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Neural Network Control of Robots and Nonlinear Systems

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Neural Network Control of Robots and Nonlinear Systems Neural Network Control of Robot Manipulators and Nonlinear Systems Neural Network Control of Robot Manipulators and Nonlinear Systems F.L. LEWIS Automation and Robotics Research Institute The University of Texas at Arlington S. JAGANNATHAN Systems and Controls Research Caterpillar, Inc., Mossville A. YES¸ILDIREK Manager, New Product Development Depsa, Panama City Contents List of Tables of Design Equations xi List of Figures xviii Series Introduction xix Preface xxi 1 Background on Neural Networks 1 1.1NEURALNETWORKTOPOLOGIESANDRECALL....... 2 1.1.1 NeuronMathematicalModel.................. 2 1.1.2 MultilayerPerceptron...................... 8 1.1.3 Linear-in-the-Parameter(LIP)NeuralNets.......... 10 1.1.4 DynamicNeuralNetworks.................... 13 1.2PROPERTIESOFNEURALNETWORKS.............. 25 1.2.1 Classification,Association,andPatternRecognition..... 26 1.2.2 FunctionApproximation..................... 30 1.3 NEURAL NETWORK WEIGHT SELECTION AND TRAINING . 33 1.3.1 DirectComputationoftheWeights............... 34 1.3.2 Training the One-Layer Neural Network— Gradient Descent 36 1.3.3 Training the Multilayer Neural Network— Backpropagation Tuning............................... 42 1.3.4 Improvements on Gradient Descent ............... 51 1.3.5 HebbianTuning......................... 56 1.3.6 Continuous-TimeTuning.................... 58 1.4REFERENCES.............................. 61 1.5PROBLEMS............................... 63 2 Background on Dynamic Systems 69 2.1 DYNAMICAL SYSTEMS ........................ 69 2.1.1 Continuous-TimeSystems.................... 70 2.1.2 Discrete-TimeSystems...................... 73 2.2SOMEMATHEMATICALBACKGROUND............. 77 2.2.1 VectorandMatrixNorms.................... 77 2.2.2 ContinuityandFunctionNorms................. 79 2.3 PROPERTIES OF DYNAMICAL SYSTEMS ............. 80 v vi CONTENTS 2.3.1 Stability .............................. 80 2.3.2 Passivity ............................. 82 2.3.3 Observability and Controllability ................ 85 2.4 FEEDBACK LINEARIZATION AND CONTROL SYSTEM DESIGN 88 2.4.1 Input-Output Feedback Linearization Controllers ....... 89 2.4.2 ComputerSimulationofFeedbackControlSystems...... 94 2.4.3 Feedback Linearization for Discrete-Time Systems . 96 2.5 NONLINEAR STABILITY ANALYSIS AND CONTROLS DESIGN 100 2.5.1 LyapunovAnalysisforAutonomousSystems......... 100 2.5.2 Controller Design Using Lyapunov Techniques ........ 105 2.5.3 LyapunovAnalysisforNon-AutonomousSystems.......109 2.5.4 Extensions of Lyapunov Techniques and Bounded Stability . 111 2.6REFERENCES.............................. 117 2.7PROBLEMS............................... 119 3 Robot Dynamics and Control 125 3.0.1 CommercialRobotControllers................. 125 3.1KINEMATICSANDJACOBIANS................... 126 3.1.1 KinematicsofRigidSerial-LinkManipulators......... 127 3.1.2 RobotJacobians......................... 130 3.2 ROBOT DYNAMICS AND PROPERTIES .............. 131 3.2.1 JointSpaceDynamicsandProperties............. 132 3.2.2 State Variable Representations ................. 136 3.2.3 CartesianDynamicsandActuatorDynamics......... 137 3.3 COMPUTED-TORQUE (CT) CONTROL AND COMPUTER SIM- ULATION.................................138 3.3.1 Computed-Torque(CT)Control................ 138 3.3.2 ComputerSimulationofRobotControllers.......... 140 3.3.3 Approximate Computed-Torque Control and Classical Joint Control.............................. 145 3.3.4 DigitalControl.......................... 147 3.4 FILTERED-ERROR APPROXIMATION-BASED CONTROL . 153 3.4.1 A General Controller Design Framework Based on Approxi- mation...............................156 3.4.2 Computed-TorqueControlVariant............... 159 3.4.3 AdaptiveControl......................... 159 3.4.4 RobustControl.......................... 164 3.4.5 LearningControl......................... 167 3.5CONCLUSIONS............................. 169 3.6REFERENCES.............................. 170 3.7PROBLEMS............................... 171 4 Neural Network Robot Control 175 4.1 ROBOT ARM DYNAMICS AND TRACKING ERROR DYNAMICS 177 4.2 ONE-LAYER FUNCTIONAL-LINK NEURAL NETWORK CON- TROLLER................................ 181 4.2.1 ApproximationbyOne-LayerFunctional-LinkNN...... 182 CONTENTS vii 4.2.2 NNControllerandErrorSystemDynamics.......... 183 4.2.3 Unsupervised Backpropagation Weight Tuning ........ 184 4.2.4 Augmented Unsupervised Backpropagation Tuning— Remov- ingthePECondition...................... 189 4.2.5 Functional-Link NN Controller Design and Simulation Example192 4.3TWO-LAYERNEURALNETWORKCONTROLLER........196 4.3.1 NN Approximation and the Nonlinearity in the Parameters Problem.............................. 196 4.3.2 ControllerStructureandErrorSystemDynamics....... 