ref_order_reduction.bib

@techreport{antoulasApproximationLargescaleDynamical2001,
  type = {Technical Report},
  title = {Approximation of Large-Scale Dynamical Systems: {{An Overview}}},
  shorttitle = {Approximation of Large-Scale Dynamical Systems},
  author = {Antoulas, A. C. and Sorensen, D. C.},
  year = {2001},
  month = feb,
  address = {Houston, Texas},
  institution = {Rice University},
  url = {https://hdl.handle.net/1911/101964},
  urldate = {2024-05-21},
  abstract = {In this paper we review the state of affairs in the area of approximation of large-scale systems. We distinguish among three basic categories, namely the SVD-based, the Krylov-based and the SVD-Krylov-based approximation methods. The first two were developed independently of each other and have distinct sets of attributes and drawbacks. The third approach seeks to combine the best attributes of the first two.},
  langid = {english}
}

@book{antoulasApproximationLargeScaleDynamical2005,
  title = {Approximation of {{Large-Scale Dynamical Systems}}},
  author = {Antoulas, Athanasios C.},
  year = {2005},
  month = jul,
  publisher = {{Society for Industrial and Applied Mathematics}},
  address = {Philadelphia},
  abstract = {Mathematical models are used to simulate, and sometimes control, the behavior of physical and artificial processes such as the weather and very large-scale integration (VLSI) circuits. The increasing need for accuracy has led to the development of highly complex models. However, in the presence of limited computational, accuracy, and storage capabilities, model reduction (system approximation) is often necessary. Approximation of Large-Scale Dynamical Systems provides a comprehensive picture of model reduction, combining system theory with numerical linear algebra and computational considerations. It addresses the issue of model reduction and the resulting trade-offs between accuracy and complexity. Special attention is given to numerical aspects, simulation questions, and practical applications.  This book is for anyone interested in model reduction. Graduate students and researchers in the fields of system and control theory, numerical analysis, and the theory of partial differential equations/computational fluid dynamics will find it an excellent reference.  Contents List of Figures; Foreword; Preface; How to Use this Book; Part I: Introduction. Chapter 1: Introduction; Chapter 2: Motivating Examples; Part II: Preliminaries. Chapter 3: Tools from Matrix Theory; Chapter 4: Linear Dynamical Systems: Part 1; Chapter 5: Linear Dynamical Systems: Part 2; Chapter 6: Sylvester and Lyapunov equations; Part III: SVD-based Approximation Methods. Chapter 7: Balancing and balanced approximations; Chapter 8: Hankel-norm Approximation; Chapter 9: Special topics in SVD-based approximation methods; Part IV: Krylov-based Approximation Methods; Chapter 10: Eigenvalue Computations; Chapter 11: Model Reduction Using Krylov Methods; Part V: SVD--Krylov Methods and Case Studies. Chapter 12: SVD--Krylov Methods; Chapter 13: Case Studies; Chapter 14: Epilogue; Chapter 15: Problems; Bibliography; Index.},
  isbn = {978-0-89871-529-3},
  langid = {english},
  annotation = {02188}
}

@article{hammarlingNumericalSolutionStable1982,
  title = {Numerical {{Solution}} of the {{Stable}}, {{Non-negative Definite Lyapunov Equation Lyapunov Equation}}},
  author = {Hammarling, S. J.},
  year = {1982},
  month = jul,
  journal = {IMA Journal of Numerical Analysis},
  volume = {2},
  number = {3},
  pages = {303--323},
  issn = {0272-4979},
  doi = {10.1093/imanum/2.3.303},
  url = {https://academic.oup.com/imajna/article/2/3/303/763034},
  urldate = {2019-05-23},
  abstract = {Abstract.  We discuss the numerical solution of the Lyapunov equation AHX+XA=-C, C=CH and propose a variant of the Bartels-Stewart algorithm that allows the Cho},
  langid = {english}
}

@article{lallSubspaceApproachBalanced2002,
  title = {A Subspace Approach to Balanced Truncation for Model Reduction of Nonlinear Control Systems},
  author = {Lall, Sanjay and Marsden, Jerrold E. and Glava{\v s}ki, Sonja},
  year = {2002},
  journal = {International Journal of Robust and Nonlinear Control},
  volume = {12},
  number = {6},
  pages = {519--535},
  issn = {1099-1239},
  doi = {10.1002/rnc.657},
  url = {https://onlinelibrary.wiley.com/doi/abs/10.1002/rnc.657},
  urldate = {2021-09-29},
  abstract = {In this paper, we introduce a new method of model reduction for nonlinear control systems. Our approach is to construct an approximately balanced realization. The method requires only standard matrix computations, and we show that when it is applied to linear systems it results in the usual balanced truncation. For nonlinear systems, the method makes use of data from either simulation or experiment to identify the dynamics relevant to the input--output map of the system. An important feature of this approach is that the resulting reduced-order model is nonlinear, and has inputs and outputs suitable for control. We perform an example reduction for a nonlinear mechanical system. Copyright {\copyright} 2002 John Wiley \& Sons, Ltd.},
  langid = {english}
}

