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

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

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

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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

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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

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

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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 9

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

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

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

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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 13

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

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

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Exercise Due to Katie Johnson or Tyrone Vincent or someone 16

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

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

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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

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

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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 22

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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 23

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

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

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

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

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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 28

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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 29

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

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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 31

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

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

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

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

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

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

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

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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 39

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

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

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Initial Value Theorem 42

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Final Value Theorem 43

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Inverse Laplace Transform 45

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Partial Fraction Idea -1 46

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Partial Fraction Idea -2 47

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Partial Fraction Idea -3 48

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

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

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

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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 52

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

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- 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) 54

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- 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

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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 56

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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 57

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

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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 59

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

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

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

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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 63

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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 64

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

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

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

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

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

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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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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 72

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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 73

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Return to example from above: 74

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

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

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

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

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

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

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