Ref: Figure 1-4, J. Sterman, Business Dynamics: Systems. • Systems Thinking
involves holistic. Thinking and Modeling for a complex world, McGraw Hill, 2000.
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 Exponential Decay 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.
