1.5 module1: fundamentalfeedbacksystemconceptsaar.faculty.asu.edu/classes/eee480f99/roadmap.pdf ·...

XXII Feedback Control System Analysis and Design: A.A. Rodriguez c©1998

1.5 Module 1: Fundamental Feedback System Concepts

Feedback Control System Analysis and Design: A.A. Rodriguez c©1998 XXIII

WEEK 1 - Module 1 (Fundamental Feedback System Concepts)

FocusThe Big Picture.

Objectives

• Establish perspective by developing the �Big Picture;� i.e. What is the central problem to be addressed in this course?What are the key issues?

• Develop basic feedback system block diagram algebra skills,

• Analyze step command properties of a pitch attitude control system (with pre-filter).

Material to ReviewComplex arithmetic, signals, differential equations, Laplace transforms, LTI system, transfer function, poles, zeros, stability,use of transfer functions for sinusoidal steady state analysis and to find a particular solution to an ode with constant coefficients.

SeeChapter 2, page 1 for a review of fundamental mathematical concepts (e.g. complex arithmetic),Chapter 3, page 15 for a review of continuous time signal concepts,Chapter 4, page 29 for an introduction to systems and the use of mathematical models,Chapter 5, page 33 for an introduction to Laplace Transforms and key system ideas,Chapter 7, page 61 for a review of LTI system concepts.

Review Examples (Must Know)

Complex ArithemticExample 2.2.1: Cube Root of -1, page 8Example 2.2.2: Generation of Exponential Sinusoids From Complex Exponentials, page 8Example 2.2.3: Addition Of Sinusoids Using Complex Phasors, page 8

Laplace TransformsExample 5.4.1: Real-Rational Function with Simple Real Poles, page 42Example 5.4.2: Real-Rational Function with Repeated Real Poles, page 43Example 5.4.3: Real-Rational Function with Complex Conjugate Poles, page 46Example 5.4.4: Improper Real-Rational Functions, page 49Example 5.4.5: Real-Rational Function with Repeated Real, Imaginary, and Complex Poles, page 51

Lecture Topics

• Big Picture (see page 139)

• Block Diagram Algebra (see Equation 10.7, page 143, Exercise 10.6.1, page 148)

• Relationship between internal signals and external (exogenous) signals (see Equation 10.15, page 144)

• Tracking Error (see Equation 10.1, page 142)

• Simple Model for an Aircraft (see Exercise 9.6.1, page 127)

Concepts and TerminologyClassical Negative Feedback Structure (see Figure 10.1, page 142), Plant, Actuator, Control, Input, Forcing Function, Out-put, Disturbances Modeled at Plant Input/Output, Measurement, Sensor Noise, Sensor, Compensator, Negative Feedback,Error Signal, Reference Command, Command Following, Disturbance Rejection, Noise Attenuation, Closed loop stability, Un-certainty, Unmodelled Dynamics, Stability Robustness, Performance Robustness, Fundamental Problem and Issues, Big Picture.

Reading AssignmentRead Chapter 8, page 75 for an overview of LTI system analysis concepts.Read Chapter 10, page 139 for an introduction to fundamental feedback control system concepts. Read Sections 10.1-10.4 tounderstand why feedback is used and to become familiar with the standard negative feedback system, its associated analysis,and the so-called Big Picture.

Recommended MaterialExample 8.2.1: An Introduction to System Design Specifications, page 75Example 8.3.1: Analysis Of A First Order System: A Multi-Concept Example, page 80

XXIV Feedback Control System Analysis and Design: A.A. Rodriguez c©1998

Exercise 8.3.1: First Order System Driven By Step and Sinusoid, page 88

Exercise 10.4.1: Why not make K Large?, page 145.Example 10.4.1: Closed Loop Sinusoidal Analysis, page 145.Exercise 10.5.1: Control Systems in the Real World, page 147.Exercise 10.6.1: Relationship Between Commonly Found Feedback Structures, page 148.

Must-know (Mandatory) MaterialExercise 9.6.1: Simplified Aircraft Dynamics, page 127.

Example 10.8.1: Fundamental Constraints and Performance Limitations, page 152Exercise 10.9.1: Sensitivity with respect to Additive Uncertainty, page 154Exercise 10.9.2: Sensitivity with respect to Multiplicative Uncertainty, page 155Example 10.10.1: Stabilization: Unstable Plant, Minimum Bandwidth Requirement, page 156Exercise 10.10.1: Aircraft Pitch Attitude Stabilization Using Proportional Control, page 160

Example 10.11.1: Our First Design: An Introduction to Design Parameter Selection, page 161Example 10.11.2: Control System Command Following and Disturbance Attenuation, page 163Exercise 10.11.1: An Introduction to Root Locus, page 169

Exercise 10.11.2: Root Locus for L = 2(s+3)(s+1)(s−3) , page 172

Homework #1EEE480 Exam #1, Fall 1997Problem 1 - Laplace, ODEs, Steady State AnalysisProblem 2 - Method of the Transfer Function, Steady State AnalysisProblem 3 - Step Response For High Order SystemProblem 4 - Steady State Analysis, Internal Model Principle

EEE480 Final Exam, Fall 1997Problem 1 - Laplace, ODEsProblem 2 - Feedback System, Laplace, ODEs, Steady State Analysis

Additional MaterialThe following reading and examples come from Franklin, Powell, Emami-Naeini, Feedback Control of Dynamical Systems, 3rdedition, 1994 [22].

