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El. knyga: Complex Conjugate Matrix Equations for Systems and Control

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The book is the first book on complex matrix equations including the conjugate of unknown matrices. The study of these conjugate matrix equations is motivated by the investigations on stabilization and model reference tracking control for discrete-time antilinear systems, which are a particular kind of complex system with structure constraints. It proposes useful approaches to obtain iterative solutions or explicit solutions for several types of complex conjugate matrix equation. It observes that there are some significant differences between the real/complex matrix equations and the complex conjugate matrix equations. For example, the solvability of a real Sylvester matrix equation can be characterized by matrix similarity; however, the solvability of the con-Sylvester matrix equation in complex conjugate form is related to the concept of con-similarity.  In addition, the new concept of conjugate product for complex polynomial matrices is also proposed in order to establish a unified approach for solving a type of complex matrix equation.

Recenzijos

The book is well organized and presents the most important notions of iterative solutions, explicit solutions and applications in systems and control. The book is suitable for senior undergraduate and graduate students as well as practical engineers, scientist and researchers interested in iterative, explicit solutions and of applications in systems and control. (Seenith Sivasundaram, zbMATH 1371.15003, 2017)

1 Introduction
1(34)
1.1 Linear Equations
2(3)
1.2 Univariate Linear Matrix Equations
5(11)
1.2.1 Lyapunov Matrix Equations
5(4)
1.2.2 Kalman-Yakubovich and Normal Sylvester Matrix Equations
9(4)
1.2.3 Other Matrix Equations
13(3)
1.3 Multivariate Linear Matrix Equations
16(11)
1.3.1 Roth Matrix Equations
16(2)
1.3.2 First-Order Generalized Sylvester Matrix Equations
18(6)
1.3.3 Second-Order Generalized Sylvester Matrix Equations
24(1)
1.3.4 High-Order Generalized Sylvester Matrix Equations
25(1)
1.3.5 Linear Matrix Equations with More Than Two Unknowns
26(1)
1.4 Coupled Linear Matrix Equations
27(3)
1.5 Complex Conjugate Matrix Equations
30(3)
1.6 Overview of This Monograph
33(2)
2 Mathematical Preliminaries
35(62)
2.1 Kronecker Products
35(7)
2.2 Leverrier Algorithms
42(4)
2.3 Generalized Leverrier Algorithms
46(3)
2.4 Singular Value Decompositions
49(3)
2.5 Vector Norms and Operator Norms
52(11)
2.5.1 Vector Norms
52(4)
2.5.2 Operator Norms
56(7)
2.6 A Real Representation of a Complex Matrix
63(10)
2.6.1 Basic Properties
64(4)
2.6.2 Proof of Theorem 2.7
68(5)
2.7 Consimilarity
73(2)
2.8 Real Linear Spaces and Real Linear Mappings
75(8)
2.8.1 Real Linear Spaces
76(5)
2.8.2 Real Linear Mappings
81(2)
2.9 Real Inner Product Spaces
83(4)
2.10 Optimization in Complex Domain
87(3)
2.11 Notes and References
90(7)
Part I Iterative Solutions
3 Smith-Type Iterative Approaches
97(22)
3.1 Infinite Series Form of the Unique Solution
98(5)
3.2 Smith Iterations
103(2)
3.3 Smith (l) Iterations
105(3)
3.4 Smith Accelerative Iterations
108(7)
3.5 An Illustrative Example
115(1)
3.6 Notes and References
116(3)
4 Hierarchical-Update-Based Iterative Approaches
119(44)
4.1 Extended Con-Sylvester Matrix Equations
121(14)
4.1.1 The Matrix Equation AXB + CXD = F
121(5)
4.1.2 A General Case
126(7)
4.1.3 Numerical Examples
133(2)
4.2 Coupled Con-Sylvester Matrix Equations
135(14)
4.2.1 Iterative Algorithms
137(2)
4.2.2 Convergence Analysis
139(7)
4.2.3 A More General Case
146(1)
4.2.4 A Numerical Example
147(2)
4.3 Complex Conjugate Matrix Equations with Transpose of Unknowns
149(9)
4.3.1 Convergence Analysis
151(6)
4.3.2 A Numerical Example
157(1)
4.4 Notes and References
158(5)
5 Finite Iterative Approaches
163(62)
