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Contents 1.Motivation for Nonlinear ControlMotivation for Nonlinear Control 2.The Tracking ProblemThe Tracking Problem 1.Feedback LinearizationFeedback Linearization 3.Adaptive ControlAdaptive Control 4.Robust ControlRobust Control 1.Sliding modeSliding mode 2.High-gainHigh-gain 3.High-frequencyHigh-frequency 5.Learning ControlLearning Control 6.The Tracking Problem, Revisited Using the Desired Trajectory 1.Feedback Linearization 2.Adaptive Control 7.Filtered tracking error r(t) for second-order systems)Filtered tracking error r(t) for second-order systems) 8.Introduction to ObserversIntroduction to Observers 9.Observers + ControllersObservers + Controllers 10.Filter Based ControlFilter Based Control 1.Filter + Adaptive ControlFilter + Adaptive Control 11.SummarySummary 12.Homework ProblemsHomework Problems 1.A1 2.A2 3.A3 4.A4 – Design observer, observer + controller, control based on filterA4 – Design observer, observer + controller, control based on filter

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2 Nonlinear Control Why do we use nonlinear control : –Tracking, regulate state setpoint –Ensure the desired stability properties –Ensure the appropriate transients –Reduce the sensitivity to plant parameters find state feedback output feedback Consider the following problem:

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Applications and Areas of Interest Mobile Platforms UUV, UAV, and UGV Satellites & Aircraft Automotive Systems Steer-By-Wire Thermal Management Hydraulic Actuators Spark Ignition CVT Mechanical Systems Textile and Paper Handling Overhead Cranes Flexible Beams and Cables MEMS Gyros Robotics Position/Force Control Redundant and Dual Robots Path Planning Fault Detection Teleoperation and Haptics Electrical/Computer Systems Electric Motors Magnetic Bearings Visual Servoing Structure from Motion Nonlinear Control and Estimation Chemical Systems Bioreactors Tumor Modeling

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The Mathematical Problem Typical Electromechanical System Model Classical Control Solution Obstacles to Increased Performance –System Model often contains Hard Nonlinearities –Parameters in the Model are usually Unknown –Actuator Dynamics cannot be Neglected –System States are Difficult or Costly to Measure

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Nonlinear Lyapunov-Based Techniques Provide –Controllers Designed for the Full-Order Nonlinear Models –Adaptive Update Laws for On-line Estimation of Unknown Parameters –Observers or Filters for State Measurement Replacement –Analysis that Predicts System Performance by Providing Envelopes for the Transient Response The Mathematical Solution or Approach Mechatronics Based Solution Transient Performance Envelopes

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6 Nonlinear Control Vs. Linear Control Why not always use a linear controller ? –It just may not work. Ex: Choose We see that the system cant be made asymptotically stable at On the other hand, a nonlinear feedback does exist : Then Asymptotically stable if Then

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7 Example Even if a linear feedback exists, nonlinear one may be better. Ex: + _ + _

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8 Example (continued) Let us use a nonlinear controller : To design it, consider the same system in the form: If Why is that especially interesting? If we could get onto that line then the system converges to the origin Both systems have interesting properties, can we combine the best features of each into a single control?

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9 Example (continued) sliding line Created a new trajectory: the system is insensitive to disturbance in the sliding regime Variable structure control HW Simulate this system and control. Be sure to plot the evolution of the states

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Example (continued) 10

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11 The Tracking Problem

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12 The Tracking Problem (continued) Feedback Linearization Exact Model Knowledge

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Example Exact Model Knowledge Dynamics: Mass Nonlinear Damper DisturbanceVelocity Control Input a, b are constants Tracking Control Objective: Open Loop Error System: Controller: Closed Loop Error System: Solution: Feedforward Feedback Assume a, b are known Drive e(t) to zero Exponential Stability

