Оглавление
Romeo Ortega. PID Passivity-Based Control of Nonlinear Systems with Applications
Table of Contents
List of Tables
List of Illustrations
Guide
Pages
PID Passivity‐Based Control of Nonlinear Systems with Applications
Author Biographies
Preface
Bibliography
Acknowledgments
Acronyms
Notation
1 Introduction
2 Motivation and Basic Construction of PID Passivity‐Based Control
2.1 ‐Stability and Output Regulation to Zero
Lemma 2.1
Proposition 2.1:
Remark 2.1:
Remark 2.2:
2.2 Well‐Posedness Conditions
Lemma 2.2
Remark 2.3:
2.3 PID‐PBC and the Dissipation Obstacle
2.3.1 Passive Systems and the Dissipation Obstacle
2.3.2 Steady‐State Operation and the Dissipation Obstacle
Proposition 2.2:
Remark 2.4:
Remark 2.5:
2.4 PI‐PBC with and Control by Interconnection
Bibliography
Notes
3 Use of Passivity for Analysis and Tuning of PIDs: Two Practical Examples
3.1 Tuning of the PI Gains for Control of Induction Motors
3.1.1 Problem Formulation
3.1.2 Change of Coordinates
‐Coordinates
State‐Space Representation
A Remark on Direct FOC
3.1.3 Tuning Rules and Performance Intervals
Passive Subsystems Decomposition
Global Stability
Proposition 3.1
Performance Intervals
3.1.4 Concluding Remarks
3.2 PI‐PBC of a Fuel Cell System
3.2.1 Control Problem Formulation
Proposition 3.2
3.2.2 Limitations of Current Controllers and the Role of Passivity
3.2.3 Model Linearization and Useful Properties. Proposition 3.3
Proposition 3.4
3.2.4 Main Result. Proposition 3.5
3.2.5 An Asymptotically Stable PI‐PBC
Proposition 3.6
3.2.6 Simulation Results
3.2.7 Concluding Remarks and Future Work
Bibliography
Notes
4 PID‐PBC for Nonzero Regulated Output Reference
4.1 PI‐PBC for Global Tracking
4.1.1 PI Global Tracking Problem
Assumption 4.1:
Remark 4.1:
4.1.2 Construction of a Shifted Passive Output
Lemma 4.1:
Remark 4.2:
Remark 4.3:
4.1.3 A PI Global Tracking Controller
Corollary 4.1:
Remark 4.4:
Remark 4.5:
4.2 Conditions for Shifted Passivity of General Nonlinear Systems
4.2.1 Shifted Passivity Definition
Remark 4.6:
Remark 4.7:
4.2.2 Main Results. Proposition 4.1:
Proposition 4.2:
Remark 4.8:
Remark 4.9:
Remark 4.10:
Remark 4.11:
4.3 Conditions for Shifted Passivity of Port‐Hamiltonian Systems
4.3.1 Problems Formulation
4.3.2 Shifted Passivity
Assumption 4.2
Assumption 4.3
Proposition 4.3:
4.3.3 Shifted Passifiability via Output‐Feedback
Proposition 4.4:
4.3.4 Stability of the Forced Equilibria
Proposition 4.5:
Remark 4.12:
4.3.5 Application to Quadratic pH Systems
Proposition 4.6:
Corollary 4.2
4.4 PI‐PBC of Power Converters
4.4.1 Model of the Power Converters
Remark 4.13:
4.4.2 Construction of a Shifted Passive Output. Proposition 4.7:
Remark 4.14:
4.4.3 PI Stabilization
Proposition 4.8:
Remark 4.15:
4.4.4 Application to a Quadratic Boost Converter
Remark 4.16:
Remark 4.17:
4.5 PI‐PBC of HVDC Power Systems
4.5.1 Background
4.5.2 Port‐Hamiltonian Model of the System
4.5.3 Main Result
Proposition 4.9:
Proposition 4.10:
Remark 4.18:
4.5.4 Relation of PI‐PBC with Akagi's PQ Method
4.6 PI‐PBC of Wind Energy Systems
4.6.1 Background
4.6.2 System Model
Wind Turbine
