Rank Invariants of Nonlinear Systems

Abstract. We introduce flat systems, which are equivalent to linear ones via a special type of feedback called endogenous. Their physical properties are subsumed by a linearizing output and they might be regarded as providing another nonlinear extension of Kalman's controllability. The distance to flatness is measured by a non-negative integer, the defect. We utilize differential algebra which suits well to the fact that, in accordance with Willems' standpoint, flatness and defect are best defined without distinguishing between input, state, output and other variables. We treat an example of non-flat system, the variable-length pendulum. A high frequency control strategy is proposed such that the averaged system becomes flat.

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“…The closed-loop averaged system (12,13,15) admits an hyperbolic equilibrium point (u sp , q sp ), which is asymptotically stable.…”

Section: 4mentioning

confidence: 99%

Differential flatness and defect: an overview

Fliesś

Lévine

Martin

et al. 1995

Banach Center Publ.

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“…They also appeared as related to the solutions of other important control problems such as disturbance decoupling. The reader is referred to the numerous papers in the literature, for example di Benedetto et al (1989); Aït Amirat (1994); Belur (2011);Laakkonen and Pohjolainen (2011); Pereira da Silva et al (2006).…”

Section: Introductionmentioning

confidence: 99%

“…referring to di Benedetto et al (1989), the structure at infinity was defined from a finite sequence of natural integers ρ 1 , ρ 2 , . .…”

Section: Introductionmentioning

confidence: 99%

On a differential algebraic system structure theory

Diop

Amirat²

2013

IFAC Proceedings Volumes

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Abstract:The structure at infinity and the essential structure are two control theory notions which were first defined for linear, then for the so-called affine, systems. And they were shown to be useful tools for the study of the fundamental problem of noninteracting control. They also appeared as related to the solutions of other important control problems such as disturbance decoupling. Their definitions are however entirely in terms of an algorithm, namely the so-called structure algorithm. The present work proposes new definitions with some advantages: they extend the class of systems from linear and affine systems to systems which may be described by algebraic differential equations, they are not tied to specific algorithms, and finally they provide more information on system structure. Let a system be a set of differential equations in variables which are grouped as m inputs, p outputs and n latent variables. To each input component is attached a rational integer, which, for a single input single output system defined by a single differential equation, is the difference between the order in the output and the order in the input of the differential equation defining the system. The m-tuple of these rational integers is the new structure at infinity of the system. Associated to the structure at infinity is also defined a p-tuple of rational integers representing a new notion of essential structure. The old structure at infinity is shown to be recoverable from the new one. Computations of system structure based upon the suggested definitions are quite complex. The present paper focuses on proofs of algorithms which attempt to reduce the complexity of these computations.

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“…Let (x) be a right inverse of (x). Letting u =-(x)(x), we find from (8) that (9) (X)-(X)(X)(X) =: (X) Note that, since each row.of B(x) is linearly dependent on the rows of B(x), (x) is independent of the choice of (x). Assume that (Xo)=0 and that (x) has constant rank s in a neighborhood of Xo in N-I.…”

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confidence: 99%

“…[22], [35]) and Singh's algorithm (cf. [34], [9]). Basically, the method used in this paper to solve the nDDDP is an extension of the method given in [17] to solve the DDDP.…”

mentioning

confidence: 99%

A Nonregular Solution of the Nonlinear Dynamic Disturbance Decoupling Problem with an Application to a Complete Solution of the Nonlinear Model Matching Problem

Huijberts¹

1992

SIAM J. Control Optim.

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Citation for published version (APA):Huijberts, H. J. C. (1992). A nonregular solution of the nonlinear dynamic disturbance decoupling problem with an application to a complete solution of the nonlinear model matching problem. SIAM Journal on Control and Optimization, 30 (2) Please check the document version of this publication:• A submitted manuscript is the version of the article upon submission and before peer-review. There can be important differences between the submitted version and the official published version of record. People interested in the research are advised to contact the author for the final version of the publication, or visit the DOI to the publisher's website.• The final author version and the galley proof are versions of the publication after peer review.• The final published version features the final layout of the paper including the volume, issue and page numbers. Link to publication General rightsCopyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright owners and it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights.• Users may download and print one copy of any publication from the public portal for the purpose of private study or research.• You may not further distribute the material or use it for any profit-making activity or commercial gain • You may freely distribute the URL identifying the publication in the public portal.If the publication is distributed under the terms of Article 25fa of the Dutch Copyright Act, indicated by the "Taverne" license above, please follow below link for the End User Agreement:www.tue.nl/taverne Take down policy If you believe that this document breaches copyright please contact us at:openaccess@tue.nl providing details and we will investigate your claim.

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Rank Invariants of Nonlinear Systems

Cited by 164 publications

References 19 publications

Differential flatness and defect: an overview

Differential flatness and defect: an overview

On a differential algebraic system structure theory

A Nonregular Solution of the Nonlinear Dynamic Disturbance Decoupling Problem with an Application to a Complete Solution of the Nonlinear Model Matching Problem

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