Introducing Digital Design Methods to System Dynamics
Education
Tzur Levin and Ilya Levin
Tel Aviv University, Schools of Education
Ramat Aviv, Israel
Tel: (972)-3-6407109, Fax: (972)-3-6407752
Email: levintzu@post.tau.ac.il
Abstract
One of the origins of system dynamics is control theory. Thus, it is worthwhile to
investigate how the growing use of digital models as means of control should affect the
field of system dynamics. In particular, it would be interesting to explore how the
introduction of digital models could enrich the learning of system dynamics in K-12
curriculum.
This paper proposes to do so through hybrid systems modeling. Hybrid systems are
dynamic systems in which a digital element controls a continuous process. The paper
illustrates the relevance of hybrid systems to K-12 education, and describes how hybrid
systems may be modeled by using icon-based software such as STELLA. Two examples
showing an implementation that is suitable for high school students are presented as
well as an outline for future research.
Keywords: hybrid systems, system dynamics, education, digital control, finite-state-
machine.
1. Introduction
One of the origins of system dynamics is control theory (Andersen & Richardson,
1980). Thus, it is worthwhile to investigate how the growing use of digital models as
means of control should influence the field of system dynamics. In particular, it would
be interesting to explore how the introduction of digital models could enrich the
learning of system dynamics in K-12 curriculum. This paper proposes to do so by
modeling a special group of system called hybrid systems.
2. Types of Control in System Dynamics
Classical system theory uses calculus as the major mathematical tool for describing
systems (Betralanffy, 1968). In particular, the system dynamics methodology makes use
of differential and difference equations to model systems and run computer simulations
(Forrester, 1968). These models are graphically expressed as stocks, flows, connectors
and converters in icon-based tools for system modeling such as STELLA (High
Performance Systems, 1985-2001). Applying these four types of icons for modeling all
system forces students to adopt a certain way of thinking (Penner, 2001). This way of
thinking may be called an analog way of conceiving systems.
a) Controlling with Feedback Loops
The fundamental assumption of System Dynamics is that dynamic behavior of a system
is mainly a consequence of feedback loops (Goodman, 1974). Feedback loops connect
elements of the system in a way that information on the system in a given time
determines its future behavior (Forrester, 1968). Thus, they control the dynamic
behavior of systems.
Feedbacks loops tend to be of a similar nature to the process they control: both obey the
dynamics of stocks and flows. Thus changes in the size of the lynx population control
the dynamics of the hares population in the classical ecological example. The two
processes are represented as stocks with inflows and outflows, and may therefore be
referred to as analog. A notable exception to the analog view of systems is the IF-
THEN-ELSE logical expression sometimes used in feedback loops. Though this logical
condition is of digital nature, we shall soon see that it is too simple to represent the
controlling power of digital models.
b) Digital Control Systems
Following the computer revolution, digital methods are now widely applied in
developing controlling systems (Dorf & Bishop, 2001). The use of the digital methods
increases the ability of control systems to handle complexity, and is popular in
computer-embedded-systems, and in particular in computer controlled robots (Brooks
1991; Mioduser, 1995).
A classical example of the power of the digital approach is the digital computer itself
(Figure 1). In the central processing unit (CPU) of the digital computer, a digital control
unit is physically distinct from an operation unit (Mano & Dime, 1997). The controlled
unit (data path) performs the arithmetic, logical and other data processing tasks. The
control unit receives input from the controlled unit, and provides signals that activate
various micro-commands to be performed by the operation unit. This kind of separation
between the units may exist also when the controlled part of the system is of an analog
nature.
Memory
toy
CPU
Controlling p Controlled
unit unit
le
Input/Output
Figure 1 Digital computer structure (Mano & Dime, 1997)
c) The Hybrid Systems Approach
Special attention is given in recent years to systems where a digital unit controls a
continuous process. Such systems are the subject of a new branch of engineering called
“Hybrid Systems Theory’ (Johansson, 2000; Kamil & Chui, 1996; Branicky, 1995). By
Hybrid systems we mean dynamic systems that are a combination of analog systems and
digital systems. The architecture of a hybrid system forms a two levels’ control structure
(Figure 2). On the low level, feedback loops are used for local control as part of the
analog system. On the high level, meta-control logic functions switch between different
modes of the analog system behavior.
