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Johansson, Stefan

Open this publication in new window or tab >>Geometry of Matrix Polynomial Spaces### Dmytryshyn, Andrii

### Johansson, Stefan

### Kågström, Bo

### Van Dooren, Paul

PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt204_0_j_idt208_some",{id:"formSmash:j_idt204:0:j_idt208:some",widgetVar:"widget_formSmash_j_idt204_0_j_idt208_some",multiple:true}); PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt204_0_j_idt208_otherAuthors",{id:"formSmash:j_idt204:0:j_idt208:otherAuthors",widgetVar:"widget_formSmash_j_idt204_0_j_idt208_otherAuthors",multiple:true}); 2020 (English)In: Foundations of Computational Mathematics, ISSN 1615-3375, E-ISSN 1615-3383, Vol. 20, no 3, p. 423-450Article in journal (Refereed) Published
##### Abstract [en]

##### Place, publisher, year, edition, pages

Springer, 2020
##### Keywords

Matrix polynomials, Stratifications, Matrix pencils, Fiedler linearization, Canonical structure information, Orbit, Bundle
##### National Category

Computational Mathematics Computer and Information Sciences
##### Identifiers

urn:nbn:se:umu:diva-163512 (URN)10.1007/s10208-019-09423-1 (DOI)000531825900002 ()2-s2.0-85068193369 (Scopus ID)
#####

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##### Funder

eSSENCE - An eScience CollaborationSwedish Research Council, E0485301
Available from: 2019-09-24 Created: 2019-09-24 Last updated: 2020-10-15Bibliographically approved

Umeå University, Faculty of Science and Technology, Department of Computing Science. School of Science and Technology, Örebro University, Örebro, Sweden.

Umeå University, Faculty of Science and Technology, Department of Computing Science.

Umeå University, Faculty of Science and Technology, Department of Computing Science.

Department of Mathematical Engineering, Université catholique de Louvain, Louvain-la-Neuve, Belgium.

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.

Open this publication in new window or tab >>Canonical structure transitions of system pencils### Dmytryshyn, Andrii

### Johansson, Stefan

### Kågström, Bo

PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt204_1_j_idt208_some",{id:"formSmash:j_idt204:1:j_idt208:some",widgetVar:"widget_formSmash_j_idt204_1_j_idt208_some",multiple:true}); PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt204_1_j_idt208_otherAuthors",{id:"formSmash:j_idt204:1:j_idt208:otherAuthors",widgetVar:"widget_formSmash_j_idt204_1_j_idt208_otherAuthors",multiple:true}); 2017 (English)In: SIAM Journal on Matrix Analysis and Applications, ISSN 0895-4798, E-ISSN 1095-7162, Vol. 38, no 4, p. 1249-1267Article in journal (Refereed) Published
##### Abstract [en]

##### Place, publisher, year, edition, pages

Society for Industrial and Applied Mathematics, 2017
##### National Category

Mathematics Computer and Information Sciences
##### Research subject

business data processing
##### Identifiers

urn:nbn:se:umu:diva-139924 (URN)10.1137/16M1097857 (DOI)000418665600009 ()2-s2.0-85022337450 (Scopus ID)
#####

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##### Funder

Swedish Research Council, E0485301Swedish Research Council, eSSENCE
Available from: 2017-09-26 Created: 2017-09-26 Last updated: 2023-03-24Bibliographically approved

Umeå University, Faculty of Science and Technology, Department of Computing Science. Umeå University, Faculty of Science and Technology, High Performance Computing Center North (HPC2N).

Umeå University, Faculty of Science and Technology, Department of Computing Science. Umeå University, Faculty of Science and Technology, High Performance Computing Center North (HPC2N).

Umeå University, Faculty of Science and Technology, Department of Computing Science. Umeå University, Faculty of Science and Technology, High Performance Computing Center North (HPC2N).

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.

