On the underlying structure of system dynamics models
Gaurav S. Chaudhari
Department of Management
College of Business Administration, University of Dubai
PO Box 14143, Dubai, United Arab Emirates
Email: gchaudhari@ ud.ac.ae
Phone: 971-050-149-8033
Robert H. Sturges
Department of Industrial and Systems Engineering, Virginia Tech
103 Durham Hall, Blacksburg, VA 24061, USA
Email: sturges@vt.edu
Phone: 1-540-231-7420
Corina Sandu
Department of Mechanical Engineering, Virginia Tech
104 Randolph Hall, Blacksburg, VA 24061, USA
Email: csandu@ vt.edu
Phone: 1-540-231-7467
ABSTRACT
The underlying structure of system dynamics models is that of a proportional feedback
controller. We propose a broader framework for system dynamics models, where systems are
modeled using a combined feedback-feedforward structure. While the traditional structure for
system dynamics models only uses proportional feedback of error for control, the proposed
structure for information feedback employs the use of proportional, integral and derivative
(PID) error. Hence, existing system dynamics models only use a small subset of the proposed
structure for modeling systems. We argue that the proposed structure provides a more flexible
framework for modeling and designing systems.
KEYWORDS
Feedback, Feedforward, Structure, System Dynamics
INTRODUCTION
Forrester (1961) argues that all “flows” within a system are integrated by information feedback
networks. He developed the initial framework for system dynamics modeling to capture the
impact of these information networks on system behavior. He describes the system dynamics
modeling framework as “one of building models of companies and industries to determine how
information and policy create the character of an organization”. He also mentions that most
mathematical models found in management and economics literature are stable steady state
models and that the practical utility of such models in dealing with economic systems has not
been significant. One of the strengths of the system dynamics modeling technique is that it is not
limited to modeling steady state conditions.
The basic principles of system dynamics are concisely summarized by Wolstenhome
(1989,1990). System dynamics is often considered to occupy a position between that of
operations research and systems thinking. Keys (1988) concluded that the exact position of
system dynamics remains unresolved. However, scientists from both domains can relate to it.
Forrester (1994) examines the methodologies followed by operations research, systems thinking
and system dynamics practitioners to determine how these approaches overlap and what their
unique contributions are to system analysis.
Even though system dynamics uses the formalisms of differential equations to simulate system
behavior, diagramming tools are used to communicate the assumptions about the structure of the
model. Most system dynamics models are represented using “causal loop diagrams” (CLDs) and
“stock and flow diagrams” (SFDs). Sterman (2000) provides an excellent reference for using
CLDs and SFDs to build system dynamics models.
Since the interaction of complex causal relationships cannot be fully explored through mental
simulation, computers are invariably used to simulate system dynamics models (Sterman 1994).
Computer simulation allows system dynamics practitioners to rigorously study the impact of
these causal relationships and determine how they lead to counter-intuitive behavior (Forrester
1970).
STRUCTURE OF SYSTEM DYNAMICS MODELS
It is the structure of the system dynamics model, which is the source of the modes of behavior
that the model demonstrates. These modes are caused by the interaction of different feedback
loops, each of which may involve non-linearities, delays, accumulation, and draining processes.
System dynamics modeling aims to explain behavior by providing a causal theory, and then
using that theory as the basis for designing policy interventions into the system structure (Lane
2008). The purpose of these policy interventions is to change the behavior and improve the
performance of the system.
Schmidt and Taylor (1970) define a system as “a collection of entities, e.g. people or machines,
which act and interact together toward the accomplishment of some logical end”. The “states” of
the system can be defined as the collection of variables necessary to define the system at given
point in time. The choice of state variables generally depends on the objectives of the study.
Systems are generally classified as either discrete or continuous in nature depending on the
behavior of the state variables of the system with respect to time. A discrete system is one in
which the state variables change instantaneously at separated points in time, while a continuous
system is one where the state variables change continuously with respect to time. System
dynamics modeling assumes that the state variables of the system are continuous in nature.
Models provide an effective means for understanding complex phenomena, which may not be
easily understood by simple observation. A model can provide information at a lower cost
quickly, as compared to the actual system. Askin and Standridge (1993) suggest that the primary
use for models include the following:
e Optimization - Finding the best values of decision variables.
Performance prediction - Predicting performance under different conditions.
Control - Selecting the desired rules to control the system.
Insight - Gaining a better understanding of the system.
Justification - Using the results as a tool to support decisions.
eecee
The primary motivation behind the development of the system dynamics methodology was to
gain insight into the operations of complex dynamic systems, which Forrester (1961) felt were a
barrier to learning. Since its inception, the system dynamics modeling technique has been used to
study complex business and social systems through the understanding of the different feedback
paths in the system.
