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E-raamat: Computational Methods for Approximation of Large-Scale Dynamical Systems

  • Formaat: 336 pages
  • Ilmumisaeg: 30-Apr-2019
  • Kirjastus: CRC Press Inc
  • Keel: eng
  • ISBN-13: 9781351028608
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Computational Methods for Approximation of Large-Scale Dynamical SystemsSuurem pilt
  • Formaat: 336 pages
  • Ilmumisaeg: 30-Apr-2019
  • Kirjastus: CRC Press Inc
  • Keel: eng
  • ISBN-13: 9781351028608
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These days, computer-based simulation is considered the quintessential approach to exploring new ideas in the different disciplines of science, engineering and technology (SET). To perform simulations, a physical system needs to be modeled using mathematics; these models are often represented by linear time-invariant (LTI) continuous-time (CT) systems. Oftentimes these systems are subject to additional algebraic constraints, leading to first- or second-order differential-algebraic equations (DAEs), otherwise known as descriptor systems. Such large-scale systems generally lead to massive memory requirements and enormous computational complexity, thus restricting frequent simulations, which are required by many applications. To resolve these complexities, the higher-dimensional system may be approximated by a substantially lower-dimensional one through model order reduction (MOR) techniques. Computational Methods for Approximation of Large-Scale Dynamical Systems discusses computational techniques for the MOR of large-scale sparse LTI CT systems. Although the book puts emphasis on the MOR of descriptor systems, it begins by showing and comparing the various MOR techniques for standard systems.

The book also discusses the low-rank alternating direction implicit (LR-ADI) iteration and the issues related to solving the Lyapunov equation of large-scale sparse LTI systems to compute the low-rank Gramian factors, which are important components for implementing the Gramian-based MOR.

Although this book is primarly aimed at post-graduate students and researchers of the various SET disciplines, the basic contents of this book can be supplemental to the advanced bachelor's-level students as well. It can also serve as an invaluable reference to researchers working in academics and industries alike.

Features:





Provides an up-to-date, step-by-step guide for its readers. Each chapter develops theories and provides necessary algorithms, worked examples, numerical experiments and related exercises. With the combination of this book and its supplementary materials, the reader gains a sound understanding of the topic. The MATLAB® codes for some selected algorithms are provided in the book. The solutions to the exercise problems, experiment data sets and a digital copy of the software are provided on the book's website; The numerical experiments use real-world data sets obtained from industries and research institutes.
About the Book v
Acknowledgments xiii
Preface xv
Author xxi
List of Acronyms and Symbols xxiii
I Preliminaries 1(120)
1 Review of Linear Algebra
3(22)
1.1 Introduction
3(1)
1.2 Matrices
4(2)
1.3 Vector space and subspace
6(2)
1.4 Orthogonalization and Gram-Schmidt process
8(1)
1.5 Krylov subspace and Arnoldi process
9(2)
1.6 Eigenvalue problem
11(3)
1.7 Matrix factorizations
14(2)
1.7.1 Eigen decomposition
14(1)
1.7.2 Singular value decomposition
15(1)
1.7.3 LU decomposition
15(1)
1.7.4 Cholesky decomposition
16(1)
1.7.5 QR decomposition
16(1)
1.7.6 Schur decomposition
16(1)
1.8 Vector norms and matrix norms
16(1)
1.9 Some important definitions and theorems
17(5)
1.10 Some useful MATLAB functions
22(3)
2 Dynamic Systems and Control Theory
25(22)
2.1 Introduction
25(1)
2.2 A brief introduction of dynamical control systems
26(1)
2.3 Representations of LTI dynamical systems
27(2)
2.3.1 Generalized state-space representation
27(1)
2.3.2 Transfer function representation
28(1)
2.4 System responses
29(4)
2.4.1 Time response
29(2)
2.4.2 Frequency response
31(2)
2.5 System Gramians
33(3)
2.5.1 Controllability Gramian
33(1)
2.5.2 Observability Gramian
34(1)
2.5.3 Physical interpretation of the Gramians
35(1)
2.6 Controllability and observability
36(1)
2.7 Stability
37(2)
2.8 System Hankel singular values
39(1)
2.9 Realizations
40(1)
2.10 The 1-12 norm and 1-1 norm
40(4)
2.10.1 The 142 norm
41(1)
2.10.2 The Woo norm
42(2)
2.11 Some useful MATLAB functions
44(3)
3 Iterative Solution of Lyapunov Equations
47(22)
3.1 Introduction
47(2)
3.2 A brief history of alternating direction implicit method
49(1)
3.3 The ADI iteration for solving Lyapunov matrix-equations
49(1)
3.4 Low-rank factor of the Lyapunov solutions
50(3)
3.5 Low-rank (LR-)ADI iteration
53(8)
3.5.1 Low-rank factors of the Gramian using ADI iteration
53(1)
3.5.2 Derivation of LR-ADI iteration
53(2)
3.5.3 Efficient handling of complex shift parameters
55(1)
3.5.4 Low-rank Lyapunov residual factor based stopping technique
56(3)
3.5.5 Reformulation of LR-ADI iteration using the low-rank factor based stopping criterion
59(1)
3.5.6 LR-ADI for generalized system
59(2)
3.6 ADI shift parameter selection
61(3)
3.7 Some useful MATLAB functions
64(1)
3.8 Numerical experiments
64(5)
4 Model Reduction of Generalized State Space Systems
69(24)
4.1 Introduction
69(1)
4.2 Goal of model order reduction
70(1)
4.3 Model order reduction methods
71(2)
4.4 Gramian-based model reduction
73(6)
4.4.1 Balancing criterion
73(3)
4.4.2 Truncation of balanced system
76(1)
4.4.3 Balancing and truncating transformations
77(1)
4.4.4 Balanced truncation by low-rank Gramian factors
78(1)
4.5 Rational Krylov subspace-based model reduction
79(5)
4.5.1 Interpolatory projections for SISO systems
80(2)
4.5.2 Interpolatory projections for MIMO systems
82(2)
4.6 Some useful MATLAB functions
84(1)
4.7 Numerical experiments
85(8)
5 Model Reduction of Second-Order Systems
93(28)
5.1 Introduction
93(1)
5.2 Preliminaries
94(6)
5.2.1 Equivalent first-order representations
95(2)
5.2.2 Transfer function of second-order systems
97(1)
5.2.3 Gramians of the second-order system
98(2)
5.3 Second-order-to-first-order reduction
100(3)
5.3.1 Balancing-based algorithm
100(1)
5.3.2 Interpolatory projection via IRKA
101(2)
5.4 Second-order-to-second-order reduction
103(5)
5.4.1 Balancing-based methods
104(2)
5.4.2 Projection onto dominant eigenspaces of the Gramian
106(2)
5.5 LR-ADI iteration for solving second-order Lyapunov equation
108(4)
5.5.1 Solution of second-order controllability Lyapunov equation
108(2)
5.5.2 Solution of second-order observability Lyapunov equation
110(2)
5.6 MOR of symmetric second-order systems
112(1)
5.7 Some useful MATLAB functions
113(1)
5.8 Numerical results
113(8)
II Model Reduction Of Descriptor Systems 121(140)
6 Introduction to Descriptor Systems
123(6)
6.1 Introduction
123(1)
6.2 Solvability
124(1)
6.3 Transfer function
125(1)
6.4 Stability
126(1)
6.5 Structured DAE system
126(1)
6.6 Some useful MATLAB functions
127(2)
7 Model Reduction of First-Order Index 1 Descriptor Systems
129(16)
7.1 Introduction
129(1)
7.2 Reformulation of dynamical system
