Modeling and Simulation for Automatic Control

Transcription

1 Modeling and Simulation for Automatic Control Olav Egeland and Jan Tommy Gravdahl Norwegian University of Science and Technology Trondheim, Norway MARINE CYBERNETICS Г~Т.! " " Trondheim, Norway

2 Contents I Modeling 1 1 Model representation Introduction State space methods State space models Second order models of mechanical systems Linearization of state space models Linearization of second order systems Stability with zero input Stability of linear systems Stability analysis using a linearized model Transfer function models Introduction The transfer function of a state-space model Rational transfer functions Impulse response and step response Loop transfer function Example: Actuator with dynamic compensation Stability of transfer functions Stability of closed loop systems Partial differential equations Network description Introduction Background Multiport Example: DC motor with flexible load Example: Voltage controlled DC motor Example: Diesel engine with turbocharger Assigning computational inputs and outputs Bond graphs Linear network theory Driving point impedance Linear two-ports Impedance of two-port with termination Example: Passive mechanical two-port Mechanical analog of PD controller Example: Transmission line model Introduction 33 ш

3 iv CONTENTS Introductory example Effort and flow model Transfer functions Transfer function for terminated transmission line Wave variables Lossless transmission line Line termination 38 2 Model analysis tools Frequency response methods The frequency response of a system Second order oscillatory system Performance of a closed loop system Stability margins Elimination of fast dynamics Example: The electrical time constant in a DC motor Nonlinear system Energy-based methods Introduction The energy function Second-order systems Example: Mass-spring-damper Lyapunov methods Contraction Energy flow in a turbocharged diesel engine Passivity Introduction Definition Examples Energy considerations Positive real transfer functions Positive real rational transfer functions Positive realness of irrational transfer functions Passivity and positive real transfer functions No poles on the imaginary axis Single poles at the imaginary axis Bounded real transfer functions Passivity of PID controllers Closed loop stability of positive real systems Storage function formulation Interconnections of passive systems Storage function for PID controller Passive plant with PID controller Example: Control of mass-spring-damper system Example: Active vibration damping Passive electrical one-port Electrical analog of PID controller Passive electrical two-port Termination of electrical two-port Passive electrical n-ports 70

4 CONTENTS v Example: Telemanipulation Passivity and gain Uncertainty in modeling General state space models Exact kinematic models Balance equations Passivity 76 II Motors and actuators 77 3 Electromechanical systems Introduction Electrical motors Introduction Basic equations Gear model Motor and gear Transformation of rotation to translation Torque characteristics The four quadrants of the motor The DC motor with constant field Introduction Model Energy function Laplace transformed model DC motor control Introduction Current controlled DC motor Velocity controlled DC motor Position controlled DC motor Motor and load with elastic transmission Introduction Equations of motion Transfer functions Zeros of the transfer function Energy analysis Motor with several resonances in the load Two motors driving an elastic load Energy analysis of two motors and load Motor and load with deadzone in the gear Introduction Elastic gear with deadzone Rigid gear with deadzone Two motors with deadzone and load Electromechanical energy conversion Introduction Inductive circuit elements Capacitive circuit elements Magnetic energy of a linear inductive element 103

5 3.7.5 Stored energy of a linear capacitive element Energy and coenergy Electromechanical two-port with inductive element Electromechanical two-port with linear flux linkage Magnetic levitation Voice coil Electromagnetic three-port Electromechanical capacitive element Ill Electromechanical two-port with linear charge Example: Capacitive microphone Piezoelectric actuator Actuator configuration DC motor with externally controlled field Model Network description DC motor with field weakening Dynamic model of the general AC motor Introduction Notation Dynamic model Induction motors Basic dynamic model Induction motor model in stator frame Dynamic model in the flux frame Lagrangian description of electromechanical systems Generalized coordinates Energy and coenergy Analogy of electrical and mechanical systems The Lagrangian Electromechanical systems Lagrange formulation of general AC motor Lagrange formulation of induction motor Lagrange formulation of DC motor 138 Hydraulic motors Introduction Valves Introduction Flow through a restriction Regularization of turbulent orifice flow Four-way valve Matched and symmetrical four-way valve Symmetric motor and valve with critical spool Symmetric motor and valve with open spool Flow control using pressure compensated valves Balance valve Motor models Mass balance Rotational motors Elastic modes in the load 154

6 CONTENTS vii Hydraulic eylinder Models for transfer function analysis Matched and symmetric valve and symmetric motor Valve controlled motor: Transfer function Hydraulic motor with P controller Symmetric cylinder with matched and symmetric valve Pump controlled hydraulic drive with P controller Transfer functions for elastic modes Mechanical analog Hydraulic transmission lines Introduction PDE Model Laplace transformed model Lossless model Linear friction Nonlinear friction Wave variables Example: Lossless pipe Linear network models of transmission lines Rational approximations of transfer function models Rational series expansion of impedance model Rational series expansion of admittance model Galerkin derivation of impedance model Galerkin derivation of the admittance model Galerkin derivation of the hybrid model Rational simulation models Lumped parameter model of hydraulic line Introduction Helmholtz resonator model Model formulation Admittance model Impedance model Hybrid model Natural frequencies Object oriented simulation models Introduction Pump controlled hydraulic motor Cylinder with balance valve Friction Introduction Background Static friction models Models for the individual phenomena Combination of individual models Problems with the static models Problems with signum terms at zero velocity Karnopp's model of Coulomb friction More on Karnopp's friction model Passivity of static models 199

