) real-tine coders is a fundamental open research problem

of this theory. Mfany more general coder types exist."Let JMC be a gi7on reel-time coder. ?ben RTC such as the shift-finitary coders (VIRey. 1930).is finite-state realizable If and only If there Current work, then. to seein to extend the aboveexists a fiats partition (P,... ,Pk) of its preliminary results to these more general code"., asdomain S auth that far ay J-l,. . .,k, anwd well as to look at other vays of modelling asynchrony&=y A. 3 in Pi. If C !n U1i is such that AC In coder design.ead 3C are bjoth in S then AC and SC are both

necessarilyr equal to J) and 2TC(AC)-ftTC(BC)." VII. Referencs

76-ere is an explicit techecI&= for constructing Kaliski, N. 1. and Lemons. K.. "Discrete-Coding. oft'-s !~Izce-s:&:* realizatica whea the partition Continuous Valued Signals," 1980 Conference. on Inform-pro~arzy holds (as in :he ;rnof of Lome 2.) tion Sciences end Systess. Princeton University,

Princeton. NJ.

An ExapI& of a vi=4t&ay leal-Tine Coder Vinpey. D. G.,* "linite-State Control of Discrete-TineContinuous Processes: An Aatomta-otivated Approach."

Z* gIev here a senpie exnpIs based upon Vinpey Ph.D. Thesis, KIT Department of Ilectrical Engineering

~192) to Illustrate; the conceptsaboe A Unitary and Computer Science, June 1962.

real-ti=e Integrator considered earlier Iq definitely Vinpey, D. G., Johnson, T. L., and Calisi, M. 3.,n2c fUnitary.. "Realization of AID and D/A Coders." 1L981 .MCC, 3

Czzse;!er the f*22owIng real-time coder. Charlottesville, VA, June 1931. I

ZC(rltl) -0 if C1l))-0; If not

0 If (ek+th) >0 0 and the numbher ofmoz-negazira suams r4l..r -l + t k- Iis even or zero. i

!s i: f!Initesc&ae realizable? And, If so, whatIs a realization for It? We proceed to answer these;-.:s:±czs.

!ue doz-aft S of this coder satisfies the partitionprpryfor the following partition {Pl. P2, W3s

79. a s-.r~zgs haw-ng am odd numer of non-negatIve rme wita the last sun negative)

72 - istring having en odd umbor of uenegative sums with the last im nnnegatiM)

sad

P3 - strings baving aa even numer (or stwo)mon-inegtive $WIms

~~1 17 our developments Above, them, 2C Is MUSiTaY.tfar:- It is stay to directly construct a I2WUU fer

based upon the Unas atvve. lthe, then do a_I~tyit is clear that, as with finitaVY. We 0tdyeE

;,"Mtisor WAd a finite state aohies (gum, 1302)IMe qU02gise' Q Outputs 0 ItfJ4& uito 3W 41 ethe

val : stuts I. The finite state m be K be*Woe sztas oemepa to n1, 1i. awn itrnQ1, Q2, and 413. 9) ise the stasting 0""t. ase

areepting Veates are 91, d4 Q). ase tgmnitime weas folows:

E ORAFDU[-: JAI;LAnY, 1983

TOUA7DS A THEOPY OF FINITARY ASYiiCI!RO[OUS CODERS

1) Generalizing the Simple Coder Eodel

The standard coders previously considered by us did notexplicitly incorporate time into their structure. Rather,tiue was implicit, synchronous, and "regular". It madesense to tall: about the kth input or output (or state), andthe value of k automatically stepped through the integers.

About the simplest way to introduce time into the coder...odel is by making it an explicit input to the coder -- itsvalue reflecting the "arrival" time of the other coderInputs. Doin- this transforms a coder C:R -->(O,1} into onewhich uaps (Rxr)+-->0,1}. However, this new coder (call itETC for "real-time coder") is defined only over a limitedsubset of (ExR)+ -- namely those finite length strings ofthe form:

(rl,tl) ... (rk,tk)

with tl < t2 < ... < tk, since time always movesfcrwards.

Due to practical restrictions inherent in physicalitiplementations of coders, this strictly monotone sequenceof time values is further constrained: there exists somereal value TI (for "Temporal Input" constraint) such that,for J=2,...,k, tj - t J-1 >= TI.

(Theoretically, at least, such constraints are ergodic-theoretic in nature -- a notion to be explored in greaterdepth during the remainder of this contract year.)

The output of the RTC may still be regarded as a binarynumber b = 0, or 1, and may be viewed as occuring within TOseconds after the last input arrives. (TO for "TemporalOutput" constraint). Note that various racing problems, andhazards, and such, must be dealt with- to physicallyimplement such a device. For example, TO may have to besmaller than TI to assure that the correct output is read.

2. An Example: A Real-Time Discrete Integrator

Consider the following "integrator-like" example. Ifthe coder input is (rl,tl), then the output is 0. If thecoder input is (rl,tl) ... (rk,tk), with k>1, then theoutput is computed as follows: We form the sum:

ri(t2-tl) + r2(t3-t2) + ... r k-1 (tk-t k-i)

L ___________________

Page 2

If the su r is Zreater than 0 the output is 1;otherwise it is 0.

One way of implementing such a coder is to extend itsdorain to all of (RxE)+ and to then employ the standardtechniques we have already developed. The most natural wayof extending it is by just using the above rule forarbitrary strings (ul,vl) ... (uk,vk), regardless of thenature of the sequence vl,v2,...,vk. That is, if thesequence is of length 1, the output is 0; if its length isgreater than 1, we compute the above sum and test itspositivity. If it is positive, then the output is 1;otherwise it is 0.

.ow this extended I/O map can be realized by thecustcmary techniques of Nerode equivalence. Let us see whatthe 'erode equivalence relation is in this case. First,consider two strings of length greater than 1. Two strings(ul,vl)...(uk,vk) and (pi,ql)...(pm,qm) will be equivalentif and only if the two sums below are equal for any(al,bl).. .(aj,bj):

ul(v2-vl) + ... + u k-1 (vk-v k-1) + uk(bl-vk)

+ al(b2-bl) + ... + a J-l(bj-b J-1) and

pl(q2-ql) + ... + p m-1 (qm-q m-i) + pm(bl-qm)

+ al(b2-bl) + ... + a J-1(bj-b J-i)

(11ote that, strictly speaking , we want POS of each sumtc be equal, where POS is the map POS(x)=1 if and only if x

is positive; otherwise it is 0. It should be clear,hovever, that if the sums differ, we can append yet anotherterm (a j+1 b J 1) which can cause differences in signs ofthese new sums, and thus differences in the PO-values.)

If j=1 wse may ignore the second line in the twoexpressions above, and, in fact, may ignore its role in theequivalence condition completely, as it is identical in bothexpressions. Thus the two original strings will beequivalent if and only if the two first lines above areequal, for any bl whatsoever. Setting bl to 0, inparticular, forces equality of the following twoexpressions:

ul(v2-vl) + ... u k-1 (vk-v k-i) - uk vk and

pl(q2-ql) + ... + p m-1 (qm-q m-1) - pm qm

Let us denote this common value by A. Then we mustalso have that A~uk bl a A n- li, for any real number bl.Since bl may be 1, it immediately follows that uk must infact equal pm.

J ",- m . ---- ,m m m i nm n a

_ -------- - . -- °

. Page 3

Summarizing, then, a necessary and sufficient conditionfor two strings (ul,vl)...(uk,vk) and (F1,qi)... (pm,qn) oflength greater than one to be Nerode equivalent is thatuc:p2 and that

ui(v2-vl) + u k-1 (vk-v k-1) - uk vk

pl(q2-ql) + ... p m-1 (qm-q n-1) - pni qm

::ow let us turn to strings of length 1. First supposethat we have two strings, both of length 1. Call themCul,vl) and (pl,ql). It must be that for any

(al,bl)...(aj,bj), J>=1,

u1(bi-vl) + ai(b2-bl) + a j-1(bj-b j-1)

pi(bl-ql) + ai(b2-bl) + . a j-i(bj-b j-1)

where the terms on the right are not present if j=1.-t is immediate that we must have -ul v1 = -pl q1, andu =p1.

