An overview of the applications and solutions of a fundamental matrix equation pair

Tsui, Chia-Chi

doi:10.1016/j.jfranklin.2004.05.004

Cited by 15 publications

(10 citation statements)

References 18 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Subsection 3.1 also shows that under separation principle of more than half of a century, satisfactory solution to this problem does not generally exist. Subsection 3.2 shows a synthesized design principle in which K is now designed based on the key observer parameter T and the system parameter C. With a now fully adjustable observer order and based on the solution of (5) as described in Section 2, exact solution to (5) and (10) exists for almost all systems [81] . The clear and fundamental reasons for the superiority [23] of this synthesized principle over the existing separation principle are discussed.…”

Section: Discussionmentioning

confidence: 99%

“…Because robustness is the most critical property of feedback control, this development improves the state space control theory so decisively that the theory becomes really and generally useful now, and becomes far more effective than classical control theory. Section 4 describes the direct application of the solution of (5) and (10) in two more special observer design problems, i.e., fault detection/identification and systems with time-delay effects [81] . The dual version of this solution of (5) and (10) also made the solution of the critical SOFC (or generalized state feedback control) design procedure from numerical to analytical, and this analytical solution made the remaining design freedom as clear as that of [9] and thus made the eigenvector assignment possible [5,10,11,81] .…”

Section: Discussionmentioning

confidence: 99%

See 1 more Smart Citation

Observer design — A survey

Tsui

2015

Int. J. Autom. Comput.

Self Cite

View full text Add to dashboard Cite

This paper surveys the results of observer design for linear time-invariant (L-T-I) deterministic irreducible open-loop systems (OLS), the most basic type of OLS. An observer estimates Kx x x(t) signal where K is a constant and x x x(t) is the state vector of the OLS. Thus, an observer can be used as a feedback controller that implements state feedback control (SFC) or Kx x x(t)-control, and observer design is therefore utterly important in all feedback control designs of state space theory. In this survey, the observer design results are divided into three categories and for three respective main purposes. The first category of observers estimate signal Kx x x(t) only with a given K, and this survey has four conclusions: 1) Function observer that estimates Kx x x(t) directly is more general than state observer that estimates x x x(t), and may be designed with order lower than that of state observer, and the additional design objective is to minimize observer order; 2) The function observer design problem has already been simplified to the solving of a single set of linear equations only while seeking the lowest possible number of rows of the solution matrix, and an apparently most effective and general algorithm of solving such a problem can guarantee unified upper and lower bounds of the observer order; 3) Because such a single set of linear equations is the simplest possible theoretical formulation of the design problem and such theoretical observer order bounds are the lowest possible, and because the general, simple, and explicit theoretical formula for the function observer order itself do not exist, the theoretical part of this design problem is solved; 4) Because the function observer order is generically near its upper bound, further improvement on the computational design algorithm so that the corresponding observer order can be further reduced, is generically not worthwhile. The second category of observers further realize the loop transfer function and robustness properties of the direct SFC, and the conclusion of this survey is also fourfold: 1) To fully realize the loop transfer function of a practically designed Kx x x(t)-control, the observer must be an output feedback controller (OFC) which has zero gain to OLS input; 2) If parameter K is separately designed before the observer design, as in the separation principle which has been followed by almost all people for over half of a century, then OFC that estimates Kx x x(t) does not exist for almost all OLS s; 3) As a result, a synthesized design principle that designs an OFC first and is valid for almost all OLS s, is proposed and fully developed, the corresponding K will be designed afterwards and will be constrained by the OFC order as well as the OFC parameters; 4) Although the Kx x x(t)-control is constrained in this new design principle and is therefore called the "generalized SFC" (as compared to the existing SFC in which K is unconstrained), it is still strong enough for most OLS s and this new design principle overcomes many fund...

show abstract

Section: Discussionmentioning

confidence: 99%

Section: Discussionmentioning

confidence: 99%

Observer design — A survey

Tsui

2015

Int. J. Autom. Comput.

Self Cite

View full text Add to dashboard Cite

show abstract

“…On the one hand, we find those who start with the solution of the Sylvester equation which appears in the existence conditions of such an observer. Let us mention for example [30,33]. On the other hand, we have the methods as those proposed in [6,7] which build, by using the result of [5], the observer of a bigger size functional, which includes of course the functional to observe.…”

Section: Introductionmentioning

confidence: 99%

Some new standpoints in the design of asymptotic functional linear observers

Rotella

Zambettakis

2015

ESAIM: Proc.