Read pp. 1-7 An Introduction Example 2.1, pp 21-22 Model for a CarExample 2.5, pp. 28 Model for a Simple Pendulum Example 2.8, pp. 37-38 Car Step Response with MATLABExample 2.14, pp. 48-49 DC Motor Model, pp. 47-49 Example 3.3 Transfer FunctionExample 3.4 Frequency Response Example 3.5 Step and RampExample 3.6 Impulse Function Example 3.7 SinusoidSection 3.1.4, pp. 95-98 Properties of Laplace Transforms Section 3.1.5, pp. 98-102 Partial Fraction ExpansionExample 3.8 Distinct Real Roots Example 3.9 Distinct Complex RootsExample 3.10 Repeated Real Roots Section 3.1.6, pp. 102-106 Laplace Transform TheoremsExample 3.11 Final Value Theorem Example 3.12 Incorrect Use of FVTExample 3.13 DC Gain Example 3.14 Initial Value TheoremExample 3.15 Homogeneous Differential Equation Example 3.16 Forced Differential Equation with ICExample 3.17 Forced Differential Equation with Zero IC Example 3.20, pp. 113-114 Transfer Function from Block DiagramSection 3.3, pp. 118-126 Response versus Pole Location Example 3.23 Impulse Response versus Pole Locations, Real RootsExample 3.24 Underdamped Impulse Response

MATLAB MacrosAn extensive library of Matlab macros and SIMULINK material can be found in Chapter 38, page 317.

Laboratory TopicsLaboratory does not start until week # 2. Learn to use MATLAB and SIMULINK as soon as possible. Examine the MAT-LAB/SIMULINK material in Chapter 38, page 317.

Feedback Control System Analysis and Design: A.A. Rodriguez c©1998 XXV


FocusTime Domain Specifications and Root Locus Ideas.

Objectives

• Understand time response (transient and steady state) specifications in terms of standard second order systems.

• Quantify sinusoidal steady state disturbance attenuation using frequency response ideas.

• Understand control system bandwidth (gain) limitations due to high frequency dynamics.

Material to ReviewReview of Laplace transforms, signals, unit step function, delta distribution (impulse function), impulse family (higher orderdeltas), real exponentials, introduction to stability ideas, unstable, stable, marginally stable, complex arithmetic, Eulers formu-lae (see Section 2.2), complex exponentials, sinusoids, ramps, parabolic signals, region of convergence.

Laplace transform properties, multiplication by an exponential - s shift (see Example 5.3.5, page 36), time multiplication -differentiation in s-domain (see Example 5.3.6, page 37), differentiation in time - multiplication by s in s-domain (see Exam-ple 5.3.8, page 38), Time delay Property (see Example 5.3.9, page 39), Convolution Property (see Example 5.3.5, page 39),Integrators (see Example 5.3.10, page 40), Initial Value Theorem (see Example 5.3.11, page 40), Final Value Theorem (seeExample 5.3.12, page 41).

Inverse transforms, Long division, Partial fraction expansions, Solution of Ordinary Differential Equations (Odes) via Laplace(see Example 8.5.1, page 98), Linear Time Invariant (LTI) Systems (Chapter 7, page 61).

Zero input response (ZIR), Initial conditions, Zero state response (ZSR), Transfer functions, Convolution, Characteristic equa-tion, Poles, Natural modes

Steady State Analysis/Calculations, Method of the Transfer Function, Sinusoidal Steady State Analysis, Particular Solutionsfor Ordinary Differential Equations (ODEs) with Constant Coefficients, Introduction to Frequency Responses

Lecture Topics

• Analysis of an aircraft pitch attitude control system, design parameter selection (Example 10.11.1, page 161)

• Transient and steady state analysis of a pitch attitude control system (with pre-filter) with respect to step commandfollowing and disturbance attenuation (see Example 10.11.2, page 163)

• Internal Model Principle (see Exercise 10.14.4, page 193)

• Step response for standard second order systems (see Example 8.4.1, page 89)

• Introduction to design specifications (see Section 8.2, page 75)

• Disturbance attenuation and frequency responses (see Example 10.11.2, page 163)

• Increasing the gain of a feedback loop, impact of high frequency dynamics - Introduction to Root Locus concepts (seeExercise 10.11.1, page 169).

Concepts and TerminologyMason’s rule, Sensitivity Function (see Equation 10.11, page 144), Complementary Sensitivity Function (see Equation 10.12,page 144), Modeling of Dynamical Systems: Simplified Aircraft Dynamics (see Exercise 9.6.1, page 127), Input/Output Models,Differential Equations, Laplace transforms. Right half plane pole-zero cancellations are not allowed, Can’t invert non-invertiblestuff! Pitch Attitude Control System: Analysis of a Feedback System (see Exercise 10.11.1, page 169), Stabilizing a systemwith a Right Half Plane Pole, Gain Stabilization (Proportional Control), Proportional plus Integral (PI) Control, MinimumGain to Stabilize, Pitch Attitude Command Following, Actuators, Sensors, High frequency dynamics, Bandwidth limitations,Introduction to Root Locus (RL), 1st Order Systems, 2nd Order Systems, Open Loop Poles, Open Loop Zeros, Poles MoveToward Zeros, Stability Robustness with respect to Gain Uncertainty, Gain Margins, Upward Gain Margin, Downward GainMargin, Imaginary (Phase) Crossovers, Phase Crossover Frequencies, Breakpoints, Complementary Root Locus (CRL).