For one string of length 1, and the other of lengthgreater than one, it can similarly be deduced that therequisite condition for equivalence is that, calling thestrings (ul,vl) and (pl,ql)...(pm,qm):

u1=pn and

-ul v1 = p1(q2-ql) + ... + p m-1 (qm-q m-i) - pm qm

As for the null string NL, a similar condition isq derived: fL is equivalent to (ul,vl).. .(uk,vk), k>=1 if and

only if:

uk=O and

ul(v2-vl) + ... u k-1(vk-v k-i) = 0

All of this suggests that the Nerode state-reducedmodel of this real time coder is the following. There isone state for each possible sum of the form:

ul(v2-vi) + ... + u k-1 (vk-v k-1) - uk vk

where the "uk" in these sums is the same. The startingstate of the coder is the congruence class of NL, the nullstring, which consists of strings whose sum, formed asabove, is 0 and whose "uk" (last term) is 0. The next stateand output maps are as follows: (Note that they arewell-defined)

Next state map:

• , i-- . . .|- ..... .

Page 4

[ (ul,vl) ... (uk,vk)], (c,d) --- >

C (ul,vl) ... (uk,vk) (c,d) ]

Output map:

[ (ul,vl) ... (uk,vk)], (c,d) --- >

POS( ul(v2-vl) + . + u k-1 (vk-v k-i)

+ uk (d- vk) )

A natural embedding of the state set into RxR exists --

[ (ul,vi) ... (uk,vk) ] --- >

( ul(v2-vl) + + u k-1 (vk-v k-1)-uk vk,

uk)

Using this embedding, a two-dimensional realization ofthi4s coder can be specified as follows:

Next state map:

--------------

(x,y), (c,d) --- > (x + (y-c)d, c)

Output map:

(x,y), (c,d) --- > POS(x + yd)

Starting state:

(0,0)

A diagram of this coder is shown in Figure 1. Notethat it is assumed in this diagram that the inputs areconnected to event-driven ideal samples and holds. Eventsoccur at the input times, and the samples and holds serve to"capture" the inputs at these times, and the timesthemselves. In this diagram we assume that the output isread TO seconds after the input occurs.

p. I . .

• "; : • ' I I i l " ' m

V Page

The Key Theoretical Problem: Structured

Extensions of Partially Specified I/O Iaps

As the above example so clearly indicates, in order toeffectively realize real-time coders, and to employ thealready developed theory for synchronous coders, we mustfind a good way to extend the ETC from its constraineddoizain to all of (RxR)+. If this extension can be made in away that "preserves" the intrinsic structure of the ETC,then we have effected the development of a useful designtool.

Towards this goal, let us begin our discussion with thefollowing lemmas , drawn from the basic ideas of automatatheory:

LemLa 1: Let f:S -- > {0,1) be a given I/O map definedon a subset S of {0,1)+. Suppose that there is a finitestate machine 1 which realizes an extension g of f to{0,11+. Then there is a finite partition P={P1,...,Pk} of Ssuch that the following property is true, for all strings Aand B in S:

If A and B are in the same block of P, and C in {0,1}+s any string for which both AC and BC are in S, then AC and_C are also in the same block of P (which is not necessarilythe block that A and B are in), and f(AC)=f(BC).

Proof of Lemma 1: We may assume without loss ofgenerality that V is a Moore machine; call its starting

state Qi. Call the remaining states of H Q2,...,Qk. Definethe partition P as follows: Block Pj of P, j=1,...,kconsists of those strings of S which take state QI to stateQJ. Clearly (P1,...,Pk) partitions S. If A and B arestrings in S that are in the same block of P, say PJ, thenfor any C whatsoever, AC and BC are in the same block of P.In are in S. Furthermore, by the very definition of "11realizes g" it must be that g(AC)=g(BC) as AC and BC lead Mto the same state. When AC and BC are both in S the g inthe above equality may be replaced by f since g is anextension of f. QED

A much more powerful "converse" to Lemma 1 is also truewhen S has the property that *once a string leaves S" itnever "gets back in", i.e. if A in (0,1)+ is not in S, thenAB is not in S for any B in (0,1)+. Call such S (for lackof a better expression) "tightly structured".

Lemma 2: Let S be a tightly structured subset of(0,1)+ and P a partition {P1,...,Pk) of S, , for which thepartition property holds for f:S -- > (0,1). Then thereexists an extension g of f to all of (0,1)+ which is finitestate realizable.

Pag e

Proof of Lemra 2: Consider the non-deterministicfinite-state 1Voore machine M defined as follows. Thestarting state of ; will be called QO. There is a specialstate of (where strings not in S will take 1i) that will becalled QiCS. The remaining states of I correspond to thenumber of blocks in 2 in the following way. For j=1,...,k,look at Pj. 1. all the strings in Pj have that propertythat f maps them to 0, then create just a single state QjO;if all the strings have the property that f maps them to 1,then create just a single state QJ1. Otherwise create twostates QjO and Qjl. Thus V! contains at least k+2 states,and at most 2k+2; in any case it is finite state.

The accepting states of 1 will be the QJ1's, ifPresent, for j=1,...,k. The transitions of 1 are defined asfollows: First let's look at QO. If 0 is in S, then 0 isin some block of P, say Pj. Direct a 0-arrow from QO to QjOif f(O)=O; otherwise direct the 0-arrow to Qjl. If 0 isnot in S then direct the 0-arrow to QNS. The 1-arrow fromQO can similarly be constructed, based upon whether or not 1is in S, and, if so, what the value of f(1) is. Next weconsider U.S. This will be considered to be a "dead state"in the sense that once entered, it cannot be left. Thusbeth the 0-arrow, and the 1-arrow, go from QNS to QNS. (Aswe will see below, all strings that are not in S lead toQI:S; by the tight structure constraint on QNS, once QNS isentered it is appropriate to get trapped there.)

Finally we turn to the transitions for the remainingstates. Consider, for a given J, the states QJO and Oil,or, if just one is present, whichever it happens to be.Consider the problem of drawing the O-arrow. Now thestrings in Pj divide into two classes -- those which, whenfollowed by 0, are still in S, and those which, whenfollowed by 0 are no longer in S. By the partition propertycn S, all of the strings so "continuable" by 0 are elementsof the same block of P, say Pm, and the continued stringshave the same f-value. We can use the above remarks to draw0-arrows from QjO and/or QJl as follows: Suppose that thereis a state QjO. This means that there are strings in PJ forwhich f is 0. If any of these strings are "continuable" by0, by the above remark, they all have the f-values whencontinued, and all belong to Pm, when continued. Call thef-value of the continued strings "b" (b=0,1). Clearly, byour very construction, state Qmb must be present. Draw a0-arrow from QJO to Qmb. If Pj also contains strings forwhich f is 0, but which are not continuable, draw a 0-arrowfrom QJO to QNS. Thus one or two 0-arrows will emanate fromQJO, going to either QNS, Qmb, or both. A similar procedurecan be used to draw 0-arrow(s) from QJ1. Furthermore thesame procedure can be used to draw the 1-arrows from QJO andQJ1.

p

Page 7

It is easy to see that if al,...,an is a string in Sthat ('ay the tight structure of S), each of the strirt-s al,al a2, ... , al .... .,a n-1 is in S an that there is a pathi n ., starting at CO, and remaining thereafter among the

's. 1' i3 also easy to see that accepts al,...,an in Sif and only if f(al ,...,an) -. ;rinrsthat are not in S, in general, but the only strings in Sthat are accepted are those for which f is 1.

