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X-765-72-315

A FEEDBACK CONTROL MODEL FOR NETWORK FLOW

WITH MULTIPLE PURE TIME DELAYS

Jacques Press

August 1972

Goddard Space Flight CenterGreenbelt, Maryland

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A FEEDBACK CONTROL MODEL FOR NETWORK FLOW

WITH MULTIPLE PURE TIME DELAYS

Jacques Press

ABSTRACT

A control model describing a network flow hind-ered by multiple pure time (or transport) delays isformulated. Feedbacks connect each desired outputwith a single control sector situated at the origin.The dynamic formulation invokes the use of differen-tial-difference equations. This causes the character-istic equation of the modelto consist of transcendentalfunctions instead of a common algebraic polynomial.A general graphical criterion is developed to evaluatethe stability of such a problem since the literature isevasive in the case of multiple coupled delays. Adigital computer simulation later confirms the valid-ity of such criterion. An optimal decision-makingprocess with multiple delays serves as an applicationof the analytical effort.

).Preceding page blank

iii

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TABLE OF CONTENTS

Page

SECTION I - INTRODUCTION .........................

A. Processes with Memory and Anticipation; The Differential-Difference Equation ...............................

B. The Delay Problem.. ..............................C. The Delay Problem and the Management Function .......D. The General Objective of this Research ..................

SECTION II - MATHEMATICAL BACKGROUND .............

A. Assumptions Leading to the Differential-Difference Equation....B. The Laplace Transform of a\Differential-Difference Equation...C. The Mathematical Form of the Pure Time Delay ............

SECTION III-- THE NETWORK FLOW GRAPH METHOD FORPROCESSES WITH PURE TIME DELAYS ........

A. From the Differential-Difference Equation to a Flow Graph.....B. The Transfer Function for a Single Pure Time Delay Problem...C. The Case of Multiple Delays. .........................

SECTION IV - THE STABILITY OF PROCESSESTIME DELAYS .............

A. Definitions ........................B. Survey of the Literature on Stability .......

(i) The Analytical Methods ....... . ..

(ii) The Graphical Methods ............

C. The Satche Technique . .............

D. Computer Implementation ..............

HAVING PURE........... ee

.............

........ *....

..... *.......

.............

..... ........

*.... ... * -.....

eeeeeeeeelee

$eeeeeeeeleee

· ··ee·eeee,

15

1515

1617

1820

SECTION V - APPLICATIONS ........................

A. An Operational Model with Two Non-Trivial Pure Time DelaysB. Application of the Generalized Satche Technique ............C. Computer Application of Section (B).....................

D. Other Non-Trivial Pure Time Delay Cases.....

Preceding page blankv

21

21232629

1

1

222

5

567

9

91011

CONTENTS- (continued)

Page

(i) Single Pure Time Delay .......................... 29(ii) Cases with Three, Four and Five Non-Trivial Pure

Time Delays ................................. 30

SECTION VI - CONCLUSION ............................ 33

A. Justification of the Differential-Difference Equation .......... 33B. Computer Time ................................... 33

C. The Choice Between the Laplace and Time Domain............ 33D. Summary ....................................... 34

BIBLIOGRAPHY ...................................... 35

APPENDIX A: BASIC DEFINITIONS USED IN THIS WORK ....... 41

APPENDIX B: TERMINOLOGY OF FEEDBACK AND CONTROL.... 43

APPENDIX C: NETWORK FLOW GRAPHS ANDMASON'S LOOP RULE ...................... 45

APPENDIX D: THE NYQUIST CRITERION............... 49

APPENDIX E: LISTING OF THE COMPUTER PROGRAM DEVELOPEDTO IMPLEMENT THE SATCHE CRITERION FOR MUL-TIPLE DELAYS AND TO PERFORM COMPUTERGRAPHICS ............................ 53

APPENDIX F: THE MATHEMATICAL FUNCTIONS USED IN THECSMP SIMULATIONS .............. ......... 65

vi

ILLUSTRATIONS

Figure Page

la The Mathematical Box as a Component of a System ......... 66lb A Generalized Feedback Control System.................. 662 Various Time Lag and Pure Time Delay Transformation Func-

tions ........... ......................... 673a The Network Graph for a Problem with One Pure Time Delay . . 683b The Algebraic Diagram for Figure 3a .............. 684a A Trivial Extension to Two Pure Time Delays ........... 694b The Reduction of Figure 4a to Two Single Delays .......... 694c The Non-Trivial Extension to Two Pure Time Delays........ 704d The General Non-Trivial Extension to N-Pure Time Delays.... 705 The Pade' Table ................................. 726a The Conformal Mapping from the s-plane onto the F(s, eS) -

plane ... ..................... 72

6b The Nyquist Plot for F(s, e ) = s + e TS = 0 ........... 72

6c The Satche Diagram ............................... 737 A Proposed Model for a Network Flow Hindered by Two Pure

Time Delays .... .................... 748 The Network Graph for the Model in Figure 7 ............. 759 Computer Output Displaying the Generation of Numerical Values

for F1 and F2 ................................ 7610 Computer Output Displaying Where F1 and F2 Intersect

Numerically ................................. 7711 Computer Output Displaying the Numerical Values at the Outer-

most Intersections ................ .-.......... 80

12 Computer Output Displaying Regions of Stability and Instability . 8113 First Satche Diagram for a Case with Two Delays .......... .8914 Second Satche Diagram for a Case with Two Delays ......... 9015 Third Satche Diagram for a Case with Two Delays......... 9116 Fourth Satche Diagram for a Case with Two Delays ......... 9217 Continuous System Modeling Program (Simulation) for the Case

of Two Delays ................................ 9318 Output to Figure 17 with K = 0.45 ..................... 9419 Output to Figure 17 with K = 0.25 ..................... 9620 Output to Figure 17 with K = 0.35 ........ .............. 98

21 First Satche Diagram for a Single Delay Case ............. 10022 Second Satche Diagram for a Single Delay Case ............ 10123 Third Satche Diagram for a Single Delay Case ............. 10224 Fourth Satche Diagram for a Single Delay Case ............ 10325 Fifth Satche Diagram for a Single Delay Case ............. 10426 First Satche Diagram for a Case with Three Delays ......... 105

vii

Figure Page

27 Second Satche Diagram for a Case with Three Delays . ....... 10628 Third Satche Diagrams for a Case with Three Delays ........ 10729 Satche Diagram for a Case with Four Delays .............. 10830 First Satche Diagram for a Case with Five Delays ... .... 10931 Second Satche Diagram for a Case with Five Delays ......... 11032 Third Satche Diagram for a Case with Five Delays .......... 111

viii

SECTION I

INTRODUCTION

A. PROCESSES WITH MEMORY AND ANTICIPATION; THEDIFFERENTIAL-DIFFERENCE EQUATION

In concept, there exists a wide class of mathematical processes which op-erate in the present using knowledge acquired from the past as well as informa-tion obtained from the future. This reveals the presence of memory, the abilityto remember the past, as well as anticipation, the ability to predict the future.If these processes are to be realizable in our world then they must obey the lawof causation: the cause of things must come before their effect. A chronologicalorder is imposed and the processes, in this case, "follow their own destiny."The capacity to anticipate is suppressed and only present and past states arenow considered. Despite this restriction, the realization of a man-made processor system* operating in the present using knowledge from the past is an elegantconcept since it simulates the way humans learn and think.

Processes with memory can be expressed mathematically by an interestingclass of equations: the differential-difference equations, involving derivativesalong with differences. The literature on such special mathematics embracesmany disciplines and spreads over three centuries in research. The earliestwork available is that of Bernoulli,lt who in 1728 was studying oscillatory mo-tion of strings. In applied mathematics, famous names like Lagrange2 , Poisson3

and Cauchy are often associated with equations of differential-difference form.They are reflected by today's contemporaries in this field such as Bellman and

5 6 7 8Cooke , Pinney , Ogutzoreli , and Wright . Phenomena possessing memory havealso been studied in the physical sciences by Bateman9 in radioactivity, Gerasimov'0in heat conduction, Lagrange1

1 in sound propagation, Arley'2 in cosmic radiations,13 14 15 16 1Minorsky 3, Nisolle , Volterra , Picard , and Satche 17all in mechanics and

elasticity. In biolo~, one finds differential-difference equations applied to therenewal process , natural and artificial selection , and the fight forsurvival .

*For convention purposes, appendices A and B provide basic definitions for the word system andits associated prefixes, such as feedback and control, in context with this work. Figure 1 illus-trates these definitions.

tA superscript indicates a reference number appearing in the bibliography.

1

Economists have also made contributions with these special equations. Tomention a few, Frish2 2 in economic dynamics, Kalecki 23 in business cycles,James and Belz,

2 4 Samuelson2 5 , Bateman 26 in retail trade theory, Theiss 2 7

in the study of savings and investments, and Tinbergen2 8 in the modeling ofship-building cycles are all notable names.

B. THE DELAY PROBLEM

In modern times, operational models with transport or time delays formthe new vocabulary for procedures governed by differential-difference equa-tions. A delay means that one or more operations are temporarily suspendedin time due to some constraints.

In engineering, one can cite numerous references on time-delay problems:Tsien2 9 in rocketry, also Smith3 0 in electrical systems, Truxal 3 1 in electricalsignals, Rogers and Connolly3 2 in analog computers, Huggins 33 in systems dy-namics, Oetker3 4 , Paul3 5 , Caughanowe and Koppe13 6 , and Tyner 3 7 all inprocess control for the chemical industry. Most recently, the time-delay prob-lem has found application in the theory of artificial intelligence. 38

C. THE DELAY PROBLEM AND THE MANAGEMENT FUNCTION

As far as management analysis is concerned, it is well-known that prac-tical trends in the U.S. are brought about by two schools of thought: those whopractice the case method, relying on past cases, and those who favor the quan-titative approach, relying on analytical knowledge gained by studying physicalsystems. The first method is reflected by the Harvard circles while the secondone is practiced by the Stanford and MIT groups. Forrester 39 in applying thelatter method at MIT has stressed the importance of time delays in industrialdynamics such as production, inventory and sales.

The new field of Operations Research, still part of the quantitative approach,renders time-delay problems imminent firstly in the network flow analysis whereminimum transition time is usually the objective and secondly in Queueing and In-ventory Theory40 . Prabhu4 t has considered similar mathematics in his work onmodels for dams and grain storage.

D. THE GENERAL OBJECTIVE OF THIS RESEARCH

In the real world, autonomous organizations are confronted with the dailytask of applying continuous improvement mechanisms in order to enhance the

2

viability of their working processes. Such strategy involves tactical efforts onthe part of decision-makers. Faced with commitments and schedules, themanagement firstly evaluates the input resources at hand, secondly applies aplanned intervention, and finally seeks feed-back, which is often delayed in time,in order to determine the effectiveness of the motion taken.

Such a philosophical management model has possible application in thefollowing:

1. Transportation networks suffering delays in schedule.

2. The administration of federal, state or local programs where a compli-cated bureaucracy often delays needed resources such as monetaryfunds.

3. Administrative control of the economy such as the imposition of ceilingson prices and wages. In this case, feedback of the results created isheavily delayed due to the complexity of the economic sector.

This paper intends to highlight the above applications through mathematicalmodeling. The effort illustrates the decision-making process involved in anetwork flow containing information and resource channels hindered by coupledpure time delays.

3

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SECTION II

MATHEMATICAL BACKGROUND

A. ASSUMPTIONS LEADING TO THE DIFFERENTIAL-DIFFERENCEEQUATION

One can interpret processes with memory and anticipation using the follow-ing general functional equation:

I [Y (x, t)] = f {Y (t), y (t + tl (t)),

Y (t i t 2 (t)), ''y (t ± t n (t)),

(1)x (t), x (t + tl (t)), x (t i t 2 (t)) ...

(t i t n (t))},

where · is a generalized n t h order (n > 0) linear or non-linear, differential ordifference, partial or ordinary operator. y and x are vectors of the- variousdependent and independent variables (functions of time) respectively. t], t 2 ,

.. t, represent backward or forward delays in time, themselves possiblefunctions of time. Terms like y (t + t n (t)) signify that y is evaluated at a timet shifted in the future by +tn (t), or in the past by -t, (t). Such a formulationmust also be accompanied with proper initial, boundary and constraint conditions.

Unfortunately, no tools are available to fully treat eq. (1). Simplifying as-sumptions must be made. If one is restricted to a realizable, i.e. non-anticipatory,linear, i.e. the principle of superposition holds, deterministic (non-probabilistic)system and furthermore, if the dependent variables are functions of time only,and having constant coefficients along with constant delays then eq. (1) reducesto the following differential-difference equation:

D [x (t)] = f {x P (t - t),(t- t2), x (t - tn), x (t), t} (2)

Preceding page blank 1

5

where

D [ I = nd + a dn- 1 1+ +'' + adt n dt n 1 I

is the linear ordinary differential operator, with the a's being constant coefficients.

If, in addition, the process is instantaneously dependent on just the presentstate, then eq. (2) further reduces to the ordinary differential equation whichtraditionally governs countless mathematical models:

D [x (t)] = f (x, t). (3)

Eq. (3) is indeed restricted when compared to eq. (1).

Equation (2), which assumes present and past states, is expressed explicitlyfor a single input-output system as:

ad "n - 1 xx d x(t) + (t) + b x(t - t) + +b 2 x (t - t2 )dan x(t) + a n-i21 dt-

(4)+ · .. + b n x (t - tn) + u (t),

where u(t) is a prescribed input activating the process. The a's and b's areconstant coefficients.

B. THE LAPLACE TRANSFORM OF A DIFFERENTIAL-DIFFERENCEEQUATION

Since Eq. (4) is linear it can be transformed to a simple polynomial formusing the well-known Laplace Transform, £. Remembering that

[ df (t) = S and £ [f (t - tl)] = e-st1

6

C

Eq. (4) becomes:

an sn 1 a + + a1

= bo + b e - tl + b2 et2 + b e - s tn

+ £ {u (t)} (5)

(e = 2.7 .. )

One obtains the characteristic equation of the system that eq. (4) governs byrearranging eq. (5) and suppressing the input. Therefore,

F(s, es ) = ansn + an 1 Sn -l + .al s - bo - ble-Stl b2e-St2 - bne-Stn = 0(6)

is called characteristic to the system because it describes the free behavior,the outside input being suppressed. This equation helps determine stability, asalient feature in any system study, as later shown.

