STATE SPACE MOMS AND STATE DESCRiPTIONS m cmomm FORM Thesis far the Degree of Ph. D. MECHEGAN STATE UNNERSITY _ IZZET GEM 96mm 1969- THEN: LIBRARY Michigan State University This is to certify that the thesis entitled 5/02? 3/9016; Iq/Y/‘OWIJ CINC/ 57/0 1Z5 flé‘if/‘I/U {lb/L; I}? [217/] 0/7/60 / 7?;de presented by fiZCZZ (6)77 KiKN/i/K) has been accepted towards fulfillment of the requirements for M degree in _é:*z gala C, [:5 fiajogproig“ Date [15“] [[111 A95? J 0-169 ()' ABSTRACT STATE SPACE AXIOMS AND STATE DESCRIPTIONS IN CANONICAL FORM By lzzet Cem Goknar Although it dates back to Newton's use of positions and momenta, the concept of "State" has only been given an abstract and rigorous definition in the last decade by Zadeh. In this thesis, starting with improved versions of Zadeh's "State Axioms," the necessity of another minor modification is shown and the different axiom sets are discussed. With the axioms modified, the important con- cept of "Equivalence Classes of Inputs" (the major tool of the behavioral approach) is used to investigate the properties of "Reduced and Half Reduced State Descrip- tions." Then, the essential properties that "State Descrip- tions" acquire when the system is "Linear" and/or "Time- Invariant" are examined, and "State-Equations" in canoni- cal form are obtained for a large class of distributed systems. The problem of approximating more general sys- tems, with only minor restrictions on the input space, by systems that possess finite dimensional "State Spaces" is given a solution. H’hflLo \/i 94", 63 9’ STATE SPACE AXIOMS AND STATE DESCRIPTIONS IN CANONICAL FORM By lzzet Cem G5knar A THESIS Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Electrical Engineering 1969 m- A. .. aw ACKNOWLEDGMENTS I would like to take this opportunity to formally acknowledge some people whose efforts, perhaps not directly related to the thesis, brought me to the state in which I am, influenced my thoughts, and therefore contributed to this work. Listed in the chronological order I met them, I wish to thank my mother, Vedide G6knar, and my father, Saim GBknar, to whom I owe my existence and my educa- tion, my wife, Aytac G6knar, whose love and support made "those moments" bearable, my Professor Tarik azker, devoted to his country and the education of his students, who caused an evolution in my thoughts, and finally to my daughter, Elif GBknar, whose addition to the family has been a source of joy and strength. I am indebted to Professor James A. Resh for his guidance of this thesis and for his unique advising, and to Professor Yilmaz Tokad for his valuable suggestions and discussions. Finally I extend my gratitude to the institutions of Michigan State University and the Technical University of Istanbul for the fine education and the support they have provided. ii "The men where you live," said the little prince, "raise five thousand roses in the same garden-- and they do not find in it what they are looking for." "They do not find it," I replied. "And yet what they are looking for could be found in one single rose, or in a little water." "Yes, that is true," I said. And the little prince added: "But the eyes are blind. One must look with the heart. . . ." The Little Prince , Antoine de Saint—Exupery iii ACKNOWLEDGMENTS LIST OF FIGURES Chapter TABLE OF CONTENTS I. INTRODUCTION. I. I.“ The Modern State Concept. Some General Concepts and Termine logy . Previous Work on the Subjec’r A Brief Summary of the Following Chapters . . . . . . . . II. AXIOM SETSA1,A‘2, *3 AND STATE DESCRIPTIONS IN GENERAL II.l II.2 II.3 II.“ Introdustion. An Example of Deficiency and the Axiom SetXA3. Interrelations of the Axioms Sets 541944, 716.. About Reduced and Half— Reduced State Descriptions. III. LINEAR, TIME-INVARIANT OBJECTS. III.l III.2 III.3 III.“ Introduction. Linear Objects and Properties of the State Description. Time—Invariant Objects and Properties of the State Description . . . . An Application of Equivalence Classes of Inputs to Lumped Objects . . . . iv Page ii vi lO 15 214 26 26 27 “2 56 71 71 73 87 96 Chapter IV. SOME CANONICAL FORMS AND PROPERTIES OF THE STATE DESCRIPTIONS FOR LINEAR, TIME INVARIANT, CONTINUOUS OBJECTS IV.l Introduction IV.2 Convolution Representation of. Linear, Time Invariant and Continuous Objects . IV.3 A Countable-Differential State Description. . IV.“ Approximation of a Large Class of Objects Having Finite Dimensional State Descriptions . . . V. CONCLUSIONS LIST OF REFERENCES APPENDIX A A. 1 About Distribution Theory. A. 2 A Brief Review, Some Definitions and Results in Distribution Theory. . . . . . A.3 Some New Results. APPENDIX B Hilbert Matrices. Page 110 135 1“? 150 15“ 15“ 158 170 185 185 LIST OF FIGURES Figure Page 1.3.2 A circuit for which some inputs may become 17 inadmissible. . . . . . . . II.2.l Input Output pairs of the object in the example of deficiency. . . . . . . . 3“ II.“.l Half Reduced Partitioning of the Input Space . . . . . . . . . . . . . 6“ vi ‘r «my .«a F-a v“ \.‘ n~§ CHAPTER I INTRODUCTION In order to present the results accomplished in this thesis and to lay down the general background for the sub- ject considered in the thesis the present chapter is divided into four sections. The first section is de— voted to the history of the concept of state and the steps toward its abstractization in the framework of modern sys— tem theory. We feel that before a meaningful discussion can be given for the findings of the thesis, some general concepts and terminology should be introduced. This is done in section 2. In section 3 the State Axioms pro- posed by various authors are outlined and some known re- sults are given. Finally, in section “ the remaining chapters of the thesis are summarized. If some idea has to be given shortly about the re— sults of the thesis, we can divide our accomplishments into three main groups. The first group of results is about the State Axioms and what can be said about the State Descriptions in gen- eral without any restriction on the system under consider- ation. An improvement on the State Axioms is given and questions about the size of the State Space and about the nature and system-independent properties of the State Description are answered. The second group is obtained by placing some re- strictions on the nature of the system and then inquiring about the State Description. The basic prOperties of the State Descriptions of linear, time-invariant systems are investigated and results are obtained by using tools developed in the first group. In the final third group, we develop analytical formulations of the State Description for some broad classes of systems. These representations can be used in the Theory of Distributed Parameter Systems, or in approxi- mating them by systems with finite dimensional State Des— criptions. Outside the main goals of this thesis, some new Theorems are obtained in the Appendix that center about Orthonormal Series Expansions of Distributions as pre- sented in [2E2]. I.l--The Modern State Concept The concept of "state," which dates back to Newton's introduction of positions and momenta as basic mechanical variables, has been used in analytical dynamics, celestial mechanics and quantum mechanics as tied to the concept of stored energy in such physical systems. The following short discussion, that stems from a 13reatment on the historical background of the ”modern con- czept of state," appeared in the literature in 1962 [ZAl]. 1X3 implied in this reference this modern concept was first Lised by Turing in his time—discrete machine. Briefly if act, ut, yt denote, respectively the state, the input and tshe output at time t, then the machine can be charac- ‘terized by x = f (x u t+l t’ t) t = o, 1, 2, —-— (1) yt = g (xtaut) Shannon [SH] in l9“8 used equations in the form (1) to characterize probabilistic systems in the sense that xt and ut determine the joint probability density function, I>(xt+l, yt/xt, ut) instead of x and yt. t Two important notions, namely, equivalent states and equivalent machines were then introduced by Moore [MO] and by Huffman [HU] independently, but in a somewhat re- stricted form by the latter. All the above work is in the discrete-state systems context. In the case of differential systems, the equa- tions (1) take the form: 52. dt X“) f (x(t), u(t)) (2) y(t) g (x(t), u(t)) where x(t), u(t), y(t) are vectors representing the "state," the "input" and the "output." Equations (2) have been used, under different forms, in such fields as ordinary differential equations, analytical dynamics, celestial mechanics, quantum mechanics, etc. Their wide use in the field of automatic control was initiated almost twenty years ago, in Russia, by A. T. Luré, M. A. Aizerman, Ya. Z. Tsypkin, A. A. Fel'dbaum, A. Ya. Lerner, A. M. Letov, N. N. Krasovskii, I. G. Malkin, L. S. Pontryagin and others, and in the United States by Bellman, Kalman, Bertram, LaSalle, Laning, Battin, Friedland and others. General methods of setting up the state equations for RLC networks were later described by Bashkow [BA] and Bryant [BR]. These methods are extended to time varying net- works by Kinarawala [KI]. Until recently, the concept of "state" was strongly connected with the specific physical identification of state variables as measurable quantities inside a specific system structure. For example, "the state vector” in an electrical network contains the variables corresponding to the branch capacitor voltages and the chord inductor currents. Thus the "initial state" at the ”initial time" is physically the initial charge and the initial flux carried by those elements, and is reflected as the "ini- tial conditions” on the differential equations modeling the network. This notion of state, namely that the state .1. ~K. v \. ~\~ «‘\ is a set of internal variables from which everything else about the system can be calculated, is referred to as the "structural approach" to the concept of state [RES]. In this approach an important property of the state that has to be singled out,is that it ensures a unique output for eeach given input. Another approach to the concept of state is in the .ffiramework of modern system theory and is introduced in 1:11e following. In Zadeh's view [ZAl], the importance of system tsraeory lies in its abstract generality and in its concern vvfi.th the mathematical properties of systems and not their lorlysical form. Such an abstraction, however, should be Ireaached from a number of known examples of such systems 8&3 physical, socio-economic, biological and others. If ‘tkie'state concept is not to be abandoned during this generalization, one has to be certain whether all the \faarious instances of the state notion that appear in Srbecific systems are sufficiently similar in meaning and uSage to be covered by a single abstract definition; if SC), what are the essentials of the notion? To elaborate ‘this point further, we consider two examples, one in SOcio-economic, the other in biological systems. For the first example, let the community of the Greater Lansing Area be our system, with the price of a certain good, say for example of Nehru jackets, and the advertisement expenditures as inputs and the demand for the same good as output. We shall concern ourselves with an important variable that affects the input—output rela— tionship of the system: "the taste” of the community. It is true that, at different times, for the same price and advertisement expenditures, our community may not have the same demand for Nehru jackets. This is due to a change in ”the taste" of the Greater Lansing Area; a taste more in favor of the good will create a larger demand for a given price and advertisement expenditures than a taste less in favor of the good. Thus if the price and the advertisement expenditures were given, as a function of time, one could determine the demand for Nehru jackets, as a function of time, if the taste of the community were lcnown. Equally important is the tie existing between the isaste and the past history of the community: the taste unill certainly vary depending on the kind and intensity of‘ the advertisement and the past fluctuations of the pIfiice. For example: fashion shows, constant T. V. com— Umircials, larger numbers of people wearing Nehru jackets tnecause of low prices, will probably push the taste to be “More in favor of the good. For the second example, we quote from Manning [MA]: It is a common observation that the same stimulus given to the same animal at different times does not always evoke the same response. Something inside the animal must have changed and we invoke an "intervening variable." This is something which comes between two things we can measure—— in this case the stimulus we give and the response we get out—-and affects the relationship between them. . . . Already in this book we have men- tioned two factors with different characteristics which alter the relationship between stimulus and response. These were "fatigue” and "maturation." To these we may add two others: "learning" and "motivation" . . . From these examples we immediately recognize the ianortant property that we noticed in the structural amiproach to the state, i.e., to make correspond a unique Otrtput to a given input, when the input and the state are .krnown. We may therefore conclude that all the various iJistances of the state notion that appear in specific sys- tsenns have a very important common property that may lead tc> a single abstract definition. What can better sum- Inaiéize "the mood of a human being (or an animal)," "the scuoial conditions of a society," "the political condi- tixons of a country" than "the state of the system"? These examples also bring light to another important Eisxoect of the state notion that was not clearly visible irl the structural approach: the strong connection between thue history of the system and the state. In fact "the ‘taéste of the society," "the fatigue, maturation, . . . of thfia animal," "the mood of the human being" at a given itinne, are all results of the past experiences of the sys— 'tefifl. Even in networks, the flux and the charge at time t are the integral of the voltage and of the current up 03 tC> time to, which certainly bear a relation to the past. To conclude, the state, in this new context makes a tinique output correspond to a given input by at least cxontaining a minimum amount of information that consists ir1 those features of the past experience of the system euffecting its future behavior. This experiential aspect is; named as "the behavioral approach" to the concept of state [RES] . In 1962, Zadeh wrote [ZAl]: Despite the extensive use of the notion of state in the current literature, one would be hard put to find a satisfactory definition of it in textbooks or papers. A reason for this is that the notion of state is essentially a primitive concept, and as such is not susceptible to exact definition. iiovvever, Zadeh in 1963 [2A2] and Kalman in 1963—6“ [KA, WEX] have independently tackled the precise formulation of T§j;ate descriptions," and Resh in [RB 1-3] ”exposed and 81:1ndnated a syndrome of shortcomings in these general fYDrunalizations of the state notion" [RBI], and offered tvvc> somewhat related though different sets of ”State -AX:Ioms." Of these syndromes, some important ones were: Kalman's formulation, besides being cumbersome (at least to this author) had the obscurity of de— fining what is to be called a "system" in terms of his state axioms, irl Zadeh's (and Kalman's) formulation, The gross properties of the "state space" of a system were not uniquely determined by the system, The states and the past histories of a system bore no necessary strong relation to one another, All systems, causal and noncausal alike, pos— sessed state descriptions. Resh, when modifying Zadeh's axioms, also intro— duced a powerful tool, "the equivalence classes of pre— to inputs" which summarizes the history of the system u;> to time to and which bears strong relations to the st:ates of the system at time to' All the works summarized above being about the gross properties of the state descriptions, some analy— ‘tixoal results are also obtained. It has first been pcxinted out in [ZA2], that for linear, time-invariant Sytstems, the state space is a finite dimensional vector Spuace iff the system can be characterized by ordinary d1JFferential equations. The finite dimensional case has ‘thuen.been extended by Balakrishnan, introducing some assumptions on the nature of the state space and the in- PLrt space in [BA 1—3] and he derived a state description starting from the input-output description with some restrictions in [BA“]. The restrictions are the linearity arm: time-invariance of the system, except in [BA3] where thuay were allowed to be time varying. The main tool Balakrishnan used was the analytical theory of semi- Srfloups of linear operators as developed by Hille—Phillips arui Yesida to obtain results of the form: lO x(t) = T(t)-x(o) + ftT(t—s)-Lu(s)-ds o wrnere T(t) is a one—parameter semigroup of linear bounded tiaansformations on the state space, and L a linear bcnanded transformation on the input space. Finally, Resh [RE“] and very recently Resh and Géiknar [REB] have given a non-reduced state description of‘ the form: dxs(t) , '“Efi7—'= s-xs(t) + u(t) sec: . . K (k) y(t) = f C(S)-xs(t)~ds + Z dk-u (t), f—l k=O wheare the dimension (sic) of the state space is a two- ditnensional continuum. I.2-—Some General Concepts and Terminology In this and the following section, we specialize in thus definitions of systems, objects, existence intervals, Urhiform objects, etc., and give the different "State ltxi_oms," discuss them more in detail and state some lcnfirwn results. It is our feeling that here is the right I3161ce, although it may not be very usual, to do this since ‘Ne talk of general concepts that underlie our work and pIV38ent some known theorems for later references. 11 Webster defines a "SYSTEM" as ". . . an aggregation 01° assemblage of objects united by some form of inter- euotion or interdependence," which is close to our en— ggineering understanding although still remaining unde- ffiined because of the use of the synonymous ”object.” The "Mathematics Dictionary" of James and James defines it as: (1) A set of quantities haVIng some common pro- perty, such as the system of even integers, the system of lines passing through the origin, etc. (2) A set of principles concerned with a central objective, as, a coordinate system, a system of notation, etc. WTLiCh has no bearing to our concept of system whatsoever. From an engineering point of view, the "system" deifinition can be given from two aspects; their main dif— feloence being the existence of the concept of "Terminals" irt one and not in the other. As an example of the first Orue, "system” in [NE] is defined by: where u = [uj(t)], y = [yJ{t)], J = 11’ . ., k are the acimissible pairs at the k terminals of the system with (38 denoting the determining constraints imposed by the System As we will be using the second definition of "sys— tefin" in our context, we will give it in its greater ‘ietails. l2 £3?F.I.2.1: T A a collection of half open intervals, (°,°], of tine real line, i.e., T g {(°,-] : (°,°]CflR}, the intervals in T are called CHBSERVATION INTERVALS. R A a set of ordered pairs of time functions de- I fined on IGT, i.e., RI g {(u,y) : Dom u = Dom y = I} A A is the family of all RI when IET, i.e., A A {RI IET}. A CONTINUOUS—TIME SYSTEM (as opposed to discrete- ‘tiJne system) is an ordered pair (T,A) where T and A are (deifined as above, satisfying: (Cl) If IET and (t0,t1]Cl then (120,1;116'1‘ (C2) If IET then RI e O (C3) If I'ET and ICI' then RI QRI,,/‘I 31f” the first member u of the ordered pair is called an INPUT, and the second one OUTPUT, the SYSTEM is then said to be ORIENTED. For an oriented system, U will denote the set of I aldl inputs whose domain is I and YI the set of all out— erts whose domain is I, YI(u) will be the set of all out- FNJIZS that can occur as a response to u. UGOU will mean truat two inputs u are in concatena- tion. 0 l,o (t,t2] DEF.I.2.2: Let a system (T,A) be given; for each observation irrterval I we define: I R A {(u,y)€RI : u,y are not the restriction to I I of pairs in some RI' 3 131'} Then: T A {I IeT and Rf r e}. Tfioe intervals in T are called the EXISTENCE INTERVALS. An oriented system is UNIFORM iff T is a unit set, i..ea., contains a unique existence interval. NCXPEZI.2.1: Thus for a uniform system it is clear that alfil pairs (u,y)6RI, for all IET except one are the re- stloictions to I of some (G,§)eRI,, ICI'. It has been shown in [REl]: . portions of a system (T,A) derived from different existence intervals lead rather inde- pendent lives. In fact, one might consider them to be different systems which it has merely been convenient to describe in language suitable for treating them in some unified way. Trnas the loss of generality that entailed by the restric— ‘tiibn of our concentration to uniform systems is very little. EEBZEngijg The description of a uniform system is com- pliately known when the unique existence interval IET and true input-output list RE is given, due to the conditions 01—, c2 and c3. l“ CON.I.2.l: From now on we will talk of OBJECTS and not of systems. No real difference exists between the two things these names describe. However, we will make the following distinction: an object is always a system, but not vice- versa. An object for us will consist of a single RA, whereas a system may consist of a combination of many objects or systems each given by a different RI' Briefly we are saying that we do not consider problems arising from the interconnection Of systems when we use the name "Object." CON.I.2.2: Def. 1 of a system allows only time functions as inputs and outputs. We think that it would cause no real difficulties, to allow distributions in our input and output spaces, excepting possibly some philosophical arguments that we will try to discuss in the Appendix (see A.l.). Thus we will refer to the elements of the input and output spaces as inputs and outputs meaning distributions or functions, and 1-0 will be an abbreviation for "input- output pair." CON.I.2.3: By an OBJECT we will always understand a "con- tinuous-time, uniform oriented,object. We will denote it by "6h" its unique existence interval by I. will be given A o I We close this section with the following important by its I-o list R definition: l5 DEF.I.2.