I'ow 11 can be converted to an equivalent deterministic.oore nachine I1' which accepts exactly the same strings asdoes i.. Let us define g:{0,1}+ -- > [0,1} by g(A)=1 if andonly if V', accepts A. By our remarks in the precedingparagraph, g is an extension of f. Obviously g is finite-ztate realizable (by 1I'). The proof is complete. QED

SUe nay summarize the above two Lemmas with thefollowing Theorem, which we call the Fundamentalrealizability Theorem for Partially-Specified I/O 1aps:

Theorem 1: (Fundamental Realizability Theorem forPartially Specified I/O Maps)

Let S be a tightly structured subset of [0,11+. Let fbe an I/O map from S into {0,1}. Then there exists ane~xtension g of f to all of 10,11+ which is finite-staterealizable if and only if S obeys the "partition property"1ith respect to f:

" There exists a finite partition P={PI,...,Pk}

of S such that for any j=1,...,k, and any A and B

in P1, if C in {0,11+ is such that AC and BC are

both in S, then AC and BC belong to the same block

of Pm of P as well (m not necessarily equal to k).

Furthermore f(AC):f(BC)."

Generalizing to I/0 maps on RxR:

In the pages that follow we will show that all of thisgeneralizes to maps on RxR, using a form of our earlierdefinition of finite-automaton defined over real alphabets.In particular this will generalize to Real-Time Coders,where, by __ ;ory nature of the domains of such coders (theset S), the tight structure criterion is met. (The input toa RTC goes "bad" as soon as the time-coordinate does notadvance sufficiently far; once it becomes bad it remainsj bad".--

Page 8

'-e will culminate, therefore, in the fcllowing variantof Theorem 1:

Theoren 2: (Fundamental Realizability Theorem for2eal-Time Coders)

Let RTC be a real-time coder. Then RTC is finite-state realizable (in the finitary coder sense) if and onlyif its domain obeys the partition property.

We beGin we an intuittve definition of a deterministicfinite-state real automaton (FSRA). 11 is a FSRA if thereexist a finite number of states Qi,...,Qk, and, associatedwith each state Qj, j=1,...,k, a partition Pj of the realnumbers. A single state of 14 is labelled as the startingztate of 1', and certain states of Y are classified asaccepting states. Thus 14 is naturally associated with acoder C1: 19 -- > {0,11. Clearly all of this generalizes toinputs that are in RxR. We call such a finite-stateautomaton a 2-dimensional FSRA, and, for brevity denote suchL device by the symbols 2DFSRA.

The notion of a non-deterministic FSRA also generalizesin a straightforward manner. In particular if oneassociates with each state Qj a cover C of R (or RxR) thenone has a non-deterministic, completely specified FSRA. Theproof that such a device is always equivalent to adeterministic FSRA is virtually identical to the one used infinite automata theory. .e sketch it below.

Cne creates a deterministic FSRA by initially defininga state for each of the non-empty subsets of [QI,.. .Qk).The starting state of this new machine is the same state(singleton subset) as that of the non-deterministic one.The accepting states are any states whose correspondingsubsets contains at least one accepting state of thenon-deterministic machine. To define the transitions of a3iven state we look at each state in the associated subset.

Ile pick a real number, say r, and ask to what states in theoriginal non-deterministic machine r takes us. Ile take theunion of all of these r-mapped states (for each state in theassociated subset) and that subset of {Q1,...,Qk) is wherewe send r to. We repeat this for each real number, therebyassociating a partition of the reals (or of RxR in thetwo-dimensional case) with each subset of {Q1,...,Qk). Therest of the proof is clear. We can easily demonstrate thatthe deterministic machine and the non-deterministic oneaccept exactly the same set of strings.

The proofs of Lemmas 1 and 2 immediately carry over forreal time coders, as their domains are tightly structured.Specifically, let us formally define a RTC:

Page 9

Definition: A Real-Time Coder (RTC) is any map from Sinto {0,11 ihere S is the following subset of RxR:

S {rl,tl) .... (rk,tk) such that

t2-t1>TI,., tk- t k-1 > TI I

where TI is some fixed real constant.

lUote that S is tightly structured. There is nothing inthe proofs of Lemmas 1 and 2 that depends upon the fact thatthe inputs to the automata are 0 or 1. Thus the Lemmas

* generalize to the real-time coder case and, thus, 'rem 2holds. We repeat its statement below and give, i. ,ection

* 2.1, a detailed example.

"Let RTC be a given real-time coder. Then RTC is

finite-state realizable if and only if there exists

a finite partition [P1,...,Pk} of its domain S such

that for any J=1,...,k, and any A,B in Pj, if C

in 2xR is such that AC and BC are both in 3 then AC

and BC are both in the same block Pm of the partition

(m not necessarily equal to J) and RTC(AC)fRTC(BC).*

There is an explicit mechanism for constructing thefinite-state realization when the partition property holds.

4. An Example of a Finitary Real-Time Coder

le give here a simple example based upon Wimpey (1982)to illustrate the concepts above.

Consider the following real-time coder.

FTC((rl,t1)) 2 0 if (r1 tl)>=O; 1 if not

RTC((r1,tl) .... (rk,tk)) z

0 if (rk+tk)>x 0 and the number of

non-negative sums rl+tl,...,

r k-1 + t k-1 is even or zero.

1 otherwise

Is it finite-state realizable? And, it so, what I a

realization for it? We proceed to answer these questions.

T ______.. .

Page 10

The domain S of this coder satisfies the partitionproperty for the following partition {P1,P2,P3}:

P1 { strings having an odd number of non-negative

sums with the last sum negative)

P2 = strings having an odd number of non-negative

sums with the last sum non-negative} and

P3 { strings having an even number (or zero)

non-negative sums}

Note that if strings A and B in Pi are such that AC andBC are in P1 then RTC(AC)=RTC(BC)=1; if they are in P2 thenFTC(AC)=RTC(BC)=0. If AC and BC are in P3 thenRTC(AC)=RTC(BC)=1. Sicilarlj if A and B are in P2 then ifAC and EC are in P1 the common output value is 1; if theyare in P2 the common output value is 0. If they are in P3the output is 1. Finally if A and B are in P3 then thecommon outputs are (for termination in P1,P2, and P3respectively) 1,0,and 1.

By our developments above, then, RTC is finitary. Infact it is easy to directly construct a 2DFSRA for RTC,based upon the arguments above. Rather than do so directlyit is clear that, as with finitary, ordinary coders, RTC mayOe realized as a cascade of a simple quantizer and a finitestate machine:

The quantizer Q outputs 0 if rj+tj is >=0; otherwiseit outputs 1. The finite state machine X has three statescorresponding to P1,P2, and P3. Call them Q1,Q2, and Q3.Q3 is the starting state. The accepting states are q1, andP'3. The transitions are as follows:

(QI,0) -- > Q3 (Q2,1) -- >Q1

(Q2,0) -- > Q3 (Q2,1) -- >Q3

(Q ,0) - Q2 (Q3,1) -- >Q3

i..I. -.;-A THEORY OF ORBITAL BEHAVIOR IN A CLASS OF NONLINEAR

SYSTEMS: "CHAOS" AND A SIGNATURE-BASED APPROACH

by M.Kaliski, Northeastern University

Boston, MA *

S.Yunkap Kwankam, Universite do Yaounde,

Yaounde, CAMEROUN

P. Halpern, Northeastern University,

Boston, MA

(s This author's work has been supported in part by tue

United States Air Foros Offioe of Scientift Research,

AFSC, Contract Number FP9620-82-C-0080.)

M ailing M&dress: Dr. Martin E. KaliskiProfessor and Director of Ciputer EngineeringNortheastern University

; Boston I NA

02115

1A,

-- - - - -- - - -

Page 2

Abstract

A theory of orbital behavior in certain autonomous

one-dimensional nonlinear systems is pursued, using a

approach based upon the concept of orbital signature.

Particular attention is paid to the fixed point structure

of such systems with the ultimate aim of using the

signature repertoires of these systems to characterize

fixed-point orders and the presence of *chaotic regimes* .

A system-theoretic approach is pursued here -- an

approach which complements other recent studies of a more

analytical nature (employing ergodic theoretic methods.)

