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This paper describes a novel function generator and multiplier using digital-to-analog conversion techniques. The device has been designed and built to supplement the existing analog computer facilities at the University of British Columbia. The function generator and multiplier, basically a digital-to-analog converter, is controlled by one ore two punched paper tapes. The device is capable of generating two independent functions specified by the input tapes, their product, and the product of an external variable and one of the tape inputs. All outputs are in analog form with a maximum range of +/- 100 volts and 0.5 % accuracy. Principle of operation, error analysis, circuit diagrams, photographs and test results are included in the paper to illustrate the operation and application of the device.

DARE II is a remarkably fast, on-line, all-digital, block-diagram-based simulation system which adds a convenient and inexpensive simulation console to a PDP-9 computer. Block statements, control logic, tabular functions, and data are entered and edited on a CRT keyboard. Solutions appear on a second CRT display and can be plotted or printed out for report preparation. A fully commented program and solution record can be printed out on a teletype and programs may be stored on punched paper tape to be run again at a later date. The DARE II system can be easily programmed to make optimal use of the computer, generating a much faster program than a FORTRAN compiler.

Simulation is the art and science of describing the behavior of a dynamic system by means of a model and using this validated model in a series of experiments designed to provide insight into the future behavior of the system under specific conditions. Therefore, the process of simulation can be visualized as consisting of four phases: (1) problem definition, (2) model specification, (3) model validation, and (4) insight acceleration^1 through experimentation. In order to facilitate the process of simulation, one develops various tools. The digital computer is one such tool. Man's interface to the computer is through the language with which he describes his problem to the computer. The ease with which he solves his problem depends upon the richness of the language as well as the ability of the language to be unambiguously specific. In turn, the structure of the language depends on the form of the model to be implemented. However, having to work with a given language may bias or control the form of the model. Therefore, one must be careful in selecting and using computer languages for system simulation. All physical systems exist in the time-space continuum. The type of model one selects depends on the degree of aggregation of individual phenomena. One may choose either a continuous or a discrete approach to model development. Continuous models are useful when the behavior of the system depends more on the aggregate flow of events than upon the occurrence of individual events Choice of the continuous-or discrete- event modeling approach depends on the nature of the system, the objectives of the simulation, and the tools available to implement the simulation. Because of the mathematical nature of the models in continuous system simulation, special languages have been evolued which are specifically adapted to expressing this class of problems. Over forty continuous-system simulation languages have been developed as of 1973. This paper will survey currently available languages (available commercially or from specific research groups) for simulating dynamic systems described by differential equations. The languages surveyed in this paper are partitioned into two groups. The first group consists of general Continuous-System Simulation Languages designed to represent dynamic models defined by sets of differential equations having primarily one independent variable. Included in the first group are MIMIC, DYNAMO, CSMP/CSMP III, CSSL-III, SL-1, and PROSE. The second group consists of languages specifically designed to represent models defined by partial differential equations. Included in the second group are SALEM, PDEL, LEANS, DSS, PDELAN and FORSIM. The objective of this survey is to describe the functional characteristics of currently available continuous-system simulation languages from a user's point of view. Emphasis will be placed on the unique features of the language, the services provided, the degree of supplier support, the degree of generality, flexibility in terms of user ability to extend the language, the ease of learning, and finally the economics of use.

Several papers present possible mechanisms for the cardiac action potential. The more suitable models are based on the Hodgkin and Huxley ionic theory with some modifications. In this paper, hybrid computation techniques are utilized to elaborate a different model applicable to Purkinje's cardiac cells.

The purpose of this paper is to show that there are many physical or problem related criteria which may be used to measure quantitatively numerical accuracy. Such criteria have been used sporadically in specialized areas, but little attempt has been made to bring them together within the mainstream of numerical analysis. It has been the case that users of analog/hybrid computers, having to worry about limitations related to the size of machines and number of computing elements in computers, have been very much interested in errors when solving partial differential equations, more than the users of digital machines. Several of the results shown in this paper originated in that field. Our interest however is to apply those concepts to numerical computations, without excluding, however, their applicability to analog/hybrid computation.

