introduction to modeling, Types of Models, Classification of mathematical models, Black box, white box, Gray box
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This document discusses the classification and modeling of systems, including static and dynamic types, and covers various modeling approaches such as black box, white box, and gray box models. It emphasizes the importance of mathematical models in system simulation, prediction, and diagnostics, as well as the advantages and disadvantages of computer simulation. Key concepts such as through and across variables, and the modeling of electrical and mechanical systems are also addressed.
![Grey Box Model
• When input and output and some information
about the internal dynamics of the system is
known.
• Easier than white box Modelling.
9
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introduction to modeling, Types of Models, Classification of mathematical models, Black box, white box, Gray box
- 1. Topics •Types of Systems •Ways to study system •Model •Types of Models •Why Mathematical Model •Classification of mathematical models •Black box, white box, Gray box •Lumped systems •Dynamic Systems •Simulation 1
- 2. Types of Systems • Static System: If a system does not change with time, it is called a static system. • Dynamic System: If a system changes with time, it is called a dynamic system. 2
- 3. Ways to Study a System 3 System Experiment with actual System Experiment with a model of the System Physical Model Mathematical Model Analytical Solution Simulation Frequency Domain Time Domain Hybrid Domain
- 4. Model • A model is a simplified representation or abstraction of reality. • Reality is generally too complex to copy exactly. • Much of the complexity is actually irrelevant in problem solving. 4
- 5. Types of Models Model Physical Mathematical Computer 5 Static Dynamic Static Dynamic Static Dynamic
- 6. What is Mathematical Model? A set of mathematical equations (e.g., differential eqs.) that describes the input-output behavior of a system. What is a model used for? • Simulation • Prediction/Forecasting • Prognostics/Diagnostics • Design/Performance Evaluation • Control System Design
- 7. Classification of Mathematical Models • Linear vs. Non-linear • Deterministic vs. Probabilistic (Stochastic) • Static vs. Dynamic • Discrete vs. Continuous • White box, black box and gray box 7
- 8. Black Box Model • When only input and output are known. • Internal dynamics are either too complex or unknown. • Easy to Model 8 Input Output
- 9. Grey Box Model • When input and output and some information about the internal dynamics of the system is known. • Easier than white box Modelling. 9 u(t) y(t) y[u(t), t]
- 10. White Box Model • When input and output and internal dynamics of the system is known. • One should know have complete knowledge of the system to derive a white box model. 10 u(t) y(t) 2 2 3 dt t y d dt t du dt t dy ) ( ) ( ) (
- 11. Mathematical Modelling Basics Mathematical model of a real world system is derived using a combination of physical laws and/or experimental means • Physical laws are used to determine the model structure (linear or nonlinear) and order. • The parameters of the model are often estimated and/or validated experimentally. • Mathematical model of a dynamic system can often be expressed as a system of differential (difference in the case of discrete-time systems) equations
- 12. Mathematical Modeling Dynamic system modeling Definition of the system and its components Formulation ofmathematical model and assumptions Represent the mathematical model by DE Solve the mathematical model Verify the solution and the assumptions Obtain the solution Otherwise
- 16. Mathematical Modeling Through and across – variables A through – variable is the variable that does not change between the ends of system element. For example, a current passing through a resistance Across – variable is the variable that changes between the ends of system element. For example, the voltage at the ends of a resistance System Through variable(FLOW) Across variable(EFFORT) Electrical Current, i Voltage diff., v Translational motion Force, F Velocity diff., V Rotational motion Torque, T Angular velocity, ω Fluids Flow rate, Q Pressure, P Thermal Heat flow, q Temp. diff., T
- 17. Simulation • Computer simulation is the discipline of designing a model of an actual or theoretical physical system, executing the model on a digital computer, and analyzing the execution output. • Simulation embodies the principle of ``learning by doing'' --- to learn about the system we must first build a model of some sort and then operate the model. 17
- 18. Advantages to Simulation Can be used to study existing systems without disrupting the ongoing operations. Proposed systems can be “tested” before committing resources. Allows us to control time. Allows us to identify bottlenecks. Allows us to gain insight into which variables are most important to system performance. 18
- 19. Disadvantages to Simulation Model building is an art as well as a science. The quality of the analysis depends on the quality of the model and the skill of the modeler. Simulation results are sometimes hard to interpret. Simulation analysis can be time consuming and expensive. Should not be used when an analytical method would provide for quicker results. 19