Control of higher-dimensional PDEs : flatness and backstepping designs

✍ Scribed by Thomas Meurer, Dipl.-Ing

• Publisher: Springer
• Year: 2013
• Tongue: English
• Leaves: 369
• Series: Communications and control engineering
• Category: Library

✦ Synopsis

This monograph presents new model-based design methods for trajectory planning, feedback stabilization, state estimation, and tracking control of distributed-parameter systems governed by partial differential equations (PDEs). Flatness and backstepping techniques and their generalization to PDEs with higher-dimensional spatial domain lie at the core of this treatise. This includes the development of systematic late lumping design procedures and the deduction of semi-numerical approaches using suitable approximation methods. Theoretical developments are combined with both simulation examples and experimental results to bridge the gap between mathematical theory and control engineering practice in the rapidly evolving PDE control area.The text is divided into five parts featuring:- a literature survey of paradigms and control design methods for PDE systems- the first principle mathematical modeling of applications arising in heat and mass transfer, interconnected multi-agent systems, and piezo-actuated smart elastic structures- the generalization of flatness-based trajectory planning and feedforward control to parabolic and biharmonic PDE systems defined on general higher-dimensional domains- an extension of the backstepping approach to the feedback control and observer design for parabolic PDEs with parallelepiped domain and spatially and time varying parameters- the development of design techniques to realize exponentially stabilizing tracking control- the evaluation in simulations and experimentsControl of Higher-Dimensional PDEs -- Flatness and Backstepping Designs is an advanced research monograph for graduate students in applied mathematics, control theory, and related fields. The book may serve as a reference to recent developments for researchers and control engineers interested in the analysis and control of systems governed by PDEs. Read more... Part 1. Introduction and Survey -- Introduction -- Part 2. Modeling and Application Examples -- Model Equations for Non-Convective and Convective Heat Transfer -- Model Equations for Multi-Agent Networks -- Model Equations for Flexible Structures with Piezoelectric Actuation -- Mathematical Problem Formulation -- Part 3. Trajectory Planning and Feedforward Control -- Spectral Approach for Time-Invariant Systems with General Spatial Domain -- Formal Integration Approach for Time Varying Systems with Parallelepiped Spatial Domain -- Part 4. Feedback Stabilization, Observer Design, and Tracking Control -- Backstepping for Linear Diffusion-Convection-Reaction Systems with Varying Parameters on 1-Dimensional Domains -- Backstepping for Linear Diffusion-Convection-Reaction Systems with Varying Parameters on Parallelepiped Domains

