The Realization Problem for Positive and Fractional Systems (Studies in Systems, Decision and Control .1) (2014. 2014. xviii, 590 S. 37 SW-Abb. 235 mm)

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The Realization Problem for Positive and Fractional Systems (Studies in Systems, Decision and Control .1) (2014. 2014. xviii, 590 S. 37 SW-Abb. 235 mm)

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  • 製本 Hardcover:ハードカバー版/ページ数 630 p.
  • 言語 ENG
  • 商品コード 9783319048338

Full Description


This book addresses the realization problem of positive and fractional continuous-time and discrete-time linear systems. Roughly speaking the essence of the realization problem can be stated as follows: Find the matrices of the state space equations of linear systems for given their transfer matrices. This first book on this topic shows how many well-known classical approaches have been extended to the new classes of positive and fractional linear systems. The modified Gilbert method for multi-input multi-output linear systems, the method for determination of realizations in the controller canonical forms and in observer canonical forms are presented. The realization problem for linear systems described by differential operators, the realization problem in the Weierstrass canonical forms and of the descriptor linear systems for given Markov parameters are addressed. The book also presents a method for the determination of minimal realizations of descriptor linear systems and an extension for cone linear systems. This monographs summarizes recent original investigations of the authors in the new field of the positive and fractional linear systems.

Table of Contents

1 Realization Problem of 1D Standard Linear    1  (66)
Systems
1.1 Basic Definitions and Problem 1 (1)
Formulation
1.2 Existence and Determination of 2 (26)
Minimal Realizations
1.2.1 Properties of Minimal Realizations 2 (3)
1.2.2 Determination of Minimal 5 (12)
Realizations for Single-Input
Single-Output Systems
1.2.3 Determination of Minimal 17 (2)
Realizations for Multi-Input
Single-Output Systems
1.2.4 Determination of Minimal 19 (1)
Realizations for Single-Input
Multi-Output Systems
1.2.5 Modified Gilbert's Method for 20 (8)
Multi-Input Multi-Output Systems
1.3 Determination of Realization in the 28 (4)
Controller Canonical Form
1.4 Determination of Realization in the 32 (4)
Observer Canonical Form
1.5 Realization Problem for Linear 36 (2)
Systems Described by Differential
Operators
1.6 Realization in the Weierstrass 38 (5)
Canonical Form
1.7 Realization of Descriptor Linear 43 (8)
Systems via Markov Parameters
1.8 Computation of Minimal Realizations 51 (8)
for Descriptor Linear Systems
1.9 Computation of Realization of 59 (8)
Discrete-Time Cone Linear Systems
2 Realization Problem of 1D Positive 67 (56)
Regular Linear Systems
2.1 Basic Definitions and Formulation of 67 (1)
Realization Problem
2.2 Existence and Computation of Positive 68 (13)
Realizations
2.2.1 Computation of Matrix D of a 68 (2)
Given Proper Rational Matrix
2.2.2 Existence and Computation of 70 (9)
Positive Realizations of Discrete-Time
Single-Input Single-Output Systems
2.2.3 Existence and Computation of 79 (2)
Positive Realizations of
Continuous-Time Single-Input
Single-Output Systems
2.3 Necessary and Sufficient Conditions 81 (7)
for Existence of Reachable Positive
