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Towards Riccati-Feedback control of Complex Flows with Moving Interfaces

PAMM 2018年第1期17卷

作者： Björn Baran Peter Benner Jan Heiland Jens Saak Computational Methods in Systems and Control Theory Max Planck Institute for Dynamics of Complex Technical Systems Sandtorstr. 1 39106 Magdeburg

Problems featuring moving interfaces appear in many applications. They can model solidification and melting of pure materials, crystal growth and other multi-phase problems. The control of the moving interface enables to, for example, influence production processes and, thus, the product material quality. We consider the two-phase Stefan problem that models a solid and a liquid phase separated by the moving interface. In the liquid phase, the heat distribution is characterized by a convection-diffusion equation. The fluid flow in the liquid phase is described by the Navier–Stokes equations which introduces a differential algebraic structure to the system. The interface movement is coupled with the temperature through the Stefan condition, which adds additional algebraic constraints. Our formulation uses a sharp interface representation and we define a quadratic tracking-type cost functional as a target of a control input. We compute an open loop optimal control for the Stefan problem using an adjoint system. For a feedback representation, we linearize the system about the trajectory defined by the open loop control. This results in a linear-quadratic regulator problem, for which we formulate the differential Riccati equation with time varying coefficients. This Riccati equation defines the corresponding feedback gain. Further, we present the feedback formulation that takes into account the structure and the differential algebraic components of the problem. Also, we discuss how the complexities that come, for example, with mesh movements, can be handled in a feedback setting. (© 2017 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)

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GR decompositions and their relations to Cholesky-like factorizations

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PAMM 2021年第1期20卷

作者： Peter Benner Carolin Penke Max Planck Institute for Dynamics of Complex Technical Systems: Computational Methods in Systems and Control Theory Sandtorstr. 1 39106 Magdeburg

For a given matrix, we are interested in computing GR decompositions A = GR , where G is an isometry with respect to given scalar products. The orthogonal QR decomposition is the representative for the Euclidian scalar product. For a signature matrix, a respective factorization is given as the hyperbolic QR decomposition. Considering a skew-symmetric matrix leads to the symplectic QR decomposition. The standard approach for computing GR decompositions is based on the successive elimination of subdiagonal matrix entries. For the hyperbolic and symplectic case, this approach does in general not lead to a satisfying numerical accuracy. An alternative approach computes the QR decomposition via a Cholesky factorization, but also has bad stability. It is improved by repeating the procedure a second time. In the same way, the hyperbolic and the symplectic QR decomposition are related to the LDL T and a skew-symmetric Cholesky-like factorization. We show that methods exploiting this connection can provide better numerical stability than elimination-based approaches.

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Numerical solution to low rank perturbed Lyapunov equations by the sign function method

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PAMM 2016年第1期16卷

作者： Peter Benner Jonas Denißen Max Planck Institute for Dynamics of Complex Technical Systems Computational Methods in Systems and Control Theory Sandtorstr. 1 39106 Magdeburg Germany

We investigate the numerical solution to a low rank perturbed Lyapunov equation A T X + XA = W via the sign function method (SFM). The sign function method has been proposed to solve Lyapunov equations, see e.g. [1], but here we focus on a framework where the matrix A has a special structure, i. e. A = B + UCV T , where B is a blockdiagonal matrix and UCV T is a low rank perturbation. We show that this structure can be kept throughout the sign function iteration but the rank of the perturbation doubles per iteration. Therefore, we apply a low rank approximation to the perturbation in order to keep its numerical rank small. We compare the standard SFM with its structure preserving variant presented in this paper by means of numerical examples from viscously damped mechanical systems. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)

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Spectral Bounds on the Solution of Linear Time-Periodic systems

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PAMM 2014年第1期14卷

作者： Peter Benner Jonas Denißen Max Planck Institute for Dynamics of Complex Technical Systems Computational Methods in Systems and Control Theory Sandtorstr. 1 39106 Magdeburg Germany

Linear time-periodic systems arise whenever a nonlinear system is linearized about a periodic trajectory. Stability of the solution may be proven by rigorous bounds on the solution. The key idea of this paper is to derive Chebyshev projection bounds on the original system by solving an approximated system. Depending on the smoothness of the original function, we formulate two upper bounds. The theoretical results are illustrated and compared to trigonometric spline bounds by means of two examples which include an anisotropic rotor-bearing system. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)