7 Vlll 5.3 Dynamic friction models Introduction The Dahl model Passivity of the Dahl model The Bristle and LuGre model Passivity of the LuGre model The Elasto-Plastic model Passivity of the Elasto-Plastic model 206 III Dynamics Rigid body kinematics Introduction Vectors Vector description The scalar product The vector cross product Dyadics Introduction Introductory example: The inertia dyadic Matrix representation of dyadics The rotation matrix Coordinate transformations for vectors Properties of the rotation matrix Composite rotations Simple rotations Coordinate transformations for dyadics Homogeneous transformation matrices Euler angles Introduction Roll-pitch-yaw Classical Euler angles Angle-axis description of rotation Introduction Angle-axis parameters Derivation of rotation dyadic The rotation dyadic Rotation matrix Euler parameters Definition Quaternions Unit quaternions The quaternion product for unit quaternions Rotation by the quaternion product Euler parameters from the rotation matrix The Euler rotation vector Euler-Rodrigues parameters Angular velocity Introduction 239

8 CONTENTS ix Definition Simple rotations Composite rotations Differentiation of coordinate vectors Differentiation of vectors Kinematic differential equations Introduction Attitude deviation Homogeneous transformation matrices Euler angles Euler parameters Normalization for numerical integration Euler rotation Euler-Rodrigues parameters Passivity of kinematic differential equations Angle-axis representation The Serret-Frenet frame Kinematics Control deviation Navigational kinematics Introduction Coordinate frames Acceleration Kinematics of a rigid body Configuration Velocity Acceleration The center of mass System of particles Rigid body Newton-Euler equations of motion Introduction Forces and torques Resultant force Torque Equivalent force and torque Forces and torques on a rigid body Example: Robotic link Newton-Euler equations for rigid bodies Equations of motion for a system of particles Equations of motion for a rigid body Equations of motion about a point The inertia dyadic The inertia matrix Expressions for the inertia matrix The parallel axes theorem The equations of motion for a rigid body Satellite attitude dynamics Example: Ball and beam dynamics 278

9 7.5 Example: Inverted pendulum Equations of motion Double inverted pendulum Example: The Furuta pendulum Principle of virtual work Introduction Generalized coordinates Virtual displacements d'alembert's principle Principle of virtual work for a rigid body Virtual displacements for a rigid body Force and torque of constraint Multi-body dynamics and virtual work Introduction Equations of motion Equations of motion about a point Ball and beam Single and double inverted pendulum Furuta pendulum Planar two-link manipulator: Derivation Planar two-link manipulator: Derivation Kane's computational scheme for two-link manipulator Manipulator dynamics in coordinate form Spacecraft and manipulator Recursive Newton-Euler Inverse dynamics Simulation 311 Analytical mechanics Introduction Lagrangian dynamics Introduction Lagrange's equation of motion Generalized coordinates and generalized forces Pendulum Mass-spring system Ball and beam Furuta pendulum Manipulator Passivity of the manipulator dynamics Example: Planar two-link manipulator Example: Planar two-link manipulator Limitations of Lagrange's equation of motion Calculus of variations Introduction Variations versus differentials The variation of a function The Euler-Lagrange equation for a general integral The variation of the rotation matrix The variation of the homogeneous transformation matrix 330

10 CONTENTS xi 8.4 The adjoint formulation Introduction Rotations Rigid motion The Euler-Poincare equation A central equation Rotating rigid body Free-floating rigid body Mechanism with n degrees of freedom Hamilton's principle Introduction The extended Hamilton principle Derivation of Lagrange's equation of motion Hamilton's principle Rotations with the Euler-Poincare equation Rigid motion with the Euler-Poincare equation Lagrangian dynamics for PDE's Flexible beam dynamics Euler-Bernoulli beam Lateral vibrations in a string Hamilton's equations of motion Introduction Hamilton's equation of motion The energy function Change of generalized coordinates Control aspects Passivity of Hamilton's equation of motion Example: Manipulator dynamics Example: The restricted three-body problem Example: Attitude dynamics for a satellite Example: Gravity gradient stabilization The Hamilton-Jacobi equation Mechanical vibrations Introduction Lumped elastic two-ports Hybrid two-port Displacement two-port Three masses in the hybrid formulation Three masses in the displacement formulation Four masses Vibrating string Linearized model Orthogonal shape functions Galerkin's method for orthogonal shape functions Finite element shape functions String element Assembling string elements Nonlinear string dynamics Kirchhoff's nonlinear string model 371