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Example Exact Model Knowledge Mass Nonlinear Damper DisturbanceVelocity Control Input a, b are constants Open Loop Error System: Control Design: Closed Loop Error System: Solution: Feedforward Feedback Assume a, b are known Exponential Stability Lyapunov Function: A different perspective on the control design

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15 Adaptive Control By Assumption 2: both f(x) and W(x) are bounded. Constant that can be factored out Yet to be designed, feed- forward term based on an estimate of the parameters

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16 Adaptive Control (continued) Lyapunov-like lemma Note: detailed in deQueiroz

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17 Adaptive Control (continued)

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Example Unknown Model Parameters Open Loop Error System: Control Design: a, b are unknown constants Same controller as before, but and are functions of time How do we adjust and ? Use the Lyapunov Stability Analysis to develop an adaptive control design tool for compensation of parametric uncertainty Closed Loop Error System: At this point, we have not fully developed the controller since and are yet to be determined. parameter error

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( is UC) Example Unknown Model Parameters Fundamental Theorem effects of conditions i) and ii) i) If ii) If is bounded iii) If is bounded satisfies condition i) finally becomes a constant Non-Negative Function: Time Derivative of V(t): is bounded examine condition ii) design and substitute the dynamics for constant effects of condition iii)

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Example Unknown Model Parameters Substitute Error System: How do we select and such that ? Update Law Design: Substitute in Update Laws: and Fundamental Theorem is bounded all signals are bounded Fundamental Theorem Feedforward Feedback control structure derived from stability analysis control objective achieved is bounded

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How Can We Use the Adaptive Controller? Design adaptive control to track a desired trajectory while compensating for unknown, constant parameters (parametric uncertainty) Adaptive control with backstepping in cascaded subsytems to track a desired trajectory while compensating for unknown, constant parameters (parametric uncertainty)

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How Can We Use the Adaptive Controller? (continued) Adaptive control with backstepping in cascaded subsytems to track a desired trajectory while compensating for unknown, constant parameters (parametric uncertainty)

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How Can We Use the Adaptive Controller? What about the case where input multiplied by an unknown parameter, can we design adaptive control to track a desired trajectory while compensating for unknown, constant parameters (parametric uncertainty) Homework A.2-2

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24 Robust Control Restriction on the structure but not the uncertainty

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25 Robust Control (continued)

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26 Robust (Sliding Mode) Control

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27 Robust (Sliding Mode) Control (continued)

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28 Robust (High-Gain) Control

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29 Robust (High-Gain) Control (continued)

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30 Robust (High-Gain) Control (continued)

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31 Robust (High-Frequency) Control =

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32 Learning Control

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33 Learning Control (continued)

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34 Learning Control (continued)

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35 Learning Control (continued)

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36 The Tracking Problem, Revisited Using the Desired Trajectory Differentiable assumption needed in analysis but not required to implement control. Feedback Linearization

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Mean Value Theorem for scalar function

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38 The Tracking Problem, Revisited (continued)

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39 The Tracking Problem, Revisited (continued) Region is adjustable (not a fixed local region)

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40 Adaptive Control 40 The Tracking Problem, Revisited (continued)

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41 The Tracking Problem, Revisited (continued)

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42 Continuous Asymptotic Tracking

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43 Continuous Asymptotic Tracking (continued)

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44 Continuous Asymptotic Tracking (continued)

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45 Continuous Asymptotic Tracking (continued)

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46 Continuous Asymptotic Tracking (continued)

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47 Continuous Asymptotic Tracking (continued)

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48 Continuous Asymptotic Tracking (continued)

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49 Continuous Asymptotic Tracking (continued)

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50 Continuous Asymptotic Tracking (continued)

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51 Continuous Asymptotic Tracking (continued)

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52 Continuous Asymptotic Tracking (continued)

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53 Feedback Linearization for Second-Order Systems

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54 Feedback Linearization Problem (continued)

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55 Feedback Linearization Problem (continued)

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56 Feedback Linearization Problem (continued)