Realizations
2.3.1 Discrete-Time Systems 81 (5)
2.3.2 Continuous-Time Systems 86 (2)
2.4 Existence and Computation of Positive 88 (10)
Realizations of Multi-Input Multi-Output
Systems
2.5 Positive Stable Realizations with 98 (17)
System Metzler Matrices
2.5.1 Problem Solution for SISO Systems 98 (9)
2.5.2 Problem Solution for MIMO Systems 107(4)
2.5.3 System with Real Poles 111(4)
2.6 Computation of Positive Realizations 115(8)
for Given Impulse Response Matrices of
Discrete-Time Linear Systems
3 Realization Problem of 1D Positive 123(74)
Regular Linear Systems with Delays
3.1 Discrete-Time Linear Systems with 123(21)
Delays
3.1.1 Single-Input Single-Output Systems 123(12)
3.1.2 Multi-Input Multi-Output Systems 135(9)
3.2 Continuous-Time Linear Systems with 144(16)
Delays
3.2.1 Single-Input Single-Output Systems 144(8)
3.2.2 Multi-Input Multi-Output systems 152(8)
3.3 Extension of Classic Gilbert Method 160(6)
for Positive Systems with Delays
3.4 State Variable Diagram Method 166(5)
3.5 Modified State Variable Diagram Method 171(13)
3.6 Positive Realizations with Reduced 184(13)
Numbers of Delays or without Delays for
Discrete-Time Linear Systems
3.6.1 Modified State Variables Diagram 188(7)
Method
3.6.2 Positive Realizations without 195(2)
Delays
4 Positive Stable Realization Problem for 197(60)
1D Regular Linear Systems
4.1 Metzler Matrices and Stable 197(2)
Polynomials
4.2 Existence and Computation of the Set 199(17)
of Metzler Matrices for Given Stable
Polynomials
4.2.1 Polynomials with Only Real 199(2)
Negative Zeros
4.2.2 Polynomials with Real and Complex 201(2)
Conjugate Zeros: Second Degree
Polynomials
4.2.3 Polynomials with Real and Complex 203(5)
Conjugate Zeros: Third Degree
Polynomials
4.2.4 Polynomials with Real and Complex 208(4)
Conjugate Zeros: Fourth Degree
Polynomials
4.2.5 Polynomials with Real and Complex 212(4)
Conjugate Zeros: General Case -- n-th
Degree Polynomials
4.3 Realization Problem Formulation for 216(3)
Discrete-Time Linear Systems
4.4 Existence and Computation of the 219(20)
Positive Stable Realizations for Given
Transfer Function
4.4.1 Systems with Only Real Positive 219(9)
Poles
4.4.2 Systems with Complex Conjugate 228(6)
Poles
4.4.3 General Case of SISO Systems 234(5)
4.5 Realization Problem Formulation for 239(1)
Continuous-Time Linear Systems
4.6 Existence and Computation of the 240(17)
Positive Stable Realizations for Given
Transfer Function
4.6.1 Particular Case 240(6)
4.6.2 General Case 246(3)
4.6.3 System with Real Poles 249(4)
4.6.4 System with Real Poles and Zeros 253(4)
5 Positive Realization Problem of ID 257(80)
Descriptor Linear Systems
5.1 Positivity of Descriptor Linear 257(11)
Systems with Regular Pencils
5.1.1 Positive Continuous-Time Systems 259(8)
5.1.2 Positive Discrete-Time Systems 267(1)
5.2 Positivity of Descriptor Linear 268(8)
Systems with Singular Pencils
5.2.1 Checking Positivity of Descriptor 269(5)
Systems
5.2.2 Extension to the Descriptor 274(2)
Discrete-Time Linear Systems
5.3 Positive Realizations for Descriptor 276(5)
Continuous-Time Linear Systems
5.4 Positive Realizations for Descriptor 281(5)
Discrete-Time Linear Systems
5.5 Positive and Asymptotically Stable 286(11)
Realizations for Descriptor Discrete-Time
Linear Systems
5.5.1 Method 1 288(5)
5.5.2 Method 2 293(4)
5.6 Weierstrass Canonical Form of 297(6)