View full text Add to dashboard Cite

Abstract. The owl of Minerva spreads its wing only with the falling of the dusk. The aim of this work is to provide some new trends in the observation for linear systems. In the general framework of designing linear functional observers for linear systems the necessary and sufficient existence conditions are well known. Whether in the O'Reilly textbook or in the recently published ones on this topic, and, roughly speaking, the design methods can be categorized in two kinds. The first one is based on the solution of a Sylvester equation and a projection of the observed linear functional. The second one, based on the recent notion of functional observability, starts from the Darouach criterion which is an Popov-Belevitch-Hautus type one. Nevertheless, the main drawback of the deduced methods is that they cannot be used for linear time-varying systems. These models are of primary importance, for instance with linearization about a trajectory. Consequently, we cope with this problem by considering a new point of view for the design of linear functional observers. We see also that Darouach observers or Cumming-Gopinath observers are particular cases of the proposed methodology. For simplicity sake we suppose the system has no unknown inputs and is not described by a distributed parameters model as well. Nevertheless, these cases can be thought as possible extensions of the presented standpoints.Résumé. La chouette de la connaissance ne vole qu'à la nuit tombée. L'objet de cet article est de proposer des pistes pour l'observation dans le cadre des systèmes linéaires. De façon plus générale, en considérant l'observation de fonctions linéaires de l'état d'un modèle linéaire, les conditions nécessaires et suffisantes d'existence de ces observateurs sont connues depuis longtemps. Que ce soit dans l'ouvrage de base de O'Reilly ou dans les ouvrages plus récents qui traitent de ce sujet, les différentes méthodes de conception d'un observateur de fonctionnelle linéaire peuvent, schématiquement,être rassemblées en deux groupes. Celles qui demandent la résolution d'uneéquation de Sylvester et d'uneéquation de projection et celles basées sur l'extension du critère d'existence de Darouach, ces dernières utilisant un critère de type Popov-Belevitch-Hautus. Ainsi les méthodes déduites de ces critères peuventêtre difficilementétendues aux systèmes linéaires non stationnaires. De façonà contourner cet inconvénient majeur lorsque l'on songeà la linéarisation autour d'une trajectoire, nous proposons une technique pratique et systématique de conception d'un observateur. On montre par exemple que les observateurs de Darouach ou de Cumming-Gopinath n'en sont que des cas particuliers. Bien sûr, pour ne pas compliquer inutilement la présentation nous n'envisagerons que des modèles linéaires stationnaires de dimension finieà entrées toutes connues, les autre situations pouvantêtre envisagéesà titre d'extension.

show abstract

“…Under the R-controllability of the matrix triple (E, A, B), Duan [19] Manuscript received March 26, 2008 gave a complete and explicit solution which uses the right coprime factorization of the input-state transfer function (sE − A) −1 B, while Duan [21] proposed a complete parametric solution which is not in a direct, explicit form but in a recursive form. The advantages of setting F in Jordan form in (3) were described again in [22,23] to enable uniquely that the corresponding solutions are decoupled. This is useful in the observer design because it enables the full realization of the critical robustness properties of state feedback control for most systems, and a systematic design of minimal order functional observers [22,23] .…”

Section: Introductionmentioning

confidence: 99%

“…The advantages of setting F in Jordan form in (3) were described again in [22,23] to enable uniquely that the corresponding solutions are decoupled. This is useful in the observer design because it enables the full realization of the critical robustness properties of state feedback control for most systems, and a systematic design of minimal order functional observers [22,23] . It is also useful in eigenvalue/vector assignment because the corresponding solutions are actually the eigenvector matrix [22,23] .…”

Section: Introductionmentioning

confidence: 99%

Closed-form solutions to the matrix equation AX − EXF = BY with F in companion form

Zhou

Duan

2009

Int. J. Autom. Comput.

View full text Add to dashboard Cite

A closed-form solution to the linear matrix equation AX − EXF = BY with X and Y unknown and matrix F being in a companion form is proposed, and two equivalent forms of this solution are also presented. The results provide great convenience to the computation and analysis of the solutions to this class of equations, and can perform important functions in many analysis and design problems in descriptor system theory. The results proposed here are parallel to and more general than our early work about the linear matrix equation AX − XF = BY .

show abstract

An overview of the applications and solutions of a fundamental matrix equation pair

Cited by 15 publications

References 18 publications

Observer design — A survey

Observer design — A survey

Some new standpoints in the design of asymptotic functional linear observers

Closed-form solutions to the matrix equation AX − EXF = BY with F in companion form

Contact Info

Product

Resources

About