Reading AssignmentRead Chapter 5, page 33 for a review of Laplace transforms and fundmanetal system concepts. Examine Chapter 9, page 123 foran introduction to modeling. Read Chapter 10, page 139 for an introduction to fundamental feedback control system concepts.

Recommended MaterialExercise 10.7.1: Naive Compensation - RHP Pole Zero Cancellations are Not Allowed!, page 149.

XXVI Feedback Control System Analysis and Design: A.A. Rodriguez c©1998

Example 10.7.1: Basic Controller Types, page 151.Example 10.8.1: Fundamental Constraints and Performance Limitations, page 152.Exercise 10.11.1: Simple Analysis and Design - An Introduction, page 169.Example 10.11.2: Closed Loop Time Response Analysis - Command Following and Disturbance Attenuation, page 163.

Must-know (Mandatory) MaterialExample 8.4.1: Step Response For Standard Second Order System, page 89Exercise 8.4.1: Standard Second Order System Step Response Formulae, page 94Exercise 8.4.2: Effect of An Additional Pole, page 95Exercise 8.4.3: Standard Second Order Feedback System: Dependence on k, page 95Exercise 8.4.4: Standard Second Order System Driven By Step and Sinusoid, page 95Example 8.4.2: Step Response Of Underdamped Second Order System With Numerator Zero, page 96

Example 8.5.1: 5th Order Linear Ordinary Differential Equation, page 98Exercise 8.5.1: LTI Step Response: URL for Understanding Effect of Poles and Zeros, page 100Example 8.5.2: Time Delay Approximations - Step Response, page 103

Example 8.6.1: An Upright Broomstick, page 104Example 8.6.2: Swinging Pendulum Within Grandfather Clock, page 105Example 8.6.3: Natural Modes Of Standard Second Order System, page 106

Exercise 8.7.1: Zeros and Steady State Output, page 112Example 8.7.1: When is the Method of the Transfer Function Applicable?, page 112Exercise 8.7.2: When Is MOTF Useful For Marginally Stable and Unstable Systems, page 113Exercise 8.7.3: Using the MOTF to Find a Particular Solution, page 115

Exercise 10.11.3: When is s = so a Closed Loop Pole?, page 173Exercise 10.11.4: Closed Loop Poles on the Imaginary Axis: Frequency Domain Criterion, page 174Exercise 10.11.5: Relating Open Loop Frequency Response to Root Locus, Stability Margins, page 175Example 10.11.3: Closed Loop Magnitude Responses: A Look At A Good Design, page 177Example 10.12.1: Feedforward Compensation To Shape Response To Step Commands, page 181Example 10.13.1: Effects of High Frequency Dynamics, Gain (Bandwidth) Restrictions, page 183

Homework #2

EEE480, Final Exam, Spring 95Problem 1 - Feedback System: ODEs via LaplaceProblem 2 - Sinusoidal Steady State Analysis

EEE480, Exam #1, Fall 95Problem 1 - CLS Analysis: ODEs, Laplace, Command Following, Disturb RejectionProblem 2 - Sinusoidal Steady State Analysis

EEE480, Final Exam, Fall 95Problem 1 - ODEs via Laplace, 2nd Order SystemsProblem 2 - Analysis of a CLS: Laplace, Steady State Analysis

Additional MaterialSee other exam problems.

Laboratory TopicsStudy the following examples and exercises for this lab and the next few labs:

Example 8.2.1: An Introduction to System Design Specifications, page 75Example 8.3.1: Analysis Of A First Order System: A Multi-Concept Example, page 80Example 8.4.1: Step Response For Standard Second Order System, page 89Exercise 8.4.1: Standard Second Order System Step Response Formulae, page 94Exercise 8.4.2: Effect of An Additional Pole, page 95Exercise 8.4.3: Standard Second Order Feedback System: Dependence on k, page 95Exercise 8.4.4: Standard Second Order System Driven By Step and Sinusoid, page 95Example 8.4.2: Step Response Of Underdamped Second Order System With Numerator Zero, page 96Exercise 8.5.1: LTI Step Response: Effect of Poles and Zeros, page 100

In this lab, we consider the analysis of the aircraft pitch attitude control system discussed in lecture. See Exercise 9.6.1, page 127

Feedback Control System Analysis and Design: A.A. Rodriguez c©1998 XXVII

for a simple aircraft model. See exercises in Section 10.11, page 161 for a fairly complete analysis of a pitch attitude controlsystem for an open loop unstable aircraft. This thorough analysis will be carried out for all of the physical systems which wewill consider throughout the semester.

In this lab, we also consider the design of a cruise control system for a car. See Exercise 9.3.1, page 123 for a simple carspeed-model. The control system is supposed to follow low frequency speed reference commands. Food For Thought: Whatcontrol strategy might you consider if you were interested in controling the precise position of the vehicle?