Positive Realizations
5.7 Computation of WCF Positive 303(2)
Realizations
5.8 Computation of Positive Realizations 305(2)
of Descriptor Linear Systems with
Unimodular System Matrix
5.9 Positive Realizations in Canonical 307(9)
Forms of Descriptor Discrete-Time Linear
Systems with Regular Pencils--State
Variables Diagram Approach
5.9.1 Method 1 309(4)
5.9.2 Method 2 313(3)
5.10 Positive Realizations of Descriptor 316(8)
Continuous-Time Linear Systems with
Delays by State Variable Diagram Method
5.10.1 Systems with Delays in State 316(4)
Vector and Output
5.10.2 Systems with Delays in State and 320(4)
Input Vectors
5.11 Positive Realizations of Descriptor 324(13)
Discrete-Time Linear Systems with Delays
in State and Input Vectors
5.11.1 Single Delays Systems 324(6)
5.11.2 Multi Delays Systems 330(7)
6 Positive Realization Problem of 1D 337(40)
Regular Fractional Linear Systems
6.1 Fractional Discrete-Time Linear 337(16)
Systems
6.1.1 Preliminaries 337(1)
6.1.2 Realization Problem 338(15)
6.2 Fractional Continuous-Time Linear 353(8)
Systems
6.2.1 SISO Systems 354(3)
6.2.2 MIMO Systems 357(4)
6.3 Fractional Continuous-Time Linear 361(8)
System with Delays
6.3.1 Preliminaries 361(4)
6.3.2 Realization Problem 365(4)
6.4 Positive Stable Realization of 369(5)
Fractional Discrete-Time Linear Systems
6.4.1 Preliminaries and Problem 369(1)
Formulation
6.4.2 Problem Solution 369(5)
6.5 Positive Stable Realizations of 374(3)
Fractional Continuous-Time Linear Systems
7 Positive Realization Problem for 1D 377(40)
Descriptor Fractional Systems
7.1 Discrete-Time Fractional Systems 377(17)
7.1.1 Preliminaries and the State 377(5)
Solution
7.1.2 State Solution Based on 382(6)
Decomposition
7.1.3 Decomposition of Discrete-Time 388(6)
Linear Systems
7.2 Continuous-Time Fractional Linear 394(15)
Systems
7.2.1 Preliminaries and the State 394(4)
Solution
7.2.2 State Solution Based on 398(2)
Decomposition
7.2.3 Descriptor Fractional Electrical 400(5)
Circuits
7.2.4 Decomposition of Continuous-Time 405(4)
Linear Systems
7.3 Realization Problem for Positive 409(8)
Fractional Linear Systems
7.3.1 Preliminaries and Positive 409(2)
Realization Problem
7.3.2 Positive Fractional Linear Systems 411(3)
7.3.3 Realization Problem for Positive 414(3)
Fractional Continuous-Time Linear
Systems
8 Positive Realization Problem of 2D 417(62)
Continuous-Discrete Linear Systems
8.1 General Model of Continuous-Discrete 417(2)
Linear Systems
8.2 Positive Realization for 419(7)
Continuous-Discrete Linear Systems in
General Model Form
8.3 Continuous-Discrete Linear Systems 426(2)
Described by Roesser Type Model
8.4 Positive Realization for 428(21)
Continuous-Discrete Linear Systems
Described by Roesser Type Model
8.4.1 Transfer Function Separation 428(7)
Approach
8.4.2 State Diagram Approach for SISO 435(7)
Systems
8.4.3 Generalization for MIMO Systems 442(7)
8.5 Positive Minimal Realization of 449(8)
Continuous-Discrete Linear Systems with
All-Pole and All-Zero Transfer Function
8.6 Positive Minimal Realization of 457(11)
Continuous-Discrete Linear Systems with
Separable Denominators or Separable
Numerators
8.7 Realization Problem for Positive 468(11)
Continuous-Discrete Systems with One
Delay in State and Input Vectors
Appendix A Elementary Operations on Matrices, 479(14)
Matrix Equations with Nonnegative Solution
and Shuffle Algorithm
