EPPO Bull. 9 (3) : 165-176 (1979)

Digital Simulation Languages : Problems and Possibilities1)

by M.S. ELZAS

Agricultural University, Wageningen (Netherlands)

ABSTRACT When faced with a model to be simulated on a computer, one is in

general faced with a series of problems, to which answers are quite often difficult to find. These problems include : 1) Can we construct a mathematical approximation of the model in such a way that the resulting formulation can be implemented as a computer for simulation purposes ? 2) Will the above mathematical formulation guarantee a result on an ideal (perfect) machine ? 3) Will the chosen method of simulation withstand the non-ideal characteristics of an actual machine ? 4) Will it be possible to compute results to the computing sequence that is created for the simulation? 5) How does one choose the right language that is to be the programming vehicle for the simulation ? 6) Can one describe every simulation problem in any of the languages for specific classes of problems ?

This paper tries to give an insight into the possible answers to these questions and endeavours to provide the reader with some basic knowledge about the possibilities available and the problems to look for when setting up simulation experiments by programming them with simulation languages for digital computers.

It has become clear that when systems are to be simulated on computers, mathematics play an important role both in the formulation stage, where the problem has to be made to fit a computing machine, and in the interpretation of the results that represent the behaviour of the simulated system. In the case of computer simulation of continuous systems, a specific subset of mathematical theory is the basis for further discussion of tools for continuous system simulation. The formal basis of continuity in a mathematical context is the theory of analytical functions, i.e. those functions, F, which fulfil the following condition :

lim F(z+&z) - F(z)

CAP7 ( z + ~ z ) - z

exists when 6z + 0 for every z in the interval under consideration and is independent of

the direction in which z is approached. This limit is usually denoted by - and called the

first derivative of F with respect to z. This definition is not only valid for the given example in one dimension, but is applicable in cases of higher dimensionality as well. For these analytical functions, a large amount of mathematical theory and tools have been developed in the past. These theories, in general, find rheir application in the field of integral and differential calculus.

1) Paper presented at the Joint EPPO/IOBC Conference on Systems Modelling in Modern

dF dz

Crop Protection, Paris, 12-14 October, 1976.

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Fortunately, apart from being important to the area of continuous system simulation, the principles of mathematical analysis have also had an important influence in formulating basic laws that describe the natural phenomena around us. For example, mathematically equally formulated laws are found for many phenomena in mechanics, electricity, chemistry, fluid dynamics, kinematics, thermodynamics, etc. Through pheno- menologic approaches, the same mathenutical formulae hold, by analogy, in less exact research areas, now called the soft sciences (economics, management science, behavioural sciences, etc.). Thus, many systems in these fields of science, which are dynamic (= time-dependent) in nature, can be formulated in the form of differential or integral equations. In general, these equations completely define the behaviour of the dependent variables of the system as a function of the independent variable (mostly time), once starting values and forcing iunctions have been defined.

Hence, the general problem confronting the researcher or engineer in this case is essentially that of finding the time dependence of the variables involved, given a set of differential or integral equations. So the equations, and therefore the problem, can be considered to be solved when all the dependent variables have been computed as functions of their independent variable(s) (e.g. time and space variables).

In approaching the problems of simulation of continuous systems on digital computers, we must discuss some basic aspects of computing machines, which govern the differences between discrete-sequential computing machines (digital computers) and their continuous-parallel counterparts (analog computers). This is especially necessary because continuous simulation practice started on the latter, while much of the work in this field is now done 011 the former type of computer. The main reason for using analog computers (also called differential analysers) in the past for continuous simulation tasks was that there are strong similarities in the mathematics that govern continuous phenomena and the basic operations that are implemented in analog computers.

The mathematical background for the results that will be presented in this paper can be found in ELZAS (1975).

When faced with a computational task to be implemented on a computer, one is, in general, faced with three basic problems :

1) Will the mathematical method that has been devised to solve the problem guarantee a result on an ideal machine (existence of the solution, convergence of the process to a unique solution) ?