Chaotic behavior in a certain subclass of these systemsais -

completely characterized in terms of the first two

iterates of a specific known point in the range of the

system transition function.

r Page3

Table of Notation

-symbol meaning

k.fpq, f a generic unimodal or subbell

function with breakpoint p and peak

value q

fk pq, fk ror k>=O, the kth Iterate of fpq or

f

sigt~x),Sig(x) the signature (infinite) of x under

the mapping f

sigfk(x),aigk(x) the k-signature of x under the map

f

ls8W, rs(x)' the left, right signatures of x

lak(x), rsk(x) the left, right k-signatures of x

aOzk that

Sk tbe k-sigzatmre repertoire of a

-A

Page 4

given map

S the (infinite) signature repertoire

of a given map

sequence oonoatenation

set interseotion

a**n the sequence a a a ... a (n times)

U set union

B'J the Jth left rotation of s

Ak, I the signature bins of a given

string a

Lj(a) the jth left shift of a

[ ... ] closed Interval

subset. of

0> not equal to

Page 5

I. Introduotion

There has been great interest in recent years in the

autonomous behavior of nonlinear one-dimensional systems

defined over the unit interval [0,11 (Kaliski and Klein,

1982; Klein and Kaliski, 1979; Collet and Rokmann, 1980;

Li and Yorke, 1975; Parry, 1966; Baillieul , Brockett, and

Washburn, 1980; and Feigenbaum, 1980, to name several)

There is, In particular, interest in the *chaotic

behavior' of the equilibrium (fixed) points of such

systems. This *random' behavior arises even in processes

describable by simple first-order difference equations of

the form:

x k+1 a f(xk)

where f:[0,1J-->[0,1].

Various approaches have been used to describe systems

of the above form, ranging from graphical methods (Hay and

Oster,1976) , to purely analytical techniques (Li and

Yorke,1975) and ergodio theoretic appoaches (Parry,1966).

A system theoretic approach, Introduced by Klein and

Kalisk-i, and cited above, Is pursued In this paper. This

approaoh treats the tuactieo t as the state transitie

function of an autonomous oae-diasnsional nonlinear

t , , . . . ... - . . a

Page 6

discrete-time system. The ooncept of Isignature' (defined

below) is used to describe the orbit of any given starting

state.

This paper serves to characterize the fixed point

s'.ructure of such systems through the use of signature

repertoires. Our development is somewhat detailed because

of the need to formalize and doflne our 'vorking tools*;

nonetheless, our results are of Intrinsic interest

because:

(i) they characterise "chaotic behavior" (the presence

of fixed points of all periods) in a subclass of such

systems strictly In terms of the first two iterates of a

specific point in the range of the system transition

function.

(ii) they demonstrate the utility of the signature

concept, a concept which complements alternate approaches

based upon #rgodic theory and -other measure theoretic

concepts.

Muoh of this material appeared in somewhat different

format In one of the authors' doctoral dissertations.

(Ewankan, 1979J

XI. Basic Cooopts. 31b#9lo# "Sanatureo aad Gra Coo

A* @rd r . ,

-. . . . .p - ,' -

Page T

As much of this development in based upon the concepts

Introduced by the earlier work of Caliski and Klein, we

merely summarize their' basic Ideas In the paragraphs

below.

11,1 We begin by describing the types ot functions

considered In this paper.

A subbell function is a continuous map f:[0,1J -

[0,1] for which (Figure 1)

(I) f(0) =f(l) s0,

(ii) f has a unique maximum q at some point p in

(0,1)

and

(111) f is strictly monotone on [0,p] and on (p,11.

We sallrefr top a th brekpont f f nd enoe-bVe sall efe to as he reakoint of anddenoe b

fpq a subbell map f with breakpoint p and peak value q.

Similarly, for k a positive integer, fk pq denotes the

kth Iterate of 'tpq (i.e. the k-told oomposition of fpqI

A subbell ftmo-tion Is a '#Veolal type, of tuaimodel"

PTvf

Page8

function:

A unimodal function in a continuous may f: [0,1] -

(0,1] tar whIcoh conditions (ii) an d (iii) above hold, but

not necessarily condition (i). CWe extend the notation

tpq and tk pq to unimodals in general. ) (Figure 2)

11,2 We introduce the concept of.(orbital) signature next. --

Let x in (0,1] be given, k a positive integer. The

k-signature of a sequence under the mapping fpq, denoted

by sigfk(x) ,is the length k string:

bO bi b2 ... b k-i

where for in Ot ... #k-1#

hi a0 if 0 <= fi pq(x) < p

- if fi pq(x) z p

and 31 If p < fi pq(x) <= I

(where tO pq(x) *xO by convention)

s a bO bi b2 ... b k-i will be oalled regular if, for

all I >a 0, bi a 0 or I but ntotr- . If a Is not regular

It will be called Irregular . In a similar fashion one

can define* ths (Infinite) signature sigf(;) of z by simply

Page 9

letting Ok" range over all the positive integers. The

notions of regularity and irregularity generalize in a

straightforward way. In the sequel when the word signature

is used without further qualification it may mean either

finite length or infinite signature, according to the

context of the discussion. Further, when the specific

function fpq is clear from this context we will often

simply write OfO and drop the f from sigft. When a point

x has a regular signature x is said to be regular;

otherwise it is said to be irregular. Note that if x is

regular then for all k, every k-signature of x is regular,

and conversely.

* 1Z,3 We can define a total ordering relationship upon

binary strings (not necessarily regular signatures) which

the reader will recognize as Gray Code order:

Let al = bO bt ... and s2 a dO dl ... denote two given

binary sequences of equal finite length, or both of

infinite length. Then al < s2 if

(i) a1 is not equal to &2, and

(Ii) if bit position j Is the first one at which al

and s2 differ, then

bO.... b J-1 0 and

1..... _ __.... ..

17

Page 10

dO ... d J-1 1

where * is the Exclusive OR funotion.

This ordering is fundamental in the theory of signatures,

in that all unimodal funotions obey a Omonotonioity of

signatures property" (monotone with respect to this

ordering.) This is explored below in section 11,5.

11,4 Let a be an irregular infinite signature of some

point x under a given unimodal map. An instance of sis

any binary string obtained from s by arbitrarily inserting

O's or 1's for each - in s. Ve refer to the least (in

the < ordering) instance of s as the left-signature of s,

la(x); we similarly define the right-signature of s,

rs(x), as the greatest Instance of a. The notion of left

and right signature trivially extends to regular points x,

where ls(x) = rs(x) = sig(x). Note that these definitions

hold equally well for finite signatures, where we write

the left and right k-signatures of x as lak(x), and

rsk(x), respectively.

We define the conoept of the exploded k-signature

ropertoire,denoted by 8k, to be

k a (rak(x) N x In [0,1]) U ilak(x) \ x In [0,11)

Thus the exploded k-signature repertoire of f

-ii

: !: Page 1 1

consists of all k-signatures of regular points, and the

greatest and least k- signatures of all irregular points

(hence the irregular k-signatures are "exploded".) A

similar interpretation holds for the (k=infinity) exploded

signature repertoire of f, denoted by S.

11,5 As indicated above all unimodal functions obey a

fundamental monotonicity property with respect to their

signatures. We cite the property without proof. Its

proof may be found elsewhere (Klein and Kaliski, 1979.)

Let f be any unimodal map. Let x, y in [0,1]

be given, x < y. Then, in the above defined

Gray code order, rsk (x) <= lak (y), for all

k, and rs (x) <X 1s (y).

III. A "Roadmap" for the Technical Developments That

Follow

To aid the reader In the developments to come let

us sketch out a "roadmap" of the remainder of the paper.

Section IV examinea certain recursions for calculating the

exploded k-aignature repertoires of unizodals and

eubbells. The role of the left signature of the peak

value q in these recursion formulas is oentral. In

Section V we examine certain necessary and sufficient

oonditiona for the existence of fixed points having given

7;. .

Page 12

finite or infinite signatures. This is followed in

Section VI by an examination of the *orders* of fixed

points and of the presence of chaotic regimes in unimodal

and subbell functions. Necessary background material is

introduced as needed, and the reader is often referred to

the existing literature for proofs of many of the basic

results.

