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Tools for Control System Design: Stratification of Matrix Pairs and Periodic Riccati Differential Equation Solvers
Umeå University, Faculty of Science and Technology, Departement of Computing Science.
2009 (English)Doctoral 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).

Abstract [sv]

Modern reglerteknik är idag i högsta grad ett interdisciplinärt forskningsområde. Lika mycket som detta kan vara problematiskt, resulterar det i en stimulerande forskningsmiljö där både praktik och teori knyts samman. Denna avhandling är utförd i gränsområdet mellan datavetenskap (numerisk analys) och tillämpad reglerteknik. Att designa och analysera ett modernt styrsystem är ett komplext problem som erfordrar högkvalitativ mjukvara. Det ideala är att mjukvaran består av robusta metoder och numeriskt stabila algoritmer som kan leverera både kvantitativ och kvalitativ information.Introduktionen till avhandlingen beskriver grundläggande styr- och reglerteori samt ger en introduktion till de huvudsakliga problemställningarna. Genom hela avhandlingen illustreras teori med exempel. Vidare belyses likheter och skillnader i terminologin som används inom matematik, styr- och reglerteori samt numerisk linjär algebra. Avhandlingen är uppdelade i två delar som behandlar två i huvudsak orelaterade problemklasser.

Del I ägnas åt den kvalitativa informationen som ges av stratifiering av mångfalder (orbits och bundles) av matriser, matrisknippen och systemknippen. Innan teorin för stratifiering introduceras beskrivs olika kanoniska former, vilka var och en avslöjar olika systemegenskaper hos den undersökta modellen. En stratifiering ger information om bl.a. vilka kanoniska strukturer av matrisknippen (systemknippen) som är nära varandra med avseende på små störningar i datat. Fundamentala koncept i styr- och reglerteori behandlas, så som styrbarhet och observerbarhet av linjära tidskontinuerliga system, och hur dessa systemegenskaper kan undersökas med hjälp av stratifiering. Nya resultat presenteras i form av relationerna för täckande (närmsta grannar) styrbarhets- och observerbarhets-par. Dessutom härleds permutationsmatriserna som tar ett matrisknippe i Kroneckers kanoniska form till motsvarande systemknippe i (generaliserade) Brunovskys kanoniska form. Två algoritmer för att bestämma dessa permutationsmatriser presenteras.

Del II avhandlar numeriska metoder för att lösa periodiska Riccati differentialekvationer (PRDE:er). De undersökta PRDE:erna uppkommer när ett linjärt kvadratiskt regulatorproblem för periodiska linjära tidsvariabla (LTV) system löses. Dessa typer av (periodiska) regulatorproblem dyker upp till exempel när man planerar rörelser för understyrda (underactuated) mekaniska system, så som en humanoid (mänsklig) robot, Furuta-pendeln och en vagn med en inverterad (stående) pendel. Konstruktionen av det icke-linjära styrsystemet är baserat på en linjär variant som bestäms via stabilisering av systemets transversella dynamik längs med cirkulära banor.

Tre metoder explicit konstruerade för att lösa PRDE:er evalueras på både artificiella system och stabiliseringsproblem av experimentella styrsystem. Metoderna är sk. en- och flerskotts metoder (one-shot, multi-shot) och SDP-metoden. Då dessa metoder använder olika tillvägagångssätt för att lösa en PRDE, beror dess numeriska egenskaper och effektivitet på det underliggande styrproblemet. Sådana metodegenskaper undersöks och sammanfattas med avseende på olika användares behov, t.ex. önskad noggrannhet och tänkbar begränsning i hur lång tid det får ta att hitta en lösning.