2) Will the method be able to withstand the non-ideal characteristics of the machine (stability) ?

3) Will it be possible to compute results to the algorithm that has been created for the implementation of the mathematical method (computability in general, sortability if a sequential machine is used) ?

Unfortunately, the theory of computation has not progressed to the point where the answers to all the above questions can be given for every problem that has to be solved on a computer. On the other hand, many problems occurring in the field of continuous system simulation can mathematically be formulated in the form of ordinary differential or integral equations, which narrows the field for which one has to answer the above questions.

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Mathematical theory allows us to prove the existence and uniqueness of the solution of all ordinary differential equations, most integral equations and some partial differential equations. Moreover, the process of iteration is, in most cases, the base for problem solving on computers, and is, furthermore, a mathematically well- defined process. The conditions for which an iteration process leads to an unique solution (i.e. is convergent) are also well known. It will suffice to remark here that these conditions are stricter for a discrete digital computer than for its continuous mathema- tical counterpart or an analog computer. This is mainly due to the fact that, on the discrete machine, the amount of computational steps that can be carried out is always finite because of finite memory and finite time limitations.

Stability of computational processes is often very difficult to prove. I t was, however, shown by ELZAS (1975) that the stability of iteration processes can be esta- blished both for the discrete and the continuous case. Again, limits of stability are narrower on the digital computer than on its counterpart.

Computability is, perhaps, the most difficult matter of all. From extensive research by many eminent scientists, ic has been possible to find sufficient conditions for computa- bility on the different types of computer ; a '' necessary " condition has, however, not been established so far. This means that proof of computability can, in many cases, not be given, while a computer user can show that his problem has been solved by checking the results of his program. Also for this case, there is a subset of the computation processes for which computability can be proven.

These so-called recursive processes exist in a discrete and a continuous version, based, respectively, on the characteristics of digital and analog computers. In the last decade, programming languages have been developed which make it possible to compute the solution to problems stated in continuous (analog) computation algorithms, on digital computers. Through these languages, an extra condition for computability on digital computers has to be fulfilled : sortability. The concept of sortability, which is a prerequisite for computability on sequential machices, requires that all computing tasks stated in a set of algorithms for a simulation task can be sorted into a sequential list (linear sequence) of digital computer operations. So, in some cases, problem formulation in these languages will have to be modified to meet this requirement.

A simple example of the implications of digital computer characteristics on continuous simulation tasks will now be given. It has already been stated that the mathe- matical formulation of many continuous systems leads to differential and/or integral equations. That is why the integration operator is of key importance to the simulation of these systems. The digital computer does not possess operators (instructions) that perform integration directly. The built-in computational abilities of these machines are limited to strictly algebraic and logic operations, like addition, subtraction, multiplication and

division. So, to perform an integration F(x) = fxf(u)du (1) on a digital computer,

this operation has to be expressed in a sequence of algebraic and logical operations. This is achieved by approximating f(u) by a finite polynomial :

aJ

N

i=O f(u) = Z: aiui

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This allows us to calculate a s” f(u)du beforehand as :

N a s” ( X a1u’)du

i=O

or

This general formulation has led to series of integral operator approximations differing by the order (magnitude of N) of the polynomial and its nature (choice of the coefficients ai). Naturally, these differences lead also to differences in the accuracy of the approximated integration. With respect to the order of the polynomial, this is illustrated in figure 1.

