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  • 1.
    Dmytryshyn, Andrii
    et al.
    Umeå University, Faculty of Science and Technology, Department of Computing Science.
    Johansson, Stefan
    Umeå University, Faculty of Science and Technology, Department of Computing Science.
    Kågström, Bo
    Umeå University, Faculty of Science and Technology, Department of Computing Science.
    Canonical structure transitions of system pencils2015Report (Other academic)
    Abstract [en]

    We investigate the changes under small perturbations of the canonical structure information for a system pencil (A B C D) − s (E 0 0 0), det(E) ≠ 0, associated with a (generalized) linear time-invariant state-space system. The equivalence class of the pencil is taken with respect to feedback-injection equivalence transformation. The results allow to track possible changes under small perturbations of important linear system characteristics.

  • 2.
    Dmytryshyn, Andrii
    et al.
    Umeå University, Faculty of Science and Technology, Department of Computing Science. Umeå University, Faculty of Science and Technology, High Performance Computing Center North (HPC2N).
    Johansson, Stefan
    Umeå University, Faculty of Science and Technology, Department of Computing Science. Umeå University, Faculty of Science and Technology, High Performance Computing Center North (HPC2N).
    Kågström, Bo
    Umeå University, Faculty of Science and Technology, Department of Computing Science. Umeå University, Faculty of Science and Technology, High Performance Computing Center North (HPC2N).
    Canonical structure transitions of system pencils2017In: SIAM Journal on Matrix Analysis and Applications, ISSN 0895-4798, E-ISSN 1095-7162, Vol. 38, no 4, p. 1249-1267Article in journal (Refereed)
    Abstract [en]

    We investigate the changes of the canonical structure information under small perturbations for a system pencil associated with a (generalized) linear time-invariant state-space system. The equivalence class of the pencil is taken with respect to feedback-injection equivalence transformations. The results allow us to track possible changes of important linear system characteristics under small perturbations.

  • 3.
    Dmytryshyn, Andrii
    et al.
    Umeå University, Faculty of Science and Technology, Department of Computing Science. Umeå University, Faculty of Science and Technology, High Performance Computing Center North (HPC2N).
    Johansson, Stefan
    Umeå University, Faculty of Science and Technology, Department of Computing Science. Umeå University, Faculty of Science and Technology, High Performance Computing Center North (HPC2N).
    Kågström, Bo
    Umeå University, Faculty of Science and Technology, Department of Computing Science. Umeå University, Faculty of Science and Technology, High Performance Computing Center North (HPC2N).
    Codimension computations of congruence orbits of matrices, symmetric and skew-symmetric matrix pencils using Matlab2013Report (Other academic)
    Abstract [en]

    Matlab functions to work with the canonical structures for congru-ence and *congruence of matrices, and for congruence of symmetricand skew-symmetric matrix pencils are presented. A user can providethe canonical structure objects or create (random) matrix examplesetups with a desired canonical information, and compute the codi-mensions of the corresponding orbits: if the structural information(the canonical form) of a matrix or a matrix pencil is known it isused for the codimension computations, otherwise they are computednumerically. Some auxiliary functions are provided too. All thesefunctions extend the Matrix Canonical Structure Toolbox.

  • 4.
    Dmytryshyn, Andrii
    et al.
    Umeå University, Faculty of Science and Technology, Department of Computing Science. School of Science and Technology, Örebro University, Örebro, Sweden.
    Johansson, Stefan
    Umeå University, Faculty of Science and Technology, Department of Computing Science.
    Kågström, Bo
    Umeå University, Faculty of Science and Technology, Department of Computing Science.
    Van Dooren, Paul
    Department of Mathematical Engineering, Université catholique de Louvain, Louvain-la-Neuve, Belgium.
    Geometry of Matrix Polynomial Spaces2019In: Foundations of Computational Mathematics, ISSN 1615-3375, E-ISSN 1615-3383Article in journal (Refereed)
    Abstract [en]

    We study how small perturbations of general matrix polynomials may change their elementary divisors and minimal indices by constructing the closure hierarchy (stratification) graphs of matrix polynomials' orbits and bundles. To solve this problem, we construct the stratification graphs for the first companion Fiedler linearization of matrix polynomials. Recall that the first companion Fiedler linearization as well as all the Fiedler linearizations is matrix pencils with particular block structures. Moreover, we show that the stratification graphs do not depend on the choice of Fiedler linearization which means that all the spaces of the matrix polynomial Fiedler linearizations have the same geometry (topology). This geometry coincides with the geometry of the space of matrix polynomials. The novel results are illustrated by examples using the software tool StratiGraph extended with associated new functionality.