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57 Previous Problem Using a Robust Approach

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58 Previous Problem Using a Robust Approach

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Nonlinear Lyapunov-Based Techniques Provide –Observers or Filters for State Measurement Replacement Observers Mechatronics Based Solution

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Estimate of the State Observers Alone xfxy (,) ygxyu (,,) u ? x Nonlinear Observer

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61 Observers Example: If angle is measured with an encoder then the velocity must be estimated, e.g. using backwards difference. Encoder Measured Position Position Velocity Estimate Backwards difference may yield noisy estimate of actual velocity

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62 Observers (continued)

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63 Observers (continued)

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64 Observers (continued)

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65 Observers (continued)

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66 Observers (continued)

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67 Observers (continued)

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68 Observers (continued)

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69 Observers (continued)

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71 Observers (continued)

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73 Observers (continued)

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74 Combining Observers & Controllers (continued)

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75 Observers (continued)

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76 Observers (continued) Mean Value Theorem (in one variable)

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77 Observers (continued)

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78 Observers (continued)

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Observer + Controller xfxy (,) ygxyu (,,) u ? x Nonlinear Controller Nonlinear Observer

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80 Combining Observers & Controllers

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81 Combining Observers & Controllers

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82 Combining Observers & Controllers (continued)

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83 Combining Observers & Controllers (continued)

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84 Combining Observers & Controllers (continued)

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85 Combining Observers & Controllers (continued)

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86 Combining Observers & Controllers (continued)

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87 Combining Observers & Controllers (continued)

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88 Filter Based Control

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89 Filtering Control (continued)

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90 Filtering Control (continued)

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91 Filtering Control (continued)

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92 Filtering Control (continued)

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93 Filtering Control (continued)

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94 Filtering Control (continued)

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95 Filtering Control (continued)

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96 Filtering Control (continued)

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97 Adaptive Approach

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98 Adaptive Approach (continued)

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99 Adaptive Approach (continued)

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100 Adaptive Approach (continued)

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101 Adaptive Approach (continued)

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102 Adaptive Approach (continued)

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103 Variable Structure Observer

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104 Variable Structure Observer (continued)

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105 Variable Structure Observer (continued)

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106 Variable Structure Observer (continued)

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107 Variable Structure Observer (continued)

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108 Filtering Control, Revisited

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109 Filtering Control, Revisited (continued)

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110 Filtering Control, Revisited (continued)

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111 Filtering Control, Revisited (continued)

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112 Filtering Control, Revisited (continued)

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113 Filtering Control, Revisited (continued)

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114 Filtering Control, Revisited (continued)

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Summary 115

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Summary 116

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Homework A.1

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Homework A.1-1 (sol)

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Homework A.1-2 (sol)

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Homework A.1-3 (sol)

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k=1 Homework A.1-3 (sol)

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Homework A.1-2 (sol)

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Homework A.1-3 (sol)

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Note that the analysis only guaranteed Ultimate Bounded tracking error. Homework A.1-3 (sol)

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Homework A.1-2 (sol)

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Homework A.1-3 (sol)

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Homework A.1-2 (sol)

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One of the advantages of the repetitive learning scheme is that the requirement that the robot return to the exact same initial condition after each learning trial is replaced by the less restrictive requirement that the desired trajectory of the robot be periodic.

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Homework A.1-2 (sol) a=1, k=kd=5

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Homework A.2

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Homework A.2-1 (sol)

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Homework A.2-2 (sol)

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Homework A.2-3 (sol)

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Homework A.3

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2.

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Homework A

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Homework A.3-1 (sol)

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Homework A.3-2 (sol) 2.

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Homework A.3-3 (sol) 3.

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Homework A.3-4 (sol) 4.

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Homework A.3-5 (sol)

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Homework A.4

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Homework A.4-1 (sol)

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But that estimate has velocity measurement in it?

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Homework A.4-2 (sol)

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Homework A.4-3 (sol)

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