Passivity-based Control of Euler-Lagrange Systems

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1 Romeo Ortega, Antonio Loria, Per Johan Nicklasson and Hebertt Sira-Ramfrez Passivity-based Control of Euler-Lagrange Systems Mechanical, Electrical and Electromechanical Applications Springer

2 Contents Notation xxxi 1 Introduction 1 1 From control engineering to mathematical control theory and back A route towards applications 3 3 Why Euler-Lagrange Systems? 4 4 On the role of interconnection 7 5 Why passivity? 8 6 What is passivity-based control? 10 7 Some historical remarks Euler-Lagrange Systems and nonlinear dynamics Passivity and feedback stabilization 12 2 Euler-Lagrange Systems 15 1 The Euler-Lagrange equations 16 2 Input-output properties Passivity of EL Systems Passivity of the error dynamics Other properties and assumptions Passive Subsystems decomposition An EL structure-preserving interconnection 26 3 Lyapunov stability properties Fully-damped Systems Underdamped Systems 28 4 Examples 30 xiu

3 xiv CONTENTS 4.1 A rotational/translational proof mass actuator Levitated ball Flexible joints robots The Duffing System A marine surface vessel 36 5 Concluding remarks 37 I Mechanical Systems 39 3 Set-point regulation 41 1 State feedback control of fully-actuated Systems A basic result: The PD Controller An introductory example Physical interpretation and literature review 46 2 Output feedback stabilization of underactuated Systems Literature review Problem formulation Euler-Lagrange Controllers Examples 51 3 Bounded Output feedback regulation Literature review Problem formulation Globally stabilizing saturated EL Controllers Examples 68 4 Set-point regulation under parameter uncertainty Literature review Adaptive control Linear PID control Nonlinear PID control Output feedback regulation: The PI 2 D Controller 85 5 Concluding remarks 91

4 CONTENTS xv 4 Trajectory tracking control 93 1 State feedback control of fully-actuated Systems The PD+ Controller The Slotine and Li Controller 96 2 Adaptive trajectory tracking Adaptive Controller of Slotine and Li A robust adaptive Controller 98 3 State feedback of underactuated Systems Model and problem formulation Literature review A passivity-based Controller Comparison with backstepping and cascaded designs A Controller without jerk measurements Output feedback of fully-actuated Systems Semiglobal tracking control of robot manipulators Discussion on global tracking Simulation results Concluding remarks Adaptive disturbance attenuation: Friction compensation Adaptive friction compensation The LuGre friction model DC motor with friction Robot manipulator Simulations State-space passifiable Systems with disturbances Background A theorem for passifiable affine nonlinear Systems Concluding remarks 131

5 xvi CONTENTS II Electrical Systems Modeling of switched DC-to-DC power Converters Introduction Lagrangian modeling Modeling of switched networks A variational argument General Lagrangian model: Passivity property Examples Hamiltonian modeling Constitutive elements LC circuits Examples Average modeis of PWM regulated Converters General issues about pulse-width-modulation Examples Some structural properties Conclusions Passivity-based control of DC-to DC power Converters Introduction PBC of stabilizing duty ratio The Boost Converter The Buck-boost Converter Simulation results Passivity based sliding mode stabilization Introduction Sliding mode control of the Boost Converter Passivity-based sliding Controller Adaptive stabilization Controller design Simulation results Experimental comparison of several nonlinear Controllers 213

6 CONTENTS xvii 5.1 Feedback control laws Experimental configuration Experimental results Conclusions 236 III Electromechanical Systems Nested-loop passivity-based control: An illustrative example Introduction Model and control problem Passivity-based control with total energy-shaping Nested-loop passivity-based control Control structure Passivity-based Controller design Output-feedback passivity-based control Comparison with feedback linearization and backstepping Feedback-linearization control Integrator backstepping control Comparison of the schemes Simulation results Conclusions and further research Generalized AC motor Introduction AC motors Review of previous work Outline of the rest of this chapter Lagrangian model and control problem The Euler-Lagrange equations for AC machines Control problem formulation Remarks to the model Examples 287

