Intro
Preface
Contents
Acronyms and Symbols
List of Figures
1 Overview
1.1 Fault Diagnosis and Fault-Tolerant Control
1.2 Differential Algebra
1.3 Fractional Calculus and Fractional-Order Systems
1.4 Scope of the Book
References
2 Fundamentals of Differential Algebra
2.1 Differential Rings
2.2 Fields and Differential Fields
2.2.1 Fields
2.2.2 Differential Fields
2.3 Differential Primitive Element
2.4 Algebraic Approach of Nonlinear Dynamics
2.5 Canonical Forms
2.5.1 Generalized Controller Canonical Form

2.5.2 Generalized Observability Canonical Form
References
3 Fault Diagnosis by Means of Invariant Observers
3.1 The Left Invertibility Condition
3.2 Invariant Observer
3.2.1 Characterization of Invariant Observers
3.3 Construction of the Observer
3.3.1 Reduced-Order Observer
3.3.2 Invariant Observer
3.4 Real-Time Application
3.4.1 Description of the Three-Tank System
3.4.2 Diagnosability Analysis
3.5 Experimental Results
3.5.1 Identification
3.5.2 Fault Estimation Results
3.6 Conclusions
References

4 Multi-Fault-Tolerant Control in Integer-Order Systems
4.1 Fault Diagnosis
4.2 Fault-Tolerant Control
4.3 Stability Analysis of the Closed-Loop System
4.4 Application
4.4.1 Numerical Example
4.4.2 Three-Tank System
4.5 Concluding Remarks
References
5 Fundamentals of Fractional Calculus and Fractional Dynamical Systems
5.1 Some Functions Used in Fractional Calculus
5.1.1 Gamma Function
5.1.2 Beta Function
5.1.3 Mittag-Leffler Function
5.2 Fractional-Order Integrals and Derivatives
5.2.1 Riemann-Liouville Fractional-Order Integral

5.2.2 Grünwald-Letnikov Fractional-Order Derivative
5.2.3 Riemann-Liouville Fractional-Order Derivative
5.2.4 Caputo Fractional-Order Derivative
5.2.5 Some Considerations for Fractional-Order Operators
5.3 Fractional-Order Differential Equations
5.3.1 Laplace Transform of Fractional-Order Functions
5.3.2 Solution of FODE by Means of the Laplace Transform
5.4 Fractional Dynamical Systems
5.4.1 Commensurate-Order Fractional Systems
5.4.2 Incommensurate-Order Fractional Systems
5.5 Fractional-Order Controllers
5.5.1 Fractional-Order PID Controller
5.5.2 CRONE

5.6 Stability Results for Fractional-Order Systems
5.6.1 Commensurate-Order Systems
5.6.2 Incommensurate-Order Systems
References
6 Observer-Based Fault Diagnosis for Fractional-Order Nonlinear Systems
6.1 Preliminary Concepts
6.1.1 Caputo Fractional Derivative
6.1.2 Mittag-Leffler Function
6.2 Statement of the Diagnosis Problem
6.3 Fractional Reduced-Order Observer
6.4 Numerical Simulations
6.5 Concluding Remarks
References
7 Fractional Integral Reduced-Order Observer
7.1 The Synchronization and Anti-synchronization Problems