@inproceedings{laubComputationSystemBalancing1986,
  title = {Computation of System Balancing Transformations},
  booktitle = {1986 25th {{IEEE Conference}} on {{Decision}} and {{Control}}},
  author = {Laub, A. J. and Heath, M. T. and Paige, C. C. and Ward, R. C.},
  year = {1986},
  month = dec,
  pages = {548--553},
  doi = {10.1109/CDC.1986.267343},
  abstract = {An algorithm is presented in this paper for computing state space balancing transformations directly from a state space realization. The algorithm requires no "squaring up." Various algorithmic aspects are discussed in detail. Applications to numerous other closely-related problems are also mentioned. The key idea throughout involves determining a contragredient transformation through computing the singular value decomposition of a certain product of matrices without explicitly forming the product.}
}

@article{laubComputationSystemBalancing1987,
  title = {Computation of System Balancing Transformations and Other Applications of Simultaneous Diagonalization Algorithms},
  author = {Laub, A. and Heath, M. and Paige, C. and Ward, R.},
  year = {1987},
  month = feb,
  journal = {IEEE Transactions on Automatic Control},
  volume = {32},
  number = {2},
  pages = {115--122},
  issn = {0018-9286},
  doi = {10.1109/TAC.1987.1104549},
  abstract = {An algorithm is presented in this paper for computing state-space balancing transformations directly from a state-space realization. The algorithm requires no "squaring up" or unnecessary matrix products. Various algorithmic aspects are discussed in detail. A key feature of the algorithm is the determination of a contragredient transformation through computing the singular value decomposition of a certain product of matrices without explicitly forming the product. Other contragredient transformation applications are also described. It is further shown that a similar approach may be taken, involving the generalized singular value decomposition, to the classical simultaneous diagonalization problem. These SVD-based simultaneous diagonalization algorithms provide a computational alternative to existing methods for solving certain classes of symmetric positive definite generalized eigenvalue problems.}
}

@article{moorePrincipalComponentAnalysis1981,
  title = {Principal Component Analysis in Linear Systems: {{Controllability}}, Observability, and Model Reduction},
  shorttitle = {Principal Component Analysis in Linear Systems},
  author = {Moore, B.},
  year = {1981},
  month = feb,
  journal = {IEEE Transactions on Automatic Control},
  volume = {26},
  number = {1},
  pages = {17--32},
  issn = {0018-9286},
  doi = {10.1109/TAC.1981.1102568},
  abstract = {Kalman's minimal realization theory involves geometric objects (controllable, unobservable subspaces) which are subject to structural instability. Specifically, arbitrarily small perturbations in a model may cause a change in the dimensions of the associated subspaces. This situation is manifested in computational difficulties which arise in attempts to apply textbook algorithms for computing a minimal realization. Structural instability associated with geometric theories is not unique to control; it arises in the theory of linear equations as well. In this setting, the computational problems have been studied for decades and excellent tools have been developed for coping with the situation. One of the main goals of this paper is to call attention to principal component analysis (Hotelling, 1933), and an algorithm (Golub and Reinsch, 1970) for computing the singular value decompositon of a matrix. Together they form a powerful tool for coping with structural instability in dynamic systems. As developed in this paper, principal component analysis is a technique for analyzing signals. (Singular value decomposition provides the computational machinery.) For this reason, Kalman's minimal realization theory is recast in terms of responses to injected signals. Application of the signal analysis to controllability and observability leads to a coordinate system in which the "internally balanced" model has special properties. For asymptotically stable systems, this yields working approximations ofX\_c, X\_{\textbackslash}baro, the controllable and unobservable subspaces. It is proposed that a natural first step in model reduction is to apply the mechanics of minimal realization using these working subspaces.}
}

@book{obinataModelReductionControl2000,
  title = {Model {{Reduction}} for {{Control System Design}}},
  author = {Obinata, Goro and Anderson, Brian D. O.},
  year = {2000},
  month = dec,
  publisher = {Springer},
  address = {New York},
  abstract = {Comprehensive treatment of approximation methods for filters and controllers. It is fully up to date, and it is authored by two leading researchers who have personally contributed to the development of some of the methods. Balanced truncation, Hankel norm reduction, multiplicative reduction, weighted methods and coprime factorization methods are all discussed. The book is amply illustrated with examples, and will equip practising control engineers and graduates for intelligent use of commercial software modules for model and controller reduction.},
  isbn = {978-1-85233-371-3},
  langid = {english},
  annotation = {00486}
}

@techreport{penzlLypackMatlabToolbox1999,
  title = {Lypack - a {{Matlab}} Toolbox for {{Large Lyapunov}} and {{Riccati}} Equations, Model Reduction Problems, and Linear-Quadratic Optimal Control Problems},
  author = {Penzl, Thilo},
  year = {1999},
  url = {https://www.tu-chemnitz.de/sfb393/lyapack/}
}