The Euler formula is based on the assumption that Au can be found, such that for all uj E [as} (ui + 1 =I uj+Au), f(u) = f(q), where uj < u < uj+hu (2). This is an approximation of order zero. The trapezium rule is based on the approximation :

f(u]+Au) A f(uj) ujf(uj+Au) - !uj+Au)p(uj) for uj < < uj+Au (3). f(u) = Au Au

It should be noticed that Simpson’s rule (formula 131) makes use, for uj < u < uj+Au, of the value off at uj+Au. In many cases, where integration is a part of a simulation setup, F(x) out of formula (1) is needed to compute f(u). This means that the integral function has to be computed before f(uj+hu) can be known. For this purpose, a modification of Simpson’s rule is often used in which f(uj+Au) from formula (3) is replaced by f(uj), while f(uj), from the same formula, is replaced by f(ub-Au). The consequences of this approach are depicted in figure 2.

U I U

Fig. 1 Integration approximations. Area \ \ \ +% = Euler Area /// i \ \ \ = trapezium rule (Simpson)

Fig. 2 Extrapolated Simpson integration.

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A way to improve this apparently erratic behaviour of the integration formula is to re-use the results F(x) of the integration, yielding f(uj+Au) in an extra integration step. In this way, the first integration can be characterized as predictive, while the second step is corrective. Repetition of this procedure often converges to a reasonably accurate estimation of the integrated function. Naturally also, the use of better (higher order) polynomials will provide better behaviour of the integral, because they use more information about the past history of the function. One could, for example, use a truncated Taylor series around f(uj) to provide a better fit :

M 1 i=o I!

f(u) = z 7(u-uj)if’(uj) (4)

dif dui

where f’(uj) denotes (-) at uI. For the modified Simpson’s rule, M would be 1.

Note that, for higher M, more and higher derivates of f at uj have to be known. This can cause start-up problems, when all the necessary derivates of f are not known at the starting point a (e.g. boundary value problems). In this case, a predictive,’ corrective approach will again have to be used to approximate the missing higher derivatives, starting from educated guesses.

AS has already been touched upon briefly in this text, another important aspect is the parallelism inherent in most continuous systems. In other words, the phenomena that can be described as subsystems in a continuous system all act together, or are simultaneous. For example, notwithstanding the fact that we can describe the bicycle with reference to its different subsystems (the movement of the wheels, the steering mechanisms, the propulsion system and the actions of the bicycle driver), this same bicycle can only be studied as a total system by simultaneously taking into account all the separate subsystems and their interaction.

By simultaneity is meant that all phenomena in the system under study progress in a strictly synchronous fashion, even if one samples the system at infinitely small intervals of the independent variable. Parallel computing machines (at this moment analog computers) are the only devices which possess all these characteristics in conjunction with their problem-solving characteristics. However, this type of computer has, for various reasons, been less widely used than its sequential counterpart : the digital computer. This type of machine is an instrument that changes in discontinuous steps from one state to the next. Moreover, the digital computer today, in general, possesses only one computing unit in which all operations take place, one at a time.

This forces us to represent essentially parallel phenomena, as they occur in continuous systems, in sequential form for solution on the digital computer. Some of the fundamental problems caused by this approach have been discussed in the previous pages and are treated in a more rigorous way in ELZAS (1975).

Notwithstanding these problems, once the elementary rules were known for approximating parallel-continuous systems by discrete-sequential procedures, many, mostly large, systems were simulated using digital tools. To provide these tools to a larger community, using the inherent programming advantages of the digital computer, some computer specialists set themselves to the task of building computer languages for this purpose. The main intent of these languages was two-fold :

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a) Make the computer easily useable for the scientist and engineer interested in simulation, but inexperienced in numerical analyses.

b) Provide means to automatically transform models stated in terms of continuity and parallelism into sequential algorithms for the digital computer.

So, in a relatively short period of time, the following digital tools came into use : (interpretive digital comupter routine for simulating diffrential analyser operations) developed by Lesh and Curl, for an IBM 704, at the California Institute of Technology in 1959.

DYSAC (digitally simulated analog computer) developed by Hurley, Rideout and Skiles, for a CDC 1604, at the University of Wisconsin in 1961.

DAS (digital analog simulator) developed by Gaskil, Harris and McKnight, for an IBM 7090, at Martin Orlando Company in 1963.