“: The object‘9'is called DETERMINATE iff for each uEUi there is a unique erf such that (u,y)GRf. It is said to be NONANTICIPATIVE ET for any IéT that starts A where I does and for any u, uTEUi satisfying u/I = u'/I there always exists pairs (u,y) and (u',y')GRf such that y/I = y'/I. Eina11y<3’is said to be CAUSAL iff it is determinate and nonanticipative. I.3--Previous Work on the Subject We start with Zadeh's state axioms [ZA 2-3]: Zadeh's STATE AXIOMS: The STATE DESCRIPTION of the chjecté?, given by the list RI of 1-0 pairs, is the pair (Z,A) that satisfies the conditions listed below. Here 2 is a set called the STATE SPACE and A a relation called the INPUT-OUTPUT- STATE—RELATION (which will be abbreviated as I-O-S—R). More precisely, A is a subset of {(I,O,u,y):ICI, 062, (u,y)ERf}. The axioms are: (Ml)-—For each ICI, (u,y)eRI iff 3062 3(I,c,u,y)eA. (Sl)--For each ICI, 092 and uEUI aexactly one N'9(I,o,u,y)EI. Denoting by AI(o,u) the unique response guaranteed by (81), we can define a family of single valued INPUT- OUTPUT—STATE FUNCTIONS AI : DI + YI, for ICI, which com— pletely characterizes the I—O—S—R. The domain of AI is DI A Z x UI' l6 (S3)--For each (to,tl]<:I and (OO’UO)ED(tO,tl]"3 at least one 0162 with the property that if (oo,u00ul) 6D then (0 u )ED and A (O u ) (t0,t2l 1’ 1 (tl,t2l (tl,t2] 1’ 1 A (O ,u u )/ - (t0,t2] O OO 1 (tl,t2] NOTE I.3.1: The MUTUAL CONSISTENCY CONDITION (M1) estab- lishes the relation of the object O'to the state descrip— tion. The first of the two SELF CONSISTENCY CONDITIONS (SI) and (S3) guarantees the uniqueness of the output for a given input and state, the prOperty that we were after, from the beginning; the second one classifies the states of the description at time t1. NOTE 1.3.2: To require DI to be 2 x UI for each I was shown to be a very important shortcoming by Resh [RE 1-2]. That D A Z x U means no matter what state the system is I I left in, one can apply any input. Many existing systems, however, do not admit this property. To the examples given by Resh, that extend from the systems of the type homosapiens and certain kinds of inputs termed propaganda, to the very technical one given by Fig. l, and that in- clude examples such as rocket engines whose fuels can be depleted by the initial input segments, we can add the example of the brain of an animal which became blind as a result of blast (an input). Any form of light, for that matter, any video—input at this state of the system (the animal) are simply not admissible to the brain. 17 In Fig. l, the switch closes exactly one second after the applied input u exceeds 1 volt in magnitude and remains closed there- IQ - flWN after. Thus, an input that + L\ was admissible before the u Vy 1 closing of the switch is 7 not admissible anymore, Figure 1'3-2 since only 0 volt can be applied once the switch has been closed. As in the above example the system is left in such a state that all inputs, except 0, are no longer admissible. NOTE 1.3.3: As it is not desirable to deny a state des- cription to such a large class of objects, since the importance of system theory lies in its abstract generality (page 5), Resh modified axiom (81), the source of the shortcoming, to read: (Sl')-—For each ICI, 062 and uEUI,,3 at most one y 3 (I,o,u,y)€A This means that the domain D of A I I consisting of pairs (O,u) for which there exists is a subset of X x UI a y such that (I,O,u,y)€A NOTE I.3.“: Unfortunately the replacement of (81) by (81'), while eliminating the above shortcoming, introduced some other inconveniences: 18 All systems had a trivial state description, where the state space 2' was the set YI of all outputs defined on the existence interval of the object, and the I—O—S-R, A' was {(I,c,u,c/I): ICI, 062' and u = fi/I for some 859(G,O)€Ri}. As a result of the state description (Z',A'), a unit resistor had two reduced (Def. II.2.“) state descriptions: one with a unit state space, the other with a gigantic state space, the set of all outputs of the resistor. To eliminate the difficulties caused by the change of ($1) to 81') Resh proposed a SECOND MUTUAL CONSISTENCY AXIOM, in two different ways that are not exactly equiva- lent, to be incorporated in the modified set. Since there are some minor changes in the language of presentation, we present the two axiom sets, proposed in [RE3] and [REl] respectively: Let for each t, a set E(t) be assigned to the object fias a conjectured state space of Gat time t. Let A, a subset of {(I,oo,u,y): I = (to,t]CI,OO€Z(tO), uEUI,y€YI}, be the conjectured I—O-S—R of<9’ meaning: (I,oo,u,y)€A implies the.object in state 00 at time tO subject to the input u from tO to t will respond by pro- ducing the output y from tO to t. (£,A) will be a valid state description iff the following four conditions are satisified: 19 FIRST AXIOM SET (denoted A1): (Ml)--For each I = (t0,t]CI, (u,y)€RI <=>3OOEZ(tO) 3(I,oo,u,y)€A (M2)--Z(t0) is a unit set, where I = (t0,tl] is the existence interval (Sl)--For each I = (tO,t;]CI, 0062(t0) and uEUI, 3 at most one y€Y19(I,oO,u,y)€A. (S2)--Letting DI = {(Oo,u): EJerl;9(I,oO,u,y)€A} then defining AI:DI+YI by AI(cO,u) = y it is re- quired that: for each IO = (t0,tll and (OO’uO) EIUtO’tl] there exists at least one 0162(tl) 3: f (01,11) €D(tl,t] and (OO’uOOu)€D(tO,t] ____-> (A(tl’tj(ol,u) \K(t0,tl(00’u00u)/(tl,t3 SECOND AXIOM SET (denoted 42): (M1), (81) and (S2) remain unaltered but (M2) takes the form: (M2')——For each to €(t0,tl] == I and each u EU A , 30 62 3 0 (t0,t01 0 (to) 20 Ku0' 0W is admissible, for EU(tO,tll (ii) In case %OW and uéow are admissible, 3 y and I l A A must equal y'/ A It is trivial to verify that "2" is an equivalence relation. Therefore we define: H [u]A{u'EU A u'=u} which are mutually exclusive, collectively inclusive EQUIVALENCE CLASSES OF INPUTS antht A{Ht0[u] O uéU ]} as the FAMILY of equivalence classes of inputs. (Eo’t DHR(tO,t] A {(OO’u)ED(tO,t] : COEZHR(%)} and KHR(tO,t] A K(to,t]/DHR(tO,t] for each (t0,t1 DEF.II 2 5 (ZHR’KHR) will be called a HALF REDUCED STATE DESCRIPTION underfidi iff it satisfies 5241, i = 1, 2, 3. NOTE II.2.3: Under State Axioms,41, there is nothing to guarantee that (ZHR’KHR) or (ZR’AR) is still a State Description under Ail. However under142 (and 43) this is not the case. M2' in142 (and S2" 111.43) guarantees us the existence of enough non singular states, so that (2R,AR) and (EHR’KHR) are still valid State Descrip- tions. NOT. II.2.“: t ut vt Zt will indicate an input (or an O l 2 3 output) which consists of segments u defined on (t0,tl], v defined on (tl,t2] and 2 defined on (t2,t3]. ut v 1 will mean t0 = to and t2 = t1 where the existence interval I = (to,tl]. Now we give a State Description for a very simple object, getting payment for the effort; the payments being discussed after the example, the effort is made right now. 21 DEF.I.3.2: Here we define a special state description (2*, K*) as follows: 2*(t0), for each t is any set of the same 0’ cardinality as the family“t . 2*(EO), for i = O (tO’ElJ is any unit set. To define K*, we choose a function Gt 0 2*(t ) 1-1 0 onto t that such a function exists is O, guaranteed by the choice of 2*(to). (I,oo,u,y)€K* with I = (to ,t] iff: for to > to, 3(u0,y0)€R(EO,tJ 'auO/(EO t Mject (oO ) and (u O/I’yO/I) = (u,y) and for t = t is the single element of 2*(tO ) and O 0’ O (u,y) is arbitrary in RI. 0' The results can be summarized in the following two theorems: THM.I.3.1: If an object has a state description under state axioms.£2 then it is causal. PROOF: [REl] THM.I.3.2: The following statements are all equivalent: (i) 9’ has a state description underfl'l (ii) 6? is causal (iii) (Z*,A*) is a state description under.4l. PROOF: (i)==9(ii) [REZ] (ii)==$(iii) [RE3] (in this reference, to show that a causal object always has state 22 description it is proved that (Z*,A*) satisfiesAl for causal objects). (iii)==$(i) is trivial. NOTE 1.3.6: Thus, the state axiomsx¢l and142, compared to Zadeh's axioms extending their domain of applicability to such important classes of existing physical systems as in the various examples of pages 16 and 17, have denied state descriptions to non—causal systems, non existing physical systems. However, this is a point much in favor of state axioms.4l and.42 since we can, without hesita— tion, qualify them as being "more realistic." I and KI. As we said earlier, AI denotes the function from DI into YI whereas AI or Al(t) denotes the values that the CON.I.3.l: Here we make the distinction between A function KI takes on i.e., y = KI(o,u) but y(t) = AI(o,u). NOTE 1.3.7: When we write f (€0,91], the case T = (-m,w) is also included. Finally, State Axioms of Kalman listed merely for completeness close this section. Kalman's STATE AXIOMS: [KA] A dynamical system is a mathematical structure de— fined by the following axioms: (Dl)-—There is a given STATE SPACE Z and a set of values of time O at which the behavior of the i.e., 23 system is defined: 2 is a topological space and O is an ordered topological space which is a subset of the real numbers. (D2)--There is given a topological space 9 of functions of time defined on O, which are the admissible INPUTS to the system. (D3)—-For any initial time tOGEO, any initial state the o 62 and any input ueQ defined for t a t O 0’ future states of the system are determined by the transition function ¢ : QxOxOxZ + 2 which is written as ¢u(t;t0,oo) = o. This function is defined only for t a to. Moreover, any to~s tl s t2 in 0, any 0060, and any fixed uEO defined over [t tIJAO the following relations hold: 0’ (D3-i)-—¢u(t03to,oo) = 00 (D3-ii)--¢u(t2;to,oo) = ¢u(t2stl,¢u(tl,t0,oo)) In addition, the system must be NONANTICIPATORY, if u,,vEQ and u E v on [t0,tl]/\O we have (D3—iii)-—¢u(t;t0,00) = ¢V(t;t0,oo) (DM)--Every output of the system is a function W O x Z + R (D5)--The functions ¢ and W are continuous, with respect to the topologies defined for Z, O and Q and the induced product topologies. 2M I.M--A Brief Summary of the Following Chapters In Chapter II, after introducing some new concepts and modifying some old ones and after presenting an example of deficiency, we conclude that a minor change is necessary in the axiom.sets.*l.and.42, obtaining axiom set A3 which is stated, for matters of presentation, at the be- ginning of Sec. II.2. Then, in Sec.II.3 we discuss the interrelations of 94-1, :42 and 43, and show that .43 is al- most equivalent to.#2. Finally, Sec. 11.4 concentrates on reduced and half reduced state descriptions, yielding im- portant results, for a given objectC?, such as: the cardi— nality of any two reduced state space is the same, or any reduced state description is nothing but (Z*,A*) obtained by use of equivalence classes of inputs (DEF.I.3.2), etc. In Chapter III, we investigate how the properties of the object<9-—its linearity, time-invariance--are reflected in the properties of its state space. We show that the state space can be constructed to possess cor- responding nice properties. An important point about this chapter is that the properties of the system are defined, not in terms of its state description, but rather in terms of its I-O pairs, and then their implications on the state space deduced. Considering linear, time invariant and continuous objects in Chapter IV, the use of convolutional :. (I) (I) 'r 1 a... ‘ U4: Qua. O 5P} '1. .- (I) I F; v._ "w- ” vu. ‘- 25 representation for such objects is justified, and a state description of the form: dx (1:) __3t__ = mElamnxm(t)+bnu(t) dfiét) = A X(t)+Bu(t) i.e. K (k) K (k) y(t) = nglcnxnwwkiodku (t) y(t) = c X(t)+kiodku (t) is given for a large class of distributed systems re- spectively in sections 2 and 3, where X(t) is a square summable sequence for each t and A an infinite Hilbert Matrix. In the final section of this chapter, the important problem of "approximating a system having a continuum of states with objects having a finite dimen- sional vector space as their state space" is discussed and solutions offered. Finally in the Appendix, Chapter V being "the conclusions" chapter, first a justification for using distribution theory, then the "Orthonormal Series Ex- pansions of Distributions," recently developed by Zemanian and others, is given in its general lines. Thirdly some new theorems that are necessary for Chapter IV, such as the convolution of distributions infil', the proof that shows certain types of functions are in 01 are presented. CHAPTER II AXIOM SETS 41, 42, A3 AND STATE DESCRIPTIONS IN GENERAL II.l-—Introduction This chapter sets the basic rules, matures the necessary background and develops some very useful tools to be used in Chapters III and IV. Many theorems are proved about State Descriptions in closed form, few of which may be considered as ends by themselves. We con- sider this chapter of prime importance for the rest of the work and apologize for some long and tedious proofs. In section 2 we define certain important concepts such as Reachable States, Singular States, Equivalent States, Reduced and Half Reduced State Descriptions, etc., some of which are new, some of which are the modifications of the old ones, in the light of the new Axiom Sets. An example in the same section shows the insuffi- ciency of the State Axioms 41 and that £41 is not equiva- lent toflv2. To remedy the situation, a modification is introduced to .41 giving rise to .43. The latter, besides being justified physically, deserves attention because of its consequences. 26 27 In section 3, we deal mostly with formalities of investigating the interrelations of the Axiom Sets and prove that most former results do still hold under543. One of these surviving results is the very useful and important State Description of an object, based on the equivalence classes of inputs. In section A, we investigate and bring to light the nice properties of Reduced and Half Reduced State Descriptions. We show that Reduced State Descriptions are basically unique and strongly related to equivalence classes of inputs. It is here that we obtain the result "any two Reduced State Spaces for a given object have the same cardinality" which is an end by itself. Thus briefly section 2 sets the basic rules, sec- tion 3 matures the necessary background and section A develops the useful tools to be used later. II.2--An Example of Deficiency and the Axiom Set 543. Because of the new State Axioms .41, 912 it is necessary to revise the definitions of some important concepts. Some of the subsequent definitions are modi- fications and some are new. No explicit reference being made with respect to which Axiom Set they are given, they remain the same for 5L1, 542 and .43 (14.3 to be introduced later). Let in the following (Z,A) be a State Descrip- tion of 6' given by R A O I 28 DEF.II.2.1: A state 0162(t1) is said to be REACHABLE FROM A STATE 0062(t0), tl > tO if there exists an input u 6U l (t t1) such that: O, (i) (o ,u )eD O l (t0,tl] (ii) (Cogulou)eD(tO’t]¢: (Ol’U)ED(tl,t]’ for uéU (t1.t1 (iii) K(t0,t](oo’u10u”Qtl,tJ = A t0 in i (ii) A(t0’t](oo',u) = X(to,t](oo",u) holds for all u described in (i). 29 Two states oo'ez(to) and co"62(t0) are EQUIVALENT iff 00' is subsumed by 00" and 00" is subsumed by 00', i.e., (i) (o ',u)€D (0 ",u)ED O (tOJCJ‘: O (t0,t] and tEI, and " _" H ' (ii) A(to,t](00"u) - A(to,t](00 ,u) is true for all u described in (i). We believe that these definitions are self ex- planatory and need no further justification or physical interpretation. We now prove a simple fact. FACT II.2.l: Let 0262(t2) be reachable from 0162(tl) and 0152(tl) be reachable from 0062(t0). Then o2€£(t2) is reachable from 0062(t0). PROOF: Cl is reachable from 00 and 02 is reachable from 0 implies respectively that there is an input u EU 1 l (t0,tl] that takes 0 into 01 and another input u2€U O that takes 01 into 02. Now we claim that: ulou2 is admissible. Since (Ol’u2)€D(tl,t2]¢:> (c ,u Ou )ED , which can happen only if 0 1 2 (to,t2] uou EU 1 2 (t0,t2]. ulou2 is an input that takes 00 into 02: (i) (GO’ulOu2)ED(tO,t from above, 2] (ii) (00’u10u20u)éD(tO,t]¢=> (Ol’uZOu)6D(t1’t] ¢==» (02,u)€D for uEU< (t2,t] t t] 2, 30 (iii) X(to’t](OO,ulOu2Ou)/(t2,t] = X(t t3(02,u) proving the fact. 2’ Another simple fact that can be proved easily is the following one: FACT 11.2.2: The equivalence of the states defined in Def.II.2.3 is an equivalence relation and thus parti- tions the state space X(t) into equivalence classes of states for each tEI. Now we proceed to the "CONSTRUCTION OF REDUCED STATE DESCRIPTION." NOT.II.2.l: Let (Z2,A2) be a State Description ofGV under the Axiom Sete#2. In this case the Reduced State Description has to be obtained in two steps, as com- pared to a State Description under.Al (or’43), because of (M2') that allows more than one state at the creation instant. We obtain a new State Description (Z,A) from (Z2,A2) proceeding as follows: E the creation instant. X(t) A 22(t) vt > t O’ O X(to) g any unit set. D(t0,t] A D2(t0,t] Vto > to D(Eo’t] g {(oE0,u) : o€0e2(t0) and ueu(go’t]} 31 A éA t>E.AA A (t0,t] — 2(to.tJ V o o (t0.t] — {((t0.tl. og0,u,y) : ogoez(t0>. eR(gO,t]} NOTE II.2.l: The definition of AI, makes sense for I = (t ,t]. For, to each u€U A , there corresponds a o (t0.t] unique yEY A since O’has a State Description under (togtj’ .A2 and must therefore be causal by Thm.I.3.l. NOTE II.2.2: The pair (E,A) obtained from (Z2,A2), as explained in Not.II.2.l, is a State Description ofG?, under #2 . (M1), (M2'), (Sl) are trivially true since (22,A2) satisfies.d2, and by Note II.2.l, (S2) is also true for I = (t0,tl], tO > to, by the same reasons. For IO = o (E O’tl] and (oto’u0)6D(tO,tl] let 0162(tl), as re- quired by (82'), to be the state 0162(tl) guaranteed by (M2'). Then: (0c u Ou)ED A u OuéU m to’ 0 (t0,t2] ¢=:> o (- ,t2] :> 3 y OyEY A 3(u00u9y00Y)€R(E ,t J O 2 a = - y A2 ‘o ’ Figure II.2.l. 2(-°°> = {0} Let our conjectured X(t) = {01,02} -m (“t g 0 state space be: X(t) = {a,B,Y,5} t > 0 together with the conjectured I—O—S—R which is defined as follows for I = (t0,t]: 35 For to = -w .t s o, A = {(I,0,0,0), (I,o,l,l)} -t > O, A = {(I,O,OOO,OOO), (I,o,OOl,OOO), (I,O,lOO,lOl), (I,O,lOl,lOl)} t0.5 O -t s O, A - {(I,ol,0,0), (I,O2,l,l), (I,Ol,l,l)} -t > O, A = {(I,o1,OOO,oOO), (I,Ol,OOl,OOO), (1,02,100,lol), (1,02,lol,lol)} tO > O , A = {(I,d,O,O), (I,B,l,0), (I,Y,O,l), (I,6,l,l)} We first bring to attention that the need for parametrization (i.e., the need for a State Description) shows up, as discussed in Chapter I, when I = (t0,t] with t > 0. If we were given, e.g., the input 0 on I = O (l, 3] we would not know what output to make correspond to it. But if we are given the input 0, with the state a, we now can say that the corresponding output is O. A--The conjectured State Description (Z,A) of(9 satisfies AA: (Ml) Let I = (t0,t], then I must be in one of the five categories, i.e., either (t0 = -w, t.