Maximum Power Extraction
Turbine Model
PMSG and Electrical Part
The Overall System
4.6.3 Control Problem Formulation
Lemma 4.2
Remark 4.19:
4.6.4 Proposed PI‐PBC
Assumption 4.4
Proposition 4.11:
Remark 4.20:
Remark 4.21:
4.7 Shifted Passivity of PI‐Controlled Permanent Magnet Synchronous Motors
4.7.1 Background
4.7.2 Motor Models
Standard Model
Incremental Model
4.7.3 Problem Formulation
Remark 4.22:
Remark 4.23:
4.7.4 Main Result
Lemma 4.3
Proposition 4.12:
4.7.5 Conclusions and Future Research
Bibliography
Notes
5 Parameterization of All Passive Outputs for Port‐Hamiltonian Systems
5.1 Parameterization of All Passive Outputs
Proposition 5.1
5.2 Some Particular Cases
Proposition 5.2
5.3 Two Additional Remarks
5.4 Examples
5.4.1 A Level Control System
5.4.2 A Microelectromechanical Optical Switch
Bibliography
Note
6 Lyapunov Stabilization of Port‐Hamiltonian Systems
6.1 Generation of Lyapunov Functions
6.1.1 Basic PDE
Proposition 6.1:
Remark 6.1:
6.1.2 Lyapunov Stability Analysis
Proposition 6.2:
Remark 6.2:
Remark 6.3:
6.2 Explicit Solution of the PDE
6.2.1 The Power Shaping Output
Proposition 6.3:
Remark 6.4:
6.2.2 A More General Solution
Assumption 6.1:
Proposition 6.4:
Remark 6.5:
6.2.3 On the Use of Multipliers
Proposition 6.5:
6.3 Derivative Action on Relative Degree Zero Outputs
6.3.1 Preservation of the Port‐Hamiltonian Structure of I‐PBC
Proposition 6.6:
6.3.2 Projection of the New Passive Output
Proposition 6.7:
6.3.3 Lyapunov Stabilization with the New PID‐PBC
Assumption 6.2:
Proposition 6.8:
Remark 6.6:
Remark 6.7:
6.4 Examples
6.4.1 A Microelectromechanical Optical Switch (Continued)
6.4.2 Boost Converter
6.4.3 Two‐Dimensional Controllable LTI Systems
Remark 6.8:
6.4.4 Control by Interconnection vs. PI‐PBC
6.4.5 The Use of the Derivative Action
Bibliography
Notes
7 Underactuated Mechanical Systems
7.1 Historical Review and Chapter Contents
7.1.1 Potential Energy Shaping of Fully Actuated Systems
7.1.2 Total Energy Shaping of Underactuated Systems
7.1.3 Two Formulations of PID‐PBC
7.2 Shaping the Energy with a PID
7.3 PID‐PBC of Port‐Hamiltonian Systems
7.3.1 Assumptions on the System
Assumption 7.1
7.3.2 A Suitable Change of Coordinates
Lemma 7.1:
Remark 7.1:
7.3.3 Generating New Passive Outputs
Lemma 7.2:
Remark 7.2:
Remark 7.3:
Remark 7.4:
7.3.4 Projection of the Total Storage Function
Assumption 7.2:
Lemma 7.3:
7.3.5 Main Stability Result
Assumption 7.3:
Proposition 7.1:
Remark 7.5:
Remark 7.6:
Remark 7.7:
7.4 PID‐PBC of Euler‐Lagrange Systems
7.4.1 Passive Outputs for Euler–Lagrange Systems
Lemma 7.4:
Remark 7.8:
7.4.2 Passive Outputs for Euler–Lagrange Systems in Spong's Normal Form
Lemma 7.5:
Remark 7.9:
7.5 Extensions
7.5.1 Tracking Constant Speed Trajectories
Proposition 7.2:
7.5.2 Removing the Cancellation of
Assumption 7.4:
Lemma 7.6:
Remark 7.10:
7.5.3 Enlarging the Class of Integral Actions
7.6 Examples
7.6.1 Tracking for Inverted Pendulum on a Cart
7.6.2 Cart‐Pendulum on an Inclined Plane
7.7 PID‐PBC of Constrained Euler–Lagrange Systems
7.7.1 System Model and Problem Formulation
Problem Formulation
Remark 7.11:
7.7.2 Reduced Purely Differential Model
Proposition 7.3:
Remark 7.12:
Remark 7.13:
7.7.3 Design of the PID‐PBC
Lemma 7.7:
Lemma 7.8:
7.7.4 Main Stability Result
Proposition 7.4:
Remark 7.14:
Remark 7.15:
Remark 7.16:
7.7.5 Simulation Results
7.7.6 Experimental Results
Bibliography
Notes
8 Disturbance Rejection in Port‐Hamiltonian Systems
8.1 Some Remarks on Notation and Assignable Equilibria
8.1.1 Notational Simplifications
8.1.2 Assignable Equilibria for Constant
8.2 Integral Action on the Passive Output
Proposition 8.1:
8.3 Solution Using Coordinate Changes
8.3.1 A Feedback Equivalence Problem
Definition 8.1
Remark 8.1:
Remark 8.2:
8.3.2 Local Solutions of the Feedback Equivalent Problem
Proposition 8.2:
Remark 8.3:
8.3.3 Stability of the Closed‐Loop
Proposition 8.3:
Remark 8.4:
8.4 Solution Using Nonseparable Energy Functions
8.4.1 Matched and Unmatched Disturbances
Matched Disturbances
Remark 8.5:
Remark 8.6:
Proposition 8.4:
Unmatched Disturbances
Proposition 8.5:
Matched and Unmatched Disturbances
Proposition 8.6:
8.4.2 Robust Matched Disturbance Rejection
Decomposition of the Damping Matrix
Remark 8.7:
Control Objective
New Closed‐Loop pH Structure
Proposition 8.7:
Stability
Proposition 8.8:
8.5 Robust Integral Action for Fully Actuated Mechanical Systems
Lemma 8.1:
Lemma 8.2:
Remark 8.8:
Lemma 8.3:
Remark 8.9:
Remark 8.10:
8.6 Robust Integral Action for Underactuated Mechanical Systems
8.6.1 Standard Interconnection and Damping Assignment PBC
Formulation of the Robust IDA‐PBC Problem
Remark 8.11:
8.6.2 Main Result
Proposition 8.9:
8.7 A New Robust Integral Action for Underactuated Mechanical Systems
8.7.1 System Model
8.7.2 Coordinate Transformation
8.7.3 Verification of Requisites
8.7.4 Robust Integral Action Controller
8.8 Examples
8.8.1 Mechanical Systems with Constant Inertia Matrix
8.8.2 Prismatic Robot
8.8.3 The Acrobot System
8.8.4 Disk on Disk System
8.8.5 Damped Vertical Take‐off and Landing Aircraft
Bibliography
Notes
Appendix A Passivity and Stability Theory for State‐Space Systems
A.1 Characterization of Passive Systems
Definition A.1
Proposition A.1
Theorem A.1
A.2 Passivity Theorem
Definition A.2
Proposition A.2
Proposition A.3
A.3 Lyapunov Stability of Passive Systems
Definition A.3
Definition A.4
Theorem A.2
Bibliography
Note
Appendix B Two Stability Results and Assignable Equilibria. B.1 Two Stability Results
Theorem B.1
Theorem B.2
B.2 Assignable Equilibria. Definition B.1
Proposition B.1
Bibliography
Appendix C Some Differential Geometric Results
C.1 Invariant Manifolds
Definition C.1
Lemma C.1
C.2 Gradient Vector Fields. Definition C.2
Lemma C.2
C.3 A Technical Lemma. Definition C.3
Lemma C.3
Bibliography
Appendix D Port–Hamiltonian Systems
D.1 Definition of Port‐Hamiltonian Systems and Passivity Property. Definition D.1
Proposition D.1
D.2 Physical Examples. D.2.1 Mechanical Systems
D.2.2 Electromechanical Systems
D.2.3 Power Converters
D.3 Euler–Lagrange Models
D.4 Port‐Hamiltonian Representation of GAS Systems
Proposition D.2
Bibliography
Index
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