# Analysis, Modeling and Stability of Fractional Order Differential Systems 2

## The Infinite State Approach

This book introduces an original fractional calculus methodology ('the infinite state approach') which is applied to the modeling of fractional order differential equations (FDEs) and systems (FDSs). Its modeling is based on the frequency distributed fractional integrator, while the resulting model corresponds to an integer order and infinite dimension state space representation. Les mer
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This book introduces an original fractional calculus methodology ('the infinite state approach') which is applied to the modeling of fractional order differential equations (FDEs) and systems (FDSs). Its modeling is based on the frequency distributed fractional integrator, while the resulting model corresponds to an integer order and infinite dimension state space representation. This original modeling allows the theoretical concepts of integer order systems to be generalized to fractional systems, with a particular emphasis on a convolution formulation.

With this approach, fundamental issues such as system state interpretation and system initialization long considered to be major theoretical pitfalls have been solved easily. Although originally introduced for numerical simulation and identification of FDEs, this approach also provides original solutions to many problems such as the initial conditions of fractional derivatives, the uniqueness of FDS transients, formulation of analytical transients, fractional differentiation of functions, state observation and control, definition of fractional energy, and Lyapunov stability analysis of linear and nonlinear fractional order systems.

This second volume focuses on the initialization, observation and control of the distributed state, followed by stability analysis of fractional differential systems.

Foreword xiii

Preface xv

Part 1. Initialization, State Observation and Control 1

Chapter 1. Initialization of Fractional Order Systems 3

1.1. Introduction 3

1.2. Initialization of an integer order differential system 4

1.2.1. Introduction 4

1.2.2. Response of a linear system 4

1.2.3. Input/output solution 6

1.2.4. State space solution 7

1.2.5. First-order system example 8

1.3. Initialization of a fractional differential equation 10

1.3.1. Introduction 10

1.3.2. Free response of a simple FDE 10

1.4. Initialization of a fractional differential system 14

1.4.1. Introduction 14

1.4.2. State space representation 14

1.4.3. Input/output formulation 15

1.5. Some initialization examples 17

1.5.1. Introduction 17

1.5.2. Initialization of the fractional integrator 17

1.5.3. Initialization of the Riemann-Liouville derivative 19

1.5.4. Initialization of an elementary FDS 21

1.5.5. Conclusion 33

Chapter 2. Observability and Controllability of FDEs/FDSs 35

2.1. Introduction 35

2.2. A survey of classical approaches to the observability and controllability of fractional differential systems 37

2.2.1. Introduction 37

2.2.2. Definition of observability and controllability 37

2.2.3. Observability and controllability criteria for a linear integer order system 37

2.2.4. Observability and controllability of FDS 39

2.3. Pseudo-observability and pseudo-controllability of an FDS 40

2.3.1. Introduction 40

2.3.2. Elementary approach 41

2.3.3. Cayley-Hamilton approach 45

2.3.4. Gramian approach 49

2.3.5. Gilbert's approach 52

2.3.6. Conclusion 57

2.3.7. Pseudo-controllability example 58

2.4. Observability and controllability of the distributed state 60

2.4.1. Introduction 60

2.4.2. Observability of the distributed state 62

2.4.3. Controllability of the distributed state 64

2.5. Conclusion 65

Chapter 3. Improved Initialization of Fractional Order Systems 67

3.1. Introduction 67

3.2. Initialization: problem statement 68

3.3. Initialization with a fractional observer 71

3.3.1. Fractional observer definition 71

3.3.2. Stability analysis 72

3.3.3. Convergence analysis 74

3.3.4. Numerical example 1: one-derivative system 76

3.3.5. Numerical example 2: non-commensurate order system 78

3.4. Improved initialization 81

3.4.1. Introduction 81

3.4.2. Non-commensurate order principle 82

3.4.3. Gradient algorithm 84

3.4.4. One-derivative FDE example 87

3.4.5. Two-derivative FDE example 91

A.3. Appendix 95

A.3.1. Convergence of gradient algorithm 95

A.3.2. Stability and limit value of 