Introduction to Engineering Systems, ESD.00
System Dynamics
Lecture 3
Dr. Afreen Siddiqi
From Last Time: Systems Thinking
•
“we can’t do just one thing” – things are
interconnected and our actions have
numerous effects that we often do not
anticipate or realize.
• Many times our policies and efforts aimed
towards some objective fail to produce the
desired outcomes, rather we often make
matters worse
• Systems Thinking involves holistic
consideration of our actions
Decisions
Goals
Environment
Image by MIT OpenCourseWare.
Ref: Figure 1-4, J. Sterman, Business Dynamics: Systems
Thinking and Modeling for a complex world, McGraw Hill, 2000
Dynamic Complexity
• Dynamic (changing over time)
• Gov
Governed
erned by feedback
feedback (actions feedback
feedback on themselv
themselves)
es)
• Nonlinear (effect is rarely proportional to cause, and what happens locally often doesn’t
apply in distant regions)
• History‐dependent (taking one road often precludes taking others and determines your
destination, you can’t unscramble an egg)
• Adaptive (the capabilities and decision rules of agents in complex systems change over
time)
• Counterintuitive (cause and effect are distant in time and space)
• Policy resistant (many seemingly obvious solutions to problems fail or actually worsen the
situation)
• Characterized
Ch
i d by
b trade‐offs
d ff (the
( h long
l
run is
i often
f
different
diff
from
f
the
h sh
hort‐run response,
due to time delays. High leverage policies often cause worse‐before‐better behavior while
low leverage policies often generate transitory improvement before the problem grows
worse.
Modes of Behavior
Exponential Growth
Time
Oscillation
Time
Goal Seeking
S-shaped Growth
Time
Growth with Overshoot
Time
Overshoot and Collapse
Time
Time
Image by MIT OpenCourseWare.
Ref: Figure 4-1, J. Sterman, Business Dynamics: Systems
Thinking and Modeling for a complex world, McGraw Hill, 2000
Exponential Growth
• Arises from positive (self‐reinforcing)
feedback.
• In pure exponential growth the state of
the system doubles in a fixed period of
time.
e
• Same amount of time to grow from
1 to 2, and from 1 billion to 2
billion!
• Self‐reinforcing feedback can be a
declining loop as well (e.g. stock prices)
• Common example: compound interest
interest,
population growth
State of the
system
Time
Net
increase
rate
R
+
State of
the
system
+
Ref: Figure 4-2, J. Sterman, Business Dynamics: Systems
Thinking and Modeling for a complex world, McGraw Hill, 2000
Image by MIT OpenCourseWare.
Exponential Growth: Examples
CPU Transistor Counts 1971-2008 & Moore’s Law
2,000,000,000
1,000,000,000
Quad-Core Itanium Tukwila
Dual-Core Itanium 2
GT200
POWER6
RV770
G80
Itanium 2 with 9 MB cache
K10
Core 2 Quad
Core 2 Duo
Itanium 2
Cell
Transistor count
100,000,000
K8
Barton
P4
Curve shows ‘Moore’s Law’;
Transistor count doubling
every two years
10,000,000
Atom
K7
K6-III
K6 PIII
PII
K5
Pentium
486
1000,000
386
286
100,000
8088
10,000
8060
2,300
4004
8008
1971
1980
1990
2000
2008
Date of Introduction
Ref: wikipedia
Image by MIT OpenCourseWare.
Some Positive Feedbacks
underlying Moore
Moore’ss Law
+
Computer performance
+
+
Design tool
capability
R2
Chip performance
relative to competitors
Differentiation
advantage
+
R5
Design bootstrap
Applications
for chips
+
R4
Design and
fabrication
experience Learning
by doing
+
Transistors per chips
+
R3
+
R1
Design and fabrication
capability
+
+
Demand for
Intel chips
New uses,
New needs
Price premium
+
Price premium
for performance
+
+
Revenue
R&D budget, Investment
in manufactutring
capability
+
+
Image by MIT OpenCourseWare.
Ref: Exhibit 4
4-7
7, J.
J Sterman
Sterman, Instructor’s
Instructor s Manual
Manual,
Business Dynamics: Systems Thinking and Modeling for a
complex world, McGraw Hill, 2000
7
Goal Seeking
• Negative loops seek balance, and
equilibrium, and try to bring the system
system
to a desired state (goal).
• Positive loops reinforce change, while
negative
ega e loops
oops counteract
cou e ac cchange
a ge o
or
disturbances.
• Negative loops have a process to
comp
pare desired state to current state
and take corrective action.