< O or t > O) or (t0 6 O, t < O or t > O) or (tO > O). In all the five cases: (u,y)ERI ¢=> 53a state (either 0, or one Of 01,02 or one of a,B,y,6) OO 9(I,Oo,u,y)éA. 36 (M2) is satisfied by definition of the state space. (81) By checking the list defining A one can see that for each 0062(t0), each uEUI 3.at most one output ya (I,oo,u,y)EA. (S2) is the most tedious one to check. We must go through all possibilities. Let I = (t0,tl] with its position indicated at the 0 beginning of each Case. t0 = -m, tl < 0 Consider (0,0)EDIO, then 01 is . A, . the state required by (S2). In t1 0 t2 fact: For t s O (0 O u )GD m u = O 2 ’ ’ t1 t2 (- ,t2] = t1 t2 = (o 0 )ED and A (o ,0) 1’ t1 t2 (tl,t2] (tl,t2] l A m (o 0 0 )/ = 0. (- ,t2] ’ t1 t2 (tl,t > 0 (0,0 2] For t u )ED .=:, either u = (-m,t2] l 2 or u = O 1 t1 0 t2 2 t t O 0 t1 0 t2 u )ED ' and again in either 1 t2 (tl’t2J In either case (cl,t u )/ = t2 (tl,t2] case A (O ,u) = A m (0,0 (t1,t2] l (- ,t2] t1 For the same IO, but for (O,l)€DIO we can go through the same arguments by replacing "0 inputs" with "1 inputs" and vice versa in the above discussion, the state required by (S2) being 02 this time. 37 t0 = -W, tl > 0 Consider (0,000tl)€DIO. The , , 1 state (S2) requires, is ‘V f I 0 t1 t2 0L€Z(tl). In fact: (0,0 u )ED m -==> u = 0 since that is the t1 t2 (- ,t2] t1 t2 only possible concatenation. Then (a, 0 )6D t t (t ,t J l 2 l 2 and A (d O) = A m (0 O O )/ _ = O. (tl,t2] ’ (— ,t2] ’ t1 t2 (tl,t2] For the three possibilities (0,0 l ),(0,1 0 ) and 0 t1 0 tl (0,101 )EDI the states required by (S2) are re- 1 O spectively B, Y and 6, the proof remaining the same. t tO < 0, t1 < 0 Let us take (Gl’tootl)EDIO° 1 The state required by (S2) is 0162(tl). Since: For t < 0, 2 )ED(toat2] =’ u = 1: 0t => (0 , O u 1 t0 t1 t2 0 and K (O ,0) = l ] (tl.t2J 1 (0 )ED l’t t2 (tl,t2 A (0 O O )/ = O. (t0,t2] l’to t1 t2 (tl,t2] (0 O u )6D 22' either 13 t0 t1 t2 (t0,t2] For t2 > O, In either case: (0l,u)ED(t J and A(t t t2](ol’u) = l’ 2 l’ A (0 O u )/ = O. (t0,t2] l’tO t1 t2 (tl,t2] For the same IO, but either for (01,1) or for (o2,l)€DI , 0262(tl) this time is the state required 0 38 by (S2). In fact for I = (t 0 t1] and (02,1)EDI , 3 O 0 both 01 and 02 work. For t2 < O, (0l,toltlu is the only choice =¢> both (01: tllt2)5D(tl,t2] ] and for both’A(t )GD 2 u = 1 t2 (to,t2] t1 t2’ and (°2’t lt )ED(tl,t (0 ,1) = l 2 t2J 1 2 >l 1t 1 (0 ,l) = A (tl.t2l 2 (t O l O,t2](°l’t For t > 0,2uEU 3(0 , l u‘ )ED , 2 (tl,t2] 1 t0 t1 t2 (t0,t2] so there is nothing to be checked and (S2) is auto- matically satisfied. tO s 0, t1 > 0 Consider (01,t OOOt )eDI . O l O _L. 1 . . Then a€2(tl) satisfies the I I l 1 t0 0 t1 t2 requirements of (S2). For: (0 , O O 0 )6D :2; u = O is the only 1 to 0 t1 t2 (t0,t2] t1 t2 possible sequel to t OOOt . Then (a,O)€D( and O l t t2] = 0. l9 t2](ol’t 0t A (d,O) = A (tl’t23 (t O l O )/ 0: t2 (tlat2j O l (O l O )9 ) 0 0 t1 ’ 2’t0 O tl lOlt )EDI the states required by (82) are re- 0 l O spectively: B, y, 6, the proof being the same as above. For the remaining (Cl,t (O2’t tO > 0 Finally for (a,t Ot )EDI , O l O , , . J a is the state that satisfies 0 t t 1 t2 (S2). In fact: 39 ] u = Ot and thus 2 2 JWith A 3y05Y(—°°,to] 9(u00u,yooy)ER(_w,t]}.n As 00 = Cl and 00 = 02 are the only two possible states at time tO < 0: Let first 0 = 0162030). Note that (01,1)ED(tO,t] O for t < O which implies y = l = A(t t](ol’l)' But 0’ since tO is negatlve it is clear that uou = Otolt¢U(-m,t] and hence there cannot be anyyO3(uOOu,yOQy)€R(_oo t]' We ’ conclude that 01 is not the state required by (M2'). Let now 00 = 0262(t0). The only possible input u such that (00,u)ED(tO,t] is u = tolt for t < O. AO But again 0t 1t is not admissible==$ 02 is not the 0 state required by (M2'). The following Notes discuss the State Axioms;4i, i = l, 2, 3 in the light of the above example, providing the promised payments. NOTE 11.2.“: It is clear, since there are no singular, and equivalent states, that the State Description of the Object(9'in the above example is a reduced State Descrip- tion. Besides that, our example serves two main pur— poses: It shows that the two sets of axiomsfll and 14—2 are not equivalent. It proves that the next Axiom Set we are going to define cannot be obtained from .41. NOTE II.2.5: One could also introduce into the I-O-S-R, A, many other quadruples, as we did for the case: tO-< O, t < O and I = (t0,t] by introducing (I,0l,l,l), that are not really necessary to get a State Description. For example, for the case, tO s O, t > O we could include in A the quadruples (I,01,100,lol), (I,0l,lol,lol) and still get a State Description which is valid under.4l. This turns out to be a deficiency ofrtl, which does not occur under State Axioms.42, as we have seen and will see. We call this a deficiency Offi4l for two reasons: 41 Even after including such "superfluous" quadruples in A, we still have a reduced State Description and can have many different ones by adding such quadruples at will. There is no way of getting rid of these quad— ruples by throwing out some states. Whenever we tried to prove the key results of this chapter (such as Thms. 11.4.2, 11.4.3, 11.4.5) we were always StOpped by the presence of such superfluous quadruples. All this trouble owes its presence to (S2) ofs4l, and was not present in142 due to (M2') which requires more than (M2). We will give a new Axiom Sets43 which will be justified by its physical interpretation and by its ends. NOTE 11.3.2: Let us suppose an observer wants to experi- ment at time t1, t1 not a creation instant, on the Object O'which is left at state 0152(tl). Let us further assume the object G'came to the state 0 from a state 0062(t0), 13 t0 < tl by an input uO, i.e., 01 is one of the states required by (S2) for the pair (0O,uo)€D(t Now if O’tl (tlgt2j’ that means u can follow uO, that also 10 the observer can apply the input uéU i.e., if (0 ,u)ED , 1 (tl,t2] means the concatenation uoou, besides being admissible, can be taken as an input pairable with the state 0062(t0). What we are trying to say is that (01,u)€D should (tl,t2] imply (OO’uOOu)ED(tO,t2] This property is not 42 reflected in (S2) ofJ4l and constitutes the only change in443. We list all of1¢3 for ease of reference. THIRD AXIOM SET 43: (Ml)-For each I (to,tJCI, (u,y)ERI 1173005sz) 3(I,00.u,y)6 A. (M2)-2(t0) is a unit set, I = (t0,tl] (Sl)-For each 1 = (tO,t], for each 0052(t0) and for each uEUI, there exists at most one output y such that (1,00,u,y)EA. (S2)-For each 10 = (to,tl] and (GO’uO)ED(tO,tl] there exists at least one 0162(tl) (i) (OO’uOOu)ED(tO,t] (=> (OI’U)ED(tl,t] (ii) X(tl,t](ol’u) = K t and VwéU t]’ uow is an admissible 3 0 (t0 input iff vow is. (ii) If for wEU (t0,t]’ uov and vow are admissible and §’2€Y(€O’t] are the corresponding outputs then y/(to’tJ Z/(t0,t] PROOF: L = A ==> et v u in U(t0,t] (i) Let uow be admissible for WEU(tO,t] ==>.HUEUE Bu = quOu/(t,tl] =:> vaOu/(t,gl] is admis- sible (since u = v) =:> vow is admissible. And vice-versa. (ii) Using the input u of (i), quOu/(t,gl] and V0W03/(t,El] are admissible. ==:- by equivalence A of u and v, y/(t t where y and z 0, = 2/ A l] (toatlj are the corresponding outputs to uowofi/ A (t,tl] and V0w0a/(t,El]' Hence clearly y/(to t] = Z/(t0,t]° 45 <2: Now let (i) and (ii) be true for u, VEU(E t J' O’ 0 Then for t = t0 we obtain the definition of u = v. FACT 11.3.2: Let D * be the domain of A *. Then I l * * A (OO’u)ED(tO,t] iff 0062 (to) and uOOUEU(tO,t] for any uOECtO(0O) and t0 > to. PROOF: (00,u)éD*(tO’t] (i, 3ygy(to,t]9y = A*(t0,t](00,u), (by def. of DI)(=? 3(u’y)€R(tO,t] Qu/(to’tjectowo) and (u’y)/(t0,t] = (u,y), (by def. of A*I). —_...:) 0062*(130) and ;=§ Let (00,u)€D*(t t] O, uOOu€U(EO,t] for any uOECtO(00), by Fact 11.3.1, Since 'any two inputs in Ct (00) are equivalent. 0 ¢== Assume 0062*(t0) with uOOuEU for any (E0,t] Note that (O,u)€D*I iff3y 3 (1,0,u,y)€A*. So by def. _* - "'* * of A , ((t0,t], 00,u,y/(t0,t]€A and hence (OO,u)ED(t t]' O, The following Thm.11.3.l is the counterpart of Thm.1.3.2 under :43. 46 THM.11.3.l: Under the State Axioms.43 the following are equivalent: (i) &has a State Description under¢13. (ii) (9 is causal. (iii) (Z*,A*) is a State Description under;43. EEOO_F: (i) =—) (ii) 9 has a State Description under 343 ==$ €?has a State Description underfiil (Note 11.3.1) ==P C?is causal (Thm.1.3.2) (ii) ==9 (iii) To show (2*,A*) is a State Description underuAG, the only axiom that needs verification is (82'), the others being verified in (Thm.1.3.2). Let 1 = (t t1] and (00 ,uO )€D* (t t be given. 0 O) 1] Then (00,uO)ED*(tO’tl] c—_—.) 0062*(t0) and O, for any u'O€C+ (00) by Fact 11.3.2. 01 = OuO EU 0 (t0 ,tlJ to CE [Ht (u'00u0)] is the state required by (82'). In 1 l ' * _.___. Y A fact. ©0,uoou)€D (t0,t2]'__9 u OOuOOUEU(tO,t2] for any V u OECtO(00) (by Fact 11.3.2). V ' _—> u 'ouEU(tO t2] for any u EHtl(u Oouo) (by definition of Ht (u'OOuO)) ===$ l (0l,u)ED* (t 2] (by Fact 11. 3. 2). l’t 47 * V i A 1 (0l,u)ED (t t2]‘=$ u OUEU(tO,t for any u ECtl(0l), 2] 1 I i.e., for any u éHtl(u OOuO) l, I A ———§u OOUOOUEU(tO,t2J and hence "' t 2] for any uoéHtO(u O) ou)6D*. (by Fact 11.3.2). (to,t2] u00u00u6U(tO,t —_? (OO’uO Let u be such that (00,u00u)éD* and (t0.t2] -l . As 0 = C [H l’ 21 0 t0 t0 Ctl[Htl(u Oou0)] we have y — A(t0,t2](00’u00u’ and y - (01,u)eD*( (u'0)] and 0 = t t l * _.. . - - I A A (tl’t2](01,u) are such that. 3373(y,u.Oouoou)eR(tO,t2] and y/ = y and ay',;(y' G ou)€R c and (t0,t2] ’ O (t0,t2] y'/(t t J = y' with GO 2 u'OouO. By definition of l’ 2 u0 = u'OOuO, the (tl,t2] portions of the outputs cor- responding to the inputs uOOu and u'OOuOOu are equal. =9y/(tl,t2] = y'. Hence (S2') is satisfied. (iii) ==9 (i) is a triviality. The next theorem is one that was promised in Note 11.2.3. It shows that the reduced State Descrip- tion of Def.II.2.4 satisfiesfiFB. THM.II.3.2: Let (Z,A) be a State Description ofl9’under A .43. Then the reduced State Description'(ZR,AR) obtained from (Z,A) always satisfieSJ43. -\ ed EEQQE; We verify each axiom to get the proof. (Ml) For each I = (to,tk:l 30052R(t0)9(l,00,u,y)€1§R => (u,.’>’)ERI since 0062(t0) and (2,23) satisfy43. (u,y)ERI =97 3(fi,§)éR(E 9(fi,§)/I = (u,y). 0’” 6D A to, ' A T." A ' D ‘ Applying (S2 ) to (ot ,u/(to t 0], w- can say 0 1‘] gooeuto) 9 (i) (“to’fi/(EO ,tolou )eD t t1<=>(00’u')5D(t t] and (11)K(EOHJ o (t0,t] [y = x(EO,t](OEO’U/(EogtOJOU)J/(to,t] = A(to,t](00’u) = y. Now GOEZS (tO ) (i.e., 00 is not a singular state) since it is reachable from 0E€Z(t0 ) by u/E( due to (1) t0 t0’ and (ii) above. Then the only way for GO¢ZR(tO) to occur ,t O], is when we throw all states in gkfto) but one. If ever 0 is thrown out there must exist a state OO'EXR(tO) 3 0 2 ' = v o 00 . In this case y X(t0,t](00 ,u) by definition ' V of equivalence of states. 80300 €2R(to) 9 (GO ’u)EDR(tO,t] and y = K 1(00',u). R(t0,t (M2) and (81) are trivially true since (2,3) satise fied them. 49 (82') Let I = (t t1] and (oO,uO)éD then 0’ and 0062(t0). R(t0,tl]’ (o ,u )ED 0 o (t0,tl] Since (2,K) satisfies A3, for I = (to,tl] and (oO,uO)EDI there exists at least one 0162(tl) such that: (i) (OO,uOOU)ED(EO,t] 4-“:9 (Ol’u)eD(tl,t] (ii) K(tl,t] (ol,u) = A(t0,t]K°O’uOOu)/(tl,t] VLlsatisfying (i). From (i) and (ii) it is obvious that Cl is reachable from o by u As 0062R(t0) it is non-singular and thus 0 O' reachable from 0t , say by u'o. Then by Fact 11.2.1 0 0162(t1) is reachable from cg . Thus the only way for O 01¢2R(tl) is that it be thrown out when we keep a single state from 201(tl). 1 ' is the state required by (82') for (ER,KR). In ' v ._~_. In that case 301 €ZR(tl) 3 01 o and Cl fact: (1) (co,u00u)éDR(tO,t] ¢==9 (OO’uOOu>ED(tO,t] since oOéZR(tO) ¢==e (01’”)eiD(tl,tl by (82') for (Z,K) (.~==9(ol',u)6D(tl’t since 50 ‘22:§ (o ',u)€D since 1 R(tl,t] (ii) A AR(tO,t](OO’uOOu)/(tl,t] proving (82'). NOTE 11.3.3: ABOUT242 end.d3. The already shown us that the two Axiom of D R(t0,t](00’u00u) = K(t0, of AR V 01 €ZR(tl) and by def. R(t t] l) t](00,u00u) by def. . Hence A(tl,t](ol’u) by by (S2') for (E,A) - A (o ', u) (tl,t] 1 since 0 ' = o l l I KR(tl,t](Ol ,u) since (01' ,u)EDR(tl,tJ example in II.2 has Sets are not obtain- able one from another. So now we concentrate our atten- tion on the relation between542 and.&3 and show that those two sets are almost equivalent, with a minor restriction on the State Descriptions satisfying1¢2. THM.II.3.3: (i) Let (2,A) be a State Description of(9'under 443. Then (LA) is a State Description ofG— under 42 . 51 (ii) Let (2,A) be a State Description of(9'under 5¢2. Then (2 KHR) obtained from (2,K) as HR’ defined in Def.II.2.5, is a State Description of (9' under 43 . PROOF: (i) (MI) and (SI) are the same for both 42 and 43, and (S2) is trivially implied by (82'). (M2') is (82') applied to the interval IO = (tO,tO]. In fact: uO€U(€O’t ; ¢==3(O“ ,u )6D A O] to O (t then (82') requires there be a state 0062(t0) such O’to] that: (i) <0g0,uOOU)€D(EO,t]“==b (Go’u)5D(tO,t] (11) x(t0,t](00’u) = [A(E0’t](oo,uoou) = yl/(t0,t] Naturally 0062(t0) is the state required by (M2'). Verifying: y = K(t0,t](OO’U) =7 (OO’U)ED(tO,t] =§ (Oto’uoou)ED(EO,tJ fia§eY(gO,t] 9W = K(Eost](0€o’u00u)J/(toat] = y ==933y0, namely yo = §/ “ .9(y oy,u OUDGZR A . (t0.tO] o o (t0,t1 3ybeY(EO,t] a(”0011’5’003’)6R(t‘o,t] ==? yO0y = A A (0A ,u Ou) ==9 (0A ,u Ou)ED A __ (t0,t] to 0 t0 0 (to,t] -47 t] ==b [yooy = A(Eo,t](ot‘o’u00u)]/(t0,t] = 52 (ii) We have shown in Note-II.2.2, on our way to the con- struction of reduced State Descriptions, that a State Description (Z,A) was obtainable from a given one, underyi2, such that X(to) was a unit set. Very little modification was necessary and indicated. Here then, we assume that this modification is already made and X(to) is a unit set. (M2') is satisfied by the above comment and (81) is automatically satisfied since(E,A) was already a State Description. (Ml) needs verification since there may not be enough states left after obtaining 2 Let I = HR' (t0,t]ci. EJOOQZHR(tO) 9(I,oo,u,y)EAHR ==9 (u,y)ERI clearly. Let (u,y)ERI => 3(fi,§)ER(€O,tJ 9(u,y)/I = (u,y). By (M2'), as X(to) is a unit set, 300€Z(t0) for u0 = u/ A .3 (l) (OEO’uOOu')€D(t0,t'](=>(oo,u')€D (OO’u')ED(tO,t'] (z) 0 and A (GO’u')€DHR(tO,t'] HR (0 u')6D <_..> - A 0’ (t ,t'] 31106U(t0’t019.< O (u) A(t0,t'](00’u') = A(€0,t'](0t ’uoou )/(t0,t'] L Applying (M2') to(o€0,uoou)€D(EO,tl], 301622(t1) 9 (5) (oto’fioouoou)eDHR(€ <=) (O‘l’u)ED(tl,t] (=9 (Ol’u)€DHR(tl,t]’ since clearly Cl is non singular and (6) [KHR(EO,t](OtO’aOOuOOu) A<€O’t](GEO,uoouOou)]/(tl,tj = A satisfies.42. PROOF: By Thm.11.3.3 \ZHR,AHR), is a State Description of G'under543. Then by Thm.11.3.2 obtained from (Z,A) the reduced State Description (ZR,AR) obtained from (2 A ) satisfies_43. Hence, again by Thm.II.3.3 HR HR’ (2 AR) is a State Description oft9'under542. R, COR.11.3.3: A conjectured, reduced State Description (2 ,A ) satisfies.42 iff it satisfie5y43. R R PROOF: This is a direct result of Thm.11.3.3 and Cor.II.3.2. NOTE 11.3.6: From now on, only Axiom SetgA3 will be used as the basic one. We will briefly say ”let (£,A) be a State Description" or "let X(t) be the State Space and A the I-O-S—R." These will mean ”satisfying543.” How- ever for reduced or half-reduced State Descriptions the set #2 can be referred to as a Theorem. 56 II.A—-About Reduced and Half—Reduced State Descriptions NOTE 11.A.l: With the following theorems we harvest the fruits of our efforts in the previous two sections. That these fruits are very nutritious will be appreciated as we proceed into the next chapters. One must have re- marked in the Example of 11.2 that to test whether a conjectured State Description is one that satisfiess43 may be a very difficult task for ~ome objects. The following theorems give us some algorithms to ease this task. THM.Il.u.l: The State Description (2*,K*>, obtained by use of equivalence classes of inputs, for an objectd}, is reduced. PROOF: (1) First we show that 2*(t), for each t, contains only non—singular states. Let 0062*(t0), then by defini- * A = . tion of Z (to), BuOEU(t ,t J act (00) Ht [uo]. This 0 O O 0 input uO, or any u'OEHt [uo] is the input that takes 0 GEO into 00. For: u ouéU c 2:.) 321 unique (unique sinceGis o (t0,t] causal by Thm.11.3.l) output it ytEEY(€ t] O 0" - A ....- —* n 3 (uOOu,ytOyt)€R(tO’tJ..u9 ((tO,t],oO,u,y)€A . Since 57 by definition of A* (I',o'0,u',y')EA* ¢==9 3(‘1'0’3"0)€R(1’50,t'] 9u'O/(‘c‘o’t'olect' (0'0) and O (u'O,y'O)/I = (u',y') which is clearly the case A __ * here. Hence uOOU€U(tO,t] .1) (oO,u)ED (. t t]° O, x : __ - . _* (003U)ED (t0,tJ —) aer(tO’t] 3((t09tJ3003u3y)€A ==>u00uEU(E t] again by definition of A*. Thus 03 ° * 11 A we have proved. (GO,u)€D (tO,t]<=é' u Ou€o(t ¢==9(0E ,uoou)ED*(E t] since Go is unique. 0 O’ [y y = A* A (0A ,u ou)]/ - y — tO t (to,t] to O (to,t] A*(t t](oo,u) from above. 0 ’ (ii) Second we prove that 2*(t) does not contain equiva- lent states for each t. We do this by showing: 1 2 n 1 = u 1 n .x. o O o 0 :=9 0 O o O for o O and o 062 (t0), t0 arbitrary. F. (O'O,u)€D*I(-—-:> (cr"o,u)€D*I and 0'0 8 o"0<==9 4 where 1 = (t0,t]. y -_- K*I(O"O,u) = A.*I(O‘"O’LI) V1.1 111(1). K y = A*I(o'0,u) ==$ 3(u'0’y'O)ER(tO,t]39u'O/(EO,tO] = Ct0(o'o) and (u'0,y'o)/I = (u,y). 3’ = Ke61(0"0’u)==9 3(u"0,y"0) R(EO,tJ 3uno/(Eo,tol = Ct0(o"o) and (u"0,y"0)/l = (u,y). 58 We will be done if we can show u'O/(A = n A u /(t0.tol equivalence classes Ct ( O equivalence classes have to be mutually exclusive. Then, as C to n 1 = Ct0(o 0) will yield 0 0 To prove u'O/<€ O’tO] = (a) Consider any wéU(to admissible. Then: 3 yéYi ' o O) and C is one—to-one by definition Ct u" / Su' / ,tll o toato] which would imply the equality of the to (0' ) O O O" O' (tO,tO] A ow is (tO’tO] 3 (u'O/(E0,to]ow’y)eRi. But, as X(EO) is a unit set y = 9(- K WE :U' w). O ’to JO o/(EO —lO :1 O" [RE3], 00 Ct (Ht [u' 0/ [(t ,t tOJJ) which is O itself, 0 t0 0 A = —* 3 (l) y/(t ,t J A (t ,t tl](0 O,w) and by state 0 l O A = -* A H equivalence (2) y/(t0,tl] A (tO’tl ](U O,w). (2)==€> A A A H 3(u03yO)ER(t ,t Jauo/(t ,to :16 Ct (0 ) and u OO/(t: ] O l O 0 t1 w. As u / A 613 (0" ), u 0/ (t0,t0] t0 0 (t0 ,to] u" / c =é> u” / c Ow is admissible. O (tO,tO] 0 (t0 to] The same proof can be used to show: u" / CV is o (t 1:00.12 J admissible ==b u' /A O (t to, t 0] By (32'):3Oc€z(to) and by is admissible. (b) Equations (1) and (2) Show that whenever u O/(EO,tO]OW and uHO/(t O’t0] (0"0), since the OW are admissible then 59 the part of the response corresponding to w is 9/(t t ] due to state equivalence. O’ 0 Hence u' / A = u" / o proving o' = O (tO,tO] O (tO,tO] O n _ 1 = n o O ——> o O o O‘ THM.II.M.2: For a given objectG}, any two reduced State Spaces have the same cardinality at time t. 33993: By Thm.II.3.l an object<9vhas a State Description iff (2*,K*) is one. By Thm.ll.u.3 2*(t0) is a reduced State Description for each to. What we will do then is to establish a well defined, one-to-one, onto corres- pondence between any reduced State Space X(to) and the equivalence classes of inputs which will imply a well- defined, one-to-one, onto correspondence between X(to) and 2*(t0). Given any 0062(t0), 00 must be reachable from at since X(to) is reduced, i.e., O H1) u uEU-A (o ,u)€D 00 (toms): 0 (120,121 E"10":U(Eo,tol=3 4(2) K A (0A u , ou)/ (t0,t] to o (t0,t] X(to,t](00’u) k We define: B : 2(t ) +39 to be : B (o ) A to o to to o Ht [no], where no is the above input guaranteed to exist 0 by reachability. 6O _ . v = H v Bto is well defined. i.e., o O o O ==> Bt0(o O) Bt0(o O) or equivalently o O o O __) Ht0(u O) H (u" ) where u' and u” are the inputs that take to O O 0 0E into 0'0 and ONO respectively. In fact: 0 u'OOu€U(EOat]<_-? (o'O,u)ED(tO’ by (l) n i. .. . = n ¢=9 (O O’u)ED(tO,t] since 0'0 0 O t] H A I g::§ u OOUEU(tO,t] by \1). K A (CA ,u' Qu)/ = A . (o' ,u) by (2). (to,tl to 0 (t0,t] (t0,t] o = - n y ._. n A(to,t](0 O,u) since 0 O o = K A (CA ,u" Ou)/ (t0,tl to o (t0,t] by (2) Hence by Fact 11.3.1, u'O : u"O. . ° v = n Bt is one to one. i.e., Bt (o O) Bt (o O)===$ O O O 1 = H ° 1 = H o O o O which is true iff Ht0(u O) Ht0(u 0) =5? _ n - z n __ y = n 0'0 - o O which is true iff u'O u 0 -—) o O o 0’ where u'O and u"O are as before. We prove o'O = o"0: ' V A (o O,u)ED(tO’t](==9 u OouEU(t 61 Bt is onto: i.e., given any Ht [uo], 30062(t0) O O 59Bt (00) = Ht [uo]. This means given any uOEU(E ,t ] O O O O is leaves the object in some state. This is guaranteed ' = A A A by (S2 ) applied to lo (tO,tO] and (oto’uO)ED(tO,tO] (or by (M2')). NOTE II.H.2: The following theorem is a simple corollary of Thm.II.A.2. But yet it is a very important one in that it shows for a given object<9'the reduced State Description is unique in a sense eliminating the ambi— guities posed by the example of Section II.2 about the reduced descriptions. THM.II.U.3: Any reduced State Description (Z,A) of<9 is nothing but (Z*,A*) defined by use of equivalence classes of inputs. PROOF: We have seen in Thm. II.U.2 the existence of a one to one, onto mapping BtO: Zkto) 9-Wto 9 Bt0(oo) A H [u 3, where u was the input that took oo into t0 0 O to 0' The existence of uO was guaranteed X(to) being reduced. Actually this makes Bt , the mapping Ct 0 O and X(to), the State Space 2*(t0) since by definition 0' 3*(to) is any set of the same cardinality as'Jft and 0 Ct any one of many one to one, onto mappings between 0 two sets of same cardinality (Def.I.3.2). 