198 4.3.3 WeightUpdatesforGuaranteedTrackingPerformance.... 200 4.3.4 Two-Layer NN Controller Design and Simulation Example . 208 4.4 PARTITIONED NN AND SIGNAL PREPROCESSING ....... 208 4.4.1 PartitionedNN.......................... 210 4.4.2 Preprocessing of Neural Net Inputs ............... 211 4.4.3 SelectionofaBasisSetfortheFunctional-LinkNN..... 211 4.5PASSIVITYPROPERTIESOFNNCONTROLLERS........ 214 4.5.1 Passivity of the Tracking Error Dynamics ........... 214 4.5.2 Passivity Properties of NN Controllers ............. 215 4.6CONCLUSIONS............................. 218 4.7REFERENCES.............................. 219 4.8PROBLEMS............................... 221 5 Neural Network Robot Control: Applications and Extensions 223 5.1FORCECONTROLUSINGNEURALNETWORKS........ 224 5.1.1 ForceConstrainedMotionandErrorDynamics........ 225 5.1.2 NeuralNetworkHybridPosition/ForceController...... 227 5.1.3 Design Example for NN Hybrid Position/Force Controller . 234 5.2 ROBOT MANIPULATORS WITH LINK FLEXIBILITY, MOTOR DYNAMICS, AND JOINT FLEXIBILITY .............. 235 5.2.1 Flexible-LinkRobotArms....................235 5.2.2 Robots with Actuators and Compliant Drive Train Coupling 240 5.2.3 Rigid-Link Electrically-Driven (RLED) Robot Arms ..... 246 5.3 SINGULAR PERTURBATION DESIGN ............... 247 5.3.1 Two-Time-ScaleControllerDesign............... 248 5.3.2 NN Controller for Flexible-Link Robot Using Singular Per- turbations............................. 251 5.4 BACKSTEPPING DESIGN ....................... 260 5.4.1 BacksteppingDesign....................... 260 5.4.2 NN Controller for Rigid-Link Electrically-Driven Robot Using Backstepping........................... 264 5.5CONCLUSIONS............................. 272 5.6REFERENCES.............................. 272 5.7PROBLEMS............................... 274 viii CONTENTS 6 Neural Network Control of Nonlinear Systems 279 6.1 SYSTEM AND TRACKING ERROR DYNAMICS .......... 280 6.1.1 TrackingControllerandErrorDynamics............281 6.1.2 Well-DefinedControlProblem................. 283 6.2 CASE OF KNOWN FUNCTION g(x)................. 283 6.2.1 ProposedNNController.....................284 6.2.2 NN Weight Tuning for Tracking Stability ........... 285 6.2.3 IllustrativeSimulationExample.................287 6.3 CASE OF UNKNOWN FUNCTION g(x)............... 288 6.3.1 ProposedNNController.....................289 6.3.2 NN Weight Tuning for Tracking Stability ........... 291 6.3.3 IllustrativeSimulationExamples................ 298 6.4CONCLUSIONS............................. 303 6.5REFERENCES.............................. 305 7 NN Control with Discrete-Time Tuning 307 7.1 BACKGROUND AND ERROR DYNAMICS ............. 308 7.1.1 NeuralNetworkApproximationProperty........... 308 7.1.2 Stability of Systems ....................... 310 7.1.3 Tracking Error Dynamics for a Class of Nonlinear Systems . 310 7.2 ONE-LAYER NEURAL NETWORK CONTROLLER DESIGN . 312 7.2.1 Structure of the One-layer NN Controller and Error System Dynamics............................. 313 7.2.2 One-layerNeuralNetworkWeightUpdates.......... 314 7.2.3 ProjectionAlgorithm...................... 318 7.2.4 Ideal Case: No Disturbances or NN Reconstruction Errors . 323 7.2.5 One-layer Neural Network Weight Tuning Modification for RelaxationofPersistencyofExcitationCondition...... 323 7.3 MULTILAYER NEURAL NETWORK CONTROLLER DESIGN . 329 7.3.1 Structure of the NN Controller and Error System Dynamics . 332 7.3.2 MultilayerNeuralNetworkWeightUpdates.......... 333 7.3.3 ProjectionAlgorithm...................... 340 7.3.4 Multilayer Neural Network Weight Tuning Modification for RelaxationofPersistencyofExcitationCondition...... 342 7.4PASSIVITYPROPERTIESOFTHENN............... 353 7.4.1 Passivity Properties of the Tracking Error System ...... 353 7.4.2 Passivity Properties of One-layer Neural Networks and the Closed-LoopSystem....................... 354 7.4.3 Passivity Properties of Multilayer Neural Networks . 356 7.5CONCLUSIONS............................. 357 7.6REFERENCES.............................. 357 7.7PROBLEMS............................... 359 8 Discrete-Time Feedback Linearization by Neural Networks 361 8.1 SYSTEM DYNAMICS AND THE TRACKING PROBLEM ..... 362 8.1.1 Tracking Error Dynamics for a Class of Nonlinear Systems . 362 8.2 NN CONTROLLER DESIGN FOR FEEDBACK LINEARIZATION 364 CONTENTS ix 8.2.1 NNApproximationofUnknownFunctions...........365 8.2.2 ErrorSystemDynamics..................... 366 8.2.3 Well-DefinedControlProblem................. 368 8.2.4 ProposedController....................... 369 8.3 SINGLE-LAYER NN FOR FEEDBACK LINEARIZATION ..... 369 8.3.1 WeightUpdatesRequiringPersistenceofExcitation..... 370 8.3.2 ProjectionAlgorithm...................... 377 8.3.3 Weight Updates not Requiring Persistence of Excitation . 378 8.4 MULTILAYER NEURAL NETWORKS FOR FEEDBACK LINEARIZA- TION..................................
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