1 Sistem Kontrol I Kuliah II Transformasi Laplace

1 Sistem Kontrol I Kuliah II : Transformasi Laplace Imron Rosyadi, ST Email: pak. imron@gmail. com

2 From Lecture #1

3 Control System Design Process • Diagram on the next page gives a flowchart of the control system design process

4 Original System - Plant - Sensors - Actuators Develop Performance Specifications Measurement Modeling Desired Performance Math Model of Plant Analysis - Modeling - Analysis - Design - Implementation Simulation Design Math Model of Controller Implementation - Physical controller - Coupling controller with plant New System Key Activities of the “MAD” Control Engineer:

5 Control System Design Process • Hidden in this chart are three important elements : 1. Modeling the system (using mathematics) 2. Analysis techniques for describing and understanding the system’s behavior 3. Design techniques for developing control algorithms to modify the system’s behavior • • Modeling, analysis, and design = the MAD control theorist A fourth key element is Implementation

6 Modeling is the key! • The single most important element in a control system design and development process is the formulation of a model of the system. • A framework for describing a system in a precise way makes it possible to develop rigorous techniques for analyzing the system and designing controllers for the system

7 Modeling • Key Point: most systems of interesting in engineering can be described (approximately) by ▫ Linear ▫ Ordinary ▫ Constant-coefficient ▫ Differential equations Requires algebra to solve • Call these LODEs • Where we are going looks like this: Physical Reality Laplace Transform LODE Requires calculus to solve

8 Review of Complex Number

9 Complex Numbers: Notation and Properties (1) • A complex number: • The complex plane Rectangular (Cartesian) coordinates Polar coordinates Due to Katie Johnson or Tyrone Vincent or someone

10 Complex Numbers: Notation and Properties (2) • Transformation between coordinates Due to Katie Johnson or Tyrone Vincent or someone

11 Complex Numbers: Notation and Properties (3) • Euler’s Formula: • Note differentiation property Due to Katie Johnson or Tyrone Vincent or someone

12 Exercise • Show Euler’s Formula is a parameterization of the unit circle Due to Katie Johnson or Tyrone Vincent or someone

13 Complex Numbers: Notation and Properties (4) • Alternate notation for polar coordinates using Euler’s Formula Compare to Note: keep track of degrees and radians! Due to Katie Johnson or Tyrone Vincent or someone

14 Complex Math – Review • Complex multiplication and division: the hard way Due to Katie Johnson or Tyrone Vincent or someone

15 Complex Math – Review • Complex multiplication and division: the easy way Given: Due to Katie Johnson or Tyrone Vincent or someone

16 Exercise Due to Katie Johnson or Tyrone Vincent or someone

Complex Math – Review • Complex conjugate: Given: Define complex conjugate as • Some key results: Due to Katie Johnson or Tyrone Vincent or someone

Complex Math – Review • A function of a complex number is also a complex number Given: • Example Due to Katie Johnson or Tyrone Vincent or someone

Complex Math – Review • Derivatives of a function of complex numbers, G(s), can be computed in the usual way • Poles/Zeros Due to Katie Johnson or Tyrone Vincent or someone

Complex Math – Review • Poles/Zeros at infinity Due to Katie Johnson or Tyrone Vincent or someone

21 Laplace Transform

22 Laplace Transform Motivation • Differential equations model dynamic systems • Control system design requires simple methods for solving these equations! • Laplace Transforms allow us to ▫ systematically solve linear time invariant (LTI) differential equations for arbitrary inputs. ▫ easily combine coupled differential equations into one equation. ▫ use with block diagrams to find representations for systems that are made up of smaller subsystems. Due to Katie Johnson or Tyrone Vincent or someone

23 The Laplace Transform Definition • Laplace Transform exists if integral converges for any value of s ▫ Region of convergence is not as important for inverting “one-sided” transforms Due to Katie Johnson or Tyrone Vincent or someone

24 Laplace Transform Example (1) • Example: Notation for “unit step” • Show that Due to Katie Johnson or Tyrone Vincent or someone

25 Laplace Transform Example (2) Due to Katie Johnson or Tyrone Vincent or someone

26 Laplace Transform of a Unit Step • Find the Laplace Transform for the following function Due to Katie Johnson or Tyrone Vincent or someone

27 Exercise • Find the Laplace Transform for the following function Due to Katie Johnson or Tyrone Vincent or someone

28 The Laplace Transform Definition (Review) • Recall: • The easiest way to use the Laplace Transform is by creating a table of Laplace Transform pairs. We can use several Laplace Transform properties to build the table. Due to Katie Johnson or Tyrone Vincent or someone

29 The function with the simplest Laplace Transform (1) • A special input (class) has a very simple Laplace Transform • The impulse function: ▫ Has unit “energy” ▫ Is zero except at t=0 Think of pulse in the limit Due to Katie Johnson or Tyrone Vincent or someone

30 The function with the simplest Laplace Transform (2) Due to Katie Johnson or Tyrone Vincent or someone

31 LT Properties: Scaling and Linearity • Proof: Both properties inherited from linearity of integration and the Laplace Transform definition Due to Katie Johnson or Tyrone Vincent or someone

32 Example 1 • Find the following Laplace Transforms ▫ Hint: Use Euler’s Formula Due to Katie Johnson or Tyrone Vincent or someone

33 Example 1 (2) Due to Katie Johnson or Tyrone Vincent or someone

34 LT Properties: Time and Frequency Shift • Proof of frequency shift: Combine exponentials Due to Katie Johnson or Tyrone Vincent or someone

35 Example 2 • Find the following Laplace Transforms Due to Katie Johnson or Tyrone Vincent or someone

36 Example 2 (2) Due to Katie Johnson or Tyrone Vincent or someone

37 LT Properties: Integration & Differentiation • Proof of Differentiation Theorem: Integration by parts Due to Katie Johnson or Tyrone Vincent or someone

38 LT Properties: Integration & Differentiation (2) Due to Katie Johnson or Tyrone Vincent or someone

39 Example 3 • Find Laplace Transform for • What is the Laplace Transform of ▫ Derivative of a step? ▫ Derivative of sine? Due to Katie Johnson or Tyrone Vincent or someone

40 Example 3 (2) Impulse! Cosine! Due to Katie Johnson or Tyrone Vincent or someone

41 Exercise • What is the Laplace Transform of -Sine! Due to Katie Johnson or Tyrone Vincent or someone

42 Initial Value Theorem

43 Final Value Theorem

44 Inverse Laplace and LODE solution

45 Inverse Laplace Transform

46 Partial Fraction Idea -1

47 Partial Fraction Idea -2

48 Partial Fraction Idea -3

49 Recall: Laplace differentiation theorem (1) • The differentiation theorem • Higher order derivatives Due to Katie Johnson or Tyrone Vincent or someone

50 Differentiation Theorem (revisited) • Differentiation Theorem when initial conditions are zero Due to Katie Johnson or Tyrone Vincent or someone

51 Solving differential equations: a simple example (1) • Consider Due to Katie Johnson or Tyrone Vincent or someone

52 Solving differential equations: a simple example (2) • Solution Summary ▫ Use differentiation theorem to take Laplace Transform of differential equation ▫ Solve for the unknown Laplace Transform Function ▫ Find the inverse Laplace Transform Due to Katie Johnson or Tyrone Vincent or someone

53 Example 1 • Find the Laplace Transform for the solution to Notation: Due to Katie Johnson or Tyrone Vincent or someone

54 - Partial Fraction Expansions • In general, LODEs can be transformed into a function that is expressed as a ratio of polynomials • In a partial fraction expansion we try to break it into its parts, so we can use a table to go back to the time domain: • Three ways of finding coefficients ▫ Put partial fraction expansion over common denominator and equate coefficients of s (Example 1) ▫ Residue formula ▫ Equate both sides for several values of s (not covered)

- Partial Fraction Expansions • Have to consider that in general we can encounter: ▫ ▫ Real, distinct roots Real repeated roots Complex conjugate pair roots (2 nd order terms) Repeated complex conjugate roots

56 Example 1, Part 2 • Given X(s), find x(t). • This Laplace Transform function is not immediately familiar, but it is made up of parts that are. • Factor denominator, then use partial fraction expansion: Due to Katie Johnson or Tyrone Vincent or someone

57 Finding A, B, and C • To solve, re-combine RHS and equate numerator coefficients (“Equate coefficients” method) Due to Katie Johnson or Tyrone Vincent or someone

58 Final Step • Example 1 completed: • Since • By inspection, Due to Katie Johnson or Tyrone Vincent or someone

59 Residue Formula (1) • The residue formula allows us to find one coefficient at a time by multiplying both sides of the equation by the appropriate factor. • Returning to Example 1: Due to Katie Johnson or Tyrone Vincent or someone

60 Residue Formula (2) • For Laplace Transform with non-repeating roots, • The general residue formula is: Due to Katie Johnson or Tyrone Vincent or someone

61 Example 2 • Find the solution to the following differential equation: Due to Katie Johnson or Tyrone Vincent or someone

62 Example 2 (2) Due to Katie Johnson or Tyrone Vincent or someone

63 Inverse Laplace Transform with Repeated Roots • We have discussed taking the inverse Laplace transform of functions with non-repeated, real roots using partial fraction expansion. • Now we will consider partial fraction expansion rules for functions with repeated (real) roots: ▫ # of constants = order of repeated roots • Example: Due to Katie Johnson or Tyrone Vincent or someone

64 Repeated real roots in Laplace transform table • The easiest way to take an inverse Laplace transform is to use a table of Laplace transform pairs. Repeated Real Roots Repeated Imaginary Roots (also use cosine term) Repeated Complex Roots (also use cosine term) Due to Katie Johnson or Tyrone Vincent or someone

65 Example with repeated roots • Example: find x(t) • Take Laplace Transform of both sides: Due to Katie Johnson or Tyrone Vincent or someone

66 Example with repeated roots (2) • Terms with repeated roots: Due to Katie Johnson or Tyrone Vincent or someone

67 Example with repeated (3) B =roots 2 C= 1 Due to Katie Johnson or Tyrone Vincent or someone

68 Exercise 1 • Find the solution to the following differential equation Due to Katie Johnson or Tyrone Vincent or someone

69 Above … Inverse Laplace and LODE solutions - Partial fraction expansions - LODE solution examples * Real roots * Real, repeated roots Next: * Complex roots

NOTE: A complex conjugate pair is actually two distinct, simple first order poles, so can find residues and combine in the usual way:

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72 Inverse Laplace Transform with Complex Roots • To simplify your algebra, don’t use first-order denominators such as • Instead, rename variables • So that Due to Katie Johnson or Tyrone Vincent or someone

73 Laplace Transform Pairs for Complex Roots • More Laplace transform pairs (complex roots): • Also, see the table in your textbook and most other control systems textbooks. Due to Katie Johnson or Tyrone Vincent or someone

74 Return to example from above:

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76 Example with complex roots • Example: find x(t) • Laplace Transform Due to Katie Johnson or Tyrone Vincent or someone

77 Example with complex roots (2) Due to Katie Johnson or Tyrone Vincent or someone

78 Example with complex roots (3) Due to Katie Johnson or Tyrone Vincent or someone

79 Example with complex roots (5) Due to Katie Johnson or Tyrone Vincent or someone

80 Exercise 2 • Find solution to the following differential equation Due to Katie Johnson or Tyrone Vincent or someone

81 Resume Lecture #2

82 • • Review of Complex Number Laplace Transform Inverse Laplace Transform Solving LODE