Numerical Methods for Large-Scale Linear Time-Varying Control Systems and related Differential Matrix Equations
Norman Lang
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Norman Lang, Numerical Methods for Large-Scale Linear Time-Varying Control Systems and related Differential Matrix Equations (2018), Logos Verlag, Berlin, ISBN: 9783832590864
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Beschreibung / Abstract
This thesis is concerned with the linear-quadratic optimal control and model order reduction (MOR) of large-scale linear time-varying (LTV) control systems. In the first two parts, particular attention is paid to a tracking-type finite-time optimal control problem with application to an inverse heat conduction problem and the balanced truncation (BT) MOR method for LTV systems. In both fields of application the efficient solution of differential matrix equations (DMEs) is of major importance.
The third and largest part deals with the application of implicit time integration methods to these matrix-valued ordinary differential equations. In this context, in particular, the rather new class of peer methods is introduced. Further, for the efficient solution of large-scale DMEs, in practice low-rank solution strategies are inevitable. Here, low-rank time integrators, based on a symmetric indefinte factored representation of the right hand sides and the solution approximations of the DMEs, are presented. In contrast to the classical low-rank Cholesky-type factorization, this avoids complex arithmetic and tricky implementations and algorithms. Both low-rank approaches are compared for numerous implicit time integration methods.
The third and largest part deals with the application of implicit time integration methods to these matrix-valued ordinary differential equations. In this context, in particular, the rather new class of peer methods is introduced. Further, for the efficient solution of large-scale DMEs, in practice low-rank solution strategies are inevitable. Here, low-rank time integrators, based on a symmetric indefinte factored representation of the right hand sides and the solution approximations of the DMEs, are presented. In contrast to the classical low-rank Cholesky-type factorization, this avoids complex arithmetic and tricky implementations and algorithms. Both low-rank approaches are compared for numerous implicit time integration methods.
Inhaltsverzeichnis
- BEGINN
- 1 Introduction
- 1.1 Motivation
- 1.2 Outline of the Thesis
- 1.3 Related own publications
- 1.4 Used Software and Hardware
- 2 Mathematical Basics
- 2.1 Linear Systems and Optimal Control
- 2.2 Model Order Reduction of Linear Systems
- 3 Optimal Control and Inverse Problems
- 3.1 Inverse Problems
- 3.2 Finite-Time Tracking-Type Optimal Control
- 3.3 Numerical Experiments
- 3.4 Summary and Conclusion
- 4 Model Order Reduction of Linear Time-Varying Systems
- 4.1 Model Order Reduction using LTI Model Approximations
- 4.2 Balanced Truncation for Linear Time-Varying Systems
- 4.3 Numerical Experiments
- 4.4 Summary and Conclusion
- 5 Time Integration Methods for Differential Matrix Equations
- 5.1 Backward Differentiation Formulas
- 5.2 Rosenbrock Methods
- 5.3 Other Implicit Methods
- 5.4 Peer Methods
- 6 Efficient Solution of Large-Scale Differential Matrix Equations
- 6.1 Classical Low-Rank Factorization
- 6.2 Symmetric Indefinite Low-Rank Factorization
- 6.3 Column Compression
- 6.4 Numerical Experiments
- 6.5 Summary and Conclusions
- 7 Conclusions and Outlook
- 7.1 Summary and Conclusions
- 7.2 Future Research Perspectives
- A Fourth-Order Rosenbrock Method
- Theses
- Bibliography