  • 5.
    Dmytryshyn, Andrii
    et al.
    Umeå University, Faculty of Science and Technology, Department of Computing Science.
    Johansson, Stefan
    Umeå University, Faculty of Science and Technology, Department of Computing Science.
    Kågström, Bo
    Umeå University, Faculty of Science and Technology, Department of Computing Science.
    Van Dooren, Paul
    Universite catholique de Louvain, Belgium.
    Geometry of spaces for matrix polynomial Fiedler linearizations2015Report (Other academic)
    Abstract [en]

    We study how small perturbations of matrix polynomials may change their elementary divisors and minimal indices by constructing the closure hierarchy graphs (stratifications) of orbits and bundles of matrix polynomial Fiedler linearizations. We show that the stratifica-tion graphs do not depend on the choice of Fiedler linearization which means that all the spaces of the matrix polynomial Fiedler lineariza-tions have the same geometry (topology). The results are illustrated by examples using the software tool StratiGraph.

  • 6.
    Elmroth, Erik
    et al.
    Umeå University, Faculty of Science and Technology, Department of Computing Science. Umeå University, Faculty of Science and Technology, High Performance Compting Center North (HPC2N).
    Johansson, Pedher
    Umeå University, Faculty of Science and Technology, Department of Computing Science.
    Johansson, Stefan
    Umeå University, Faculty of Science and Technology, Department of Computing Science.
    Kågström, Bo
    Umeå University, Faculty of Science and Technology, Department of Computing Science. Umeå University, Faculty of Science and Technology, High Performance Compting Center North (HPC2N).
    Orbit and Bundle Stratification for Controllability and Observability Matrix Pairs in StratiGraph2004In: Proceedings of the 16th International Symposium on Mathematical Theory of Networks and Systems (MTNS), 2004, p. 1-9Conference paper (Refereed)
    Abstract [en]

    The canonical structures of controllability and observability pairs (A,B) and (A,C) associated with a state-space system are studied under small perturbations. We show how previous work for general matrix pencils can be applied to the stratification of orbits and bundles of matrix pairs. A stratification provides qualitative information about the closure relation between canonical structures.We also present how the new results are used in StratiGraph, which is a software tool for computing and visualizing closure hierarchies.

  • 7.
    Elmroth, Erik
    et al.
    Umeå University, Faculty of Science and Technology, Department of Computing Science.
    Johansson, Pedher
    Umeå University, Faculty of Science and Technology, Department of Computing Science.
    Johansson, Stefan
    Umeå University, Faculty of Science and Technology, Department of Computing Science.
    Kågström, Bo
    Umeå University, Faculty of Science and Technology, Department of Computing Science. Umeå University, Faculty of Science and Technology, High Performance Compting Center North (HPC2N).
    Orbit and bundle stratification of controllability and observability matrix pairs in StratiGraph2004In: Proceedings MTNS 2004Article in journal (Refereed)
  • 8.
    Elmroth, Erik
    et al.
    Umeå University, Faculty of Science and Technology, Department of Computing Science.
    Johansson, Stefan
    Umeå University, Faculty of Science and Technology, Department of Computing Science.
    Kågström, Bo
    Umeå University, Faculty of Science and Technology, Department of Computing Science. Umeå University, Faculty of Science and Technology, High Performance Compting Center North (HPC2N).
    Stratification of controllability and observability pairs: theory and use in applications2009In: SIAM Journal on Matrix Analysis and Applications, ISSN 0895-4798, E-ISSN 1095-7162, Vol. 31, no 2, p. 203-226Article in journal (Refereed)
    Abstract [en]

    Cover relations for orbits and bundles of controllability and observability pairs associated with linear time-invariant systems are derived. The cover relations are combinatorial rules acting on integer sequences, each representing a subset of the Jordan and singular Kronecker structures of the corresponding system pencil. By representing these integer sequences as coin piles, the derived stratification rules are expressed as minimal coin moves between and within these piles, which satisfy and preserve certain monotonicity properties. The stratification theory is illustrated with two examples from systems and control applications, a mechanical system consisting of a thin uniform platform supported at both ends by springs, and a linearized Boeing 747 model. For both examples, nearby uncontrollable systems are identified as subsets of the complete closure hierarchy for the associated system pencils.

