Introduction to the application of graph nets

04th Proceeding of the

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Manner of proceedings:
Announcement in the special edition of the magazine SAMS into 9/94 (statement)

Topic:
The Interdisciplinary Simulation of Continuous and Discontinuous Technical Systems

Author:
Christian E. Jacob

Table of contents: (0) ABSTRACT (1) RESOLUTION LEVEL WITH REGARD TO MODELING AND GRAPHIC INPUT (2) TYPICAL STRUCTURES OF GRAPH NETWORKS (3) CLOSED GRAPH NETWORKS (4) A TYPICAL OPEN GRAPH (5) EXECUTION SEQUENCE OF NETWORK ELEMENTS (6) ACQUISITION OF EXTREME VALUES (MEASURING TASK) (7) EVALUATION OF PERIODS (MEASURING TASK) (8) PULSE-WIDTH MODULATOR (CONTROL APPLICATION) (9) Bibliography

(0) ABSTRACT
The simulation of interdisciplinary interrelationships in technical systems requires technical languages which are necessary for the modelling at component, functional and behavioural levels. In these process continuous systems (network-oriented) will be described by means of network structures with concentrated components and discontinuous systems (event-oriented) by means of graph networks respectively. The simulator should be able to reconcile the two description languages and service simultaneously. Interdisciplinary simulation makes it possible to model continuous electrical systems expanded by switches, hysteresis qualities and the like. Yet it also makes it possible to model continuous mechatronic formations containing non-linear coefficients of friction, transmission trips and the like. The application of event-oriented structures (in the form of state graphs or Petri-networks) has not (yet) become everyday practice for modelling. This article is intended to give an introduction in this respect. It presents several examples for application which have been modelled with the IDAS simulation programme (Company SIMEC in Chemnitz). The notation and implementation on the computer are briefly presented.

(1) RESOLUTION LEVEL WITH REGARD TO MODELING AND GRAPHIC INPUT
The discussion on which resolution level modelling should take place is not yet finished. Starting from the lowest level, on which (semiconductor) physics is described in the continuum, modelling increasingly lifts off the level of components and the functional level up to the behavioural level (see chart 1). The behavioural simulation facilitates the investigation into complex systems, such as micro systems, mechatronic formations, propulsion, electric power supply etc. on the computer. Behavioural models are, however, "lifted off" of physics or the actual technical formation. This presupposes an appropriate experience on the part of the user. That is why simulation systems should only contain the behavioural simulation if they also support the modelling of lowlying levels.

Chart 1: Levels of abstraction of analogue and discrete simulation

Level of abstraction	Characteristic	Analogue mode	Discrete mode
Behavioral level	Equations describing a "black box"	Signal-flow diagrams with free of retroaction blocks	Graph networks for process monitoring and control of calculations
Functional level	(Macro) models with idealized components	Network with concentrated ideal components	Instantaneous logic circuit (noted at Boolean algebra)
Component level	1:1-simulation of structure	Network interconnected with block programmes and/or graph networks	Graph networks to control delay times of logic gates
Physical level	Block programmes for bordered formations	Simulator-independent semiconductor models	Simulator-independent semiconductor models

In spite of structured or object oriented programming it is difficult to comprehend the algorithmic description of physical and technical facts. Programmes are frequently only understood by the author himself. Outsiders prefer to "programme anew" rather than to acquaint with the existing programme.

Yet the description of the respective physical or technical fact by means of graph networks (state graphs, Petri-networks, modified B/E-networks or process networks) provides remedy in this respect. It is also possible to enter these graph networks directly and visually into the computer (with a graphics editor / CAD programme).

As far as the structure elements are reduced to two-terminal, three-terminal and four-terminal net-works the graphic entry should be considered as settled. In this system the information flow leads from the pattern (normally via several conversions) to the simulator. In most of the simulation systems the opposite way (backtracking) is open: Entries into the simulator or intermediate stages of the translation are not automatically updated in the pattern. This state still requires a manual matching of data. Yet irrespective of this fact the advantages of a graphic entry are evident.

For the modelling of physical or technical formations graph networks are not yet widely used. At many colleges they are neither part of the training of students of technical science. They are used for the direct entry of control sequences in control engineering, in support of systems analysis (prototyping), the notation of parallel-siding models (chaotic problem definition) and for non-linear models (ideal and non-ideal switches, hysteresis of the characteristic magnetisation line, non-linear coefficient of friction, transmission trips etc.). They are especially suitable for behavioural simulation since they ensure the requisite freedom of modelling and do not require intensive computation work.

