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ContentsMotivation for Nonlinear ControlThe Tracking ProblemFeedback LinearizationAdaptive ControlRobust ControlSliding modeHigh-gainHigh-frequencyLearning ControlThe Tracking Problem, Revisited Using the Desired TrajectoryFeedback LinearizationAdaptive ControlFiltered tracking error r(t) for second-order systems)Introduction to ObserversObservers + ControllersFilter Based ControlFilter + Adaptive ControlSummaryHomework ProblemsA1A2A3A4 Design observer, observer + controller, control based on filter 1Applications and Areas of InterestMobile Platforms UUV, UAV, and UGV Satellites & Aircraft Automotive Systems Steer-By-Wire Thermal Management Hydraulic Actuators Spark Ignition CVTMechanical 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 EstimationChemical Systems BioreactorsTumor Modeling The Mathematical ProblemTypical Electromechanical System Model Classical Control Solution

Obstacles to Increased PerformanceSystem Model often contains Hard NonlinearitiesParameters in the Model are usually UnknownActuator Dynamics cannot be NeglectedSystem States are Difficult or Costly to Measure

Nonlinear Lyapunov-Based Techniques ProvideControllers Designed for the Full-Order Nonlinear ModelsAdaptive Update Laws for On-line Estimation of Unknown ParametersObservers or Filters for State Measurement ReplacementAnalysis that Predicts System Performance by Providing Envelopes for the Transient ResponseThe Mathematical Solution or ApproachMechatronicsBased Solution

Transient Performance Envelopes6Nonlinear Control Vs. Linear ControlWhy 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

Then56Feedback Linearization Problem (continued)

56Nonlinear Lyapunov-Based Techniques ProvideObservers or Filters for State Measurement ReplacementObserversMechatronicsBased Solution

Observer + Controllerxfxy=(,)ygxyu=(,,)u?xNonlinearControllerNonlinearObserver88Filter Based Control

8811

The Tracking Problem12The Tracking Problem (continued)

Feedback LinearizationExact Model KnowledgeExample Exact Model KnowledgeDynamics:Mass

Nonlinear DamperDisturbanceVelocityControl Inputa,b are constantsTracking Control Objective:Open Loop Error System:Controller:Closed Loop Error System:Solution:FeedforwardFeedbackAssume a,b are knownDrive e(t) to zeroExponential Stability

Example Exact Model KnowledgeMass

Nonlinear DamperDisturbanceVelocityControl Inputa,b are constantsOpen Loop Error System:Control Design:Closed Loop Error System:Solution:FeedforwardFeedbackAssume a,b are knownExponential Stability

Lyapunov Function:

A different perspective on the control design15Adaptive Control

By Assumption 2: both f(x) and W(x) are bounded.Constant that can be factored outYet to be designed, feed-forward term based on an estimate of the parameters16Adaptive Control (continued)

Lyapunov-like lemma

Note: detailed in deQueiroz

1617

Adaptive Control (continued)

Example Unknown Model ParametersOpen Loop Error System:Control Design:a,b are unknownconstantsSame 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 uncertaintyClosed Loop Error System:At this point, we have not fully developed the controller since and are yet to be determined.parameter error

( is UC)Example Unknown Model ParametersFundamental Theorem

effects of conditions i) and ii)i) Ifii) If

is boundediii) Ifis bounded satisfies condition i)

finally becomes a constant

Non-Negative Function:Time Derivative of V(t):is boundedexamine condition ii)design andsubstitute the dynamics forconstanteffects of condition iii)

Example Unknown Model ParametersSubstitute Error System:How do we select and such that ?Update Law Design:Substitute in Update Laws:andFundamental Theoremis bounded

all signals are boundedFundamental TheoremFeedforwardFeedbackcontrol structurederived fromstability analysiscontrol objective achievedis bounded

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)

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)

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.2-224Robust Control

Restriction on the structure but not the uncertainty2425Robust Control (continued)

2526

Robust (Sliding Mode) Control

2627Robust (Sliding Mode) Control (continued)

2728

Robust (High-Gain) Control

2829

Robust (High-Gain) Control (continued)

2930Robust (High-Gain) Control (continued)

3031

Robust (High-Frequency) Control

=3132

Learning Control

3233Learning Control (continued)

3334Learning Control (continued)

3435Learning Control (continued)

3536The Tracking Problem, RevisitedUsing the Desired Trajectory

Differentiable assumption needed in analysis but not required to implement control.