  • 9.
    Gusev, Sergei
    et al.
    Department of Mathematics and Mechanics, St. Petersburg State University, St. Petersburg, Russia.
    Johansson, Stefan
    Umeå University, Faculty of Science and Technology, Department of Computing Science.
    Kågström, Bo
    Umeå University, Faculty of Science and Technology, Department of Computing Science.
    Shiriaev, Anton
    Umeå University, Faculty of Science and Technology, Department of Applied Physics and Electronics.
    Varga, Andras
    Institute of Robotics and Mechatronics, German Aerospace Center, DLR, Germany.
    A numerical evaluation of solvers for the periodic riccati differential equation2010In: BIT Numerical Mathematics, ISSN 0006-3835, E-ISSN 1572-9125, Vol. 50, no 2, p. 301-329Article in journal (Refereed)
    Abstract [en]

    Efficient and accurate structure exploiting numerical methods for solvingthe periodic Riccati differential equation (PRDE) are addressed. Such methods areessential, for example, to design periodic feedback controllers for periodic controlsystems. Three recently proposed methods for solving the PRDE are presented andevaluated on challenging periodic linear artificial systems with known solutions and applied to the stabilization of periodic motions of mechanical systems. The first twomethods are of the type multiple shooting and rely on computing the stable invariantsubspace of an associated Hamiltonian system. The stable subspace is determinedusing either algorithms for computing an ordered periodic real Schur form of a cyclicmatrix sequence, or a recently proposed method which implicitly constructs a stabledeflating subspace from an associated lifted pencil. The third method reformulatesthe PRDE as a convex optimization problem where the stabilizing solution is approximatedby its truncated Fourier series. As known, this reformulation leads to a semidefiniteprogramming problem with linear matrix inequality constraints admitting aneffective numerical realization. The numerical evaluation of the PRDE methods, withfocus on the number of states (n) and the length of the period (T ) of the periodicsystems considered, includes both quantitative and qualitative results.

  • 10.
    Gusev, Sergei
    et al.
    Department of Mathematics and Mechanics, St. Petersburg State University, St. Petersburg, Russia.
    Johansson, Stefan
    Umeå University, Faculty of Science and Technology, Department of Computing Science.
    Kågström, Bo
    Umeå University, Faculty of Science and Technology, Department of Computing Science.
    Shiriaev, Anton
    Umeå University, Faculty of Science and Technology, Department of Applied Physics and Electronics.
    Varga, Andras
    Institute of Robotics and Mechatronics, German Aerospace Center, DLR, Oberpfaffenhofen, Germany.
    A Numerical Evaluation of Solvers for the Periodic Riccati Differential Equation2009Report (Other academic)
    Abstract [en]

    Efficient and robust numerical methods for solving the periodic Riccati differential equation (PRDE) are addressed. Such methods are essential, for example, when deriving feedback controllers for orbital stabilization of underactuated mechanical systems. Two recently proposed methods for solving the PRDE are presented and evaluated on artificial systems and on two stabilization problems originating from mechanical systems with unstable dynamics. The first method is of the type multiple shooting and relies on computing the stable invariant subspace of an associated Hamiltonian system. The stable subspace is determined using algorithms for computing a reordered periodic real Schur form of a cyclic matrix sequence, and a recently proposed method which implicitly constructs a stable subspace from an associated lifted pencil. The second method reformulates the PRDE as a maximization problem where the stabilizing solution is approximated with finite dimensional trigonometric base functions. By doing this reformulation the problem turns into a semidefinite programming problem with linear matrix inequality constraints.

