Simulation & Modelling

1. Simulation & Modelling Lecture-1: Simulation & Modelling Fundamentals Systems Engineering Training Programme for DRDO Scientists at IAT, Pune 1st July 2003

2. Lecture Outline • What is Systems Modelling & why is it needed? • Engineering Vs. Non-Engineering Systems. • Modelling Techniques for Engg. Systems. • What is Simulation? • Offline Vs. Real-time Simulations. • Simulation Methodology. • Examples of Modelling. • Examples of Offline Simulations.

3. Why is systems approach needed? • “Need” for better products/services pushes technology into frontier areas. (e.g. Telecom, MEMS, Computers) • “Cost” decides success or failure of innovations/ new developments. (e.g. Concorde) • This puts pressure on design & development process. • Short Development Cycle; Long Product Life • Small Development Cost; Large Benefits • Environment Friendly; Socially Acceptable • Multi-disciplinary nature of products makes the development process extremely complex & tough. • Systems Approach is an important aid to this process.

4. What is systems approach? • “Systems Approach” is a systematic tool for Integrated Technology and Product Development. • “Systems Approach” is a methodology for looking at a complex product/ process as a single unit, composed of many sub-systems functionally related to each other. • “Systems Approach” provides a mechanism for assessing the “performance” of System, against overall “design objectives”. • “Systems Approach” is an extremely useful philosophy in the design and development process of Multi-disciplinary products.

5. What is the role of modelling? • “Design” is a process of “repeated generation of responses to inputs”, to arrive at an acceptable product configuration, meeting stated requirements. • “Concept” is usually what is available at early design stages, from which “responses to inputs” are generated. • “Modelling” is a process by which a concept is translated into a more tangible form, quite often, but not necessarily, a mathematical one. • “System Modelling” is a process of converting “Sub-Systems” into their corresponding tangible forms.

6. What is a system model? • “System Model” is a specific assembly of “Sub-system” models, arrived at for meeting specified objectives. • “Systems Modelling” is the process of determining (a) relationships of all “Sub-systems” and (b) quantitative features describing operations of all “Sub-systems”. • “Relationships” among “Sub-systems is usually through ‘inputs’ to them and ‘outputs’ from them. • “Quantitative Features” are usually derived from the laws governing the operation of “Sub-systems”.

7. Example of a system model Input System Output Sub - 1 Sub - 2 Sub - N-1 Sub - N

8. How is a system model created? • Identify the system and its boundary. • Define the environment. • Decompose system into sub-systems till desired level of details, and identify various interactions between them. • Define aspects of environment pertinent to the system. • Identify laws that govern the operation of each of the sub-systems. • Decide the required complexity level of the model. • Choose a model philosophy. (Math./Physical/Hybrid)

9. Categories of systems & models Abstract System Physical Engineering Non-Engg. Continuous Vs. Discrete Linear Vs. Nonlinear Time Invariant Vs. Time Varying Deterministic Vs. Stochastic Causal Vs. Non-Causal

10. Engg. Vs. Non-Engg. Systems

11. Engg. Vs. Non-Engg. Systems

12. Engineering System Model Types • Purely Mathematical Models • Most commonly used in initial design stages • Fairly inexpensive and usually less accurate • Hybrid Models (Software-Hardware Models) • Used when some sub-systems are realized/frozen • More expensive and more accurate • Useful for modelling interfaces between sub-systems • Scaled/Prototype Models • Used when actual system behaviour is needed • Quite expensive, but close to real system • Possible only towards end of the development process

13. How to choose a model type? • Availability & Completeness of Modelling Information • Depends on the stage of product development • Accuracy Required Vs. Accuracy Possible • Low fidelity models Vs. high fidelity models • Multi-disciplinary modelling capability • Essential physical effects to be included in model • Availability of Resources for Response Generation • Available computational resources • Available software tools & trained manpower • Available infrastructure for testing & evaluation

14. Mathematical Modelling Procedure • Definition & Characterization of Fundamental Variables • Mass, Momentum & Energy as basic quantities • Idealized Element & Constitutive Relation Evolution • Known physical effects as idealized element • Specific to each engineering discipline • Mathematical relations describing energy exchange • Continuity & Compatibility Condition Prescription • Translation of physical boundary conditions • Prescription of interface behaviour • Continuity on variables through an element • Compatibility on variables defined on element boundaries

15. Discipline-wise Idealized Elements

16. Miscellaneous Idealized Elements

17. Idealized Mechanical Elements

18. Idealized Electrical Elements Resistance Element Voltage Source v v = i R R i Current Source Capacitance Element i C v = C i dt L1 i Magnetic Element Inductance Element v = L di/dt L L2

19. Idealized Thermal Elements

20. Idealized Fluidic Elements

21. Idealized Fluidic Elements

22. Idealized Electro-Mechanical Elements

23. Mechanical System Modelling • Spring-Mass-Damper System • Support Motion Excitation • Force Equilibrium • Newton’s Law Md2y/dt2 + Bdy/dt + Ky - Bdu/dt - Ku = 0 • System (I/O) Form y(t) = (Bd/dt+K) u(t) (Md2/dt2+Bd/dt+K) M y(t) B K u(t)