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Solving Differential Matrix Equations using Parareal

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PAMM 2016年第1期16卷

作者： Martin Köhler Norman Lang Jens Saak Max Planck Institute for Dynamics of Complex Technical Systems Magdeburg Computational Methods in Systems and Control Theory Technische Universität Chemnitz Faculty of Mathematics

Differential matrix equations appear in many applications like optimal control of partial differential equations, balanced truncation model order reduction of linear time varying systems and many more. Here, we will focus on differential Riccati equations (DRE). Solving such matrix-valued ordinary differential equations (ODE) is a highly time consuming process. We present a Parareal based algorithm applied to Rosenbrock methods for the solution of the matrix-valued differential Riccati equations. Considering problems of moderate size, direct matrix equation solvers for the solution of the algebraic Lyapunov equations arising inside the time intgration methods are used. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)

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Model reduction of controlled fokker-planck and liouville-von neumann equations

arXiv

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arXiv 2017年

作者： BENNER, PETER BREITEN, TOBIAS HARTMANN, CARSTEN SCHMIDT, BURKHARD Computational Methods in Systems and Control Theory Max Planck Institute for Dynamics of Complex Technical Systems Sandtorstr. 1 MagdeburgD-39106 Germany Institut fur Mathematik Karl-Franzens-Universitat Heinrichstr. 36/III GrazA-8010 Austria Institut fur Mathematik Brandenburgische Technische Universitat Konrad-Wachsmann-Allee 1 CottbusD-03046 Germany Institut fur Mathematik Freie Universitat Berlin Arnimallee 6 BerlinD-14195 Germany

Model reduction methods for bilinear control systems are compared by means of practical examples of Liouville-von Neumann and Fokker-Planck type. methods based on balancing generalized system Gramians and on minimizing an H2-type cost functional are considered. The focus is on the numerical implementation and a thorough comparison of the methods. Structure and stability preservation are investigated, and the competitiveness of the approaches is shown for practically relevant, large-scale examples. Copyright © 2017, The Authors. All rights reserved.

关键词： Lyapunov functions

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Reorthogonalized Pythagorean variants of block classical Gram-Schmidt

arXiv

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arXiv 2024年

作者： Carson, Erin Lund, Kathryn Ma, Yuxin Oktay, Eda Department of Numerical Mathematics Faculty of Mathematics and Physics Charles University Sokolovská 49/83 Praha 8186 75 Czech Republic Computational Methods in Systems and Control Theory Max Planck Institute for Dynamics of Complex Technical Systems Sandtorstr. 1 Magdeburg39106 Germany Computational Mathematics Theme Building R71 STFC Rutherford Appleton Laboratory Harwell Oxford Didcot OxfordshireOX11 0QX United Kingdom Department of Mathematics Chemnitz University of Technology Reichenhainer Str. 41 Chemnitz09126 Germany

Block classical Gram-Schmidt (BCGS) is commonly used for orthogonalizing a set of vectors X in distributed computing environments due to its favorable communication properties relative to other orthogonalization approaches, such as modified Gram-Schmidt or Householder. However, it is known that BCGS (as well as recently developed low-synchronization variants of BCGS) can suffer from a significant loss of orthogonality in finite-precision arithmetic, which can contribute to instability and inaccurate solutions in downstream applications such as s-step Krylov subspace methods. A common solution to improve the orthogonality among the vectors is reorthogonalization. Focusing on the "Pythagorean" variant of BCGS, introduced in [E. Carson, K. Lund, & M. Rozložník. SIAM J. Matrix Anal. Appl. 42(3), pp. 1365–1380, 2021], which guarantees an O(Ε)κ2(X) bound on the loss of orthogonality as long as O(Ε)κ2(X) MSC Codes 65-04, 65F25, 65G50, 65Y20 Copyright © 2024, The Authors. All rights reserved.