This paper presents a tutorial analysis of a hybrid computer model of a missile reentering the atmosphere. An important aspect of the paper is that an example of how to go about analyzing the suitability of a hybrid computer configuration for a particular large- scale simulation is presented from an overall systems point of view. Equations describing a missile (six- degrees of freedom) reentering the atmosphere are given, as is a hybrid computer configuration for solving them. This configuration is then analyzed to establish its suitability for simulation.

The paper deals with the dynamic aseismic design of an elevated irrigation water-tower, 150 ft (45 meters) high, capacity approximately 500 tons, to be erected in series in Northern Greece. The water capacity of the tower's reservoir amounts to approximately 500 tons. The dynamic investigation of the tower's behaviour under seismic action was carried through in the writer's Laboratory in Athens, with the use of a properly equipped electronic analog computer, model TR48-3 of Electronic Associates Inc., Long Branch, New Jersey. In addition to the assumption of a rigid foundation, the influence of the rotation of the foundation's footing in proportion to an estimated foundation modulus (Bettungsziffer) was investigated. The solution is given in the form of diagrams, recorded on the computer's plotter, representing the deformations and stress components along the tower's height with respect to time, due to a harmonic seismic earth motion of a given maximal acceleration but of variable frequency. The comparison between rigid and yielding foundation leads to inferences which are exposed at the end of the paper.

After discussing the already-known means for solving optimization problems by the maximum gradient principle, this paper then contributes several modifications to this method which permit solution by analog iteration of a large number of problems comprising several variables; including those which are limited by parameter constraints, eventually accompanied by sensitivity constraints. The factors which make up the performance index are generated simultaneously by an analog method. This paper examines as well the effects of noise and perturbations on the convergence rate of the iterative process of optimization.

This paper is concerned with the concepts of the computer simulation of cybernetic systems. Engineers, sociologists, business men, economists, architects and most recently health scientists have found a need for simulating systems so that their behavior may be predicted and finally controlled. This paper points out the importance of simulation and how simulation is carried out. In particular, it discusses the control of man and machine systems which Norbert Wiener and his colleagues at M.I.T. called Cybernetic systems. With regard to man controlled systems, the simulation of a tumer cell in water is considered on an Analog Computer. Particular attention is given to man's direct control of the biological system on the computer and the organization of the differential equations. Also discussed is the simulation of a feedback oscillator having a non-linear active element. This simulation was carried out on both an analog and digital computer. Emphasis is placed on the use of the Hytran Simulation Language (HSL) used in the simulation. To give greater depth to the discussion, a simulation of a traveling repairman is given. This simulation may be applied to health systems also, with some imagination. Although the examples used to demonstrate simulation are in the areas of biology, electronics, and business, there is no reason why these computers cannot be utilized in many other areas from sociology to health. This fact is continually pointed out in the paper with the hope that the reader will create an analogy between those systems considered and his particular system.

It is convenient to consider analog computer programs in two parts, patching and amplifier input factors. Scaling is accomplished by the specification of the input factors. In this paper a simple method is presented for imposing both magnitude and time scaling factors. Negative time scaling is also considered. Automated scaling for hybrid computers is the logical sequel to this work and is reported in a companion paper.

This paper considers the possibility of using incremental integrators based on the third and fourth order Runge-Kutta integration algorithms in computer systems. This work is based on the study of more recent proposals for Digital Differential Analysers (DDA's). Performance curves for the solution of the second order harmonic equation on a DDA over. a set time and a range of frequencies are extracted from error curves taken from General Purpose Digital Computer (GPDC) solutions of the equation. The design principles of the new integrators are explained with block diagrams and the associated register arithmetic is derived from the standard form of the algorithms. The applicability of this work to DDA's and fast digital solution of differential equations is also discussed.

The electrical analogue provides a useful approach to the solution of Partial Differential Equations. A general purpose analogue for PDEs would be a useful peripheral for a small digital computer, enabling it to handle quickly, cheaply and with reasonable accuracy problems which otherwise require substantial computational time on large machines. In this paper the author discusses the design of an analogue for @?^2U + U"x = 0, suitable for further extension as a computer controlled general analogue. It is shown that AC networks, using capacitors as passive elements, offer substantial advantages over other analogues. The use of buffer amplifiers at each node greatly simplifies the unilateral impedance problem, but introduces unidirectional error components. These errors are analysed, and specifications for suitable amplifiers are obtained Experimental results on small (20 node, 90 node) models confirm the satisfactory performance of the analogue, and the author predicts that large scale (> 900 nodes) models will yield results accurate to better than 99.9%, with less than 100 ms settling times

In spite of the power of digital computers, the solution of some problems remains difficult by numerical methods. An example is given in this paper: the difficulties arise not from the equation, but from the geometrical complexity of the three dimensional region. The rheoelectric analog constitutes, in this case, an extremely well suited method of solution.