The structure of system dynamics models stems from servomechanism theory. A
servomechanism is a system that uses information feedback to control the performance of the
system. This concept of an information feedback system provides the underlying structure
system dynamics models. Forrester (1961) mentions, “The first and most important foundation
for industrial dynamics is the concept of servomechanisms (or information feedback systems) as
evolved during and after World War II.” He also adds that “the information feedback system
will become a principal basis for an underlying structure to integrate the separate facets of the
management process”. An information feedback system is said to exist whenever the
environment leads to a decision that results in action which affects the environment and thereby
influences future decisions (Forrester 1961).The study of information feedback systems reveals
how information is used for control. In order to achieve a desired system response, it is necessary
to understand how the amount of corrective action and the associated time delays impact the
performance of the system. The behavior of an information feedback system is governed by its
structure, delays within the system, and the amplification of the system. Hence, the design of an
information feedback system must consider these three characteristics if it is to be successful.
PROPOSED FEEDBACK STRUCTURE
The Proportional-Integral-Derivative (PID) structure is a generic feedback control structure that
is widely used in industrial control systems (Ogata 2005). The PID control algorithm
continuously measures a process variable and compares it with a desired set point. The error
between the two quantities is used to adjust the process. The algorithm used to control the
process involves three separate adjustments:
Proportional adjustment: This is the adjustment based on the current error
Integral adjustment: This is the adjustment based on the sum of recent errors
Derivative adjustment: This is the adjustment based on the rate of change of error
The weighted sum of these three adjustments is used to control the process. The weights chosen
are dependent on the requirements of the process. Some applications of this control algorithm
may not require all three modes for control. In such cases, the weights for the modes that are not
desired can be set to zero.
The following are combinations of these three modes, which are commonly used in practice:
Proportional (P) control
Proportional- Integral (PI) control
Proportional-Integral- Derivative (PID) control
(Ogata 2005)
The PID control algorithm continuously monitors a process variable and compares it with a
reference point. The reference point can be fixed or variable over time. Any difference between
the measured state variable and its reference point represents an error. This error is then used to
control a flow control variable, which is sometimes referred to as the “manipulated variable”
(MV). It should be noted that there can exist an error term for each state variable in the model.
The proportional, integral and derivative component of each error term can then be used to
determine the value of the manipulated variable. This provides a large design space for the
feedback control system. The choice of which error values and their corresponding modes are
used to control the system depends on the desired response characteristics for the system. If the
error associated with each state variable is used for control, it can lead to an increase in system
responsiveness. However, such a control scheme can also increase the amplification of the
system (Chaudhari 2008).
When using the PID feedback structure, the manipulated variable is calculated as the sum of the
proportional, derivative and integral adjustment, as shown below in Equation 1.
MV (t) = Pag tag +Dagj (1)
The proportional adjustment, Pag; makes an adjustment that is proportional to the error, E (t). Itis
defined as a product of the error and a proportional gain constant, Ky. The proportional
adjustment is defined as shown in Equation 2
Pag = Kp* E(t) @)
A high proportional gain shall lead to a large change in the manipulated variable for a given
change in error. While this may make the process responsive to change, it can also make the
process unstable if the proportional gain is made very high. The integral adjustment depends on
both the magnitude and duration of the error term.
The integral adjustment, Iagj is defined as a product of integral of the error and an integral gain
constant, K;. Hence, the integral adjustment is defined as shown in Equation 3.
I
t
adj = K,*[E(elde (3)
The integral adjustment, when used in addition to the proportional adjustment can enable a
system to react in a more responsive manner to disturbances. The impact of the integral
adjustment depends on the integral gain constant that is used. However, since the integral
adjustment is based on the accumulation of past errors, it can cause the system to overshoot the
desired level.
The derivative adjustment, Dag depends on the rate of change of the error term. It is defined as a
product of the derivative of the error with respect to time, and a derivative gain constant as
shown below in Equation 4.
dE (t)
Dig =Kg*—— (}
adj dt a)
The impact of the derivative adjustment depends on the choice of the derivative gain constant.
This mode of adjustment is often used in conjunction with integral adjustment as it reduces the
overshoot that may be caused by the integral adjustment. This mode of adjustment is sensitive to
noise and may cause the system to become unstable if a large derivative gain is used.
The choice of gain constants for the PID feedback system reflects the policies of the
organization. If the organization has an aggressive policy for correcting discrepancies between
the state variables and their ideal values, then the gain constants would be set to large values.
Conversely, if the organization has a mild policy for correcting discrepancies between the state
variables and their values, then the gain constants would be set to small values. Ultimately, the
choice of gain constants is dependent on the desired response characteristics of the system.
