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Higher order feedback loops One stock – first order system More stocks – higher order Behavior Second order negative feedback – oscillating system Second order positive feedback – exponential growth and oscillations (unstable)

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Higher order feedback loops Second order negative feedback pendulum

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Higher order feedback loops

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Generalization

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Higher order feedback loops

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Second order positive feedback loop

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Higher order feedback loops

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Delays Delays occur frequently in many systems. When a pollutant is dumped into a river, it takes time to dissipate. Delays are divided into two types: (a) Material delays - resulting from the time involved in processing materials. (b) Information delays - resulting from time in perceiving and acting upon information.

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Material delay example Construction of apartments in a large city. Builders construct apartments in response to the gap between the total number of apartments desired and available. Start with 10,000 apartments. Increase to 15,000. Delay - completion time 2 years.

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Material delay example

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Initiation time = 1 Apartments in construction = 0 Completion rate=Apartments in construction/Completion time Total number of apartments =10,000 Desired apartments = 10,000 + STEP(5000,1) Difference= Desired apartments – Total number of apartments Completion time = 2

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Material delay example

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Information delay example Joes Cement Factory makes and sells cement blocks Monday through Friday. Each morning Joe has to decide how many blocks to make, and he relies on the average sales over the past five days in making his decision.

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Information delay example

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Cement blocks to make = STEP (20,10) Sales = STEP (20,5) Making rate = SMTH (Sales, 5) – Cement blocks to make

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Information delay example

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Solution interval DT System Principle 11 Solution interval is in all level equations and no others

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Solution interval DT DT Solution interval Period of measurement Delta time Time step DT is the time period in which the level is changed by the rate

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DT in Vensim The best size of TIME STEP is determined by the following considerations: TIME STEP should allow test inputs to be accessed regularly. TIME STEP should allow data to be accessed with appropriate regularity. TIME STEP should be smaller than 1/3 of the shortest time constant in the model (not applicable with automatic step size adjustment in Runge-Kutta integration). TIME STEP should be smaller than the shortest period for which a significant change in model behavior is at all likely.

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DT in Vensim An inappropriately long TIME STEP leads to incorrect behavior. In general if you see oscillation with a frequency that is near to twice TIME STEP you should test TIME STEP to see if it has an appropriate value. If you are using Runge-Kutta integration with automatic step size adjustment, the third and forth considerations do not apply. Vensim will automatically determine how small it needs to make TIME STEP in order to achieve the desired accuracy or issue an error if it is unable to do so.

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