C. THE MATHEMATICAL FORM OF THE PURE TIME DELAY

Terms containing e - s in eq. (6) correspond to the various transport lags ordelay terms such as y (t - ti) present in eq. (4). The syntax of the word "time-delay" is often misleading due to the existence of numerous synonyms. Depend-ing on the field of study, the following definitions occur in the literature:

(1) distance-velocity lag (Mechanics)

(2) retarded action (Mechanics)

(3) dead time (in Electrical Engineering)

(4) transport lag or delay (in Transportation Science)

(5) pipe-line delay (in Chemical Engineering)

(6) iddle period (Operations Research)

7

(7) pure time delay (Applied Mathematics)

(8) retarded argument (Applied Mathematics)

(9) translation operator (Applied Mathematics)

They all refer to the same transformation:

pure-time delayy (t) --*I~ Y (t - tl)

L of period t 1 l

The confusion arises when another function is mentioned: the phase or time lag.The latter is called, at a loss, an exponential delay.3 9 Various such transforma-tions are illustrated in Figure 2. The first-order time lag 1/(s T + 1), for in-stance, produces an output at the instant the input activates it. The pure timedelay e- s t , on the other hand, does not respond for a time period tl after theinput activates it, and only then, the response comes out as the exact image ofthe input.

This work will adopt the following notation:

[f (t - tl)]= e = pure time delay (which concerns

most of this research)

1[et/l] = simple time lag (1S t order)s 1 + 1

8

SECTION III

THE NETWORK FLOW GRAPH METHOD FORPROCESSES WITH PURE TIME DELAYS

A. FROM THE DIFFERENTIAL-DIFFERENCE EQUATIONTO A FLOW GRAPH

Instead of working directly with a differential-difference equation it is pos-sible to manipulate its Laplace equivalent, in a graphical form. As an illus-tration, consider the following coupled system of differential-difference equationshaving a pure time delay and a simple time lag:

xl (t)= a prescribed function, f (t), (7)

serving as an input.

x2 (t) = (t) - a x6 (t) (8)

X3 (t) =K k2 (t) (9)

x4 (t) = e - t/ X3 (t) (10)(time lag)

X (t)= X4 (t) = 4 (t)dt (11)

x6 5 ( t - tl) (12)(pure time delay)

(') signifies d/dt. K, A, T and t I are constants. The Laplace transforms toequations (7-12) are:

X1 (s) = F (s) (13)

9

s X2 (s) = s X1 (s) - a X6 (s)

s X3 (s)= K s X2 (s)

1S X4 (S) · s X3 (s)

1X5 (s) = X4 (s) _'s X4 (s)

X6 (s) =e 1 X5 (s)

(15)

(16)

(17)

(18)

Utilizing the flow graph method exposed in Appendix C, eqs. (13-18) can berepresented by the network flow appearing in Figure 3a. Such a graph consistsof three major terms, shown in a brief version in Figure 3b as:

he series product of allH (s) = H = -= erms in cascade excluding

T s + 1 Sthe pure time delay,

-st 1

D (s) = D = e ' = the pure time delay,

G (s) = G = - a = the feedback loop.

the}

(19)

(20)

(21)

B. THE TRANSFER FUNCTION FOR A SINGLE PURETIME DELAY PROBLEM

Applying Mason's Loop Rule (see Appendix C) it is possible to obtain thetransfer function (T.F.) defined in Appendix B for the system in Figure 3b asfollows:

10

(14)

Pi =HD

L1 =-HDG

A -L= - 1 +HDG

A 1 =

where P1 is the only forward path, L1 is the onlydeterminant, and A1 is the cofactor. Thus

N= P Ai

T. F. F. P i ii

Setting the denominator of equation (26) equal toequation for the process:

feedback loop factor, A is the

HD1 +HDG

(26)

zero, yields the characteristic

1 + H D G = 1 + H (s) D (s) G (s)

(27)a K .1 -stl =

-1 - +K e 0 e(-s + 1) s

(C) THE CASE OF MULTIPLE TIME DELAYS:

Consider a more complicated case. One can allow for two pure time delaysas shown in Figure 4a. For this case one has, again for Mason's Loop Rule:

P1 = H1 D1

P2 = H2 D2

L1 = - H1 D1 G1

(28)

(29)

(30)

11

(22)

(23)

(24)

(25)

(31)L2 = - H2 D2 G2

A = 1 - (L1 + L2 ) = 1 + H1 D1 G1 + H2 D2 G2

A = 1

A2 =1

(32)

(33)

(34)

and the T.F. is:

T. F. =H1 H2 D1.D2

(35)1 + H1 D1 G1 + H2 D2 G2

This problem can be said to be a trivial extension of the single delay case. Thisis so because it can be separated into two single delay problems as shown inFigure 4b. The two loops are not coupled since the output of the first system isdirectly used as the input to the second one and the latter has no feedback in-fluence on the first one. If coupling is present however then the system is nottrivial. Figure 4c shows the non-trivial case for two delays. For this flowgraph, one has:

P1 = H1 H2 D1 D2 (36)

P2 =0

L1 = - H1 D1 G1

(37)

(38)

L2 = - H1 H2 D1 D2 G2 (39)

A = 1 - (L1 + L 2 ) = 1 + HI D1 (G1 + H2 G2 D2)(40)

(41)A1 =

12

and therefore,

T. F. = i H= H2 D1 D2 (42)A 1 + Hi D1 (H2 G2 D2 + G1 )

Extending the work to a non-trivial problem with N delays, the flow isshown in Figure 4d. Its general T.F. can be shown to take the form:

N

TTHi Di

T. F. = i=l (43)

1+, [Gjf Hi

D]j=l i=l

whereN

2. = summation of terms over the j index from 1 to N,j=1

NT- = product of terms over the i index from 1 to N.i=1

-stDi = e i (pure time delay)

G. = feedback terms

Hi = product of all terms in each i t h forward path between nodes, ex-cluding each Di .

If one .sets the denominator of equation (43) equal to zero then the result yieldsthe characteristic equation for the general non-trivial case:

N j

F (s, eS) =1 1+1 [GJT H Hi 0 (43a)j=

with Gj and Fi containing algebraic terms in s, and Di containing the exponentialterms e st' As an illustration, if three non-trivial pure time delays arepresent then equation (43a) takes the form for N = 3:

F (s, e s ) = 1 + G1 H1 D1 + G2 Hi H2 D1 D2 + G3 H1 H2 H3 D1 D2 D3

1+Gl~s)H 1 (s · -st 1 -s (tl+t2)= 1 + G1 (s) H1 (s) e 1+ G2 (s) Hi

(s) H2

(s)' e

-s 3 s H(tl+t 2+ t3) (43b)+ Ga (s) H1 (s) H2 (s) Ha

(S)'.e 0

13

pr1lN(; PAGE BANK NOT Mfi1

SECTION IV

THE STABILITY OF PROCESSES

HAVING PURE TIME DELAYS

A. DEFINITIONS

A linear system is defined as stable if the time function representing itsresponse to a simple impulse input remains bounded in amplitude as t -co , andunstable otherwise. More important, a stable system is asymptotically stableif its impulse response tends to zero as t - co . Since the transient responseresulting from an arbitrary input, or disturbance, can contain only time func-tions of the type which occur in the impulse function, it is a corollary to thisdefinition that an asymptotically stable linear process will eventually returnto equilibrium for any transient disturbance. It can also be shown that the out-put of an asymptotically stable process will remain bounded if the input isbounded. In the real world, processes are not created to simply be stable fora short period. They must be stable in the long run. Therefore asymptoticstability (t - co) is the prime objective. To shorten notation, in the rest of thiswork the word stable will signify asymptotically stable. Finally if a system isstable one desires to know how close it is to being unstable. This is the addi-tional concept of relative stability as compared to absolute stability, previouslydiscussed.

B. SURVEY OF THE LITERATURE ON STABILITY:

Numerous criteria based on the previous definitions are available toprovide an answer to the question of stability for linear systems. These methodsare either analytical or graphical. Basically, they all start with the character-istic equation derived by Laplace transformation. Traditionally, the analyticalapproach is reflected in the Routh and Hurwitz criteria and the Root-Locusmethod. The first two answer the question of absolute stability while the lastone determines relative stability. On the other hand, names like Nyquist, andBode often appear when the graphical approach is selected. Numerous othercriteria have also been derived. 42

Preceding page blank

15

(i) The Analytical Methods:

The Routh and Hurwitz criteria give fast results 43 for the followingcharacteristic equation:

ao + al s + + a2

S2 + a+ a + a * .a n = 0 (44)

where the a.'s are constants. This equation corresponds to a process governedby ordinary differential equations with no differences, and where n, the number ofroots of equation (44), is finite.

The exponential terms in equation (43a), on the other hand, are periodic bydefinition:

-st 1 -(Cr+ic)tl -= t1

-i*t1

e e e

-Crt 1 'i(otl 1+2n7Tr)= t e-i (n = 0 1, ±2...) (45)

(i = -1 )

and thus equation (43a) is transcendental with an infinite number of roots.

Although the stability requirements are the same for a purely algebraiccharacteristic equation as well as for one with exponential terms, the finiteanalytical criteria fail to give exact results for the latter. This is due to thepresence of the infinite order polynomial which represents the exponential asfollows:

-st t s2 ts2 3 s 3

e = 1 - t 1 s + 2!- + (46)2 3 !

Attempts have been made to approximate e -s

t ', by truncating this Taylor seriesexpansion. This approximation has been shown to be contradictory. Choksy4 4

demonstrated that, depending on where one truncates the series, the processcan be found to be both stable and unstable when the Routh criterion is later ap-plied, for instance. Other approximations to e- s ti, have been derived in orderthat the Routh criterion can handle it. Truxal3 l discusses various onesin increasing order of sophistication. Firstly, he starts with the exponentialfunction expressed as a limit:

16

e I = 1 im (47)

but mentions its weakness at n 3. Secondly, he proposes the n series

but mentions its weakness at n = 3. Secondly, he proposes the McLaurin seriesapproximation as:

-tls 1 (48)

t s2 t3 s 3

1 + t s + --- + ...2! 3!

which is derived from the previous Taylor series expansion, again with limi-tations. Thirdly, the Pade approximation table is presented. This is a ratio oftwo finite polynomials with selected coefficients (see Figure 5). One can chooseany one fraction in this table as an approximation to e- s . The higher order poly-nomials in the numerators or denominators produce a better accuracy. ThePade approximation and its extension, the technique of Single and Stubbswhich produces a similar ratio, are favorite methods in industry.3 2 ' 3 5

Instead of customizing the troublesome exponential to the Routh criterion,an effort has been directed to the root locus technique. Kral 4 6 ' 4 7 was initiallyinterested in the roots of transcendental equations from an involved mathematicalpoint of view. Finally in 1967, he produced a digital algorithm4 8 used to generatethe root locus diagram for an equation having an e - s term. Such studies werepioneered by Chu4 9 in 1952.

All of these works, so far, have considered only the case of the single puretime delay, thus providing restricted results. Contributions to the problem ofmultiple delays are very rare, probably due to the fact that most real-worldapplications assume that the multiple case is a trivial extension of the singlecase. This is not always so, as discussed in Section III. Yuan-Yun, Quing andLian5 0 produced a paper on multiple delays from a sophisticated analyticalview. In addition, Shaughnessy and Kashiwagi 5' have derived a stability indica-tive function for the multiple delay case.

(ii) The Graphical Approaches:

These methods, such as the Nyquist criterion, can determine exactly therelative stability of systems with pure time delays. They handle the latterwithout any numerical approximations. The Nyquist criterion and its applicationto the case of a single delay are outlined in Appendix D. Figures 6a and 6b referto the text in this appendix.

17

C. THE SATCHE TECHNIQUE:

It can be seen that the simple example used in Appendix D produces a Nyquistplot difficult to draw. The diagram is even more troublesome to evaluate forhigher order polynomials, not to mention the multiple non-trivial cases.Satche 17 derived the following elegant procedure which greatly simplifies theNyquist plot. One supposes that the characteristic polynomial in equation (43a)can be separated into pure algebraic and pure exponential parts. Therefore,F(s, eS) can be written as:

F (s, eS) = F1 (s) - F2 (es) (49)

The loci for Fl and F2 are separately drawn in Figure 6c according to the Ny-quist method. Looking at that figure, the F, (s) curve is a closed form algebraicpolynomial in s. It closes on itself and part of it resides at an infinite distancefrom the origin. The area it encloses is cross-hatched. The F2 (eS) locus, onthe other hand, describes a finite contour around the origin. The area enclosedby F2 is covered with dots. In the Nyquist Diagram, one recalls that it isnecessary to determine the number of rotations performed by a vector, originat-ing at the origin, whose head follows the contour F as shown in figure 6a. Thisnumber of revolutions is identical to the number of roots with positive realparts (unstable roots). In the Satche diagram (Figure 6c), a similar vector isdrawn for each contour in F 1 and F2 . Then, according to equation (49), F(s, es)is the resultant vector for F1 - F2 o It joins the points A and B in the diagram.Both ends of the vector AB are now free to move. The reader can follow themotion of AB as (A and OB are rotated clockwise in Figure 6c. AB revolveson itself while its magnitude changes. At the lower intersection of Fl and F2

the magnitude is zero and as soon as the intersection is passed, AB emergeswith a complete half rotation. The same phenomenon occurs at the other inter-section. Thus the intersection points are critical. Each point forces AB torotate half a revolution. The Nyquist criterion can now be modified to imposeon AB the condition that if the vector performs one or more complete revolutionon itself then the system is unstable. In summary, three cases arise in this newcriterion called the Satche criterion:

(1) If the two curves in the Satche diagram are completely disjoint, thesystem is stable. This case is obvious since the angle of the rotationof AB is bounded and always less than 360 ° .

(2) If one curve completely encircles the other without intersection, thesystem is unstable; this is another obvious case, since at least onerevolution of AB is inevitable.

18

(3) If the two curves intersect, then stability can be determined as follows.Recalling that s = ios in the Nyquist plot, let cw co -< W2 be the param-eter range of two successive outermost intersection points on thecurve of F1 during which F1 lies within the area bounded by theF2 curve, (i.e. the area where instability can occur correspondingconformally to the right-half of the complex s-plane in Figure 6b.)Then if that part of the curve of F2 corresponding to the same co rangelies completely outside the area bounded by F1 , then the vector ABsuffers no net rotation and the system is stable.

This case is demonstrated by Figure 6c as follows: One notes that thevector A2 B 2 just above the first intersection proceeds to become A B3without any rotation, thus indicating stability. Once the upper inter-section is crossed, A3B 3 rotates 180° to become A4B 4

. The latterproceeds clockwise, and somewhere after AB, it eventually has thesame direction as A:B 3 indicating one complete revolution. Anyfurther rotation dictates instability.