• Pure exponential decay is characterized
byy its half life – the time it takes for half
the remaining gap to be eliminated.
Goal
State of
the system
Time
+
State of
the system
Goal (desired state
of system)
-
B
Discrepancy
Corrective
action
+
+
Image by MIT OpenCourseWare.
Ref: Figure 4-4, J. Sterman, Business Dynamics: Systems
Thinking and Modeling for a complex world
world, McGraw Hill,
Hill 2000
Oscillation
• This is the third fundamental mode of
behavior.
beha
vior.
State of
the system
Goal
• It is caused by goal‐seeking behavior,
but results from constant ‘over‐shoots’
over‐shoots
and ‘under‐shoots’
Measurement, reporting
and perception delays
Action delays
Delay
+
State of
the system
lay
De
B
Goal (desired state
of system)
-
Discrepancy
+
Del
ay
• The over
over‐shoots
shoots and under‐shoots
under shoots result
due to time delays‐ the corrective action
continues to execute even when system
reaches desired state giving rise to the
oscillations.
Time
Corrective
action
+
Administrative and decision
making delays
Image by MIT OpenCourseWare.
Ref: Figure 4-6, J. Sterman, Business Dynamics: Systems
Thinking and Modeling for a complex world, McGraw Hill, 2000
Interpreting Behavior
• Connection between structure and behavior helps in generating hypotheses
• If exponential growth is observed ‐> some reinforcing feedback loop is dominant
over the time horizon of behavior
• If oscillations are observed, think of time delays and goal‐seeking behavior.
behavior, the
the
future maybe different. Dormant
• Past data shows historical behavior,
underlying structures may emerge in the future and change the ‘mode’
• It is useful to think what future
•
future‘modes
modes’ can be,
be how to plan and manage them
• Exponential growth gets limited by negative loops kicking in/becoming dominant
l t on
later
10
Limits of Causal Loop Diagrams
•
Causal loop diagrams (CLDs) help
– in capturing mental models, and
– showing interdependencies and
– feedback processes.
•
CLDs cannot
– capture accumulations (stocks) and flows
– help in determining detailed dynamics
Stocks, Flows and Feedback are central
concepts in System Dynamics
Stockss
Stock
• Stocks are accumulations, aggregations,
summations over time
Inflow
Outflow
Stock
Image by MIT OpenCourseWare.
• Stocks characterize/describe the state of the
system
• Stocks change with inflows and outflows
Stock
• Stocks provide memory and give inertia by
accumulating past inflows; they are the sources
of delays.
• Stocks, by accumulating flows, decouple the
inflows and outflows of a system and cause
variations such as oscillations over time.
Flow
Valve
(flow regulator)
Source or Sink
Image by MIT OpenCourseWare.
Mathematics
Ma
thematics of Stocks
Stocks
• Stock and flow diagramming were based
on a hydraulic metaphor
Inflow
Outflow
Stock
Image by MIT OpenCourseWare.
• Stocks integrate their flows:
Inflow
Stock
•
The net flow is rate of change of stock:
Outflow
Image by MIT OpenCourseWare.
Ref: Figure 6-2, J. Sterman, Business Dynamics: Systems
Thinking and Modeling for a complex world, McGraw Hill, 2000
Stocks and Flows Examples
Field
Stocks
Flows
Mathematics, Physics and Engineering
Integrals, States, State
variables, Stocks
Derivatives, Rates of
change, Flows
Chemistry
Reactants and reaction products
Reaction Rates
Manufacturing
Buffers, Inventories
Throughput
Economics
Levels
Rates
Accounting
Stocks, Balance sheet items
Flows, Cash flow or Income
statement items
Image by MIT OpenCourseWare.
Ref: Table 6-1, J. Sterman, Business Dynamics: Systems
Thinking and Modeling for a complex world, McGraw Hill, 2000
Snapshot Test:
Freeze the system in time – things that are measurable in
the snapshot are stocks.
stocks
Example
Net flow
(units/second)
20
10
0
Image by MIT OpenCourseWare.
Ref: Figure 7-2, J. Sterman, Business Dynamics: Systems Thinking and Modeling for a complex world
world, McGraw Hill,
Hill 2000
15
Example
Flows (units/time)
100
50
0
0
5
10
In flow
15
20
Out flow
Image by MIT OpenCourseWare.
Ref: Figure 7-4, J. Sterman, Business Dynamics: Systems Thinking and Modeling for a complex world
world, McGraw Hill,
Hill 2000
16
Flow Rates
• Model systems as networks of stocks
and flows linked by information
information
feedbacks from the stocks to the
rates.
