Non-linear Control for Underactuated Mechanical Systems

This book deals with the application of modern control theory to some important underactuated mechanical systems, from the inverted pendulum to the helicopter model. It will help readers gain experience in the modelling of mechanical systems and familiarize with new control methods for non-linear systems.

Contenu

1 Introduction.- 1.1 Motivation.- 1.2 Outline of the book.- 1.2.1 Energy-based control approaches for several underactuated mechanical systems.- 1.2.2 The hovercraft model, the PVTOL aircraft and the helicopter.- 2 Theoretical preliminaries.- 2.1 Lyapunov stability.- 2.2 Lyapunov direct method.- 2.3 Passivity and dissipativity.- 2.4 Stabilization.- 2.5 Non-holonomic systems.- 2.6 Underactuated systems.- 2.7 Homoclinic orbit.- 3 The cart-pole system.- 3.1 Introduction.- 3.2 Model derivation.- 3.2.1 System model using Newton's second law.- 3.2.2 Euler-Lagrange's equations.- 3.3 Passivity of the inverted pendulum.- 3.4 Controllability of the linearized model.- 3.5 Stabilizing control law.- 3.5.1 The homoclinic orbit.- 3.5.2 Stabilization around the homoclinic orbit.- 3.5.3 Domain of attraction.- 3.3 Stability analysis.- 3.4 Simulation results.- 3.5 Experimental results.- 3.6 Conclusions.- 4 A convey-crane system.- 4.1 Introduction.- 4.2 Model.- 4.3 Passivity of the system.- 4.4 Damping oscillations control law.- 4.5 Asymptotic stability analysis.- 4.6 Simulation results.- 4.7 Concluding remarks.- 5 The pendubot system.- 5.1 Introduction.- 5.2 System dynamics.- 5.2.1 Equations of motion via Euler-Lagrange formulation.- 5.3 Passivity of the pendubot.- 5.4 Linearization of the system.- 5.5 Control law for the top position.- 5.5.1 The homoclinic orbit.- 5.5.2 Stabilization around the homoclinic orbit.- 5.6 Stability analysis.- 5.7 Simulation results.- 5.8 Experimental results.- 5.9 Conclusions.- 6 The Furuta pendulum.- 6.1 Introduction.- 6.2 Modeling of the system.- 6.2.1 Energy of the system.- 6.2.2 Euler-Lagrange dynamic equations.- 6.3.3 Passivity properties of the Furuta pendulum.- 6.3 Controllability of the linearized model.- 6.4 Stabilization algorithm.- 6.5 Stability analysis.- 6.6 Simulation results.- 6.7 Conclusions.- 7 The reaction wheel pendulum.- 7.1 Introduction.- 7.2 The reaction wheel pendulum.- 7.2.1 Equations of motion.- 7.2.2 Passivity properties of the system.- 7.2.3 Linearization of the system.- 7.2.4 Feedback linearization.- 7.3 First energy-based control design.- 7.4 Second energy-based controller.- 7.5 Simulation results.- 7.6 Conclusions.- 7.7 Generalization for Euler-Lagrange systems.- 8 The planar flexible-joint robot.- 8.1 Introduction.- 8.2 The two-link planar robot.- 8.2.1 Equations of motion.- 8.2.2 Linearization of the system.- 8.2.3 Passivity of the system.- 8.3 Control law for the two-link manipulator.- 8.3.1 Equivalent closed-loop interconnection.- 8.4 Stability analysis.- 8.5 Simulation results.- 8.6 The three-link planar robot.- 8.7 Control law for the three-link robot.- 8.8 Stability analysis.- 8.9 Simulation results.- 8.10 Conclusions.- 9 The PPR planar manipulator.- 9.1 Introduction.- 9.2 System dynamics.- 9.2.1 Equations of motion via Euler-Lagrange formulation.- 9.2.2 Passivity properties of the planar PPR manipulator.- 9.3 Energy-based stabilizing control law.- 9.3.1 Equivalent closed-loop interconnection.- 9.4 Convergence and stability analysis.- 9.5 Simulation results.- 9.6 Conclusions.- 10 The ball and beam acting on the ball.- 10.1 Introduction.- 10.2 Dynamical model.- 10.2.1 Mechanical properties.- 10.3 The control law.- 10.3.1 Stability analysis.- 10.4 Simulation results.- 10.5 Conclusions.- 11 The hovercraft model.- 11.1 Introduction.- 11.2 The hovercraft model.- 11.2.1 System model using Newton's second law.- 11.2.2 Euler-Lagrange's equations.- 11.2.3 Controllability of the linearized system.- 11.3 Stabilizing control law for the velocity.- 11.4 Stabilization of the position>.- 11.4.1 First approach.- 11.4.2 Second approach.- 11.4.3 Third approach.- 11.5 Simulation results.- 11.6 Conclusions.- 12 The PVTOL aircraft.- 12.1 Introduction.- 12.2 The PVTOL aircraft model.- 12.3 Input-output linearization of the system.- 12.4 Second stabilization approach.- 12.5 Third stabilization algorithm.- 12.6 Forwarding control law.- 12.6.1 First step: a Lyapunov function for the altitudeangle (y, ?)-subsystem.- 12.6.2 Boundedness of ?(t).- 12.6.3 Second step: forwarding design.- 12.6.4 Third step: last change of coordinates.- 12.7 Simulation results.- 12.8 Conclusions.- 13 Helicopter on a platform.- 13.1 Introduction.- 13.2 General considerations.- 13.2.1 Flight modes.- 13.2.2 Aerodynamic forces and torques.- 13.3.3 Inertia moments and products.- 13.4.4 The general model.- 13.3 The helicopter-platform model.- 13.4 Dissipativity properties of the 3-DOF model.- 13.5 Control design.- 13.5.1 Passivity-based control of the rotational part.- 13.5.2 Take-off.- 13.5.3 Altitude control.- 13.6 Simulation results.- 13.6.1 Simulation 1.- 13.6.2 Simulation 2.- 13.7 Conclusions.- 14 Lagrangian helicopter model.- 14.1 Introduction.- 14.2 Helicopter model.- 14.3 Energy-based control design.- 14.4 Analysis and simulations.- 14.5 Conclusions.- 15 Newtonian helicopter model.- 15.1 Introduction.- 15.2 Modeling a helicopter using Newton's laws.- 15.3 New dynamic model for control design.- 15.4 Lyapunov-based tracking control design.- 15.5 Analysis.- 15.6 Simulations.- 15.7 Conclusions.