62 Let now ((tO,t],oO,u,y)6A( t]’ consider that u t O 0’ taking OE into 00. By definition of reachability O uOOuEU(tO,tO] and 3y09(u00u,y00y)€R(€0’t]. Clearly = u 6B (0 OJ 0 to = 3* (uOOu’yOOy)/(to,t] = (u,y) and uOOu/(€O,t O). The definition of A* I being satisfied K(t t] (t t] O’ O’ and hence (£,A) = (E*,A*). COR.II.A.l: A State Description of<9 is reduced iff it is the description (Z*,A*) obtained by use of equivalence classes of inputs. PROOF: The corollary is a direct result of Thms.II.A.l and II.U.3. NOTE II.A.3: As we go along, in Chapter IV especially, we will see that some results about Half Reduced State Des— criptions can be conveniently used to prove some descrip- tions are State Descriptions and half reduced. We first give the definition of a particular State Description (already defined in [RE5]) then prove it is a half re- duced description. But unfortunately the converse will not necessarily be true as will be explained in Note 11.4.6. 63 DEF.II.u.l: For t 6,3, a partitioning of U A into classes 0 (tO,tO] H't [uO] of inputs is called a HALF EQUIVALENCE PARTITION— O ING iff: A _._' ' (i) uOE U(t0’t03 _) uOEH tOEuO] ' Y A __ ' = (ii) uO,u O€U(t ,t 1") either H t [uo] O O O t v v v v = H tOEu O] or H O[u0](\H tOEu 0] ¢. ” ' ' 1 9; (iii) u 06H to[uo] -._=) u O uO. t The family 39' is: W A. {H' [u 1 : u eU ,. }, to to to o o (tO,tO] for tO > EO‘ As before we take Z'(tO) to be any set with the same cardinality as fl?t , for to > £0 and assign O t . 1 ! [uo] by C t . Z (to) + H t [uO], where O O O C'to is one of the many one to one, onto mappings between two sets of same cardinality. 1 I 0062 (to) to H t The I-O-S—R is defined as before: for IO = (t0,t], t > t Cu is admissible for some 0 O’ O 1%)ECUtO(oO) and y = yO/I, where (uoou,yo)€RI for t0 = t Z'(to) is any unit set with ((E0,t],o€0,u,y) being (I,oO,u,y)€A' iff u 03 such that 0€552(t0) and (u,y)éR(EO,t]. EQTE II.4.M: What we did in Def.II.u.l was to partition the equivalence classes of inputs into some mutually 6A exclusive classes of pre—tO inputs, and then take these classes as describing the states, everything else remain- ing the same° EQUIVALENCE CLASSES OF PRE—tO INPUT SPACE This is a parti— tion based on a sufficient condi— tion, rather than a necessary and sufficient condi- tion for equiva— lence of pre-tO inputs' That we HALF REDUCED PARTITIONING are now provided with a Half Re- Figure II.M.l duced State Description is the why of the next theorems. THM.II.4.4: The description (Z',A') of Def.II.u.l is a Half Reduced State Description (EHR’AHR)° PROOF: First of all it is a State Description since: (Ml) Let I = (t0,t]ci (the case to = 80 being trivial we consider tO > to): (U,y)€RI # 3(U,Y)€R(€O’t] 9 (u,y)/I = (u,y)- l _ :- Consider GO — Ct (H't O O A', (I,oO,u,y)€A'. [u/(E t ])' Then by definition of O’ O 65 (I,oO,u,y)6A' ——=)(u,y)€RI is obvious. (M2) is fulfilled by definition of 2*(EO). (81) is automatically satisfied, since for I = (t0,t], COEZ'(tO) and uEU we have a unique output that can I t ___ 1 correspond to uOOu, where Cto(oo) H tOEuO], if uOOu is admissible. (82') As in Fact 11.3.2 the domain D' of A' is: I I v ___ . l A D (t0,t] {(oo,u) . 0062 (to) and uOOu€U(tO,t] for v A t A = uOGC to(00)}, tO > to, and D (t0,t] {(ogo,u) : oEéEZ'(tO) and uEU<€O’t]}. For IO = (t0,tl] and (CO,uO)6D'I the state required by (82') becomes: -1 A _ I V I 1 ' -1 A = ' ' = I (i) (Go,uoou)6D'(tO,t](=2)oro€£'(t0) and u700(uOOU)EU(’£O’t] for u'OEC'tO(oO) <—__:> (U'OOUO)OUEU(EO’t] and 01 = c';:ez' <:=£>(ol,u)éD'(tl’t] for to > to. 66 A ' A — A ' A (oto,uoou)ED (t0,t]<"> otégz (to) and t t] O, <=:>u00u€U(EO,t] and Cl = "'1 I v (H tlEuO])EZ(tl) C' tl for (ii) A'(t0’t](oo,uoou) = yO/(t (u'00(u00u),yO)ER(EO’tJ with u'OEC'tO(CO). v = ~ A (tl,t](ol’u) yO/(tl,t] where (uoou,yO)ER(E for some uOEC' (o 0’ t1 1 v ~ z 1 ~ = H tltu oOuoJ ==? uo u OOuO =55 yo/(tl,tJ t] l) y / by definition of equivalence. Thus 0 (t1,t] . —t _ we have. A (t0,t](00’u00u>/(tl,t] _ A'(tl,t](ol,u) for tO > to. For t0 = to, t](0t ,uOOu) = y0 where (uOOu’yO)ER(tO,t] . A' O u = ~ 0 o) (tl,t]( 1’ ) yo/(tl,t] where (uoou,yo)€R for some, hence for (tom any uoéc'tl(ol) = ~ 7 hence yO/(tl,t] yO/(tl,t] giving us (S2 ). H'tlEuO] 2:; GO 2 uO and Now we prove that (Z',A') is half reduced. To show there are no singular states in Z'(t0) for any to, we 67 consider COEZ'(tO). Then 3u06U(EO,t] 9 C't0(oo) = H't [no]. uO is the input that takes 0“ into 0 . For: 0 to O (i) Clearly (CA u )6D' A . tO’ O (t0,tol A ' A A (ii) (oto,u00u)éD (120,,“ (2:) uOOueU(tO,t] and o€€2(to) by def. of O ¢=;> u00u6U(EO,t] for I uOEC to(CO) from above (74'? (<30,u)€D'(t t] by 0, definition of D'I. _' A = (iii) A (to,t](0t ,uoou) yo, for 0 yo 59(“00u’yo)53(€0,t] and A'(to’t](oo,u) , . . yO/(t0,t] since u OOu is admissible for any to O the (t0,t] portions of the outputs correspond— u’OQC't (00) = H' [uo] and since u' = uO, 0 ing to u' Ou and uoou are equal. 0 NOTE II.U.5: That the State Description (£‘,A') is not a reduced one can be observed by the following fact, if (Z',A') is based on a QPt that is finer than m% (the O 0 condition "finer" is necessary because equivalence Classes of inputs are by definition a Half Reduced Par— titioning). The fact now is there are states that are 68 equivalent, namely the ones corresponding to the classes I I I H tOEuO], H tOEul], . . . , H to[un], . . . such that a u z 0 O O z 2 O uo 1 un NOTE 11.4.6: The converse of Thm.II.u.A (the counterpart of Thm.II.A.2) is unfortunately not true unless some extra hypothesis is added. Any Half Reduced State Description is not based on a Half Reduced Partitioning for the following reason. Consider a Half Reduced State Description (Z AHR) and define 200(t0) A HR’ v . v :2 {o OezHR(tO) . o 0 do} for a fixed oOezHR(tO). If we had thrown all of Z (t ) but 0 GO 0 0 would correspond to an equivalence class Ht [uo] for 0 some uO, making 20 (to) correspond to the same class. 0 from £HR(tO) then 00 Now we compare card 20 (to) with card Ht [no]. We can 0 0 always assume card 2 (t ) > card H [u J (if it were 00 O to 0 not we could always add as many equivalent states to 00 as we wish without disturbing the State Description) for the purposes of our note. Then card 200(t0) > card HtOEuO] would make impossible the existence of a one to one, onto correspondence between X(to) and any Half Reduced Partitioning ’JI't , since equivalent states 0 can come only from the partitioning of an equivalence class Ht ['1 and the most Ht E-] can be partitioned to, 0 O is into its individual inputs. 69 NOTE II.H.7: We close this section, and the chapter, with a theorem that constitutes an answer to a problem posed in [RE2], concerning a property that State Des- criptions Should have. THM.II.A.5: The following is true for a Half Reduced State Description (ZHR’AHR) (i) (Co’uoou)eDHR(tO,t] =fi> (oo’uo)eDHR(tO,tl] where tl€(t0,t] is arbitrary. (ii) AHR(tO,t](OO’uOOu)/(t0,t = AHR(tO,tl](OO’uO) ll PROOF: As all states are non singular, Bu'o 300 is reachable from CA by u' and t0 0 I (GO’uOOu)€DHR(tO,t]‘==°'u OOuOOueU(tO,t] by definition of reachability I A =>u OOuOEU(tO,tl] the restriction of an input to ICE is admissible. =9 (CO’uO)eDHR(tO,t by reachability lJ again. AHR(to,t](OO’u00u) > both by reachability A (o ,u ) HR(tO,t1] o o A A (CA ,u' Ou )/ HR(tO,tl] t O O (t0,tl] J O 70 A A (0A u' ou ou)/ = A A (0A u' ou ) HR(t0,t] to’ O O (tO’tlJ HR(tO,tl] tO’ O O by causality (Thm.II.3.l). Hence: A (o ,u 0u)/ = A (o ,u ). HR(tO,t] o o (tO,tl] HR(tO,tl] o o CHAPTER III LINEAR, TIME—INVARIANT OBJECTS III.l--Introduction Many authors including Zadeh and Balakrishnan give the definition of "Linearity," ”Time-Invariance," etc., for objects, in terms of the State Descriptions of objects [ZA2], [BAA]. However, whether an object has these properties or not, does not depend on its State Description, a machinery introduced by us. In fact state descriptions are ambivalent: even if an object is linear or time—invariant, there are state descriptions in which the State Space and the I-O—S-R are not linear or time—invariant. Consider: db] [’51 “SIAM y(t) [:0 1]l-:%]+ u(t) ; This certainly is a Linear Object, but the state descrip- tion is non-linear (for an example of time-invariant case, substitute x by t in the 2 x 2 matrix). 1 71 72 In Sections 2 and 3 of this chapter, we start with the basic definitions of "Linearity" and "Time-Invariance" and proceed to show that state descriptions can be choosen to provide the object with "Linear" and "Time Invariant” Reduced Descriptions. Then Zadeh's definitions are ob- tained as results of these natural definitions. Also some nice results, such as "Separation Property of the I-O-S-R" for linear systems, "X(t) is the same set at any time t" for time-invariant objects, are attained among others. As another application of equivalence classes of inputs, the linear, time invariant system given by the equations: Qgé—Q = Ax(t) + Bu(t) y(t) = Cx(t) + Du(t) is investigated and conditions are found for its State Space to be reduced when A is in Jordan form. These conditions will prove useful in the last section of Chapter IV. To Close this section we would like to add that the whole chapter illustrates the importance of "the equiva- lence classes of inputs" and exhibits how much can be accomplished with the help of this concept without going into deep mathematical analysis. 73 III.2—-Linear Objects and Properties of the State Description NOTE III.2.l: We start the basic definition of "Linear Objects" and after stating some facts, we show that equivalence classes of inputs can be given some linear structure, which will be very useful in showing the "Linearity of the State Description." As usual, we have to stand some tedious Lemmas. DEF.III.2.1: The objectG}, given by the list Rf of I-O pairs, is a LINEAR OBJECT iff: (ul,yl)€Ri( and k 2:: (ul + au2,yl + ay2)ERA, for aER . (”2’y2)5Ri) FACTS III.2: (u ,y )ER 1. (9 is linear iff 1 l I (u2,y2)ERI (ul + au2,y1 + ay2)eEI vici, aenz. 2. (0,0)ERI, VICI. But (0,0)€R is the unique (19.12] pair with 0 first element, for all t due to causality. 3. (9 is Linear 2:? UI and YI are linear function spaces VI. DEF.III.2.2: l , l aHt [uO] g {auO . uOEHtOEuO] and aEIR} for a 7‘ O aHtOEuO] HtOEO] for a = O. ||l> O n ___ 2.1 : J + aH [u 0] {uO + auO . quHt [u 0], O O O 2 n uoéHtOEu O] and aéml} I u o t LEMMA III.2.l: Let(9 be linear, u and u'OEU and 0 (t0 ,tOJ ' A A WEU(tO’t1] Then (uO + u O>OWEU(tO,tl] 2:; gwl and A ' ’ A A w26U(tO’tlJ auoowl and u OOW2EU(tO,tl]’ and uoowl + u'OOw2 = (uO + u'O)OW. PROOF: "<:::" is trivial. H H 2:: u EU A ¢::; 3w éU A 3L1ereU o A . O (tO,tO] l (tO’tlJ O l (tO’tlJ As the input space is linear (Fact III.2.l), (uO + u'0)0w — Now if we let Owl 6U ::$u Mo(w wl )EU uo (to ,tl] (1:O ,tll — A o ' A W2 A w wleU(t ,t 3 then obv1ously u 00w2.€-U(t at J and O l O l v = v (uO + u O)ow uOOWl + u OOWZ' LEMMA III.2.2: For a linear objectd}, u'O : u”O and 1 ~ 2 , l N n 2 , n l u 0 - u 0 :=> u 0 + u 0 — u 0 + u 0’ for u 0’ u 0’ u 0 2 and u O€U(t0,t0]' PROOF: (i) Let (u'O + ulo)ow be admissible, by Lemma III.2.l. l ,1 1 _ 3w' and w EU(t ,E ] 3 \u'O + u O)Ow - u'Oow' + uloowl (ii) 75 1 u OwiéU A A iff O (to,tl] R C , since u' u" ow'eU A A o (t0,tl] 0 o > Using linearity and l l . u 00w EU(EO’€l] iff 2 l l 2 u 00W €UOWEU(tO,tl] 2:; we can show (u" l t A A We must now show that the portion corresponding to w, of the response to (u" + u20)0w is identical with O the portion corresponding to w of the response to (u'O + ulo)ow. In the following all concatenations occur at to. There exists unique, since causal l l (Thm.1.3.2), y'ooy' and y 00y €YU.Oo(a W)€U( O O tofill n “l = n A <;:3 a[u O0(a w)] (au O)OwEU(t by linearity. Then there exists a unique y'Ooy5Y(gO,€l] 3 Y ' A A - ((au O)ow,y 00y)ER(tO,tl] Now. —1 —l _ v ' A A Linearity :::>(u O0(a w), a (y 00y))6R(tO,tl] t N n H ‘1 “l n A A u u 0 _ u 0 ::s (u O0(a w), a (y OOy))€R(tO’tl] where y 0 is 3 (au" o’y"o (E t] 0’ o . n H A A Linearity 22$ ((au O)OW: y 00y)ER(tO,tl]° That the t to £1 portions of the responses, to (au'0)0w O and (au" Ow, corresponding to w are equal proves this 0) Lemma” 77 NOTE III.2.2: Our aim is to show the linearity of the collection of equivalence classes of inputs and all the machinery of the previous three Lemmas has been intro- duced for this purpose. Although Ht [u'o] + aHt [u"0] is O 0 well defined for each t in Def.III.2.2, there is nothing 0, that guarantees it is an equivalence class. Once this is established, the linearity ofiflt for each tO is then 0 reached with respect to the operations defined in Def. + au" III.2.2. On the other hand, u' 0 being an input 0 for aEfi , H [u' + au"0] is a well defined equivalence t O 0 class. The next theorem is central, in that it estab- lishes the linearity of Rt 0 THM.III.2.l: For a linear object6>, Ht [u'o + au"O] = 0 Ht [u'o] + aHt [u"0] Vtoél. Hence the equivalence 0 0 classes of inputs form a linear space'fl£ VtOEI. O PROOF: (1) First we show H [u' + u" = H [u'O] + H J Eu" 0 O 0 t0 0 V H 2: I H Let uOEHt [u 0 + u 0] then uO u 0 + u 0' We 0 _ l 2 can write uO as follows. uO - u 0 + u 0 where l _ 2 _ l , u 0 — u'0 and u 0 — uO - u'O. Then u OEHtOEu 0] ~ n _ v 2 n obviously. Also uO - u'O + u 0 :2; uO u 0 u 0 t t O]° Eu" 0 we obtained u = u1 + u2 where u1 EH [u' J O O O O to O ’ 2 ~ . by Lemma III.2.2 ::$ u 0 - uO - u OEHt 0]. So (ii) (iii) 78 2 u v u OEHtOEu OJ. By Def.III.2.2 uOEHtOEu O] + H t H 1 H Ht [u OJ :29 Ht [u 0 + u OJC:Ht [u OJ + Ht [u OJ. 0 O O O 1 n = Now we let u E HtOEu OJ + HtOEu OJ. Then uO l 2 1 ~ 2 ~ n u 0 + u 0 Bu 0 - u'O and u 0 - u 0’ (Def.III.2.2). _ l 2 z 1 H By Lemma III.2.2 uO — u 0 + u 0 u 0 + u 0 :2: t n y n = uOEHtOEu O + u OJ. Thus HtOEu 0 + u OJ H [u' J + H [u" J. to O to 0 Second we show Ht [aqu = aHt [uOJ for O # aER , 0 0 since for a = O,Ht0[auOJ = HtOEOJ and aHtOEuOJ = Ht [OJ by Def. III.2.2. 0 ~ 1 Let u'OEHtOEauOJ :29 u'O - auo. Define u 0 A -l , , _ l l = -l , z a u 0' Then u 0 - au 0 and u 0 a u 0 uO by Lemma III.2.3. Hence u' EaH [u J 2:; 0 t0 0 HtOEauOJCIaHtOEuOJ. 1 7 = Now let u OEaHtOEuOJ :::>u O au 0 for l 1 ~ u OEHtOEuOJ. Using Lemma III.2.3 u 0 - u0 :=; l g Y I: ' au 0 auo : u 0 8.1.10 fi U. OéHtOEauOJ . Thus H [au J = aH [u J. to O to O t n = 1 n = Finally H [u 0 + au OJ Ht [u 0] + Ht [au OJ 0 0 O H [u' J + aH Eu" J proving the theorem. to 0 t0 0 t 79 DEF.III.2.3: 2*(t0) will be called a COMPATIBLE STATE SPACE for the linear objecté> iff 2*(t0) is a vector space iso- morphic to Rt for each toéi. 0 NOTE III.2.3: When defining 2*(t0), the one to one, onto mapping Ct : 2*(to) +‘H was chosen to be any 0 . t 0 one of such mappings among many others existing between two sets of the same cardinality (Def.I.3.2). To get a compatible state space we also require Ct be chosen in 0 such a way that, it be an isomorphism between 2*(t0) and Hit , which is always possible due to Thm.III.2.l. 0 Now we are sure, at least, that a linear object can be provided with a reduced state space that is linear. NOTE III.2.4: The answer to the question of whether a half reduced state space, based on a half reduced parti- tioning (Def.II.u.l), can be chosen to be a linear space, may not be affirmative. The reason is the lack of a theorem similar to Thm.III.2.l providing us with some linear structure for'X'to. For example, if we take as half reduced partitioning, the equivalence classes of inputs‘xeo with only one equivalence class partitioned into two nonempty subclasses (the others remaining the Sanka), then this‘M't0 has not a linear structure in the sense of Def.III.2.2. However, we can with no difficulty assert that there are special ways of partitioning 3% O 80 in a manner that can provide us with a linear structure. One way of doing it is to take each input as a half reduced partitioning; other ways will be seen in Chapter Iv. The next theorem is a result about the dimension of the state space and illustrates the usefulness of equivalence classes of inputs. THM.III.2.2: If the input space of(9’is finite dimensional of dimension n, then the dimension of the compatible state space 2*(to), dim 2*(t0), is such that dim 2*(t0) S dim U(t ,t J S n, for all to. O O PROOF: It is enough to show dimfik AS dim U A _____ t (t ,t l O O 0 VtOET, since 2*(t0) is isomorphic to It :2) dim 2*(t0) O dhngfi . It is also clear that dim U A is less than t (t ,t l O O O dim U6 ,{3‘ J VtOEI, since every u OEU(tO J is the O 1 t0 restriction of some input uEU to (t ,t J. Let (to ,tlJ O 0 1L1, u2, , u nEU(tO J be a basis for U(to to] Then the classes HtOEulJ, HtOEu2J, . . . , Ht0[unJ spanszmto For any equivalence class Ht [uOJ in let we have: 0 O u 6U A and u can be expressed in terms of the basis 0 (t0 ,tOJ O 1%) = g a1. Using Thm.III.2.l Ht [uOJ = Ht [ Z aiu iJ - i=1 0 t0 i=1 ? II [ J H di 1% di U a u . ence m < m A . i=1 i to i to (t0,tol 81 NOTE III.2.5: Fortunately, dim 2*(t ) = dim U A does 0 (tO,tOJ not generally hold, since there is a multitude of known examples where the dimension of the state space is less than the dimension of the input space. In order to show dim H = dim U(A we need to show that the set 0 to’to] HtOEulJ, . . . , HtOEunJ is linearly independent. Equivalently then, we must show that Ht [ukJ = O t n n E ath [uiJ for at least one a1 7 O :2; uk = E aiui. i—l O i-l ifik ifik n However we can only infer uk = 2 aiui, which does not i=1 i¢k necessarily give equality. NOTE III.2.6: Thm.III.2.3 that follows is as close as we can get to Zadeh's definition of linearity [2A2] without further assumptions on our object. It also demonstrates a linearity property of the I—O—S-R, AI. THM.III.2.3: The object @—is linear iff it can be given a reduced state description (ZR’KR) such that the follow- ing are true: (i) 2R(t0) is the compatible state space (Def.III.2.3), vtOeI. (11) D is a linear space for to > to, i.e., R(t0,tJ ' Y H H (0 O,u O)EDR(tO,tJ and (o O,u O>EDR(tO,tJ 2:; (iii) PROOF: 82 ' H I H (o O + ao O,u O + au O>EDR(tO,tJ for (t0,tJ<:(tO,tlJ and for any aGR . g AR(t tJ DR(t t] 15 a linear trans- O’ 0’ " g n t H _ formation, i.e., AR(tO,tJ(O O + ao O,u + au ) _ A ,u') + aA ,u") for (o'o,u') (01 (0." R(t0,tJ 0 R(t0,tJ o and (0" ,u")ED for (t ,tJ C:(t ,t J and for O O # O l R(t0,tJ all aEIR . "<=: " Only (ii) and (iii) are enough to imply that G?is linear. In fact (ii) implies that the input space is linear and (iii) implies that the object is linear. (i) (ii) ":=;" is somewhat tedious to prove. is true by Thm.II.2.l and Def.III.2.2. By Cor.II.u.l any reduced state description is nothing but (Z*,A*) and its properties will be used in the proof of (ii) and (iii). ' H H Let (o'O,u )EDR(tO,tJ and (o O,u O>eDR(tO,tJ' By Fact 11.3.2 we can write (O'O’u'O)EDR(tO,tJ ¢:j'°'OEZR(tO) and uoou'6U(EO,t] for any uOECt (0'0) 0 élj 3u'06U(€0,tO] 901:0(0'0) = V V ' A H [u OJ and u Oou eU(tO,tJ to (iii) 83 (0H0,u"0)€DR(tO,t] @ auner(E0,tO] 3Ct0(0"0) = u u n A HtOIu OJ and u Oou'eU(tO,tJ The object being linear u'OOu' + a(u”Oou“) = (u'O + au"0)o(u' + au")6U(tO,tJ° The equivalence ! n = v n _ class HtOEu O + au OJ Ht0[u OJ + aHtOEu OJ cor responds to the state 0'0 + ao"0 since C : Z (t ) + H is defined to be an isomorphism. tO R 0 t0 Hence 3 uOEU ’to], namely u0 = u'O + au"O 3 (t0 f '1 .- Ct0(o O + ao O) — HtOEu' (u' + au"ohflu' + au”)6U(€ tJ° This implies C) 3 v n‘ v n (o O + ao O’ u 0 + au O>EDR(tO,tJ by Fact II.3.2. 1! O + au OJ and O ' 9 l '1 I! I! Let y A KR(tO,tJ(O O,u ) and y A KR(tO,tJ(O O,u ). i I' ! 