✦ Table of Contents

Cover......Page 1
Front matter......Page 2
Introduction......Page 19
Feedback Stabilization of PDE Systems......Page 20
Trajectory Planning and Tracking Control for PDE Systems......Page 22
Objectives of this Book......Page 25
Outline and Structure......Page 27
References......Page 29
Non–Convective Heat Transfer......Page 37
Convective Heat Transfer in Single Phase Flow......Page 40
Thermal Battery Management......Page 45
Building Climate Control......Page 47
References......Page 48
Model Equations for Multi–Agent Networks......Page 50
Agent Models — Discrete and Continuous Formulations......Page 51
Communication Topology by Discretization......Page 57
Consensus and Stabilization......Page 59
Leader–Enabled Formation Deployment......Page 60
References......Page 61
Continuum Mechanical Preliminaries......Page 63
Flexible Plate with Distributed MFC Actuators......Page 70
Preparations......Page 72
Potential Energy, Kinetic Energy, and Virtual Work of Non–Conservative Forces......Page 74
Strong Form of the Equations of Motion......Page 78
Weak or Variational form of the Equations of Motion......Page 81
Motion Planning and Transient Elastic Shaping of Structures......Page 84
Vibration Suppression and Elastic Motion Tracking......Page 85
References......Page 86
General System Setting......Page 88
Trajectory Planning and Tracking Control......Page 90
References......Page 91
Spectral Approach for Time–Invariant Systems
with General Spatial Domain......Page 92
Abstract Formulation and Spectral Analysis......Page 94
Admissible Control and Observation Operators......Page 95
Abstract Boundary Control Systems......Page 96
Bases of Hilbert Spaces, Riesz Bases, and Spectral Operators......Page 100
Formal Parametrization of Riesz Spectral Systems......Page 107
Finite–Dimensional In–Domain and Boundary Control......Page 108
Infinite–Dimensional In–Domain and Boundary Control......Page 112
Convergence in Gevrey Classes......Page 118
Operational Convergence......Page 119
Convergence of the Parametrized Fourier Series......Page 124
Finite Time Transitions between Stationary States......Page 127
Finite Time Transitions between Non–stationary States......Page 132
Heat and Wave Equation on 1–Dimensional Domain......Page 136
Boundary Controlled Linear Diffusion–Reaction Equation on r–Dimensional Riemannian Manifold......Page 142
Boundary Controlled Linear Diffusion–Convection–Reaction Equation on Parallelepiped Domain......Page 152
Spectral Properties and Spectral System Representation......Page 175
Formal State and Input Parametrization......Page 179
Convergence Analysis for Special Plate Configurations......Page 180
Semi–Numeric Finite–Dimensional Realization and Numerical Convergence Indicator......Page 182
Experimental Results for Feedforward and Closed–Loop Tracking Control......Page 184
References......Page 194
Formal Integration Approach for Time Varying
Systems with Parallelepiped Spatial Domain......Page 198
Trajectory Planning Problem......Page 199
Transformation into Standard Form......Page 200
Boundary Control Problem......Page 202
Formal State and Input Parametrization......Page 203
Construction of a Basic Output......Page 204
Uniform Series Convergence in Gevrey Classes......Page 205
Stationary Profiles......Page 213
Admissible Trajectories for the Basic Output......Page 214
Construction of Admissible Trajectories for the Basic Output......Page 215
Extension to Multiple Input Configurations......Page 219
Application Examples and Simulation Results......Page 222
Isotropic Diffusion and Reaction......Page 224
Orthotropic Diffusion and Reaction......Page 225
References......Page 228
Backstepping for Linear
Diffusion–Convection–Reaction Systems with Varying Parameters on 1–Dimensional Domains......Page 229
Stabilization and Tracking Control Problem......Page 230
Selection of the Target System......Page 233
Determination of the Kernel–PDE......Page 236
Solution of the Kernel–PDE......Page 238
Backstepping–Based State–Feedback Controller......Page 246
Inverse Backstepping–Transformation and Exponential Stability of the Closed–Loop System......Page 247
State–Observer with Exponentially Stable Error Dynamics......Page 250
Selection of the Target System......Page 251
Determination of the Kernel–PDE and the Observer Gains......Page 252
Solution of the Kernel–PDE......Page 254
Inverse Backstepping–Transformation and Exponential Stability of the Observer Error Dynamics......Page 257
Separation Principle and Exponential Stability of the Closed–Loop System......Page 259
Flatness–Based Trajectory Planning......Page 261
Trajectory Assignment in Gevrey Classes Using the Backstepping Transformation......Page 264
Combining Backstepping and Differential Flatness for Exponentially Stabilizing Tracking Control......Page 266
Application Examples and Simulation Results......Page 267
Stabilization and Tracking......Page 269
References......Page 272
Backstepping for Linear
Diffusion–Convection–Reaction Systems with Varying Parameters on Parallelepiped Domains......Page 274
Stabilization and Tracking Control Problem......Page 275
Transformation into Standard Form......Page 277
Boundary Control Problem......Page 278
Determination of the Kernel–PDE and Selection of the Target System......Page 279
Solution of the Kernel–PDE......Page 284
Backstepping–Based State–Feedback Controller......Page 285
Inverse Backstepping–Transformation and Exponential Stability of the Closed–Loop System......Page 286
Approximate Finite–Dimensional Realization of Backstepping–Based State–Feedback Control......Page 288
State–Observer with Exponentially Stable ErrorDynamics — The Single Output Case......Page 289
Selection of the Target System......Page 291
Determination of the Kernel–PDE and the Observer Gains......Page 292
Solution of the Kernel–PDE......Page 294
Inverse Backstepping–Transformation and Exponential Stability of the Observer Error Dynamics......Page 295
Separation Principle and Exponential Stability of the Closed–Loop System......Page 296
Approximate Realization of the State–Observer by means of Spatial Output Interpolation......Page 299
Tracking Control — The Single Input and Output Case......Page 301
Exponentially Stabilizing State–Feedback Control — The Multiple Input Case......Page 303
Multi–linear Backstepping–Transformation......Page 304
Determination and Solution of the Kernel–PDEs......Page 305
Backstepping–Based State–Feedback Controller......Page 308
Inverse Multi–linear Backstepping–Transformation and Exponential Stability of the Closed–Loop System......Page 310
Approximate Finite–Dimensional Realization of Backstepping–Based State–Feedback Control......Page 311
State–Observer with Exponentially Stable ErrorDynamics — The Multiple Output Case......Page 312
Multi–linear Backstepping–Transformation......Page 314
Determination of the Kernel–PDEs and the Observer Gains......Page 315
Inverse Backstepping–Transformation and Exponential Stability of the Observer Error Dynamics......Page 324
Separation Principle and Exponential Stability of the Closed–Loop System......Page 325
Approximate Realization of the State–Observer by means of Spatial Output Interpolation......Page 332
Tracking Control — The Multiple Input and Output Case......Page 333
Exponential Feedback Stabilization and State Estimation for an Unstable Time Varying Diffusion–Reaction System......Page 334
Synchronization of Large Scale Multi–Agent Network......Page 339
References......Page 351
Back matter......Page 352