Set
Points
-———>| P,,, = Kk, * E(t)
+ +
Error, E(t) t ep
+———»| I1.,=K, * [EC t)dt Process > State
0 Outputs
2 +
_ps dE(t)
Diy =Ka at
Figure 1: Proportional-Integral-Derivative (PID) feedback structure
COMBINED FEEDBACK-FORWARD STRUCTURE
A feedback system is one that reacts to changes in its environment, usually trying to maintain
some desired state. On the other hand, a feedforward system reacts to a measured disturbance in
a pre-defined manner. Hence, the disturbance is measured and action is taken before the
disturbance affects the system. The difference between the two schemes for control can be
discussed in the context of an example. Consider the cruise control mechanism of an automobile,
a well-known feedback system. The purpose of a cruise control mechanism is to maintain the car
at a steady speed. When the car encounters an uphill slope, the car would slow down. This
difference between the actual and desired speed would generate an error signal, which would
cause the throttle to open further, and hence bring the car back to the desired speed. If the cruise
control mechanism had been using a feedforward mechanism for control, it would have used a
sensor to detect the uphill slope and opened the throttle in anticipation of the decrease in speed.
Hence, the car would not lose any speed before a correction was made. In the context of this
example, it should be noted that there are several other factors such as temperature, wind,
altitude etc., which can impact the speed of the car. Since the relationship of these variables with
the speed of the car cannot be modeled accurately, it would not be possible for a cruise control
mechanism to operate solely with feedforward control. Feedback and feedforward control
structures are not mutually exclusive. One could adopt a combined feedback feedforward control
structure. Feedforward control would lead to a quick response, while feedback control would
correct any error that arose due to the pre-set response.
A standalone feedforward controller is a good choice if the following conditions can be met:
Disturbances are known before they impact the system
Disturbances are measureable
There are no significant unmeasured disturbances
Heylighen and Joslyn (2001) discuss the advantages and disadvantages of feedback and
feedforward control. These are summarized below in Table 1. It can be seen that the two control
structures are actually complementary to each other. By using a combined structure, one can
increase responsiveness to known disturbances, while being robust to unknown disturbances.
Feedforward C ontrol Feedback Control
Advantages 1. Compensates for| 1. Zero steady state offset
disturbance before it| 2. Is appropriate for use with
affects the system all disturbances.
2. Does not impact the|3. Does not require any
stability of the control additional sensor for each
system disturbance
Disadvantages 1. Shall require a sensor and| 1. Requires the disturbance
model for each disturbance to impact the system
2. Can’t eliminate steady before any response is
state offset made
3. The controlled state | 2. Affects stability of control
variable is not monitored. system.
Hence, no error correction
is possible
4. Tends to require more
calculation/analysis in
design phase
Table 1: Comparison of feedback and feedforward control
Figure 2 shows the proposed structure for the combined feedback-feedforward control. The
feedback control effort focuses on ensuring that one or more states remain at specified set points.
The process output is continuously monitored to compare the values of the state variables with
their desired set points. If there is a discrepancy, the error is multiplied by an appropriate gain
value and fed back to the flow control variable for the process. If there is a measurable
disturbance, its value is measured and multiplied by an appropriate feedforward gain. This
information is transmitted to the flow control variable for the process. Hence, the combination of
the feedback and feedforward information flows determines the setting for the flow control
variable of the process.
It should be noted that in the above framework, the feedback control structure chosen depends on
the nature of the process and the state variables of interest. One could use a PID control structure
if appropriate. If not, one could use a P, or PI structure. A combined feedback-feedforward does
not restrict this choice in any way.
The formulation of the feedforward control loop also depends on the nature of the disturbance
and its impact on the process. Depending on the specifics, the disturbance value can be
multiplied by an appropriate gain value to determine its effect on the flow value for the process.
The choice of gain depends on the units of the disturbance.
Disturbance
Set Feedforward Control
Points
+ +
Error, E(t) ie
PID Feedback
Process
Control > State
Outputs
\¢
Figure 2: Combined feedback- feedforward control structure
CONCLUSIONS
The primary contribution of this work is to highlight the limitation of the existing structure of
system dynamics models. Existing stock and flow control models only use proportional feedback
of error to control the process. Hence, existing models only use a subset of the proposed PID
structure. We hypothesize that by using integral and derivative feedback of error in addition to
the proportional feedback of error, one could better model the dynamics of systems. Such a
structure also provides a much larger design space for policy design, as compared to traditional
models which only utilize proportional feedback of error.
Traditionally, system dynamics models use a feedback control structure to explain the dynamics
of systems. We examine the possibility of using a combined feedback-feedforward structure to
model the behavior of systems. The current practice of modeling systems as strictly feedback
control systems is based on the premise that systems do not respond to disturbances before they
impact the system. This is contrary to the behavior of several business and social systems, which
actively respond to disturbances in anticipation of the impact of the disturbance on the system.
Further, since feedback and feedforward structures are complementary to each other, a combined
structure forms a much broader and robust framework for modeling and designing system
behavior.
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