Selected Problems of Fractional Systems Theory (Lecture Notes in Control and Information Sciences) 〈Vol. 411〉

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Selected Problems of Fractional Systems Theory (Lecture Notes in Control and Information Sciences) 〈Vol. 411〉

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  • 製本 Paperback:紙装版/ペーパーバック版/ページ数 346 p.
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基本説明

Covers selected problems of positive fractional 1D and 2D linear systems.

Full Description


This monograph covers some selected problems of positive fractional 1D and 2D linear systems. It is an extended and modified English version of its preceding Polish edition published by Technical University of Bialystok in 2009. This book is based on the lectures delivered by the author to the Ph.D. students of the Faculty of Electrical Engineering of Bialystok University of Technology and of Warsaw University of Technology and on invited lectures in several foreign universities in the last three years.

Table of Contents

Preface                                            v
List of Symbols ix
List of Figures xi
1 Fractional Discrete-Time Linear Systems 1 (26)
1.1 Definition of n-Order Difference 1 (1)
1.2 State Equations of the Fractional 2 (2)
Linear Systems
1.2.1 Fractional Systems without Delays 2 (1)
1.2.2 Fractional Systems with Delays 3 (1)
1.3 Solution of the State Equations of 4 (3)
the Fractional Discrete-Time Linear
System with Delays
1.3.1 Fractional Systems with Delays 4 (2)
1.3.2 Fractional Systems without Delays 6 (1)
1.4 Positive Fractional Linear Systems 7 (2)
1.5 Externally Positive Fractional Systems 9 (1)
1.6 Reachability of Fractional 10 (2)
Discrete-Time Linear Systems
1.7 Reachability of Positive Fractional 12 (3)
Discrete-Time Linear Systems
1.8 Controllability to Zero of the 15 (3)
Fractional Discrete-Time Linear Systems
1.9 Controllability to Zero of Positive 18 (1)
Fractional Discrete-Time Linear Systems
1.10 Minimum Energy Control of Positive 19 (3)
Fractional Systems
1.11 Fractional Different Orders 22 (2)
Discrete-Time Linear Systems
1.12 Positive Fractional Different Orders 24 (3)
Discrete-Time Linear Systems
2 Fractional Continuous-Time Linear Systems 27 (26)
2.1 Definition of Euler Gamma Function 27 (1)
and Its Properties
2.2 Mittag-Leffler Function 28 (1)
2.3 Definitions of Fractional 28 (3)
Derivative-Integral
2.3.1 Riemann-Liouville Definition 28 (2)
2.3.2 Caputo Definition 30 (1)
2.4 Solution of the Fractional State 31 (4)
Equation of Continuous-Time Linear System
2.5 Positivity of the Fractional Systems 35 (1)
2.6 External Positivity of the Fractional 36 (1)
Systems
2.7 Reachability of Fractional Positive 37 (3)
Continuous-Time Linear System
2.8 Positive Continuous-Time Linear 40 (2)
Systems with Delays
2.9 Positive Linear Systems Consisting of 42 (11)
n Subsystems with Different Fractional
Orders
2.9.1 Linear Differential Equations 42 (4)
with Different Fractional Orders
2.9.2 Positive Fractional Systems 46 (1)
2.9.3 Fractional Linear Electrical 47 (6)
Circuits
3 Fractional Positive 2D Linear Systems 53 (28)
3.1 Definition of (Backward) Fractional 53 (1)
Difference of 2D Function