Hydrologic systems, like most environmental systems, are usually characterized by partial differential equations. In order to develop a suitable mathe matical model for the analysis of such systems, it, is first necessary to make use of observed data in order to identify or estimate the system parameters. The digital simulation language PDEL has been ex tended so as to facilitate this type of parameter identification. The extended version of the lan guage has been named PDEL-ID. In this paper, two hydrologic modeling problems taken from recent water-resource studies are used as examples to illustrate the parameter identification facilities of PDEL. The solution of these two problems demon strates that little programming effort and virtually no knowledge of optimization techniques are required.

Conventional methods of inventory control in most retail enterprises emphasize minimization of cost while maintaining an appropriate degree of service to the clientele. This is done without regard to the effects produced on the distributor who is supplying the goods or the factory which is producing them. Such policies characteristically result in the amplification of a small disturbance (small change in sales volume) at the retailer level into a large disturbance at the factory level producing large cyclic swings in factory production, inventory, and operating efficiency. This paper discusses the concept of digital control of inventory and requisitioning procedures for a marketing system of three levels representing the retailer, the distributor, and the factory. A model consisting of three interacting control loops (one for each level), to simulate the flow of requisitions from retailer to distributor, and from distributor to factory, and a corresponding flow of goods in the reverse direction has been constructed on a hybrid machine. Digital control algorithms have been applied in a manner designed to reduce the undesirable amplification and to maintain factory production consistent with sales volume. Results indicate that the inventory and requisitioning responses have conflicting requirements. A procedure for obtaining any desired trade-off between the retailer inventory and factory requisitions by means of digital networks is suggested.

Mathematical description of flow in underground reservoirs gives rise to partial differential equations that are often non-linear and that contain parameters which are position dependent. The solutions of these equations provide useful information for the evaluation and prediction of reservoir performance. The complexity of these equations is such that analytical solutions are not readily available and numerical techniques implemented using digital computers have been employed. The hybrid technique developed by Bishop and Green ^[^1^]has been extended to non-linear partial differential equations with space dependent parameters and implemented using modern hybrid computing equipment. Application of the technique is made, in this paper, to the simulation of ground water flow in an unconfined aquifer.

In this paper a comprehensive theoretical basis is formulated concerning the hybrid computer approximation of partial differential operators. The investigation leads to a theorem that enables the identification of consistency and error properties of general hybrid methods. The theorem is applied to several well-known hybrid methods and the results presented can be directly used as a guide for selecting the optimal implementation.

This paper explains the principles of operation of an interpolation device, which transforms a discretely defined signal into a continuous one. The conditions for designing an interpolating parabola, which is going through (m + 1) discrete points for the m-th order of interpolation, are given. By the selection of a parameter 1 it is possible to ohoose the allocation of a working interval, along which the created parabola is valid. The case of cubic interpolation, using the third interval as the working interval, is used to illustrate the synthesis of the required overall transfer function and the transfer function of the forward and the feedback block. These transfer functions are implemented by rather simple circuitry, which may be designed, for example, on general purpose analog computer. The circumstances limiting the order of interpolation in these types of interpolating circuits are explained. At the end, a survey of transfer functions, together with actual circuits is given for first, second and third order interpolation. Some practical results achieved are discussed and an example of utilization is presented.

It has always been a tremendous task for numerical analyses to solve non-linear partial differential equations. The modal technique presented in this paper, approaches the problem on his own specific way transforming the partial differential equation into two infinite sets of ordinary differential equations, very well suited to be solved on a hybrid computer. Since this technique gives rise to only differential equations, it can be classified as a continuous time continuous space method. Application of this technique to the moisture-flow equation shows that this method gives extremely good results, which are in close agreement with experimental data.

