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.
Part 1. Initialization, State Observation and Control
1. Initialization of Fractional Order Systems.
2. Observability and Controllability of FDEs/FDSs.
3. Improved Initialization of Fractional Order Systems.
4. State Control of Fractional Differential Systems.
5. Fractional Model-based Control of the Diffusive RC Line.
Part 2. Stability of Fractional Differential Equations and Systems
6. Stability of Linear FDEs Using the Nyquist Criterion.
7. Fractional Energy.
8. Lyapunov Stability of Commensurate Order Fractional Systems.
9. Lyapunov Stability of Non-commensurate Order Fractional Systems.
10. An Introduction to the Lyapunov Stability of Nonlinear Fractional Order Systems.
Jean-Claude Trigeassou is Honorary Professor at Bordeaux University, France, and has been associated with the research activities of its IMS-LAPS lab since 2006. His main research interests include the modeling of fractional order systems, based on the infinite state approach.
Nezha Maamri is Associate Professor at Poitiers University, France. Her research activities concern the method of moments, robust control using integer order and fractional order controllers, plus the modeling, initialization and stability of fractional order systems.
Table of Contents
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