1' From part (ii) (0 O + ao 0’ u + au )EDR(tO,tl]. Therefore we will be done if we can show y' + ay" = A + a0" u' + au"), i.e., we have R(t0,tJ(G'O o: to show: 3(fi,y)6R(EO’t]9 fi/(EO’tOJECtOW'O + ao"o) and (fi’§)/(t0,tJ = (u' + au", y' + ay"). y' = §R(t0,tJ(O'O’u') ¢==$ 3(3',§')€R(€O’t] air/(g 1:35 c:t (0'0) and (fi"§')/(t0,tj = (u',y'). y” = KR = (8' + afi",§' + a§">1/ , (tO’tlJ (u' + au”,y' + ay") which is what we had to show. NOTE III.2.7: As we said in the previous note Thm.II.2.3 is the closest result to Zadeh's definition of linear objects under State Axioms A3 and the basic definition of "Linear Objects." One extra condition on the nature of 6; namely: 85 (Cl) "U A is so that u 0 is admissible " (t 11 E0 to E1 for any uEU A and any t , as well as O uA is (to,tOJ 0 o to t1 admissible for any u€U(o,u)EDR(tO,t] by definition of reachability, for any uEU(t t]° 0, From all these discussions for a linear object 6» satisfying (Cl), we obtain the separation prOperty and the linearity of the I-O—S-R as defined by Zadeh [ZA2]. This is summarized in the next theorem. THM.III.2.M: Let G}be a linear object that satisfies ((01) Note III.2.7). Then the object Gycan.be given a reduced state description (ZR’KR) with the following properties: (1) ER has "the separation property" i.e. KR(tO,t](G’u) = KR(t t ’ —R(t O, for all 062(t0) and for all uEU (t0,t]Cl is arbitrary. (ii) K is "zero input linear” i.e. R AR(tO,t](Ol + ao2,0) = AR(t aAR(tO,tJ(G2’O) (iii) ER is "zero state linear" i.e. KR(tO,t](O’ul + au2) = KR(tO,tJ(O’ul) + aKR(tO,t](O’u2) both (ii) and (iii) are for all 01, o2EER(tO), for all ul, u2eU(tO,t] and for all aETR. 87 PROOF: The proof follows directly from Thm.III.2.3, the condition (Cl) and the discussion of Note III.2.7. NOTE III.2.8: That (Cl) has to be assumed is shown by the example of the object "with complete memory." Let G'be given by: U A {u(t) : u(t) = aER VtE(E0’E1] and {8 :3u(t)€U(EO,€l]9u(t) = a} =18} ‘ A {(u,ku) : uEU A A and kEHiis fixed}. The object (9 defined by R(A t J is linear but clearly 1 guOéU(E0,t J 3u0006U(E t]’ t > tO unless u0 = O. R0903 III.3--Time-Invariant Objects and Properties of the State Description NOTE III.3.1: According to Zadeh and Balakrishnan, and although Zadeh defines the concept of "weak time- invariance," the same way we define our "time invariant objects," the definition of a time invariant object is based on the I—O-S-R. That X(t) is the same set for all t is part of this definition [BAH]. Our task here is again to start with the more basic definition of time- invariant objects and get the afore—mentioned definition as a result under .43. Contrary to the definition of linearity, Def.III.2.l, where the existence interval could be any finite or semi-finite interval and not hurt linearity, the concept of time invariance requires from 88 the object, 3 = (—m,W) as the existence interval. From a time invariant object we at least expect that it does not change its main properties as regard to the inputs and outputs, i.e., for example an input u admissible from tl on, must be admissible from any t2 on and must yield the same output y whenever applied. As the existence interval is finite means that all basic I—O pairs are defined on i=(E t1], where both E and t1 are finite, one cannot 0’ O speak of an input being admissible for t2 < to or for t2 > t1. properties reflecting the time varying aspect of anything Starting to exist and dying are such important that even theoretical objects possessing these properties must be expected to be time varying. Thus, although some ”semi time-invariance" can be defined for objects with finite existence interval, the only real (expected) defi- nition of time invariance can be given for objects that exist forever. NOT.III.3.l: From now on, the existence interval T will be (—w,w) for the objects under consideration (this was already mentioned in Note 1.3.7). NOT.III.3.2: Let f(-) be a function defined on the domain DCHR. AT is the operator defined on the space of func— tions with domain D by ATf(t) g f(t-T) VtED and where T can be any finite real number. The domain of ATf is the set D + T g {t + T : tED}. 89 DEF.III.3.l: An object G’whose existence interval T = (-w,w) is called TIME—INVARIANT iff (u,y)ERi <:_—_> (ATu,ATy)€-Ri. FACT III.3.l: The object (915 time invariant iff (u,y)ER(tO,tlJ :==$(ATu,ATy)€R(tO + T,t + T] for all 1 intervals (t0,tlJ. PROOF: "¢:::" is trivial. " => " If G is time invariant then: (u,y)€R(tO,tl] :1» 3(fi,§)ERi 3 (mm/(120,1: ] =(u,y). But 1 we have (u,y)ERi :::>(A1fi’AT§)€Ri by time invariance. Hence (ATu’ATy)/(t0 + T,tl + r] = (ATu’ATy)eR(tO +I,tl + 11° DEF.III.3.2: The TRANSLATE AT OF AN EQUIVALENCE CLASS is defined o ' to be. A-THt [uo] A {u EU 0 u' = ATuO} (—°°,tO + T] NOTE III.3.2: The minus sign in A—THt [uo] is strictly O notational, we could as well have used AT. The reason which made us choose A_T is, when AT is applied to a function f(t) it changes its argument to f(t-T), however as we shall soon see in Thm.III.3.l, A-THt [no] is the 0 same equivalence class as H [ATuo] where this time 0 t! t'O = t0 + T, i.e., the argument has been modified by T instead of -T. 90 NOTE III.3.3: It will make more sense to speak of "the translate of an equivalence class" once the next theorem is proved. We have to show that ATH [no] is an equi- t valence class, however, it may be clegr intuitively for a time invariant object. So we prove that when an equivalence class is shifted, it still contains nothing but the shifted version of the inputs it had before the translation. THM.III.3.1: For a time invariant object (9, u'OEHt [uo] <=a o v = ATu O€A_THt [uO], i.e., ATHt [uo] Ht + T[ATuO , for O 0 0 any TETR . 0 V PROOF. Let u OEHtOEuO]. By definition of HtOEuO], u'OEHt [uo]‘==$ u'O = uo. Then 0 <1) (ATu'O)Ou€U(_m’t](=>u'00(A_Tu)EU(_m’ 1H] by time invariance. @ uOO(A—TU)EU(-°°,t—T] since u = u' 0 O ¢::>(A1u0)0ueU(-agt] by time invariance for uEU( is t > t + T, proving (ATu'O)Ou is t + T,t]’ O O admissible iff (Aru0)0u is. 91 (ii) Since they are admissible there exists y' and y such that: ((ATu'O)Ou,y')ER(_m’t] @223 1 (u O0(A_Tu), A_Ty')ER and (-°°,t-T] ((ATuO)Ou,y)ER(_m,tJ ¢::b (uOO(A_ u),A_Ty)€R both by time in— T (-w,t—T] ! = variance. But A_Ty /(to,t-T] A—Ty/(to,t-T] since u' = u ' = ° ' z ' y/(to + T,t] which proves ATu O ATu and 9 therefore ATu € A-THtOEuO] . OE A-THtO[uO]’ which is true iff (Def.III.3.2), we can proceed as Letting ATu' ' a: ATu O ATuO v 2 1 above to show u 0 u giving us u OEHtOEuO]. That A-THtOEUO] = Hto +T[ATuO] clearly follows from above. NOTE III.3.U: For a time invariant object(9, a shifted equivalence class is still an equivalence class, justify— ing Def.III.3.2. It would be nice to show that this prOperty alone makes G'timm invariant, that is a converse to Thm.III.3.l. However, this is not true in general as the following counter example shows: Let G—be given by the unique pair, R(_m,w) = (u(t),e—|t|) where u(t) = C, Vt. Then HtOEuO] = {c/(_m,to]} is the unique ~\b EA ‘ a . Vl- N‘s sill 92 equivalence class Vtoé l, and trivially uEHt [uO] (:2) O ATuEAqHt [no]. But the object is not time invariant, O ltl is not an admissible output. for A e- T Of the following two corollaries, the second one is a result we were aiming for. COR.III.3.l: A—T(ATHtO[uOJ) = AT(A-THt0[uO]) = HtOEuO]. COR.III.3.2: The reduced state space X(t) for a time invariant object can be taken the same set Vté(-W,w). PROOF: All we need to show is that Ht and Ht have the O 1 same cardinality for any t0 and t1. That there exists a well—defined and l-l (since invertible) mapping T :tho + th defined by T(HtOEuOJ) A A-THtOEuOJ’ where T = t1 - tO is clear by Thm.III.3.l and Cor.III.3.l. It is also clear that T is onto, since any Ht [uo]€]% is O l the image of the equivalence class H [A u 36H because tO -T 0 t0 of T(HtO[A_TuO]) = A_THtO[A_Tu0] = HtlEuO]. This proves card Wt = card 3ft . As by Thm.II.M.3 any reduced state 0 1 space is nothing but 2*(t), the one obtained by use of equivalence classes of inputs, the same set can be put into l—l, onto correspondence with both3€ and 36,5 for t O 1 any tO # t1, i.e., a unique set suffices to be taken the state space VtEl. (I) ll: --.1 \"‘" 93 NOTE III.3.5: Of course we are completely free to select the set 2R(t) for each t as long as it has the same cardinality as H but any other choice, than the same t’ set Vt seems to be artificial, unless there be a neces- sity. We achieve our next goal with Thm.III.3.2 by showing the form the I-O—S—R takes when the object is time- invariant. It is here that we have to remember the dis- tinction made between AI and AI in Con.I.3.l. NOT.III.3.3: Let 00(t0)62(t0) denote the state corres— ponding to the class Ht [uo] for the time—invariant O object(93 where X(t) is reduced Vtél. Then 00(tO +T)EZ(t04-T) will denote the state corresponding to the class A_THtO[uO],‘VT€(—w,w), i.e., 00(tO + T) = c—1 (A_THt [u01>. tO + T O THM.III.3.2: An object(9'existing over (-w,m) is time invariant iff it has a reduced state description (2 AIR) such that: R) (1) (00(t0)’U)€DR(tO,t] e==a (00(t0 + T),ATu)€DR(tO TE(_oo,oo) (ii) A (O (t ),u) R(t0,tl] 0 o A (00(tO + T),ATU) Vt€(t0,tl] R(t + T,t + T] O l 9U PROOF: ":::;" Let Gfbe time invariant. Then: (i) Using Thm.II.H.3, as usual, we do the proof for (11) (Z*,A*). By Fact 11.3.2 (0(t0),u)ED*(tO’tl] (:2, uOOu€U(-w,t1] for any u act (00) = HtOEu]. 0 By Fact III.3.1 uOOu€U(-w,tl] ¢::;AT(uOou) = O uOEHtO[uJ <2) ATuOEA-THtOD‘IO:l A C 1 Thus we have (ATu O)Q(ATU)EU( T] and by Thm.III.3.l to + T(o(tO + 1)). for any u' -m,tl] 0’ the quantifier "for any" being well-placed due to Thm.III.3.l. Thus, again by Fact II.3.2 (c(tO + T),ATu)€D*( t + T,t + T]‘ O l y = K*(t0,tl](o(t0),u)<::> 3(fi’§)ER(-°°,tl] 9 U/(-m’t0]ECtO(OO) and (u,y)/(130,123 =(uay) by definition of A* By Fact III.3.l I' (u,Y)€R(_w,tl]<::; (Aru’ATy)ER(-m,tl + T] and hence + T] = (ATu,ATy). Moreover, (ATu,ATy)/(t0 + T,t l afieu m 33/ 00 ac (o ) =H [a]. By (- 31:0] (- 31:0] to O to Thm.III.3.l, AT(u/(_m’to])€A_THtO[u] = Ct + T(o(tO + T)). And again by definition of ’* = ‘x - A I’ ATy A (t + T,t + T](o(t0 + T),ATu) which 0 l 95 * implies A (to,tl]3°€z*(to) 3 y(t) = A*(t0,tl](°(to)’u) by axiom (Ml). Then (ii) :::>A* (0(t ),u) (t0,tl] O A* + T](o(t0 + T),ATU) for any TE (-w,W), or (t + T,t H O = -* that ATy A (toatl](0(t0 + T),ATu) :3 (ATu,ATY)ER proving the theorem. (tO + T,t + T] NOTE III.3.6: A word about half reduced state descrip- tions closes this section. A proof in the same lines as Thm.III.3.l can be given to show that the familyIR't 0 based on any half reduced partitioning can be translated by T, to yield the family 18": + T which has the same 0 structure as W't . More precisely, any H't [uo]€ I't O O O can give rise to a A_TH'tO[uO], which can be shown to be ' to + TEAIuOJEQEtO + T and to constitute a half reduced partitioning of U(_0° t 3 equal to H' O + T]. This then will allow us to keep the same half reduced state space ZHR(t),‘VtE(-m,m) and to have a property of AIHR that is similar to the one in Thm.III.3.2. NOTE III.“. 96 III.U--An Application of Equivalence Classes of Inputs to Lumped Objects which have where Anxn’ 1: In this section we shall deal with objects a representation of the form: dfiét) = AX(t) + Bu(t) III.A.l y(t) = CX(t) + Du(t) anl’ Clxn’ Dlxl are constant matrices. Our concern here will be to see under what conditions on A, B, C and D equations III.A.1 yield a reduced state description with minimal dimension. This result will be useful in section IV.5. We start with the precise defi- nition of t he object under consideration. DEF.III.4.1: A linear, time—invariant object will be calledC}L iff it sati (i) (ii) sfies: UL(_m,m) g {u(t) : u(t) is a regular distribu- tion with support bounded at left and which is summable on (-w,b), for all finite béfl?}, is the input space. RL(—oo,oo) A {(uo’yo) L(—w,w) satisfies III.A.1 for this uo}. For a given uOEU and y0 uEUL, we will denote by T, the point such that u(t) 5() Vt < T. 97 FACT III.A.1: It follows from our definition of UL that any input can follow and can be followed by any other, i.e., all concatenations are permissible. For: uO€UL(-w,t0] and uEUL(t t] implies both are summable 0’ making uoou summable on (-w,t]. LEMMA III.u.1: Let (uoou,yOOY)6RL(_m,tl] where (uO’y0)€RL(-w,t0] with tO < t1. Then: y(t) = CeA(t—t A(t-T) O)X(t0) + toftCe BU(T)dT + Du(t) III. Vt€(t0,tl]where the integrals are in the Lebesgue sense and X(to) = _wftOeA(tO'T)BuO(T)dT III. PROOF‘ Qgéil = AX(t) + B(u00u)(t) III.A.l gives: III. (yooy)(t) = CX(t) + D(uOQU)(t) Since we are talking of distributions we can write (see [2E1] or [8C2]): I6'(t)*X(t) = A6(t)*X(t) + B(u00u)(t) III. where 6(t) is the delta-distribution, I the identity— matrix and * denotes the convolution Operation (A.2.9). Then III.N.5 yields: [I6'(t) - A6(t)]*X(t) = B(u00u)(t) III. 98 1(t)eAt, where 1(t) is the unit step distribution, is the convolution inverse of 16'(t) - A6(t) as one can easily verify. Therefore: . At t A(t—T) X(t) = l(t)e B*(u00u)(t) = TI e B(u00u)(T)dI III.A.7 since the convolution of two locally summable regular distributions can be written as the right hand side of III.A.7 (Thm.A.2.7). Moreover: (yooy>(t) = CeAtBl(t)*(uOOu)(t) + D(u00u)(t) = CTfteA(t-T)B(u00u)(r)dr + D(u00u)(t), Vt€(—°°,tl] = CeAtTJtOe-ATBuO(T)dT + CtoarteMt-flBuHMI + Du(t) , Vt€(t0,t1] = CeA(t-tO)Tft0eA(tO-T)BuO(T)dT + CtdfteA(t-T)BU(T)dT + Du(t). III.A.8 Using III.H.7 at t = t0 we obtain III.H.2 and III.A.3. NOTE III.U.2: The expression III.U.8 can also be written as: y(t) = [CeAtBl(t) + D5(t)]*u(t) for any (u.y)6RL, 1:1.u.9 AtB1(t) + D6(t), can be viewed where this form, namely Ce as the convolution representation of the objectCPL. This gave us the initial idea in Chapter IV about how to find a state description for more general objects of the form y = w*u. 99 NOTE III.4.3: The restriction in Def.III.N.l that any input has support bounded at left, is necessary for the existence of the integral in III.A.7 and alike, since A may have positive eigenvalues. Another alternative then would be to assume that A has negative eigenvalues and then let the input space be the space of regular distributions that are summable on (-w,b] for all finite b. We would also like to note that III.A.2 and III.A.3 are true for any t€(t0,w) when (uoou,y00y)ERL(_m,m) are such that (u,y)éRL(tO,m). o ' g ' LEMMA III.A.2. Let uO,u OEUL(-m,to]° Then no u 0 $22; t _mf|OCeA(t-T)B[uO(T)-u'0(r)]dT = O, Vt> t III.A.lO O, PROOF: uOOu and u'Oou are admissible for any uEU L(t0,w) by Fact III.A.1. ' Y Now let (uoou,y0) and(u Oou,yO )éRL(_m’m). Then A expression III.A.9 gives: y0(t) = Ce tBl(t)*(u00u)(t) + D6(t)*(u00u)(t) and y'O(t) = CeAtBl(t)*(u'OOu)(t) + D6(t)*(u'oou)(t) b’t. As the object is linear: CeAtBl(t)*[(uoou)(t)-(u'oou)(t)] + D[(u00u)(t)-(u'oou)(t)], VtE(-w,m). CeAtBl(t)([(uO-u'O)OO](t) + Dt x/te<-w,w). lOO _._. t _ In our case, as indicated above, uO u 0 QyO/(tom) I 3’ o/(to,°°)’ i‘e" u0 = u'O (2:)CeAtBl(t)*[(uO-u'O)OO](t) = o Vt>t0 Ill.u.1l A(t—T)B[(uO-u'O)OO](T)dT t z ' uO u 0 <-_—_>_°J Ce 0 Vt > to or A(t-T) ¢:=?_mftOCe B[u0(T)-u'O(T)]dT O Vt > t O which is III.4.10 THM.III.A.l: Let 2L(t0) A {x(to)emn :BuOEU(__m’tO]3X(tO) t -mj OeA(t0'T)BuO(T)dT} for t > _m and 2L(t0) A {0} for 0 t0 = -m. Also let y(t) = AL(tO,t](X(tO)’u)’ for X(tO)EZL(tO) and uEU e the expression III.A.2. L(t0,t]’ b Then (XL,AL ) is a half reduced state description ofG}L I and ZL(tO) is a linear subspace offRn. PROOF: To prove the theorem we show that (ZL’AL) is based on a half reduced partitioning. Then Thm.II.A.A com- pletes our task. v = v . ConsiderIW t H t [no] . uOEUL(-m,t0] where we define: t H't [uo J A {u' OEU(- m t O] :_wI’OeA(tO-T)Bu'o(1)dt = 0 if 0 eA _mJtOCGA(t-T)BEUO(T)-u'0(T)JdT = O in par- ticular Vt > tO :::>uo = u' by Lemma III.H.2 O proving that “Ht is indeed a half reduced 0 partitioning. Moreover each X(tO)EZL(tO) represents one and only one H't [no] due to expressions III.U.3 and III.U.12. O This is to say that there exists a one to one, onto mapping 0' : 2 (t ) +‘H' [u ]. Thus using Thm.III.U.4, to L O to O 2L(t0) qualifies for the state space ofc9L. The I—O-S—R, y(t) = AL(t0,t](X(tO)’u)’ defined by III.M.2 is such that clearly y(t) is the (to,w) portion of the response to uoou, for any uOEC't (X(to)), due to O III.H.8. Hence A qualifies for the I-O—S-R based L(t0,t] on the half reduced partitioning III.N.12. 102 Finally it is simple to see that £L(t0) is a linear space since: Xl(t0), X2(t0)6£L(tO) :::> t _ O A(t —T) 3u0, u'0€U(—w,t0] 9Xl(t0) - _mf e O BuO(T)dT and t _ O A(t -T) X2(t0) — _mf e O Bu'O(T)dT. AsC9L is linear uO + au'OEU Fact III.2.3, which implies Xl(t0) + (-mgtoj, aX2(tO)E£L(tO), for aEFR. NOTE III.U.4: However it is not generally true that 2L(t0) as defined in Thm.III.H.l is a reduced state space. The necessary and sufficient condition III.H.10 does not require the defining relation of III.M.12 to to be equivalent. Con- hold, for the inputs u and u' 0 O dition III.H.12 is a sufficient one for III.N.10 or for that matter for u0 = u'o. The following theorem and its corrollaries tell us when III.U.l2 becomes also neces- sary for III.H.lO, i.e., when £L(t0) becomes reduced, or if not, what is the dimension of the reduced state description, etc. NOTE III.N.5: We assume that in the equations III.4.1 the matrix A is in, or has been brought to, its Jordan Canonical form. This is no restriction at all, at least theoretically, since every matrix has a Jordan equivalent [HO]. We further assume that: 103 - - F (k) ‘ - Al 0 ----- 0 J1 o o Ak 1 o ----- o (k) 0 fiéu.g 0 J2. ...... o o 5k 1 ...... o .2 : k - :-. A: i ..s ’ Ax“ ’ JJ( ) § 1 o o~~ AK 0 0 Jn (k) c 0 AK _ A _ k 4 — for k = 1, 2,...,k for J = 1, III.4.l3 In Ak the size of JJ(k) Ak corresponds to a different eigenvalue Ak' If the matrix A has a diagonalizable part, with or without dis- decreases as J increases and each tinct eigenvalues, we view each entry on the diagonal as a l x 1 Jordan block. We also partition the row and column vectors C and B conformal to the partition of A, i.e. P - C = [ClC2 . . . Ck] and B = 2 III.H.12 such that the product C A B k k k is defined for k = 1, 2,...,K. THM.IIT.H.2: Suppose that: (i) Ak is formed of a unique elementary Jordan block J(k), and (ii) Ck(l)bk(dk) # 0, where Ck(l) is the first element of 0k and bk(dk) is the last element 10“ of B d being the multiplicity of A be— k’ k k cause of (i). Then (ZL’KLI) is a reduced state description, where zL(tO) as defined in Thm.III.A.1 islzn VtOE (-°°,°°), and AL is as defined in III.N.2. I PROOF: Using Lemma III.H.2 for u and u' EU , the ———-—— O O <‘w3t03 expression III.U.10 takes the form u0 = u'O :22; t k 0 A (t— , _ _ I kElcke k T)Bk[uO(T) - u 0(1)]dT - o, Vt>t0 III.u.1u At Akt since the matrix e has the submatrices e on its diagonal and 0 submatrices elsewhere, like A had the Ak's on its diagonal. CkeAktBk is conformal by Note III.A.5. It was pointed out in the same note that each A corresponds to a different eigenvalue A so that two k k’ different terms of the summation in III.4.l3 corresponding Aklt to, say k and k and l 2 A t e k2 as factors. By summing up such terms there is no A t A t k1 by another e k2 will yield terms containing e chance of cancelling one e for all t > to. Thus: Ak(t-T) ~ v mftoc B [u (1)-u' (T)]d - 0 U0 “ u o @229— ke k 0 o T ” Vt;> t III.A.15 t (k) <:::;_mJbOCkeJ (t“T)Bk[uO(T)-u'O(T)]dT = O O Vt > t III.4.16 0 since by (1) each A is constituted of a single elementary k Jordan block. 105 (k) Using the matrix form of eJ t as given by [CO], III.U.16 reduces to: dk j- -i u0 = u'O $22: 2 ck(i gjt0(t_i I)! exk(t T)[uO(I) _ J,i=l 1 tO k=l,2,...,K III.u.17 We consider now the term which contains the highest power of t in the summation of III.U.16. t has the highest power when 3 = (1k and k = l, yielding the term (l)b (dk) to (t_1) dk'leAk(t-T)[uO(T)_qu(T)]dT, Ck k -m (dk—15' where c (l)b (dk) # O by hypothesis. k k If we expand (t-T)dk-l and consider the term that d -l . contains t k , it is of the form dk—l A t to -A T d t e k I. e k [u (T) - u' (1)]dr, a # 0. d -l -m 0 0 d —l k k and it is the only term in 111.u.17 with tdk'l as factor. Thus if the left hand side of III.H.17 has to be zero Vt > t the only way this can happen is to 0, have: t -mr-Oe-AKTEUO(T) - u'O(T)]dT = 0 III.A.18 106 Proceeding in this manner at each step we will be left with a term of the form _dftOTme-AKTEuO(T)-u'0(t)JdT that must equal zero, vious steps. ~ uo Thus we will obtain: for m k- 11' Kw"; Itorme_kkT[u (T)—u' (T)]dT = O O V”’ -w 0 O considering the results of the pre- III.N.18 O,l...,dk—l l,2,...,K The defining relation 111.4.12 for H't [uo], after cancellation of eAtO H! t = {u'OEU O (—°°,t0] to -J i.e., 11' EH' [u] <: I e O for k = J 1,2,... since Ak JtOe—J(k)T -oo u'OEH'tO[uO]<::> t _mI-Oe-XKT I... for k ,K F t I Oe-XkT i by hypothesis. dk E b =1 , can be rewritten as: (k) TBk[u0(T)-u'O(T)ldT O _mftOe-ATBEUO(T)-U'O(T)JdT = O} III.N.19 III.U.2O Writing the column vector Bk[uO(T)-u'O(T)]dT, we get k (1115: t _mf’Oe-AKT gkbk(i)(—T)i-2[u i=2 i"? ! gk i=d (i—d 5! k k = 1,2,...,K (i)<-r>i‘1[uO(T)-u'o<¥w>i'dk[u0-ubJdr (T)-u'O(T)]dT III.U.21 107 Starting with the last row of III.N.21 which has the t unique term b (dk)_wj-Oe-kaEuO(T)-u'0(r)]dr with k bk(dk) # O by hypothesis, we see that 0 “ART ' . _m e [uO(T)-u 0(1)]dr must equal zero. Moving up- wards, each previous step eliminating all terms except, b (dk)_mJtOe-AkTrm—l[u0(r)-u'O(T)JdT at the m—th step, k m = l,...,dk we finally obtain (changing m-l to m): u' GH' [u 1 :22; jtotme—AKTEu (1)-u' (T)]dT III.H.22 0 t0 0 -w 0 O for m = O,l,...,dk-l k= l,2,...,K Combining III.M.18 with III.H.22 we see that v v t 2 u 06H tO[u0]<:==$,>u O uO, making the classes H'tOEuO] equivalence classes of inputs, i.e., the sufficient condition has turned out to be a necessary condition in this case, as it was indicated by Note III.H.H. Thus under the hypothesis (ZL’ALI) becomes based on equivalence classes of inputs. As equivalence classes are also a half reduced partitioning Thm.III.u.l proves that (XL’ALI) is a reduced state description under (1) and (ii). That 2L(t0) is the n-dimensional Euclidean Space V’tO€(-w,w) is quite obvious. It was proved in Thm. III.M.1 that 2L(t0) was linear (it had dimension n by 108 definition), it became reduced here, giving these pro- perties to the reduced state description Vtoe(-m,w), sinceG'L is time invariant (Cor.III.3.2). NOTE III.H.6: The following corollaries follow Thm. III.h.2. The proofs follow the same lines as the proof of Thm.III.H.2 and are not given. The results are for more general cases, the last one being the most general. Let the vectors C and B be partitioned as in III.u.l2. and B The vectors C that pre- and post-multiply A k k are partitioned into submatrices: k C = [C C III.U.23 k k,l k,2'°'Ck, nk] and Bk = _Bk’md COR.III.u.l: If the first entry 0(1) of C and the k,l k,l are nonzero for k = l, last entry of the vector Bk 1 ’ K . 2,...,K then: dim z (t ) = 2 size J (k) III.u.2u LB 0 k=1 1 i.e. the dimension of the reduced state space is equal to the sum of the sizes of the largest elementary Jordan blocks for different eigenvalues. COR.III.H.2: Let again each A consist of a single k elementary Jordan block and let the Yk + 1 th element of Ck be nonzero, the first Yk being zero and the 109 Bk + 1 th element of Bk be nonzero, the last Bk elements being zero, i.e. (1) 9k 5 (Bk+l) C = [00 ..... o c (Yk+ll.nnc (dk)] and B = k 111.“.25 k k k k 0 Yk-zeroes ; 0 g _l c (Yk+l) and b (Bk+1) # o. k k K * u 6 Then. dim 2LR(tO) — kiltdk—Yk-Bk} Ill. .2 where [dk-yk-Bk} A dk—Yk-Bk if dk‘Yk‘Bk > O 0 otherwise. COR.III.4.3: In the most general case, let for each be the size of Jj(k) such that ' (k) J (1) . C J B . is non-zero and C e 3 tB ives rise k,3 J k,J k,J k.J g to the highest power of t. If the numbers yk and Bk k, k = 1,2,...,K, d'k denote the number of first consecutive and last con- secutive zeros in C and Bk 3’ as in Cor.III.u.2 then: 3 k,J K dim 2LR(tO) k§l[d'k—yk—Bk} III.u.27 CHAPTER IV SOME CANONICAL FORMS AND PROPERTIES OF THE STATE DESCRIPTIONS FOR LINEAR, TIME INVARIANT, CONTINUOUS OBJECTS IV.l--Introduction In the previous chapters we have only dealt with the gross properties of the state description, without trying to generate any analytic description of the I—O-S—R, except maybe in section III.“. So, Chapter IV gives us some analytical forms for the I-O-S-R and a good knowledge about the interesting properties of the state space, when, as the title indicates, the object under consideration is a linear, time—invariant and continuous one. Of the two strategy procedures available to reach the goal, the less mathematically sophisticated and more engineering approach, of first guessing what the I-O—S-R and the state space might be and then showing that they satisfy the axioms, is chosen, rather than building up to the result by using the state axioms and mathematical tools as does Balakrishnan in [BA 1—u]. However, we would also like to point out that by proceeding as such, it should not be understood that we are being 110 lll mathematically irrigorous. Our main tool in these investi— gations is the theory of distributions and their orthogonal series expansion as developed by Zemanian in [ZE2],eabrief expose of which is given in Appendix A. The next section of the present chapter tries to justify the use of convolutional objects as our starting point by means of arguments that stem from the refer- ences [ZEl, A]. In section 3, we give an infinite but countable state description of a large class of convolutional ob- jects, namely the ones with an impulse response which has an infinite series representation. Then we investigate and prove some very important properties of the I-O-S-R and the state space such as: "The infinite A—matrix associated with the I—O-S-R is a Hilbert Matrix," "The state space X(t) is a closed linear subspace of the Hilbert Space [2," etc. The last section deals with the most general con- volutional objects and shows that it is possible to approximate any such object with objects that have a finite dimensional reduced state space. This, as noted in [ZAl],happens to be a very important problem in that it may provide us with some tools of approximating a large class of distributed systems with passive, lumped RLC'networks. ll2 IV.2--Convolution Representation of Linear, Time Invariant and Continuous Objects NOTE IV.2.l: In Fact 1.2.1 we noted that a uniform ob- ject was completely defined when the input—output list RI over the existence interval I was known. Furthermore, Thm.1.3.2 stated that the object G’had a state descrip- tion iff it was causal. We thus have a single valued mapping from the input space UI into the output space YA I due to causality. Taking I = (-W,W), if we restrict our attention, for the moment, to objects with inputs in the spacef? of testing functions and outputs from the space 9” of distributions overnB, we then have a single valued mapping from .9 into 3' (for the definition of 2,.9' and notions related to distributions see Appendix A). Moreover we have a linear, time invariant mapping from 9 into 9' if we let our object be linear and time invariant. To these properties of single valuedness, linearity and time invariance possessed by many systems we will add one more property, "continuity," which is more difficult to interpret physically and which can crudely be described by: "in the input-output list Ri of the object 9', to two different inputsthat are almost the same, correspond two outputs that are almost the same." A precise definition of "almost the same" would require a discussion of the 113 neighborhood concept in.9-and.9”, and that would lead us to Topological Vector Spaces (see, e.g., [TR], [HOR]). To avoid that, and as we already have a concept of con- vergence in.9”, we define "continuity" as follows (a slightly modified version of the definition in [ZEA]). DEF.IV.2.1: An object is said to be CONTINUOUS (or a CON- TINUOUS MAPPING FROM b-INTO.9') iff the convergence of my to u, in.B':=;>the convergence of {yn}co to y, n=l n=l in.$” with (u,y)ERi. THM.IV.2.1: SCHWARTZ'S KERNEL THEOREM [TR]. The mapping fromb into 9', that the object (5’ given by RI describes, is single valued, linear and continuous iff there exists a unique w(t,T)€$” defined on the real plane such that (u,y)ERi<—::>y(t) = W(t,T)Xu(T) vueg, where w(t,T)xu(T)6£' is‘defined by é vase. THM.IV.2.2: [s01, vol.II, pp. 53—5u] The object satisfies the hypothesis of Thm.IV.2.l and is time in- variant iff there exists a unique w(t)E£f such that (u,y)ERi {:2} y(t) = w(t)*u(t) V1169, where w(t)*u(t) is defined in Def.A.2.7. NOTE IV.2.2: [ZEA, p. 8] Now, because of this convolu- tion representation, the input space UI of<9-can be 11“ extended to the space £' of distributions with compact support, i.e., for uGUi = 2' (u,yXERi ¢::>y(t) = w(t)*u(t). As Jiis dense in £' this extension of the convolution representation is unique. Moreover, if w happens to be suitably restricted the input space Uf can be further extended to larger spaces of distributions. If for example wébYR the space of distributions with support bounded at left then UI can be taken as all of gflR, or for that matter any subspace of it.‘ Also, if wei', then UI can be taken to be all of.9”. In both cases, as 9 is dense in9'R and9', the extensions are unique. In case the object<9'is not time invariant, the same extensions can be made by using the kernel represen— tation ofG>. THM.IV.2.3: [ZEA, p. 9, THM.3] Let(9’be defined by y(t) = W(t,T)Xu(T), where u belongs to the extended U“; then.®'is causal (Def.I.2.A) iff supp y(t,T) is contained in the half plane {(t,T) : t > T}. If in addition (9' is time invariant then G is causal iff supp w(t)C[O,w). NOTE IV.2.3: In the light of the above discussions, the following two sections concentrate on convolutional ob- jects with supp w(t)C[O,w). These will be defined more precisely at the beginning of each section. 115 IV.3—-A Countable-Differential State Description NOTE IV.3.l: In this section we will obtain the state description of a linear, time invariant continuous object whose impulse response w(t) is a distribution in CH' (Def.A.2.9). This state description can be viewed as the generalization of the familiar state equations in III.A, to include state descriptions for distributed systems. In fact they ressemble very much the form in III.A.l except that the matrices and the vectors involved are infinite in size. The idea in developing the state description is simple and its root lies in the fact that for a lumped network we obtain a state description via the decomposi— tion of w(t) into different exponomial terms (see Def. IV.A.2 for a description of "exponomial term"), as we have already remarked in Note III.A.2. First we precisely define the object, for which the description can be given then proceed to obtain the state description. DEF.IV.3.1: The convolutional object under consideration is called an 0D object iff its input space UDf A {u(t) : U(t) is real and square summable over (-w,b] for any real finite b}. We further assume that w(t) 116 is a real distribution in Cn' and that the I-O list is: RDf g {(u,y) : uéUDi and y(t) = w(t)*u(t)}. NOTE IV.3.2: The convolution of w(t)E(1' with u(t)EUDf is well defined (Thm.A.3.l). Furthermore supp w(t)CI[O,w) due to causality by Thm.IV.2.3. As the input space, w(t) and the eigenvalues kn (Not.A.2.l) are 00 real, we can take {‘i’n(t)}n=l to be real. THM.IV.3.l: Any G}D object can be given the following dynamic description (a conjectured I-O—S-R): dxn(t) 0° dX(t) ___— : Z anmxm(t) + bnu(t) —a-t— : AX(t) + BU(t) IV.3.1 dt m=l n = 1’2’ i.e. _ e K (k) _ y(t) — cnxn(t) + z dku (t) y(t) — CX(t) + DU*(t) “=1 k‘0 IV.3.2 where in IV.3.l the convergence is pointwise and in IV.3.2 in 3' together with r _ ‘F __ ._ _ _ Xl(t)[ aliaié """""" F51 Ci 32(t) P21§22 ---------- P2 ?2 3 I f I T ' X(t) = I 3 A = : a B = j a C = ; a _ _ _. : _J .1 _J __ _ ._ fl ._ d Fu(t) 1 (1) d2 u (t) DT = 3 and U*(t) = : ‘ 3(k) d u (t) _ kJ ~__ J 117 PROOF: As w(t)ecm', w(t) § wn(t) by Thm. n=l A.2.15 where {Tn} m are as given by Not.A.2.l. Using n=l Thm.A.3.3, y(t) = W(t)*u(t) = [ E Tn(t)l(t) + E d 6(k)(t)]*u(t). IV.3.3 n=l k= 0 k By Thm.A.3.?, the infinite summation in IV.3.3 can be taken outside and Thm.A.2.8 can be used to yield: m K y(t) = n§l[wn(t)1(t)*u(t)J + kEOdku(k)(t)° IV.3.u Now we define xn(t) A Wn(t)l(t)*u(t) for n=l,2,... IV.3.5 Differentiating IV.3.5 according to Thm.A.2.9 we obtain dxn(t) [g%>1*u w'n1*u + dt Wn(t)6(t)*u(t) W'n(t)l(t)*u(t) + Wn(O)u(t). IV.3.6 But T'n(t)€OI(Lemma A.3.3) and can be expressed as v = °° t . T n(t) m£1\Pm(t), Thm.A.3.ll. This time the convergence being in<3[, and certainly in L2(_oo w). So 3 for each n we can write dxn(t) dt [m§lWm(t)l(t)]*u(t) + vn(o)u(t) IV.3.7 t _mJ.milWm(t-T)u(T)dT + Tn(O)u(t) 118 dx (t) 11 _ °° l for each t. IV.3.8 Now as convergence in L2 defined by the norm is continuous with respect to the inner product, i.e., as strong con— vergence implies weak convergence [R1, p. 69], the infinite summation can be taken outside the inner product in IV.3.8 and dxn(t) dt m£l + Wn(0)u(t) for each t. t ._. m£l__i Wm(t—T)u(T)dT + wn(o)u(t) = mgl[vm(t)1(t)*u(t)J + Wn(0)u(t) for each t, IV.3.9 can be obtained. Using IV.3.5 in IV.3.A and IV.3.9 and defining the coefficients arm A , bn A wn(0) and on g IV.3.lO dxn(t) m = I at m£1xm(t) + Tn(0)u(t) = milanmeCt) + bnu(t) IV.3.l 119 xn(t) + § d u(k)(t) 1 k=0 k y(t) Z n K cnxn(t) + E. d u(k)(t). IV.3.2 1 k 0 k 2 n That the convergence, in IV.3.1 is pointwise is clear since IV.3.9 converges for each t and that in IV.3.2 it is in.9” is clear by IV.3.A, where the convergence is in 5?. NOTE IV.3.3: The following theorem obtains a certain half reduced partitioning (Def.II.A.1) that is compatible with the dynamic description of Thm.IV.3.l. The theorem after that using this half reduced partitioning and Thm.II.A.A shows that the expressions IV.3.1 and IV.3.2 provide us with a half reduced state description. o ' Y Y . THM.IV.3.2. The family ”to A {H to[uO] . quUD(-°°,t01} of classes of inputs where for any t0€(—W,w) H't [uO] g 0 {u' eU - JtOv (t T)u' (T)dT = O D(-m,tO] ° —e n O‘ O t O — _mf wn(tO'T)uo(T)dT for uOEU and n — l,2,...} D(—°°,toj IV.3.11 is a half reduced partitioning. PROOF: We verify (Def.11.u.1): (i) H't [uO] # O, if nothing else one such class may 0 contain only the defining input. 120 ‘ t 11 1 1 1 11 (ii) Let u 0 and u OEUD(-m,t0] and H to[u dV\H to[u O) # (111) ch. Then BUOEUD(—w,t0] 3u0€H1t0[qu]/\Hvto[unoj . 1 1 o For any uOEH t0[u O] we have. t t _mf'OWn(tO-T)u0(1)d1 _wf OWn(tO-T)u'0(r)dr = t t _mj own(tO-T)GO(T)dT _oof OWn(tO_T)unO(T)dT n = 1,2,... 1 11 1 1 1 11 ° Thus uOEH t0[u O] and H to[u OJCH to[u 0]. Similarly 1 11 1 1 H t0[u OJCH to[u O] can be shown, giving H' [u' J = H' [u" 0 0 130 Let u1 01. 2 1 ~ 2 OEHtO[u 0]. We have to show u 0 - u 0’ i.e., as all concatenations are allowed all we need to prove is: for any UEUD(to:m)’ to < m such that 1 l 2 2 (u Oou,y ) and (u OOu’y )éRD(-W,m) we must have 1 _ 2 . y /(t m) - y /(t w). Going back to express1on O’ O’ IV.3 A y1(t) = w(t>* m i = ngl[Tn(t)l(t)*(u Oou)(t)] + K o z dk(uloou)(k)(t) k=O 00 t i = Z [—mf Wn(t—T)(u Oou)(T)dT + n=l K i (k) . k20dk(u OOu) (t) for 1 = 1,2. IV.3.12 121 yi(t)/(t0,“) = n§1[—mftovn(t—T)uiO(T)dT + t tdf-Wn(t—T)U(T)dTJ + K 2 dku(k)(t) for t > t =O IV.3.13 k 0 _ ' 2 _ As Wn(t T) 18 in L (_m,m) and as Tn(tO T) forms a complete orthonormal basis for L2(_oo m) we can use 3 Fact A.3.2 in IV.3.13 to obtain: i y (t)/ m (to, ) t O 1 [_wf m Wm(tO—T)u O(r)dr + IIM8 IIM8 n l l t . K (k) Tn(t—T)u(T)dT] + z dku (t) O k=O for i=1,2 and t > t IV.3.1A 0 Using once more the continuity of inner product with respect to convergence in L2 t oo .102 m-l (—°°,°°) Tm(tO—T)uiO(T)dI = ‘._ a 1 ..Wm(tO-T), (u OQO)(T)> 122 m i mEl w O i mE1<‘Pn(t--T),‘Pm(tO—T)>_OJ-t Tm(tO-T)u O(T)dT IV.3.15 Using IV.3.1A in IV.3.lA we finally obtain: 2 [ z _mj Tm(tO-T)u O(T)dT + d u(k)(t) IV.3.16 tgt‘l’n(t—T)u('r)dr] +— k "MW k 0 t As _mJ.OTn(tO-T)ulo(1)dt = _mftownho-r)u20(t)dt for n=1,2,... by IV.3.11, it follows that 1 _ 2 y O/(t0,w) ‘ y O/(t0,w)' DEF.IV.3.2: We define the set XD(t) (a conjectured state space) by: 2D(t0) g {X(to) : X(to) = (xl(t0),x2(t0),...) where t O xn(t0) _mJ- Wn(tO-T)uO(T)dT for each n and for some quUD(_m,tOJ} IV.3.16 123 The mapping (0' )'l :38' e-z (t ) is defined as t0 t0 D o I t . v '1 1 _ follows, for any class H t0[u01€x’t . (C t ) (H t [uo]) - O O O X(tO) where the defining input u'O for X(tO) is any 1 1 u 06H to[no]. DEF.IV.3.3: We define the INFINITE TRANSITION MATRIX by T(t,t0) A [Jnm and the conjectured I-O-S—R, where O A X(t0)62'(t0) and C is the infinite 0 vector whose components are defined by the expression IV.3.10, with: AD(tO,w)(GO’U) A C¢(t,tO)X(tO) + w(t)*u(t) on (t0,w). IV.3.17 THM.IV.3.3: (ZD’KD) as given by Def.IV.3.2 and IV.3.3 is a half reduced state description of the object(}D. PROOF: We verify Def.II.A.1 where the half reduced partitioning is that of Thm.IV.3.2. (i) For £D(t0) to be a half reduced state description —1 all we have to show is that (C't ) (Def.IV.3.2) O is one to one and onto. It is onto: by definition any X(to)€£(t0) is such that there exists a uOEU for which D(-m’tO] t - O - xn(t0) - _ 1 Tn(tO-T)uO(T)dT for n—l,2,... : "l 1 thus X(to) = (c to) (H to[110]). (11) 12A . 1 11 It is one to one. let u 0 and u OEUD(-w,to] be such that H' [u' ] # H' [u" ]. By definition to O to O of H't [uO] this means that there exists nl such 0 t . 0 _ 1 that. _mf Tnl(tO T)u 0(T)dT ¢ t O _mj. wnl(tO-T)U"O(T)dT, which then implies: x'‘1(H' [u' 1) a (c' >‘1 = O t t O t t O O O O O X"(tO). With the above proof the use of the inverse notation for C't is also justified. 0 That AD( as in Def.IV.2.3 is the I-O—S-R t0,w) can easily be shown. In the previous theorem, expression IV.3.16 gave us the (to,w) portion of the response to uoou. m < w > ” _mJ Wm(tO-T)uo(r)dr + t t; Tn(t—T)u(1)dt] + O k 0 k K g d u(k)(t), t > t =0 = nEl[m£lxm(to)] + P [wn(t)1(t):u(t)1 + n=l § d 6(k)(t)*u(t) k=1 k 125 = :10 [mElxm(t0)] + E m K (k) z wn(t)l(t) + Z d 6 (t)]*u(t) n—l k=0 k can be achieved using the definitions of xm(t0) and on, and Thm.A.3.2. Finally using the in- finite matrix notation and Thm.A.3.3: y/(t0,w) = C¢(t,tO)X(tO) + w(t)*u(t) = A IV.3.17 m (0 ,u). D(t0, ) 0 Clearly for any uOECtO(OO)/OO=X(tO)’ uoou is admissible and A is the (t0,w) portion of D(to,w) the response to uOOu, thus making IV.3.17 an I—O-S-R by Def.II.A.A and (2D,AD) a half reduced state description by Thm.II.A.A. NOTE IV.3.A: Thm.IV.3.2 and Thm.IV.3.3 have shown that (ZD’KD) constitutes a state description and in their light, the dynamic equations IV.3.1 and IV.3.2 can be viewed as the state equations of the object<9D. To make the tie between the state description and the state equa- tions stronger and to justify the name "Infinite Transi— tion Matrix" for T(t,t0), we prove the following two theorems which also improve the I-O-S—R, IV.3.17. THM.IV.3.A: The infinite transition matrix T(t,t0), Def.IV.3.3, is the FUNDAMENTAL MATRIX of the infinite 126 dx(t) differential equation system dt = AX(t), with ®( tO,tO ) = I. PROOF: T(t O,tO ) A [<‘Pn (t- T), T m(t W—T)>] =[3mn], where amn is the Kroenecker delta, Since {Tn(tO—T)}n=l forms a complete orthonormal system (Fact A.3.2). To show T(t,t0) is a fundamental matrix: (1) we first show that every column of T(t,t0) is a vector solution of the infinite differential system. To pick a column of T(t,t0) = []mn we fix the column index m at an arbitrary m0; then we substitute the vector so obtained by X(t) dX in 55 = AX(t) to get: d __ _ _ _ v _ - dt<‘¥n(t T),Wm (tO T)>: z O m= l O IV.3.18 using IV.3.lO, definition of A. Now we have to verify the identity IV.3.18. In fact: — Eé%-_ :f T n(t- T)Wm (t O-T)dT mO mO _meT’n(t—'0Tmo(tO-T)dt since Tn(-) is infinitely d dt smooth. T'n(t—T),Wmo(tO-T)> E ' -— _- — — 1 v n(t t),wm(t t)>wm(t t),TmO(tO r)> IV.U.19 A IIM8 m (ii) 127 Using the continuity of the inner product and noting that: = _wfg'n(t—T)Wm(t~0d1 = _«S T'n(I)Wm(I)dT = d ' ___. _ .. : <¥ n,W >, IV.3.19 becomes dt<‘Pn(t T),Wmofib T)> _ § verifying the m=l 0 identity IV.3.18. We now show that the columns of ¢(t,t0) are linearly independent for all t. To do this select any k (k also arbitrary) columns of ©(t,t0) and suppose for some t, there exists scalars al,a2,...,ak such that not all ak are zero and al + a2 + ... + ak = O or k = O for i=l,2,... IV.3.2O 3:1 n. O J J As {Wi(t—T)}i: is a complete orthonomal basis, 1 k IV.3.2O gives ; ajwn (t -T) = 0; but this cannot . . o J=l J be true since {WJ(tO—T)}j:l also forms a complete orthonormal basis. This contradiction implies the columns of ©(t,t0) are linearly independent for any t. 128 THM.IV.3.5: The I—O—S—R, K of the object 9D is also D given by: t y/(t0,°°) = C(t,tO)X(tO) + ctof (t,tO)Bu(T)dT + DU*(t) IV.3.2l PROOF: Combining IV.3.16 with IV.3.l7 we can write: t y/(t0,m) = C®(t,tO)X(tO) + § t~{ Wn(t-T)u(r)dt + K n—l O z dku(k)(t) IV.3.22 k=O For fixed n consider Wn(t—T+z) which can be written for 00 each t,T as: Wn(t-T+z) = mElWm(z) where s is the dummy variable, i.e., m w Wn(t—T+z) = m:l[_mf Wn(t-T+s)wm(s)ds]Wm(z) IV.3.23 by Thm.A.2.12, Wn(t—T+z) and the convergence being inCfl . Then by Cor.A.2.l it follows that the convergence is uniform or compact subsets of (-w,m); therefore we can evaluate IV.3.23 at z = 0. Also doing the change —x = -T+S of variables [_mfmwn(t—X)Wm(T—x)dXJWm(O) Wn(t—r) 1 ll IIM8 m m§l¥m(0) IV.3.2“ follows from IV.3.23. Substituting IV.3.2“ in IV.3.22: 129 tf °§ <‘Yn(t—x), y/ m = C®(t,t )X(t ) + (to’ ) O O o m=l "M8 n l K Tm(I—x)>Tm(O)u(T)dI + kEodku(k)(t) and using IV.3.lO and Def.IV.3.3 t y/(tO’m) = C®(t,tO)X(tO) + Ctar.¢(t’tO)Bu(T)dT + DU*(t) IV.3.21 is finally obtained. NOTE IV.3.5: We, thus, have shown the strong resemblance A(t-t0) between the exponential transition matrix e for a square A of finite size and our transition matrix ©(t,t0). Now we will investigate the nature of our state space and an essential property of our infinite A matrix that may bear strong relation to the stability of the objectC?D under consideration. The following theorem is the main reason for all the work we had to go through in A.3 when defining the convolution of a distribution in Cl': with an input from UD; it makes it possible to show that X(to) is a closed linear subspace of the classical Hilbert space {2, which could not be proved if U was D not taken as the space of square summable functions over (—m,b) for any finite b. THM.IV.3.6: For each tOE(—m,M), the state space ED(tO) of the object<9b is a closed linear subspace of the classical Hilbert space Z2. 