IV. Recursions for Determining Signature Repertoires

We begin by discussing two recursions for obtaining

the realizable k-bit signatures of a given unimodal

function. By *realizable" signature we mean a signature

of a regular point or the left or right signature of an

irregular point, i.e. a signature in the exploded

repertoires of f. The first is drawn from the cited

references. Recall that the exploded k-signature

repertoire is denoted by 8k.

Theorem 1: (Klein and Kaliski, 1979) For all k >31,

S k+1 u ((0 . Sk) Cl[ k.1(0), la k+l(p)]}

U ((1 M Sk E[rs kel(p), rs k+1(1)1)

where . denotes conoatenation, denotes interseotion,

and the square brackets represent closed intervals In the

4 ordering on the (k.l)-bit sequences.

a

Page 13

Since the k-bit signatures of the points 0 and 1 under a

subbell function are respectively, 00k and 10*k-1, the

recursion is greatly simplified when f is a subbell. This

simplification leads to an alternate method of determining

realizable signatures for subbell mappings:

Theorem 2: Suppose that f is a subbell. Then for k>=1 :

(i) S1 = (0,11 and,

(ii) S k+1 = 0. S'k U 1. S'k

where S'k consists of those strings s in Sk for which s

is <= lsk(q).

Proof: We need to cite the following Lemma without proof:

Lemma 1: When f Is a subbell the exploded repertoires 3k -

are given by:

S k 1 m 0 . (Sk [O*k, Isk(q)])

U" 1. • k [ [099k, lsk(q)]}

1 __e

Page 114

(The proof of Theorem 2 resumes) Clearly, 31 is as given.

As for the expression for S k+1, observe that 3k " [O&k ,

lsk(q)] a 3'k. The expression for 3 k+1 then follows

immediately from the Lemma. QED

The last theorem underscores the significance of the left

signature of the peak q.In *signature space* a subbell

becomes a one-parameter family of sequences determined

completely by the left signature of the peak value.

V. On The Existence of Fixed Points of fk pq Having Given-

Finite and Infinite Signatures

In this section we are concerned with the existenoe

of fixed points of fk pq for various values of k, having

given finite and infinite signatures.Thes* points are

points xO for which fk pq(xO) a xO and their

characterisation is essential if one is to obtain a theory

of the orbital behavior of the function f in question.

Fixed points eventually map into themselves. We are thus

motivated to oonsider rotations of finite signatures. Our

disoussion begins here.

K

Page 15

V,1 Let a be a k-bit sequence . Let .sJ be the sequence

obtained by rotating a circularly j bits to the left,

0<=j< k. We call s'J the jth left rotation of s. We view

s 0 to be equal to a. We say that sfJ is a rotation

maximal of a, denoted ru(s), if sei <= sfj for ± =

0,1,...,k-1. Note that the existence of a rotation

maximal of any finite binary sequence is guaranteed since

<= is a linear ordering of finite binary sequences of

equal length. The value of J may not be unique however

(for example in the string s= 011011).

V92 Let us now examine sets which are such that any two

points in it have the same k-signature, for some fixed k,

under a. &.Lvo.nLtmtda 3 map,,ni. ( If the points are

irregular then they are in the set if either their left or

right signatures is the signature in question) . From a

system theoretic point of view the common k-eignatures may

be seen as defining a type of isomorphiss. It turns out

that such points form a continuum; i.e.,the sets are

intervals. This stems from the smonotonicity of

signatures' property cited earlier. In faot, the

Intervals are closed. (Klein and Kaliski, 1979).

V3 Let f be a given unimodal map. Also let a a bO bl...

be an arbitrary Infinite binary sequence. The k-sIgnature

bin of a (kfl), with respect to t# Is the set Ak a

Page 16

{x\lsk(x) and/or rak(x) a bO bl ... b k-1). The signature

bin of a with respect to f is the set A = (x\ls(x) and/or

ra(x) = s). As stated in V,2 , above, the sets Ak, if

non-empty, are closed intervals in [0,1]. It is also true

that A is a closed interval, and in fact is equal to " Ak

(taken over all values of k.) (Klein and Kaliski, 19791.

V,4 Infinite Sequences and Fixed Points of fk pq:

When considering infinite binary sequences, the concepts

of rotation and rotation maximality need to be replaced by

their infinite sequence analogues, shift and shift

maximality, respectively. Let al and s2 be infinite

binary sequences. We call s2 the jth left shift of &1

denoted LJ(sl) , if 32 is obtained from 31 by deleting the

leftmost j bits. Let S be the set of all left shifts (or

tails) of the infinite binary sequence s. Then, the shift

maximal of a is sup(S), with respeot to the < order. We

denote the shift maximal of a by sm(s). Note that, In

general, sm(s) is not an element of S.

We now turn to three key questions that arise in

exploring the Issue of tixed point existence.

QCISTIOI 1: When Is as Iatinite binary sequenoe s In the

Page 1T

exploded signature repertoire S of a given subbel. mapping

f pq?

Theorem 3: Let a be an infinite binary sequenoe. Then a is

in the exploded signature repertoire 3 of a subbell fpq if

* and only if

iiLi(s) <z 1s(q), for J>u.

Proof (-)): Suppose a is in S, and is the left and/or

right signature of x in [0,1]. Then every shift of s Is In

S , and is the left and/or right signature of a point in

the orbit of x. Suoh points are, from the form of f pq,

Fat most q. If q is not in the orbit of x then, from the

monotonioity of signatures principle, Lj(s) <= ls(q) for

all J>:l, and we are done. If q Is in the orbit of x

It Is easy to argue that all shifts of a corresponding to

Iterates of x equal to q are equal to le(q); all others,

by monotonioity of signatures, are less than or equal to

ls(q). The oonclusion follows.

(<-): .To prove the oconverse, we shall need the following:

Loma 3: Let a be a given iatinte binary sequence. Then a

Is In the exploded signature repertoire 3 of a unimodal r

pq if and only If for all k every k-trunscation of a Is In

Page 18

Sk. CBy the k-trunoation or a we mean Its first k-bits.)

Proof of Lonma 3: The necessity of the condition Is

obvious. Sufficiency requires some disocussion. Suppose

every k- truncation of a Is In Sic. Lot (LIE) ,kui,2,...

be the k-signature bins based an a. Thus the Ak are

nested and closed, and non- empty by hypothesis. Ve have

noted that A s Ak. Thus A Is non-empty also, I.e. a Is

in S. QED.

(The proof of Theorem 3 resumes) We exploit the fact that

f pq is a subbell by Invoking Theorem 2. Suppose Uj(s) <=

le(q) for .1 1,2.... Let a a bO bi.. Since bO Is

either 0 or 1, bO is in 31. Similarly bi is In S1. And

as bi is the first bit of Li(s) and Li(s) <a 3s(q),by

hypothesis,it follows that bi <a lsl(q). It is immediate

that hO hi Is in 32# by Theorem 2. Now coider bO-bi b2.

It Is easy to prove that hi b2 is In 32 by an argument

similar to that given above; further hi b2 represent the

first two bits of Li(s). Therefore < sLis a l2s(q),

bl b2 <= 1sV(q.Thua it is readily shown that bO bl b2 is

In S3, again by Theorem 2 .by repeated use of the

arguments above 'we - an, shiny that ,hO- hi ...b k-.1 the

k-truncation of a. Is a realizable finite signature of f

pq flor all k. ueaca, by hona 3, V is In S. OM3

Ve turn next to; t- 1

- ~- - -- ... -- ~T

1 -Page 19

QUESTION 2: When is an infinite signature a in S the

signature of a fixed point of a unimodal fk pq?

We have the following result:

Theorem 4: Let a u bO bl b2 ... be a given infinite binary

sequenoe in S , for some unimodal f pq . Let k>=l be

given. If a is periodic with period k then there is a

fixed point x of f pq of period k for which ls(z) and/or

rs(z) = a.

(Note that the Theorem applies, in particular, to

subbells, as well.)

Proof: We state the following lemma without proof.