• Rates can be influenced by stocks,
other constants (variables that change
very slowly) and exogenous variables
(variables outside the scope of the
model).
• Stocks only change via inflows and
outflows.
Net rate of change
Stock
Image by MIT OpenCourseWare.
Net rate of change
Stock
Constant
Exogenous variable
Image by MIT OpenCourseWare.
17
Auxiliary Variables
Net birth rate
Population
+
?
?
Ref: Figure 8-?, J. Sterman, Business Dynamics: Systems
Thinking and Modeling for a complex world, McGraw Hill, 2000
Food
Image by MIT OpenCourseWare.
• Auxiliary variables are
neither stocks nor flows,
but intermediate
concep
pts for clarityy
• Add enough structure to
make polarities clear
18
Aggregation
Aggreg
ation and Boundaries ‐ I
• Identifyy main stocks in the syystem
and then flows that alter them.
• Choose a level of aggregation and
b
boundaries
d i for
f the
th system
t
• Aggregation is number of internal
stocks chosen
• Boundaries show how far
upstream and downstream (of the
flow)) the
fl
h system is modeled
d l d
19
Aggregation
Aggreg
ation and Boundaries ‐ II
II
• One can ‘challenge the clouds’, i.e.
make previous sources or sinks explicit.
explicit
• We can disaggregate our stocks further
to capture additional dynamics.
• Stocks with short ‘residence time’
relative to the modeled time horizon
can be lumped together
• Level of aggregation depends on
purpose of model
• It is better to start simple and then add
details.
20
From Structure to Behavior
• The underlying structure of the
system defines the time‐based
behavior.
• Consider the simplest case: the
state of the system is affected by its
rate of change.
Ref: Figure 8-1, & 8-2 J. Sterman, Business Dynamics:
Systems Thinking and Modeling for a complex world
world, McGraw
Hill, 2000
Net
increase
rate
+
+
State of
the
system
R
Net rate of change
Stock
Images by MIT OpenCourseWare.
Population Growth
•
Consider the population Model:
Population
+
•
The mathematical representation of this
structure is:
Net birth rate
+
R
Fractional net
birth rate
Image by MIT OpenCourseWare.
Net birth rate = fractional birth rate * population
Ref: Figure 8-2, J. Sterman, Business Dynamics: Systems Thinking and Modeling for a complex world
world, McGraw Hill,
Hill 2000
2000
Note: units of ‘b’, fractional
growth rate are 1/time
Phase‐Plots
Phase
Plots for Exponential Growth
• Phase plot is a graph of system state
vs. rate of change of state
• If the state of the system is zero,
zero the
rate of change is also zero
Net inflow rate (units/time)
• Phase plot of a first‐order, linear
positive feedback system is a straight
straight
line
dS/dt = Net Inflow Rate = gS
g
1
0
• The origin however is an unstable
equilibrium.
Ref: Figure 8-3, J. Sterman, Business Dynamics: Systems
g and Modeling
g for a comp
plex world,, McGraw Hill,,2000
Thinking
0
State of the system (units)
Unstable equilibrium
Image by MIT OpenCourseWare.
23
Time Plots
Plots
• Fractional growth rate g =
0.7%/time unit
Net inflow (units/time)
10
• Initial state state = 1.
8
t = 1000
6
t = 900
4
t = 800
2
• State doubles every 100 time units
0
t = 700
0
128
256
512
1024
State of system (units)
Ref: Figure 8-4, J. Sterman, Business Dynamics: Systems
g and Modeling
g for a comp
plex world,, McGraw Hill,, 2000
Thinking
State of the system (units)
10.24
State of the system
(left scale)
512
5.12
256
2.56
128
0
Net inflow
(right scale)
0
200
400
600
800
Net inflow (units/time)
• Every time state of the system
doubles, so too does the absolute
rate of increase
Behavior
1024
1.28
0
1000
Time
Images by MIT OpenCourseWare.
24
Rule of 70
• Exponential growth is one of the most powerful processes.
• The rate of increase grows as the state of the system grows.
• It has the remarkable property that the state of the system doubles in fixed
period of time.
• If the doubling time is say 100 time units, it will take 100 units to go from 2
to 4, and another 100 units to go from 1000 to 2000 and so on.