3.2 State Equation of Fractional 2D 53 (1)
Linear Systems
3.3 Solution of the State Equation of the 54 (2)
Fractional 2D Linear System
3.4 Extension of the Cayley-Hamilton 56 (1)
Theorem
3.5 Positivity of Fractional 2D Linear 57 (1)
Systems
3.6 Reachability and Controllability of 58 (3)
Positive Fractional 2D Linear Systems
3.7 Controllability to Zero of Positive 61 (1)
Fractional 2D Linear System
3.8 Models of 2D Linear Systems 62 (9)
3.8.1 Positive 2D Linear Systems 62 (1)
3.8.2 Positive Fractional 2D Linear 63 (6)
Systems
3.8.3 Positive 2D Linear Systems with 69 (2)
Delays
3.9 Positive Fractional 2D Linear System 71 (10)
of Different Orders
3.9.1 Definition of (Backward) 71 (1)
Difference of (α, β) Order
of 2D Function
3.9.2 State Equations of Fractional 2D 71 (1)
Linear System
3.9.3 Solution of the State Equations 72 (5)
of the Fractional 2D Linear Systems
3.9.4 Extension of the Cayley-Hamilton 77 (1)
Theorem
3.9.5 Positivity of the Fractional 2D 78 (3)
Linear Systems
4 Pointwise Completeness and Pointwise 81 (22)
Degeneracy of Linear Systems
4.1 Standard Discrete-Time Linear Systems 81 (1)
4.2 Standard Continuous-Time Linear 82 (1)
Systems
4.3 Standard Discrete-Time Linear Systems 83 (1)
with Delays
4.4 Positive Discrete-Time Linear Systems 84 (1)
4.5 Positive Continuous-Time Linear 85 (1)
Systems
4.6 Positive Discrete-Time Linear Systems 86 (2)
with Delays
4.7 Fractional Discrete-Time Linear 88 (1)
Systems
4.8 Fractional Continuous-Time Linear 89 (3)
Systems
4.9 Positive Fractional Discrete-Time 92 (1)
Linear System
4.10 Positive Fractional Continuous-Time 93 (1)
Linear Systems
4.11 Pointwise Completeness and Pointwise 94 (2)
Degeneracy of Electrical Circuits
4.12 Standard Continuous-Discrete Linear 96 (3)
System Described by the General Model
4.13 Positive Continuous-Discrete Linear 99 (4)
System Described by the General Model
5 Pointwise Completeness and Pointwise 103(14)
Degeneracy of Linear Systems with
State-Feedbacks
5.1 Standard Discrete-Time Linear Systems 103(5)
5.2 Standard Continuous-Time Linear 108(1)
Systems
5.3 Positive Standard Discrete-Time 108(3)
Linear Systems
5.4 Positive Standard Continuous-Time 111(3)
Linear Systems
5.5 Fractional Discrete-Time Linear 114(1)
Systems
5.6 Fractional Continuous-Time Linear 114(1)
Systems
5.7 Positive Fractional Discrete-Time 115(1)
Linear System
5.8 Positive Fractional Continuous-Time 116(1)
Linear Systems
6 Realization Problem for Positive 117(24)
Fractional and Continuous-Discrete 2D
Linear Systems
6.1 Fractional Discrete-Time Linear 117(13)
Systems
6.1.1 SISO Systems 119(4)
6.1.2 MIMO Systems 123(7)
6.2 Fractional Continuous-Time Linear 130(11)
Systems
6.2.1 SISO Systems 131(2)
6.2.2 MIMO Systems 133(8)
7 Cone Discrete-Time and Continuous-Time 141(20)
Linear Systems
7.1 Basic Definitions 141(5)
7.1.1 Cone Discrete-Time Systems 141(1)
7.1.2 Cone Continuous-Time Systems with 142(2)
Delays
7.1.3 Cone Fractional Discrete-Time 144(1)
Systems
7.1.4 Cone Fractional Continuous-Time 145(1)
System
7.2 Reachability of Cone Fractional 146(3)
Systems
7.2.1 Cone Fractional Discrete-Time 146(2)
Systems
7.2.2 Cone Fractional Continuous-Time 148(1)
System
7.3 Controllability to Zero of Cone 149(1)
Fractional Discrete-Time Systems
