This book is about nonlinear observability. It provides a modern theory of observability based on a new paradigm borrowed from theoretical physics and the mathematical foundation of that paradigm. In the case of observability, this framework takes into account the group of invariance that is inherent to the concept of observability, allowing the reader to reach an intuitive derivation of significant results in the literature of control theory.
The book provides a complete theory of observability and, consequently, the analytical solution of some open problems in control theory. Notably, it presents the first general analytic solution of the nonlinear unknown input observability (nonlinear UIO), a very complex open problem studied in the 1960s. Based on this solution, the book provides examples with important applications for neuroscience, including a deep study of the integration of multiple sensory cues from the visual and vestibular systems for self-motion perception.
Observability: A New Theory Based on the Group of Invariance is the only book focused solely on observability. It provides readers with many applications, mostly in robotics and autonomous navigation, as well as complex examples in the framework of vision-aided inertial navigation for aerial vehicles. For these applications, it also includes all the derivations needed to separate the observable part of the system from the unobservable, an analysis with practical importance for obtaining the basic equations for implementing any estimation scheme or for achieving a closed-form solution to the problem.
This book is intended for researchers in robotics and automation, both in academia and in industry. Researchers in other engineering disciplines, such as information theory and mechanics, will also find the book useful.
Author(s): Agostino Martinelli
Series: Advances in Design and Control
Edition: 1
Publisher: Society for Industrial and Applied Mathematics
Year: 2020
Language: English
Pages: 259
City: Philadelphia
Front Matter
Title Page
Copyright Page
Dedication
Contents
List Of Algorithms
Preface
Chapter 1 Introduction
1.1 Elementary example of an input-output system
1.2 Observability
1.3 Main features of a complete theory of observability
1.3.1 Definition of observability
1.3.2 The role of the inputs
1.3.3 The link between observability and continuous symmetries
1.3.4 The concept of observable function
1.3.5 The group of invariance of the theory
1.3.6 The assumptions of the theory
1.4 The twofold role of time
1.5 Unknown input observability
1.5.1 An example of an unknown input system
1.5.2 The role of the inputs in the case of unknown inputs
1.5.3 Extension of the observability rank condition
1.6 Nonlinear observability for time-variant systems
Part I Reminders on Tensors and Lie Groups
Chapter 2 Manifolds, Tensors, and Lie Groups
2.1 Manifolds
2.1.1 Topological spaces
2.1.2 Map among topological spaces
2.1.3 Manifolds and differential manifolds
2.2 Tensors
2.2.1 Formal definition of vectors and covectors
2.2.2 Einstein notation
2.2.3 Tangent and cotangent space
2.2.4 Tensors of rank larger than 1
2.2.5 Lie derivative
2.3 Distributions and codistributions
2.3.1 Distribution
2.3.2 Codistribution
2.3.3 Frobenius theorem
2.4 Lie groups
2.4.1 Examples of Lie groups
2.4.2 Group representation
2.4.3 Lie algebra
2.4.4 Correspondence between a Lie group and a Lie algebra
2.4.5 Examples of Lie groups and Lie algebras
2.5 Tensors associated with a group of transformations
Part II Nonlinear Observability
Chapter 3 Group of Invariance of Observability
3.1 The chronostate and the chronospace
3.2 Group of invariance in the absence of unknown inputs
3.2.1 Output transformations’ group
3.2.2 Input transformations’ group
3.2.3 The Abelian case
3.2.4 Simplified systems in the absence of unknown inputs
3.3 Group of invariance in the presence of unknown inputs
3.3.1 The SUIO for driftless systems
3.3.2 The SUIO in the presence of a drift
3.3.3 The Abelian case
3.3.4 Simplified systems in the presence of unknown inputs
Chapter 4 Theory of Nonlinear Observability in the Absence of Unknown Inputs
4.1 Theory based on a constructive approach
4.2 Observability rank condition for the simplified system
4.3 Observability rank condition in the general case
4.3.1 Analytic derivation
4.3.2 Extension to the case of multiple outputs
4.3.3 Examples of applications of the observability rank condition
4.4 Observable function
4.5 Theory based on the standard approach
4.5.1 Indistinguishability and observability
4.5.2 Equivalence of the approaches
4.6 Unobservability and continuous symmetries
4.6.1 Local observable subsystem
4.6.2 Continuous symmetries
4.7 Extension of the observability rank condition to time-variant systems
Chapter 5 Applications: Observability Analysis for Systems in the Absence of Unknown Inputs
5.1 The unicycle
5.2 Vehicles moving on parallel lines
5.3 Simultaneous odometry and camera calibration
5.4 Simultaneous odometry and camera calibration in the case of circular trajectories
5.5 Visual inertial sensor fusion with calibrated sensors
5.5.1 The system
5.5.2 Observability analysis
5.6 Visual inertial sensor fusion with uncalibrated sensors
5.6.1 The system
5.6.2 Observability analysis
5.7 Visual inertial sensor fusion in the cooperative case
5.7.1 The system
5.7.2 Observability analysis
5.8 Visual inertial sensor fusion with virtual point features
5.8.1 The system
5.8.2 Observability analysis
Part III Nonlinear Unknown Input Observability
Chapter 6 General Concepts on Nonlinear Unknown Input Observability
6.1 A constructive definition of observability
6.1.1 Observability for a specific unknown input
6.1.2 Observability independent of the unknown input
6.2 Basic properties to obtain the observable codistribution
6.2.1 Augmented state and its properties
6.2.2 An achievable upper bound functions’ set
6.3 A partial tool to investigate the observability properties in the presence of unknown inputs
6.3.1 Examples
6.3.2 Main features and limits of applicability
6.4 Theory based on the standard approach
6.4.1 Indistinguishability and observability
6.4.2 Equivalence of the approaches
Chapter 7 Unknown Input Observability for Driftless Systems with a Single Unknown Input
7.1 Extension of the observability rank condition for the simplified systems
7.1.1 Observable codistribution
7.1.2 The analytic procedure
7.2 Applications
7.2.1 Unicycle with one input unknown
7.2.2 Unicycle in the presence of an external disturbance
7.2.3 Vehicle moving in 3D in the presence of a disturbance
7.3 Analytic derivations
7.3.1 Separation
7.3.2 Convergence
7.3.3 Extension to the case of multiple known inputs
Chapter 8 Unknown Input Observability for the General Case
8.1 Extension of the observability rank condition for the general case
8.1.1 Observable codistribution
8.1.2 The analytic procedure
8.2 Applications
8.2.1 Visual-inertial sensor fusion: The planar case with calibrated sensors
8.2.2 Visual-inertial sensor fusion: The planar case with uncalibrated sensors
8.2.3 Visual-inertial sensor fusion: The 3D case with calibrated sensors
8.2.4 Visual-inertial sensor fusion: The 3D case with uncalibrated sensors
8.3 Analytic derivations
8.3.1 Observable codistribution
8.3.2 Convergence
8.3.3 Extension to the case of multiple known inputs
8.4 Extension to time-variant systems
Appendix A Proof of Theorem 4.5
Appendix B Reminders on Quaternions and Rotations
Appendix C Canonic Form with Respect to the Unknown Inputs
C.1 System canonization in the case of a single unknown input
C.1.1 Driftless case
C.1.2 The case with a drift
C.2 System canonization in the general case
C.2.1 The recursive procedure to perform the system canonization in the general case
C.2.2 Convergence of the recursive procedure
C.2.3 Nonsingularity and canonic form
C.2.4 Remarks to reduce the computation
Bibliography
Index
Back Cover
