Ebook: Distributed-Order Dynamic Systems: Stability, Simulation, Applications and Perspectives

Distributed-order differential equations, a generalization of fractional calculus, are of increasing importance in many fields of science and engineering from the behaviour of complex dielectric media to the modelling of nonlinear systems.
This Brief will broaden the toolbox available to researchers interested in modeling, analysis, control and filtering. It contains contextual material outlining the progression from integer-order, through fractional-order to distributed-order systems. Stability issues are addressed with graphical and numerical results highlighting the fundamental differences between constant-, integer-, and distributed-order treatments. The power of the distributed-order model is demonstrated with work on the stability of noncommensurate-order linear time-invariant systems. Generic applications of the distributed-order operator follow: signal processing and viscoelastic damping of a mass–spring set up.
A new general approach to discretization of distributed-order derivatives and integrals is described. The Brief is rounded out with a consideration of likely future research and applications and with a number of MATLAB® codes to reduce repetitive coding tasks and encourage new workers in distributed-order systems.

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Distributed-order differential equations, a generalization of fractional calculus, are of increasing importance in many fields of science and engineering from the behaviour of complex dielectric media to the modelling of nonlinear systems. This Brief will broaden the toolbox available to researchers interested in modeling, analysis, control and filtering. It contains contextual material outlining the progression from integer-order, through fractional-order to distributed-order systems. Stability issues are addressed with graphical and numerical results highlighting the fundamental differences between constant-, integer-, and distributed-order treatments. The power of the distributed-order model is demonstrated with work on the stability of noncommensurate-order linear time-invariant systems. Generic applications of the distributed-order operator follow: signal processing and viscoelastic damping of a mass–spring set up. A new general approach to discretization of distributed-order derivatives and integrals is described. The Brief is rounded out with a consideration of likely future research and applications and with a number of MATLAB® codes to reduce repetitive coding tasks and encourage new workers in distributed-order systems. Table of Contents Cover Distributed-Order Dynamic Systems ISBN 9781447128519 e-ISBN 9781447128526 Preface Acknowledgements Contents Acronyms 1 Introduction 1.1 From Integer-Order Dynamic Systems to Fractional-Order Dynamic Systems 1.2 From Fractional-Order Dynamic Systems to Distributed-Order Dynamic Systems 1.3 Preview of Chapters 1.4 Chapter Summary References 2 Distributed-Order Linear Time-Invariant System (DOLTIS) and Its Stability Analysis 2.1 Introduction 2.2 Stability Analysis of DOLTIS in Four Cases 2.3 Time-Domain Analysis: Impulse Responses 2.4 Frequency-Domain Response: Bode Plots 2.5 Numerical Examples 2.6 Chapter Summary References 3 Noncommensurate Constant Orders as Special Cases of DOLTIS 3.1 Introduction 3.2 Stability Analysis of Some Special Cases of DOLTIS 3.2.1 Case 1: Double Noncommensurate Orders 3.2.2 Case 2: N-Term Noncommensurate Orders 3.3 Numerical Examples 3.4 Chapter Summary References 4 Distributed-Order Filtering and Distributed-Order Optimal Damping 4.1 Application I: Distributed-Order Filtering 4.1.1 Distributed-Order Integrator/Differentiator 4.1.2 Distributed-Order Low-Pass Filter 4.1.3 Impulse Response Invariant Discretization of DO-LPF 4.2 Application II: Optimal Distributed-Order Damping 4.2.1 Distributed-Order Damping in Mass-Spring Viscoelastic Damper System 4.2.2 Frequency-Domain Method Based Optimal Fractional-Order Damping Systems 4.3 Chapter Summary References 5 Numerical Solution of Differential Equations of Distributed Order 5.1 Introduction 5.2 Triangular Strip Matrices 5.3 Kronecker Matrix Product 5.4 Discretization of Ordinary Fractional Derivatives of Constant Order 5.5 Discretization of Ordinary Derivatives of Distributed Order 5.6 Discretization of Partial Derivatives of Distributed Order 5.7 Initial and Boundary Conditions for Using the Matrix Approach 5.8 Implementation in MATLAB 5.9 Numerical Examples 5.9.1 Example 1: Distributed-Order Relaxation 5.9.2 Example 2: Distributed-Order Oscillator 5.9.3 Example 3: Distributed-Order Diffusion 5.10 Chapter Summary References 6 Future Topics 6.1 Geometric Interpretation of Distributed-Order Differentiation as a Framework for Modeling 6.2 From Positive Linear Time-Invariant Systems to Generalized Distributed-Order Systems 6.3 From PID Controllers to Distributed-Order PID Controllers References Appendix A MATLAB Codes A.2 Bode Plots of Distributed-Order LinearTime-Invariant Systems A.3 Impulse Responses of Distributed-Order LinearTime-Invariant Systems A.4 Solution of Distributed-Order Relaxation Equation A.5 Solution of Distributed-Order Oscillation Equation Index