To the author's knowledge the above Satche criterion has been applied onlyto single pure time delay cases. It is possible to extend the criterion for generalcharacteristic equations with numerous exponentials provided that the algebraicand exponential terms can be separated. The proof of this extension involves asimple axiom from vector algebra, namely the law of uniqueness in vector ad-dition. This states that to every pair of vectors, there is a unique vector, calledtheir sum. As an illustration, equation (49) can be written in detail as:

i

F (s, es) = F (s) - F. (eStij - ) (50)

j=2

where F2 , F3 , ... Fi form a series of curves on the Satche diagram, each onecorresponding to a different exponential in equation (43a). One can add up thevector joining a point on the Fi locus to a corresponding one on the Fi -_1 locus.The result is a single vector (the addition law) which corresponds to a new locus,Fi + Fi_ l . The new vector is then added to the appropriate vector for Fi 2

·The procedure is repeated until a final curve and corresponding vector are ob-tained containing all of the exponentials. This curve is then compared to F1 (s)as done in the simplified Satche diagram, in Figure 6c. For the case of numer-ous exponentials, the F2 curve has more than one intersection with F1 . Forstability, one should consider only the outermost crossovers.

19

D. COMPUTER IMPLEMENTATION

The Generalized Satche criterion is readily adaptable to computer pro-gramming. Such effort is performed in order that the computer can determinenumerically whether a system is stable or not without recourse to a diagram.The Satche diagram thus becomes secondary and serves only as a graphicalsummary of the numerical work.

Proper instructions in FORTRAN are set up for an IBM 360/91 system.The program, accepts any number of pure time delays, within the capability ofthe machine. The graphical capabilities of CALCOMP (Computer Graphics) arethen used and the final result is a Satche diagram for a non-trivial multiplepure time delay problem drawn by the machine. The application that followsillustrates the procedure.

20

SECTION V

APP LICATIONS

A. AN OPERATIONAL MODEL WITH TWO NON-TRIVIAL PURETIME DELAYS:

Consider a real network possessing both an information and a material, orresource, flow. Information flow is almost instantaneous since it usually in-volves electrical or electronic means such as telecommunications. Resourceflow, on the other hand, suffers delays in schedule such as those in transpor-tation and in office bureaucracy. The process is shown in Figure 7 and con-sists of:

1. A management sector made up of:

(a) A single junction point where feedback information from two desti-nation points is subtracted from the forward resource flow rateentering, (data processing),

(b) A control having the decision-making option to vary a factor, K,which, in turn , takes on fractional values between 0 and 1 andmultiplies the resource flow rate such as to increase or diminishit.

2. An undesirable forward time lag (1/ [Tl s + 11] ) which deforms the char-acteristics of the resource rate function. The time-lag occurs quiteoften in the real world, as demonstrated by Forrester.

3. An integrator which changes the resource flow rate into a flow ofindividual units.

4. Two pure time (or transport) delays, hindering the flow which eventuallyreaches two destination points.

5. An information line connecting each of the two destinations with theoriginal junction, bringing back knowledge on the volume level in time-past due to the presence of the delays.

The input takes the form of a rate (units/time) since the management sectortraditionally deals with rates. The destination points on the other hand, situated

21

at a lower level in the organizational hierarchy, usually report on the volumelevel of actual units received. An integration is therefore required to transformthe rate of flow into a level of flow. Figure 8 represents the mathematical flowgraph for Figure 7. One notes that the model constitutes a non-trivial case sincethe two destination points report to the same centralized data processor. Thedifferential-difference equations extracted from Figure 8 are:

x2 (t) = xl (t) - b 1 x6 (t) - b 2 x7 (t) (51)

x3 (t) = K X2 (t) (52)

X4 (t) =e-t/r1

3 (t) (53)

x5 (t) = 4 (t) d t (54)

x6 (t) = x5 (t - tl) (55)

x7 (t) x= 6 (t - t2) (56)

where

= ddt

xl (t) = an arbitrary input rate (units/time) originating from the environ-ment.

x2 (t) = the resultant rate exiting from the junction point (units/time)

x3 (t) = the rate of resource flow (units/time) resulting from the manage-ment control on x 2 (t).

x 4 (t) = the rate of resource flow (unit/time) transformed by the undesirabletime-lag.

22

x5 (t) = the level of resource flow (units), once the above rate (units/time)is integrated.

x6 (t), x 7 (t)= the levels (units) at the two destinations during which the flowsuffers pure time delays of t

1 and t2 , respectively.

K = the management control (dimensionless) which is to be varied.

bl, b2 = the feedback constants (1/time).

Equation (52) describes the transformation at the central junction; feedbackterms such as b1 x6 (t) and b

2 x7 (t) (units/time) represent the rate at which

the resource units entering the destination points are being rejected due tooverflow created by constraints. A linear relationship has been assumed be-tween the rejection rates b 1 x6 (t) and b

2 x7 (t) and the levels x6 (t) and X7 (t),since a higher "crowding" of units at these receiving stations causes the latterto produce a higher rejection rate. The central junction processor (such as acomputer) is programmed to take the above into account and thus pre-plan theschedule by subtracting the rejection rates from the entering resource rate asshown in equation (52). That difference enters the management control wherean optimal value of K has to be determined. The latter should allow for maxi-mum entering flow rate and, at the same time, provide asymptotically stablelevel fluctuations at the destination points.

B. APPLICATION OF THE GENERALIZED SATCHE TECHNIQUE:

To determine such optimum value of K, the generalized Satche technique isused. Applying first the Laplace Transform to equations (51-56), six equationsare obtained in the s domain. Combining these into one and rearranging termsas in equation (5), the characteristic equation becomes:

F (s, es ) = s (r 1 s + 1) + b1 K e + b2 K e(1t2) = (57)

The same result can be obtained directly from the graph in Figure 8 alongwith the general equation (43a) for N = 2. The latter becomes:

F (s, es) = 1 + G1 H1 D1 + G2 Hl D1 H2 D2

(58)

=0

23

and since in Figure 8,

G1 = - b1

1 1Hi=K'1H S + un1 S

H2 = unity

D1 = e l-t 1Die

-St2e

then equation (58) eventually becomes equation (57). Since equation (57) takesthe form of equation (50), one has:

F (s, es) = F1 - F2

F 1 = s (T 1 S + 1)

(59)

(60)

(61)F 2 = - b l K e 1 - b2 K e (t 1 2)

Substituting s = i w (see Appendix D) into equations (60) and (61) one obtains(i= /-):

'(62)F1 (w) = - T 1 w2 + i w

24

with

and

-icot -icow(tl+t2)F

2(Wc) = - b

1K e

= - b1

K {cos (cw tl) - i sin (co tl)}

- b2 K {cos [(t1 + t2).o] - i sin [(t1 + t 2 ).c]} (63)

Separating real and imaginary parts yields:

X1 (w) = Re F1 () = - 1 2 (64)

Y1 (w) =Im F1 (co) = c (65)

X2 (c) = Re F 2 (c) = - b1 K cos (co tl) - b2

K cos [(t 1 + t 2 ).w] (66)

Y2 (c) Im F2 (co) =bl K sin (c tl) + b 2 K sin [(tl + t2).)o] (67)

where Re = "real part of" and Im = "imaginary part of." The constants inequations (64-67) are arbitrarily selected as:

b1 = b2 = 1 (68, 69)

t 1 =1 (70)

t 2 = 2 (71)

-1 = 1.0 (72)

K 0.45 (73)

remembering that 0 • K • 1.

25

C. COMPUTER APPLICATION OF SECTION (B)

The general program described in Appendix E is utilized to obtain the neces-sary numerical and graphical results for stability. The procedure consists ofthe following five major steps.

(i) Generation of Values for X1 , Y 1, X2 , Y2 in Equations (64-67):

The parameterw is varied over an interval of -u to + iT with an incre-mental value arbitrarily chosen as 7/256. Thus w takes on 513 values. Cor-responding numerical values for X1 , Y1 , Y2 , and Y2 are generated by the com-puter utilizing equations (64-67). The output is shown in Figure 9. There, thetop headings state the numerical values of the constants as in equations (68-73)."B(1) TO B(M)" and "T(1) TO T(M)" represents equations (68-71). Subscriptsappear in parantheses, with M = 2, the number of pure time delays in this case.Due to the presence of sine and cosine in equations (66-67), X2 and Y2 areperiodic and thus redundant outside the + -T to -7r interval in w. The incrementfor ) is selected such as to produce satisfactory smooth curves when computergraphics are later used.

The nature of equations (64-65) dictates that F, is a parabola symmetricabout x-axis, open to the left and going through the origin on the Satche diagram.It closes on itself at infinity as in Figure 6c. The area of the right of this parab-ola is enclosed by F1 . Equations (66-67) indicate that F2 is a closed curvearound the origin and thus F, and F2 intersect once or more in the Satchediagram. Therefore, of the three possible cases outlined in the Satche criterionin Section IV, only the last and most general one will be examined here.

(ii) Locating the Regions of Intersection of F1 and F2 :

The set X, and Y1 along with the set X2 and Y2 each corresponds to anordered pair (a point) on the F, and F2 curves, respectively. It is first neces-sary to determine numerically the points where these two curves intersect.The SCAN SUBROUTINE, a numerical algorithm (see Appendix E), performs thisrequired step as follows. Starting with the first line of Xi and Yi in Figure 9,the entire ordered X2, Y2 set is scanned for that pair on F2 closest in distanceto that point (X , Y,) on F1 . The algorithm utilizes the distance definition fortwo ordered pairs:

D = Y(X1 - X2)2 + (Y1

- Y2 )2 (74)

26

The pair X2 , Y2 found closest is at a distance identified as DMIN. Further-more, if DMIN is less than or equal to the value 0.025 (an estimated very smallterm) then X 2, Y2 is not only the closest to X1 , Y1 but at these four valuesDMIN is near zero and F i and F2 intersect. The next pair (X1 , Y1) on F 1 inFigure 9 is selected and the above procedure is repeated until all of the pointson F have been examined. Figure 10 gives the results of the work done bysuch algorithm. The code N = 0 signifies that X 2 = Re F2 and Y2 = Im F2 , on thatline, form the closest non-intersection point resulting from the scanning of theentire set of X2, Y2 . Furthermore, N = 1 indicates a region of intersection. Onenotes that F1 and F2 do not have each the same w at N = 0 or 1. The examinationof the column for N in Figure 10 reveals that a long series of O's first appearfollowed by a few l's, and then more O's. These 1's indicate a region of inter-section, with probably the middle line, within these l's, as being the desiredpoint. The fact that a region and not a point of intersection can be determinedis explained by the small value 0.025 taken as criterion. The smaller it is, thesmaller the 1's region becomes and the better the resolution. However, anestimated value that is too small, such as less than 0.025, can cause the scanningalgorithm to miss some intersections depending on how close to orthogonal(perpendicular) the meeting of F1 is to F2 .

(iii) Locating the Outermost Intersections:

The computer program next examines which two regions of intersectionsare the outermost ones with an increasing w for F . These two points areidentified in Figure 11. The last column in that figure gives the bounds for wcorresponding to F 2 . The numerical values under "OMEGA OF Fl" in that figurecorrespond to the lower and upper bound on ac for F2 and refer to wl and C2,respectively, in the Satche criterion.

(iv) Determining if the Values of F2 (l,) through F2 (w) are Enclosed by F 1:

The set of pairs X 2 (wI), Y 2 (°cl) through X 2 (w2), Y2 (@2) represent thecritical part of F2 and must be examined for possible intersection with F 1 .Subroutine SCAN is again utilized, but this time F, and F2 are replacing eachother in the algorithm, namely, the entire F1 is scanned for each selected pointX 2 , Y2 within F2 (w,) to F2 (002 ). If an intersection occurs then a section of thatcritical part of F

2must be to the right of F, and instability is dictated. If no

intersection occurs then two possible cases arise, either F (w l) through F2 (aw2)is completely to the right or completely to the left of F 1. If it is to the right,the system is unstable, if to the left, stability is declared, according to theSatche criterion. The computer program performs the above logic and pro-duces output of this step in Figure 12. One should read the explanatory N andL codes in the figure.

27

(v) The Satche Diagram:

The program next stores the preceding results (Figures 9 through 12)on a magnetic tape. Subroutine SATCHE performs this step such that the tapecan be later mounted on a Calcomp (California Computing) System. This hard-ware electronically deciphers the code and mechanically draws the Satche dia=gram. The latter is shown in Figure 13 and summarizes the complete effortfor the case of K = 0.45. One can observe five intersections between the para-bola for F1 , identified with ++++, and F2 . F1 closes on itself at infinity in thefar right plane. That part is not drawn since it is of secondary importance.The critical part of F2 (wl) through F2 (C 2), identified by **** goes to the rightbeyond the outermost intersections and therefore the case is unstable. Figure14 shows a highly unstable case with K = 1.0. In this instance the resource flowrate is at a maximum with no management restraint. Figure 15 indicates a stablesystem for K = 0.25. Figure 16 shows a case where for K = 0.35 the criticalpart of F2 (wc) through F2 (w2) is to the left of F1 and terminates at the outerintersection points. The value K = 0.35 is therefore an optimal case since it isthe largest K for which the process is still stable.

It is possible to confirm the integrity of the generalized Satche criterion, thecomputer program and the plotting routine by performing a relatively fast digitalsimulation. Since this approach falls outside of the stability theory presentedso far it becomes a fairly rigorous test. The Continuous System Modeling Pro-gram (CSMP) package provided by IBM 52, performs such simulation. Figure 17shows the CSMP model which corresponds to the flow graph in Figure 8. The simulationproduces the time response of the process when an input is applied. It solves thesystem of differential-difference equations (51-56) utilizing a digital integrationmethod, the Adams technique, provides a discretized evaluation of the two puretime delays, and finally produces a numerical and graphical display of the out-put to the system. In this model simulation, a unit step function is arbitrarilychosen as input. One is interested in the fluctuations as time progresses of x7,the final destination level, representing the output. Figure 18 shows the resultsfor K = 0.45. First of all the response of x 7 does not appear until time = 3.00.This represents the overall initial pure time delay of t l (= 1) + t 2 (= 2) = 3.Furthermore the level at x 7 in Figure 18 shows growing oscillations indicatingan unbounded output thus causing the overall process to be unstable. Figure 19shows a stable condition when the above simulation is repeated with K = 0.25,this time. Here, the final destination level fluctuates significantly at the startbut such transient behavior eventually decays to a settled level as dictated bythe control. The stability for this process is evident at the end of 80 time units.The simulation is performed a final time with K = 0.35. The oscillations forthis case are shown in Figure 20. The oscillation heights slighly decreasefrom one peak to the other revealing eventual asymptotic stability but with a"settling" time that is much longer than for K = 0.25. The decision-making con-trol is confronted with two alternatives. Depending on the trade-off objective,

28

the management can apply a tight control on the resource flow such as K = 0.25and obtain a fast settling but low, level at the destination; or else liberally in-crease K to the allowed maximum value of 0.35 for stability, causing levelfluctuations which take a longer time to settle but obtaining a higher finallevel. In summary, the graph in Figure 18 reveals an unstable system, forK = 0.45. For K = 0.25, one has a definite stable case. And for K = 0.35, thesystem, though stable, is very close to be uncontrollable. The simulation istherefore in complete agreement with the predictions of the Satche techniqueapplied in the two-delay case.

D. OTHER NON-TRIVIAL PURE TIME DELAY CASES:

Since the computer program is a general one, numerous delay cases canbe examined.

(i) Single Pure Time Delay:

This situation is the simplified version with no time lag illustrated inAppendix D, (Equation D-5). Figures 21 through 23 present three cases forK = 2, 1, 7/2 and which are unstable and optimal, respectively. In the last case7/2 has been approximated by 1.57. The Satche functions Fl (cc) and F2 ~) forequation (D-5) are iw, a straight line through the imaginary axis and -Ke - i ', acircle of radius K. Furthermore, the fact that F2 in Figures 21 and 23 lookslike an ellipse rather than a circle is no surprise since it is simply due to thegraphic subroutine trying to optimize the area where the diagram is to be drawn;numerically, Figures 21 and 23 represent a circle. Figures 21 through 23 arein agreement with the results in Refs. .42 and 43.

Figures 24 and 25 show an unstable and a stable single pure time delaycase with time lag, respectively. The characteristic equation for these twocases is taken as equation (57) with b2 = 0. These two figures illustrate,somewhat, how to stabilize a process by manipulating K and T

1. A decrease in

T in Figure 25 flattens to the right the parabola of Fig. 24 as dictated byequations (64-65). This allows a larger region of Fl to the left of F2 to be ex-posed and on which the critical part of F. I(***) can possibly fall, thus producinga stable case. Another manipulation involves a decrease in K thus concentrat-ing the critical part of F2 (***) to the left of F1 , as seen in going from Figure24 to 25. Here, one also notes that a decrease in the time delay improves thestability.

29

(ii) Extension to Cases with Three, Four and Five non-trivialPure Time Delays:

The preceding model is extended to incorporate up to five pure timedelays. The general equation (43a) and Figure 4d are utilized with N = 3, 4, and 5.The characteristic equation for three delays is an extension of equation (57). Itis given as:

F(s, es) = S(T 1 S + 1) + blKe-Stl + b 2 Ke-s(tl+t2) + b3

Ke -s (tl+t2+t3) = 0 (75)

with

F1 (s) = S(T 1 S + 1) (76)

and

F2

(es) = - [blKe-Stl + b2Ke-s(tl+t2) + b

3Ke-S(tl+t2+t3)]

For four delays equation (57) becomes:

F(s, e s ) = s(T1S + 1) + blle-Stl

+ b 2 Ke-s(tl+t2) + b3Ke-S(tl+t2+t3)

+ b 4 Ke-S(tl+t2+t3+t4) - O

(77)

(78)

with

F,(s) = s(T7S + 1)

F2 (es) = - [blKe-Stl + b 2 Ke-S(tl+t2)

+ b3Ke-s(tl+t2+t3+t 4 )]

30

and

(79)

(80)

And for five delays, one has:

F(s, eS) = S(T 1S + 1) + blKe-Stl + b 2 Ke-S(tl+t2)

+ b3Ke-S(tl+t2+t3) + b 4 Ke-s(tl+t2+t3+t4)

+ bKe-s( tl+t2+t3+t4+t5)]

Fl(s) = S(T 1 S + 1)

and

F 2 (eS) = - [blKe-Stl + b 2 Ke-S(tl+t2) + b 3 Ke-s(t l + t 2 +t 3 )

(83)

+ b 4 Ke-s(tl+t2+t3+t4) + b5 Ke -s (tl+t 2 + t3+t4+tS)]

Figures 26 through 28 illustrate near optimum cases (optimal K) for the three-pure time delay problem. Figures 29, 30, 31 and 32 presents diagrams for thecase of four and five pure time delays, respectively. Figures 29 and 30 shows astable situation while Figure 31 indicates instability. Figure 32 gives a stablecase for five pure time delays.

31

with

(81)

(82)

NG PAGE BANK NOT M1Am

SECTION VI

CONCLUSION

A. JUSTIFICATION OF THE DIFFERENTIAL-DIFFERENCE EQUATION:

The models forged so far, containing a combination of continuous and dis-crete equations (with derivatives and differences) can be simplified to a purelydiscrete form involving only difference equations. Such a discrete model posesno difficulties in the stability analysis. In this case, the characteristic poly-nomials involve the Z transform which acts efficiently on transcendental terms.4 2

As seen, this is not so in the case of the continuous-discrete formulation. Injustification of the continuous approach taken here, the adjectives discrete andcontinuous are relative terms depending on the application. As an illustration,feedback information flow entering the decision-making headquarters in theform of a telephone call once a day, for instance, appears as a discrete eventamong other daily office activities. However, if these daily phone reports serveto inform the management on a transportation network suffering time delays ofmonthly magnitude, then they form a continuous stream while the transportationflow is of much more discrete nature.

B. COMPUTER TIME:

The total computer time for analyzing the stability of the models presentedso far requires at most 30 seconds of processing per case. Therefore the con-templation of developing more efficient routines is not pursued.

C. THE CHOICE BETWEEN THE LAPLACE APPROACH AND THETIME DOMAIN APPROACH:

The Laplace transform method used here differs noticeably from the theorydeveloped in the time domain, often called the state variable theory. The firstone is rigorous since it optimizes a model by insuring its stability. The secondapproach, instead, heuristically optimizes an objective function. This functionis often left at the analyst discretion to formulate, based on what he feels areimportant variables.

Preceding page blank

33

D. SUMMARY

/ R. Bellmans3 in a survey on the literature related to Kalecki's Model2 2

discusses two characteristic equations, one of the form:

Xs 2 _ s + b - ce - S = 0 (84)

where X, b and c are constants, and the other one as:

s = b(e- stl - e - s t2) (85)

where b is a constant and t2 = t, + 1. The above author comments that thedetermination of the roots of eq. (84), the one containing a simple delay, hasbeen fully examined for stability. Equation (85) on the other hand, containingtwo coupled delays is, quoting the author, "considerably more difficult" andwhere '"the details are much more perplexing" in the analytical stability ap-proach. The extended Satche method developed herein, can easily determinethe stability of eq. (85) from the graphical point of view, thus by-passing theanalytical determination of the roots. In addition, since this technique iscomputer-based, it serves as a fast and accurate method for a particular user.Furthermore, the extension to multiple delays is quite impossible without a

computer, justifying the use of a machine. The complexity of Figure 31 supportssuch statement.

One notes that, in general, as the number of pure time delays increases, theallowable value for K decreases allowing for less resource flow. The undesir-ability of delays in the real world is thus mathematically illustrated. Most im-portant is the total agreement between the simulation and the criterion predic-tion. This Satche extension must undergo other tests if it is to become an es-tablished technique. Finally, the validity of the model for multiple inputs andoutputs and for non-deterministic random variables remains to be demonstrated,thus providing grounds for further research.

34

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19. Lotka, A. J., "Relation between Birth Rates and Death Rates," Science,Vol. 26, (1907), p. 21.

20. Haldane, J. B. S., "A Mathematical Theory of Natural and Artificial Selec-tion," Proc. Cambridge Philos. Soc., Vol. 23, (1927), pp. 607-631.

21. Volterra, V., "Lemons sur la th6orie mathematique de la lutte pour la vie,"Gauthier-Villars & Cie., Paris, (1931).

22. Frish, R., Holme, H., "The Characteristic Solutions of a Mixed Differenceand Differential Equation occurring in Economic Dynamics," Econometrica,3, (1935), pp. 225-239.

23. Kalecki, M., "A Macrodynamic Theory of Business Cycles," Econometrica,Vol. 3, (1935), pp. 327-344.

24. James, R. W., Belz, M. H., "The Significance of the Characteristic Solutionsof Mixed Difference and Differential Equations," Econometrica, Vol. 6,(1938), pp. 326-343.

25. Samuelson, P., "Foundation of Economic Analysis," Harvard U. Press,Cambridge, Mass. (1947).

36

26. Bateman, H., "An Integral Equation Occurring in a Mathematical Theoryof Retail Trade," Messenger of Math., Vol. 49, (1920), 134-137.

27. Theiss, E., "Dynamics of Savings and Investment," Econometrica, Vol. 3,(1935), pp. 213-224.

28. Tinbergen, J., "Annual Survey: Suggestions on Quantitative Business CycleTheory," Econometrica, Vol. 3, (1935), pp. 241-308.

29. Tsien, H. S., "Engineering Cybernetics," McGraw-Hill Book Co., N.Y.,(1954).

30. Smith, O. J. M., "Feedback Control Systems," Chap. 10, (1958) N.Y.:McGraw-Hill.

31. Truxal, J. G., "Feedback Theory and Control System Synthesis," N.Y.:McGraw-Hill, (1955).

32. Rogers and Connolly, "Analog Computation in Engineering Design," McGraw-Hill, (1960).

33. Flagle, C. D., Huggins, W. H., "System Dynamics," "Operations Researchand Systems Engineering," John Hopkins Press, Baltimore (1960).

34. Oetker, R., "On the Control of Sectors with Dead Time," Proceedings ofFirst International Congress of the Int. Fed. of Auto. Control, Moscow,1960, Vol. 1, Butterworth (1961).

35. Paul, J., "Analog Simulation," San Fernando State College, Calif., Coursetaught at the Civil Service Commission, Washington, D.C. (1972).

36. Coughanour, D., and Koppel, L., "Process Systems Analysis and Control,"N.Y.: McGraw-Hill, (1965).

37. Tyner, M., May, F., "Process Engineering Control," N.Y.: Ronald Press,(1968).

38. McNaughton, R., and Papert, S., "Counter-Free Automata," MIT Press,Cambridge, Mass., (1971), pp. 61-66.

39. Forrester, J. W., "Industrial Dynamics," MIT Press and J. Wiley and Sons,N.Y., (1961).

37

40. Gue, R. L., Thomas, M. E., "Mathematical Methods in Operations Research,"The MacMillan Co., N.Y., (1968).

41. Prabhu, N. V., "Queues and Inventories, A Study of Their Basic Theory,"Wiley, N.Y., (1965).

42. Stoker, B., "Stability Criteria for Linear Dynamical Systems," N.Y.:Academic Press, (1968).

43. Distefano, Stubberud, and Williams, "Feedback and Control Systems,"McGraw-Hill, (1967).

44. Choksy, N. H., "Time Lag Systems - A Bibliography," IRE Trans. Autom.Contr., Vol. AC-5, No. 1, Jan (1960), pp. 66-70.

45. Single, C. H., and Stubbs, G. S., "Transport Delay Simulation Circuits,"Westinghouse Electric Corp., Atomic Power Division, Rept. WAPD-T-38,(1956).

46. Krall, A. M., "An Extension and Proof of the Root Locus Method," J.SIAM 9 (1961), pp. 644-653.

47. , "A Closed Expression for the Root-Locus Method," J. SIAM,11, (1963), pp. 700-704.

48. , "An Algorithm for Generating Root-Locus Diagrams," Com-munic. of the ACM, Vol. 10, No. 3, March (1967), pp. 186-188.

49. Chu, Y., "Feedback Control System with Dead-Time Lag or Distributed Lagby Root-Locus Method," Trans. Amer. Inst. Elect. Engrs., Vol. 71, 291,(1952), pp. 291-296.

50. Yuan-Xun, Q., Iong-Quing, L., and ian, U., "Effect of Time-Lags on Sta-bility of Dynamical Systems," Proceedings of First International Congressof the Int. Fed. of Autom. Control, Moscow, (1960), Vol. 1, Butterworth,(1961).

51. Shaughnessy, J. D., and Kashiwagi, Y., "The Determination of a StabilityIndicative Function for Linear Systems with Multiple Delays," NASA Tech.Rpt. TR R-301, (1969).

52. IBM, "System/360 Continuous System Modeling Program," GM20-0367-2,(1969).

38

53. Bellman, R., "A Survey of the Mathematical Theory of Time-Lag, RetardedControl, and Hereditary Processes," Project Rand, R-256, (1954), The RandCorp., Santa Monica, Calif., p. 88.

39

PR1iC)WG PAGE BIANX NM FMFM

APPENDIX A

BASIC DEFINITIONS USED IN THIS WORK

1. A system or process: an arrangement of elements connected or related insuch a manner as to form a working entirety.

2. A control system: a system able to regulate or command itself or othersystems.

3. A feedback control system: a control system where the output of somecontrolled system variable is compared with the input to the system so thatthe appropriate control action may be formed as a function of the output orinput. Feedback increases the accuracy of a control system and favorsoverall system stability.

4. Modeling: a mathematical method used to express the real world through:

(a) Functional or differential equations.

(b) Block diagrams, similar to flow charting.

(c) Flow graphs.

Block diagrams and flow graphs are schematic representations of the model'sequations and real components. Appendix B explains the block diagram notation.The flow graph method appears in Appendix C.

41

Preceding page blank

PRCEDGNG PAGE BLANK NOT FILMED

APPENDIX B

TERMINOLOGY OF FEEDBACK AND CONTROL: THE BLOCKDIAGRAM AND THE TRANSFER FUNCTION

A. THE BLOCK DIAGRAM:

The block diagram notation is a graphical representation of the overallcause and effect relationship between input and output of a process or system.The components or elements of a system is each characterized by a box (orblock) with an arrow penetrating and exiting, representing an input and output,respectively. The box usually contains a mathematical transformation such as adifferentiation, integration, division or multiplication which acts on the input toproduce the desired output. Figure la illustrates such a representation. All

'the blocks are related to one another and eventually form the overall system.Figure lb illustrates a generalized beedback control system. In this diagram,the control block function can be varied, the process is fixed and the feedbackis subtracted (negative feedback) or added (positive feedback) from an inputoriginating outside the system.

B. THE TRANSFER FUNCTION (T.F.):

The ratio of the output to the input of a system constitutes the transferfunction. This ratio can be determined analytically by equating the output to thetransformed input for each block of the system. This gives a series of coupledequations which can be combined into a single equation containing the originalinput and final output. The ratio of these two is then readily obtained.

The method in Appendix C provides a more elegant and rigorous way ofdetermining the transfer function for linear, deterministic systems.

