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A numerical comparison of solvers for large-scale, continuous-time algebraic Riccati equations

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arXiv 2018年

作者： Benner, Peter Bujanović, Zvonimir Kürschner, Patrick Saak, Jens Max Planck Institute for Dynamics of Complex Technical Systems Computational Methods in Systems and Control Theory Magdeburg Germany University of Zagreb Department of Mathematics Zagreb Croatia

In this paper, we discuss numerical methods for solving large-scale continuous-time algebraic Riccati equations. These methods have been the focus of intensive research in recent years, and significant progress has been made in both the theoretical understanding and efficient implementation of various competing algorithms. There are several goals of this manuscript: first, to gather in one place an overview of different approaches for solving large-scale Riccati equations, and to point to the recent advances in each of them. Second, to analyze and compare the main computational ingredients of these algorithms, to detect their strong points and their potential bottlenecks. And finally, to compare the effective implementations of all methods on a set of relevant benchmark examples, giving an indication of their relative performance. Copyright © 2018, The Authors. All rights reserved.

关键词： Riccati equations

Cross-gramian-based dominant subspaces

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arXiv 2018年

作者： Benner, Peter Himpe, Christian Computational Methods in Systems and Control Theory Max Planck Institute for Dynamics of Complex Technical Systems Sandtorstr. 1 Magdeburg39106 Germany Faculty of Mathematics Otto von Guericke University Magdeburg Universitätsplatz 2 Magdeburg39106 Germany Computational Methods in Systems and Control Theory Max Planck Institute for Dynamics of Complex Technical Systems Sandtorstr. 1 Magdeburg39106 Germany

A standard approach for model reduction of linear input-output systems is balanced truncation, which is based on the controllability and observability properties of the underlying system. The related dominant subspaces projection model reduction method similarly utilizes these system properties, yet instead of balancing, the associated subspaces are directly conjoined. In this work we extend the dominant subspace approach by computation via the cross Gramian for linear systems, and describe an a-priori error indicator for this method. Furthermore, efficient computation is discussed alongside numerical examples illustrating these findings. in93A15, 93B11, 93B20 Copyright © 2018, The Authors. All rights reserved.

关键词： Observability

Peer methods for the solution of large-scale differential matrix equations

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arXiv 2018年

作者： Benner, Peter Lang, Norman Max Planck Institute for Dynamics of Complex Technical Systems Computational Methods in Systems and Control Theory MagdeburgD-39106 Germany Technische Universitat Chemnitz Faculty of Mathematics Mathematics in Industry and Technology ChemnitzD-09126 Germany

We consider the application of implicit and linearly implicit (Rosenbrock-type) peer methods to matrix-valued ordinary differential equations. In particular the differential Riccati equation (DRE) is investigated. For the Rosenbrock-type schemes, a reformulation capable of avoiding a number of Jacobian applications is developed that, in the autonomous case, reduces the computational complexity of the algorithms. Dealing with large-scale problems, an efficient implementation based on low-rank symmetric indefinite factorizations is presented. The perfor- mance of both peer approaches up to order 4 is compared to existing implicit time integration schemes for matrix-valued differential *** Codes 15A23, 34A09, 65L99, 93A15 Copyright © 2018, The Authors. All rights reserved.

关键词： Ordinary differential equations

Towards riccati-feedback control of complex flows with moving interfaces

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Towards riccati-feedback control of complex flows with movin...

88th Annual Meeting on the International Association of Applied Mathematics and Mechanics, GAMM 2017

作者： Baran, Björn Benner, Peter Heiland, Jan Saak, Jens Computational Methods in Systems and Control Theory Max Planck Institute for Dynamics of Complex Technical Systems Sandtorstr. 1 Magdeburg39106 Germany

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.

关键词： Feedback

Inexact methods for the low rank solution to large scale Lyapunov equations

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arXiv 2018年

作者： Kürschner, Patrick Freitag, Melina A. Max Planck Institute for Dynamics of Complex Technical Systems Computational Methods in Systems and Control Theory Magdeburg Germany Department of Mathematical Sciences University of Bath Claverton Down BA2 7AY United Kingdom

The rational Krylov subspace method (RKSM) and the low-rank alternating directions implicit (LR-ADI) iteration are established numerical tools for computing low-rank solution factors of large-scale Lyapunov equations. In order to generate the basis vectors for the RKSM, or extend the low-rank factors within the LR-ADI method the repeated solution to a shifted linear system is necessary. For very large systems this solve is usually implemented using iterative methods, leading to inexact solves within this inner iteration. We derive theory for a relaxation strategy within these inexact solves, both for the RKSM and the LR-ADI method. Practical choices for relaxing the solution tolerance within the inner linear system are then provided. The theory is supported by several numerical examples.65F10, 65F30 Copyright © 2018, The Authors. All rights reserved.