MIDAS (modified integration DAS) developed by Harnett, Sanson and Warshawsky, for an ISM 7094, for the US Dept of Commerce in 1964.

FORBLOC (Fortran compiled block oriented simulation program) developed by Rideout, Skiles and WebLer, €or a CDC 3600, at the University of Wisconsin in 1964.

PACTOLUS (named after the river in Asia Minor in which the mythological, ambitious, king MIDAS is said to have come to his end by drowning) developed by Brennan, for an IBM 1620, at IBM - Pabo Alto in 1964. (digitaal integratie system voor de analoge rekenwijze) developed by Bos and Elzas, for a Telefunken TR4, at the Delft University of Technology in 1965. (an improved version of MIDAS that mimicked an analog computer) deve- loped by Peterson, Sanson and Warshawsky, for an IBM 7094, at the Wright Patterson Airforce Base in 1965. (digital simulittion language for the 7090) developed by Lyn and Wyman, for an IBM 7090. a t IBM in 1966.

DEPI

DISAR

MIMIC

DSL 70

This confusing proliferation formed the motivation for Simulation Councils (a professional organization devoted to the advancement of simulation) to set up a committee that, by the end o f 1967, came up with definitions and rules for Continuous System Simulation Languages (CSSL's) that were to be a basis for new languages that would be developed for this application field.

Since 1967, three languages have been based on these definitions, and have found a widespread use for programming continuous system simulation on digital computers. These languages are, in order of popularity :

- CSMP (Continuous System Modelling Program) available on IBM 360/370, DEC

- SLl (Simulation Language 1) available on SDS-machines that, unfortunately, are

system 10, some Burroughs and some CDC computers.

out of production now.

- CSSL I11 available on CDC-CYBER machines.

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This does not mean that other languages (not compatible with the CSSL defini- tions) do not exist today. In particular, languages derived from DSL-90 still enjoy a reasonable popularity (a.0. Block-CSMP for IBM 1130 and DEC-PDP 9 and 11, DSLH, developed by Wageningen University, for CDC 3000 and CYBER series computers).

How does one program a model into a CSSL-compatible language ? In the first place, the CSSL definitions cater for a specific organisation in the model description for the computer (fig. 3 ) .

INITIAL REGION [Defines initial state of the processes for every computational run)

I

DVNAMIC REGION [Contains the description of the main body of the model. Dynamic because all differential and integral equations appear in this region. In the real world, these equations describe the dynamics of the system under study. In this region also, proce- dural [or sequential) aspect of the model can be described in Fortran)

I I I

TERMINAL REGION [Controls end-state of processes - in the real time sequence - for New Run or Continue Conditions)

9. 3 Typical CSSL-program organization.

An example will be taken to illustrate the use of such a language. The technical system to be modelled consists of a simple servo mechanism that serves to bring a heavy body into the right position by rotation (e.g. a servo-controlled ships’ rudder). The system is illustrated in figure 4.

RUDDER

S E N S I N G P O S I T I O N ) I D E V I C E

A P O S I T I O N POWER E*(+D-+M) (MEASURED AMPLIFYING DIFFERENCE SUfigf lC- RUDDER

SIGNAL) , DEVICE . I

(RUDDER I DRIVE ” 1

CONTROL I

S I G N A L

a.e

I

3

Fig. 4 System under study.

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The mathematical equation for this system is

-- In this equation, IR denotes the moment of inertia of the rudder ; ID : idem of the drive. D(h,@) is the damping ratio of the system, which has been taken to be a function of angular position and rotational speed. @M is the measured angular rudder position. OD is the desired angular position and a is the power amplifier amplification coefficient.