A central objective of the lab is to connect mathematical and engineering concepts to real-world engineering problems. Anotherobjective is to become proficient in using MATLAB and SIMULINK to solve real-world design problems.

XXVIII Feedback Control System Analysis and Design: A.A. Rodriguez c©1998


FocusRelationship Between Open Loop Frequency Response (Bode Plot) Information and Root Locus Plots.

ObjectivesTo relate open loop frequency response information to root locus (and complementary root locus) plots.

Lecture Topics

• Typical open loop frequency responses,

• Relationship between open loop frequency response and imaginary crossovers on the root locus (see Exercise 10.11.4,page 174 and Exercise 10.11.5, page 175),

• Stability Margins: gain, phase, and delay margins, gain and phase crossover frequencies (see Exercise 10.11.5, page 175),

• Polar (Nyquist) plots.

Concepts and TerminologyWhy Feedback? Stabilization, Command Following, Disturbance Rejection, Sensitivity Reduction, An Example: Stabilizationof an Unstable Plant, Proportional Control, Minimum gain (bandwidth) to stabilize, Example Continued, Low Frequency Com-mand Following and Disturbance Rejection

Reading AssignmentRead Section 10.14, page 188 on the Internal Model Principle.

Must-know (Mandatory) MaterialExercise 8.8.1: Magnitude and Phase Response Properties, page 115Example 8.8.1: Magnitude And Phase For Standard Second Order System, page 116Exercise 8.8.2: Standard Second Order System: Peak Magnitude Response Formula, page 118Exercise 8.8.3: Standard Second Order System: Bandwidth Formula, page 118Example 8.9.1: Series RLC Circuit Descriptor and State Space Representations, page 119

? Exercise 10.11.5: Relating Open Loop Frequency Response to Root Locus, Stability Margins, page 175Example 10.11.3: Closed Loop Magnitude Responses: A Look At A Good Design, page 177Example 10.12.1: Feedforward Compensation To Shape Response To Step Commands, page 181Example 10.13.1: Effects of High Frequency Dynamics, Gain (Bandwidth) Restrictions, page 183

Homework #3EEE480, Exam #1, Fall 95Problem 3 - Balancing an Inverted Pendulum in the Presence of DisturbancesProblem 4 - Internal Model PrincipleProblem 5 - Root Locus for a Second Order System

EEE480 Exam #1 Fall 1997Problem 5 - An Introduction to Root Loci: 2nd Order Systems

EEE480 Exam #1 Spring 1998Problems 1 - 5

EEE480 Exam #1 Spring 1999Problems 1 - 5


Laboratory TopicsIn this lab, we consider the modeling and control of a single robotic link; i.e. the classical fixed-base inverted pendulum problem.See Exercise 9.5.1, page 125 for a model of a single degree-of-freedom (dof) robotic link. See EEE480 Exam #1, Fall 1995, Prob-lem # 3: Balancing an Inverted Pendulum in the Presence of Disturbances. The goal here is to design a pendulum angle controlsystem which follows low frequency pendulum angle reference commands and attenuates low frequency disturbances. This classi-cal problem appears in many application areas (e.g. precision pointing of a surgical robot, weapon system, space telescope, etc.).

Feedback Control System Analysis and Design: A.A. Rodriguez c©1998 XXIX


FocusInternal Model Principle

ObjectivesTo learn the Internal Model Principle (IMP) and how it can be used in the design of feedback control systems.

Lecture Topics

• Closed loop zeros in terms of open loop poles and zeros

• Internal Model Principle (see Exapmle 10.14.4, page 193, Example 10.14.2, page 194)

• Steady State Analysis.

Concepts and TerminologyCruise Control Example, Integral Controller, Step Command Following, Ramp Command Following, Steady State TrackingError, Internal Model Principle, Command Following, Disturbance Rejection, System with a Right Half Plane Zero, Destabi-lization from Increasing Gain, Time Delays.

Reading Assignment

Must-know (Mandatory) MaterialExample 10.14.1: Internal Model Principle, Command Following, Disturbance Rejection, page 188Exercise 10.14.1: Stable Aircraft: Feedback Concepts, page 191Exercise 10.14.2: Step Command Following for an Unstable Aircraft, page 191Exercise 10.14.3: Step Command Following for a Single Degree-of-Freedom Robotic Link, page 192Exercise 10.14.4: Closed Loop Zeros and the Internal Model Principle, page 193Example 10.14.2: Internal Model Principle: Ramps, Steps, and Sinusoids, page 194

Homework #4EEE480, Final Exam, Fall 94Problem 1 a, b - Design for Downward Gain Margin, Phase crossover frequency Problem 2 - Design for Upward Gain Margin,Root Locus, Bode, Margins, Polar Plot

EEE480, Final Exam, Spring 95Problem 3 - Compensator Design, Internal Model Principle, Pole Placement, Tracking Error

EEE480, Final Exam, Fall 95Problem 3 - Compensator Design, Internal Model Principle, Root Locus

EEE480, Final Exam, Spring 99Problem 1 - Laplace Transforms, Sinusoidal AnalysisProblem 2 a, b - Analysis of a Semiconductor Process Temperature Control System

Master the following material from Franklin, et. al.:Section 3.4 Time Domain Specifications pp. 126-130, Formula 3.39 pp. 127 Time to peakFormula 3.40 pp. 127 Overshoot, Figure 3.17 pp. 128 Overshoot versus damping ratio zetaFormula 3.41 pp. 128 Settling Time, Figure 3.20 pp. 132 Second Order System with a ZeroExample 3.26 pp. 134-136 Boeing 747 Aircraft, Effect of RHP Zero


Laboratory TopicsIt is well known that helicopters are unstable near hover (see Exercise 9.7.1, page 129). In this laboratory, we consider thedesign of a control system for a Sikorsky UH-60A Helicopter near hover. Real world autopilots offer a pilot various options(e.g. speed hold, attitude hold, altitude hold, etc.). In this lab, we consider the design of a horizontal speed cruise control system- similar to that for a car. We also consider the design of a pitch attitude control system. You will see that proportional controlis not sufficient in either case. In either case, derivative information is essential. Dynamic compensation is thus unavoidable.See Section 8.9, page 118 on state space methods. To obtain a state space representation for a SISO transfer function, seeSection 20.5, page 263.

XXX Feedback Control System Analysis and Design: A.A. Rodriguez c©1998


FocusSimple Control System Design Examples, Prepare for Exam#1

ObjectivesDesign several simple feedback control systems. Prepare for Exam #1.

Material to ReviewPrepare for Exam #1.

EXAM #1 TopicsBlock diagram algebra, Laplace transfroms, solution of ode’s with constan coefficients, design specifications, first and secondorder time responses and frequency responses, time constant, settling time, overshoot, time-to-peak, rise time, root locus (RL)and complementary root locus (CRL) plots for low order systems (e.g. first, second, third, fourth), realting open loop frequencyresponse ideas and RL/CRL ideas, stability margins, simple design.

Lecture Topics

• Control Design for single-degree-of-freedom robotic link (see Exercise 9.5.1).

• Design for settling time, rise time, overshoot.

• PI controller with pre-filter.

Concepts and TerminologyPerformance Specifications, Transient Specifications, 1st Order Systems, Impact of Pole, Time constant, Step response, Settlingtime, Rise time, Impulse response, Examples of 1st order systems: RC circuit - voltage input/voltage output, temperature in asemiconductor fabrication furnace - heat input/temperature output, automobile - force input/speed output, dc motor - voltageinput/shaft speed output, airplane - elevator input/altitude output, Standard 2nd Order System: Transient Specifications, Im-pact of Poles, Damping Factor, Undamped Natural Frequency, Damped Natural Frequency, Nature of roots and dependence onDamping Factor, Overdamped Poles, Critically damped Poles, Underdamped Poles, Undamped Poles, Overshoot, Visualizationin complex s-plane, time constant, settling time, Time-to-Peak, Overshoot Formula, Examples of 2nd order systems: Car-enginesystem, DC motor, inverted pendulum, Performance Specifications, Steady State Specifications, Final Value Theorem, Analysisof Feedback Systems, Example: Cruise Control with Integral Controller, Design for Overshoot

Reading AssignmentPrepare for Exam# 1. Also start reading Chapter 11 on frequency response construction, Bode approximation ideas, and polarplots.

Must-know (Mandatory) MaterialMust be able to design a simple control system for an aircraft, car, single degree-of-freedom robotic link, and helicopter.

Homework #5EEE480, Final Exam, Spring 95Problem 5 - Compensator Design: Internal Model Principle, Disturbance Rejection, Pole Placement

EEE480 Final Exam, Fall 97Problem 4 - Control Design, Overshoot, Noise Attenuation (Just part a)

EEE480 Exam #1 Review, Fall 95Problem 1 - Step Response of a High Order LTI SystemProblem 2 - Analysis of a 2nd Order CLSProblem 3 - Analysis of a Liquid Level Control SystemProblem 4 - Internal Model Principle (IMP)Problem 5 - Root Loci for 2nd Order System

Additional MaterialSee other exam problems!

Laboratory TopicsIn this lab, we consider the design of an altitude capture-and-hold control system for a stable aircraft. The simple dynamicalmodel developed in Exercise 9.6.1, page 127) will be used. Also see EEE480 Exam #2, Problem # 4, Spring 1999: AltitudeHold Control System for a Stable Aircraft.

Feedback Control System Analysis and Design: A.A. Rodriguez c©1998 XXXI

1.6 Module 2: Analysis and Design Tools

XXXII Feedback Control System Analysis and Design: A.A. Rodriguez c©1998

WEEK 6 - Module 2 (Analysis and Design Tools)

FocusExam #1, Bode Plot Construction Methods

ObjectivesTake Exam#1 and Learn Bode’s frequency response construction (approximation) methods.

Material to ReviewNone.

Lecture Topics

• Stable systems with real poles

• Unstable systems

Concepts and TerminologyExample: Cruise Control with PI Control

Reading AssignmentChapter 11 on frequency response construction, Bode approximation ideas, and polar plots.