Lemma* 4: Let (Bi) , izl,2,... be a collection of subsets

of 0,1] and r:[0,1 -> [0,1] a given map. Then

r( AB ) E r(Bi)

(where the intersection is taken over all 1)

(The proof of Theorem 4 esumes) Lot (Al) , Il,2,.i. be

the i-signature bins based on a. Then A "Ai. is a

nonempty olosed antepVal-.as are-the Al , aAd A a (y\la(y)

and/or ra(y) a )}. Note that Al 1,,A2 I A ... Aad beeaV6

-7

Page 20

of this nesting Amk zA also, mz1,2,...

NownotethatfkCAmk) E A (m-l)k, for all a )u 2,

because of the periodicity of a. 3o afk(Ank) ^ A (m-l)k,

where the first intersection begins with mal; the second

with m=2. but the latter is (beginning with mini) OAmk

A,so we have, ^fk(Auk) I A. But by Leoas 4, fk(aAak) E

^fk(Ank) . Therefore fk(A) ufk(Amk) E A and it is

readily shown that f' has a fixed point x of period k In A.

Clearly this point has ls(x) and/or ra(x) = a. QED

In the case where the unimodal is, In fact, a subbell an

even more powerful statement can be made.

Theorem 5: Let k>zl and finite, and s an infinite binary

sequence and fpq ,a subbell, be given. Then there is a

fixed point of fk pq with left or right signature s if

(1) am(s) <= ls(q.), an4

(ii) a is periodic, with period n such that-divide k.

Proof: The proof we give Is based on the fact that-every

periodic Infinite binary sequenoe, contains Its shift.

maximal. Suppose s is periodic with period. n which

divides k# ad am(s) (o.s(q). As. s(s) to esatalned. &s at

Lis als(q), torjs1, 24. RUeno,by_ Theorem 3. a Is

in the signare, prepevte~vse S.* *fpq s ad'thea'esu2I-

follows Imed*atSy';fi* Theeea s..

Page 21

V,5 Finite Sequences and Fixed Points of fk pq.

Before posing *QUESTION 38 we need to address several

issues. Let us begin with the following finite sequence

analogue of Theorem 3:

Theorem 6: Let f be a subbell function with peak value q.

Let a be a k-bit sequence such that rm(s) <= lak(q). Then

every rotation of a is in Sk.

Proof: In order to prove the theorem, we shall need the

following result, stated without proof: (Kwankau, 19791

Lemma 5: Let f be a subbell function with break point p.

Let a be a binary sequence of length k, such that for 3.

Op ... ,pk-1

(i) a'j <X lak(p), if sfj begins with 0,

(it) stj >= rsk(p), if stj begins with I

Then s*J Is in 3k for j 0,1,2,...k-1

(The proof of Theorem 6 resumes.) Let f be a subbellwfunotion with breakpOi~nt pe sad peak Yalue q. V. Show by a

Ut

rage 2

reductio-ad-absurdum method that if a is as defined in the

Theorem, every rotation al, of s, satisfies the following

conditions:

31 <= lsk(p), if al begins with 0

and

51 ) rsk(p), if al begins with 1.

We will then invoke Lemma 5 to complete the proof.

Assume, then, that there is some rotation &l, of s for

which either al 0 bl...b k-1 > lk(p) or sl I bl...b k-1

< rsk(p).As p 'maps into q, we know that lsk(p) = 0 is

•k-l(q) and rsk(p) a 1 Is k-l(q) Thus in the first ease 0

bl ... b k-1 > 0 Is k-1(q) and so bl ... b k-1 > is

k-i(q). Note that it z1 and z2 are binary sequenoes of

equal length with s1 > x2, then for any binary sequenoes

z3 and z4 of equal length, z1 s3 > z2 %4. Thus bl ... b

k-1 0 a s121 > lsk(q). (In the above notation al a bl

*b k-i, :2% is k-l(q), 83=0, and z4a the kth bit of ls(q).)

In the scoond ease we have that I bl ... b k-1 < I I&

k-i(q) and so bl... b-k-1 > I k-l(q.). Therefore, by the

note bl ... b k-1 I &I1l > lskCq).

"ae ths hev& tM '.1tha :forl %h& , case 81.i >'

Page 23

lsk(q), which cannot be since s1*1 <a ru~sl) <a Isk(q).'

*Therefore the original assumptions aunt be false, I.e.

every rotation sl of a must be such that al <a lsk(p) If

it begins with a 0; otherwise it must be >z rak(p). By

Lemma 5, then, every rotation of a is a realizable

k-signature of f. QED

Remark 1: Let s be a binary sequence of finite length.

Then rm(s'0 n) =rm(s)"On.

We shall need the following lemma as well in the

developments below.

Lemma 6: Let xO be a fixed point of a unimodal fk pq.

Then either ls(xO) and/or re(xO) is periodic of period k.

Let a denote the first k bits of this periodic signature.

Then there exists x1 In the orbit of xO whose left

k-signature is equal to ru(s). Furthermore, rm(s)9*2

<= lo2k(q), and, when It is equal to la2k(q), then ls(q)

Is, in fact, periodic, and equal to ru(s) rm(s)..

Proof: Clearly *ig(xO) is* periodic with period k. if

sig(xo) is regular then a is equal to sigk(zO) and it is

equally apparent that the rotation maximal of a occurs as

a subsequence of sig~xO). If this subsequence begins at

position Jin *L4(xO), J)*O, then ra(s) is the k-signature

Page 24

of xl =fj pq(xO). Furthermore sig(xl) =ru(s) ru(s)

Note that when sig(xo) is regular p and hence q cannot

occur in the orbit of xO, and thus all points in the orbit

of xO are less than q. It is immediate, from the

uonotonicity of signatures principle, that r3(&)*2(

ls2k(q). If rm(s)6"2 is < ls2k(q) there is nothing left

to prove. Assume then that ru(s)662 is equal to ls2k(q).

For ease of notation, write rm(s) as 81. Thus sig(xl) val

sl .... Again, from the montonicity of signatures

principle, sig~xl) <= 13(q). Assume that it is less than"

ls(q). We know by hypothesis that its first 2k bits agree

with those of ls(q). It must be, then , that 13(q) is of

the form slyln z, where n>=2 and where the Infinite

sequence z does not begin with sl and obeys sl**n z > al

31.. Suppose that aig(xl) and ls(q) differ for the

first time at the nlc+j th bit, with O<J<k. Let anc, If n

is even, and (n-l)k if n is odd. Consider L'm(a1 *I..

and L'm(ls(q)). The former Is again al al ... ,# whereas

the latter is z if a i.s *ven, and al z if n is odd. Note

thatr In both cases (n even, n odd) an even number of alls

were deleted, and hence al sl ... < L'm(la~q)). Since

these two sequences differ within the first 2k bits (since

a does not begin with W1 It must be that the first 2k

bits of L'm(lo(q)) are greater than sI sI w ls2k~q). So

L'u(ls(q)) > 18(q).. This Is not possible by Theorem 3,

for Js(q) is In 3. The conalusioa then is that lo(q) Is

equal to sIg(x1) and is thus periodic and equal to ru~e)

Page 25

rm(s) ...

When xO is irregular we argue as follows. p and q must

occur in the orbit of xO, and no point greater than q is

in its orbit. It is easy to demonstrate that ls(q) will be

periodic of period k and that either la(xO) or rs(xO) will

be also, depending on the parity of the initial portion of

sig(xO) before the first "-" occurs. (After this 0-8 the

remainder of both ls(xO) and rs(xO) is ls(q).) So s is the

initial k-bits of this periodic signature and rm(s) is

lsk(q), by the monotonicity of signatures principle.

Furthermore the point xl having rm(s) as its left

k-signature is q itself, and thus ls(xl) is equal to

s(q). With ls(q) periodic it is immediate that rm(s)**2

is equal to ls2k(q) and 1s(q) = rm(s) rm(a) ... QED

We are now in a position to address the final question of

section V:

QUESTION 3: When is a given k-bit sequence a the k-bit

left and/or right signature of a fixed point of fk pq?

