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# System dynamics discovering of how's and why’s

What is System Dynamics.
Application in various fields

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### System dynamics discovering of how's and why’s

1. 1. System Dynamics -Discovering of How's and Why’s Ms. Princy Randhawa Assistant Professor Department of Mechatronics Engineering Thursday, October 27, 2016 1
2. 2. Outline  System Dynamics Discipline Need for studying Dynamic System. Key concepts and Goals of the system dynamics discipline. Highlight the breadth of the System dynamics discipline. The mathematical foundation of system dynamics.  What is a system? What makes the system Dynamic?  State Space Representation Thursday, October 27, 2016 2
3. 3. Introduction-System Dynamics Principle Thursday, October 27, 2016 3 Image from Wikimedia Commons Dynamics Modeling Parameters Identification Stability Forces/load Motion Control Feedback Feed-forward Robustness Sensors/Actuators
4. 4. Breadth and Mathematical foundation of System Dynamics Discipline Thursday, October 27, 2016 4 Aerospace Engineers Automotive Engineers Electrical Engineers Bio Engineers Population Scientists System Biologists Kinesiologists Economists Computer Scientists Video game industry Mathematician Electrochemists SYSTEM ENGINEERS Calculus Differential Equations Linear Algebra Complex Numbers
5. 5. System Dynamics Define the term “system” Define what makes the system “dynamic” Highlight the defining role of “memory” in dynamic systems Thursday, October 27, 2016 5
6. 6. What is a System? Thursday, October 27, 2016 6 Cruise Control System Autopilot System Taxation System Cardiopulmonary System Economic System Communication System Governance System Health Care System Grading System Tropical Storm System “Complex Systems” “System of Systems”
7. 7. Contd… Collection of components Non Trivial Interactions Well defined boundary separating the system from environment. Thursday, October 27, 2016 7
8. 8. What is a system? (Mathematical Definition) Thursday, October 27, 2016 8 Inputs u(t) Outputs y(t) A mapping from time dependent inputs to time dependent outputs (Casual Definition)
9. 9. What makes a system Dynamic?  Inputs changes with time?  Outputs changes with time? Thursday, October 27, 2016 9 Currency Exchange System500 Rupees 32,500 \$ 100 Rupees 6,500 \$ 1000 Rupees 65,000 \$ Currency Exchange System Currency Exchange System
10. 10. Contd… Thursday, October 27, 2016 10 Food Song ? A system is “dynamic” if and only if it has a memory
11. 11. Static, Dynamic and clairvoyant systems  A System is static if its output now is a function of its input now.  A System is dynamic (causal) if its output now is a function of its input both now and in the past.  A System is clairvoyant if its output now is a function of its input in the future. Thursday, October 27, 2016 11 In order for a system to be dynamic, it must have a memory
12. 12. State Space Representation  Introduce State Variable  Introduce Memory Operator  Introduce integration w.r.t. time as a memory operator  Introduce the continuous time state space model Thursday, October 27, 2016 12
13. 13. Variables in a Dynamic Systems Model Thursday, October 27, 2016 13  Input Variables Control Inputs u(t) Exogenous w(t) (“Disturbances”)  Output Variables y(t)  “Memory” Variables z(t) (State Variables)
14. 14. “ Memory Operators”  Needed for dynamic system modeling.  No unique choice of memory operator.  Different popular choices of memory operators different bodies of fundamental knowledge. Thursday, October 27, 2016 14
15. 15. Integration w.r.t. Time as a Memory Operator: Example 1- Wishing Well Thursday, October 27, 2016 15 u(t) : Money influx (S/second) y(t): Money accumulated (Rs) State Variable: z(t): Money accumulated (Rs) z(t) = ∫ u(w)dw Differentiating both side w.r.t. time gives: ż(t) =u(t) Sate Equation y(t) = z(t)
16. 16. Example 2- Wishing Well…With Interest Thursday, October 27, 2016 16 a: Interest rate u(t) : Money influx (S/sec) y(t): Money accumulated (Rs) State Variable: Z(t): Money accumulated (Rs) ż(t) =az(t)+u(t) Sate Equation y(t) = z(t) Output Equation
17. 17. Example 3- Wishing Well…With Deposit –Dependent Interest a: Interest rate a=a0+a1z(t) u(t) : Money influx (S/sec) y(t): Money accumulated (Rs) State Variable: Z(t): Money accumulated (Rs) ż(t) =a0z(t)+a1z2(t)+u(t) Sate Equation y(t) = z(t) Output Equation Thursday, October 27, 2016 17
18. 18. Example 3- What if output was Equivalent Property? a: Deposit dependent interest rate a=a0+a1z(t) β: Fixed cost for purchase of property Γ:Variable cost per unit property u(t) : Money influx (S/sec) y(t): Money accumulated (Rs) State Variable: Z(t): Money accumulated (Rs) ż(t) =a0z(t)+a1z2(t)+u(t) Sate Equation y(t) = (z(t)-β)/γ Output Equation Thursday, October 27, 2016 18
19. 19. State Space Representation Ż(t)= f(z(t),u(t),t) State equation(s) y(t) = g(z(t),u(t),t) Output Equation(s) U(t): Input variable(s) y(t): Output variable(s) Z(t): State(“memory”) variable(s) f(): Sate function(‘equation’) g(): Output function(‘equation’) Time invariant vs. Time varying systems Thursday, October 27, 2016 19