5.1 Generalized Con-Sylvester Matrix Equations
163(16)
5.1.1 Main Results
164(8)
5.1.2 Some Special Cases
172(3)
5.1.3 Numerical Examples
175(4)
5.2 Extended Con-Sylvester Matrix Equations
179(19)
5.2.1 The Matrix Equation AXB + CXD = F
179(13)
5.2.2 A General Case
192(3)
5.2.3 Numerical Examples
195(3)
5.3 Coupled Con-Sylvester Matrix Equations
198(23)
5.3.1 Iterative Algorithms
198(1)
5.3.2 Convergence Analysis
199(7)
5.3.3 A More General Case
206(1)
5.3.4 Numerical Examples
207(2)
5.3.5 Proofs of Lemmas 5.15 and 5.16
209(12)
5.4 Notes and References
221(4)
Part II Explicit Solutions
6 Real-Representation-Based Approaches
225(50)
6.1 Normal Con-Sylvester Matrix Equations
226(15)
6.1.1 Solvability Conditions
226(4)
6.1.2 Uniqueness Conditions
230(3)
6.1.3 Solutions
233(8)
6.2 Con-Kalman-Yakubovich Matrix Equations
241(9)
6.2.1 Solvability Conditions
241(2)
6.2.2 Solutions
243(7)
6.3 Con-Sylvester Matrix Equations
250(9)
6.4 Con-Yakubovich Matrix Equations
259(8)
6.5 Extended Con-Sylvester Matrix Equations
267(3)
6.6 Generalized Con-Sylvester Matrix Equations
270(3)
6.7 Notes and References
273(2)
7 Polynomial-Matrix-Based Approaches
275(60)
7.1 Homogeneous Con-Sylvester Matrix Equations
276(8)
7.2 Nonhomogeneous Con-Sylvester Matrix Equations
284(10)
7.2.1 The First Approach
285(8)
7.2.2 The Second Approach
293(1)
7.3 Con-Yakubovich Matrix Equations
294(13)
7.3.1 The First Approach
295(10)
7.3.2 The Second Approach
305(2)
7.4 Extended Con-Sylvester Matrix Equations
307(14)
7.4.1 Basic Solutions
308(3)
7.4.2 Equivalent Forms
311(5)
7.4.3 Further Discussion
316(2)
7.4.4 Illustrative Examples
318(3)
7.5 Generalized Con-Sylvester Matrix Equations
321(13)
7.5.1 Basic Solutions
322(2)
7.5.2 Equivalent Forms
324(5)
7.5.3 Special Solutions
329(3)
7.5.4 An Illustrative Example
332(2)
7.6 Notes and References
334(1)
8 Unilateral-Equation-Based Approaches
335(20)
8.1 Con-Sylvester Matrix Equations
336(7)
8.2 Con-Yakubovich Matrix Equations
343(6)
8.3 Nonhomogeneous Con-Sylvester Matrix Equations
349(5)
8.4 Notes and References
354(1)
9 Conjugate Products
355(34)
9.1 Complex Polynomial Ring (C[ s], +)
355(4)
9.2 Division with Remainder in (C[ s], +)
359(3)
9.3 Greatest Common Divisors in (C[ s], +)
362(3)
9.4 Coprimeness in (C[ s], +)
365(1)
9.5 Conjugate Products of Polynomial Matrices
366(5)
9.6 Unimodular Matrices and Smith Normal Form
371(6)
9.7 Greatest Common Divisors
377(2)
9.8 Coprimeness of Polynomial Matrices
379(3)
9.9 Conequivalence and Consimilarity
382(3)
9.10 An Example
385(1)
9.11 Notes and References
385(4)
10 Con-Sylvester-Sum-Based Approaches
389(16)
10.1 Con-Sylvester Sum
389(5)
10.2 Con-Sylvester-Polynomial Matrix Equations
394(6)
10.2.1 Homogeneous Case
394(3)
10.2.2 Nonhomogeneous Case
397(3)
10.3 An Illustrative Example
400(2)
10.4 Notes and References
402(3)
Part III Applications in Systems and Control
11 Stability for Antilinear Systems
405(34)
11.1 Stability for Discrete-Time Antilinear Systems
407(3)
11.2 Stochastic Stability for Markovian Antilinear Systems
410(13)
11.3 Solutions to Coupled Anti-Lyapunov Equations
423(12)
11.3.1 Explicit Iterative Algorithms
424(4)
11.3.2 Implicit Iterative Algorithms
428(4)
11.3.3 An Illustrative Example
432(3)
11.4 Notes and References
435(4)
11.4.1 Summary
435(1)
11.4.2 A Brief Overview
436(3)
12 Feedback Design for Antilinear Systems
439(32)
12.1 Generalized Eigenstructure Assignment
439(3)
12.2 Model Reference Tracking Control
442(8)
12.2.1 Tracking Conditions
443(2)
12.2.2 Solution to the Feedback Stabilizing Gain
445(1)
12.2.3 Solution to the Feedforward Compensation Gain
446(1)
12.2.4 An Example
447(3)
12.3 Finite Horizon Quadratic Regulation
450(11)
12.4 Infinite Horizon Quadratic Regulation
461(6)
12.5 Notes and References
467(4)
12.5.1 Summary
467(1)
12.5.2 A Brief Overview
468(3)
References 471(14)
Index 485
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