Preceding page blank

43

APPENDIX C

NETWORK FLOW GRAPHSAND MASON'S LOOP RULE

A. NETWORK OR SIGNAL FLOW GRAPHS:

Basically, the block diagram notation in Figure lb can be simplified to analgebraic graph as follows:

INPUT - -H-- OUTPUT

H=NET FORWARD FLOW] G=NET FEEDBACK FLOW]

where the transfer function is

OUTPUT GH.= T. F. (C-1)INPUT 1 - GH

The arrows in the above diagram are called branches while the destination dotsare called nodes. A path is a branch or a continuous sequence of branches whichcan be traversed from one signal (node) to another signal (node). A loop is aclosed path which originates and terminates on the same node and, along which,no node is met twice.

B. MASON'S LOOP RULE:

Equation (C-1) is obtained by equating the incoming flow of values to theoutgoing one at each node, since no accumulation can occur in a flow graph.This produces a set of equations which reduce to equation (C-1). A shorterway to derive the transfer function is to utilize Mason's Loop Rule (see ref.43). The Rule states that, given a flow graph, the transfer function betweentwo nodes, i and j respectively, is given by:

Preceding page blank45

Pi k i j k

kT. F. (C-2)

A

where

P..J k

= k th forward path from variable nodes i to j,

A = determinant of the graph,

A.. = cofactor of the path P..J k x j k

and the summation is taken over all possible k paths from the k to j nodes. Thecofactor Ai j k is the determinant with the loops touching the kth path removed.The determinant A is:

N

A = 1 -1 LL +n=l

M,Q

m=l, q=l

Lm Lq -T L LLLt + · · ·

where L q equals the value of the qth loop transmittance. Therefore, in words,equation (C-3) signifies that A = 1 - (sum of all different loop branches) + (sumof branch products of all combinations of two non-touching loops) - (sum of thebranch products of all combinations of three non-touching loops) + .... Twoloops are non-touching if they do not have a common node.

C. ILLUSTRATION OF MASON'S LOOP RULE:

Consider the following flow graph:

H5

G3

INPUT OUTPU

46

(C-3)

There are five forward paths connecting the input to the output:

P1 = H1 H H3 (C-3)

P2 = H1 G1 H3 G3 (C-4)

P3 H1 G1 H3 H (C-5)

P4 = H1 H2 H3 G3 (C-6)

P- = H5 (C-7)

There are four self loops:

L1 = H1 H. 2 (C-8)

L 2 = H4 G4 (C-9)

L3 = H1 H2 3 G5 (C-10)

L 4 =- H2 H 3 G 6 (C-ll)

Loops L1 and L 2 do not touch. Therefore,the determinant is:

A = 1 - (L1 + L2 + L4 ) + (L1 L2 ) (C-12)

The cofactor along P1 is obtained by removing from A the loops that touch P1:

Al = 1 - O = 1 (C-13)

47

The same applies for P2 , P3 , P4 and Ps:

2 = 1-0= 1

A3= 1 - 0= 1

A4=1-0=1A4 =1 -o = 1

As = 1 - L4

and therefore,

P1 A1 + P2 A2 + P3 A3 + P4 A4 + P5 AS

(C-14)

(C-15)

(C-16)

(C-17)

A

= (H1 H2 H 4 ) + H1 G1 H3 G3 + H I G1 H3 H4 + H5 (1 - H2 H3 G6 )

1 - (Hi H2 G2 +.H4 G4 + H i H2 H3 G5 + H2 H3 G6 ) + H1 H2 H4 G2 G4

(C-18)

48

I. r. =

APPENDIX D

THE NYQUIST CRITERION

A. CONFORMAL MAPPING, THE ENCIRCLEMENT THEOREM:

If the variable s, in the Laplace domain, assumes real and imaginary partsas follows:

s = a + i c (D-l)

(where i = 'T), then the characteristic equation in s also assumes real andimaginary parts given by:

F (s, eS) = U (o-, 0) + i V (cr, co) (D-2)

According to the theory of complex variables, a conformal mapping from s toF is defined as follows. To any point (a, c ) in the s-plane, there correspondsa unique point (U, V) in the F-plane. If the point (o-, co) traverses a closed con-tour C in s then (U, V) will trace a corresponding contour, F. Figure 6a illus-trates such mapping. Accordingly, the Encirclement Theorem states that whenC is traversed once clockwise by a point (ao, co) then the corresponding mappedpoint (U, V) revolves N times counter-clockwise on F in Figure 6a, such that,

N = -Z (D-3)

where Z is the number of those roots of equation (D-2) which lie within C pro-vided that none of them lie on C itself. A negative N indicates that F is tra-versed in clockwise fashion, while a positive N indicates a counter-clockwisedirection. A complete proof is available in Reference 42.

49

B. THE CONDITION ON THE s-PLANE FOR ASYMPTOTIC STABILITY

The condition for asymptotic stability, exposed in Section IVA, requires thatall the roots of the characteristic equation to have negative real parts and there-fore to lie in the left-half of the s-plane in Figure 6a. This is briefly explainedas follows. The response of a linear stable system to an impulse function musttend to zero as t - co, as stated in Section IVA. This response is generalized as:

(t) A1 e ( l+ i l ) t + A e (0 2 +i 2 )t + An e( n n)t (D-4)

(i = v-i-)

where A1 , A A2 . . . A n are constants and (a 1 + ico ), (c2 + i W2 ) . . . , (-n + icn)are the complex roots of the characteristic equation (D-2). It can be seen fromequation (D-3) that if the o-' s are negative, then y(t) decays exponentially thussatisfying the requirement for asymptotic stability. Abiding to such prerequisite,one cancels all possibilities for unstable roots to lie in the left-hand plane. Thisleaves only the co -axis and the right-hand s-plane in Figure 6a to examine forunstable roots. The general contour C then specializes to a newcontour, C',which is shown in the left portion of Figure 6b. It consists of the whole imaginaryaxis between w = - Xc to c = + co, along with the infinite semi-circular arc en-compassing an area to the right of the direction of the arrow, namely, the com-plete, bounded right-hand plane.

The conformal mapping of this semi-circular closed contour C' onto theF-plane produces a closed F contour. First, one considers the semi-circulararc portion (excluding the w axis) of C' on which s can be written in terms ofpolar coordinates as:

s = Re' i (7T/2 > 6 > - 7T/2) (D-5)

where R = infinity, i = v-1. The mapping of this portion onto the F-plane gene-rates a clockwise F portion which also bounds the entire right hand F-planewith the imaginary F axis. The equation of this F portion is immaterial sinceevery point on it is at an infinite distance from the origin. It can be taken as asemi-circle with infinite radius. It is referred to as part of the F-locus at in-finity. Returning to the s-plane, the imaginary s-axis remains to be examinedfrom s = w = -co to s = w = + co. Depending on F, the mapping produces variousloci which then connect with the part of the F-locus at infinity to form a closedcontour. The area enclosed depends again on the direction of the arrows on F.

50

C. THE NYQUIST CRITERION:

The Criterion basically combines the requirements of section (A) with thoseof section (B) by determining the number of roots which reside inside C' of thes-plane in Figure 6b, (i.e. those roots with positive real parts and thereforethose which cause instability). It can be stated as follows:

According to the Encirclement Theorem, the number of times, N, that the F-locus encircles the origin in a clockwise sense corresponds to the number ofroots within C' in the s-plane, i.e. those roots with positive real parts and whichcause the system to be an unstable one, according to the stability condition onthe s-plane. Therefore, if the F-locus never encircles the origin completely,then the system is (asymptotically) stable, and unstable otherwise.

D. ILLUSTRATION:

The previous criterion is best illustrated as follows. Consider the charac-teristic equation containing one pure time delay as:

F (s, eS ) = s + K e - s = 0 (D-6)

where K is an arbitrary constant.

The mapping of s onto F(s, e s) consists of two parts. First, the semi-circular portion of C' in the s-plane is represented by Equation (D-4). Aspreviously stated, the F(s, e s) contour corresponding to every point (oc, ow) onthat section is itself a semi-circular curve traced at an infinite distance fromthe origin. It is shown in the F-plane drawn in the right portion of Figure 6band labeled I F(s, eS) =co . The second part of the mapping traces F (iwo, ei ° )for " varying from - into + o. This tracing is done by firstly substituting iLo fors in equation (D-5). The result is

i + K e-iw = 0 (D-7)

Since e - i = cos co - i sin o, one separates the real and imaginary parts ofequation (D-6) to yield:

51

X1 (co) = Real F = K, [cos co]

Y1 (c) = Imaginary F = K [i (co- sin co)] (D-9)

For every co on the imaginary axis of the s-plane, there corresponds an orderedpair (X1 (co), Y1 (co)) on the F-plane. Numerical values are calculated for X1

and Y1 for a varying co. The result is the mapping of w onto F. It is shown asthe "wiggle-like" portion along the imaginary axis of the F-contour in the right-hand portion of Figure 6b. One notes that three curves appear each correspond-ing to a different value of K, namely 1.0, 7T /2, and 2.0, respectively. All threecurves connect with the semi-circular portion of F labeled IF(s, eS)I = 0o atinfinity. The mapping, called the Nyquist plot, therefore completely enclosesan area to the right of the direction of the arrows (shaded in Figure 6b) in F.

It can be seen that for K = 2 the F-locus encloses the origin twice in theclockwise sense, once very close to it and another time very far from it atR = co so that N = -2 and therefore Z = -N = 2, indicating two roots with positivereal parts in C'. The cases for K = 1.0 and 7r/2 on the other hand reveal noencirclement and therefore constitute stable cases. The value K = mr/2 is at thethreshold of stability. It represents the largest allowed value for stability.

52

(D-8)

APPENDIX E

LISTING OF THE COMPUTER PROGRAM DEVELOPED TO IMPLEMENTTHE SATCHE CRITERION FOR MULTIPLE DELAYS AND TO PERFORM

COMPUTER GRAPHICS

A. THE SUBROUTINES USED:

The following program is written in FORTRAN IV. The implementationutilized an IBM 360/91 operating under Release 20, a relatively high speed,large memory machine.

Subroutine SCAN determines the intersection points while subroutineSATCHE converts the numerical results into a special code which is transferredon a 9-track magnetic tape. This tape is then mounted on a CALCOMP 780(California Computing) System which in turn draws the desired plots on a drumusing ink and paper.

The subroutines PLOTS, PLOT, LINE, NUMBER, SCALE, and AXIS aremade available by the CALCOMP company and assist in drawing the plots ac-cording to a desired scale, format and size. CALCOMP also provides arepertory of symbols, letters, and other characters which label thediagram.

B. LIST OF COMPUTER SYMBOLS IN CONTEXT WITH THE MAINWORK (IN APPROXIMATE ORDER OF APPEARANCE):

X1 = X1 (l) = Re F1

X2 2 (co) = Re F2

Y1 = Y1 (= ) = Im F1

Y2 = Y2 (c) = Im F2

BUFFER = A dummy vector used as a "scratch pad" by subroutineSATCHE.

53

OMEGA = a vector containing all the values for w.

B = a vector containing the feedback factors, bi, in order of ap-pearance in the network forward flow.

T = a vector containing the exponents of the delays in order of ap-pearance in the network forward flow.

PI = 7

PIB = selected maximum value for w.

DPIB = selected increment value for o .

I = subscript for OMEGA, XI Y1, X2, Y2 initially set at 1 andfinally set at 513.

NFMAX = selected maximum number of computer runs per batch.(one run per delay case)

NFILE = file number assigned to each case, initially set at 1 and finallyset at NFMAX.

M = number of pure time delays in the particular problem beingprocessed.

XK = K= management control

TAU1 = IT = time lag

MM = dummy subscript

N1, N2, N3, N1iMAX, N2MAX, N3MAX = various pointers of secondary im-portance used to produce a convenient output format.

W = an intermediary value for OMEGA.

XDPI = an increment smaller than DPIB (of secondary importance)

XT = a temporary value for T

XX2 = a temporary value for X2 initially set at 0

YY2 = a temporary value for Y2 initially set at 0

54

K = a dummy subscript

TEMPX2 = another temporary value for X2

TEMPY2 = another temporary value for Y2

WP = a value used to transform w into a fractional part of 7 to be-come readable in the output printout.

N = a code indicating "intersection" or "no intersection."

IMAX = 513 = maximum number of values for X1, YI and of OMEGAused to evaluate X1 and Y1.

JMAX = 513 = maximum number of values for X2, Y2 and of OMEGAused to evaluate X2 and Y2.

IIMIN = subscript of the OMEGA corresponding to o , the smallervalue at which one of the outermost intersections occurs.

IIMAX = subscript of the OMEGA corresponding to w 2 the larger valueat which the other outermost intersection occurs.

J = temporary subscript

ID = the number of values of OMEGA for F1, X1 and Y1 on F1between the outermost intersections.

IIIMIN = subscript of a reconstructed OMEGA for F1, X1 and Y1 withthe subscript ID = 1 to ID = IIMAX - IIMIN. This is necessaryso that the curve F1 and F2 have the same order of magnitudewhen the Satche diagram is performed.

SUBROUTINE SCAN: (additional symbols to the above)

D = distance between (X1, Y1) and (X2, Y2).

DMIN = smallest distance between (X1, Y1) and (X2, Y2).

JX = subscript identifying the point (X2, Y2) closest to the givenpoint (X1, Y1).

55

SUBROUTINE SATCHE: (additional symbols to the above)

XL, YL = values specifying the paper area on which the Satche diagramis to be drawn.

XSUBX, YSUBY, YSUBX, XSUBY = CALCULATED origin of the Satche dia-gram. (Optimized values such that the diagram is centeredwithin the area specified by XL, YL).

D = a temporary value used in the above optimization.

JJ, NN = dummy subscripts.

XN = a value used for proper format (of secondary importance)

56

DATE = 72214 16/08/18

C

C NYQUIST-SATCHE DIAGRAMDIMENSION X1(600), X2(600).YI(600).Y2(600) ,3UFFR (7500).OMEGA(600).IB(I0),T(10)

C

CC SETTING UP A WORK AREA FOR PLOTTING

CALL PLOTS(BUFFER,7500.0)

CC

C

C

C CONSTANTS DEFINEDPI=3.14,159PIB=PIDPI=16.*16.

C

C

C READING THE NUMBER OF COMPUTER RLNS DESIRE)READ (5977) NFMAX

77 FORMAT(12)C

C

DO 888 NFILE=I,NFMAXC READING IN ORDER THE DESIRED NUMEER OF RUNS.C FEEDBACK MULTIPLICATIVE FACTORS,AND THE EXOONEhTS OF THEC PURE TIME DELAYS.