Let us refer to the k-signature a of Lemma 6 (i.e. the

initial k bits of ls(xO) and/or rs(xO), whichever is

periodic) as the *periodic" k-signature of xO.

Theorem 7: Let a be a given k-bit binary sequenoe and fpq

Page 26

I l n

a subbell function. Then there is a fixed point of fk pq

with Operiodic" k-signature s if and only if rm(s) "*2 <=

ls2k(q), with ls(q) = rm(s) ru(s) ... if rm(s) "2 a

ls2k(q).

Proof: (-)) This is immediate from Lemma 6.

(<-) Conversely, write rm(s) as z1 to simplify the

discussion below. Consider the sequence z2 = a **n, for

any n > =2. From Remark 1, rm(z2) = z1 **n. Now, by

hypothesis, z1 902 <a la2k(q), with ls(q) = z1 z1 ... in

the case of equality. Thus, rm(z2) <= lsnk(q). By

Theorem 6, then, z2 is a realizable finite signature of

fpq . Thus every truncation of z2 is also a realizable

finite signature of fpq. Note that this is true for all n

> 1. Therefore by Lemma 3, the infiniteNsequence s s ...,

made up of repetitions of a is a realizable signature of

fpq. As a 3 ... is periodic with.'eriod k (or possibly a

factor of k), Theorem 4 implies that there is a fixed

point x, of fk pq with 1s(x) and/or rs(x) a a a s .... s

is clearly its "periodic" k-signature. QED

Signature-Distinct Uniuodals and Subbells:

A unimodal Fn fpq is said to be signature-distinot

if sig(x) = sig(y) implies that x u y. A wide-class of

signature distinot unimodals have been shown to exist

Page 27

(Klein and Kaliski, 1979) and include all those unimodals

which are "piecewise strictly expansive", i.e. for which

there exists E > I such that for all x,y in [Op],

IF(x)-F(y)I >= E Ix-y-1, and similarly for all x,y in

[p,11. A consequence of signature-distinctness is that if

x is not equal to y then no instance of sig(x) is equal to

an instance of sig(y), either. (Klein and Kaliski, 1979)

When attention is restricted to the

signature-distinct subbells a more powerful form of

Theorem 7 can be proven:

Theorem 8: Let s be a given k-bit binary sequence and f

pq a signature-distinct subbell function. Then there is a

fixed point of fk pq with "periodic" k-signature s if and

only if rm(s)0"2 <= ls2k(q) with the firther proviso,in

the case of equality, that q be a fixed point of fk pq.

1 ".i.

Proof:

(-)) If such a fixed point exists then by Theorem 7

rm(s)'*2 < a ls2k(q). In the case of equality ls(q) is

periodic with period k and equal to rm(s) rm(s) ... So, by

Theorem 4 there exists a fixed point x of fk pq whose left

and/or right signature is equal to ls(q). Since f pq is

signature-distinct x must, in fact, be equal to q.

jU-) If rm(s)e*2 is < ls2k(q) the result is immediate from

Page 28

Theorem 7. If equality holds and q is a fixed point of fk

pq, it is easy to see that ls(q) must be periodic with

period k. Since ls(q) begins with ra(s) ru(s) it must, in

fact, be the case that lo(q) a ru(s) ru(s) ... The result

is immediate from Theorem 7 then too. QED

VI On Fixed Points and Their Orders

Our attention turns in this final section to understanding

the nature of the fixed point regimes of unimodals and

subbell functions. To simplify the notation in this

section we omit the wpqw from fpq and fk pq, when

convenient to do so.

Let us begin our development by discussing the notion of

the order of a fixed point. First let us define the order

of a sequence.

A finite sequence s is of order k if there exists k

such that a can be written as al al ...81, where &l Is a k

bit sequence ,and where al cannot be similarly decomposed.

Thus a sequence of only ones (or xeros) is of order one.

The five bit sequence 10110 is of order five, whereas the

six bit sequence 010101 Is of order two and the eight bit

sequence 10111011 is of order four.*

.. . . . ..... . ......... . ....... . .... . .. ... .. ... ! L ! _ . :

Page 29

xO is a fixed point of order k of the mapping f if,

(i) fk (O) xO

and

(ii) fj (xO) <> xO, for j 1,2,...,k-1

i.e., a fixed point of order k first maps into itself

after k iterations under the given mapping. Note that if

x is a fixed point of fk for some k and if the Operiodic e

k-signature of x is of order k then x Is a fixed point of

order k.

Certainly every uniaodal map f with peak value

greater than its break point has a regular fixed point

with one-bit signature 1. This is clearly a fixed point

of order one which we call the nontrivial fixed point of

*order one . Thus the k-signature of such a point is 1'k,

and is in Sk. We are naturally motivated to ask if a

k-bit sequence containing but a single zero might be too.

Our next result shows that 101"0k-2 is, indeed, a

realizable signature provided that f obeys the

restrictions below.

Theorem 9: Let f be a unimodal map with peak value q and

break point p such that 0 and 1 are regular and such that

p

'.L

Page 30

(i) 132(q) z 10

(ii) sig2(o) - 00

and (iii) sig3(1) * 100

Then 1Ol'k-2 is a realizable k-signature of f, for k >v 2.

Proof: Let f, p, and q be as stated. Clearly the Theorem

is true for k=2. Assume then that k > 2. With 1a2(q) s

10, then q>p, Is(p) a 010... and rs(p) a 110... Now

1ek-2 is a realizable signature, by our remarks above.

Therefore, since sig k-1(0) < 01ek-2 <z Is k-1(p) we have

that 010"k-2 is too, by Theorem 1. Next onsider 101"*k-2.

Since rsk(p) < 1010ek-2 < sigk(1),.by the sane reoursion

101ek-2 is a realizable signature of f. QED

A subell satisfies all three conditions of the

theorem provided 1*2(q) a 10. Therefore,for such

subbells,101*0k-2 Is always a realizable k-signature. It

is also easily veri.fied that this sequence is of order k.

Let us restrict our attention to these subbella In the

remainder of this section, referring to then as

weoll-structured* subbells.

Theorem 10: Let f be a Ovoll-struotured* subbell. Then f

Page 31

has fixed paints at order k, k odd, and >= 3, if and only

if it has a fixed point, xO, of fk, whose *periodic*

k-signature is equal to =1Ol'0k-2.

Proof: The proof of this theorem hinges on three

properties of the sequence 101'**c-2. First, as already

nated,it is at order k. Secondly, it is Its rotation

maximal. The third property is stated as a lemma, also

given without proof. (Kwankam, 1979)

Lemma 7: Let s be a k-bit sequence other than the trivial

sequences 09*k and 1"*k. If k is odd, then

ru(s) >= 1O1'*k-2

(The proof of Theorem 10 resumfes:)

(C)By our earlier remarks .4tdoncerning orders of

sequences and orders of fixed :pOints, the result is

immediate, since 1019"k-2 is of order k.

C-)Conversely, suppose f has fixed points of order k.

Let yO be one 'such fixed point. Assume that Its

*periodic" k-signature s is not equal to 1010 *k-2. (If It

Is, we are done). Then by Lemma 7, ru(s) > 11*k-2.

Now, by Theorem 7# ra(s)*92 Is less than or equal to

1s2k(q). Thus, It In Immediate that 1O1*sk-2 1O10*k-2 <

ls2k(q). Using Theorem 7again, the result follows. ORD

kp

Page 32

We have seen, in the foregoing, conditions governing

the existence of fixed points of odd orders for

*well-structured* aubbells. We shall now show how the

existence of fixed points of order k, implies the

existence of fixed points of order k + 1, k + 2 , and k -

1, as long as k is odd and greater than one. Formally, we

have:

Lemma 8: Let f be a "well-structured" subbell. If f has a

fixed point of order k, k odd and k > 1, then f has fixed

points of orders k + 1, k + 2, and k - 1.

Proof: Let k, an odd integer greater than one be given.