关键词： Matrix algebra

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On the loss of orthogonality in low-synchronization variants of reorthogonalized block classical Gram-Schmidt

arXiv

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arXiv 2024年

作者： Carson, Erin Lund, Kathryn Ma, Yuxin Oktay, Eda Department of Numerical Mathematics Faculty of Mathematics and Physics Charles University Sokolovská 49/83 Praha 8186 75 Czech Republic Computational Mathematics Theme Building R71 STFC Rutherford Appleton Laboratory Harwell Oxford Didcot OxfordshireOX11 0QX United Kingdom Computational Methods in Systems and Control Theory Max Planck Institute for Dynamics of Complex Technical Systems Sandtorstr. 1 Magdeburg39106 Germany Department of Mathematics Chemnitz University of Technology Reichenhainer Str. 41 Chemnitz09126 Germany

Interest in communication-avoiding orthogonalization schemes for high-performance computing has been growing recently. This manuscript addresses open questions about the numerical stability of various block classical Gram-Schmidt variants that have been proposed in the past few years. An abstract framework is employed, the flexibility of which allows for new rigorous bounds on the loss of orthogonality in these variants. We first analyze a generalization of (reorthogonalized) block classical Gram-Schmidt and show that a "strong" intrablock orthogonalization routine is only needed for the very first block in order to maintain orthogonality on the level of the unit roundoff. Then, using this variant, which has four synchronization points per block column, we remove the synchronization points one at a time and analyze how each alteration affects the stability of the resulting method. Our analysis shows that the variant requiring only one synchronization per block column cannot be guaranteed to be stable in practice, as stability begins to degrade with the first reduction of synchronization points. Our analysis of block methods also provides new theoretical results for the single-column case. In particular, it is proven that DCGS2 from [Bielich, D. et al. Par. Comput. 112 (2022)] and CGS-2 from [Świrydowicz, K. et al, Num. Lin. Alg. Appl. 28 (2021)] are as stable as Householder QR. Numerical examples from the BlockStab toolbox are included throughout, to help compare variants and illustrate the effects of different choices of intraorthogonalization *** Codes 65-04, 65F25, 65G50, 65Y20 Copyright © 2024, The Authors. All rights reserved.

关键词： Subroutines

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Constructing the Optimal Strategy for a Drone in an Obstacle Avoidance Problem

Constructing the Optimal Strategy for a Drone in an Obstacle...

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International Conference on control systems, Mathematical Modelling, Automation and Energy Efficiency (SUMMA)

作者： Oleg Malafeyev Irina Zaitseva Mark Akbashev Tatiana Smirnova Sergei Miheev Evgeny Kolpak Department of Modelling in Social and Economical Systems Saint Petersburg State University St. Petersburg Russia Department of Higher Mathematics and Physics Russian State Hydrometeorological University St. Petersburg Russia Department of Modeling of Socio-economic systems Saint Petersburg State University St Petersburg Russia Department of mathematical theory of control systems modeling Saint-Petersburg State University St. Petersburg Russia Department of computational methods of mechanics of deformable body Saint-Petersburg State University St. Petersburg Russia

ISBN: (数字)9798331532178

ISBN: (纸本)9798331532185

This article discusses the problem of building an optimal strategy for a drone in the task of chasing a target and avoiding obstacles. The paper analyzes a mathematical model of a game with incomplete information, in which players make decisions based on limited information about each other's condition. Theorems related to the existence of successful strategies for players in this task are presented. The study presents strategies for a drone in conditions of limited information and identifies optimal approaches to solving the problem of harassment. A strategy has been formulated for the drone in conditions of limited information and determine the optimal approaches to solving the task of pursuit. The results demonstrate in the MATLAB/SCRIPT application program the existence of an optimal strategy for unmanned aerial vehicles, which ensures effective overcoming of obstacles under various conditions and constraints. The results obtained can be applied to develop algorithms for controlling unmanned systems in pursuit scenarios.

关键词： Autonomous systems Buildings Games control systems Autonomous aerial vehicles Mathematical models Energy efficiency MATLAB Collision avoidance Drones

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Numerical Solution of Large Scale Sparse Matrix Equations in Python

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PAMM 2014年第1期14卷

作者： Björn Baran Martin Köhler Nitin Prasad Jens Saak Computational Methods in Systems and Control Theory Max Planck Institute for Dynamics of Complex Technical Systems Sandtorstr. 1 39106 Magdeburg Department of Electrical Engineering Indian Institute of Technology Madras Chennai - 600036.

The M.E.S.S. software suite for solving large scale matrix equations and related problems is the successor of the obsolete LyaPack MATLAB® toolbox. The software suite consists of a new MATLAB toolbox and a separate C library C-M.E.S.S. which works independent from MATLAB. Due to the fact that many scientists use Python with NumPy and SciPy for their computations, we want to provide the key algorithms of M.E.S.S. there as well. In this paper, we describe and compare two possible approaches for the implementation, with special focus on their multicore performance. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)

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