  • 11.
    Johansson, Stefan
    Umeå University, Faculty of Science and Technology, Department of Computing Science.
    Reviewing the Closure Hierarchy of Orbits and Bundles of System Pencils and Their Canonical Forms2009Report (Other academic)
    Abstract [en]

    Using a unifying terminology and notation an introduction to the theory of stratification for orbits and bundles of matrices, matrix pencils and system pencils with applications in systems and control is presented. Canonical forms of such orbits and bundles reveal the important system characteristics of the models under investigation. A stratification provides the qualitative information of which canonical structures are near each other in the sense of small perturbations. We discuss how fundamental concepts like controllability and observability of a system can be studied with the use of the stratification theory. Important results are presented in the form of the closure and cover relations for controllability and observability pairs. Furthermore, different canonical forms are considered from which we can derive the characteristics of a system. Specifically, we discuss how the Kronecker canonical form is related to the Brunovsky canonical form and its generalizations. Concepts and results are illustrated with several examples throughout the presentation.

  • 12.
    Johansson, Stefan
    Umeå University, Faculty of Science and Technology, Departement of Computing Science.
    Tools for Control System Design: Stratification of Matrix Pairs and Periodic Riccati Differential Equation Solvers2009Doctoral thesis, comprehensive summary (Other academic)
    Abstract [en]

    Modern control theory is today an interdisciplinary area of research. Just as much as this can be problematic, it also provides a rich research environment where practice and theory meet. This Thesis is conducted in the borderline between computing science (numerical analysis) and applied control theory. The design and analysis of a modern control system is a complex problem that requires high qualitative software to accomplish. Ideally, such software should be based on robust methods and numerical stable algorithms that provide quantitative as well as qualitative information.

    The introduction of the Thesis is dedicated to the underlying control theory and to introduce the reader to the main subjects. Throughout the Thesis, the theory is illustrated with several examples, and similarities and differences between the terminology from mathematics, systems and control theory, and numerical linear algebra are highlighted. The main contributions of the Thesis are structured in two parts, dealing with two mainly unrelated subjects.

    Part I is devoted to the qualitative information which is provided by the stratification of orbits and bundles of matrices, matrix pencils and system pencils. Before the theory of stratification is established the reader is introduced to different canonical forms which reveal the system characteristics of the model under investigation. A stratification reveals which canonical structures of matrix (system) pencils are near each other in the sense of small perturbations of the data. Fundamental concepts in systems and control, like controllability and observability of linear continuous-time systems, are considered and it is shown how these system characteristics can be investigated using the stratification theory. New results are presented in the form of the cover relations (nearest neighbours) for controllability and observability pairs. Moreover, the permutation matrices which take a matrix pencil in the Kronecker canonical form to the corresponding system pencil in (generalized) Brunovsky canonical form are derived. Two novel algorithms for determining the permutation matrices are provided.

    Part II deals with numerical methods for solving periodic Riccati differential equations (PRDE:s). The PRDE:s under investigation arise when solving the linear quadratic regulator (LQR) problem for periodic linear time-varying (LTV) systems. These types of (periodic) LQR problems turn up for example in motion planning of underactuated mechanical systems, like a humanoid robot, the Furuta pendulum, and pendulums on carts. The constructions of the nonlinear controllers are based on linear versions found by stabilizing transverse dynamics of the systems along cycles.

    Three different methods explicitly designed for solving the PRDE are evaluated on both artificial systems and stabilizing problems originating from experimental control systems. The methods are the one-shot generator method and two recently proposed methods: the multi-shot method (two variants) and the SDP method. As these methods use different approaches to solve the PRDE, their numerical behavior and performance are dependent on the nature of the underlying control problem. Such method characteristics are investigated and summarized with respect to different user requirements (the need for accuracy and possible restrictions on the solution time).

  • 13.
    Johansson, Stefan
    et al.
    Umeå University, Faculty of Science and Technology, Department of Computing Science.
    Kågström, Bo
    Umeå University, Faculty of Science and Technology, Department of Computing Science. Umeå University, Faculty of Science and Technology, HPC2N (High Performance Computing Centre North).
    Shiriaev, Anton
    Umeå University, Faculty of Science and Technology, Applied Physics and Electronics.
    Varga, Andras
    Institute of Robotics and Mechatronics, German Aerospace Center, DLR, Oberpfaffenhofen, Germany.
    Comparing One-shot and Multi-shot Methods for Solving Periodic Riccati Differential Equations2007In: Proceedings of the third IFAC Workshop on Periodic Control Systems (PSYCO’07), International Federation of Automatic Control , 2007, p. 1-6Conference paper (Refereed)
    Abstract [en]