关键词： Lyapunov functions

Synchronization and aggregation of nonlinear power systems with consideration of bus network structures

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arXiv 2018年

作者： Mlinarić, Petar Ishizaki, Takayuki Chakrabortty, Aranya Grundel, Sara Benner, Peter Imura, Jun-Ichi Computational Methods in Systems and Control Theory Max Planck Institute for Dynamics of Complex Technical Systems Magdeburg39106 Germany Department of Systems and Control Engineering School of Engineering Tokyo Institute of Technology 2-12-1 Meguro Tokyo Japan Electrical & Computer Engineering North Carolina State University RaleighNC27695

We study nonlinear power systems consisting of generators, generator buses, and non-generator buses. First, looking at a generator and its bus' variables jointly, we introduce a synchronization concept for a pair of such joint generators and buses. We show that this concept is related to graph symmetry. Next, we extend, in two ways, the synchronization from a pair to a partition of all generators in the networks and show that they are related to either graph symmetry or equitable partitions. Finally, we show how an exact reduced model can be obtained by aggregating the generators and associated buses in the network when the original system is synchronized with respect to a partition, provided that the initial condition respects the partition. Additionally, the aggregation-based reduced model is again a power *** Codes 93A15, 93C10, 94C15 Copyright © 2018, The Authors. All rights reserved.

关键词： Synchronization

ON THE EXISTENCE AND UNIQUENESS OF THE SOLUTION OF A PARABOLIC OPTIMAL control PROBLEM WITH UNCERTAIN INPUTS

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arXiv 2018年

作者： Benner, Peter Onwunta, Akwum Stoll, Martin Computational Methods in Systems and Control Theory Max Planck Institute for Dynamics of Complex Technical Systems Sandtorstrasse 1 Magdeburg39106 Germany Department of Computer Science University of Maryland College ParkMD20742 United States Technische Universität Chemnitz Faculty of Mathematics Professorship Scientific Computing Chemnitz09107 Germany

In this note, we consider the existence and uniqueness of the solution of a time-dependent optimal control problem constrained by a partial differential equation with uncertain inputs. Relying on the Lions’ Lemma for deterministic problems, we furthermore characterize the optimal control of the stochastic problem. Copyright © 2018, The Authors. All rights reserved.

关键词： Optimal control systems

Fast low-rank empirical cross gramians

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Fast low-rank empirical cross gramians

88th Annual Meeting on the International Association of Applied Mathematics and Mechanics, GAMM 2017

作者： Himpe, Christian Leibner, Tobias Rave, Stephan Saak, Jens Computational Methods in Systems and Control Theory Max Planck Institute for Dynamics of Complex Technical Systems Sandtorstr. 1 Magdeburg39106 Germany Applied Mathematics University of Münster Einsteinstr. 62 Münster48149 Germany

The cross Gramian matrix encodes the input-output coherence of linear control systems and is used in projection-based model reduction. The empirical cross Gramian is a data-driven variant of the cross Gramian which also extends to nonlinear systems. A drawback of the empirical cross Gramian for large-scale systems is its full order and dense structure;yet, it may be computed column-wise. Using the hierarchical approximate proper orthogonal decomposition (HAPOD), this partial computability can be exploited to obtain a truncated projection for model order reduction without assembling a full cross Gramian. © 2017 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim.

关键词： Linear control systems

On an Invariance Principle for the Solution Space of the Differential Riccati Equation

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PAMM 2018年第1期18卷

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

The differential Riccati equation appears in different fields of applied mathematics like control theory and systems theory. For large-scale systems the numerical solution comes with a large amount of storage requirements. This motivates the use of Krylov subspace and projection based methods [1–3]. In the present paper we apply an invariance theorem for ODEs to the differential Riccati Equation. We show that the solution is contained in a Krylov like subspace and extend our results to certain time-varying cases.

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