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Maxplus algebra and system theory: Where we are and where to go now
 Annu. Rev. Control
, 1999
"... Abstract: More than sixteen years after the beginning of a linear theory for certain discrete event systems in which maxplus algebra and similar algebraic tools play a central role, this paper attempts to summarize some of the main achievements in an informal style based on examples. By comparison ..."
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Abstract: More than sixteen years after the beginning of a linear theory for certain discrete event systems in which maxplus algebra and similar algebraic tools play a central role, this paper attempts to summarize some of the main achievements in an informal style based on examples. By comparison with classical linear system theory, there are areas which are practically untouched, mostly because the corresponding mathematical tools are yet to be fabricated. This is the case of the geometric approach of systems which is known, in the classical theory, to provide another important insight to systemtheoretic and controlsynthesis problems, beside the algebraic machinery. A preliminary discussion of geometric aspects in the maxplus algebra and their use for system theory is proposed in the last part of the paper. Résumé: Plus de seize ans après le début d’une théorie linéaire de certains systèmes à événements discrets dans laquelle l’algèbre maxplus et autres outils algébriques assimilés jouent un rôle central, ce papier cherche àdécrire quelques uns des principaux résultats obtenus de façon informelle, en s’appuyant sur des exemples. Par comparaison avec la théorie classique des systèmes linéaires, il existe des domaines pratiquement vierges, surtout en raison du fait que les outils mathématiques correspondants restent à forger. C’est en particulier le cas de l’approche géométrique des systèmes qui, dans la théorie classique, est connue pour apporter un autre regard important sur les questions de théorie des systèmes et de synthèse de lois de commandes àcôté de la machinerie purement algébrique. Une discussion préliminaire sur les aspects géométriques de l’algèbre maxplus et leur utilité pour la théorie des systèmes est proposée dans la dernière partie du papier.
Scherpen, “Eigenstructure of nonlinear Hankel operators
 in Nonlinear Control in the Year 2000, ser. Lecture
"... Abstract. This paper investigates the eigenstructure of Hankel operators for nonlinear systems. It is proved that the variational system and Hamiltonian extension can be interpreted as the Gâteaux differentiation of dynamical inputoutput systems and their adjoints respectively. We utilize this diff ..."
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Abstract. This paper investigates the eigenstructure of Hankel operators for nonlinear systems. It is proved that the variational system and Hamiltonian extension can be interpreted as the Gâteaux differentiation of dynamical inputoutput systems and their adjoints respectively. We utilize this differentiation in order to clarify the eigenstructure of the Hankel operator, which is closely related to the Hankel norm of the original system. The results in the paper thus provide new insights to the realization and balancing theory for nonlinear systems. 1
On Adjoints and Singular Value Functions for Nonlinear Systems
, 2000
"... this paper we study the nonlinear extension of such adjoint operators, and apply the results to Hankel theory. Nonlinear adjoint operators can be found in the mathematics literature, e.g. [1], and they are expected to play a similar role in the nonlinear systems theory. So called nonlinear Hilbert a ..."
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Cited by 3 (2 self)
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this paper we study the nonlinear extension of such adjoint operators, and apply the results to Hankel theory. Nonlinear adjoint operators can be found in the mathematics literature, e.g. [1], and they are expected to play a similar role in the nonlinear systems theory. So called nonlinear Hilbert adjoint operators are introduced in [5, 11] as a special class of nonlinear adjoint operators. The existence of such operators in an inputoutput sense was shown in [6], but their statespace realizations are only preliminary available in [4], where the main interest is the Hilbert adjoint extension with an emphasis on the use of portcontrolled Hamiltonian system methods. Here, we consider these adjoint operators from a variational point of view and provide a formal justification for the use of Hamiltonian extensions by using Gateaux derivatives. We investigate whether one can use the statespace realizations given by the Hamiltonian extensions to characterize singular values of nonlinear operators, and, in particular, for the Hankel operator. We also consider the relation with the previously defined singular value functions that have been defined entirely from the controllability and observability functions corresponding to a state space representation of a nonlinear system [10]. In Section 2 we present the linear system case as a paradigm, in order to present the line of thinking for the nonlinear case. In Section 3 we present the statespace realizations of nonlinear adjoint operators, in terms of Hamiltonian extensions. In Section 4 we provide the formal justification of the use of Hamiltonian extensions for nonlinear adjoint systems. In Section 5 we concentrate on the Hankel operator, and correspondingly on the controllability and observability operators for nonlinear sy...