11 Xll CONTENTS Marine cables Euler Bernoulli beam Model Boundary conditions Energy Orthogonal shape functions Clamped-free Euler Bernoulli beam Beam fixed to an inertia and a mass Orthogonality of the eigenfunctions Galerkin's method for orthogonal mode shapes Finite element model of Euler Bernoulli beam Introduction Beam element Assembling a structure Finite element model and Galerkin's method Motor and Euler Bernoulli beam Equations of motion Assumed mode shapes Finite elements Irrational transfer functions for beam dynamics Introduction Clamped-free beam Motor and beam 395 IV Balance equations Kinematics of Flow Introduction Kinematics The material derivative The nabla operator Divergence Curl Material coordinates The dilation Orthogonal curvilinear coordinates General results Cylindrical coordinates Reynolds' transport theorem Introduction Basic transport theorem The transport theorem for a material volume The transport theorem and balance laws Mass, momentum and energy balances The mass balance Differential form Integral form Control volume with compressible fluid 418

12 CONTENTS xiii Mass flow through a pipe Continuity equation and Reynolds' transport theorem Multi-component systems The momentum balance Euler's equation of motion The momentum equation for a control volume Example: Waterjet Example: Sand dispenser and conveyor Irrotational Bernoulli equation Bernoulli's equation along a streamline Example: Transmission line Liquid mass flow through a restriction Example: Water turbine Example: Waterhammer Angular momentum balance General expression Centrifugal pump with radial blades Euler's turbomachinery equation Pump instability The energy balance Material volume Fixed volume General control volume The heat equation Transfer function for the heat equation Viscous flow Introduction Tensor notation The velocity gradient tensor Example: The velocity gradient for a rigid body The stress tensor Cauchy's equation of motion Newtonian fluids The Navier-Stokes equation The Reynolds number The equation of kinetic energy The energy balance for a viscous fluid Fixed volume General control volume Gas dynamics Introduction Energy, enthalpy and entropy Energy Enthalpy Specific heats Entropy The entropy equation Internal energy equation in terms of temperature Energy balance in terms of pressure 471

13 xiv CONTENTS Piston motion Isentropic conditions Isentropic processes Stagnation state Energy balance for isentropic processes The speed of sound Helmholtz resonator Acoustic resonances in pipes Dynamic model Pipe closed at both ends Pipe closed at one end Pressure measurement in diesel engine cylinder Gas flow Gas flow through a restriction Example: Discharge of gas from tank The Euler equation around sonic speed Compressor dynamics Introduction Compressors Surge and rotating stall Centrifugal Compressors Introduction Shaft dynamics Compressor system Mass balance Momentum equation Compressor characteristic Derivation The compressor characteristic at zero mass flow Compressor surge The Greitzer surge model Linearization Passivity of the Greitzer surge model Curvefitting of compressor characteristic Compression systems with recycle 504 V Simulation Simulation Introduction The use of simulation in automatic control The Moore Greitzer model The restricted three-body problem Mass balance of chemical reactor Preliminaries Notation Computation error The order of a one-step method 517

14 CONTENTS xv Linearization The linear test function Euler methods Euler's method The improved Euler method The modified Euler method Explicit Runge-Kutta methods Introduction Numerical scheme Order conditions Some explicit Runge-Kutta methods Case study: Pneumatic spring Stability function FSAL methods Implicit Runge-Kutta methods Stiff systems Implicit Runge-Kutta methods Implicit Euler method Trapezoidal rule Implicit midpoint rule The theta method Stability function Some implicit Runge-Kutta methods Case study: Pneumatic spring revisited Stability of Runge-Kutta methods Aliasing A-stability, L-stability Stiffly accurate methods Pade approximations Stability for Pade approximations Example: Mechanical vibrations Frequency response AN-stability B-stability Algebraic stability Properties of Runge-Kutta methods Automatic adjustment of step size Estimation of the local error for Runge-Kutta methods Adjustment algorithm Implementation aspects Solution of implicit equations Dense outputs Event detection Systems with inertia matrix Invariants Introduction Linear invariants Quadratic functions Quadratic invariants 569

15 xvi CONTENTS Symplectic Runge-Kutta methods Rosenbrock methods Multistep methods Explicit Adams methods Implicit Adams methods Predictor-Corrector implementation Backwards differentiation methods Linear stability analysis Stability of Adams methods Stability of BDF methods Frequency response Adams methods BDF methods Differential-algebraic equations Implicit Runge-Kutta methods for index 1 problems Multistep methods for index 1 problems Computational fluid dynamics Introduction Governing equations Classification Hyperbolic equations Parabolic equations Elliptic equations Diffusion Introduction Finite volume method for stationary diffusion Solution of equations Worked example on stationary diffusion Stability issues Finite volume method for diffusion dynamics Finite volumes for Convection-Diffusion Introduction Finite volume method for ID diffusion and convection dynamics Pressure-velocity coupling Introduction The staggered grid The momentum equations The transient SIMPLE algorithm Von Neuman stability method 620