Digital controller +_ Analog
<<" controlled
yi system
Figure 2 Hybrid system model
The analog subsystem is modeled by differential and/or difference equations. The
digital controller is usually described in the form of a Finite State Machine (FSM),
which reacts to events occurring in the controlled system. (Bencze & Frnanklin, 1994;
Branicky, 1995; Mosterman & Biswas, 2000). The Finite State Machine receives input
from the control system, and computes the output according to the state of the system,
and its state transition rules. The model of the Finite State Machine will be described
below. The general properties of hybrid systems are summarized in Table 1.
Table 1. Properties of the Hybrid System Components
Digital
Analog
Mathematical model
Finite state machine
Differential/difference
equations
Behavior
State transition according to
input and transition rules
Change as a
derivative/integral function
Graphical Model
State diagram
Stock-Flow diagram
Type of control
Event driven control
Local feedback loops
d) Relavance to k-12 Education
The inclusion of hybrid system modeling in k-12 system dynamics education has two
advantages. First, it improves the understanding of specific systems and provides
efficient means to solve control problems. Second, it provides a meaningful context for
learning concepts from discrete math, computer science and digital design.
The systems suitable for hybrid modeling belong to one of two types: 1. Systems
containing simple microprocessors such as climate control systems, elevator control
panels, traffic controllers, drugs controllers etc. 2. System involving decision making,
which effects a natural process, i.e. government decision concerning the market
mechanism, hunting regulations to control prey-predator equilibrium, municipal council
decisions to control urban development and so on.
While modeling such system, students get an opportunity to learn the principles of
digital design. This knowledge is now limited to a small group of students, mainly from
engineering and computer science classes. However, the influence of these concepts on
the modern world should make them part of general education, in the framework of
science and technology education for all. By applying the hybrid approach, students
with different interests will find relevant contexts to learn the subject of digital control —
be it a physical, medical, environmental or economical context.
3. Modeling FSM with System Dynamics Tools
Digital system may be as complex as the hardware of a digital computer, or simple as an
On/Off switch. But in its core digital design consists of simple building blocks, which
can be taught to high school students in relatively small number of lessons. This section
describes what it takes to teach student to create simple digital controllers, modeled as
Finite State Machines.
a) Fundementals of Digital Design
The root of digital design is in Boolean algebra, invented by the English mathematician
George Boole in the 19" century. Boolean Algebra enables a convenient notations for
manipulation of logic statements, which is equivalent to truth tables (Katz, 1994). To
allow designing digital machines, Boolean expressions are translated to logical gates. A
logic gate is a device, which represents a Boolean function, and can be implemented in
a variety of ways, the most useful of which are electrical circuits with switches and
relays (Hillis, 1998).
A network of logical gates creates a logical device that executes logical functions. There
are two basic types of logical devices (Mano & Dime, 1997): Combinational logic
devices are devices that receive input values and calculate output values based on the
arrangement of logic gates. Sequential logic devices are devices that calculate output
according to the inputs and the history of the system. Sequential devices contain a
memory, and can therefore execute time-varying function (Hillis, 1998). The most
important model of the sequential device is the Finite State Machine (Katz, 1994). In
the context of digital control, Finite State Machine is the main object of study for k-12
students and beyond.
b) Finite State Machine Representations
The Finite State Machine (FSM) as a fundamental model of finite digital systems is
commonly used in computer science and digital control engineering (Hopcroft et al,
2001; Mano & Dime, 1997; Varshavsky & Pospelov, 1988). As a controller, the FSM
receives input and calculates output according to the state of the systems. Didactic
examples of FSMs are automatic vending machines, combination locks, traffic-light
controllers and elevator control panel (Levin & Mioduser, 1996).
A graphical representation of the Finite-State Machine is the state diagram. The tabular
form of the state diagram is that of state table. The state diagram and the state table
describe the behavior of the FSM. The inner structure is modeled by the canonical
representation, also referred to as “abstract synthesis” (Levin & Mioduser, 1996). This
representation is useful in design and implementation of logical circuits. The structure
of FSM is that of two interacting components: a combinational scheme and a memory
register (Figure 3).
if J
Combinational
Scheme Memory
| | a(t+)}
Y; Yy
Figure 3. Structure scheme of the controlling FSM
c) Modeling in STELLA
For didactical reasons, the FSM should be modeled in the same software tool in which
the controlled system is constructed. The following section shows how the FSM can be
modeled and integrated with the controlled system using STELLA (High Performance
Systems, 1985-2001). The representation of the FSM in STELLA is based on the
canonical representation, which requires implementation of both the combinational
scheme and the memory register.