Quasi-linear hyperbolic partial-differential equations of the second order are easily transformed into their characteristic form, and hybrid computer algorithms based on the Continuous Space Discrete Time method of lines, taking advantage of such transformations are conveniently implemented. This is illustrated in this paper, together with considerations of the minimization of spurious diffusion effects which are associated with difference approximations of such equations. The example taken is that of the transient simulation of electrical transmission lines, but the concepts presented apply equally well to other problems in this class (e.g. one dimensional hydraulic simulations).

This paper gives a brief overview of some of the present problems and achievements in cardiovascular modeling. Typical of most of the simulations described is a level of complexity such that all-analog simulation often requires too much equipment and all-digital simulation is often too slow. Thus, with today's technology, a hybrid computer is an essential simulation tool. Fortunately, in both the case of multiple model simulation, and that of parameter estimation, the describing equations may be logically divided into groups suitable for solution on the two parts of a hybrid computer, with the overall result that real time operation, or faster, is possible.

In Vansteenkiste and De Schutter propose a modified modal method to simulate diffusion processes with non-homogeneous boundary conditions. Moreover, the hybrid implementation of the modal simulation is discussed by the authors from an exclusively theoretical point of view. The modification of the modal simulation consists of a transformation of non-homogeneous boundary conditions into homogeneous conditions. Since the modified method considered in presumes the analytic solution of the original partial differential equation possible only under certain restrictions, this paper is concerned with two other proved modal methods for the simulation of non-homogeneous boundary value problems. In addition to this, some critical remarks are made both on the analog/hybrid modal simulation and on the hybrid implementation shown in.

Starting from recent computer display technique developments, the paper illustrates some examples of graphic representation of dynamic systems for cases where the model is partly simulated on a parallel computer and partly described in a digital program, in order to take full advantage of the range of a hybrid computation unit.

Transfer functions are an aid in the analysis of many linear systems. This paper presents an application of the Matric Computor, a revolutionary new analytic computing device, to the solution of certain problems described by transfer functions, with examples. The mathematical basis for transfer functions are differential equations. A canonic matric formulation of transfer functions from this viewpoint is presented which can be entered directly into the Matric Computor and yields system responses for given inputs. This, furthermore, leads naturally to formulations for the interconnection of sub-systems - described by transfer functions - in cascade and in parallel, and thus also for feedback systems.

In the present paper, we shall develop the analog simulation of electric systems involving saturable core coils. Two cases are analyzed here. In the first one, we consider simulation of the behaviour of a saturable core coil, fed by a sinusoidal voltage generator; we use a mathematical model of the hysteresis loop that enables to select the loop branches without the help of any switching device. In the second case, a more complex one, simulation of a ferroresonant circuit is performed, which contains non-linearities about inductance and losses.

This paper describes a Monte Carlo technique for estimating the lowest eigenvalues of certain elliptic and hyperbolic partial differential equations with Dirichlet boundary conditions. (2) A stochastic process whose output conditional probability density distribution satisfies a partial differential equation similar to the partial differential equation under consideration is, along with the boundary conditions, implemented on ASTRAC II, a fast repetitive analog computer. Eight different boundaries on the scalar Helmholtz equation in one, two, and three-dimensional space were implemented. Each of the resulting eight lowest-eigenvalue estimates is then compared to its corresponding and expected true lowest eigenvalue. Computer hardware errors, along with the error resulting from the mathematical approximation employed in deriving the estimate, are indicated. Corrective measures are included when necessary. Applications result from a presentation of analogies relating the partial differential equations under consideration to partial differential equations modeling physical processes.

The purpose of this paper is to describe an analog simulation of the problem of the dynamic behavior of stressed thin ferromagnetic films and to compare the solution obtained with the results of a digital solution already available and with a digital computer study performed by the authors for the same problem.

This paper describes a FORTRAN IV digital computer program for solving ordinary differential equations. Two versions are described, one without graphical display for use in standard batch-processing. The second version is written for the DEC PDP-9, and features a self-scaling display-driving subroutine (for scope or plotting board) and on-line keyboard entry.

This paper describes the use of a fast iterative analog computer to study a sophisticated communications system. This system uses amplitude modulated orthogonal polynomial waveforms as carrier signals. The Arizona Statistical Repetitive Analog Computer (ASTRAC-I) setup includes the polynomial waveform generator and a matched filter receiver. Accuracy and crosstalk are measured using a sample-hold-memory-pair module. The system is also tested with additive Gaussian noise with the aid of the ASTRAC-I hybrid analog-digital statistics computer (sample averaging unit and distribution analyzer).