@article{rowleyModelReductionFluids2005,
  title = {Model Reduction for Fluids, Using Balanced Proper Orthogonal Decomposition},
  author = {Rowley, C. W.},
  year = {2005},
  month = mar,
  journal = {International Journal of Bifurcation and Chaos},
  volume = {15},
  number = {03},
  pages = {997--1013},
  publisher = {World Scientific Publishing Co.},
  issn = {0218-1274},
  doi = {10.1142/S0218127405012429},
  url = {https://www.worldscientific.com/doi/abs/10.1142/S0218127405012429},
  urldate = {2021-09-03},
  abstract = {Many of the tools of dynamical systems and control theory have gone largely unused for fluids, because the governing equations are so dynamically complex, both high-dimensional and nonlinear. Model reduction involves finding low-dimensional models that approximate the full high-dimensional dynamics. This paper compares three different methods of model reduction: proper orthogonal decomposition (POD), balanced truncation, and a method called balanced POD. Balanced truncation produces better reduced-order models than POD, but is not computationally tractable for very large systems. Balanced POD is a tractable method for computing approximate balanced truncations, that has computational cost similar to that of POD. The method presented here is a variation of existing methods using empirical Gramians, and the main contributions of the present paper are a version of the method of snapshots that allows one to compute balancing transformations directly, without separate reduction of the Gramians; and an output projection method, which allows tractable computation even when the number of outputs is large. The output projection method requires minimal additional computation, and has a priori error bounds that can guide the choice of rank of the projection. Connections between POD and balanced truncation are also illuminated: in particular, balanced truncation may be viewed as POD of a particular dataset, using the observability Gramian as an inner product. The three methods are illustrated on a numerical example, the linearized flow in a plane channel.}
}

@article{safonovSchurMethodBalancedtruncation1989,
  title = {A {{Schur}} Method for Balanced-Truncation Model Reduction},
  author = {Safonov, M. G. and Chiang, R. Y.},
  year = {1989},
  month = jul,
  journal = {IEEE Transactions on Automatic Control},
  volume = {34},
  number = {7},
  pages = {729--733},
  issn = {0018-9286},
  doi = {10.1109/9.29399},
  abstract = {It is shown that a not-necessarily-balanced state-space realization of the Moore reduced model can be computed directly without balancing via projections defined in terms of arbitrary bases for the left and right eigenspaces associated with the large eigenvalues of the product PQ of the reachability and controllability Grammians. Two specific methods for computing these bases are proposed, one based on the ordered Schur decomposition of PQ and the other based on the Cholesky factors of P and Q. The algorithms perform reliably even for nonminimal models.{$<>$}}
}

@misc{sorensenDirectMethodsMatrix2003,
  type = {Research Article},
  title = {Direct Methods for Matrix {{Sylvester}} and {{Lyapunov}} Equations},
  author = {Sorensen, Danny C. and Zhou, Yunkai},
  year = {2003},
  journal = {Journal of Applied Mathematics},
  doi = {10.1155/S1110757X03212055},
  url = {https://www.hindawi.com/journals/jam/2003/245057/abs/},
  urldate = {2019-05-27},
  abstract = {We revisit the two standard dense methods for matrix Sylvester and Lyapunov equations: the Bartels-Stewart method for A1X},
  langid = {english}
}

@book{tanAdvancedModelOrder2007,
  title = {Advanced {{Model Order Reduction Techniques}} in {{VLSI Design}}},
  author = {Tan, Sheldon and He, Lei},
  year = {2007},
  month = jun,
  publisher = {Cambridge University Press},
  address = {Cambridge},
  abstract = {Model order reduction (MOR) techniques reduce the complexity of VLSI designs, paving the way to higher operating speeds and smaller feature sizes. This book presents a systematic introduction to, and treatment of, the key MOR methods employed in general linear circuits, using real-world examples to illustrate the advantages and disadvantages of each algorithm. Following a review of traditional projection-based techniques, coverage progresses to more advanced MOR methods for VLSI design, including HMOR, passive truncated balanced realization (TBR) methods, efficient inductance modeling via the VPEC model, and structure-preserving MOR techniques. Where possible, numerical methods are approached from the CAD engineer's perspective, avoiding complex mathematics and allowing the reader to take on real design problems and develop more effective tools. With practical examples and over 100 illustrations, this book is suitable for researchers and graduate students of electrical and computer engineering, as well as practitioners working in the VLSI design industry.},
  isbn = {978-0-521-86581-4},
  langid = {english},
  annotation = {00126}
}

@article{willcoxBalancedModelReduction2002,
  title = {Balanced {{Model Reduction}} via the {{Proper Orthogonal Decomposition}}},
  author = {Willcox, K. and Peraire, J.},
  year = {2002},
  journal = {AIAA Journal},
  volume = {40},
  number = {11},
  pages = {2323--2330},
  publisher = {{American Institute of Aeronautics and Astronautics}},
  issn = {0001-1452},
  doi = {10.2514/2.1570},
  url = {https://doi.org/10.2514/2.1570},
  urldate = {2021-09-03}
}