Must-know (Mandatory) MaterialAll examples and exercises in Chapter 11, page 207:

Example 11.2.1: First Order System: Bode Magnitude Approximation Ideas, page 208Example 11.2.2: First Order System: Bode Phase Approximation Ideas, page 210Example 11.2.3: First Order Unstable System: Bode Approximation Ideas, page 210Example 11.3.1: Integrator, page 212Example 11.3.2: Approximate Integrator at High Frequencies, page 213Example 11.3.3: Differentiator, page 214Example 11.3.4: Approximate Differentiator at Low Frequencies, page 215Example 11.3.5: Simple Second Order System with High Frequency Pole, page 216Example 11.3.6: Second Order System with High Frequency Pole and Zero, page 218Example 11.3.7: Second Order System with High Frequency Pole and Two High Frequency Zeros, page 219

Example 11.3.8: First Order Unstable System, page 220Example 11.3.9: First Order Unstable System with High Frequency Pole, page 221Example 11.3.10: First Order Unstable System with Two High Frequency Poles, page 222Exercise 11.3.1: Lead System: Phase Characterisitc, page 222Exercise 11.3.2: Lag System: Magnitude Characterisitc, page 223Exercise 11.3.3: Time Delay Approximations, page 223Exercise 11.4.1: Polar Plots, page 224

Homework #6Redo Exam #1. Due next class.

Additional Material

Laboratory TopicsIn this lab, we consider the design of a motor shaft displacement controller. The model for an armature controlled dc motordeveloped in Exercise 9.12.1, page 135 will be used.

Feedback Control System Analysis and Design: A.A. Rodriguez c©1998 XXXIII


FocusBode Plot Construction Methods, Routh Table Methods

ObjectivesApply Bode Plot Construction Methods and learn Routh Table Methods.


Lecture Topics

• Nonminimum phase systems

• Second Order Systems

• Routh Table

Concepts and TerminologyBode Plots for Systems with 2nd Order Underdamped Terms, Resonances due to lightly damped poles, Notches due to lightlydamped zeros, Bode Plots for higher order systems, Fitting a Stable Minimum Phase Transfer Function to a Magnitude Re-sponse. Routh Table and Routh Stability Criterion, Determining roots symmetric with respect to origin via Routh, Computationof Imaginary Crossovers via Routh.

Introduction to Frequency Response Methods: Bode Plots, Magnitude approximation ideas, Bode asymptotes for magnitude,systems with 1st order and 2nd order (Overdamped and Critically damped) terms, break frequencies, corrections, gain crossoverfrequencies, estimation of gain crossover frequencies

Stable Minimum Phase Systems

Phase approximation ideas, Bode asymptotes for phase, phase crossover frequencies, Approximation of imaginary crossoversusing Bode asymptotic phase ideas

Stability Robustness: Gain Margins - measured at phase crossovers, Phase Margin - measured at gain crossover, Delay Margin

Root Locus and Bode Plots: Determining Imaginary crossovers from a Bode Plot

Reading AssignmentChapter 12 on Routh Table Methods. Chapter 13 on Root Locus method.

Must-know (Mandatory) MaterialAll examples and exercises in Chapters 12, page 227:

Example 12.2.1: A Cubic Polynomial, page 228Example 12.2.2: Stability of a Cubic Polynomial, page 229Example 12.2.3: Zero Element in the First Column of Routh Table, page 229Example 12.2.5: Routh Analysis of a PI Control System for an Unstable Plant, page 231Example 12.2.6: Routh Analysis of PI Control System with High Frequency Pole, page 232Exercise 12.3.1: Stable Plant: Impact of LHP Pole and RHP Zero, page 234Exercise 12.3.2: Unstable Plant: Impact of LHP Pole and RHP Zero, page 235Exercise 12.3.3: Integrator Plant: Impact of Second Order Parasitics and RHP Zero, page 236

Homework #7Redo EXAM #1. Due next class.


Laboratory TopicsIn this lab, we consider the design of a vertical displacement controller for a magnetically suspended high speed train. Themodel developed in Exercise 9.10.1, page 133 will be used.

XXXIV Feedback Control System Analysis and Design: A.A. Rodriguez c©1998


FocusRoot Locus Rules

ObjectivesLearn Evans’ Root Locus rules.

Material to ReviewChapter 14 on Nyquist Stability Criterion.

Lecture Topics

• Asymptotes, cg, angle of asymptotes

• Real Axis Rule

• Break points

• Imaginary crossovers

Concepts and TerminologyIntroduction to Root Locus Method, open loop poles and zeros, closed loop poles, poles move toward zeros, angle criterion,real-axis rule, positive gain rule, negative gain rule, number of loci, number of asymptotes, imaginary crossovers, imaginarycrossovers, phase crossover frequecies, angle of asymptotes, center of gravity, break points, magnitude condition, stability sum-mary. 1st, 2nd, and 3rd Order Root Locus Examples Revisited, Root Locus Deformation Concepts, Root Loci Examples,Low and high frequency approximation ideas, Root loci for complex systems, unmodeled dynamics, High frequency actuatordynamics and structural modes, Effects of High Frequency Dynamics

Reading AssignmentChapter 13 on Root Locus.

Must-know (Mandatory) MaterialProblem 13.1.1: Standard �Root Locus� Problem Statement, page 237Example 13.2.1: Sample �Root Locus� Problems, page 238Exercise 13.2.1: Root Locus and Complementary Root Locus Plots, page 242

Homework #8Derive expression 3.39 for the peak time tp. See page 127 of Franklin, et. al..Derive expression 3.40 for the overshoot Mp. See page 127 of Franklin, et. al..