130 00 g lai|2<”}- To prove i=1 i=1 this let X(tO)EZ'(tO),then by Def.IV.3.2 xn(to) = 2 PROOF: ZD(tO)Cfl AHai} t 0 _ _mj Tn(tO-T)uo(r)dr for some quU(_ n-l,2,... oo,t0:| ’ where xn(t0) is a component of X(to). Thus: xn(t0) = _ ' 2 Since uoOOEL (_m’w). Thus Xn(t0) is the Fourier coefficient of u 00 for each n O with respect to the complete orthonormal basis {Tn(tO—T)}n:l. Invoking a classical theorem (see e.g., [P0, p- 361) we can write: nglkflfiétO-T),(u000)(r)>|2 = m 2 2 . 2 . nEllxn(t0)| < IIuOOOll l2 = lim I]u(r) - (uiOO)(T)||2 i+oo by Parseval's equality. 132 = lim _mfmlu(T) — (uioo)(T)|2dT 1900 by definition of L2 <-oo,oo> norm . = lim _mjtolu(T) — ui(r)|2dt + i->oo t—f Iu(T)|2dT = O 0 Therefore u(t) must equal zero i.e., for t > t so that O, the convergence in IV.3.2“ holds. Thus, the function u(t) = u(t)/(_oo t J will certainly be in U< and ’ O -oo,to] _ 0 will be such that xn(t0) - _mjt Tn(tO-T)u(1)dt, n=l,2,... proving that ZD(tO) is closed for any tOE(—°°,°°) - NOTE IV.3.6: In concluding this section, our aim now is to show that the infinite matrix A in the state equations IV.3.l is a Hilbert matrix (Def.B.3). As our infinite state vectors are from {2, that makes A a bounded operator mapping {2 into £2 (Note 8.2), thus enabling us in the future to investigate about the spectrum of A and its other properties and carry out some important analysis of the object GD such as its stability. However we could only prove that A is a Hilbert matrix, under an assumption for the eigenvalues {An}n:l of the operator 11, in Not.A.2.l used to generate the testing function space G>(Def.A.2.8) and their dual space 0(' of distributions, to which w belongs. The eigenvalues 133 {An}n:l were already real and no value of An was assumed more than a finite number of times. The numbering was so chosen that |A1I<|A2I< ... . This clearly implies that Iknleco as n +m. Now, although we conjecture that Thm. IV.3.7 is true without any further assumption, we assume that there exists a finite integer pO such that nglllnl—p0 O for which n21 IA An¢0 I‘2qOIc I2, n=l,2,... To show that A is a Hilbert n — n matrix we proceed in five steps: °‘ : V i = _ (l) _wf w n(t)wm(t)d Tn(t)Tm(t)_i I = _ V «3 w = '1 _mf ?n(t)w m(t)dt since Tn(: ) 0 fol n=l,2,... by Fact A.3.3. (ii) From (i) it follows that E Tm(t) converges m 1 Z 1 in(].as well as m=lTm(t). Then using m i Thm.A.2.l3 we have that m=llAm a 2k I I converges nml (iii) 13“ for every k = 1,2,... and for every n = 1,2,... 00 E 2k n=llxm| Ia whereas converges for every nml k = 1,2,... and for every m = 1,2,..., since . ! arm A ' Z 6 ° ' _ m=llanm| M, where M is inde Now we will show that pendent of n. By Note IV.3.6 there exists a finite . ' ' 2 k0 such that m=l IAmI Anfio —2k0 converges. But m 2 “k0 . , , m=l|xml lamn converges for every n, since it is convergent for any integer power of the Am's. Thus given 6 = 1, there exists a finite mO such that E IA Iukola I2 no, mEllamn|ku IV.U.8 with these new coefficients, remembering that i denotes a superscript not an exponent the summation over v in IV.H.7 takes the form: i i i v-l ku i ku b t 2 CivtV-leyu tl(t) = I: 2 v1.11) ' + v=l \)=1 kui b1 \)t"‘2 t i z “2 ' +...+ b i ]e Y“ l(t) v=2 V_ ' 11k u IV.A.9 kui kui bi t"'3 i _ UV tYp - z z (-:—y, ]e l(t) Iv.u.io j=l v=j v 3 ° To verify that the right hand side of IV.A.9 with IV.A.8 gives the left hand side of IV.A.9, we note that for every coefficient of every power of t, i.e., the coeffi- cients of tv-1 from v = l to v = kui, we have: 1ND tJ‘l [bi +bi i tj—l , i O . ° + . . .+ . = C — . .— (J-l)! uJ u(J+l> b uklu] 0—15!“3 1) C uJ i . i j!c u(j+l)+J!C LKJ+1)_ i (3+1)!c u(j+2)+"" (klu-l)!cluki ] _ i 3-1 c ujt for v = j, j arbitrary, all the terms in the brackets except (j—l)!ci cancelling each other. Now, for each pi i again, we define: v-J kui b1 v t i Z VEJ ! 6 Yu 11(t)*U(t) IV.U.11 i x Llj(t) A E J V Differentiating IV.A.ll, as given by Thm.A.2.9, in the distributional sense ___£i___ = [ 2 _° 1 a? nu tv JetY ul(t)]*u(t) = dt v—j (V—j)! v—° t i z E€[(v—j)1t Je Y “l(t)]*u(t) iv.u.l2 kui bi t"‘J‘l = z [T5:3:I71 etyl“l(t)l*u(t) + v=j+l i Y i z [(3:§7, etY Ul(t)]*u(t) + i 2 [73:37! etY u5(t)1*u(t) IV.A.l3 1U1 In IV.u.l3 the first and second summations are easily 1 i . recognized to be x u(j+l)(t) and x Uj(t) respectively due to IV.A.11. In the third summation of IV.M.l3 all . v-j tyi the terms are zero Since t e L16(t) = 0, except for v = j in which case we have the term ° 1 blujetY u6(t)*u(t) = biuju(t). Thus IV.U.13 becomes: 1 dx (t) uj = i + i i + i dt x u(j+l)(t) y ux u3(t) b “ju(t) where j=l,2,...,klu for n=l,2,...,qi iv.u.5 Substituting IV.U.lO into IV.H.7 we obtain: 1 v-J q. kui kui b t 1 . l mu ty u y(t) = 11m 2 E z z _ , e l(t)*u(t)l + i+m u=l 3:1 v=j (V 35' K 2 dku(k)(t) IV.H.1A k=l Finally using the definition for xiuJ(t), IV.A.ll in IV.A.1H we get IV.H.6. NOTE IV.A.A: The next theorem, the main result of this section, is the one advertised much earlier. It shows that any object(%,, can be approximated with objects having a finite dimensional state space. We think this result is important because we are thus given the possi- bility of approximating closely distributed systems, with lumped RLC networks that have a finite number of 1H2 elements. However, what subset ofCZ objects can be approximated by such RLC networks is still an important question that remains open. DEF.IV.U.3: Let the objects G’and.@3 for i=l,2,... be given by their I—O list RA and R-A Then G’is said to be the I 11‘ LIMIT of the objects Gi’ G= lim (91, iff: i+oo (i) Band 91 for i=l,2,... all have the same input space Uf. (ii) If uEUf and (u,Y)ERi, (U,yi)éRiE for i=l,2,... then y = lim yi in.9'. i+oo THM.IV.H.3: Every(%; object is the limit of objects 91’ which have a finite dimensional reduced state descrip- tion of the form IV.U.l and IV.U.2. PROOF: For each i, the expressions IV.M.5 and IV.U.6 can be written in the matrix form: r" i ‘ r i ....... "r ‘ 1 i .- i ‘ x pl Y H 1 O O X 111 b 111 i i ...... i k 3 u2 9 Y P la 9 § “2 2 “2 £1 . = E I ':. 2- 3 3 + 3 u t dt ; ; -. xi : : ( )’ i . ii 3 x i - X ki b i IJk u 0 O O ...... Y1“ U U Mk U __ a —— —J — — — _ IV.A.15 1M3 We write IV.A.15 in the more compact matrix form: “ = J1 x1u + B u(t) Iv.u.16 where the definitions of the involved entities is self— explanatory. Now combining IV.A.l6 for different values of u, we obtain: P "' " . "1 _- . 1 — ‘1 i i i i X 1 J 1 O O X 1 B 1 i i_, i i X 2 O ;.2 O X 2 5 2 d _ I l I : : a? _ . I z_ : E + I u(t), for x1 o O-- qu xiq Biq each i. _ qL c. i _ i_ _ i 111.4.17 Equation IV.H.6 can also be written with matrix notation, as: F i‘T X l i X 2 . : K (k) y(t) = lim [lki lki ...lki l . + z dku (t) iv.u.18 i+00 l 2 u E k=l xi . i . where l'kj A [l l....l]lka is a row vector. Now, defining the I—O list Bil by IV.u.l7 and . . K (k) y.(t) = [l l i ...l i J : + Z d u (t) IV.A.l9 l kil k 2 k u : k=1 k qi h we obtain the objects Gi’ such that @’= limt9i, since i+oo y(t) = lim yi(t) in B“. Moreover, the state description i+m obtained from the state equations IV.U.l7 and IV.U.19 is a reduced description for eaché}i by Thm.III.A.2, since: (1) every matrix Jiu in IV.4.l7 is an elementary Jordan block, and (ii) the leading entry of each lki (ck(l) in J Thm.III.A.2) is l, and the last entry of each kl A B (bk(dk) in Thm.iii.u.2) is biU L1 L1 , where Cuki is the coefficient (kin-l)!cLJ U U Ri of the term with the highest power of t in each simple exponomial and therefore assumed to be non—zero. NOTE IV.A.5: Similar to the development in section IV.3, it can again be shown that: 1A5 t o ' A . o O p TYlUA - H to[uO] A {uOEU(—W,t0] . for each l,_wj. T e uO(T)dT - t i _wfioipeTY uuO(T)dT where p = 1,2,...,k and u = 1,2,...,qi} iv.u.2o or equivalently that: HtOEuO] A {quU(_m,tO] : for each i, kui bi . i Z HY ftO(T-t )V—'Je(T_tO)Y uu (T)dT = _ (v- ) -m 0 o V-J k 1b1 . 1 u “v ’t _ _ A for JJ=l,2,...,kiu and u = 1,2,...,qi} Iv.u.2l and M'to = {Hvt0[u0] : uOEU(-°°,tO]} constitutes a half reduced partitioning. The equivalence of IV.U.2O to TV.A.21 follows from CU i # O and parts of the proof of k u Thm.III.U.2. Now defining successively: . kibiuv . x:L (t ) a E .W_wltoV‘3e(to‘T)Y1Uu(r>dt i,u,j as in IV.A.21, then 1U6 i i i i T . X u(to) A [x u (to) X n (to)....x u i (tO)] for 1,u as l 2 k U . i i i i T in IV.H.2l, then X (to) A [X l(to) X 2(to)....x qi(’60)) for i=l,2,... and finally the set Z'G(t0) A {X(t ) - X(t ) = [Xl(t ) X2(t ) 1T} IV u 22 O . O O , O ,.... , . . we can show the existence of a one to one, onto mapping c't between Z'L(t0) and H't due to Iv.u.21. Z'G(t0) and o o y(t) A AG tO IV.H.23 constitute a half reduced state description of the object 8C (IV.U.23 is easily derivable from IV.u.ll and IV.U.1U). An important point of the state description (E'G’KG) is the countable dimension of its state vectors, Def.IV.U.22. CHAPTER V CONCLUSIONS It is our hope that with the discussion in Chapter III, the state axioms have reached their final form. The nnain.contributions of this chapter have been this final fflorm of the state axioms and the establishment of the stxrong connection between equivalence classes of inputs arui reduced state descriptions. We have shown that the Iweduced state description of a causal object is almost uruique. This chapter has also provided us with means of ccnqstructing state descriptions that are half reduced. Using the concepts and the results of Chapter II iri Chapter III, We proved that linear and/or time- ileariant objects can always be provided with linear audd/or time—invariant state descriptions; a result of Ifiither academic value which shows that properties of otujects need not, and it is our belief that they should not, be given in terms of their state descriptions. One of the contributions of Chapter IV has been to ob tain a half reduced state description for a large class of distributed objects based on the construction in Chapter II, and to generalize concepts such as "Funda— mental Matrix" used in lumped systems. The other main 147 1&8 x°esult of this chapter was to approximate very general Ciistributed systems by lumped objects possessing reduced estate descriptions using results from Chapter IV. Many open questions that constitute a rich basis for f”urther research arose during the development of the thesis. A few important ones, starting with the obvious cyuestion about the state description of non—linear and/or tzime varying objects, are: l. The reflection in the state description of properties other than linearity and time- invariance, such as continuity, of the system. The formstflxzstate descriptions will take after interconnections of different objects necessi- tating a study of the equivalence classes of in— puts from the individual classes of each system. Studies about the stability of the system using the Hilbert matrix representation obtained in Chapter IV and spectral theory. The approximation of distributed systems by stable and lumped (or lumped RLC) ubjects by placing restrictions on their convolutional representation. Finally, the synthesis procedures obtained in [DA], SX>r the state description in [RES] (given in Chapter I), constitute another solid justification and application of 1149 tshe State Space Theory. It is strongly possible that some ssyntehsis procedures can also be derived from the state ciescriptions in Chapter IV of this thesis. LIST OF REFERENCES [ZIXR] Arsac, J. Fourier Transforms and the Theory of Distributions. Englewood Cliffs, N.J.: Prentice— Hall, 1966. EIBAI] Balakrishnan, A. V. "Linear Systems with Infinite Dimensional State Space." Symposium on System Theory, Polytechnic Institute of Brooklyn, April 20, 21, 22, 1955. E13A2] . "On the Problem of Deducing States and States Relations from Input-Output Relations for Linear Time Varying Systems." SIAM Journal on Control, Vol. 5, No. 3 (1967), pp. 309-325. [I3A3] . "On the State Space Theory of Linear Systems." Journal of Mathematical Analysis and Applications, Vol. 1“ (1966): pp. 371-391. [IiAA] . "Foundations of the State Space Theory of Continuous Systems 1." Journal of Computer and System Sciences, Vol. 1 (19675 pp. 91-116. [EBA] Bashkow, T. "The A Matrix, New Network Description." 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APPENDIX A A.l——About Distribution Theory It has been some twenty years since Schwartz intro- duced and develOped his theory of distributions, a theory that owes its birth to physicists, who have used the delta function since the nineteenth century [ZEl preface]. Mathematicians have plunged into it and a large body of mathematical literature has been published in areas such as ordinary and differential equations, operational calculus, transformation theory and functional analysis. This impetus mathematics has gained from physics did not prevent the more and more abstractization of distribution theory, which is now going the entangled paths of topology and topological vector spaces [TR, HOR]. In mathematical sciences, the most notable applica- tion of distribution theory has been to quantum field theory [ZEA, p. 1]. In network and system theory it has been extensively used in the axiomatic foundation of system theory [ZE1, A], in the time-domain theory of linear n-ports, in obtaining a frequency-domain criterion for the causality of active networks [ZEA, secs. A, 5, 6], in the theory of generalized Bode equations and in the Characterization of various broad classes of systems by 15A 155 their real frequency behavior, [AR], [BEL], [WO], [GU], etc. Distributions have also been used in an essential way in the analysis and synthesis of time varying net- works, see e.g. [NE1-2], [DOl-2]. In other subjects, various classical problems which had been solved in terms of classical mathematics, become open problems once again when reformulated in terms of distribution theory [ZEA, pp. 1]. There exists a serious drawback to distribution theory and this is its uselessness in the theory of non— linear systems, which is due to the fact that the product 5,! of two distributions cannot be defined in general, but only when one of the distributions is a Special one. How— ever, efforts are being made to generalize the product of distributions which are used in quantum field theory [BRE]. This may render possible the application of distributions, at least to special classes of nonlinear systems. Despite this extensive use of distributions, some applied scientists are reluctant to accept the description of physical quantities by a concept that is not an ordinary point function, but is something of functional nature [PA]. That this objection of philosophical nature is not justifiable, can be shown as follows. First of all we can take the attitude of Newcomb when, he assumes physical variables are infinitely differentiable, and justifies it with: "since no physical measurement can 156 prove otherwise" [NE2, pp. 6]. We can equally well say "all physical quantities are distributions since no physical measurement can prove otherwise." However we shall try to do more, since such reasonings can prove anything (i.e. nothing). To begin with, the assumption that a physical variable F can be characterized with an ordinary function “.i f(t) is a convenient idealization [PA]. Why should we not characterize F with a distribution, if that is also a convenient idealization (which we think it is)? A propos, , Zemanian writes: "It is impossible to observe the instan- l_A taneous values f(t) of F. Any measuring instrument would merely record the effect that F produces on it over some nonvanishing interval of time" [ZEl]. Now, it may be that this uncertainty about F being representable by a function, is due to the imperfection of our measuring instruments. The fact is that the imperfection will always be there since indicators will always be subject to parasitic effects such as mass, and we will always justify our theories with measurements using such instruments. Thus, it is a more realistic assumption, as we can infer that much from the physical measurements, to characterize F by a distribution. Finally, Liverman in a recent Article [LI], gives a physical motivated definition of distributions, by showing that one obtains the same space of distributions when one 157 confines himself to the testing functions that are pro- bability densities, i.e. p(x):>0, p(x) is infinitely smooth and fp(x)dx =1, instead of considering the space.9 of all testing functions. As physical background Liverman roughly says: if f(x,t) is the characterization of the physical variable F, where x and t are the space and time variables, a measurement of F yields a quantity f(x,t)+en(x,t), where ffi ek's are error functions and a particular one en is in effect during the experiment. Furthermore, a measurement of the location (x,t), actually occurs in (§,§+d€)x(T,T+dT) with probability p(£,T)d€.dT . Then the expected value of ;__ a measurement of F, intended to be at (x,t) is the weighted average: = + . The functions ek are random and we assume the expected value of over various k, to be zero. Thus: = = ff(£,T)p(€.T)d€.dI. Finally Liverman points that, to say lim fv(x,t) exists pointwise or uniformly becomes a physically non—verifiable, mathematical assertion. The statement lim exists for every probability density is operationally much more relevant, and consistency requires that we include into our list all functionals f such that = lim for all probability densities p. This leads us to gen- eralized functions, which turns out to be a better pencil 158 and paper depiction of physical phenomena, in the presence of errors in the experimental determination of physical variables. In the following sections of the Appendix we intro- duce the necessary definitions and already proven results in A.2., and prove some new results in A.3. mainly about the orthonormal series expansions of distributions, that are needed in Chapter IV. A.2--A Brief Review; Some Definitions and Results in Distribution Theory NOTE A.2.l: The definitions and notations used are con- sistent with those used in [ZE1,2]. Known results are ,given without proof, where a reference to the proof is Inade with the page and theorem number of the corresponding theorem in the literature. IZEF.A.2.1: A function is INFINITELY SMOOTH ON A SET iff it has Guantinuous derivatives of all orders on that set. The Space of all complex valued functions p(t) that Eixse infinitely smooth and zero outside some finite interval is called THE SPACE or TESTING FUNCTIONS, and is denoted by .9. QIEIP.A.2.2: A sequence of testing functionS'EpY(t)}:)=l CONVERGES In] 1? iff the pY(t) are all inii, are all zero outside some nae-o: -... 0.. n—.. u—. «RU n. . \‘r .IV A. TO 16A THM.A.2.5 [ZEl, pp. 115, Thm. 5.2.1] The direct product f(t)xg(1) of two distributions f(t) and g(r) is a distribution in $Jt T. 3 DEF.A.2.5: The CONVOLUTION of two distributions f and g over R is given by the expression A g > A.2.3 NOTE A.2.5: A problem arises in the definition of the con- volution. In A.2.2, p(t,T) and thus had bounded support, but in A.2.3 p(t+T) is infinitely smooth lvithout having bounded support and therefore it is not a izesting function. However a meaning can be attached to £1.2.3 if either the supports of f and g are suitably Icestricted or some conditions are placed on the behavior C>f the distributions as their arguments approach infinity (Ive will not give the theorems related to this last Siituation because definitions of new testing function and (ij_stribution spaces are required; they may be found in [801, vol. II]). The following theorem illustrates when tries convolution process can be given a meaning. In section A-Z3 we investigate another case that is not given in the liftéerature where the convolution can be defined. ”If!" ‘ ...UA a; n: .5» CU Auls In.» Q» 159 fixed finite interval I and for every fixed nonegative interger k, the sequence {pY(k)(t)}:=l converges uniformly for —w a DISTRIBUTION is a functional on 33 such that: = + a for pl, p269' and dEC if {py(t)}:=1 converges to O in£? then the numbers converge to O. The Space of all distributions on .9, denoted by 9', is called the DUAL SPACE OFQ. NOTE A.2.2: In most of our discussions we will deal with ‘ distributions that are defined over the real linelR. How- ever for some theorems, especially the ones about the convolution of distributions, we will have to use distri- butions over n—dimensional Spaces. Thus we have to expand our definitions to multi—dimensional cases. For this 160 11 let xé(xl,x2,...,anHR. The TESTING FUNCTIONS are those that vanish outside a compact set in Rn'and for which all partial derivatives exist and are continuous for all x. Denoting the partial derivative by 000+ kl+k2+ kn k 8 D p(x)$ lc lc k: p(xl,x2,...,xn) n where k A k1 + k2 + ... + kn a sequence of testing functions {DY(X)}:=1 CONVERGES IN B‘TO ZERO iff all py(x) are zero outside a fixed compact subset of’Rn and {kaY(x)}:=1 con- verges to zero for any choice of k. Again, a DISTRIBUTION ONIRn is a linear, continuous functional on.9 defined overlRn (continuous in the sense py + O in ::; + O in.C). DEF.A.2.A: Two distributions f and g are said to be EQUAL iff = , vpeg, The SUPPORT of a testing function p69 is the closure of the set of all points where p(t) is different than zero, and is denoted by supp p(t). Two distributions f and g are EQUAL OVER THE OPEN SET 9 iff = for every testing function p, with supp p(t)co. The complement of the union of all open sets, over each of which a distribution f equals zero, is called the 161 SUPPORT of f, denoted supp f(t). If a set O contains the support of a distribution, that distribution is said to be CONCENTRATED ON O. THM.A.2.1: [ZEl, pp. 30, Thm. 1.8.1.] If a distribution is equal to zero on every set of a collection of open sets, then it is equal to zero on the union of these sets. THM.A.2.2: [TR, pp. 266, Thm. 2A.6] The distributions ian which are concentrated on a point, are the finite linear combinations of the 6- functional and its derivatives. DEF.A.2.5: A sequence of distributions {fy}:= CONVERGES INS?‘ 1 iff for every p69 the sequence of numbers {}:=1 converges. The LIMIT of {}:=1 defines a functional on£>, and the next theorem proves that f is a distribution. A series y=lfY of distributions CONVERGES in19' iff m the sequence hmé¥-lfY of partial sums converges in 3'. THM.A.2.2: [ZEl, pp. 37, Thm. 2.2.1] CD If a sequence of distributions {f } Y Y=l converges in9' to the functional f, then f is also a distribution i.e. the Space 9” is closed. M I l 162 NOTE A.2.3: One way of generating an important class of distributions is to imbed locally summable functions into 3” through the convergent integral [ZE2, pp. 26A] =g_£f(t)op(t5edt Vpeb' A.2.l More precisely, the distribution T because of A.2.l, 1", represents the equivalence class of functions that equal f almost everywhere. It is also worthwhile to note that if Tf = Tg in 9” then f:;:fg is also true. Thus we shall denote Tf by f, any function in the equivalence class that T represents, and call such distributions REGULAR f DISTRIBUTIONS. A.2.l is not the only way to generate distributions from functions. Another standard procedure that leads to the concept of PSEUDOFUNCTION is given in [ZEl], [TR]. l/t which do However there also are functions such as e not define distributions no matter what procedure one tries on them [TR, pp. 226]. The above simple discussion is useful since we con- sider regular distributions frequently in our work and is necessary for the next theorem that is of importance in section IV.A. THM.A.2.3: [TR, pp. 30A, Thm. 38.3] Let Q be an open subset of Rn. Any distribution in Q is the limit of a sequence of polynomial functions in£>'. 163 NOTE A.2.A: Now we concentrate on the convolution of dis- tributions which is a very general process. Various types of differential equations, difference equations and integral equations are all Special cases of convolution equations [ZEl, pp. 11A]. The convolution is also a very general way of characterizing linear, time-invariant and continuous systems that we use in our developments of Chapter IV. THM.A.2.A: [ZEl, pp. 7A, Cor. 2.7.2a] Let x be an n—dimensional real variable and y an m- dimensional real variable. Also, let p(x,y) be a testing 1Rn+m. If f(x) is a distribution function inB defined over defined overIRn, then C(y) A is a testing function of y in£> and an arbitrary partial derivative D§O(y) with respect to the components of y is given by: DkO(y) = . DEF.A.2.6: Let p(t,T) be a testing function infilt I, defined 3 overlRZ, and let f(t)€£% g(T)€.9"T be distributions over m1. Then by THM.A.2.A. is a testing function 11’19t and the DIRECT PRODUCT f(t)xg(T) is defined by g > A.2.2 16A THM.A.2.5 [ZEl, pp. 115, Thm. 5.2.1] The direct product f(t)xg(r) of two distributions f(t) and g(T) is a distribution in $Jt 1‘ 3 DEF.A.2.5: The CONVOLUTION of two distributions r and g over R is given by the expression g A > A.2.3 NOTE A.2.5: A problem arises in the definition of the con- volution. In A.2.2, p(t,T) and thus had bounded support, but in A.2.3 p(t+T) is infinitely smooth without having bounded support and therefore it is not a testing function. However a meaning can be attached to A.2.3 if either the supports of f and g are suitably restricted or some conditions are placed on the behavior of the distributions as their arguments approach infinity (we will not give the theorems related to this last situation because definitions of new testing function and distribution spaces are required; they may be found in [SC1, vol. II]). The following theorem illustrates when the convolution process can be given a meaning. In section A.3 we investigate another case that is not given in the literature where the convolution can be defined. 165 THM.A.2.6 [ZEl, pp. 12A, Thm. 5.A.l] Let f and g be two distributions over“?. Then fxg exists as a distribution overlR, under any one of the following conditions: (i) Either f or g has a bounded support (ii) Both f and g have supports bounded on the left (or on the right). THM.A.2.7 [ZEl, pp. 12A, Ex. 5.A.1] If f and g are locally summable functions whose supports satisfy one of the conditions stated in THM.A.2.6, then their distributional convolution h(t) = f(t)*g(t) is given almost everywhere by the regular distribution cor- responding to the locally integrable function h(t) = im f(T).g(t-T)dT. co THM.A.2.8: [ZEl, pp. 127, Exc 5.A.3] r The convolution of 8m)(t-a) with any distribution in 9', is given by: 6(m)(t—a)*f(t) = f(m)(t—a) m-1,2,... THM.A.2e9: [ZEl, pp. 132] A convolution may be differentiated, by differentiating either one of the distributions in it, i.e. (f(t)*g(t))(m) = f(m)*g(m)(t) 166 THM.A.2.10: [ZEl, pp. 136, Thm. 5.6.1] Let the sequence of distributions {fy}:=l converge in9' to f. Then {fy*g}:=ionverges in 9' to fxg if f and g all have supports bounded from the left. {fY}Y=l’ NOTE A.2.6: The remaining part of this section is devoted to the orthonormal series expansion of certain distribu— tions as given by Zemanian [ZE2] and which constitutes the main tool in obtaining an infinite dimensional state description of a large class of systems in section IV.3. We first give the necessary notation, then state the theorems that we use later. NOT.A.2.1: [ZE2, pp. 262-265] I = (a,b) denotes an open interval on the real line and the case a = -“3 b =G°is not excluded. L? is the space of square summable functions on I with the usual inner product = Zf(t)g(€7dt for f,gEL§. g: denotes the space of all testing functions infi}, whose supports are contained in I..$"I is the space of distributions defined on $1. With Ok(t)?§0 and infinitely smooth on I, n denotes n the linear differentmmion Operator: n A OD Dnl 01 D 2 ... n D V Ov, where the nk are nonnegative integers and k n Dk = d The O and n are so chosen that n = O (-D) v th' k k v n n (-D) 2 O§(-D) l 56. Moreover it is assumed that n Iwi 5. MI 0. n .1 and ... 1 {\r \o“ 167 possesses real eigenvalues "n and normalized eigenfunctions Tn, n=l,2,... with the properties that {Wn}z=l is a com- plete orthonormal sequence in L? and the "n are real, have no finite point of accumulation (this also means no value of "n is assumed more than a finite number of times) and are so numbered that llllsllzl The use of the symbol <.,.> to denote the inner pro- duct in Li conforms with its use to denote the number, a distribution makes correspond to a testing function, Since for a regular distribution f, =_f f(t)3(t)dt (Note A.2.3). DEF.A.2.8: The set of all infinitely smooth, complex valued p functions on I, such that Yk(p)A[élnk0(t)l2dt12 Tn(t) where the series converges in(l . THM.A.2.13: [ZE2, pp. 268, Lem.2] Let {an}:=1 denote a sequence of complex numbers. Then, a T converges in 01 iff I |An|2k|a I2 n n n=l 5M8 :1 n converges for every k. DEF.A.2.9: The set of all linear, continuous functionals on(m- is the SPACE OF DISTRIBUTIONS OP, and the number that fECH' assigns to any péOLis denoted by . (By a continuous functional onCX we again mean if py+O in 01 then the numbers +O). A sequence of distributions {fy}:=l CONVERGES INcn' iff for every pém.the sequence of numbers {}:=l converges i.e.(X' has the weak topology generated by the (f) = | seminorms n ¢ 169 THM.A.2.1A: [ZE2, pp. 269, Thm. 2] 0U is a sequentially complete space. FACT A.2 [ZE2, pp. 269] 5: By FACT A.2.3 the restriction of feov to BI is infi”, and convergence inCMJ-implies convergence in.9;. 6: By the above fact, Li and thereforecn is imbedded into<fi' by defining the number féL2 I assigns to peoz as a A éf(t)6(t)dt. NOTE A.2.7: Another subspace of(l' is the space of all distributions with compact support in I. This with FACT A.2.6 give us an idea about the size ofCl'. FACT A.2.6 also confirms us of the consistency to use the symbol <.,.> in DEF.A.2.9. The next theorem is the result which required all this preparation. THM.A.2.15 [ZE2, pp. 2 O, Thm. 3] 7 If fEOU then f==§ wn(t) where the series con- =1 verge in 01.‘ . THM.A.2.16 [ZE2, pp. 270, Thm.5] Let bn denote complex numbers. Then g=lbn wn(t) converges inCn' iff there exists an integer q a'O such 170 that i IAnI-2qlbnl2 converges. Moreover, if f denotes n#O (D the sum g=1 bn wn(t) in(l' then bn = . A.3-—Some New Results NOTE A.3.l: In this section we start by stating some facts, which are already well known, then we continue with some lemmas and theorems that are necessary to define the convolution of a distribution in(n', with inputs from the input space U of section IV.3. The interval of interest is (-00300) I = (~w,w), Not.A.2.1 and the O 'S in the definition of k the differential operator n are assumed to be bounded on (Tk,m) for some Tk’ k=O,1,.... FACT A.3.l: u(t)€U :::eu(t) is locally summable. U D(_m’oo) D(-°°,°°) is as in DEF.IV.3.1. _F_ACT A.3.2: {wr§t)}:;l is a complete orthonormal sequence for 2 (_oo’oo) L ¢:${wT§T—t)}:;l is one for any finite T. 171 FACT A.3.3: co J If(t)|2dt <0° and f(t) is infinitely smooth :;f(t) _oo is bounded everywhere and ]1T f(t) = O. t+oc FACT A.3.A: J lf(t)|2dt (I) i | ———-—dka 12 _oo dtk dt IV.3.1. EEMMA A.3.1: Let u(t)EUDA, 0(t)Ebvand let C(t) be infinitely I smooth, bounded with supp OC(b,w) for some finite b. Then 00 h(t) A I O(t).u(T). p(t+T)dI A.3.1 oo 172 is a testing function in<3Land 00 (X) I|h(t)|2dt s K [ |p(t)|2dt A.3.2 J —(XJ .00 K a constant. PROOF: First we note that h(t) is well defined for each t since p(t+I) has compact support for each t and u(I) is locally integrable by fact A.3.1. We have three things to be Shown for h(t) to be inCX . (i) That h(t) is infinitely smooth; which is true since C(t) and p(t+t) are infinitely smooth. (D (ii) That { Inkh(t)|2dt <0° ; which will be shown as J — = is shown as n n follows: 00 g Jh(t) n wn(t5.dt ...oo oo n1 n2 nu Jh(t) [OO(t)D Ol(t)D ...D Ov(t)wn(t)]dt A.3.8 _(X) Since the Ok's and wn are functions of the real variable t, A.3.8 can be written as: (I) “1-1— “2 “v— _— = J h(t). 90(t) D[D - D G ..D ov.wn]dt _oo A.3.9 To integrate by parts we let v(t) = h(t)OO(t5 and ri-l Dnt du = D[D 9v wn] dt in A.3.9. To get 176 —— r1l—l— r12 "u— = h(t)OO(t)[D OlD ...D Ov(t)wn(t)] -m nl-l-’n2 “v HAD ...D Ov(t)wn(t)]dt A.3.1O — {D[Oo(t)h(t)]D But h(w) = O by FACT A.3.3 Since h(t) is square summable and infinitely smooth, h(—w) = 0, since h(t) has support bounded at left. So the first expression on the right side of A.3.lO is zero and: 00 n n n _ 1-1 —— 2 v (h,n¢n> _ J(-D) [Oo(t)h(t)]D [olD ...D Ov(t)wn(t)]dt ..oo A.3.ll As in (ii) -D[O;(t)h(t)] is composed of two terms each of which, satisfying the hypothesis of the present lemma, is square summable making -D[O;Tg]h(t)] square summable. Thus integration by parts can be used for A.3.11 again, with the same reasoning as for A.3.9, to yield: 00 n1 n _ 2 ————— "1-2 —— c . - (-D) [Oo(t)h(t)]D [OlD ...D Ov(t)wn(t)]dt R —00 With exactly the same arguments, repeating this process nl + n2 + ... + nv times we will end up with: n 00 n n . v 2—— l J wn(t)[6v(-D) ...(-D) Ov(-D) Oo(t5h(t)]dt (nh, ¢n> 177 due to the assumed form for n, Not. A.2.l. In order to show = we note that the Operator nk has k nl the same form as n i.e. n = [OOD n n l...D vOV] where the bracketed term occurmdlctimes in nu 000D evjooo [GOD succession. The integration by parts can be repeated as many times as we want yielding = for finite k. LEMMA A.3.2 Let u(t)€UD and C(t) be as in Lemma A.3.1, and ("'00,”) let {py(t)}:=lC9’ converge to zero infb. Then 00 {hy(t) A [9(t)u(T)pY(t+T)dT}Y=l converges to zero inCX . _oo PROOF: From Lemma A.3.1 we have that: J IhY(t)I2dt < K JIpY(t)|2dt y=l,2,.... We note that K is independent of y, due to expressions A.3.A and A.3.6, and due to the definition of convergence in $'(Def. A.2.2) ivhich requires supp pyc:[a,8] for =1,2,.... Thus 8 oo lim ley(tfl2dt = o, implying lim thy(t)|2dt = o. y+oo 0L Y"‘°° _ Again, nkh(t) is the sum of a finite number of terms each 178 (X) of the form y(t)f u(T)¢Y(t+T)dT with y(t) and ¢y as in -00 Lemma A.3.1 for y=l,2, .... Moreover m m B J|Y(t)|2[ I|u(t)¢Y(t+¢)|2dt]dt s C [IOY(1)(t)I2dt for -m -w a some i and for y=l,2,.... AS the convergence of py(t)'s is in9,py(i) converges to zero for any 1. So 8 lim Jlo (i)(t)l2dt = o I) Y+°° a Y lim J |y(t)|2[ JIU(T)¢Y(t+T) |2dT]dt = 0. Using Minkowski's y+m —m —m inequality, as we have a finite number of terms we conclude oo oo lim [Ink{O(t)Ju(T)py(t+T)dT}I2dt = O for each k. y+oo -oo —00 NOTE A.3.3: Now we define the convolution of a distribution inCX' having support bounded from the left with u(t)E UD(_m,w) of Def. IV.3.1. We need the two previous lemmas to prove the outcome of the convolution to be in1>'. This defini- tion coincides with the usual definition of convolution if supp u(t) is bounded from the left. QEF.A.3.1: Let w(t)60v be such that supp w(t)(:[b,w], b finite, and let u(t)EU Choose an infinitely smooth C(t) (-oo,oo>° 179 such that it equals one over some neighborhood of supp w(t) and zero outside this neighborhood. Finally let p(t)€9- be arbitrary. The CONVOLUTION of w(t) with u(t), denoted w(t)*u(t) is defined by: A A.3.12 —oo THM A.3.1: Let w(t), u(t), O(t), p(t) be as in Def. A.3.1. Then w(t)*u(t) as given by A.3.12 is well defined and is a distribution in 9”. PROOF: First we note that as C(t) is infinitely smooth w(t)O(t) is well defined and w(t)O(t) = w(t). Then by Lem. A.3.1 O(t)[ u(I)p(t+I)dI is a testing function inCfl. and as w(t)E(I', is well defined. —oo Moreover: -CD A = —oo = (w(t), > A.3.l3 Since u(t) is locally summable it is imbedded in.9'. The expression A.3.l3 verifies that A.3.12 is indeed a convolu- tion where C(t) is necessary in making h(t) a function inCm . 180 Since p(t)€9'was arbitrary, we will be done if we can show w(t)*u(t) is linear and continuous on.B w(t)*u(t) is linear, since for pl and p269 and aETR we have: (D A oo 00 + -00 a + w(t)*u(t) is continuous onMSz If {pY(t)}:=1 is a zero convergent sequence in.9'then: CD _ converges —oo to zero as y+w since O(t) Tu(T)pY(t+T)dT converges to zero iIIOIDy Lemma A.3.2 and.w(t)EOU. NOTE A.3.A: The next lemma, may be one that does not require a proof. Although it is not explicitly mentioned in [ZE2], it must be true for OL' to be a distribution Space. Since we use it in our proofs we felt to prove it briefly would be adequate. 181 LEM.A.3.3: W(t)€Ol =7'JJ' (H601 PROOF: (i) w'(t) is infinitely smooth. (ii) flw'(t)|2dt = we can proceed exactly as we did in Lem. A.3.1, i.e. using integration by parts. THM.A.3.2: Let w(t) be in OL‘ with supp w(t)C[b,°°] and let u(t)EU(_m m). Suppose w(t) = lim WY(t) in(]} also with 3 y+oo supp wY(t)c:[b,w] for y =1,2,...,b finite. Then: w(t)*u(t) = lim [wY(t)*u(t)] in .9'. y-mo 182 PROOF: w(t)*u(t) is well defined by Thm. A.3.1 and so is wy(t)*u(t) for each y. Then for p(t)€$rand an infinitely smooth O(t) which equals one over a neighborhood of supp w(t) and zero outside we have: 00 A by Def.A.3.l. —(X) 00 Y+m _m lim A.3.1A +oo Y -m (D Since O(t) IU(T)p(t+T)dT and by definition of convergence in 01' i.e. lim w (t) = w(t) inOL' iff Y+w Y lim = V¢EOLo Thus A.3.1A gives: y-mo lim V069 y+oo VpEb y-roo _EfiM.A.3.3: Let w(t)EOU with supp w(t)c:[b,w]. Then w(t) can be written as: K (k) wn(t)l(t-b) + 2 d d (t-b) A.3.15 w(t) = k=0 k 3M8 =0 Where K is finite. 183 PROOF: Since w(t)€0fl we can write w(t) = wn(t) 5M8 =0 by Thm. A.2.ll. Then we define f(t) A w(t) — Z wn(t)l(t-b). In order to prove n=O the theorem all we have to Show is: f(t) can at most have its support concentrated at the point b. Let p(t) be any testing function with supp pCK—w,b). We can easily write = - <2 wn(t)1(t-b),p(t)> =0 n=O since both terms defining f have their support in [b,w). Let f(t) have its support in (b,w). Then: ,p> - <§ wn(t)1> n=0 - <2 wn(t),p(t)> = 0 n=0 Since l(t) = l and l(t-b) is infinitely smooth on (b,w). Thus f(t) = O on (—w,b) and on (b,w) hence it is zero on the union of these open sets Thm. A.2.l. There- fore f(t) has support concentrated to the origin. As only finite linear combinations of the delta functional and its derivatives are concentrated at a point, Thm. A.2.2, f(t) = i=0 dk.6(k)(t) and A.3.15 follows. 18A NOTE.A.3.5: The last two theorems are of importance in section IV.3 when obtaining the state description of a large class of objects. Another result, exactly similar to Thm. A.3.3 is useful in section IV.A and is stated in this note. Let a sequence of infinitely smooth functions Oy(t) converge in b" to the distribution w(t)€$fi,£?' defined on (-m,”), with supp w(t)CI[b,W]. Then K w(t) = lim O (t)l(t) + Z d 6 (t), K finite. A.3.16 y-Hao Y k=0 k k The proof is exactly in the lines of the proof of Thm. A0303. APPENDIX B HILBERT MATRICES NOTE 8.1: Although they are a natural extension of finite matrices, infinite matrices, i.e. matrices with infinite rows and columns, do not occupy much place in today's literature, possibly because they are preempted by the theory of abstract transformations and operators. A good book available on the subject is Cook's Infinite Matrices and Sequence Spaces [CO] written in 1950 from where stems the following short discussion. As the theorems will Show a Hilbert Matrix, a name that seems to be abandoned in general but for the matrix [(p+q) llpq. sequence space. Bounded Operators are important and much is nothing but a bounded Operator on a is known about them. Moreover as every linear Operator on the Hilbert space R2 of square summable sequences can be written as an infinite matrix [P0, pp. A12, Ex.l], that makes the Hilbert Matrices important, especially if we encounter them in technical Characterizations as we did in section IV.3. 185 186 DEF.B.1: Let a double series 2 C be given. We form the m,n m,n sequence Sp q of partial sums by finite rectangles, i.e. 3 Sp q is obtained by adding all terms whose first index 3 is s p and whose second index-s q. Then 8,n Cm,n is said to be PRINGSHEIM-CONVERGENT iff for every e>O there is a number 3, independent of s, and two numbers P(e) and Q(e) such that p ;.P(s), q_acxe) implies IS sl s e. The p.q" number S is called the INNER (or PRINGSHEIM) LIMIT of the double sequence S . ps9 DEF.B.2: For an infinite matrix A = [amnl a BILINEAR FORM is 8 T T defined as x Ay A Z a y x where xT=(x x ) y m,n=1 m mn 1’ 2"" ’ n (yl,y2,...) and the convergence is Pringsheim. DEF.B.3: Let E denote the unit hypersphere, i.e. co 1 E A {x = (xl,x2,...): IIxIIA [Z lIx I216‘§ 1}- n: An infinite matrix A = [amn] is Called a HILBERT MATRIX T iff x Ay is Pringsheim convergent on E. THM.B.1: [00, pp. 253, Cor. 2] A necessary and sufficient condition that A should be a Hilbert matrix is that: 187 THM.B.2: [00, pp. 260, Thm. 9.5.V] A = [a ] is a Hilbert matrix if f mn m independent of n and if i=llamnl< N independent of m. NOTE B.2: Thm.B.l shows that a Hilbert Matrix is a bounded Operator on 22 and THM.B.2 is the one we use to show that the infinite matrix A in IV.3.1 is a Hilbert Matrix. That Hilbert matrices are not compact operators (bounded linear operators that map bounded sets into relatively compact sets) is easily seen since the identity matrix I is a Hilbert matrix but not a compact operator since it maps the unit hypersphere, whose closure is not compact, into itself. MICHIGAN STATE UNIVERSITY LIBRARIES 0 306’ 5078 3 1293