In the second section the author would like to give a brief introduction to the use of graph networks, and in the third section he would like to present a few easily comprehensible examples.

(2) TYPICAL STRUCTURES OF GRAPH NETWORKS
The simplest state graph consists of a starting and a final state (see figure 1). The two states are linked with each other by an edge (directional connection). The arrangement of the starting and final state and of the edge is also called network element (ubNE). The starting state passes its mark on to the final state once the transition condition on edge bNE = true is reached. The mark then remains for ever in the final state.

If it is intended to mark several states simultaneously a way needs to be found to increase the number of marks. A network element based on figure 2 will be used for this purpose. Once the input state zE1/m is marked and the transition condition becomes bNE = true the mark will leave state zE1 so that the states zA1, zA2, ..., zAj will be marked simultaneously.

If several marks are to be put together in one mark a network element based on figure 3 will be lodged. This network element will switch when all input states zE1, zE2, ..., zEi are marked and the transition condition becomes bNE = true. The marks will merge in one and travel to state zA1.

The general definition of a Petri-network-element can be derived from figures 2 and 3. It combines the qualities described there: Once all input states zE1,zE2,..., zEi are marked and the transition condition becomes bNE = true all output states zA1,zA2, ..., zAj will be marked. The transition condition will be set bNE = true by an event that commences in a monitored process and/or a hierarchically "higher-order" state of a graph network.

If there is only one input state zE1 and one output state zA1 this network element is also called state graph [1]. This means that the network element of a state graph is a special case of a Petri-network element.

(3) CLOSED GRAPH NETWORKS
A Petri-network element stands out by a directional movement of marks, i.e. it can only transport marks from the input to the output states. Yet in the physical and technical structures there is, strictly speaking, neither a beginning nor an end. In simple words, the marks are always moving "in a circle" through the graph networks. Figure 5 shows the simplest closed state graph. In this state graph the mark is travelling between the two states z1 and z2.

Comes the transition condition bNE1 = true the mark travels from z1 to z2. And when the transition condition bNE2 = true is reached the mark travels from z2 to z1. The transition conditions bNE1 and bNE2, however, must not simultaneously become true: Considering that the mark changes infinitely rapidly between the states this implies that its "whereabouts" would be uncertain. This is, however, expressly ruled out by the Petry-network theory [2]. It is therefore the duty of the programmer of graph networks to work with special care in this area!

The above-mentioned also applies analogously to directional ring structures, e.g. based on figure 6 and free ring structures based on figure 7. Although the visual ring structures as described in figures 6 and 7 is simple, the notation in the IDL technical language is labourintensive. It is necessary to describe each individual network element with all input and output states. An on-line conversion, which generates the IDL source text directly from the graphic representation, is helpful in this respect. What still needs to be entered in this system is the Petri-network with a graphic editor. The dedicated symbols are used as well supplied by the manufacturer.

(4) A TYPICAL OPEN GRAPH
A Petri-network usually consists of a great number of closed graph networks as they have, inter alia, been described under item 2.2. In the following we only want to look at the section of a Petri-network. This structure shall be described as an open - i.e. incomplete - graph network.

A typical field of application of graph networks is the monitoring of analogous signals. It usually takes place within major graph networks in order to "find out" certain process states. Figure 8 contains a section of states z0, z1, z2, ... zk. Starting from the state z0 the mark is to be passed on as a function of the amount of the analogous value Signal to one of the respective states z1, z2, ... zk. This structure contains the network elements ubNE1, ubNE2, ..., ubNEk.

The transition conditions bNE1, bNE2, ..., bNEk are selected by the programmer in such a way that precisely one transition condition is always true. Supposing that two transition conditions would simultaneously become true and z0/m is marked the transmission of the mark is accidental. This is ruled out by the Petri-network theory [2]. In other words, the flow of the mark must be unambiguous!

In case that none of the transition conditions bNE1, bNE2, ..., bNEk is true the following should be taken into consideration: The flow of the mark would be interrupted at network elements ubNE1, ubNE2, ..., ubNEk. This may have considerable consequences on the function of the succeeding network elements, which are dependent on the availability of the mark.

(5) EXECUTION SEQUENCE OF NETWORK ELEMENTS
Section 2.1 has already pointed out that network elements monitor events in other network elements and can switch as a function of these events. The execution sequence of Petri-networks should be controlled in this case by a chain of priorities. Higher-order Petri-networks, state graphs and/or Petri-network elements control the subordinate ones. This applies both to all control tasks in automatic control engineering and to the use of corresponding models within the scope of a simulation task.

It is fundamental that the execution sequence set forth in chart 2 is adhered to. On the basis of appropriate simulations typical tasks of process monitoring, analysis and control are supplemented by models for the generation of test signals (set-point sources, test vectors, ...) connected ahead with regard to the execution sequence. In addition models advantageous for simulation are also used for the on-line evaluation of simulation data (recording of extreme values, formation of integrals, averaging, variation in time of products of diverse signals, ...). They are connected behind with regard to the execution sequence.

Chart 2: Execution sequence of Petri-networks, state graphs and/or Petri-network elements (applies to all computing steps)

Execution sequence	Function	Examples
1	Generation of test signals	Generation of nominal values; test vectors
2	Process monitoring and analysis	Monitoring and filtering of actual values
3	Process control	Adaptation and regulating
4	On-line evaluation of simulation dates	Recording of extreme values, formation of integrals

The functions mentioned in chart 2 have the following tasks:

1.) At first reference input variable, disturbance variable and other input variables will be composed. By means of the generation of test signals the process control will be synchronized with input signals.

2.) Process monitoring and analysis serve the extraction of signals for process control. Process states, process inputs, switching states of converter valves etc. will be ascertained.

3.) Process control implies the programming of technical control applications and particularly with regard to parallel calculations the notation of dependent on events [3].

4.) The on-line evaluation of simulation data deserves special mention. As an anticipation of the post process it makes it possible to ascertain characteristic values of simulated time-dependent values. In the course of simulation on-line evaluation constitutes a simulation programme into a calculation and/or optimization programme [4].

It is last but not least the execution sequence of Petri-networks, state graphs and/or Petri-network elements which is decisive for the exact function of an event-oriented model. In chart 2 we compare the functions assigned to the 2nd and 4th execution sequence. The process monitoring and analysis (2nd) and the on-line evaluation of simulation data (4th) have to fulfill taks of the same kind. In this respect they are almost interchangeable. Yet it is the execution sequence which, for each model, differs in the calculation states:

a) Models on process monitoring and analysis (2nd) supply data and events required for further calculations and/or control applications in process control (3rd). This means that these data and events should be made available at the beginning of each computing step.

b) Models on the on-line evaluation of simulation data (4th) require data and events of the antecedent models (1st), (2nd) and (3rd). In the simplest case evaluated simulation data can be displayed. It should be pointed out that multitudes of data of worst case, multisimulation and optimisation sequences are evaluated, too. The evaluation models should be irreversible with regard to the simulation sequence; otherwise systematic measuring errors will arise in our view. These data should only be extracted at the end of each computing step.

The four-stage sequence plotted in chart 2 can later be subdivided even finer. The IDAS simulation programme of the company SIMEC in Chemnitz contains a technical language, inter alia, for the description of graph networks. Linear, branched and parallel structures of control graphs can be logged with the aid of the corresponding network elements. The execution sequence of network elements will be given by the user [5].

Today it is expected that the tasks of a simulation programme will only have to be set as detailed as necessary [6]. This concerns both the resolution of the structure and also the used time resolution. This requirement is expressly supported by the IDAS simulation programme [5][7]. In the following three event-oriented models will be presented which were tested with the IDAS simulation programme.

(6) ACQUISITION OF EXTREME VALUES (MEASURING TASK)
In section 2.4 we have already pointed out the necessity to ascertain extreme values as early as in the run time of simulation. That is why we will have a closer look at a state graph described in figure 9 which will serve as an example for the acquisition of maximum values.

It contains two network elements which switching, as function of the peak value of the exploit value Signal. In the first network element the mark /m will be transported from state zHLT to zWZW. If Signal is greater than MaxValue the respective current peak value in state zWZW will be passed to the variable MaxValue. Figure 10 shows a section of a test run: The extreme value MaxValue is tracked without delay.

This state graph will be acquired for once each logged maximum value. It goes without saying that the variables Signal and MaxValue have to relate to each of the corresponding values. For the acquisition of minimum values the starting value (MinValue:= +E37) and the transition condition (Signal < MinValue) will be changed.

(7) EVALUATION OF PERIODS (MEASURING TASK)
Oscillations (with periodic variations in time) are generated by resonances in physical and technical formations with energy storage devices (e.g. moving and/or revolving masses, inductivities, capacities, ...). Amplitude Â and period T permit conclusions concerning values, distributions and possible other qualities (linearity) of these storage devices.