Laboratory TopicsIn this lab, we consider the design of a ball displacement controller for the ball-and-beam system considered in Exercise 9.11.1,page 134.

Feedback Control System Analysis and Design: A.A. Rodriguez c©1998 XXXV


FocusNyquist Stability Criterion.

ObjectivesLearn the Nyquist Stability Criterion - how to assess the stability of a closed loop system based on a polar plot of open loopfrequency response data. An analytic expression for L(jω) need not be known - only open loop frequency response data L(jω)is needed. Such data can be obtained empirically; i.e. from experiments performed on the real system.

Lecture Topics

• Polar plots in the L-plane

• Cauchy’s Principle of the Argument, Rouche’s theorem

• Nyquist stability criterion (see Theorem 14.2.1, page 243)

• Stability Summary

• Integration of Bode, RL/CRL, Routh, and Nyquist - Imaginary crossover and phase crossover frequencies

Concepts and TerminologyOpen loop frequency response L, Polar plots, closed loop stability, encirclements of the -1 point in the L-plane, distance to -1point in L-plane, stability robustness, stability margins.

Reading AssignmentChapter 14 on Nyquist Stability Criterion.

Must-know (Mandatory) MaterialEquation 14.1, page 243 for number of unstable closed loop poles.Exercise 14.2.1: Good Margins Is Not Enough For Robustness, page 244

Homework #9Handout on Root Locus Method - Including Examples

Master the following material from Franklin, et. al.:Example 4.6 - Proportional Control for DC Motor, Root Locus


Laboratory Topics

XXXVI Feedback Control System Analysis and Design: A.A. Rodriguez c©1998


FocusIntermediate Control System Design Examples.

ObjectivesDesign and evaluate several feedback control systems of intermediate complexity.

Lecture Topics

• Polar plots in the L−1-plane

• Inverse Nyquist stability criterion

Concepts and TerminologyEncirclements of the -1 point in L−1 plane, open loop right half plane zeros.Compensator Implementation Issues: Derivative Action - Feedback, Series, and Feedforward Compensation Structures.

Design Problems

Reading Assignment

Must-know (Mandatory) MaterialNyquisit Stability Analysis. Design via root locus techniques. All exercises in Chapter 15.

Homework #10

EEE480, Exam #1, Spring 95Problem 1 - IMP, Pole Placement, Uncertainty Via RouthProblem 2 - IMP, Sinusoidal Steady State Analysis - ReviewProblem 3 - Feedback System: Transfer Function, ODE, Laplace, Steady State - ReviewProblem 4 - Root Locus: Approximation of Critical ParametersProblem 5 - IMP, Stability, Steady State Tracking Error

EEE480, Exam #2, Spring 95Problem 1 - Stability Margins: Upward/Downward Gain MarginsProblem 2 - Sinusoidal Steady State Analysis using Bode Asymptotic ApproximationsProblem 4 - Bode Plots, Phase Margin, Root LocusProblem 5 - Root Locus Shape: Effect of Moving a Pole

EEE480, Final Exam, Spring 95Problem 4 - Routh Table: Stability Analysis, Single Parameter VariationsProblem 6 - Bode Plots: System Identifcation from a Frequency ResponseProblem 7 - Bode, Root Locus, Stability Robustness MarginsProblem 8 - Root Locus (Higher Order System)

Master the following material from Franklin, et. al.:Section 4.4.3, pp. 215-223 - Routh’s Stability CriterionExamples 4.17-4.20 - Routh ExamplesExamples 5.2-5.6 - Root Loci Examples

Master the following material from Franklin, et. al..Figure 6.2, page 342 - 2nd Order System Frequency ResponseFigure 6.5-6.7 - Simple Bode PlotsExample 6.3, page 352, Figure 6.8 - Bode Plot, Real Poles and ZerosExample 6.4 - Bode Plot, Integrator and Complex PolesExample 6.5 - Bode Plot, 2 Integrators, Complex Poles, Complex ZerosFigure 6.11, page 357 - Nonminimum Phase System


Feedback Control System Analysis and Design: A.A. Rodriguez c©1998 XXXIX

WEEK 11 - Module 3 (Control System Design)

FocusNyquist Concepts

ObjectivesReexamine Nyquist stability concepts, stability margins, and design considerations.

Lecture Topics

• An Introduction to Robust Performance

Concepts and TerminologyOpen loop frequency response L, Polar plots, closed loop stability, encirclements of the -1 point in the L-plane, distance to -1point in L-plane, stability robustness, stability margins.

Reading AssignmentChapter 18 on Small Gain Theorem.