A.1 Elementary Operations on Numerical 479(2)
Matrices
A.2 Elementary Operations on Polynomial 481(3)
Matrices
A.3 Matrix Equations with Nonnegative 484(2)
Solutions
A.4 Transformation of the State Equations 486(7)
by the Use of the Shuffle Algorithm
A.4.1 Continuous-Time Linear Systems 486(4)
A.4.2 Discrete-Time Systems 490(3)
Appendix B Positive Regular Continuous-Time 493(18)
and Discrete-Time Linear Systems
B.1 Externally Positive Continuous-Time 493(2)
Linear Systems
B.2 Externally Positive Discrete-Time 495(2)
Linear Systems
B.3 Internally Positive Continuous-Time 497(1)
Linear Systems
B.4 Internally Positive Discrete-Time 498(1)
Linear Systems
B.5 Asymptotic Stability of Positive 499(2)
Continuous-Time Linear Systems
B.6 Asymptotic Stability of Positive 501(3)
Discrete-Time Linear Systems
B.7 Asymptotic Stability of Positive 504(2)
Continuous-Time Linear Systems with
Delays in State
B.8 Reachability of Positive 506(1)
Discrete-Time Linear Systems
B.9 Reachability of Positive 507(1)
Continuous-Time Linear Systems
B.10 Controllability and Controllability 508(1)
to Zero of Positive Discrete-Time Linear
Systems
B.11 Controllability and Controllability 509(1)
to Zero of Positive Continuous-Time
Linear Systems
B.12 Observability of Positive 510(1)
Discrete-Time Linear Systems
Appendix C Fractional Linear Systems 511(14)
C.1 Fractional Discrete-Time Linear 511(2)
Systems
C.2 Reachability of Fractional 513(1)
Discrete-Time Linear Systems
C.3 Reachability of Positive Fractional 514(1)
Discrete-Time Linear Systems
C.4 Controllability to Zero of Fractional 515(1)
Discrete-Time Linear Systems
C.5 Controllability to Zero of Positive 516(1)
Fractional Discrete-Time Linear Systems
C.6 Practical and Asymptotic Stability of 516(2)
Fractional Discrete-Time Linear Systems
C.7 Fractional Different Orders 518(2)
Discrete-Time Linear Systems
C.8 Positive Fractional Different Orders 520(1)
Discrete-Time Linear Systems
C.9 Fractional Continuous-Time Linear 520(2)
Systems
C.10 Positivity of Fractional 522(1)
Continuous-Time Linear Systems
C.11 Reachability of Fractional Positive 523(2)
Continuous-Time Linear Systems
Appendix D Minimal and Cyclic Realizations 525(30)
and Structure Stabilityof Normal Transfer
Matrices
D.1 Basic Notions and Problem Formulation 525(1)
D.2 Existence of the Minimal and Cyclic 526(2)
Realizations
D.2.1 Existence of the Minimal 526(1)
Realizations
D.2.2 Existence of the Cyclic 527(1)
Realizations
D.3 Computation of Cyclic Realizations 528(5)
with the Matrix A in the Frobenius
Canonical Form
D.4 Computation of Cyclic Realizations 533(12)
with the Matrix A in the Jordan Canonical
Form
D.5 Structural Stability of Cyclic 545(10)
Matrices and Cyclic Realizations
D.5.1 Structural Stability of Cyclic 545(1)
Matrices
D.5.2 Structural Stability of Cyclic 546(2)
Realization
D.5.3 The Impact of the Coefficients of 548(7)
the Transfer Function on System
Description
Appendix E Positive 2D Continuous-Discrete 555(32)
Linear Systems
E.1 General Model of Continuous-Discrete 555(1)
Linear Systems and Its Solution
E.2 Positive General Model of 556(2)
Continuous-Discrete Linear Systems
E.3 Reachability of the Standard and 558(4)
Positive General Model
E.4 Stability of the Positive General 562(6)
Model
E.5 Continuous-Discrete Linear Systems 568(19)
with Delays
References 573(14)
Subject Index 587