Theoretical Developments and Applications of Non Integer Order Systems 7th Conference on Non Integer Order Calculus and Its Applications, Szczecin, Poland 1st Edition by Stefan Domek 3319230399 9783319230399

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ISBN-10 :  3319230399 

ISBN-13 :  9783319230399

Author: Stefan Domek 

This volume is devoted to presentation of new results of research on systems of non-integer order, called also fractional systems. Their analysis and practical implementation have been the object of spontaneous development for a few last decades. The fractional order models can depict a physical plant better than the classical integer order ones. This covers different research fields such as insulator properties, visco-elastic materials, electrodynamic, electrothermal, electrochemical, economic processes modelling etc. On the other hand fractional controllers often outperform their integer order counterparts. This volume contains new ideas and examples of implementation, theoretical and pure practical aspects of using a non-integer order calculus. It is divided into four parts covering: mathematical fundamentals, modeling and approximations, controllability, observability and stability problems and practical applications of fractional control systems. The first part expands the base of tools and methods of the mathematical basis for non-integer order calculus. Part two focuses on new methods and developments in process modeling and fractional derivatives approximations. In the third part a bunch of papers which raise problems of controllability, observability and stability of non-integer order systems is provided. Part four is devoted to presentation of different fractional order control applications. This book was created thanks to many experts in the field of fractional calculus: authors, anonymous referees whose comments allowed us to improve the final form of the papers and active and inspiring discussion of the participants of RRNR’2015, the 7th Conference on Non-Integer Order Calculus and Its Applications that was organized by the Faculty of Electrical Engineering, West Pomeranian University of Technology, Szczecin, Poland.

Theoretical Developments and Applications of Non-Integer Order Systems: 7th Conference on Non-Integer Order Calculus and Its Applications, Szczecin, Poland 1st Table of contents:

Part IMathematical Fundamentals
1 Fractional Sturm-Liouville Problem in Terms of Riesz Derivatives
1 Introduction
2 Preliminaries
3 Main Results
4 Conclusions
References
2 Multidimensional Discrete-Time Fractional Calculus of Variations
1 Introduction
2 Preliminaries
3 Main Result
4 Example
References
3 On a New Symmetric Fractional Variable Order Derivative
1 Introduction
2 Fractional Variable Order Grunwald-Letnikov Type Derivatives
3 Main Result—symmetric Variable Order Derivatives
4 Numerical Example
5 Conclusions
References
4 Linearization of the Non-linear Time-Variant Fractional-Order Difference Equation
1 Introduction
2 Mathematical Preliminaries
3 Linearization of the Non-linear Time-Variant FOS
4 Numerical Example
4.1 Non-linear Time-Invariant FODE of the Restricted Growth
4.2 Linearized Time-Variant FODE of the Restricted Growth
4.3 Non-linear Time-Variant FODE of the Restricted Growth
5 Final Conclusions
References
The Z-Transform Method for Sequential Fractional Difference Operators
1 Introduction
2 Preliminaries
3 Solutions of Sequential Fractional Difference Equations Derived by mathcalZ-transform method
4 Conclusions
References
Part IIModeling and Approximations
6 An Estimation of Accuracy of Charef Approximation
Abstract
1 An Introduction
2 A Transfer Function of Fractional Order Plant
3 The Charef Approximation
4 Cost Functions Describing the Accuracy of Approximation
5 An Example
6 Final Conclusions
Acknowledgements
References
7A New Method for the Integer Order Approximation of Fractional Order Models
1 Introduction
2 Non–Integer Order Linear Systems Recap
3 Fractional Order Model Simplification
4 Model Reduction
5 Examples
6 Conclusions
References
8 Stability Properties of Discrete Time-Domain Oustaloup Approximation
1 Introduction
2 Oustaloup Method
3 Discretization of Transfer Function
4 Time Domain Approximation
4.1 Realization
4.2 Structure Preserving Discretization
4.3 Method of Analysis
4.4 Numerical Experiments
4.5 Method Limitations
5 Conclusion
References
9 Digraphs Minimal Positive Stable Realisations for Fractional One-Dimensional Systems
1 Introduction
2 Notion and Digraph Basic Definitions
2.1 Notion
2.2 Digraph
3 Fractional Order System
3.1 Model and Representation
3.2 State-Space Representation
3.3 Problem Formulation
4 Problem Solution
4.1 Complexity
5 Numerical Examples
5.1 Example I
5.2 Example II
6 Concluding Remarks
References
10 Modeling of Discrete-Time Fractional-Order State Space Systems Using the Balanced Truncation Metho
1 Introduction
2 System Representation
2.1 Expanded State Model of a Fractional-Order State Space System
3 Model Order Reduction
4 Simulation Example
5 Conclusion
References
11 Experimental Results of Modeling Variable Order System Based on Discrete Fractional Variable Order S
1 Introduction
2 Fractional Variable-Order Grunwald-Letnikov Type Derivatives
3 Analog Model
3.1 Realization of the Multi-switching Integral System
3.2 Realization of the Fractional Variable-Order Inertial System
4 Discrete Variable Fractional Order State-Space System
4.1 Parametric Identification
5 Experimental Results
5.1 Identification Results
5.2 Parameter Estimation Results
6 Conclusions
References
Part IIIControllability, Observability and Stability
12 Positivity and Stability of a Class of Fractional Descriptor Discrete-Time Nonlinear Systems
Abstract
1 Introduction
2 Fractional Descriptor Discrete-Time Nonlinear Systems and Their Solution
3 Positive Fractional Descriptor Nonlinear Systems
4 Stability of Positive Nonlinear Systems
5 Concluding Remarks
Acknowledgment
References
13 Stability of Linear Discrete–Time Systems with Fractional Positive Orders
1 Introduction
2 Preliminaries
3 Stability of Systems with Positive Order
4 Conclusions
References
14 Robust Hinfty Observer-Based Stabilization of Disturbed Uncertain Fractional-Order Systems Using a
1 Introduction
2 Preliminaries
3 Main Results
3.1 Hinfty-norm Computation for FOS with Uncertainties
3.2 Robust Hinfty Observer-Based Controller Parametrization
3.3 Robust Hinfty Observer-Based Controller Design
4 Numerical Example
5 Conclusion
References
15 Relative Observability for Fractional Differential-Algebraic Delay Systems Within Riemann-Liouville
1 Introduction
2 Preliminaries
3 Representation of Solutions into Series of Determining Equations Solutions
4 Observability
5 Duality
6 Example
7 Conclusions
References
Part IVControl and Applications
16 Minimum Energy Control of Linear Fractional Systems
Abstract
1 Introduction
2 Fractional Systems
3 Controllability
4 Minimum Energy Control
5 Concluding Remarks
Acknowledgment
References
17 Use of Alpha-Beta Filter to Synchronization of the Chaotic Ikeda Systems of Fractional Order
Abstract
1 Introduction
2 Preliminaries and Problem Formulation
3 Main Results
4 Concluding Remarks
References
18 On Dynamic Decoupling of MIMO Fractional Order Systems
1 Introduction
2 Basics of Calculus and Modeling of Fractional Order Systems
3 Static Decoupling of Fractional MIMO Plants
4 Dynamic Decoupling of Fractional MIMO Plant
4.1 Dynamic Decoupling of TITO Plants
4.2 Dynamic Decoupling of Fractional MIMO Plants with a State Feedback
4.3 Interconnection Transmission Zeros in a Dynamic Decoupling of a Fractional MIMO Plant
5 Summary
References
19 Perfect Control for Fractional-Order Multivariable Discrete-Time Systems
1 Introduction
2 Fractional-Order State Space System [9, 10]3 Fractional-Order Perfect Control
4 Simulation Research
5 Conclusion
References
20 Implementation of Non-Integer Order Controller Using Oustaloup Parallel Approximation for Air Heatin
1 Introduction
2 Air Heating Process
3 Design Non-Integer Controller
3.1 Non-Integer Order PID Controller
4 Oustaloup Approximation
5 Oustaloup Parallel Approximation
6 Experiments Results
7 Conclusion and Further Research
References
21 Classical Versus Fractional Order PI Current Controller in Servo Drive
1 Introduction
2 Basics of Fractional Order Calculus
2.1 Fractional Order Controller
3 Servo Drive
4 Research Assumptions
5 Results of the Simulations
6 Summary
References
22 Parametric Optimization of Non-Integer Order PDμ Controller for Delayed System
1 Introduction
2 Problem Description
2.1 Analysed System
2.2 Optimization Problem
3 Approximation Method
4 Optimisation Algorithm
5 Results
5.1 Convergence of μ
6 Conclusion and Further Research
References
23 Adaptive Non-Integer Controller for Water Tank System
1 Introduction
2 Laboratory Hydraulic Cascaded Three-Tank System
2.1 Mathematical Model of the System
3 Oustaloup Method
4 Time Domain Approximation
4.1 Realization
5 Adaptive Non-Integer Controller
6 Simulation Result
7 Conclusions
References
24 Model-Plant Mismatch in Fractional Order Model Predictive Control
Abstract
1 Introduction
2 Fractional Order Model Predictive Control
3 Model-Plant Mismatch
4 Simulation Experiments
5 Conclusions

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