7.4 Cone Realization Problem for Linear 150(11)
Systems
7.4.1 Discrete-Time Linear Systems 150(3)
7.4.2 Cone Realization Problem for 153(8)
Continuous-Time Systems with Delays
8 Stability of Positive Fractional 1D and 161(28)
2D Linear Systems
8.1 Asymptotic Stability of Discrete-Time 161(15)
Linear Systems
8.1.1 Positive Discrete-Time Systems 161(4)
8.1.2 Positive 2D Linear Systems 165(3)
8.1.3 Relationship between Asymptotic 168(3)
Stability of 1D and 2D Linear Systems
8.1.4 Positive Fractional Discrete-Time 171(2)
Linear Systems with Delays
8.1.5 Positive Fractional 2D Linear 173(3)
Systems
8.2 Practical Stability of Fractional 176(6)
Systems
8.2.1 Positive Fractional 1D Systems 176(4)
8.2.2 Positive Fractional 2D Systems 180(2)
8.3 Asymptotic Stability of 182(7)
Continuous-Time Linear Systems
8.3.1 Positive Continuous-Time Linear 182(3)
Systems
8.3.2 Asymptotic Stability of Positive 185(4)
Continuous-Time Systems with Delays
9 Stability Analysis of Fractional Linear 189(32)
Systems in Frequency Domain
9.1 Fractional Continuous-Time Systems 189(9)
9.2 Fractional Continuous-Time Systems 198(7)
with Delays of the Retarded Type
9.3 Fractional Discrete-Time Systems 205(6)
9.4 Robust Stability of Convex 211(10)
Combination of Two Fractional Polynomials
10 Stabilization of Positive and Fractional 221(24)
Linear Systems
10.1 Fractional Discrete-Time Linear 221(3)
Systems with Delays
10.2 Fractional Continuous-Time Linear 224(3)
Systems with Delays
10.3 Application of LMI to Stabilization 227(18)
of Fractional Linear Systems
10.3.1 Fractional 1D Linear Systems 227(4)
10.3.2 Positive 2D Linear Systems 231(2)
10.3.3 Positive 2D Linear Systems with 233(5)
Delays
10.3.4 Fractional 2D Roesser Model 238(7)
11 Singular Fractional Linear Systems 245(22)
11.1 Singular Fractional Continuous-Time 245(2)
Linear Systems
11.2 Singular Fractional Electrical 247(4)
Circuits
11.3 Singular Fractional Discrete-Time 251(7)
Linear Systems
11.4 Reduction of Singular Fractional 258(5)
Systems to Equivalent Standard Fractional
Systems
11.5 Decomposition of Singular Fractional 263(4)
System into Dynamic and Static Parts
12 Positive Continuous-Discrete Linear 267(40)
Systems
12.1 General Model of Continuous-Discrete 267(2)
Linear Systems and Its Solution
12.2 Positive General Model of 269(3)
Continuous-Discrete Linear Systems
12.3 Reachability of the Standard and 272(6)
Positive General Model
12.4 Stability of the Positive General 278(8)
Model
12.5 Robust Stability of Linear 286(6)
Continuous-Discrete Linear Systems
2.6 Positive Realization Problem for 292(15)
Continuous-Discrete Linear Systems
12.6.1 Problem Formulation 292(2)
12.6.2 SISO Systems 294(7)
12.6.3 MIMO Systems 301(6)
A Laplace Transforms of Continuous-Time 307(6)
Functions and Z-Transforms of Discrete-Time
Functions
A.1 Convolutions of Continuous-Time and 307(1)
Discrete-Time Functions and Their
Transforms
A.2 Laplace Transforms of 308(1)
Derivative-Integrals
A.3 Z-Transforms of Discrete-Time 309(4)
Functions
B Infinite Long Cable with Zero Inductance as 313(4)
an Example of Fractional System
C Right Inverse of Matrices 317(4)
D Elementary Operations on Matrices 321(2)
References 323(18)
Index 341