Control Engineering Chap3

Chapter 3 Laplace Transform 31 Chapter 3 Overview 3.1 Review of Laplace Transform 3.1.1 Laplace transform of the step function3.1.2 Laplace transform of the exponential function3.2 Laplace Transform of Commonly Used Functions3.3 Inverse Laplace Transform 3.4 Laplace Transform Properties 3.5 Solving ODEs Using Laplace Transform3.5.1 Example 1 3.5.2 Example 2 (spring-mass system) 3.6 Finding Transfer Function of a Single-Input Single-Output System Using Laplace Transform3.6.1 Example 1 3.6.2 Example 2 3.6.3 Example 3

32 Chapter 3 Overview 3.7 Block Diagrams and Signal Flow Graphs 3.7.1 Series connection 3.7.2 Parallel connection 3.7.3 Closed loop negative feedback connection 3.8 Simplifying Connected Subsystems 3.9 Masons Gain Formula for Signal Flow Graphs 3.9.1 Example 1 3.9.2 Example 2 3.9.3 Example 3 3.10 Alternative Method for Block Diagram Simplification3.10.1 Example 1

33 Laplace transform can be used to solve ODEs and study stability. Laplace transform can be used to represent a system in the transfer function form. 3.0 Laplace Transform 3.1 Review of Laplace Transform 34 We denote Laplace transform ofby . We say thatis the frequency-domain representation of . The frequency variable is a complex number: where ,are real numbers with units of frequency (i.e., Hz)3.1.1 Laplace transform of the step function35 3.1.2 Laplace transform of the exponential function3.2 Laplace Transform of Commonly Used Functions36 3.3 Inverse Laplace Transform 37 3.4 Laplace Transform Properties 38 Linearity Differentiation Integration 3.4 Laplace Transform Properties 39 Final value theorem (FVT) Initial value theorem (IVT) 40 3.5 Solving ODEs Using Laplace Transform3.5.1 Example 1 where 3.5 Solving ODEs Using Laplace Transform41 3.5.2 Example 2 (spring-mass system) where 3.6 Finding Transfer Function of a Single-Input Single-Output System Using Laplace Transform42 Consider the input-output relation of a linear time-invariant system described by the following nth-order ODE: whereTo obtain transfer function take the Laplace transform on both sides and assume initial conditions are zero: 3.6 Finding Transfer Function of a Single-Input Single-Output System Using Laplace Transform43 3.6.1 Example 1 3.6 Finding Transfer Function of a Single-Input Single-Output System Using Laplace Transform44 3.6.2 Example 2 3.6 Finding Transfer Function of a Single-Input Single-Output System Using Laplace Transform45 3.6.3 Example 3 3.7 Block Diagrams and Signal Flow Graphs 46 3.7.1 Series connection 3.7.2 Parallel connection 3.7 Block Diagrams and Signal Flow Graphs 47 3.7.3 Closed loop negative feedback connection 3.7 Block Diagrams and Signal Flow Graphs 48 3.7 Block Diagrams and Signal Flow Graphs 49 3.8 Simplifying Connected Subsystems 50 3.9 Masons Gain Formula for Signal Flow Graphs 51 Input-output relations of a signal flow graph can be determined by: where input node variable. output node variable. gain betweenand .total number of forward paths between and .gain of the forward path betweenand . 1-(sum of the gains of all individual loops)+(sum of products of gains of all possible combinations of two nontouching loops)-(sum of products of gains of all possible combinations of three nontouching loops)+ thefor that part of the signal flow graph that is nontouching with theforward path. 3.9 Masons Gain Formula for Signal Flow Graphs 52 3.9.1 Example 1 3.9 Masons Gain Formula for Signal Flow Graphs 53 3.9.2 Example 2 3.9 Masons Gain Formula for Signal Flow Graphs 54 3.9.3 Example 3 3.10 Alternative Method for Block Diagram Simplification55 3.10 Alternative Method for Block Diagram Simplification56 3.10 Alternative Method for Block Diagram Simplification57 3.10.1 Example 1 Step 1: 3.10 Alternative Method for Block Diagram Simplification58 3.10.1 Example 1 Step 2: Step 3: Step 4: