notes - Notes for ECE-320 Fall 2004 by R Throne The...

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Notes for ECE-320 Fall 2004 by R. Throne The following pages contain a first attempt at writing notes for ECE-320. The topics we cover in ECE-320 are not covered in any single book. These notes are not complete, especially the sections on root locus design and design using Bode plots. The major sources for these notes are Analog and Digital Control System Design , by C. T. Chen. Sanders College Publishing. 1993. Linear Control Systems , by Rohrs, Melsa, and Schulz. McGraw-Hill, 1993. Modern Control Engineering , by Ogata. Prentice-Hall, 2002. Modern Control Systems , by Dorf and Bishop. Prentice-Hall, 2005. 1
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Contents 1 Table of Laplace Transforms 4 2 Laplace Transform Review 5 2.1 Poles and Zeros . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.2 Proper and Strictly Proper Transfer Functions . . . . . . . . . . . . . . . . . . . 5 2.3 Impulse Response and Transfer Functions . . . . . . . . . . . . . . . . . . . . . 5 2.4 Partial Fractions with Distinct Poles . . . . . . . . . . . . . . . . . . . . . . . . 6 2.5 Partial Fractions with Distinct and Repeated Poles . . . . . . . . . . . . . . . . 9 2.6 Complex Conjugate Poles: Completing the Square . . . . . . . . . . . . . . . . . 14 2.7 Complex Conjugate Poles-Again . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 3 Final Value Theorem 20 4 Step Response and Position Error, Ramp Response and Velocity Error 21 4.1 Step Response and Position Error . . . . . . . . . . . . . . . . . . . . . . . . . . 21 4.2 Ramp Response and Velocity Error . . . . . . . . . . . . . . . . . . . . . . . . . 24 4.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 5 Response of a Ideal Second Order System 28 5.1 Step Response of an Ideal Second Order System . . . . . . . . . . . . . . . . . . 28 5.2 Time to Peak, T p . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 5.3 Percent Overshoot, PO . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 5.4 Settling Time, T s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 5.5 Constraint Regions in the s -Plane . . . . . . . . . . . . . . . . . . . . . . . . . . 33 5.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 6 Characteristic Polynomial, Modes, and Stability 42 6.1 Characteristic Polynomial, Equation, and Modes . . . . . . . . . . . . . . . . . . 42 6.2 Characteristic Mode Reminders . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 6.3 Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 6.4 Settling Time and Dominant Poles . . . . . . . . . . . . . . . . . . . . . . . . . 44 7 Basic Feedback Configuration 46 8 Model Matching 47 8.1 ITAE Optimal Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 8.2 Quadratic Optimal Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 8.3 Summary and Caveates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 9 System Type and Steady State Errors 54 9.1 Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 9.2 System Type For a Unity Feedback Configuration . . . . . . . . . . . . . . . . . 54 9.3 Position and Velocity Errors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 9.4 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 2
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10 Controller Design Using the Root Locus 59 11 Pole Placement By Matching Coefficients: Diophantine Equations 61 11.1 Theoretical Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 11.2 Pole Placement with Robust Tracking . . . . . . . . . . . . . . . . . . . . . . . . 65 11.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 12 System Sensitivity 68 12.1 Sensitivity to Parameter Variations . . . . . . . . . . . . . . . . . . . . . . . . . 68 12.2 Sensitivity to External Disturbances . . . . . . . . . . . . . . . . . . . . . . . . . 73 12.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74 13 State Variables and State Variable Feedback 75 13.1 State Variable to Transfer Function Model . . . . . . . . . . . . . . . . . . . . . 77 13.2 State Variable Feedback . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80 13.3 Controllability for State Variable Systems . . . . . . . . . . . . . . . . . . . . . 85 13.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86 14 Controller Design Using Bode Plots 87 15 Linearization 90 15.1 Linear Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90 15.2 Taylor Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90 15.3 Linearization Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 A Matlab Commands i A.1 Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . i A.2 Transfer Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . i A.3 Feedback Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ii A.4 System Response to Arbitrary Inputs . . . . . . . . . . . . . . . . . . . . . . . . ii A.5 Changing the Line Thickness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii A.6 Poles and Zeros . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iv A.7 Roots and Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iv A.8 Root Locus Plots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v A.9 Bode Plots, Gain and Phase Margins . . . . . . . . . . . . . . . . . . . . . . . . v 3
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1 Table of Laplace Transforms f ( t ) F ( s ) δ ( t ) 1 u ( t ) 1 s tu ( t ) 1 s 2 t n - 1 ( n - 1)!
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