READ (5,99)NFtLEM. XK.TAUI,(B(MM).MM=1,5),(T(t ),MM=1 5)'99 FORMAT(12,Ii,1-2(F5.2))

C

C

1=1771 WRITE(6,1 )NFILEMXK.TAUI (B(MM),MM=I'5) ,(T(k¥)'4MM=1.5)

I FORMAT('{l",///3X,2'COMPUTER RUN NUMBER 'I12, 9X,'SATCHE DOTA GENERATION'/%3X,7'THE NUMBER OF DELAYS IS ',12./*3X.8'THE CONSTANTS ARE:' 9/3X,I'K= ' FB8.3,CX,'TAUI=' ,F8.,3 //X3X,3'8(1) TO 'B(M):'"5(F8.3.2X)*/.3X,

5'T(1) TO T(M):"' 5(F8.3,2X))

772 WRITE(6,3)3 FORMAT(//X4X, 'OMEGA OF Fl',8X.'FE Fl,1IOX.1'IM Fe1.lOX*eRE F2',1OX,'IM F2'. 9X,'CMEGA IF F2')

CC THE NEXT TWO STATEMENTS ARE USED FOR AC DESIRED PRINT-FORMAT.

N1=1N 1MAX=26-( M*2)

773 W--PIXDPI=(PI/DPI)/2.

C GENERATION OF F(S).4 Xl(I)=-TAU1*W**2

61 Yl(I)=WC GENERATION OF F(EXP(S)),THE CURVE WITH DELAYS. -

XT=T( )XX2=0.

57

kN IV G LEVEL 20 MAIN

DATE = 72214 16/08/18

YY2=0.

DO 67 K=I,MTEMPX2=B(K )*XK*COS(XT*W)TEMPY2=B(K)*XK*SIN(XT*W)IF(K-M) 69,68e65

69 XT=XT+T(K+1)68 X2(I)=-(XX2+TEMPX2)

XX2=-X2(I)Y2( I )=YY2+TEMPY2YY2=Y2(I)

67 CONTINUEWP=256.*W/PIOMEGA(I)=WP

777 WRITE(6.2) WP.XI(I).YI1¥I)X2(I),Y2(1).WP

2 FORMAT(3X,F6.1,'/256 PI '.5X,4(1PEIO.3.5X),PF6.1,'/256 Pi')IF(NI-NlMAX)8910,1C

10 NI=0WRITE(6.1 )NFILE.MXKTAU1.(B(MM).MM=1.S),(T(M),tMM=1,5)WRITE(6*3)

8 NI=NI+188 IF((.W+XOPI )-PI8)12 ,9 912 1=1+17 W=W+PI/DPIGO TO 4

9 CONTINUE

19 WRITE(6,1 ')NFILE.M.XK.TAUI1(B(MM).MM=1,5).(T(IvN),MM=1,5)97 WRITE(6,23)'23 FORMAT(/,3XemFINDtNG ALL POINTS CF INTERSECTICN')96 WRITE(6,201)20 FORMAT(/.3X.'OBSERVE THE FOLLOWI G CODES BLOW'./3X,

1'N=O : NOT A REGION OF INTERSECTION'./,3X,

2'N=1 : A REGION OF INTERSECTION')95 WRITE(6.21)21 FORMAT(//,4X, 'OMEGA OF F1',6X.'FE FI',8X.

1'4IM Fl', 8X,'RE F2'.C8X,'IM F2', 7X,'OMEGA JF F2',4X.'N')98 IMAX=I

JMAX=II IMAX=OIIMIN=999N2=1N2MAX=19-( M*2)DO. 32 I=I,IMAXCALL SCAN(IXIX(I),YI(I).X2,Y2,N,_,;JMAX)

24 WRITE(6.26) OMEGA(I),Xl(I),Y(Il ) X2(J) Y2(J).CnEG4(J),N26 FORMAT(3XFe6.,'/256 PI' 3X,4(IPEIO.3,3X) ,)PF6.1.'/256 PI'92X.12)

IF(N2-N2MAX)30.31.jl

31 N2=0

WRITE(6,1 )NFILE,M,XKTAU1,(B(MM),MM=1 5) .(T(tM.).MM=1,5)WRITE (6.23)WRITE(6,20)WRITE (6.21)

30 N2=N2+1C LOCATING THE OUTERMOST INTERSECTIONS.

45 IF(N) 32,32,27-27 IF(I-IIMIN)29,29,34

29 IIMIN=I

58

4N TV G LEVEL 20 MAIN

DATE = 72214 16/08/18

J1=J

34 IF(I-IIMAX) 32,33.3333 IIMAX=I

J2=J32 CONTINUE

WRITE(6,1 )NFILE.MXK,TAUI,(B(MM),MM=I.S),(T(Mt),MM=1.5)WRITE(6.35)

35 FORMAT(//,3X,' THE OUTERMOST INTERSECTIONS OCCJS AT')

WRITE(6.3)WRITE(692) CMEGA(I IMIN),XI(IIMIN),YI(IIMIN),

I X2(JI),Y2(JI),CMEGA(J1)WRITE(6'2)OMEGA(II MA X).XI(IIMAX) .Yl(I]IMAX).

IX2(J2).Y2(J2) CMEGA('J2)C CHECKING IF F2(w) IS OUTSIDE F1(1) FOR THE RANGEC OF OMEGA OF F2 VARYING FROM OMEGA SUB IIMI'4 TOC OMEGA SUB IIMAX

WRITE(6*1 )NFILEt.MXKTAUIC(B(MM).MM=,5) ,(TC(MW),MM=1,5)WRITE (6·40)

40 FORMAT(/ ,3X,'**CHECKING IF F2 CF OMEGA SJS I IMIN THROUGH .·/3X.1'F2 OF OMEGA SUB IIMAX LIES OUTSIDE OR INTERSECTS Fl**')WRITE(6620)W.RITE(6,62)

62 FORMAT(3X,1'L=-I : X2 IS T0 THE RIGHT OF FI·UNSTABLE .ASE'./.3X·2'L=1 : F2 INTERSECTS Fl,UNSTABLE CASE'./,3X.

3'L=0 : F2 IS OUTSIDE OF Fl,STABLE CASE')WRITE(6,55)

55 FORMAT(//,4X, 'OMEGA OF Fl',6X,'RE Fl',8X,I'IM Fl'. 8X,'RE F2OS08X,'IM F2', 7X,'OMEGA 3F F2'.4X*.N',3X.'L')N3=1N3MAX=18-(M*2)

DO 51 J=IIMIN,IIMAX44 CALL SCAN(J.X2(J).Y2(J).xl Y, N.I,IMAX)

IF(N) 46.4464747 L=l52 WRITE(6.53) OMEGA(I),XI(I),YI(I),X2(J),Y2(J).40MEGA(J).N·L

53 FORMAT(3X·Ff.1' /2E6 Pl'.3X,4(1PE10.3,3X),3J'F6.1,'/256 PI',2X,12,12X.12.' UNSTABLE')GO TO 56

46 IF(X2(J)-X1(I))7.71.17272 L=-l

WRITE(6,53) OMEGA(I)X1(I).YI(I) ·X2(J).Y2(J),40MEGA(J).N·L

GO TO 5671 L=O49 WR-ITE(6,48) OMEGA(I)·Xl(I)Y14I )·X2(J).Y2(J),40MEGA(J)·N·L

48 FORMAT( 3X.Ff.1. · I'/2E6 PI ' ,3X,4(1PE10.3 ,3 X) , ))F6.1,'/256 PI' 2X,12,12X.12,' STABLE')

56 IF(N3-N3MAX)54·f4,6464 N3=0

wRITE(6·1 )IFILEeM·XK.TAUI.(B(MM),MM=I,5)t(T(IV~)eMM=1·5)WRITE(6.40)WRITE(6.20)WRITE(6.62)

59

,N IV G LEVEL 20 MAIN

DATE = 7221 16/08/ 18

WRITE(6.55)54 N 3=N 3+151 CONTINUE

C PREPARING THE CALCOMP DIAGRAMI IM IN= IIM INID=IIMAX--I IMINDO 50 1=1 IDXI( 1)=XI( I IMIN)Yt(I)=YI(IIMIN)IIMIN=I IMIN+I

50 CONTINUECALL SATCHE(XI YtI. X2 ,Y21ID JM AX,BUFFERt'4,XKTAUI ,NFILEo,T, llIMINIIMAX)

888 CONTINUECALL PLOT(O0.,. 99 9g)

999 STOPEND

60

AN IV G LEVEL 20 MAIN

DATE = 72214 16/08/18

SUBROUTINE SCAN(I . XI .YI *X2.Y2 .N.J (KJMAX)

DIMENSION X2(1) Y2( )C. SUBROUTINE USED TO FIND POINTS OF INTERSECTION OFC TWO CURVES.

DMIN=999.DO 3 J=1.JMAXD=SORT(( XI-X2(J))*2+( Y-Y2(J))*. 2)IF(D-DMIN)2,2.3

2 DMIN=DJX=J

3 CONTINUEIF(DMIN-O.025) , 7

7 N=ORETURN

6 N=1RETURNEND

61

AN IV G LEVEL 20 SCAN

DATE = 72214 16/08/18

SUBROUTINE SATCHE (XI eYI.X2*Y2J* IJtBUFFER,M,XK,TAUINFILE*8,IT IIIMINIIMAX)DIMENSION XI(I) YI,(l).X2(1) Y2(1 )BUFFER(II)B(I).T(I)XL=9.YL=7.

CALL PLOT (.5..5.23)CALL SCALE(X2,XL,1 ,1)XI(J+I)=X2(1+1)Xt(J+2)=X2(1+2)CALL' SCALE(Y2.YL 1 ,1)YI(J+I)=Y2(1+1)YI(J+2)=Y2(I+2)

C DETERMINING ORIGIN OF AXES FOR X2.Y2(LOCATING HEIGHT OF X-AXIS)XSUBX=O.IF(Y2(I+1)--0) 1.2,2

2 XSUBY=O.0GO TO 5

1 D=Y2(1+1)+YL*Y2(I+2)IF (D-O.) 3,3,4

4 XSUBY=-Y2( I+1)/Y2( 1+2)GO TO 5

3 XSUBY=YLCCC LOCATING WIDTH OF Y-AXIS

5 YSUBY=O.IF(X2(I+1)-O.)e,7.7

7 YSUBX=O.GO TO 10

6 D=X2(I+1)+XL*X2(I+2)IF(D-0.)8. 811

11 YSUSX=-X2( I+1)/X2( 1+2)GO TO 10

-8 YSUBX=XL10 CALL AXIS(kSUBX.XSUBY,'

IXLO.tX 2(I+0X2( 2(1+2))CALL AXIS( YSUBX.YSLBY.'

1Y2(1+1).Y2(1+2))CALL LIN;E( X2Y21I.1,0,0)CALL LINE(XIY1,J,1,+3.03)IID1IIMAX-IIIMIN

DO 12 JJ=I.IIDX2(JJ)=X2( IIIMIN)Y2(JJ)=Y2( IIIMIN)I IIMIN=IIIMIN+I

12 CONTINUEX2(JJ+ 1 )=X2( I+1)X2(JJ+2)=X2(I+2)Y2(JJ+I)=Y2(1+1)Y2(JJ+2)=Y2(1+2)CALL LINE( X2,Y2.JJ .1.--3.-11)CALL PLOT(YSUBX+O. 20 .YL.23)CALL PLOT(4.8,0.·22)CALL PLOT( 0..-1I.l22)CALL PLOT(-4'.8.0.,22)

REAL F1 F2'.-32.

IMAG Fi ,F2'25,YL,90 .,

62

AN IV G LEVEL 20 SATCHE

DATE = 72214 16/08/18

CALL PLOT(0.,1.1.22!

CALL SYMBOL(0.15.-0.25,0.1522HSATCHE DIAGRAM ,0..22)CALL NUMBER ( 16.- .4.0. 1XK0O. 2)CALL NUMBER(2.7.-0.4.'0.1,TAUI 0. 2)CALL SYMBOL (2.8.-0.25,0. 10,17H-ELAY(S) PAOBLEM,O.*17)CALL NUMBER(2.700,-0.25,0.10OFLOAT(M) O.·-I)CALL SYMBOL(0.15- 0.4.0.1 30HCON ST ANTS: K= ,TAUI= .9.30)CALL SYMBOL(0.15,-0.55,0.10,43HTHE EXPONENTS- FC THE PURE TIME DEL

lAYS ARE:.0.,43)CALL SYMBOL(0. 15,-0.85,0.1,45HTHE-FEECBACK :OEFFICIENTS FOR THE DE2LAYS ARE:,0.e45)XN=O00014 NN=1.5CALLNUMBER(0.15+XN,-0.7,0.1,T(NN),O.' 1)XN=XN+O 8

14 CONTINUEXN=O

DO 15 NN=I.ECALL NUMBER(O.I5+XN,-I.,O .IB(NN).O.,I)XN=XN+0 .8

15 CONTINUECALL SYMBOL(1.2.-6.7*0.1.27HCALCCMP CCMPUTI-R-DRAWN PLOT.O0.27)CALL SYMBOL (1.2e-. 85,0. 137HPRCGRAM-PLOT DESIGN : JACQUES PRES

1S.0-.37)CALL SYMBOL (1'.2,-7.00.0.1,12HDATE: 8/1/72.0.,12)

CALL PLOT(-4.0,-0.2.3)CALL PLOT(-3.7C,-0.2.2)CALL SYMBOL( ~3.659, -0.25,0.I10C27HF(EXP(S)):CURVE WITH DELAYS,

10.·27)CALL SYMBOL(-4.,-0.65,0.1,30H++++ F(S):CURE WITH NO DELAYSO.,30)CALL SYMBOL(-4.,.-1.0S.0.1.37H**** PART 'OF (EXF(S)) CORRESPONDING

1,0..37)CALL SYMBOL(-3.65,-1.20.0.1.28HTC OMEGA OF F(S) BCUNDEO BY .0.,28)CALL SYMBOL(-3.65.--1t35.0. 128HITS OUTER I4TERSECTIONS WITH.0-.28)

CALL SYMBOL(-3.65.'-1.50O.1,9HF(EXP(S)) 0.-I)CALL PLOT(10.--7.5.-23)RETURN

END

63

AN IV G LEVEL 20 SATCHE

PRECEDING PAGE BLANK NOT wT M1ME

APPENDIX F

THE MATHEMATICAL FUNCTIONS USED IN THE CSMP SIMULATION(SEE FIG. 17)

Computer Notation

A. Y = STEP (P)(STEP FUNCTION)

Mathematical Notation

Y=0 t<PY-1 t2P

B. Y = REALPL (IC, P. X) PY= Y= X

Y(O) = IC ( 1

(SIMPLE LAG)

C. Y = INTGRL (IC, X) Y = X dt + IC

Y (0) = IC

(INTEGRAL)

D. Y= DELAY (N, P, X)

P = DELAY TIME

N = NUMBER OF POINTSSAMPLED IN INTERVAL P(INTEGER CONSTANT)

DEAD TIME (DELAY)

Y(t) = X(t - P) t 2 P

Y= t<P

(e- sp)

Preceding page blank -

65

BLOCK

dxr INPUT FOR OUTPUT(a rate) A L Jx(t)dt EXAMPLE

fdt

Figure la. The mathematical box as a component of a system.