• To find doubling time:
Negative Feedback and
Exponential Decay
Decay
• First‐order linear neggative feedback
systems generate exponential decay
Net outflow rate
S state of the
system
+
the
• The net outflow is proportional to the
size of the stock
• The solution is given by: S(t) = So
• Examples:
+
B
Fractional decay
rate d
Image by MIT OpenCourseWare.
dt
e‐dt
Net Inflow = -Net Outflow = -d*S
d: fractional decay rate [1/time]
Reciprocal of d
Ref: Figure 8-6, J. Sterman, Business Dynamics: Systems
Thinking and Modeling for a complex world, McGraw Hill, 2000units in stock.
is average lifetime
Phase Plot for Exponential
Phase
ExponentialDecay
Decay
• In th
he ph
hase‐pllot, th
he net rate off
change is a straight line with negative
slope
• The origin is a stable equilibrium, a
minor perturbation in state S increases
the decay rate
th
t to b
bring
i systtem backk to
t
zero – deviations from the equilibrium
are self‐correcting
Net inflow rate (units/time)
Net Inflow Rate = - Net Outflow Rate = -dS
0
State of the system (units)
Stable equilibrium
1
-d
• The goal in exponential decay is implicit
Image by MIT OpenCourseWare.
and equal to zero
Ref: Figure 8-7, J. Sterman, Business Dynamics: Systems
Thinking and Modeling for a complex world
world, McGraw Hill,
Hill 2000
Negative Feedback with Explicit Goals
General Structure
• In general, negative loops have
non‐zero goalls
dS/dt
S state of the
system
• Examples:
Net inflow rate
-
S* desired state
of the system
+
• The corrective action determining
net flow to the state of the
AT adjustment time
system is : Net Inflow = f (S, S*)
B
-
+
Discrepancy (S* - S)
Image by MIT OpenCourseWare.
plest formulation is:
• Simp
Net Inflow =
Discrepancy/adjustment time =
(S*‐S)/AT
AT: adjustment time is also known
as time constant for the loop
Ref: Figure 8-9, J. Sterman, Business Dynamics: Systems
Thinking and Modeling for a complex world, McGraw Hill, 2000
Phase Plot for Negative Feedback
with Non‐Zero
Non Zero Goal
• In the phase‐plot, the net rate of change is
a straight line with slope ‐1/AT
Net inflow rate (units/time)
• The behavior of the negative loop with an
explicit goal is also exponential decay, in
which the state reaches equilibrium when
S=S*
Net Inflow Rate = -Net Outflow Rate = (S* - S)/AT
1
-1/AT
S*
0
State of the system (units)
• If the initial state is less than the desired
Stable equilibrium
state, the net inflow is positive and the
state increases (at a diminishing rate) until
S=S*. If the initial state is greater than S*,
Image by MIT OpenCourseWare.
the net inflow is negative and the state
falls until it reaches S*
Ref: Figure 8-10, J. Sterman, Business Dynamics: Systems
Thinking and Modeling for a complex world
world, McGraw Hill,
Hill 2000
Half Lives
Half‐Lives
• Exponential decay cuts quantity remaining in half in fixed period of time
• The ‘half‐life’ is calculated in similar way as doubling time.
• The system state as a function of time is given by:
State of system
Initial Gap
S (t )
S − (S − S0 ) e
) =Desired
*
*
−t /τ
State
Gap Remaining
• Th
he exponentiiall term decays from 1 to zero
as t tends to infinity.
• Half life is given by value of time th :
Time Constants and Settlingg Time
Fraction of Initial
Gap Corrected
0
e‐00 = 1
1 1= 0
1‐1
τ
e‐1 = 0.37
1‐e‐1 = 0.63
2τ
e‐2 = 0.14
1‐e‐2 = 0.87
3τ
e‐3 = 0.05
1‐e‐3 = 0.95
4τ
e‐4 = 0.02
1‐e‐4 = 0.98
5τ
e‐5 = 0.007
1‐e‐5 = 0.993
0.6
0.4
0.2
exp(-1/AT)
exp(-2/AT)
0
1 - exp(-3/AT)
0.8
1 - exp(-2/AT)
1.0
1 - exp(-1/AT)
• The steady‐state
•
steady‐state is not reached
technically in finite time because the
rate of adjustment keeps falling as
the desired state is approached.
Fraction of Initial Gap
Remaining
Fraction of initial gap remaining
• For a first order, linear system with
negative feedback,
feedback the system
reaches 63% of its steady‐state value
in one time constant, and reaches
98% of its steady state value in 4
time constants.
Time
0
1AT
2AT
exp(-3/AT)
3AT
Time (multiples of AT)
Ref: Figure 8-12, J. Sterman, Business Dynamics: Systems
Thinking and Modeling for a complex world, McGraw Hill, 2000
Image by MIT OpenCourseWare.
MIT OpenCourseWare
http://ocw.mit.edu
ESD.00 Introduction to Engineering Systems
Spring 2011
For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms.