# Sistem Kontrol I Kuliah II : Transformasi Laplace Imron Rosyadi, ST 1.

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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

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

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

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

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

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

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

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

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

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

Exercise Due to Katie Johnson or Tyrone Vincent or someone 16

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

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

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

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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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Initial Value Theorem 42

Final Value Theorem 43

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Inverse Laplace Transform 45

Partial Fraction Idea -1 46

Partial Fraction Idea -2 47

Partial Fraction Idea -3 48

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

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

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

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

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

- 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

- 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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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