State-space realization of nonlinear control systems: unification and extension via pseudo-linear algebra

summary:In this paper the tools of pseudo-linear algebra are applied to the realization problem, allowing to unify the study of the continuous- and discrete-time nonlinear control systems under a single algebraic framework. The realization of nonlinear input-output equation, defined in terms of the pseudo-linear operator, in the classical state-space form is addressed by the polynomial approach in which the system is described by two polynomials from the non-commutative ring of skew polynomials. This allows to simplify the existing step-by-step algorithm-based solution. The paper presents explicit formulas to compute the differentials of the state coordinates directly from the polynomial description of the nonlinear system. The method is straight-forward and better suited for implementation in different computer algebra packages such as \textit{Mathematica} or \textit{Maple}

Nabla derivatives associated with nonlinear control systems on homogeneous time scales

The backward shift and nabla derivative operators, defined by the control system on homogeneous time scale, in vector spaces of one-forms and vector fields are introduced and some of their properties are proven. In particular the formulas for components of the backward shift and nabla derivative of an arbitrary vector field are presented

Une approche par l’analyse algébrique effectivedes systèmes linéaires sur des algèbres de Ore

The purpose of this paper is to present a survey on the effective algebraic analysis approach to linear systems theory with applications to control theory and mathematical physics. In particular, we show how the combination of effective methods of computer algebra - based on Gröbner basis techniques over a class of noncommutative polynomial rings of functional operators called Ore algebras - and constructive aspects of module theory and homological algebra enables the characterization of structural properties of linear functional systems. Algorithms are given and a dedicated implementation, called OreAlgebraicAnalysis, based on the Mathematica package HolonomicFunctions, is demonstrated.Le but de ce papier est de présenter un état de l’art d’une approche par l’analyse algébrique effective de la théorie des systèmes linéaires avec des applications à la théorie du contrôle et à la physique mathématique.En particulier, nous montrons comment la combinaison des méthodes effectives de calcul formel - basées sur lestechniques de bases de Gröbner sur une classe d’algèbres polynomiales noncommutatives d’opérateurs fonctionnels appelée algèbres de Ore - et d’aspects constructifs de théorie des modules et d’algèbre homologique permet lacaractérisation de propriétés structurelles des systèmes linéaires fonctionnels. Des algorithmes sont donnés et uneimplémentation dédiée, appelée OREALGEBRAICANALYSIS, basée sur le package Mathematica HOLONOMIC-FUNCTIONS, est présenté

Minimal realizations of nonlinear systems

International audienceThe nonlinear realization theory is recasted for time-varying single-input single-output nonlinear systems. The concept of realization has been extended to cover also the realizations with order greater than the order of input–output equation. The minimal realization problem is studied. The state realization is said to be minimal if it is either accessible and observable or its state dimension is minimal. In the linear case the two definitions are equivalent, but not for nonlinear time-invariant systems. It is shown that the two definitions remain equivalent for nonlinear systems under certain technical assumptions. Two alternative methods are presented for finding the minimal realization

Port-Hamiltonian framework in power systems domain: A survey

Many nonlinear physical phenomena, including various power systems, can be modeled, analyzed, and controlled using the framework of port-Hamiltonian systems. Moreover, this framework can offer more advanced methods to cope with modern challenges in the energy sector. This paper presents a comprehensive and systematic survey of recent studies on the application of the port-Hamiltonian approach to power systems. Over a hundred of relevant research works are reviewed to show the vast capabilities of this approach and point out its possible gaps. The works are classified according to the type of power systems under study. The analysis of the articles shows that the vast majority of them are dedicated to controller design, a much smaller part of the works deals with the modeling and stability issues, and only a few consider the problem of optimal control. Moreover, the paper discusses current challenges and future trends in this direction

Remarks on realization of time-varying systems

International audienceThe realization problem of nonlinear time-varying input–output equations is considered. Differentials of the statecoordinates, necessary for realization, are determined by the vector space of differential one-forms, spanned over the field of meromorphic functions. Formulas for computing the basis one-forms are given, based on the Euclidean division of non-commutative polynomials. Moreover, it is shown that in the case of a reducible system, the subspace admits a basis with certain structure, explicitly related to reduced input–output equations

Feedback linearization of an active magnetic bearing system operated with a zero–bias flux

Input-output linearization by state feedback is applied to a flux-controlled active magnetic bearing (AMB) system, operated in the zero-bias mode. Two models of the AMB system are employed. The first one is described by the third-order dynamics with a flux-dependent voltage switching scheme, whereas the second one is the fourth-order system, called self-sensing AMB, since it does not require the measurement of the rotor position. In the case of that system we had to find the flat outputs to guarantee its stability. The proposed control schemes are verified by means of numerical simulations performed within the Matlab environment