Fault-tolerant control and diagnosis for interger and fractional-order systems : fundamentals of fractional calculus and differential algebra with real-time applications

This book is about algebraic and differential methods, as well as fractional calculus, applied to diagnose and reject faults in nonlinear systems, which are of integer or fractional order. This represents an extension of a very important and widely studied problem in control theory, namely fault diagnosis and rejection (using differential algebraic approaches), to systems presenting fractional dynamics, i.e. systems whose dynamics are represented by derivatives and integrals of non-integer order. The authors offer a thorough overview devoted to fault diagnosis and fault-tolerant control applied to fractional-order and integer-order dynamical systems, and they introduce new methodologies for control and observation described by fractional and integer models, together with successful simulations and real-time applications. The basic concepts and tools of mathematics required to understand the methodologies proposed are all clearly introduced and explained. Consequently, the book is useful as supplementary reading in courses of applied mathematics and nonlinear control theory. This book is meant for engineers, mathematicians, physicists and, in general, to researchers and postgraduate students in diverse areas who have a minimum knowledge of calculus. It also contains advanced topics for researchers and professionals interested in the area of states and faults estimation. 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 7.2 Fractional Calculus Tools 7.3 Fractional Synchronization and Anti-synchronization Problems 7.3.1 Fractional Algebraic Observability Condition 7.3.2 Fractional Integral Reduced-Order Observer 7.3.3 Fractional Synchronization Problem 7.3.4 Fractional Anti-synchronization Problem 7.4 Application to Fractional Chaotic Systems 7.4.1 Fractional Lorenz System 7.4.2 Fractional Rössler System 7.5 Concluding Remarks References 8 Multi-fault-tolerant Control in Fractional-Order Systems 8.1 Fault Diagnosis 8.2 Fault-Tolerant Control 8.3 Stability Analysis of the Closed-Loop System 8.4 Application 8.4.1 Fractional van der Pol Oscillator 8.4.2 Fractional Model of a DC Motor 8.4.3 Comparison of the Fractional-Order DC Motor with the Integer-Order Case 8.5 Concluding Remarks References 9 Fractional-Order Controller Based on a Robust PIα Observer for Uncertain Fractional-Order Systems 9.1 Preliminaries 9.2 Robust PIα Observer for Fractional Uncertain Systems 9.2.1 Stability Analysis 9.3 Fractional-Order Sliding Mode Control 9.3.1 Convergence Analysis 9.4 Case Study: Nα-Differentiator 9.4.1 Generalized Nα-Differentiator 9.5 Conclusions References Appendix Appendix A.1 Appendix to Chap. 5摥映數爠eflinkchapt555 A.2 Appendix to Chap. 8摥映數爠eflinkchap888 Index