Feedback Linearization36

Mean Value Theorem for scalar function

3738The Tracking Problem, Revisited (continued)

3839

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

3940Adaptive Control40

The Tracking Problem, Revisited (continued)41The Tracking Problem, Revisited (continued)

4142Continuous Asymptotic Tracking

4243

Continuous Asymptotic Tracking (continued)

4344Continuous Asymptotic Tracking (continued)

4445Continuous Asymptotic Tracking (continued)

4546

Continuous Asymptotic Tracking (continued)

4647

Continuous Asymptotic Tracking (continued)

4748

Continuous Asymptotic Tracking (continued)

4849

Continuous Asymptotic Tracking (continued)

4950Continuous Asymptotic Tracking (continued)

5051Continuous Asymptotic Tracking (continued)

5152Continuous Asymptotic Tracking (continued)

5254Feedback Linearization Problem (continued)

5455Feedback Linearization Problem (continued)

5557

Previous Problem Using a Robust Approach5758Previous Problem Using a Robust Approach

58Estimate of the StateObservers Alonexfxy=(,)ygxyu=(,,)u?xNonlinearObserver

61Observers

Example: If angle is measured with an encoder then the velocity must be estimated, e.g. using backwards difference.Encoder Measured PositionPositionVelocity EstimateBackwards difference may yield noisy estimate of actual velocity6162Observers (continued)

6263Observers (continued)

6364Observers (continued)

6465Observers (continued)

6566Observers (continued)

6667

Observers (continued)

6768

Observers (continued)

6869

Observers (continued)

69

Observers (continued)

7071

Observers (continued)

Observers (continued)

7273

Observers (continued)

74Combining Observers & Controllers (continued)

7475Observers (continued)

76

Observers (continued)

Mean Value Theorem (in one variable)

7677Observers (continued)

7778

Observers (continued)

7880Combining Observers & Controllers

8081Combining Observers & Controllers

8182Combining Observers & Controllers (continued)

8283Combining Observers & Controllers (continued)

8384Combining Observers & Controllers (continued)

8485Combining Observers & Controllers (continued)

8586Combining Observers & Controllers (continued)

8687Combining Observers & Controllers (continued)

8789Filtering Control (continued)

8990Filtering Control (continued)

9091Filtering Control (continued)

9192Filtering Control (continued)

9293Filtering Control (continued)

9394

Filtering Control (continued)

9495Filtering Control (continued)

9596Filtering Control (continued)

9697Adaptive Approach

9798Adaptive Approach (continued)

9899Adaptive Approach (continued)

99100Adaptive Approach (continued)

100101Adaptive Approach (continued)

101102Adaptive Approach (continued)

102103Variable Structure Observer

103104Variable Structure Observer (continued)

104105Variable Structure Observer (continued)

105106Variable Structure Observer (continued)

106107Variable Structure Observer (continued)

107108Filtering Control, Revisited

108109Filtering Control, Revisited (continued)

109110Filtering Control, Revisited (continued)

110111Filtering Control, Revisited (continued)

111112

Filtering Control, Revisited (continued)112113Filtering Control, Revisited (continued)

113114Filtering Control, Revisited (continued)

114Summary115

115Summary116

116Homework A.1

117Homework A.1-1 (sol)

118

Homework A.1-1 (sol)119

Homework A.1-1 (sol)120Homework A.1-2 (sol)

121Homework A.1-2 (sol)

122Homework A.1-2 (sol)

123

Homework A.1-2 (sol)

124

Homework A.1-2 (sol)125Homework A.1-2 (sol)

126Homework A.1-3 (sol)

127k=1

Homework A.1-3 (sol)128Homework A.1-2 (sol)

129Homework A.1-3 (sol)

130Homework A.1-3 (sol)

131

Note that the analysis only guaranteed Ultimate Bounded tracking error.Homework A.1-3 (sol)132

Homework A.1-3 (sol)

133Homework A.1-2 (sol)

134Homework A.1-3 (sol)

135

Homework A.1-3 (sol)136

Homework A.1-3 (sol)137Homework A.1-2 (sol)

138

Homework A.1-2 (sol)One of the advantagesof the repetitive learning scheme is that the requirement thatthe robot return to the exact same initial condition after each learningtrial is replaced by the less restrictive requirement that the desired trajectoryof the robot be periodic.Homework A.1-2 (sol)a=1, k=kd=5

Homework A.2

141Homework A.2-1 (sol)

142Homework A.2-2 (sol)

143Homework A.2-3 (sol)

144Homework A.2-3 (sol)

Homework A.2-3 (sol)

Homework A.3Homework A.3

2.Homework A.3

3.

4.

Homework A.3-1 (sol)

Homework A.3-2 (sol)2.

Homework A.3-3 (sol)3.

Homework A.3-4 (sol)4.

Homework A.3-5 (sol)

Homework A.3-5 (sol)

Homework A.3-5 (sol)

Homework A.3-5 (sol)Homework A.4

Homework A.4-1 (sol)

Homework A.4-1 (sol)

But that estimate has velocity measurement in it?

Homework A.4-2 (sol)

Homework A.4-2 (sol)

Homework A.4-2 (sol)

Homework A.4-2 (sol)

Homework A.4-3 (sol)

Homework A.4-3 (sol)