Our proposal for implementing the combinational scheme in STELLA is by using a
logical device called multiplexer. The multiplexer is a combinational logic device,
which receives several input variables and selects one of them to be its output (Mano &
Dime, 1997). For example, consider a case where the multiplexer receives two inputs
from the controlled system, x; and x2, and that the current state of the system is either a,
or a2. The multiplexer returns the new state of the system — a; or a2 - according to the
values of x), x2 and the current state.
The multiplexer-based implementation of the combinational scheme of the controller
can be represented in STELLA as tree-graph with converters as nodes. The nodes on the
lowest level of the graph represent values of input variables. The graph executes a
function of selection between the variables based on conditional statements (if-then).
The output of the multiplexer affects a corresponding flow (Figure 4).
Memory is implemented in Stella by means of a stock with two flows. Each state has a
numerical value (a;=1 a2=2 etc.), and the value of the current state is kept in the stock
(Figure 6). With each “move” of the controlling systems, the value of stock nulls via the
outflow, and a new value is added via the inflow, to be used as the value for the next
calculation. Thus, the effect of delay, which is essential to the FSM, is achieved.
Memory Output
State
ae al ee yl “|
New: iw Clear State imi
y2
\
Combinational scherne Input
x2
Figure 4. FSM implementation in STELLA
4. Examples for Use in Class
After the fundamentals have been taught, the class is now ready for hybrid modeling. In
the system dynamics tradition, the context for the modeling should be a certain problem
(Andersen& Richardson, 1980). Two examples are hereby presented.
a) Artificial Climate Control
A simulation of an artificial climate system is an example associated with both physics
and technology education. The physical process is that of air cooling and heating. The
technological aspect concerns the way an air-condition functions. To gain basic
understanding of the subject we simplify both aspects of the systems. First, we model
the physical process as change in temperature (a side effect of energy change). Second,
we focus solely on the controlling element of the air-condition system, ignoring the way
cold and hot air is produced. Further improvement of the accuracy and richness of the
model may be achieved at a more advanced stage of the teaching.
The problem presented to the students is a typical control problem:
Create a digital thermostat that will regulate the temperature of a room
according to the chosen value.
The traditional way to model this system is by using a proportional-integral controller,
which turns a radiator on and off. An alternative approach is based on a digital
controller, represented as FSM. The controller will monitor the range of the room
temperature, and will turn the radiator on and off according to the state of system. The
system is described by its continuous and discrete subsystems, input and output vectors
and a state vector (Figure 5):
y2 x
it ee
yi
Controlling FSM Controlled room
Memory
state Temp of Room
Coating ‘temp lossigain
all wa
New! Empty State
outside temp
Gaoling time eonstant
temp loss time constant
tum onott
Cool ir Temp
Combinational scheme fanosehSoon
xe
Thermostat
Figure 5. Controlling FSM and implementation in STELLA of a climate system
Input vector: x; = (Room Temp 2 Thermostat+2) ; x2 = (Room Temp < Thermostat-2)
Output vector: y; means turn on the radiator; y2 means turn off the radiator.
System states: a; means radiator on, while a, means radiator off.
The graph of the room temperature shows its regulated behavior over time (Figure 6).
The simulation may be used for what-if scenarios, to evaluate the effects of the level of
insulation, and to explore changes in the thermostat settings. More advanced models
may describe the flow in the physical system in terms of energy changes and include
more features of control such as humidity and smoke control.
1: Temp of Room
40.004
26.004
NS
10.00 T T T 1
0.00 26.00 60.00 75.00 100.00
Figure 6. Graph of temperature
b) Drug Control
Medical treatment can be viewed as adopting control strategies to regulate the behavior
of natural processes. Consider this example: a patient is to receive a dangerous drug.
The drug enters the blood stream and then moves on to the stomach, where its
effectiveness can be measured. A minimum concentration of the drug in the stomach is
required to have therapeutic value. However a high concentration of the drug might be
dangerous and even kill the patient.
Students are asked to model the process of drug absorption in the blood, and to design a
digital control unit to regulate the consumption of the drug in real time. The controller
has to monitor the amount of drug in the patient’s body, and to determine whether to
give the patient an additional dosage. The tricky point in this exercise is the issue of
delay. Since the drug has to go first through the blood stream and only then reaches the
stomach, it takes time until control decisions are felt in the body.
