Computational Methods for Approximation of Large Scale Dynamical Systems 1st Edition by Mohammad Monir Uddin 9781351028608 135102860X
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ISBN 10:135102860X
ISBN 13:9781351028608
Author:Mohammad Monir Uddin
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.
Computational Methods for Approximation of Large Scale Dynamical Systems 1st Table of contents:
Part I: PRELIMINARIES
1. Review of Linear Algebra
1.1 Introduction
1.2 Matrices
1.3 Vector space and subspace
1.4 Orthogonalization and Gram-Schmidt process
1.5 Krylov subspace and Arnoldi process
1.6 Eigenvalue problem
1.7 Matrix factorizations
1.7.1 Eigen decomposition
1.7.2 Singular value decomposition
1.7.3 LU decomposition
1.7.4 Cholesky decomposition
1.7.5 QR decomposition
1.7.6 Schur decomposition
1.8 Vector norms and matrix norms
1.9 Some important definitions and theorems
1.10 Some useful MATLAB functions
2. Dynamic Systems and Control Theory
2.1 Introduction
2.2 A brief introduction of dynamical control systems
2.3 Representations of LTI dynamical systems
2.3.1 Generalized state-space representation
2.3.2 Transfer function representation
2.4 System responses
2.4.1 Time response
2.4.2 Frequency response
2.5 System Gramians
2.5.1 Controllability Gramian
2.5.2 Observability Gramian
2.5.3 Physical interpretation of the Gramians
2.6 Controllability and observability
2.7 Stability
2.8 System Hankel singular values
2.9 Realizations
2.10 The H2 norm and H1 norm
2.10.1 The H2 norm
2.10.2 The H1 norm
2.11 Some useful MATLAB functions
3. Iterative Solution of Lyapunov Equations
3.1 Introduction
3.2 A brief history of alternating direction implicit method
3.3 The ADI iteration for solving Lyapunov matrix-equations
3.4 Low-rank factor of the Lyapunov solutions
3.5 Low-rank (LR-)ADI iteration
3.5.1 Low-rank factors of the Gramian using ADI iteration
3.5.2 Derivation of LR-ADI iteration
3.5.3 Efficient handling of complex shift parameters
3.5.4 Low-rank Lyapunov residual factor based stopping technique
3.5.5 Reformulation of LR-ADI iteration using the low-rank factor based stopping criterion
3.5.6 LR-ADI for generalized system
3.6 ADI shift parameter selection
3.7 Some useful MATLAB functions
3.8 Numerical experiments
4. Model Reduction of Generalized State Space Systems
4.1 Introduction
4.2 Goal of model order reduction
4.3 Model order reduction methods
4.4 Gramian-based model reduction
4.4.1 Balancing criterion
4.4.2 Truncation of balanced system
4.4.3 Balancing and truncating transformations
4.4.4 Balanced truncation by low-rank Gramian factors
4.5 Rational Krylov subspace-based model reduction
4.5.1 Interpolatory projections for SISO systems
4.5.2 Interpolatory projections for MIMO systems
4.6 Some useful MATLAB functions
4.7 Numerical experiments
5. Model Reduction of Second-Order Systems
5.1 Introduction
5.2 Preliminaries
5.2.1 Equivalent first-order representations
5.2.2 Transfer function of second-order systems
5.2.3 Gramians of the second-order system
5.3 Second-order-to-first-order reduction
5.3.1 Balancing-based algorithm
5.3.2 Interpolatory projection via IRKA
5.4 Second-order-to-second-order reduction
5.4.1 Balancing-based methods
5.4.2 Projection onto dominant eigenspaces of the Gramian
5.5 LR-ADI iteration for solving second-order Lyapunov equation
5.5.1 Solution of second-order controllability Lyapunov equation
5.5.2 Solution of second-order observability Lyapunov equation
5.6 MOR of symmetric second-order systems
5.7 Some useful MATLAB functions
5.8 Numerical results
Part II: MODEL REDUCTION OF DESCRIPTOR SYSTEMS
6. Introduction to Descriptor Systems
6.1 Introduction
6.2 Solvability
6.3 Transfer function
6.4 Stability
6.5 Structured DAE system
6.6 Some useful MATLAB functions
7. Model Reduction of First-Order Index 1 Descriptor Systems