ENVIRONMENT

/j- DECISION-MAKING FORWARDFLOWSECTOR I

REFERENCEI(INPUT-FEEDBACK I CONTROLLEDCONTROLLING · PROCESS OUTPUTAEDAK VARIABLES WITH DELAYS

FEEDBACK FEEDBACK FLOW ELEMENTS

Figure lb. I A generalized feedback control system.

66

AN INTEGRATOR

INPUTAl

1TS+1

1st ORDERTIME LAG

+1

T, S2+T S+1

2nd ORDERTIME LAG

0

¢0

CDELAY

OUTPUT

A

OUTPUT

A____ _ _ _

OUTPUT

O-tT-

)

Figure 2. Various time lag and puie time delay transformation functions.

67

0t

INPUTA

A

t

0 t

INPUT

t

0 t t

--------- O.-

1~~~~~~~~ .1

.I

INPIIT

x1 x 1 x x x el

x-F(s) ------ P-c00- em~ ~ 00--c X6(s)

· ~ ~-~ r+la~. / ,,

1/s = INTEGRATOR

e-Stl= PURE TIME DELAY

OUTPUT

K,a = MULTIPLICATIVE CONSTANTS

1 =TIME LAGTS+1

Figure 3o. The network graph for 6 problem with one pure time delay.

F(s) X6 (S)_- -H (s) - O- D(s) --- 10O

-G(s) ~ SlhORAPL

ix6F-- -cHD - 0

/

Figure 3b. The algebraic diagram for Figure 3a.

68

INPUT

Figure 4a. A trivial extension to two pure time delays.

INPUT

Figure 4b. The reduction of F'igure 4a to two single delays.

69

OUTPUT

SINGLE FEEDBACKJUNCTION \

INPUT 1- 00H1 D1 H2 D2INPUT be % -- 0DI b OUTPUT

Figure 4c. The non-trivial extension to two plure time delays.

SINGLE FEEDBACKJUNCTION, ,.

70

Figure 4d. The general non-trivial extension to N-pure time delays.

V)

IJx4;.0

-Vor-0)l---c~

71

C/

ri2

iV

C

6 ev)-PLANE

L, Xo ) s -PLANE E

Figure 6a. The conformal mapping fro'm the s plone onto the F(s, eS)-plane.

(

(SPECIALIZED s-PLANE)

ImF(s,es )

ReF(s,es )

Figure 6b. The Nyquist plot for F(s, es) =s + e - ' =0.

72

W =-+ o

=- 00

(A,B are moving points)

F2 (es): CURVE WITH DELAYS

-.-.-.-- FF1 (s): CURVE WITH NO DELAYS

*(an area is enclosed if it lies to theright of the arrow on the locus)

---- RESULTANT VECTOR AB

>XX AREA ENCLOSED BY F1

"::Z" AREA ENCLOSED BY F2

Figure 6c. The Satche diagram.

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I-oUJ

I

-F (EXP(S3)tCURVE WITH DELAYS

+**+ FI(SICURVE WITH NO DELATS

ANNN PART OF F(EXP(SJI CORRESPONDINGTO OMEGA OF F(S) 80UNDED BYITS OUTER INtERSECTIONS WITHFa(EXP (S)

o

SRTCHE IRGRRM 2-DELART(S) PROBLEMCONSTANTSi K= 0.45 .TAUI= 1.00THE EXPONENTS FOR THE PURE TIME DELAYRS AREs1.0 2.0 0.0 0.0 0.0

o THE FEEDBACK COEFFICIENTS FOR THE OELATS AREsG_. 11.0 1.0 0.0 0.0 0.0

REALF , F2

CRLCOMP COMPUTER-ORAWN PLOTPROGRAM-PLOT OESIGN s JACOUES PRESSORTE, 8/1/72

Figure 13. First Satche diagram for a case with two delays.

89

-I

-F,(EXP(Sll)CURVE WITH DELRTS

**+* Fe(S sCURVE WITH NO DELAYS

**-u PRRT OF F,(EXP(S)) CORRESPONTO OHEGR OF F(S) BOUNDED BYITS OUTER INTERSECTIONS WITHFz(EXP (S )

SATCHE DIRGRRM 2-OELAY(TIS) PROBLEMCONSTANTS, K= 1.00 .TRUI= 1.00THE EXPONENTS FOR THE PURE TIHE DELAYS AREs1.0 2.0 0.0 0.0 0.0THE FEEDBACK COEFFICIENTS FOR THE DELAYS AREi1.0 1.0 0.0 0.0 0.0

2.50

CALCOHP COHPUTER-DRAWN PLOTPROGRAMR-PLOT DESIGN s JACQUES PRESSDATEs 8/1/72

Figure 14. Second Satche diagram for a case with two delays.

90

O

.,;

O

-F,I(EXP (Sl,CURVE WITH DELAYS SATCHE DIAGRAM 2-DELAYTS) PROBLEMCONSTANTSt K= 0.25 .TAUI 1.00THE EXPONENTS FOR THE PURE TIHE DELAYS AREs

++++ F(S),CURVE WITH NO DELAYS 1.0 2.0 0.0 0.0 0.0Xo THE FEEDBACK COEFFICIENTS FOR THE DELAYS ARE,

wmmm PART OF F&(EXPIS)) CORRESPONOING- 1.0 1.0 0.0 0.0 0.0TO OMEGA OF 6FS) BOUNDED BYITS OUTER INTERSECTIONS WITHF (EXP (S) !

o0

CALCOMP COHPUTER-DRRNN PLOTPROGRAM-PLOT DESIGN ; JACQUES PRESSDATEs 8/1/72

Figure 15. Third Satche diagram for a case with two delays.

\ 91

- F(EXP (S) CURVE WITH DELAYTS

+*. F,(S)sCURVE WITH NO DELAYS

wNN. PART OF F,(EXP(S)) CORRESPONDINGTO OMEGA OF F(S) BOUNDED BTITS OUTER INTERSECTIQ#.%l-TW...Fj(EXP (5)) r.\",

SATCHE DIAGRAM 2-DELAT(S) PROBLEMCONSTANTSI K= 0.35 ,TAUI= 1.00THE EXPONENTS FOR THE PURE TIME DELAYRTS ARE1.0 2.0 0.0 0.0 0.0

O THE FEEDBACK COEFFICIENTS FOR THE DELATS ARE:1.0 1.0 0.0 0.0 0.0

0

00 o020 0.10 0o 0.80 1.00oo' o; 20 0. 0 0.60 1 o.a o L oo

Figure 16. Fourth Satche diagram for a case with two delays.

92

****CONTlINUnr,S SYSTFM MnOELI N' Pi- CGDA4***e

*** VFRSY.O) 1.3 ***

LABEL tTMF DFLAY MnDFL UNIT ST'F INPUTCfrNSTAsT F1=1 . ,2=1 .DAQAMETFP XK=0.45· TAIJI =l1 · .*

T= I . T=2 .DYNAM T'X 1= STF3 ( O )

X 2=-- X 1- 1 X6-2*t X7X 3= XK* X 2X4=QEA.L L( 0., TAU .X3 X F= IN TGRL ( 0 . X4)Xf=O=FL Y( 10. T I X5)X7 =DELAY( 10 T2 :X6)T IE4P ')ELT=0. 05,FINT[M=20. ·PPR)FL=1.0 ·O()Tr)FL= 1. MT HOD ADAMSPRTPLT XKChln

PAPAMFTFP XK=O.25 (PROGRAM IS RERUN AT THIS STEP)FNDPAPAMcTFO X=O.*35 (PROGRAM IS RERUN AT THIS STEP)FNDSTOP

OUTPUT VAPTARLE SEQUF4CFX6 X7 Xl XP X3 Z7ZOC .3 xa

.JTDUTS TNPJTS APRAMS INTFGS + MFM qLKS F7 r Tn DN oATA C );12(500) 3A(1400) 10(400) 2+ 2= 4( 300) (50 C) 1

FNDJOR

NOTATION:XK= K (AS IN TEXT)REALPL, INTGRL, DELAY, STEP = FUNCTIONS DESCRIBED IN APPENDIX F.METHOD ADAMS = INTEGRATION METHOD USED. (SEE REF. 52)TIMER = DESCRIBES THE SELECTED TIMING SEQUENCE.

Figure 17. Continuous System Modeling Program (simulation) for the case of two delays.

93

TIME DELAY MD0EL UN4T STEP INPUT

-1,X7

0.00.0

0.00.01 .6572F-015. 0530E-018.5712E-01

1.1128E 001.184RE 001.0273E 006.745SE-012.3200E-01

-1.5673E-01-3.5369E -01-2.7464E-017.7014E-026.0409E-011.1408F 001.5039E 001.5538E 00

1.2464E 006.5637E-01

-3. 105E-02-6.0776E-01-8.4637F-01

-6. 4355E-01-3.0878E-02

8.1673E-011.6286F 002.1232E 002.1034E 001.5317E 00

5.5676E-01-5.2163F-01-1.3424E 00-1.6016E 00-1.1620E 00-1.1433F-011.2344E 002.4473E 003.0965F 002.9106E 00I.eP16F 00

2.89C0E-01-1.3669F 00-2.5256E 00-2.7461E 00

-1t.717? 00-1.081QE-OI

2.015,F 007.e02AF 00

MINIMUM

.2101E 01I

X' 7 VERSUS TIME

A GE I

MAX IMlJ

I1.0575SE 01I

---------------------------------------------------- 4-

----------------------------.

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

------------ -------------.

…------------- --------- .------------- ------------ ----.

-4------------.---------- ----------- …--------… ----- 4-------------------------- 4.

------------------------- 4.---------- …-------------4

------ …….'.. ......--- 4..

……---------- -…-…- - --- -- 4-- .

-------------------------------------- ---.--------- --

____ _------------ _

--------- --------- ----.--

….'…...-…-…--… . . -- …-- ----- ..

…__ _ _ _…_4 .__ _ _ _

Figure 18. Output to Figure 17 with K =0.45.

94

TIME0.0I.OOOOE 002.0000E 003.0000E 004.OOOOE 005.OOOOE 006.OOOOE 00

7.0000E 00

9.OOOOE 00I.OCOOE 011.1000E 011.,2000E 011.3000E 01

1.4000F 011.5000E 011.6000E 011.7000E 011.8000E 01

1.9000E 01

2.0000E 012.1000E 012.2000E 012.3000E Cl2.4000E 012.5000E 012.6000F 012.7000E 012.8000E 012.9000E 013.OOOOE 013.1000E 013.2000E 013.3000F 013.4000E 013.5000E 013.6000E 013.7000e 013.8000E 013.9000E 014.0000E 014.1000E 014.2000E 014.3000E 014.4000E 014.5000E 014.6000E 014.7000e 014.8000E 014.9000E 015.0000E 01

L

I

TIME DELAY MODEL UNIT STEP INPUT

MINIMUM

-1.2101E 01

X74.6103E 004.076OE 002.2703E 00

-2.S934E-01-2.o160E 00-4.411 7 E 00-4.4594E 00-2.8095E 00

1.1811E-OI3.4289E 00

6.01S8E 006.Q399E 00

5.7295F 00

2.6204E 00-1.4840E 00-5.2611E 00-7.3869E 00

-6.9893E 00

-3.9877E 008.1219E-01

5.9184E 009.6052E 00

1.0489E 01

q..-0148E 002.7531F 00

-3.7349E 00-9.3286E 00-1.2032F 01-1.0667E 01-5.3538E 00

X7 F RSUS TIME MAX I'UM

1.0575E 01

I

…---------------- -

….....-… -.--- .... 4-4

…------------------ ----- ……------ -------- -- ----------- --- _------- - -s

----------------..-----

-------------.…----------

------------- .

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

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

…---…----4-

_- - -- - -_ _____ _ ___

Figure 18. (Continued)

95

7

TIME5.1000E 01

5.2000E 015.3000E 015.4000E 01

5.5000E 01

5.6000E 015.7000E 015.8000E 015.9000E 016.0000E 016.1000E 016.2000E 016.3000E 016.4000E 016.5000E 016.6000E 016.7000E 016.8000E 016.9000E 017.OOOOE 01

7.1000E 01?.2003F 017.3000E 017.4000E 017.5000E 017.6000E 017.700F 017.8000E 017.9000F 018.OOOOE 01

OA GE 2

TIME DELAY M3DEL UNIT STEP INPUT

MINIMUM0.0

X70.0

0.00.00.0

9.2065E-022.8214E-01

4.9230E-01

6.7s32E-01

8.0689E-01

8.5620E-018,2467E-01

7.2930E-015.9470E-01

4. 623E-01

3.4264E-01

2.7509E-012.6257E-013,.0153E-013,7794E-01

4.7144E- 015.6039E-01

6.2655E-01

66,5355E-01

6.5350E-016.1666E- 015.5939E-01

4.9607E-01

4.4074E-014.0416E-013.9192E-014.0385E-01

4.3470E-014.7584'- 015. 1741E-01

5.5062E-015.6940E-01

5.7142E-01

5.5816E-01

5.341 3E-05.0563E-01

4.7916F-01

4.6007E-014,515'F-01

4.5423E-01

4.6625E-014.e402E-01

5.0315E-015.1945E-015.2981F- 01

5.3271E-01

5.2834E-01

X7 %ERSUS TIM- MAX IMUM8.5620E-01 I

I

…--------------- … ----_ _ _ _ _ _ _ _ _ _ _ __%_... _ %

---.------- -------- ----- ........----------------- ------- '

---------- …--------------------.--------

..........-- …............ --+…-4---------- .----------- ---------

_--_________-_-_-_-- - _____ ____- --- ___- -- ___ _____

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

.----------------------

-----------.----- ,

------------------------- 4

------------------------ …-----4

-------------------------------- …

---------------------- …--…

------------------------- 4.

------------------------ 4--

---------------------------- 4

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

---------- …----------…--------…

…-……_________.__ 4___ __ __ _ __ __ __ _

…__ _ __ _ _ _ __ _ _ _ 4.__ _ __ __%

…___ ___ ___ _…_____ _ _ ___ __%

_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _

..........................

_ _ _ _ _ _ _ _ _ _ _

Figure.19. Output to Figure 17 with K = 0.25.