Open this publication in new window or tab >>Canonical structure transitions of system pencils### Dmytryshyn, Andrii

### Johansson, Stefan

### Kågström, Bo

PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt204_2_j_idt208_some",{id:"formSmash:j_idt204:2:j_idt208:some",widgetVar:"widget_formSmash_j_idt204_2_j_idt208_some",multiple:true}); PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt204_2_j_idt208_otherAuthors",{id:"formSmash:j_idt204:2:j_idt208:otherAuthors",widgetVar:"widget_formSmash_j_idt204_2_j_idt208_otherAuthors",multiple:true}); 2015 (English)Report (Other academic)
##### Abstract [en]

##### Publisher

p. 26
##### Series

Report / UMINF, ISSN 0348-0542 ; 15.15
##### Keywords

linear system, descriptor system, state-space system, system pencil, matrix pencil, orbit, bundle, perturbation, versal deformation, stratification
##### National Category

Mathematics Computer and Information Sciences Electrical Engineering, Electronic Engineering, Information Engineering Civil Engineering
##### Identifiers

urn:nbn:se:umu:diva-111632 (URN)
#####

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#####

##### Funder

eSSENCE - An eScience CollaborationSwedish Research Council, E048530
Available from: 2015-11-18 Created: 2015-11-18 Last updated: 2018-06-07Bibliographically approved

Umeå University, Faculty of Science and Technology, Department of Computing Science.

Umeå University, Faculty of Science and Technology, Department of Computing Science.

Umeå University, Faculty of Science and Technology, Department of Computing Science.

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.

Open this publication in new window or tab >>Geometry of spaces for matrix polynomial Fiedler linearizations### Dmytryshyn, Andrii

### Johansson, Stefan

### Kågström, Bo

### Van Dooren, Paul

PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt204_3_j_idt208_some",{id:"formSmash:j_idt204:3:j_idt208:some",widgetVar:"widget_formSmash_j_idt204_3_j_idt208_some",multiple:true}); PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt204_3_j_idt208_otherAuthors",{id:"formSmash:j_idt204:3:j_idt208:otherAuthors",widgetVar:"widget_formSmash_j_idt204_3_j_idt208_otherAuthors",multiple:true}); 2015 (English)Report (Other academic)
##### Abstract [en]

##### Publisher

p. 28
##### Series

Report / UMINF, ISSN 0348-0542 ; 15.17
##### National Category

Mathematics Computer and Information Sciences
##### Identifiers

urn:nbn:se:umu:diva-111639 (URN)
#####

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#####

##### Funder

Swedish Research Council, E0485301eSSENCE - An eScience Collaboration
Available from: 2015-11-18 Created: 2015-11-18 Last updated: 2018-06-07Bibliographically approved

Umeå University, Faculty of Science and Technology, Department of Computing Science.

Umeå University, Faculty of Science and Technology, Department of Computing Science.

Umeå University, Faculty of Science and Technology, Department of Computing Science.

Universite catholique de Louvain, Belgium.

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.

Open this publication in new window or tab >>Codimension computations of congruence orbits of matrices, symmetric and skew-symmetric matrix pencils using Matlab### Dmytryshyn, Andrii

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).PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt204_4_j_idt208_some",{id:"formSmash:j_idt204:4:j_idt208:some",widgetVar:"widget_formSmash_j_idt204_4_j_idt208_some",multiple:true}); PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt204_4_j_idt208_otherAuthors",{id:"formSmash:j_idt204:4:j_idt208:otherAuthors",widgetVar:"widget_formSmash_j_idt204_4_j_idt208_otherAuthors",multiple:true}); 2013 (English)Report (Other academic)
##### Abstract [en]

##### Place, publisher, year, edition, pages

Umeå: Umeå Universitet, 2013. p. 41
##### Series

Report / UMINF, ISSN 0348-0542 ; 13.18
##### Keywords

Congruence; *congruence; Symmetric matrix pencils; Skew-symmetric matrix pencils; Orbits; Codimension; MATLAB
##### National Category

Computer Sciences Computational Mathematics
##### Research subject

Numerical Analysis; Computer Science
##### Identifiers

urn:nbn:se:umu:diva-80524 (URN)
#####

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Available from: 2013-09-19 Created: 2013-09-19 Last updated: 2018-06-08Bibliographically approved

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.

