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Mathematical Models of Systems O bjectives

Utilize quantitative mathematical models for designing and analyzing control systems. Include nonlinear systems and linearization methods using Laplace transform for system analysis.

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Mathematical Models of Systems O bjectives

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  1. Mathematical Models of Systems Objectives We use quantitative mathematical models of physical systems to design and analyze control systems. The dynamic behavior is generally described by ordinary differential equations. We will consider a wide range of systems, including mechanical, hydraulic, and electrical. Since most physical systems are nonlinear, we will discuss linearization approximations, which allow us to use Laplace transform methods. We will then proceed to obtain the input–output relationship for components and subsystems in the form of transfer functions. The transfer function blocks can be organized into block diagrams or signal-flow graphs to graphically depict the interconnections. Block diagrams (and signal-flow graphs) are very convenient and natural tools for designing and analyzing complicated control systems

  2. Define the system and its components Formulate the mathematical model and list the necessary assumptions Write the differential equations describing the model Solve the equations for the desired output variables Examine the solutions and the assumptions If necessary, reanalyze or redesign the system Introduction Six Step Approach to Dynamic System Problems

  3. Differential Equation of Physical Systems

  4. Differential Equation of Physical Systems Energy or Power Electrical Inductance Describing Equation Translational Spring Rotational Spring Fluid Inertia

  5. Differential Equation of Physical Systems Electrical Capacitance Translational Mass Rotational Mass Fluid Capacitance Thermal Capacitance

  6. Differential Equation of Physical Systems Electrical Resistance Translational Damper Rotational Damper Fluid Resistance Thermal Resistance

  7. Differential Equation of Physical Systems

  8. Differential Equation of Physical Systems

  9. Differential Equation of Physical Systems

  10. Differential Equation of Physical Systems

  11. Linear Approximations

  12. Linear Approximations • Linear Systems - Necessary condition • Principle of Superposition • Property of Homogeneity • Taylor Series • http://www.maths.abdn.ac.uk/%7Eigc/tch/ma2001/notes/node46.html

  13. Linear Approximations – Example 2.1

  14. The Laplace Transform Historical Perspective - Heaviside’s Operators Origin of Operational Calculus (1887)

  15. Historical Perspective - Heaviside’s Operators Origin of Operational Calculus (1887) v = H(t) Expanded in a power series (*) Oliver Heaviside: Sage in Solitude, Paul J. Nahin, IEEE Press 1987.

  16. The Laplace Transform

  17. The Laplace Transform

  18. The Laplace Transform

  19. The Laplace Transform

  20. The Partial-Fraction Expansion (or Heaviside expansion theorem) Suppose that + The partial fraction expansion indicates that F(s) consists of s z1 F ( s ) a sum of terms, each of which is a factor of the denominator. + × + ( s p1 ) ( s p2 ) The values of K1 and K2 are determined by combining the individual fractions by means of the lowest common denominator and comparing the resultant numerator or coefficients with those of the coefficients of the numerator before separation in different terms. K1 K2 + F ( s ) + + s p1 s p2 Evaluation of Ki in the manner just described requires the simultaneous solution of n equations. An alternative method is to multiply both sides of the equation by (s + pi) then setting s= - pi, the right-hand side is zero except for Ki so that + × + ( s pi ) ( s z1 ) s = - pi Ki + + + ( s p1 ) ( s p2 ) The Laplace Transform

  21. The Laplace Transform

  22. The Laplace Transform Useful Transform Pairs

  23. The Laplace Transform Consider the mass-spring-damper system

  24. The Laplace Transform

  25. The Transfer Function of Linear Systems

  26. The Transfer Function of Linear Systems Example 2.2

  27. The Transfer Function of Linear Systems

  28. The Transfer Function of Linear Systems

  29. The Transfer Function of Linear Systems

  30. The Transfer Function of Linear Systems

  31. The Transfer Function of Linear Systems

  32. The Transfer Function of Linear Systems

  33. The Transfer Function of Linear Systems

  34. The Transfer Function of Linear Systems

  35. The Transfer Function of Linear Systems

  36. The Transfer Function of Linear Systems

  37. The Transfer Function of Linear Systems

  38. The Transfer Function of Linear Systems

  39. The Transfer Function of Linear Systems

  40. The Transfer Function of Linear Systems

  41. The Transfer Function of Linear Systems

  42. The Transfer Function of Linear Systems

  43. The Transfer Function of Linear Systems

  44. The Transfer Function of Linear Systems

  45. Block Diagram Models

  46. Block Diagram Models

  47. Block Diagram Models Original Diagram Equivalent Diagram Original Diagram Equivalent Diagram

  48. Block Diagram Models Original Diagram Equivalent Diagram Original Diagram Equivalent Diagram

  49. Block Diagram Models Original Diagram Equivalent Diagram Original Diagram Equivalent Diagram

  50. Block Diagram Models

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