Must-know (Mandatory) Material

Homework #11EEE480 Exam #2, Spring 95Problem 3 - Bode Asymptotic Magnitude PLot: Lightly Damped Zeros

EEE480 Exam #2, Fall 95Problem 1 - 2nd Order Frequency ResponsesProblem 2 - Stability Analysis of a Feedback System Via Routh Table, Stability Margins, RL and CRL SketchesProblem 3 - Bode Magnitude and Phase, Crossovers, Routh, Root LocusProblem 4 - Bode to Transfer Function: System Identification

EEE480, Final Exam, Fall 95Problem 4 - Compensator Design, Internal Model Principle, Routh, Bode, Root Locus, Stability Summary, MarginsProblem 5 - Bode Magnitude Plots, Gain Crossovers

EEE480 Exam #2 Review, Fall 97Problem 1 - Compensator Design, Overshoot, MarginsProblem 2 - Compensator Design, Stabilization, IMPProblem 3 - Controller Design, Pole PlacementProblem 4 - Root Locus: The Building of a Cool ProblemProblem 5 - Bode and Nyquisy: The Building of a Cool Problem

EEE480 Exam #2, Fall 97Problem 1 - Feedback System: Command Following, Disturbance Rejection, Noise Attenuation, Overshoot SpecificationProblem 2 - Analysis of a Feedback System: Root Locus, Bode, Crossovers, Stability MarginsProblem 3 - Compensator Design: Command Following, Disturbance Rejection, IMP, Stability Margins, Pole Placement, RootLocus


Laboratory Topics

XL Feedback Control System Analysis and Design: A.A. Rodriguez c©1998


FocusSmall Gain Concepts (Don’t raise the bandwidth too high!), Prepare for Exam#2

ObjectivesLearn small gain theorem and �small gain concepts� - fundamentally, we should never increase the bandwidth too high. TheSmall Gain Theorem quatifies how high is high.

Material to ReviewPrepare for Exam#2

EXAM #2 TopicsPerformance Specifications, Transient Specifications, Steady State Specifications, Second Order Systems, Overshoot, Root Lo-cus Rules, Imaginary Crossovers, Phase Crossover Frequencies, Routh Table, Bode Plots, Approximation Ideas, Gain CrossoverFrequencies, Simple Design, PID Controllers, Design Implementation, Feedback Compensation, Series Compensation, Feedfor-ward Compensation, Stability Margins

Lecture Topics

• Small Gain Theorem: A Stability Robustness Tool

• High Frequency Unmodeled Dynamics (Unstructured Uncertainty)

Concepts and TerminologyBandwidth restrictions, unstructured uncertainty stability robustness tests.

Reading Assignment


Homework #12

EEE480 Final Exam, Fall 97Problem 3 - Compensator Design, Command Following, Disturbance Rejection, IMP, StabilizationProblem 4 - Control Design, Overshoot, Noise Attenuation (Complete Problem)Problem 5 - Combining Root Locus, Bode, and Nyquist

Master the following material from Franklin et. al.:Figure 6.32, page 375 - GM and PMFigure 6.33, page 376 - GM and PMFigure 6.34, page 377 - GM and PM


Laboratory Topics

Feedback Control System Analysis and Design: A.A. Rodriguez c©1998 XLI


FocusLead and Lag Design using Root Locus plots, Exam#2

ObjectivesLearn how to design lead and lag controllers using Root Locus Methods.

Material to Review

Lecture Topics

• Lead design - improvement of transient response.

• Lead design using Root Locus techniques.

• Lag design - improvement of steady state response.

• Lag design using Root Locus techniques.

• Lead-Lag design using Root Locus plots.

Concepts and TerminologyNyquist Stability Criterion

Stability Summary

Reading AssignmentChapter 15 on design via Root Locus Plot techniques.Chapter ?? on design via Bode Plot techniques.

Must-know (Mandatory) MaterialExercise 15.2.1: Lag-Lead Controller Design using Root Locus Techniques, page 245Exercise 15.2.2: Lag-Lead Controller Design using Root Locus Techniques, page 245Exercise 15.3.1: Lag Controller Design using Root Locus Techniques, page 246

Exercise 14.2.1: Good Margins Is Not Enough For Robustness, page 244

Homework #13Mandatory material above.Redo Exam #2. Due next class.


Laboratory Topics

XLII Feedback Control System Analysis and Design: A.A. Rodriguez c©1998


FocusLead and Lag Design using Bode plots.

ObjectivesLearn to design lead and lag controllers using Bode plot methods.

Lecture Topics

• Lead design using Bode plots.

• Lag design using Bode plots.

• Lead-Lag design using Bode plots.

Concepts and TerminologyDesign via Bode techniques.

Reading AssignmentChapter ?? on design via Bode plot techniques.Pepare for Exam # 3 - a review for the final exam.

Must-know (Mandatory) MaterialAll exercises in Chapter ??:

Exercise 16.2.1: Lag-Lead Controller Design Using Bode Plot Techniques, page 247Exercise 16.3.1: Lag Controller Design Using Bode Plot Techniques, page 247Exercise 16.3.2: Lag Controller Design Using Bode Plot Techniques, page 248Exercise 16.4.1: Lead-Lag Controller Design Using Bode Plot Techniques, page 248

Homework #14Mandatory material above.


Laboratory Topics

Feedback Control System Analysis and Design: A.A. Rodriguez c©1998 XLIII


FocusControl System Design Examples, Exam#3 (take home).

ObjectivesDesign several feedback control systems.


Lecture Topics

• Design examples.

Concepts and Terminology

EXAM #3Topics: Feedback System Transient and Steady State Analysis, Compensator Design, Root Locus, Bode, Nyquist. This examis intended to be a review for the final exam!

Reading Assignment


Homework #15Prepare for final exam!


Laboratory Topics

1.5 module1: fundamentalfeedbacksystemconceptsaar.faculty.asu.edu/classes/eee480f99/roadmap.pdf ·...

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