24. Electrical System Modelling • Resistance-Inductance-Capacitance Circuit • Sinusoidal Voltage Excitation L • Charge, Potential Conservation • Kirchoff’s Laws Ldi/dt + Cidt + Ri = v(t) • System (I/O) Form i(t) = (d/dt) v(t) (Ld2/dt2+Rd/dt+C) C R i(t) v(t)

25. Fluidic System Modelling • Two-Tank Liquid Flow System • Hydrostatics & Hydrodynamics p1 =  g h1; p2 =  g h2 Tank1: q-q1= C1dp1/dt Tank2: q1-q2 = C2 dp2/dt Pipe-1: g(h1-h2)=R1 q1 Pipe-2: gh2 =R2 q2  Q + q H2+h2 H1+h1  q1  q2 R1C1C2 d2h2/dt2 + [C1(1+R1/R2)+C2]dh2/dt + h2/R2 = q/g h1 = (1+R1/R2)h2 + R1C2dh2/dt

26. Modelling Aircraft FCS Auto-Pilot Sensors Servo Elevator Stick Servo Ailerons Airframe Pedal Human Pilot Servo Rudder Motion Throttle Fuel Engine Cockpit Instruments

27. Summary of Modelling • Modelling depends on knowledge base. • Type of model depends on resources. • Accuracy of model depends on technology. • Complexity of model depends on the overall objective.

28. What is simulation? on a Modelling Exercise Physical System gives System Response to Expected Inputs for Mathematical, Physical or Hybrid Model No Acceptable? acts on Solution Exercise Yes End

29. What is simulation? Solution Exercise; • Is a Process for generating information about the state of a system at any desired instant of time. • Consists of impressing inputs on the system model and recording its behaviour as time progresses. Solution Exercise can be Mathematical or Experimental

30. What is simulation? Dictionary Definitions of Simulation; • Appearing to be Something Else. • Imitation of a Condition. • Act or an Object that is Counterfeit or Feigned. • Activity Producing Conditions Unreal but having Appearance of being Real. Simulation is an activity which makes system model imitate the behaviour of actual system.

31. What is simulation? Engg. Interpretation of Simulation; • System model can be looked upon as an object expected to pretend to be the real Engg. system. • Behaviour (or response of model to expected inputs), as time progresses, is expected to indicate how the actual system would have behaved under the same conditions. • Therefore, “simulation” is a “step-by-step solution process” by which the behaviour (or response) of a model is generated as time progresses.

32. What is simulation? Engg. Interpretation of Simulation; • System model simulation gives an impression of an activated system, going through the various possible stages of response evolution. • An important aspect of response evolution in system simulation is its “visual” representation.

33. Why simulation? US Undersecretary of Defence Acquisition & Technology, “Study on Effectiveness of Modelling & Simulation in the Weapon System Acquisition Process”, Oct. 1996.

34. Why simulation? US Undersecretary of Defence Acquisition & Technology, “Study on Effectiveness of Modelling & Simulation in the Weapon System Acquisition Process”, Oct. 1996.

35. Simulation as Visualization R RC (dv(t))/dt + v(t) = eo v(t) = eo [ 1 – e-(t/RC) ] C v(t) i(t) eo For Step-by-step approach; N waypoints at t = T/N v(k+1) = (1 - t/RC) v(k) + eo t/RC v(t) t

36. Simulation Characteristics Simulation provides a “feel” for the system behaviour under differing inputs. In case of constraints on system response, simulation quickly provides the exceptions to be noted . Simulation methodology can mix software and hardware, to provide higher fidelity responses. However, simulation is highly problem specific and needs to be repeated as many number of times as there are test cases. This makes simulation a time consuming exercise.

37. Off-line Vs. Real-time Simulation In case of time scale mismatch between “simulation” and actual system, we get “offline simulation”. In case “simulation” proceeds at the same speed as the actual system, we get “real-time” simulation. Time scale of Spring-Mass-Damper System; d = (1 - 2) . (K/m) Time for solving the corresponding equation. a few Milliseconds M y(t) K B u(t)

38. Simulation Techniques Continuous simulation techniques are used when model is a set of differential equations (e.g. Analog Computer Simulation). Numerical simulation techniques are used when either system model is discrete or a powerful digital computer is available (e.g. Queueing Process). Hybrid simulation combines both Continuous and Numerical simulation techniques. Hardware-in-loop Simulation technique is used when both mathematical equations and hardware are part of the system model description. (e.g. Iron Bird).

39. Monte’ Carlo Simulation System “simulation” with Random Nos. Developed at Los Alamos Lab. For Nuclear Research. Random Nos., satisfying constraints, used in very large trials. (Name borrowed from famous gaming city). (xp,yp): a < xp < b; 0 < yp < ymax A large No. of Points Chosen & Tested for the conditions Area fraction; ratio of selected points to total points used. ymax y a x b

40. Summary of Simulation Simulation is tool to improve team cohesiveness. Simplified models are easier to simulate. Development/Life Cycle are evaluated quickly. Simulation accelerates IPPD Process. Verification & Validation issues are handled adequately in simulation.