7.1 Introduction
7.2 Reformulation of dynamical system
7.3 Balancing-based MOR
7.4 Solution of the Lyapunov equations by LR-ADI iteration
7.5 Tangential interpolation via IRKA
7.6 Some useful MATLAB functions
7.7 Numerical results
8. Model Reduction of First-Order Index 2 Descriptor Systems
8.1 Introduction
8.2 Reformulation of dynamical system
8.3 Balancing-based MOR and low-rank ADI iteration
8.4 Solution of the projected Lyapunov equations by LR-ADI iteration and related issues
8.4.1 LR-ADI for index 2 systems
8.4.2 ADI shift parameters selection
8.5 Interpolatory projection method via IRKA
8.6 Numerical results
9. Model Reduction of First-Order Index 2 Unstable Descriptor Systems
9.1 Introduction
9.2 BT for unstable systems
9.3 BT for index 2 unstable descriptor systems
9.4 Solution of the projected Lyapunov equations
9.5 Riccati-based feedback stabilization from ROM
9.6 Numerical results
10. Model Reduction of First-Order Index 3 Descriptor Systems
10.1 Introduction
10.2 Equivalent reformulation of the dynamical system
10.2.1 Projector for index 3 system
10.2.2 Formulation of projected system
10.3 Model reduction with the balanced truncation avoiding the formulation of projected system
10.4 Solution of projected Lyapunov equations
10.4.1 Initial residual factor
10.4.2 Solutions of linear systems and update of residual factors
10.4.3 Computation of ADI shift parameters
10.5 Interpolatory method via IRKA
10.6 Numerical results
11. Model Reduction of Second-Order Index 1 Descriptor Systems
11.1 Introduction
11.2 Second-order-to-first-order reduction techniques
11.2.1 Balancing-based method
11.2.2 Interpolatory projections via IRKA
11.3 Second-order-to-second-order MOR techniques
11.3.1 Conversion into equivalent form of ODE system
11.3.2 Balancing-based method
11.3.3 PDEG-based method
11.4 Solution of Lyapunov equations using LR-ADI iteration
11.4.1 Computation of low-rank controllability and observability Gramian factors
11.4.2 ADI shift parameter selection
11.5 Symmetric second-order index 1 system
11.6 Numerical results
11.6.1 Second-order-to-first-order reduction
11.6.2 Second-order-to-second-order reduction
12. Model Reduction of Second-Order Index 3 Descriptor Systems
12.1 Introduction
12.2 Reformulation of the dynamical systems
12.3 Equivalent finite spectra
12.4 Second-order-to-first-order reduction
12.4.1 Balancing-based technique
12.4.2 Interpolatory method via IRKA
12.5 Second-order-to-second-order reduction
12.5.1 The BT method
12.5.2 The PDEG method
12.6 Solution of the projected Lyapunov equations
12.7 Numerical results
12.7.1 LR-ADI iteration
12.7.2 Second-order-to-first-order reduction
12.7.3 Second-order-to-second-order reduction
Part III: APPENDICES
Appendix A: Data of Benchmark Model Examples
A.1 Introduction
A.2 First-order LTI continuous-time systems
A.2.1 CD player
A.2.2 FOM
A.3 Second-order LTI continuous-time systems
A.3.1 International Space Station
A.3.2 Clamped beam model
A.3.3 Triple chain oscillator model
A.3.4 Butterfly Gyro
A.4 First-order LTI continuous-time descriptor systems
A.4.1 Power system model
A.4.2 Supersonic engine inlet
A.4.3 Semi-discretized linearized Navier-Stokes model
A.4.4 Semi-discretized linearized Stokes model
A.4.5 Constrained damped mass-spring system
A.5 Second-order LTI continuous-time descriptor systems
A.5.1 Piezo-actuator based adaptive spindle support
A.5.2 Constrained damped mass-spring (second-order) system
A.5.3 Constrained triple chain oscillator model
Appendix B: MATLAB Codes
B.1 Algorithm 1
B.2 Algorithm 2
B.3 Algorithm 3
B.4 Algorithm 6
B.5 Algorithm 7
B.6 Algorithm 8
B.7 Algorithm 9
B.8 Algorithm 10
B.9 Algorithm 15
B.10 Algorithm 16
B.11 Algorithm 19
B.12 Algorithm 21
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