Assume f has a fixed point of order k. Then by Theorem

10, there is a fixed point xO, of fk whose *periodic"

k-signature is 101ek-2. From Theorem 7, it follows that

1010#k-2 1010k-2 <a ls2k(q)

Consider the sequence 101ek-1 which we denote

by *I. It In of order k + 1. Moreover, it is Its

rotation maximal. It we can show that al a1< Is 2k+2(q) ,

then by Theorem 7, there is a fixed point of f k+1 with

'poriodio' (k.1) -SLgnature i. Ixamine the leftmost k +

2 bits of l *I and of (10106k.2)(lO1tkk-2). With k odd,

10leek-1 1 < 101Ok-2 lO.Thus the.lettmoet 1k bits of s1

I are < (10100k-2)(COi*k-2) cad hence C slk(q). so &I

Page 33

So, itel Theore 7, n mntined there Is a fixed point at

fk whose 'periodic3 k-signature is 1O1"k-1. Since this

is of order k+1 it is immediate that the fixed point In

also.

Consider ntext the sequence 10104k ,which we denote by s2.

It is at order k+2, and is its own rotation maximal also.

By considering the leftmost k+.2 bit& of s2 s2 and those of

101##k-2 1O10*k-2 and by arguing as we have done for al,

jit can be readily shown that &2 s2 < 1s 2k+4(q). Henoe f

has fixed points of order k + 2.

Finally consider the sequence 10109k-3 which we denote by

93. It is of order k - 1, and it is rotation maximal.

This time, we examine the leftmost k + I bits of s3 83 and

of (1O1**k-2)(1O100k-2) By arguments similar to -those

presented above for sl and a2, it can be shown that s3 33

< 1s 2k-2(q). Theictore, f has fixed points-of order kc -

* 1. This completes the proof. QED

Therefore when a *wejll-struatured" subbell ha~s fixed

points of o"d .ordev kc, kc > 1, i:t also has fixed points or.

orders kc - 1, kc *., and I. 2., Note that If kc SA 0*44, so

is kc + 2. ftb emistnacwot tixfed potafts of ordt k* Il

thus implies the eith.of' ftix04 40ints vt 0#40118 I' + 3and k + c 4 is, 16*r o. Alia ~c gaet bib

Page 34

repeated, ad infinttum. This leads, naturally, to theI following result:Theorem 11: Let f be as above. Then if f has a fixed

point of order kO, with k.0 > 1 and kG odd, then f has

fixed points of every order k >= (kG 1).

We conclude, from this theorem, that a

woll-structured" subbell with fixed points of

order three

has fixed points of all orders. (Every subbell has,&

fixed point of order one.) This conclusion is very such

like that of Li and Yorke (LI and Yorke, 1975). If, we

restrict our attention to signature distinct

vwell-struoturedO subbella, we can propose a method of

determining the existence of fixed points of all orders

which is considerably easier than the method implied by Li

and Yorke's theorem.

Theorem 12: Lot f be a aignature-distinct

*well-structured* subbell. Then f has fixed points of all

orders if and only if r33(q)zl00.

Proof:(<-) The sufficiency of the condition in readily

proved. Suppose r$3(q) Is 10&. If $i83(q) has no ftsh in

it, then 1s3(q) io also 100. Therefore as 101,101(

156(q), we have by ?hoores 7, that fS has a -fix.dpointt

with three-bit Uperio#1e' signature 101. This L oloarly

Bob - -----..-.-

Pass 35

*a fixed point of ordor three, which by fteorea I t means f

has fixed points of all ardoir greati' than one. As every

* subbell has a fixed point ot order one, f has fixed points

of all ordera.

If sig3Wq has a da init he ig3(q) skuit be 10-

or else 1.2(q) <> 10, a inedeaty for 'vell-utruaturedO

subbells. Theiietore- q is a, fixed point of f3

Furthermore, q It a fixed point' of- ordor 3. Thus f has

fixed points of all orders.

(->) To prove that ra3,6) a t",O is -a "necessary oontudito,

i~t ins ufficient t6 show' that If it is not not" therwe is

at least one drder oif ixed poiji~f -bich Is not-part- O~f

the 10 fixed' oit, repike '4Wr -of f-I souch or~der IS-

three. Since 100 is the Itrgesi th)-if -bit iaeno., It

must, be that rs3,(q) < 100. Since ls2(q)a10, It follows

that as2(q) is 10 and, rs3(q) and 1.3(q) must both-btV

101. Thus siC3(q) does. not contain a dash. Therefore q

.1 ot a fixed, point off3. Now, from Theorem Sp lo(q) Is

shift maia. v ia x )-Uz 101t, ti eay,** pio,'

that toe(q) 4V iM4tti

the. rr 1*.e~i~i~ia leti 04tOWSIltta-60 Li

at~ Mwe 691416 Sirt& ~ U4e~~.L ~S~~

-Ot t441 VdMP4,4tsitr ~~ .&ar zt s *e-

ip

A -1 .-

Page 36

immediate that f has no fixed points of order three from

Theorem 10. This completes the proof. QED

We therefore need examine only the right signature of

the peak value of a "well-struotured" signature distinct

subbell to determine whether or not the subbell has fixed

points of all orders. What is more important --from

considerations of ecomomy of computation-- is that we need

look at only its first three bits i.e., we need to compare

with p the values of q, f(q) and f2(q)1

The family of "symmetric tent' maps shown in Figure 3 will

exhibit this 'chaotic" behavior (fixed points of all

orders) provided that a>= (1+sqrt(5))/4, a fact that Is

readily demonstrated by showing that for such values of a,

and only for such values, will rs3(q)= 100.

VII. Conclusions

This paper has presented a variety of results

concerning the fixed point structure of certain mape on

the unit interval. The underlying common factor of the

developed theory has been that of the signature of a point

and of its role in characterizing the map's orbital

behavior. From an expositional point-of-viev the

signature-basod, theory is appealing due to its minimal

dependence on advanoed measure-theoretio concepts

A

Page 3T'

-typically found in the current literature.

iv.77

References

Kalisici, M. and Klein, Q. 'B ehavior at a Class of

Nonlinear Discrete Time Systems.*, submitted

to Mathematical Systems Theory, 1982

Klein, Q. and Kaliski, M. "Functional Equivalence in a

Class of Autonomous One-Dimensional Nonlinear

Discrete-Time Systems', Information a-ad

Control,J42,2, 1979

Collett,P. and Eckmann,J. "Iterated Maps on the Interval

as Dynamical Systems', Birkhausr, 1980.

Li,T. and !orke,J. "Period Three Implies Chaos', American

Mathematical Monthly, 82, 1975.

Parry, W. "Symbolic Dynamics and Transformations of the

Unit Interval*, Trans. American Mathematical

Society,122, 1966.

Ballieul,J., Brockett,R. and Wash%.ur., R. 'Chaotic Motion

in Nonlinear Feedback Systems*, IEEE Trans. on Circuits

* and Systems, CAS-27,11,1980.

Felgenbau,4. 'Universal Behavior in Nonlinear Systems',

Page 39

Los Alamos Science, 1980.

May, R. and Oster,G. "Bifurcations and Dynamic Complexity

in Simple Ecological Models", American Naturalist, 1976.

Kwankam, S. *A Structure Theory for the Fixed Points of

the Iterates of Certain One-Dimensional Nonlinear

Functions Defined Over the Unit Interval", Ph.D.

Dissertation, Northeastern University Dep't of

Electrical Engineering,Boston, MA, 1979.

4

At 0 17 A YNCIF NS DI SRI If COMIRO[ Uf CONT INIJIUS PROCI(I)NOR1 A;T RAN INI V BIOSTON MA M I KAIL SK I I I At

10 J H It AtOFFIR 3 87 F 49620 F82 0081

I lt ( 1 1,1 ,11 1 DI , I N

F --

A Lit

lj 2''' - "

Ut> K MW-

J.

U /k

A~~~~~n -.,., .~~

- - ~Wwo- w

zt

~-..~--- ~ - - -~ - - - - - - -.------ - - -~- - -----

IA - - -

I

'I; e

II I

I II

I

I V I

9.

.- ~

1! ~>i7i Kj <:~~4II.