    One-shot methods and recently proposed multi-shot methods for computing stabilizing solutions of continuous-time periodic Riccati differential equations are examined and evaluated on two test problems: (i) a stabilization problem for an artificially constructed time-varying linear system for which the exact solution is known; (ii) a nonlinear stabilization problem for a devil stick juggling model along a periodic trajectory. The numerical comparisons are performed using both general purpose and symplectic integration methods for solving the associated Hamiltonian differential systems. In the multi-shot method a stable subspace is determined using recent algorithms for computing a reordered periodic real Schur form. The results show the increased accuracy achievable by combining multi-shot methods with structure preserving (symplectic) integration techniques.

  • 14.
    Johansson, Stefan
    et al.
    Umeå University, Faculty of Science and Technology, Department of Computing Science.
    Kågström, Bo
    Umeå University, Faculty of Science and Technology, Department of Computing Science. Umeå University, Faculty of Science and Technology, High Performance Computing Center North (HPC2N).
    Van Dooren, Paul
    Department of Mathematical Engineering, Université catholique de Louvain.
    Stratification of full rank polynomial matrices2013In: Linear Algebra and its Applications, ISSN 0024-3795, E-ISSN 1873-1856, Vol. 439, no 4, p. 1062-1090Article in journal (Refereed)
    Abstract [en]

    We show that perturbations of polynomial matrices of full normal-rank can be analyzed viathe study of perturbations of companion form linearizations of such polynomial matrices.It is proved that a full normal-rank polynomial matrix has the same structural elements asits right (or left) linearization. Furthermore, the linearized pencil has a special structurethat can be taken into account when studying its stratification. This yields constraintson the set of achievable eigenstructures. We explicitly show which these constraints are.These results allow us to derive necessary and sufficient conditions for cover relationsbetween two orbits or bundles of the linearization of full normal-rank polynomial matrices.The stratification rules are applied to and illustrated on two artificial polynomial matricesand a half-car passive suspension system with four degrees of freedom.

  • 15.
    Kågström, Bo
    et al.
    Umeå University, Faculty of Science and Technology, Department of Computing Science. Umeå University, Faculty of Science and Technology, High Performance Computing Center North (HPC2N).
    Johansson, Stefan
    Umeå University, Faculty of Science and Technology, Department of Computing Science. Umeå University, Faculty of Science and Technology, High Performance Computing Center North (HPC2N).
    Johansson, Pedher
    Umeå University, Faculty of Science and Technology, Department of Computing Science. Umeå University, Faculty of Science and Technology, High Performance Computing Center North (HPC2N).
    StratiGraph Tool: Matrix Stratifications in Control Applications2012In: Control and Optimization with Differential-Algebraic Constraints / [ed] Lorenz T. Biegler, Stephen L. Champbell, Volker Mehrmann, Philadelphia: Society for Industrial and Applied Mathematics, 2012, p. 79-103Chapter in book (Refereed)
    Abstract [en]

    In this contribution, the software tool StratiGraph for computing and visualizing closurehierarchy graphs associated with different orbit and bundle stratifications is presented. Inaddition, we review the underlying theory and illustrate how StratiGraph can be used toanalyze descriptor system models via their associated system pencils. The stratificationtheory provides information for a deeper understanding of how the dynamics of a controlsystem and its system characteristics behave under perturbations.

  • 16.
    Kågström, Bo
    et al.
    Umeå University, Faculty of Science and Technology, Department of Computing Science. Umeå University, Faculty of Science and Technology, High Performance Computing Center North (HPC2N).
    Johansson, Stefan
    Umeå University, Faculty of Science and Technology, Department of Computing Science.
    Johansson, Pedher
    Umeå University, Faculty of Science and Technology, Department of Computing Science.
    StratiGraph Tool: Matrix Stratifications in Control Applications2011Report (Other academic)
    Abstract [en]

    In this contribution, the software tool StratiGraph for computing and visualizing closure hierarchy graphs associated with different orbit and bundle stratifications is presented. In addition, we review the underlying theory and illustrate how StratiGraph can be used to analyze descriptor system models via their associated system pencils. The stratification theory provides information for a deeper understanding of how the dynamics of a control system and its system characteristics behave under perturbations.

1 - 16 of 16
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