Open this publication in new window or tab >>Stratification of full rank polynomial matrices### Johansson, Stefan

### 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

PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt204_5_j_idt208_some",{id:"formSmash:j_idt204:5:j_idt208:some",widgetVar:"widget_formSmash_j_idt204_5_j_idt208_some",multiple:true}); PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt204_5_j_idt208_otherAuthors",{id:"formSmash:j_idt204:5:j_idt208:otherAuthors",widgetVar:"widget_formSmash_j_idt204_5_j_idt208_otherAuthors",multiple:true}); 2013 (English)In: Linear Algebra and its Applications, ISSN 0024-3795, E-ISSN 1873-1856, Vol. 439, no 4, p. 1062-1090Article in journal (Refereed) Published
##### Abstract [en]

##### Place, publisher, year, edition, pages

Elsevier, 2013
##### Keywords

polynomial matrices, matrix pencils, linearization, perturbations, stratification, closure hierarchy, cover relations, StratiGraph
##### National Category

Computational Mathematics Computer Sciences
##### Research subject

Numerical Analysis; Automatic Control
##### Identifiers

urn:nbn:se:umu:diva-71154 (URN)10.1016/j.laa.2012.12.013 (DOI)000321084700021 ()2-s2.0-84879881922 (Scopus ID)
#####

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##### Funder

Swedish Foundation for Strategic Research , A3 02:128
Available from: 2013-05-21 Created: 2013-05-21 Last updated: 2023-03-24Bibliographically approved

Umeå University, Faculty of Science and Technology, Department of Computing Science.

Department of Mathematical Engineering, Université catholique de Louvain.

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.

Open this publication in new window or tab >>StratiGraph Tool: Matrix Stratifications in Control Applications### 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).### 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).PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt204_6_j_idt208_some",{id:"formSmash:j_idt204:6:j_idt208:some",widgetVar:"widget_formSmash_j_idt204_6_j_idt208_some",multiple:true}); PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt204_6_j_idt208_otherAuthors",{id:"formSmash:j_idt204:6:j_idt208:otherAuthors",widgetVar:"widget_formSmash_j_idt204_6_j_idt208_otherAuthors",multiple:true}); 2012 (English)In: 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]

##### Place, publisher, year, edition, pages

Philadelphia: Society for Industrial and Applied Mathematics, 2012
##### Series

Advances in Design and Control ; 23
##### Keywords

Stratification, differential-algebraic equations, descriptor systems, Kronecker structures, orbit, bundle, closure hierarchy, cover relations, StratiGraph
##### National Category

Computational Mathematics Computer Sciences
##### Research subject

Numerical Analysis; Automatic Control
##### Identifiers

urn:nbn:se:umu:diva-61417 (URN)978-1-611972-24-5 (ISBN)
#####

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##### Funder

eSSENCE - An eScience CollaborationSwedish Foundation for Strategic Research , A3 02:128
Available from: 2012-11-13 Created: 2012-11-13 Last updated: 2019-06-26Bibliographically approved

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.

Open this publication in new window or tab >>StratiGraph Tool: Matrix Stratifications in Control Applications### 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).### Johansson, Stefan

### Johansson, Pedher

PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt204_7_j_idt208_some",{id:"formSmash:j_idt204:7:j_idt208:some",widgetVar:"widget_formSmash_j_idt204_7_j_idt208_some",multiple:true}); PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt204_7_j_idt208_otherAuthors",{id:"formSmash:j_idt204:7:j_idt208:otherAuthors",widgetVar:"widget_formSmash_j_idt204_7_j_idt208_otherAuthors",multiple:true}); 2011 (English)Report (Other academic)
##### Abstract [en]

##### Place, publisher, year, edition, pages

Umeå: Department of Computing Science, Umeå University, 2011. p. 24
##### Series

Report / UMINF, ISSN 0348-0542 ; 11.12
##### Keywords

Stratification, differential-algebraic equations, descriptor systems, Kronecker structures, orbit, bundle, closure hierarchy, cover relations, StratiGraph
##### National Category

Computer Sciences Computational Mathematics
##### Research subject

Numerical Analysis; Automatic Control
##### Identifiers

urn:nbn:se:umu:diva-50774 (URN)
##### External cooperation:

#####

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Available from: 2012-01-02 Created: 2011-12-21 Last updated: 2019-06-26Bibliographically approved

Umeå University, Faculty of Science and Technology, Department of Computing Science.

Umeå University, Faculty of Science and Technology, Department of Computing Science.

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.