Amplitude Â can, inter alia, be determined with the aid of the measurement of extreme values described in item 3.1. The period T is frequently determined by means of a frequency measurement T=1/f. This is equivalent to an averaging of the period, since measuring is carried out over a certain time (e.g. f oscillations in one second). If a precise statement on period T is to be obtained this presupposes a linear behaviour of the physical or technical formation. Yet nature is practically non-linear. This means that period T is not constant. It depends on various factors including amplitude Â itself.

Figure 11 contains a model for testing a state graph intended for direct period measurements with-out averaging. The model uses a resonant circuit. Inductivity decreases after the function Magnet. Along with the value of inductivity the period is decreasing as well.

Figure 12 shows the variation in time of the metered period of the a periodic resonant circuit in figure 11. The exact measurement will be carried out after a transient time corresponding to the maximum length of T. The simulator of the IDAS programme tries to determine the transition condition u"cKonst" = 0 as exactly as possible. In this process the computing step size h is reduced to the minimum step size h_min, i.e. the user indicates the precision of the period measurement by giving the minimum step size h_min.

(8) PULSE-WIDTH MODULATOR (CONTROL APPLICATION)
Inverted rectifiers require a control unit in order to control the static switch deployed there. The duration of the turn-on and turn-off time for the respective static switch is given by a pulse-width modulator (PWM) in figure 13.

The pulse-width modulator (PWM) is given the frequency f_a(1)nom and amplitude U_a(1)nom of a voltage u_a(t) as nominal values. On this basis it generates a logical value bZP for the pair of arms control (PAC). The pair of arms control (PAC), on its part, produces signals iTr1 and iTr2 for the semi-conductor switch (SCS). In transistorized inverted rectifiers signals iTr1 and iTr2 are equivalent to the base currents of the switching transistors.

The following static status applies to static switches Tr1 and Tr2: bZP = true means: Tr1 = on and Tr2 = off and bZP = false means Tr1 = off and Tr2 = on. The switching frequency of the static switches is given by the so-called pulse frequency f_p (synchronous pulse method). Actual values of the control application are frequency f_a(1) and the amplitude of the 1st harmonic of output voltage U_a(1).

Figure 14 shows the model of a pulse-width modulator for one phase _L1. The logical value bZP will be formed on the basis of a comparison of an analytical control function U_a(1)nom(sUA10), whose amplitude is constant

with a triangular function drei of frequency f_p. The model is, inter alia, used to ascertain the level of content of basic oscillation v_gua of the chained voltage U_a12.

Figure 15 shows the entire course of the above-mentioned level of content of basic oscillation. Small n produce the best ratio between fundamental and harmonic oscillations. With n diminishing, the control function approaches a rectangular function. This means that a rectangule-triangule-comparison is better for the pulse-width modulation of an inverted rectifier than a sine-triangle comparison.

(9) Bibliography
[1] Grunde, Ullrich: Ausführungen zu Petri-Netzen; Firmenschrift PSI Berlin 1989.

[2] Wolfgang Reisig: Petrinetze - eine Einführung; Springer-Verlag Heidelberg 1982.

[3] Christian Jacob; Dietrich Möller: Anwendung des Simulationssystems IDAS (LENE) für den Entwurf von Stromrichtern; Lecture at 10th conference "Industrielle Automatisierung - Automatisierte Antriebe" TU Chemnitz 14./16.02.89.

[4] Christian Jacob: Anwenderbezogene Architekturen bei Mehrsprachensimulatoren; Lecture at 8th Workshop "Simulationsmethoden und -sprachen für verteilte Systeme und parallele Prozesse" of ASIM in Dresden 27./28.04.92.

[5] Birgitt Knorr; u.a.: Referenzhandbuch für das Simulationssystem IDAS zur Simulation elektronischer und leistungselektronischer Schaltungen und Anlagen mit Steuer- und Regeleinrichtungen; Fa.SIMEC GmbH & Co.KG Chemnitz, 1991.

[6] Christian Jacob: Entwicklungstendenzen auf dem Gebiet CAE für Leistungselektronik und Weiterentwicklungen am Simulationssystem IDAS; Lecture at Workshop of the usergroup IDAS, TU Chemnitz 5./7.06.1990 and publication in "ELEKTRIE" 4(1991), pages 127-131.

[7] Christian Jacob: CAD-Lösungen für Entwurf, Konstruktion und Technologie von Leistungselektronik; publication in "VEM-Technische Information" (1989)19, pages 39-41