If equation (5) is tramformed into an equation in the error signal E = Q ~ M

one obtains: d2E . de (IA+ID)--+ D(E,E& - ae = 0 dt2 dt (6).

de dt

4ssume that, in this equation, D is of the form K(1-(- )' -E "), and that a = -u

and b = - then (6) reduces to : K

IA+ID II+ID' d2E dE' de dt2 dt da --+ a( 1-( -)2-~2) -+ be = 0 (7)

Then (7) is a known type of differential equation : Van der Pol's equation.

In view of the ship's handling characteristics, we are interested in the angle error correction rate as a function of the angle error (in terms of non-linear differential equation theory, this is known as a phase-plot). The CSMP-program for the simulation of this system, based on equation (7), is given in figure 5. Because CSSLs are based on the operation of integration, the equation has been re-organized to be solved into two state- ments involving integration operators. Moreover, we have had to replace the symbol E (that is not available as a character in the set known to the computer) by Y.

TITLE V.D. POL - EQUATION PARAM YO = 0.2. DERYO = 0.2, A = 1 Y = INTE:GRL[YO,DERVl DERY = INTEGRL(DERVO,DER2Y) DERPY = A*(I-Y* *P-DERV* *2]*DERY-Y TIMER FINTIM = 12.0, PRDEL = 1.0, OUTDEL = 0.05

PRINT DELT. DERPY. DERY. Y PLOT Y(DERV1 [20,20) METHOD RKS END PARAM YO = 1,5 END STOP ENDJOB

Fig. 5 CSMP program for V.D. POL - EQUATION.

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In figure 5, more variables are seen than in equation (7). These are essential for the solution of the problem :

YO,DERYO are the initial values of E and - at the moment that we start to study the

model. This moment has been taken, arbitrarily, to be t = 0 (time zero).

FINTIM defines the extent of time wanted to study the model (0 to 12 s).

dE dt

PRDEL is the sampling period, i.e. the intervals at which one wants to see the values of the dependent variables ( lsecond interval).

OUTDEL is an estimate of the maximum integration step in time which allows accurate results to be obtained (step = 0.05 s).

The time needed to perform this simulation on a digital machine is significantly longer than 12 s (i.e. slower than real time). Because the statement ” time is money” is certainly valid for computers, many institutes instruct their mathematicians to program the simulations directly in a machine-oriented language which guarantees a more efficient use of the digital computer both in time and in memory requirements. Moreover, this approach opened up the possibility of using numerical techniques optimally suited for the problem under study.

A well-known disadvantage of this approach is, however, the interposition of a go-between in the formulation stage of the model and the interpretation of the results. Moreover, the necessary specialists for this approach are expected to have an extreme multidisciplinary schooling, in order to understand the implications of their programs in terms of the model under study. It goes without saying that professionals of this type are few and far between and that, because of their small number, they can easily become the main handicap against the urge to carry out many different simulations in a short period of time. In general, however, computing capacity is cheaper than highly skilled labour. This situation has given rise to further developments in software for digital computers, which simplifies even further the process of solving simulation problems on these machines for the inexperienced user. These developments, however, have only been proved to be remunerative in those fields where the problems to be solved show a great uniformity. In the area of crop study and protection, however, this has not yet been the case.

Apart from the tools that are available on digital computers to easily program continuous systems, other approaches are also possible. In general, it can be said that tools for continuous simulation apply for those systems that can be analyzed in a deterministic fashion or, in other words, systems that can be described on the basis of laws, or analogies of laws, derived from mathematical physics. It is also assumed to be clear that, in such cases, strictly causal relationships can be established. That is to say that any change in conditions is directly related on a one-to-one basis with the conse- quences in the behaviour of the system under study.

It often happens that the systems that we can detect around us are not entirely (or not at all) amenable to such an approach, either because the nature of the phenomenon

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is discrete (for example, uniquely determined by a sequence of events that occur in a fashion that cannot a pdori be related to a time-history), or because the phenomenon acts on the basis of random stochastic relationships.