The stray capacities in the input and feedback paths of an operational amplifier, its finite bandwidth and d.c. gain effectively narrow the bandwidth of an amplifier causing amplitude and phase errors in the computer solutions. A given degree of accuracy consequently becomes more difficult to obtain with an amplifier when used in a high-speed repetitive computer than when used in a slower machine. In a high-speed computer, phase distortion is more serious than the amplitude distortion. An attempt has been made in this paper to analyze an operational amplifier with its associated input-feedback networks and a practical method has been suggested for reducing the undesired phase shift.

The stability of solutions of systems of algebraic equations on arithmetic machines by iterative or indirect methods and directly on analytic machines, is considered in this paper. The concept of an Explicit Operation Matrix is introduced which simplifies the analysis of stability of solutions. A thorough study of the various existing procedures for solutions of systems of linear equations is made on this basis, and useful criteria have been obtained for the stability. The analytic machine is analysed as a particular case of the iterative arithmetic machine wherein the iterative sequences become continuous rather than being discrete integral, which as a consequence results in the transformation of the Explicit Operation Matrix into an Implicit Operation Matrix. These analyses show that arithmetic machines using iterative methods, and analytic machines giving directs solutions, have similar constraints as to the positive definiteness of the matrix of the algebraic system. However, it has been found that for stability, an analytic machine imposes less stringent conditions on the nature of Implicit Operation Matrix, unlike arithmetic machines, which impose more stringent conditions on the nature of Explicit Operation Matrix. Some experimental results obtained along these lines using the Matrix Computor are also included.

This paper describes the working of the APACHE (Analog Programming And CHEcking) system for the production of computer programs using a digital computer, which has been developed jointly by digital and analog programmers at the computation center of EURATOM in Ispra, Italy. The language which has been formulated for the writing of analog problems is discussed and its rules are set out in the appendix, while the processes involved in the utilization of the program are described and illustrated by an example of the listing of the digital computer input and output for a range of equation types. Finally, the significance of this new programming system in terms of economic exploitation of the analog machine, the use of trained personnel, and the design of new computers, are considered briefly.

The problem of determining optimum temperature profiles for tubular reactors is solved by the calculus of variations in its classical form. This leads to a two-point boundary value problem, which may be solved, in the simpler cases on an ususal analog computer and in more complex cases preferably on an iterative type of analog computer. In a tubular reactor there may be bounds both on the temperature and the temperature gradient. It is shown how such inequality constraints can be handled. At the end of the paper there are given a few examples to illustrate the procedure developed.

The APE II computer system is a small solid-state iterative differential analyser designed for laboratory instruction and is capable of both repetitive and real-time operation. This paper describes the APE II digital module, which includes a master clock for generating reset and sampling pulses, random and pseudo-random noise generators, digital counters and free patchable digital logic. This module, when used with the APE II analog module or similar small analog computers, permits a very wide variety of hybrid simulation and statistical experiments.

This paper describes an analog computer method for designing dead beat controllers for linear and nonlinear sampled data system. In this method the output of the digital controller is replaced by the output from a special function generator. Thus the plant input consists of a number of step signals (with variable magnitude and sign) applied at the sampling instants. By adjusting the magnitudes and polarities of the input signals the plant output and the derivatives of the output signal are adjusted so that they have the desired values at the required sampling instant, the signals being observed on a multitrace oscilloscope. The coefficients of the dead beat controller are completely determined by the successive values of the step input signals and the system outputs at the sampling instants. This design method is particularly useful when the plant is nonlinear.

In the optimization of a non-linear system one is generally faced with a complicated two point boundary value problem. This two point boundary value problem in the case of a linear system can be solved by breaking it into two one point boundary value problems, by going into the Riccati equation. However in case of a non-linear or bilinear system the solution of the Riccati equation is not a simple undertaking. Thus various iteration methods are often tried. In this paper a method of solving this two point boundary value problem on analog computer has been presented. The steepest descent method is applied here on boundary condition iteration. This method overcomes a past hesitation of using the analog computer for the optimization problems, which require a large computer memory. This method in fact is applicable to any system linear or non-linear. However the size of the computer and the degree of accuracy definitely limit its application.

This paper describes a relay-operated one-amplifier memory-pair circuit, which frees a much more expensive track-hold pair of electronically switched integrators in sampled-data system simulation. The circuit is intended for experiments with sampled-data control systems and difference equations on inexpensive educational analog computers. The periodically operated relays can be driven by any laboratory oscillator if no logic circuits are available. Three typical experiments are discribed.

The use of the analog computer in teaching engineering and science has become increasingly widespread. The reason for this vogue is that the analog computer can easily solve differential equations, linear and non-linear, which characterize so many physical systems. However when the mathematical model requires decision making logic switching, timing, memory or individual mode control of analog components, the need to add parallel logic to the analog computer becomes self-evident. This paper intends to show how it is possible to extend the capabilities of the analog computer by the addition of parallel logic.

This paper describes the application of an analogue computer to the design of control systems when constraints are imposed on the variables of the system. The operation of the computer is fully automatic, so that manual adjustment of parameter settings is completely eliminated. The performance index can be any function of the variables of the control system, and this index is automatically optimised by the technique developed. Practical examples are included.

Increased interest in high-speed hybrid analog/digital computation has led to the development of the Arizona Statistical Repetitive Analog Computer (ASTRAC I), which employs inexpensive digital logic to control a high-speed repetitive analog computer. The design of the machine has been described elsewhere; this paper reports the results of the error analysis performed on the linear analog computing elements and digital timing circuits. The results are of particular interest for the design of new computing systems of this type. The results of several statistical experiments are also included to further illustrate the operation and application of the various ASTRAC I computing elements.

This paper gives the theoretical basis of a method allowing one to design a variable delay simulator using analogue computer elements. It is shown that, if conventional methods of approximation of a constant delay are extrapolated to time-dependant delays by simply making the coefficients become time-dependant as well, the circuit obtained will be mathematically wrong, unless a special condition on the location of those coefficients is fulfilled.

This paper describes a hybrid computer simulation of a simple signal detection problem. A signal of unknown phase plus white Gaussian noise is detected with a correlation receiver. Measured detection probabilities and false-alarm probabilities for various signal to noise ratio are compared to the theoretical curves for the Bayes optimum receiver. Good results are obtained using a lowcost educational computer, APE II.

Dynamic errors and destabilizing effect in hybrid loops, introduced by sampling of the analog variables before digital processing, have been studied in the most general case in a previous paper published by Elmer G. Gilbert and Ghislain C. Vansteenkiste. If the system treated can be represented satisfactorily by a linear model, the error expressions, as well as system stability considerations take a reduced form. This has been investigated in this study.

The purpose of the paper is to establish a bridge between iteration methods in mathematics and analog computing networks. Special attention is given to some new analog iteration procedures which can form an intermediate step between iteration processes as used in mathematics and the continuous solution of equations by analog means.

The interconnection of dynamo-electric machines and analytic computers yields interesting and useful properties not available from either of the devices singly. Moreover, the homogeneity between Kron's modern matric-mathematical description of dynamo-electric machines on the one hand, and the mathematics pertinent to Matric Computors on the other hand, yields unity of concepts and notation. This is additionally exhibited by the ease with which dynamos and Matric Computors can rationally be combined into single electric networks. The objective of this didactic paper is to introduce the elemental concepts of the supermatric equation of state describing the interconnections of dynamo-electric amplifiers with Matric Computors, the latter being the control elements whose mathematical entries dominate the performance of the composite interconnected system.

This paper describes the construction of an instrument for measuring the impedance using a physical model. The model has to be adjusted in such a way that its differential equation corresponds with that of the unknown impedance. For the implementation of model and adjustment mechanism, standard hybrid computer components are used. Also the testsignal is easily implemented and is composed in such a way that it is possible to measure three parameters of the impedance simultaneously with little dependence between the parameter adjustments. Some attention is paid to the stability-problem.

A system of partial differential equations can be represented on an analog computer only approximately by using finite differences. Considering a time-space system with only one space dimension, the mostly used method is to simulate a lumped system where the distributed parameters along the space dimension are replaced by concentrated parameters. The investigation described in this paper shows how it in several cases should be possible to gain higher accuracy by replacing the original equations with difference equations in time t, instead of with difference equations in space dimension x.