Analysis and optimization of certain qualities of controllability and observability for linear dynamical systems

doi:10.1016/0005-1098(72)90044-1

Automatica

Volume 8, Issue 3, May 1972, Pages 237-246

https://doi.org/10.1016/0005-1098(72)90044-1 Get rights and content

Abstract

Controllability and observability criteria provide an answer to the binary question of whether or not a linear dynamical system is completely controllable and/or completely observable. However, these criteria do not associate or identify a degree of quality of complete controllability or complete observability. Therefore, the problem of assigning physically meaningful measures of the quality of controllability and observability is considered here. Three physical measures—determinant, trace, and maximal eigenvalue of the inverse characteristic controllability or observability matrix—are imbedded in a set of measures which is defined as the set of certain means related to the eigenvalues of the characteristic matrix. By this measure concept, the qualities of controllability and observability are maximized with respect to certain adjustable structural parameters of the dynamical system. Some illustrative applications in satellite attitude control problems are discussed in detail.

Résumé

Les critères d'aptitude à la commande et d'aptitude à l'observation fournissent une reponse à la double question de savoir si un système dynamique linéaire peut ou non être complètement commandé et/ou être complètement observé. Toutefois, ces critères ne comportent pas et n'identifient pas un degré de qualité d'aptitude complète à la commande ou d'aptitude complète à l'observation. L'article considère en conséquence le problème d'assigner des mésures ayant une signification physique à la qualité d'aptitude à la commande et d'aptitude à l'observation. Trois mesures physiques—le déterminant, le tracé et la valeur propre maximale de la matrice caractéristique inverse d'aptitude à la commande ou de l'aptitude à l'observation—sont contenues dans une série de mesures qui est définie comme une série de certaines moyennes liées aux valeurs propres de la matrice caractéristique. A l'aide de ce concept de mesure, les qualités d'aptitude à la commande et d'aptitude à l'observation sont maximalisées par rapport à certains paramètres règlables de la structure du système dynamique. Certaines applications, prises à titre d'exemple, dans les problèmes de commande d'altitude de satellites sont discutées en détail.

Zusammenfassung

Kriterien der Kontrollierbarkeit und der Beobachtungsmöglichkeit geben die Antwort auf die binäre Frage, ob oder ob nicht ein lineares, dynamisches System völlig kontrollierbar und/oder völlig zu beobachten ist. Trotzdem verbinden oder bezeichnen diese Kriterien nicht einen Qualitätsgrad für völlige Kontrollierbarkeit oder völlige Möglichkeit der Beobachtung. Daher wird das Problem hier betrachtet, physische bedeutungsvolle Messungen der Qualität der Kontrollierbarkeit und der Möglichkeit der Beobachtung festzulegen. Drei physische Masstäbe—Determinante, Spur und maximaler Eigenwert der inversen, charakteristischen Kontrollierbarkeit—oder Beobachtungsmöglichkeitmatrix—sind in einem Satz von Messungen eingeschlossen, der als der Satz bestimmter, auf die Eigenwerte der charakteristischen Matrix bezogenen Mittel definiert wird. Durch diese Auffassung der Bemessung werden die Eigenschaften der Kontrollierbarkeit und Beobachtungsmöglichkeit mit Bezug auf bestimmte, verstellbare, strukturelle Parameter des dynamischen Systems maximalisiert. Einige illustrierende Anwendungen in Kontrollproblemen des Verhaltens von Satelliten werden in Einzelheiten besprochen.

Реферат

Кpитepии yпpaвляeмocти и нaблюдaeмocти дaют oтвeт нa двoйнoй вoпpoc o вoзмoжнocти или нeвoзмoжнocти для линeйнoй динaмичecкoй cиcтeмы быть впoлнe yпpaвляeмoй и/или впoлнe нaблюдaeмoй. Oднaкo, эти кpитepии нe включaют и нe oпpeдeляют cтeпeнь кaчecтвa пoлнoй yпpaвляeмocти или пoлнoй нaблюдaeмocти. Бcлeдcтвии этoгo, cтaтья paccмaтpивaeт пpoблeмy пpилoжeния к кaчecтвy yпpaвляeмocти и нaблюдaeмocти измepeний имeющич физичecкий cмыcл. Tpи физичecкиe измepeния—дeтepминaнт, тpaccиpoвкa и мaкcимaльнoe coбcтвeннoe знaчeниe oбpaтнoй мaтpицы чapaктepизyющeй yпpaвляeмocть или нaблюдaeмocть—coдepжaтcя в pядe измepeний кoтopый oпpeдeляeтcя кaк pяд cpeдныч знaчeний cвязaнныч c coбcтвeнными знaчeниями чapaктepизиpyющeй мaтpицы. C пoмoщью этoгo измepитeльнoгo кoнцeптa, кaчecтвa yпpaвляeмocти и нaблюдaeмocти пpивoдятcя к мaкcимyмy пo oтнoшeнию к нeкoтopым измeняeмым cтpyктypным пapaмeтpaм динaмичecкoй cиcтeмы. Cтaтья дeтaльнo oбcyждaeт, в кaчecтвe пpимepa, нeкoтopыe пpимeнeния в пpoблeмe yпpaвлeния пoлoжeниeм opбитныч кocмичecкич кopaблeй.

References (23)

R.E. Kalman
Contributions to the theory of optimal control
Bol. Soc. Mat. Mex.
(1960)
R.E. Kalman
On the general theory of control systems
R.E. Kalman et al.
Controllability of linear dynamical systems
Contrib. Differential Equations
(1962)
R.E. Kalman
Mathematical description of linear dynamical systems
J. SIAM Control
(1963)
R.G. Brown
R.A. Monzingo
A note on sensitivity of system observability
IEEE Trans. Aut. Control
(1967)
C.D. Johnson
Optimization of a certain quality of complete controllability and observability for linear dynamical systems
ASME Trans. J. Basic Engng
(1969)
R.E. Kalman et al.
New results in linear prediction and filtering theory
ASME Trans. J. Basic Engng
(1961)
E.F. Beckenbach et al.
F.R. Gantmacher
Matrizenrechnung I, II
(1965)

P. Lancaster

Theory of Matrices

(1969)

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^☆: The original version of this paper was presented at the 2nd IFAC Symposium on Multivariable Technical Control Systems which was held in Düsseldorf, Germany during October 1971. It was recommended for publication in revised form by Associate Editor P. Dorato.

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Abstract

Résumé

Zusammenfassung

Реферат

Contributions to the theory of optimal control

Bol. Soc. Mat. Mex.

On the general theory of control systems

Controllability of linear dynamical systems

Contrib. Differential Equations

Mathematical description of linear dynamical systems

J. SIAM Control

A note on sensitivity of system observability

IEEE Trans. Aut. Control

Optimization of a certain quality of complete controllability and observability for linear dynamical systems

ASME Trans. J. Basic Engng

New results in linear prediction and filtering theory

ASME Trans. J. Basic Engng

Matrizenrechnung I, II

Theory of Matrices