96

TIME0 .0

I.000OF 002.0000E 003.OOOOF 00

4.0000E 00

5.0000E 006.0000E 007.0000F 00

9.00OO00 009.0000E 00

1.0000E 01I.100OE 011.2000E 01

1.3000E 01

1.4000E 011.5000E 011.6000E 01

1.7000E 011.8000E 01

1.900OE 012.0000E 01

2.1000E 012.2000E 01

2.3000E 012.4000E 01

2.5000E 01

2.6000E 012.7000E 012.8000 012.9000E 013.0000E 01

3.1000F 013.2000E 013.3000E 013.4000E 013.5000F 01

3.6000E 01

3.7000E 013.8000E 013.900OE 01

4.0000E 01

4.1000E 014.2000F 01

4.3000E 014.4000F 01

4.5000E 01

4.6000E 014.7000E 014.8000E 01

4.9000E 01

5.0000F 01

De GE

I

TIME DELAY M30EL UNIT STEP INPUT

MINIMUM X7 .. RSUS TIM- MAXIMUM

0.0 A .5620E-01

TIME X7 I

5.10OOE 01 5.1857E-01 -----------------------------.

5.2000E 01 5.0594E-01 ------------------------------

5.3000E O1 4.9349E-01 ---------------------------+

5.4000E 01 4.8382E-01 ----------------------------4

5.5000E 01 4.7?65E-01 ---------------------------5.6000E 01 4.7857E-01 -

5.7000E 01 4.e301E-01 ----------------------------+

5.8000E 01 4.9054E-01 ----------- ----------------5.9000E 01 4.9920E-01 ----------6.0000F 01 5.0706E-01 -----------------------------.

6.1000E 01 5.1254E-01 ---------------------------- +

6.2000E 01 5.1475E-01 --------------------'----------

6.3000F 01 5.1361E-01 ----------------------------- +

6.4000E 01 5.0972F-01 ------------------------------6.5000E 01 5.0423E-01 -------------------------------6.6000E 01 4.9846E-01 ----------------------------

6.7000E 01 4.9367E-01 ---------------------------+

6.8000E 01 4.9076E-01 ---------------------- ---- +6.9000F 01 4.9012E-01 -- 4

7.0000E 01 4.9164E-01 ----------… -----------_----+

7.100OE 01 4.9475E-01 ------------------------ ---

7.2000F 01 4.9861F-01 ----------------------------

7.3000E 01 5.0233E-01 ------------ ------------- +

7.4000E 01 5.0514E-01 -------------

7.5000E 01 5.0655E-01 ----------------------------- +

7.6000E 01 5.0641F-01 ---------------------------+7.7000E 01 5.0494E-01 ----------------7.800E 01 5.0259E-01 -----------------------------

7.9000E 01 4.9997E-01 --------------- +

5.0000E 01 4.9764E-01 ----------------------------

Figure 19. (Continued)

97

3A GE 2

TIM' DELAY M)DEL UNIT STEP INPUT

X70.0

0.00.0

. 01.2889E- 013.9400E- 016.7793E-019.0743E-01

1.0245F 009.9422E-01

9.2593E-015. 674E-012.9262E-017.6824A-02

-1.9587E-022.9534E-022.0957E-01

4.6950o-017.3631F-O19.3556F-01

1.0121F 009.4527e-01

7.5469E-014.9426E-012.3704E-015.4775E-02

-2.1728E-038.1304E-02

2.9105E-015.4062E-01

7.872sE-01

9.5222E-019.9003E-018.9092F-01

6. 339e-01

4.2599E-011.O084E-014.3390O-022.4215E-021 3787--0o3.5182E-016.05758-018.2b65F-O1

9.5846E-01

9.5959F-018.3252E-01

6.1 34E- 013.6416E-01

1.5428E-01

4.2132E-02

5.8395F-02

MINIMUM-2.1531E-02

X7 ¥EPSUS TIME 4AXIMUM

1.0306E 00

I-4

-+

-4

------- 1

_---------------… +

-- -------- - --------- --- -- ------ -+

---------- - ----------- -------_ -_-------------

------------------------ _-----

-----------

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

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

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

_---------------------------------- -

_------------- --- _-

…------------

---_ ---------- -

----------------------- ----- ___-- +

_--------- -----------

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

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

- - - - - - - - - --_-_- - - --_- ---_-- - - - -- - - - --- - - - - --_-...--- _. ----.------ --.--------------- _ __

--_ ---- _--- ----------- _------

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

-.. _ _.. .

---…--

---_ -_

_-----4-

Figure 20. Output to Figure 17 with K =0.35.

98

TIME0.01.OOOOE 002.OOOOE 00

3.OOOOE 00

4.OOOOE 005.0000F 006.0000E 007.0000O 009.0000F 00

.0000E 00

.0000E 011.1000E 011.2000E 01

1.3000E 011.4000E 011.5000E 01

1.6000E 01

1.7000E 011.O0E 011.9000E 012.0000E 012.1000E 012.2000F 012.3000E 012.4000E 012.5000E 012.6000F 012.7000E 012.8000E 012 .9000E 013.0000E 01

3.1000E 013.2000E 013.3000E 013.4000E 013.5000E 013.6000E 013.7000E 013.8000E 013.900F 014.0000E 014.1000E 014.2000E 014.3000E 014.4000E 014.5000E 014.6000E 014.7000E 014.8000E 014.9000E 015.00001 01

DAGE

I

TIME DELAY M3DEL UNIT STEP !NPUT

MIN-2.15

X71.C773F- 01

4.2051- 016.6408E-018.6037E-019.5491 -01

9.2198E-017.7156E-01

5.4632F-01

3.0 55F-011.2740E-015.0297E-029.9109E-02

2.5944E-01

4.8587E-017.1483F-01

8.8245E-01

9.4235E-019.7849E-01

7.0946E-01

4.8305E-01

2.6278E-011. 100F-016.7037E-02

1.4506E-013.2159E-01

5.4679E-017.5754E-01

8.9514E-01

9.2167F-01

8.3042F-01

NIMUM

531E-02!

X7 . RSUS TIMI MAXIMUMt1.0306E 00

I---------- …

--------------------- …

----

............------------ ----

------------------------ -------- -----. -- -

…+........ _. -__ _ _ …............+'1-

…...…._ _

…...+4 -

…+ _

…+..... . . . . . .. .. _

----....................... __

…__ _ _ _ _ _ _ _ _ _ _ _ _

…+ _ _ _ _ _ _ _

…-- --_ _- -_

…__- - -+

…_ _…+

…4 _ _ _ _ _

Figure 20. (Continued).

99

T IME5.1000E 01

5.2000E 015.3000E 015.4000E 01

5.5000E 01

5.6000E 015.7000E 01

5.8000E 01

5.9000E 016.0000E 01

6.1000E 01

6.2000E 016.3000F 01

6.4000E 016.5000E 016.6000E 01

6.7000E 016.9000E 016.9000E 01

7.0000E 017.1000E 017.2000E 01

7.3000E 017.4000E 017.'i00O 01

7..0005E 01

7.7000E 017.R000E 01

7.9000E 019.OOOOE 01

OA GE 2

0(Uj

tm01-F,(EXP S)IsCURVE WITH DELAYS

++++ FISItCURVE WITH NO OELAYS

-NNH PART OF F(EXP(S)) CORRESPONDING -TO OMEGA OF F(S) BOUNOED BY ITS OUTER INTERSECTIONS WITHF,zEXP (SI .

SATCHE DIAGRAM 1-DELAT(S) PROBLEMCONSTANTSt K= 2.00 TAUI= 0.00THE EXPONENTS FOR THE PURE TINE OELAYS RREi1.0 0.0 0.0 0.0 0.0THE FEEDBACK COEFFICIENTS FOR THE DELAYS AREt1.0 0.0 0.0 0.0 0.0

!.50

CALCOMP COHPUTER-ORAWN PLOTPROGRAN-PLOT DESIGN t JACQUES PRESSOATEl 8/1/72

Figure 21. First Satche diagram for a single delay case.

100

o

-F.(EXP(S)IsCURVE WITH DELATS

+*t+ FIS!:CURVE WITH NO DELATS0

rm#n PART OF F&(EXP(SI) CORRESPONDING -TO OMEGR OF F(SI BOUNDED BTITS OUTER INTERSECTIONS WITHF,(EXP IS) )

SATCHE OIAGRRM I-DELAY(S) PROBLEMCONSTANTSs K= 1.00 TAUI= 0.00THE EXPONENTS FOR THE PURE TIRE DELAYS ARE.1.0 0.0 0.0 0.0 0.0THE FEEDBACK COEFFICIENTS FOR THE DELAYS AREs1.0 0.0 0.0 0.0 0.0

CALCOMP COMPUTER-DRAWN PLOTPROGRAM-PLOT DESIGN t JACQUES PRESSORATEs 8/1/72

Figure 22. Second Satche diagram for a single delay case.

101

0

-F,(EXP(S)) CURVE WITH DELARTS

**+ F(S) lCURVE WITH NO DELRTS

.... PART OF F(EXP(SI) CORRESPONOING*-TO OMEGA O F,(S) BOUNDED BYITS OUTER INfERSECTIONS WITHF, (EXP (S) !

u.

SATCHE DIAGRRM I-DELAY(SI PROBLEMCONSTRNTS K= 1.57 .TARUI= 0.00THE EXPONENTS FOR THE PURE TIME OELARTS AREs1.0 0.0 0.0 0.0 0.0THE FEEDBACK COEFFICIENTS FOR THE DELAYS AREs1.0 0.0 0.0 0.0 0.0

2.00

CALCOMP COPUTER-ORRAWN PLOTPROGRAR-PLOT DESIGN * JRCQUES PRESSORATE 8/1/72

Figure 23. Third Satche diagram for a single delay case.

102

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*++ F,(S)sCURVE WITH NO DELAYS

uNNN PART IF F,(EXP(S)) CORRESPONDINGTO OMEGA OF F CS) BOUNDEO BTITS OUTER INTERSECTIONS WITHFz(EXP (SI))

SATCHE DIAGRARM 1-DELAY(S) PROBLEMCONSTANTS, K= 1.00 ,TAUI= 1.50THE EXPONENTS FOR THE PURE TIME DELAYS ARE,1.0 0.0 0.0 0.0 0.0THE FEEDBACK COEFFICIENTS FOR THE DELAYS ARE81.0 0.0 0.0 0.0 0.0

CALCOMP COMPUTER-DRAWN PLOTPROGRAM-PLOT DESIGN , JACQUES PRESSDATEt 8/1/72

Figure 24. Fourth Satche diagram for a single delay case.

103

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-F,(EXP(S)hICURVE WITH DELAYS SATCHE OIRGRRM L-DELAY(S) PROBLEMCONSTANTS K= 0.50 ,TAUI= 1.00THE EXPONENTS FOR THE PURE TIRE DELAYS AREt

++** F,(S)3CURVE WITH NO DELAYS 2.0 0.0 0.0 0.0 0.0o THE FEEDBACK COEFFICIENTS FOR THE DELAYS AREsto

HWII PART OF F,(EXP(SI) CORRESPONDING 1.0 0.0 0.0 0.0 0.0TO OMEGA OF F(S) BOUNDED BYITS OUTER INtERSECTIONS WITHFz(EXP ISI )

00

CALCOMP COMPUTER-ORRWN PLOTPROGRAM-PLOT DESIGN i JRCQUES PRESSDATEr 8/1/72

Figure 25. Fifth Satche diagram for a single delay case.

104

-F,(EXP (S))CURVE WITH DELAYS

*++, F,(S)tCURVE WITH NO DELAYTS

SATCHE DIAGRRM S-OELAY(S) PROBLEMCONSTRNTS. K- 0.17,TAUID 1.00THE EXPONENTS FOR THE PURE TIRE DELARTS ARE1.0 2.0 3.0 0.0 0.0THE FEEOBACK COEFFICIENTS FOR THE DELAYS AREI1.0 1.0 1.0 0.0 0.0

00 0.40 0.60 0.80REI Fl, Ft

CALCOMP COHPUTER-DRAHN PLOTPROGRRA-PLOT OESIGN t JACOUES PRESSOATE. 8/1/72

Figure 26. First Satche diagram for a case with three delays.

105

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**.. (3S)iCURVE WITH NO DELAYS

0.;0-

RT OF IEXPSI CRRESPNING0mamm PART OF F.IEXP(S)) CORRESPONDINGO-

TO OMEGA OF F(SI BOUNDED 8YITS OUTER INTERSECTIONS WITHFIEXP(SI)

SRTCHE OIRGRRM S-DELAY(S) PROBLEMCONSTANTS1 K- 0.20 .TAUI- 1.75THE EXPONENTS FOR THE PURE TIME DELATYS AREt1.0 1.0 2.0 0.0 0.0THE FEEDBACK COEFFICIENTS FOR THE DELAYS AREs1.0 1.0 1.0 0.0 0.0

;00 _ 0.20 _ 0.40 0.0 0.80

CALCOMP COMPUTER-ORAIN PLOTPROGRAM-PLOT DESIGN I JACQUES PRESSORATE 8/1/72

Figure 27. Second Satche diagram for a case with three delays.

106

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o-FI(EXP IS))sCURVE NITH DELAYS

+**+* F(S)sCURVE WITH NO DELAYS

*WNN PART OF F(EXP(SI) CORRESPONDINGc-TO OMEG OF F (S) BOUNOEO BYITS OUTER INTERSECTIONS HITHFJIEXP (S)1

SRTCHE ODIRGRRM 3-OELRAY (S' PROBLEMCONSTRNTS7 K= 0.15 .TAUI= 2.00THE EXPONENTS FOR THE PURE TIME DELAYS AREs1.0 2.0 3.0 0.0 0.0THE FEEOBACK COEFFICIENTS FOR THE DELAYS AREs1.0 1.0 1.0 0.0 0.0

0.aO

CRLCOMP COMPUTER-ORAWN PLOTPROGRAM-PLOT DESIGN I JARCUES PRESSDATEt 8/1/72

Figure 28. Third Satche diagram for a case with three delays.

107

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*..+ F(S)1CURVE WITH NO OELAYS

mmuN PART OF F(EXP(S)I CORRESPONDING -TO OMEGA OF F(S) B0UNDOE BY ITS OUTER INtERSECTIONS WITHF, (EXP (SI)

SRTCHE OIRGRRM U-DELAY(S) PROBLEMCONSTANTS: K= 0.11 .TAUI= 1.00THE EXPONENTS FOR THE PURE TIME DELAYS AREt1.0 2.0 1.0 2.0 0.0THE FEEOBACK COEFFICIENTS FOR THE DELAYS AREi1.0 1.0 1.0 1.0 0.0

i 00 0 0.2 Z 0.30 0 40

CALCOHP COHPUTER-DRAWN PLOTPROGRAM-PLOT DESIGN t JACQUES PRESSDATE: 8/1/72

Figure 29. Satche diagram for a case with four delays.

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Figure 31. Second Satche diagram for a case with five delays.

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