Place, publisher, year, edition, pages
Umeå: Print & Media, Umeå universitet , 2009. , 59 p.
Series
Report / UMINF, ISSN 0348-0542 ; 09.04
Keyword [en]
control system design, stratification, controllability, observability, matrix pair, canonical forms, linear quadratic regulator, periodic systems, periodic Riccati differential equation
National Category
Computer Science
Research subject
Numerical Analysis
Identifiers
URN: urn:nbn:se:umu:diva-18509ISBN: 978-91-7264-733-6 (print)OAI: oai:DiVA.org:umu-18509DiVA: diva2:160135
Public defence
2009-03-06, MA121, Umeå universitet, MIT-huset, Umeå, 13:15 (English)
Opponent
Supervisors
Available from: 2009-02-13 Created: 2009-02-12 Last updated: 2009-02-13Bibliographically approved
List of papers
1. Reviewing the Closure Hierarchy of Orbits and Bundles of System Pencils and Their Canonical Forms
Open this publication in new window or tab >>Reviewing the Closure Hierarchy of Orbits and Bundles of System Pencils and Their Canonical Forms
2009 (English)Report (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.

Place, publisher, year, edition, pages
Umeå: Institutionen för datavetenskap, Umeå universitet, 2009. 88 p.
Series
Report / UMINF, ISSN 0348-0542 ; 09.02
Keyword
Stratification, Jordan canonical form, Kronecker canonical form, Brunovsky canonical form, orbit, bundle, closure relations, cover relations, state-space system, system pencil, matrix pencil, matrix pair
National Category
Computer Science
Research subject
Numerical Analysis
Identifiers
urn:nbn:se:umu:diva-18475 (URN)
Available from: 2009-02-12 Created: 2009-02-10 Last updated: 2011-12-28Bibliographically approved
2. Stratification of controllability and observability pairs: theory and use in applications
Open this publication in new window or tab >>Stratification of controllability and observability pairs: theory and use in applications
2009 (English)In: SIAM Journal on Matrix Analysis and Applications, ISSN 0895-4798, E-ISSN 1095-7162, Vol. 31, no 2, 203-226 p.Article in journal (Refereed) Published
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.

Place, publisher, year, edition, pages
Philadelphia: Society for Industrial and Applied Mathematics, 2009
Keyword
Stratification, matrix pairs, controllability, observability, robustness, Kronecker structures, orbit, bundle, closure hierarchy, cover relations, StratiGraph
National Category
Computer Science
Research subject
Numerical Analysis
Identifiers
urn:nbn:se:umu:diva-18480 (URN)10.1137/080717547 (DOI)
Available from: 2009-12-18 Created: 2009-02-10 Last updated: 2011-03-16Bibliographically approved
3. Comparing One-shot and Multi-shot Methods for Solving Periodic Riccati Differential Equations
Open this publication in new window or tab >>Comparing One-shot and Multi-shot Methods for Solving Periodic Riccati Differential Equations
2007 (English)In: Proceedings of the third IFAC Workshop on Periodic Control Systems (PSYCO’07), International Federation of Automatic Control , 2007, 1-6 p.Conference paper, Published 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.

Place, publisher, year, edition, pages
International Federation of Automatic Control, 2007
Keyword
periodic systems, reordered periodic Schur form, Riccati differential equations, stabilizing controllers, linear quadratic regulators
National Category
Computer Science
Research subject
Numerical Analysis
Identifiers
urn:nbn:se:umu:diva-18477 (URN)978-3-902661-30-2 (ISBN)
Available from: 2009-02-10 Created: 2009-02-10 Last updated: 2009-07-09
4. A Numerical Evaluation of Solvers for the Periodic Riccati Differential Equation
Open this publication in new window or tab >>A Numerical Evaluation of Solvers for the Periodic Riccati Differential Equation
Show others...
2009 (English)Report (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.

Place, publisher, year, edition, pages
Umeå: Institutionen för datavetenskap, Umeå universitet, 2009. 34 p.
Series
Report / UMINF, ISSN 0348-0542 ; 09.03
Keyword
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 Science
Research subject
Numerical Analysis
Identifiers
urn:nbn:se:umu:diva-18476 (URN)
Available from: 2009-02-12 Created: 2009-02-10 Last updated: 2012-01-09Bibliographically approved

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