7 CONTENTS 3 A passivity-based approach for Controller design Passive Subsystems feedback decomposition Design procedure A globally stable torque tracking Controller Strict passifiability via damping injection Current tracking via energy-shaping From current tracking to torque tracking PBC for electric machines PBC of underactuated electrical machines revisited Realization of the PBC via BP transformability A geometric perspective Examples Conclusions Summary Open issues 308 Voltage-fed induction motors Induction motor model Dynamic equations Some control properties of the model Coordinate transformations Remarks to the model Concluding remarks Problem formulation A nested-loop PBC A Systems "inversion" perspective of the torque tracking PBC Observer-less PBC for induction motors Remarks to the Controller Integral action in stator currents Adaptation of stator parameters A fundamental obstacle for rotor resistance adaptation A d^-implementation 337

8 CONTENTS xix 3.8 Definitions of desired rotor flux norm Simulation results A PBC with total energy-shaping Factorization of workless forces Problem formulation Ideal case with füll State feedback Observer-based PBC for induction motors Remarks to the Controller A dg-implementation Simulation results Concluding remarks Field-oriented control and feedback linearization Rationale of field-oriented control State estimation or reference values Shortcomings of FOC Feedback linearization Experimental results Experimental setup Outline of experiments Observer-less control Observer-based control Comparison with FOC Concluding remarks Current-fed induction motors Model of the current-fed induction motor Field orientation and feedback linearization Direct field-oriented control Indirect field-oriented control Observer-based feedback-linearizing control Remarks to OBFL and FOC Passivity-based control of current-fed machines 392

9 3.1 PBC is downward compatible with FOC Stability of indirect FOC for known parameters Experimental comparison of PBC and feedback linearization Experimental setup Selection of flux reference in experiments Speed tracking Performance Robustness and disturbance attenuation Conclusion Robust stability of PBC Global boundedness Coordinate changes and uniqueness of equilibrium Local asymptotic stability Global exponential stability Off-line tuning of PBC Problem formulation Change of coordinates Local stability A Performance evaluation method based on passivity Illustrative examples Discrete-time implementation of PBC The exact discrete-time model of the induction motor Analysis of discrete-time PBC A new discrete-time control algorithm Discussion of discrete-time Controller Experimental results Conclusions and further research Feedback interconnected Systems: Robots with AC drives Introduction Cascaded Systems Robots with AC drives General problem formulation 446

10 CONTENTS xxi 3 Assumptions Realizability of the Controller Other assumptions Problem Solution Proof of Theorem Application to robots with AC drives Model Global tracking Controller Simulation results Concluding remarks Other applications and current research Other applications Current research Power electronics Power Systems Generation of storage functions for forced EL Systems Performance 471 A Dissipativity and passivity Circuit example and 2 e Spaces Passivity and finite-gain stability Feedback Systems Internal stability and passivity The Kaiman-Yakubovich-Popov lemma 481 B Derivation of the Euler-Lagrange equations Generalized coordinates and velocities Hamilton's principle Prom Hamilton's principle to the EL equations EL equations for non-conservative Systems List of generalized variables 489

11 xxii CONTENTS 6 Hamiltonian formulation 489 C Background material 493 D Proofs Proofs for the PI 2 D Controller Properties of the storage Tisiq, q,d) Lyapunov stability of the PI 2 D Proof of positive definiteness of f(q p ) defined in (3.43) The BP transformation Proof of Proposition A Lemma on the BP Transformation Proof of Eqs. (10.41) and (10.77) A theorem on positivity of a block matrix Proof ofeq. (10.77) Proof ofeq. (10.41) Derivation of Eqs. (10.55) and (10.56) Derivation of Eq. (10.55) Derivation of Eq. (10.56) Boundedness of all Signals for indirect FOC Proof of Proposition Bibliography 515 Index 539

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