This problem may be given to students in biology, medical and paramedical classes, as
well as control students in technology and engineering classes. The problem can be
stated as follows:
Design an automatic controlling mechanism to treat a patient using a dangerous
drug. The concentration of the drug in the stomach should only be values
between the minimum effective and the toxic levels.
Students are expected to design the physiological model first, and then to create the
digital controlling element. Both the continuous and the discrete subsystems are to be
implemented in Stella as shown in Figure 7.
Input vector:
x, = (Blood concentration < Minimum therapeutic concentration+0.05)
X2 = (Blood concentration = toxic concentration-0.25)
Output vector: y; means set the dosage to 20 mg; y2 means set the dosage to 5 mg.
System states: ay means 20 mg dosage is being taken; a7 means 5 mg dosage is being
taken.
Cantralling FS mM
State
Os
Clear state
feu State
ya
x1
2
Drug absorption process
Drug in Stemaeh
absorption eonstant
metabolism constant
Drug in Bloodstream
Lay onoit
ynin therapeutic sone
blood concentration
toxie coneentration
absorption
metabolism
stomach volume’
blood cancentration
food volume
Body Weight
Figure 7. Drug control implementation in Stella
The graph in Figure 8 describes the drug concentration in the blood over time for the
duration of the simulation.
1: min therapeutic cone 2:toxie concentration blood concentration
1.004
0.504
0.00
0.00 26:00 60.00 76.00 100.00
Figure 8. Concentration graph
5. Summary and Research Agenda
Despite the advance made in the theory of hybrid systems, efficient teaching oriented
models of hybrid systems have not been developed. This paper is a first step towards
filling this hiatus within the system dynamics approach to k-12 education. Future
research on hybrid system modeling in education should follow two directions:
1. On the theoretical level, more examples need to be constructed and
analyzed in diverse subject matters.
2. On the didactical level, empirical research on the pedagogical aspects of
solving control problem in the hybrid approach is required.
Reference
1. Andersen David F.& Richardson George. 1980. Toward Pedagogy of System
Dynamics. TIMS Studies in the Management Sciences, vol. 14, 91-106.
2. Beneze William J.& Franklin, Gene F. 1995 A Separation Principle for Hybrid
Control System Design. JEEE Control Systems Magazine, vol.15, no.2, p. 80-
4.
3. Branicky, Michael 1995. Studies in Hybrid Systems: Modeling, Analysis, and
Control. Unpublished doctoral dissertation. MIT, Cambridge.
4. Dorf, Richard C. & Bishop Robert H. 2001. Modern Control Systems
(ninth edition). New Jersey: Prentice-Hall.
5. Forrester, W. Jay. 1968. Principles of Systems (second edition). Cambridge,
10.
11.
12.
13.
14.
15.
Mass: Wright-Allen Press.
Forrester, W. Jay. 1994. Learning through System Dynamics as Preparation to
the 21" Century. Keynote Address for System Thinking and Dynamic
Modeling Conference for K-12 Education. Concord Academy. Concord, Ma,
USA.
Hillis, W. Daniel. 1998. The pattern on the stone: the simple ideas that make
computers work. New York: Basic Books.
Hopcroft J.E, Motwani R. & Ullman J.D. 2001. Introduction to Automata
Theory, Languages, and Computation. Second Edition. New York: Addison-
Wesley.
Johansson, Karl. 2000. Hybrid Systems. Spring 2000 Lecture notes. Berkeley.
http://robotics.eecs.berkeley.edu/~johans/ee291e.html
Kamil, Ahmad & Chui Heung Wing. 1996. Hybrid Control Systems and
Optimal Control. Surprise, vol. 4.
http://www.doc.ic.ac.uk/~nd/surprise_96/journal/vol4/ahak/report.html
Katz, Randy H. 1994. Contemporary logic design. Redwood City, CA :
Benjamin Cummings.
Mano, M. Morris & Dime, Charles R. 1997. Logic and Computer Design
Fundamentals. New Jersey: Prentice-Hall.
Mosterman Pieter J. & Biswas Gautam B. 2000. A comprehensive
methodology for building hybrid models of physical systems. Artificial
Intelligence, 121, 171-209.
Penner, David. 2001. Cognition, Computers, and Synthetic Science: Building
Knowledge and Meaning Through Modeling. In Review of Research
Education 25, Secada Walter G. (Editor). American Education Research
Association: Washington; 1-35.
Varshavsky, V.I. & Pospelov, P.A. 1988. Puppets without Strings. Trans. By
A. Dandarov. Moscow: Mir Publishers.
Back to the Top
