A nyquist criterion for time-varying periodic systems, with application to a hydraulic test bench

In this paper, stability results dedicated to sampled periodic systems are applied to a mechanical system whose stiffness exhibits quick variations: a hydraulic test bench used to achieve mechanical test on complex structures. To carry out this application, time-varying w transformation representation of sampled periodic systems are first introduced. An extension of the Nyquist Criterion to sampled periodic systems is then given. Finally, this theorem is applied to evaluate the stability degree of the hydraulic test bench controlled using CRONE control methodology

Analyse et commande CRONE de systèmes linéaires non stationnaires

Le thème génèral des travaux qui font l'objet de ce mémoire est l'extention de la commande CRONE à la commange de systèmes non stationnaires tel que les systèmes non stationnaires à coefficents périodiques et les systèmes non staionnaires à coefficents asymptotiquement constants. Ces extensions, menées à la fois dans le domaine continu et dans le domaine discret, ont été permises grâce à la représentation des systèmes considérés au moyen de transmittances en p non stationnaires (pour les systèmes continus) et de transmittances en z non stationnaires (pour les systèmes discrets). Ces représentations ont également permis d'étendre des théorèmes bien connus tels que le théorème de la valeur finale et de la valeur initiale aux systèmes non stationnaires, ainsi que de formuler plusieur extensions du critère de Nyquist. Chaque théorème proposé et chaque extension de la commande CRONE réalisée dans le cadre de ces travaux a été validé par son application à la commande d'un banc d'essai composé de 2 moteurs à courant continu.This work deals with the extension of CRONE Control (robust control method based on fractional calculus) to the control fo time-varying systems such as time-varying systems with periodic coefficients and time-varying systems with asymptotically constant coefficients. These extensions, carried out both in the continuous and in the discret-time domain, have been feasible thanks to the representations of the considered systems using time-varying p-transfer functions (for continuous-time systems) and z-transfer functions (for discrete-time systems). These representations have also allowed to extend several well-known theorems such as initial and final value theorems to time-varying systems and have also allowed several extensions of Nyquist theorem. Each theorem and each extension fo CRONE control carried out in this work has been validated through their application to control of a testing bench with two DC motors.BORDEAUX1-BU Sciences-Talence (335222101) / SudocSudocFranceF

A Nyquist criterion for time-varying periodic systems, with application to a hydraulic test bench

Abstract: In this paper, stability results dedicated to sampled periodic systems are applied to a mechanical system whose stiffness exhibits quick variations: a hydraulic test bench used to achieve mechanical test on complex structures. To carry out this application, time-varying w transformation representation of sampled periodic systems are first introduced. An extension of the Nyquist Criterion to sampled periodic systems is then given. Finally, this theorem is applied to evaluate the stability degree of the hydraulic test bench controlled using CRONE control methodology. Key words: Time-varying system, Periodic systems, CRONE Control, stability degree, hydraulic actuator In this paper, extensions of Nyquist criterion developed in [Garcia 01] is applied to the analysis of the stability degree of a testing bench constituted of a hydraulic actuator used to achieve mechanical deformations on complex structures. Given parametric variations of the parameters of the structure and given quick variations of the structure stiffness during the test, the testing bench whose velocity is controlled using a robust CRONE controller, behaves as a sampled time-varying system with parametric uncertainties. Given that the velocity of the actuator is controlled on a finite time interval, this time-varying system can be artificially considered as a sampled periodic system. The paper is organized as follows. Section 2 deals with the representation of continuous time-varying systems using TVWTs and TVPFRs. Section 3 is dedicated to the stability analysis of sampled time-varying systems with periodic coefficients. Section 4 first gives a presentation of the testing bench and of its control using CRONE control methodology. Then this section presents the stability degree analysis of the testing bench using the extension of Nyquist criterion given in section 3. -Introduction -Continuous-time periodic systems -Definitions and hypothesis A linear time-varying continuous-time system with periodic coefficients, also called periodic continuous-time system, is described by the state variable equation: where A(t), B(t), C(t) and D(t) are supposed to be A(t), B(t), C(t) and D(t) and their first derivatives are also supposed to be respectively continuous and piecewise continuous on [0, T]. They can thus be developed in uniformly convergent Fourier series of the form

CRONE control of continuous linear time periodic systems: application to a testing bench

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