Open this publication in new window or tab >>A numerical evaluation of solvers for the periodic riccati differential equation### Gusev, Sergei

### Johansson, Stefan

### Kågström, Bo

### Shiriaev, Anton

PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt204_8_j_idt208_some",{id:"formSmash:j_idt204:8:j_idt208:some",widgetVar:"widget_formSmash_j_idt204_8_j_idt208_some",multiple:true}); ### Varga, Andras

PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt204_8_j_idt208_otherAuthors",{id:"formSmash:j_idt204:8:j_idt208:otherAuthors",widgetVar:"widget_formSmash_j_idt204_8_j_idt208_otherAuthors",multiple:true}); Show others...PrimeFaces.cw("SelectBooleanButton","widget_formSmash_j_idt204_8_j_idt208_j_idt222",{id:"formSmash:j_idt204:8:j_idt208:j_idt222",widgetVar:"widget_formSmash_j_idt204_8_j_idt208_j_idt222",onLabel:"Hide others...",offLabel:"Show others..."}); 2010 (English)In: BIT Numerical Mathematics, ISSN 0006-3835, E-ISSN 1572-9125, Vol. 50, no 2, p. 301-329Article in journal (Refereed) Published
##### Abstract [en]

##### Place, publisher, year, edition, pages

Springer, 2010
##### Keywords

Periodic systems, Periodic Riccati differential equations, Orbital stabilization, Periodic real Schur form, Periodic eigenvalue reordering, Hamiltonian systems, Linear matrix inequalities, Numerical methods
##### National Category

Computer Sciences
##### Research subject

Numerical Analysis
##### Identifiers

urn:nbn:se:umu:diva-39652 (URN)10.1007/s10543-010-0257-5 (DOI)000277283100005 ()2-s2.0-77952010194 (Scopus ID)
#####

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Available from: 2011-02-03 Created: 2011-02-03 Last updated: 2023-03-24Bibliographically approved

Department of Mathematics and Mechanics, St. Petersburg State University, St. Petersburg, Russia.

Umeå University, Faculty of Science and Technology, Department of Computing Science.

Umeå University, Faculty of Science and Technology, Department of Computing Science.

Umeå University, Faculty of Science and Technology, Department of Applied Physics and Electronics.

Institute of Robotics and Mechatronics, German Aerospace Center, DLR, Germany.

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.

Open this publication in new window or tab >>A Numerical Evaluation of Solvers for the Periodic Riccati Differential Equation### Gusev, Sergei

### Johansson, Stefan

### Kågström, Bo

### Shiriaev, Anton

PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt204_9_j_idt208_some",{id:"formSmash:j_idt204:9:j_idt208:some",widgetVar:"widget_formSmash_j_idt204_9_j_idt208_some",multiple:true}); ### Varga, Andras

PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt204_9_j_idt208_otherAuthors",{id:"formSmash:j_idt204:9:j_idt208:otherAuthors",widgetVar:"widget_formSmash_j_idt204_9_j_idt208_otherAuthors",multiple:true}); Show others...PrimeFaces.cw("SelectBooleanButton","widget_formSmash_j_idt204_9_j_idt208_j_idt222",{id:"formSmash:j_idt204:9:j_idt208:j_idt222",widgetVar:"widget_formSmash_j_idt204_9_j_idt208_j_idt222",onLabel:"Hide others...",offLabel:"Show others..."}); 2009 (English)Report (Other academic)
##### Abstract [en]

##### Place, publisher, year, edition, pages

Umeå: Institutionen för datavetenskap, Umeå universitet, 2009. p. 34
##### Series

Report / UMINF, ISSN 0348-0542 ; 09.03
##### Keywords

Periodic systems, periodic Riccati differential equations, orbital stabilization, periodic real Schur form, periodic eigenvalue reordering, Hamiltonian systems, linear matrix inequality, numerical methods
##### National Category

Computer Sciences
##### Research subject

Numerical Analysis
##### Identifiers

urn:nbn:se:umu:diva-18476 (URN)
#####

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Available from: 2009-02-12 Created: 2009-02-10 Last updated: 2018-06-09Bibliographically approved

Department of Mathematics and Mechanics, St. Petersburg State University, St. Petersburg, Russia.

Umeå University, Faculty of Science and Technology, Department of Computing Science.

Umeå University, Faculty of Science and Technology, Department of Computing Science.

Umeå University, Faculty of Science and Technology, Department of Applied Physics and Electronics.

Institute of Robotics and Mechatronics, German Aerospace Center, DLR, Oberpfaffenhofen, Germany.

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.