130(2)
7.3 Balancing-based MOR
132(2)
7.4 Solution of the Lyapunov equations by LR-ADI iteration
134(1)
7.5 Tangential interpolation via IRKA
135(2)
7.6 Some useful MATLAB functions
137(1)
7.7 Numerical results
138(7)
8 Model Reduction of First-Order Index 2 Descriptor Systems
145(18)
8.1 Introduction
145(1)
8.2 Reformulation of dynamical system
146(3)
8.3 Balancing-based MOR and low-rank ADI iteration
149(2)
8.4 Solution of the projected Lyapunov equations by LR-ADI iteration and related issues
151(5)
8.4.1 LR-ADI for index 2 systems
152(2)
8.4.2 ADI shift parameters selection
154(2)
8.5 Interpolatory projection method via IRKA
156(1)
8.6 Numerical results
157(6)
9 Model Reduction of First-Order Index 2 Unstable Descriptor Systems
163(18)
9.1 Introduction
163(1)
9.2 BT for unstable systems
164(2)
9.3 BT for index 2 unstable descriptor systems
166(2)
9.4 Solution of the projected Lyapunov equations
168(3)
9.5 Riccati-based feedback stabilization from ROM
171(1)
9.6 Numerical results
172(9)
10 Model Reduction of First-Order Index 3 Descriptor Systems
181(20)
10.1 Introduction
181(1)
10.2 Equivalent reformulation of the dynamical system
182(3)
10.2.1 Projector for index 3 system
183(1)
10.2.2 Formulation of projected system
184(1)
10.3 Model reduction with the balanced truncation avoiding the formulation of projected system
185(2)
10.4 Solution of projected Lyapunov equations
187(6)
10.4.1 Initial residual factor
188(1)
10.4.2 Solutions of linear systems and update of residual factors
188(3)
10.4.3 Computation of ADI shift parameters
191(2)
10.5 Interpolatory method via IRKA
193(2)
10.6 Numerical results
195(6)
11 Model Reduction of Second-Order Index 1 Descriptor Systems
201(30)
11.1 Introduction
201(1)
11.2 Second-order-to-first-order reduction techniques
202(6)
11.2.1 Balancing-based method
204(1)
11.2.2 Interpolatory projections via IRKA
205(3)
11.3 Second-order-to-second-order MOR techniques
208(6)
11.3.1 Conversion into equivalent form of ODE system
209(1)
11.3.2 Balancing-based method
210(2)
11.3.3 PDEG-based method
212(2)
11.4 Solution of Lyapunov equations using LR-ADI iteration
214(5)
11.4.1 Computation of low-rank controllability and observability Gramian factors
214(4)
11.4.2 ADI shift parameter selection
218(1)
11.5 Symmetric second-order index 1 system
219(1)
11.6 Numerical results
219(12)
11.6.1 Second-order-to-first-order reduction
220(2)
11.6.2 Second-order-to-second-order reduction
222(9)
12 Model Reduction of Second-Order Index 3 Descriptor Systems
231(30)
12.1 Introduction
231(2)
12.2 Reformulation of the dynamical systems
233(2)
12.3 Equivalent finite spectra
235(2)
12.4 Second-order-to-first-order reduction
237(6)
12.4.1 Balancing-based technique
238(2)
12.4.2 Interpolatory method via IRKA
240(3)
12.5 Second-order-to-second-order reduction
243(3)
12.5.1 The BT method
243(2)
12.5.2 The PDEG method
245(1)
12.6 Solution of the projected Lyapunov equations
246(4)
12.7 Numerical results
250(13)
12.7.1 LR-ADI iteration
250(1)
12.7.2 Second-order-to-first-order reduction
251(2)
12.7.3 Second-order-to-second-order reduction
253(8)
III Appendices 261(28)
Appendix A: Data of Benchmark Model Examples
263(12)
A.1 Introduction
263(1)
A.2 First-order LTI continuous-time systems
264(1)
A.2.1 CD player
264(1)
A.2.2 FOM
264(1)
A.3 Second-order LTI continuous-time systems
265(2)
A.3.1 International Space Station
265(1)
A.3.2 Clamped beam model
265(1)
A.3.3 Triple chain oscillator model
266(1)
A.3.4 Butterfly Gyro
266(1)
A.4 First-order LTI continuous-time descriptor systems
267(5)
A.4.1 Power system model
267(1)
A.4.2 Supersonic engine inlet
268(1)
A.4.3 Semi-discretized linearized Navier-Stokes model
268(1)
A.4.4 Semi-discretized linearized Stokes model
269(1)
A.4.5 Constrained damped mass-spring system
270(2)
A.5 Second-order LTI continuous-time descriptor systems
272(3)
A.5.1 Piezo-actuator based adaptive spindle support
272(1)
A.5.2 Constrained damped mass-spring (second-order) system
273(1)
A.5.3 Constrained triple chain oscillator model
273(2)
Appendix B: MATLAB Codes
275(14)
B.1 Algorithm 1
275(1)
B.2 Algorithm 2
275(1)
B.3 Algorithm 3
276(1)
B.4 Algorithm 6
277(1)
B.5 Algorithm 7
277(1)
B.6 Algorithm 8
278(1)
B.7 Algorithm 9
278(1)
B.8 Algorithm 10
278(1)
B.9 Algorithm 15
278(3)
B.10 Algorithm 16
281(3)
B.11 Algorithm 19
284(3)
B.12 Algorithm 21
287(2)
Bibliography 289(16)
Index 305
Dr Mohammad Monir Uddin is an Assistant Professor in the Department of Mathematics and Physics at the North South University, Bangladesh. His research interests are Model Order Reduction,Systems and Control Theory, Iterative Methods for Large Sparse Matrix Equations, Numerical Linear Algebra, Optimization and Scientific Computing.
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