Based on the main characteristics of such systems, computer models of these systems can be divided into the following important classes :

a) Models that are built upoa the principle of fixed periods. According to this principle, the independent variable (e.g. time) is divided into fixed periods (equal intervals). The dependent variables of the model can only change at the end of such a period, either as a consequence 0,' a causal relation (in a mathematical sense, discretisation of a continuous model, represented by difference equations), or as a consequence of some stochastical rule.

b) T h e so-called discrete-event models. In these models, the independent, as well as the dependent variables, are discrete magnitudes, which describe characteristic state variables of the systems. Interactions (either causal or stochastic) between these state variables can only take place at discrete (but not necessarily predetermined) points in time, which are separated by time periods in which no interaction occurs.

Such interactions are normally known as events. In most cases, the relation between dependent variables of the system and the relation of event occurrence to real elapsed time is influenced by stochastic factors. For these types of computer simulations, specific languages have also been developed over the past years (GPSS, GASP, SIMSCRIF'T, SIMULA '67, etc.). In the following section, their characteristics will be briefly described, based on SIMULA '67, which is a language for this purpose, derived from ALGOL '60 (HILLS, 19708).

The problems outlined above are concerned with systems of discrete entities, such as vehicles or parts or activities, moving from event to event (place to place or stage to stage). For instance, in a customer service model, one is concerned with the arrivals and departures of customers, the way they join queues, the movement of individuals within queues to reach service points and the nature and properties of the proferred service. In this case, the entities in the process are the customers, and the events are arrivals, departures and the servicing of customers. The events are governed by probability distributions (pd), such as the p.d. of time between arrivals of customers or the p.d. of the length of time needed to serve a customer. The points of interest for results would be, for example, the number of customers that can be served per unit time, the average amount of waiting time per customer, the service point idle time per unit time.

SIMULA (HILLS, 1970) programs of such models will fall into two parts: a definition part, which specifies the entities and their attributes, then the activities and their way of operation, followed by a control section and report part. Definition of entities is carried out with link class definitions. This allows entities to be created during the running of the program and automatically destroyed as soon as they are no longer required in the model. Further, because of their link properties, they may be held in queues (represented by sets) when they are not taking part in the activities.

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Activities are each defined by a process class, and this will have the following parts : - A specification of attributes and procedures local to the activity. - A set of conditions necessary for the activity to be able to start, usually the availabi-

lity of the right entities in appropriate queues. - A series of instructions starting the activity, taking entities into it from the

queues and making any other necessary changes of state. - A hold statement which puts the activity down into the event list, where it will

remain suspended until the simulation time progresses to its termination time. - A series of instructions to end the activity, releasing the entities taking part into

their next queues and making other appropriate state changes.

The control part of the program cycles over a list of the activities, creating a new instance of each in turn every time a change of simulation time occurs. As each instance is created, its conditions are tested. If the test proves successful, the new instance of the activity is started up and left in the event list. If the test is unsuccessfu!, the instance is destroyed. At the end of each cycle, the event list will hold one or more activities (unless the whole simulation goes dormant). The activity at the top of the list is then automatically ended and the cycle begins again testing each activity in turn to see if it can start as a result of the changes consequent on the last ending.

In such a way a language has been constructed that closely follows, in its formulation, the actual history of the process and reflects closely the logical relation of all events.

This close resemblance of the program to the model, or even the actual system, is the goal that any simulation language tries to achieve. In this way these languages will become more than simply means af programming: their main benefit will be to serve as a universal communication vehicle for model descriptions.

Bull. OEPP 9 ( 3 ) : 165-176 (1979)

RESUME

Langages de simulation digitale : problemes et possibilit6s par M.S. ELZAS

Agricultural University, Wageningen (Pays-Bas)

La simulation d’un modi.le B l’aide d’ordinateurs pose une sbrie de problcmes auxquels il peut &re difficile de trouver des solutions. 11s concernent notamment:

1) la construction d’un modde mathbmatique qui puisse Ctre transcrit sur ordinateur pour la simulation;

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