On the output feedback control of a synchronous generator H. Sira-Ram´ ırez †and M. Fliess ‡ †CINVESTAV-IPN. Departamento de Ing. El´ ectrica.Secci´ondeMecatr´onica. Av. IPN No. 2508. Col. San Pedro Zacatenco. A.P. 14740, 7300 M´ exico, D.F., M´ exico. Phone: + 52(55)50613794, email: hsira@mail.cinvestav.mx ‡Laboratoire Stix, Ecole Polytechnique 91128 Palaiseau, France email: Michel.Fliess@stix.polytechnique.fr Abstract — In this article, we propose an output feed- back regulation scheme for the electrical angle trajectory tracking in a synchronous generator. Exploiting the flat- ness of the system we obtain a trajectory tracking er- ror linearizing feedback controller which depends on the available output and its time derivatives. Using a variant of a recently introduced signal time derivative estimation approach based on the “algebraic derivative method ”, we demonstrate, through computer simulations the effec- tiveness of our method even in the presence of computer generated stochastic perturbations affecting the dynam- ics of the system and the available measurements. I. Introduction Feedback control through state estimation has a long story ever since the original contributions of Kalman and Luenberger within the linear systems setting. The extensions of asymptotic observers to the case of nonlin- ear systems for successfully completing the loop via a de- signed state feedback control law, has been the topic of extensive research in the last 25 years. Just to mention but a few approaches, we have: Approximate lineariza- tion, Extended linearization or pseudo-linearization, the geometric approach and its exact reconstruction error linearization through state coordinate transformations and output injection, and the several attempts involv- ing the algebraic approach. A complete survey of the field is prohibitive for a conference paper. The idea of taking time derivatives of a given output signal is not en- tirely new and some rather interesting approaches have been proposed in the past (See Diop et al [2], Pelestan and Grizzle [9] and Diop et al [4]). What is new in our approach is the involving of a non-commutative algebra approach to state estimation, based on the “algebraic derivative method”, presented by the authors in [7]. In that article the approach pertained to the computation of states in a linear time-invariant system and its impli- cations in, on line, unknown constant parameter calcu- lations (see also, Fliess and Sira-Ram´ ırez [6] ). In this article, we take an algebraic viewpoint for the *This research was supported by the Centro de Investagaci´ on y Estudios Avanzados del IPN (Cinvestav-IPN), M´ exico City, M´ exico. and by Conacyt under Research Grant 42231-Y. The work of M. Fliess was partially supported by the action sp´ ecifique (CNRS, RTP 24) M´ ethodes alg´ ebriques pour les syst` emes de com- munications num´ eriques. state estimation problem, associated to the feedback law synthesis, in general dynamical finite dimensional sys- tems (linear or non linear). The article emphasizes an algebraic method for the efficient and fast computation of successive time derivatives of the output signal in a differentially flat system example (see, for instance [10]). Instead of attempting the construction of a nonlinear observer, a set of formulae is developed for the required time derivatives of the output signal which is based on a Taylor series approximation of the signal and the use of the “algebraic derivative method” introduced in [7]. The key issue is to initially view the output signal as a time signal, with no other systems oriented view of func- tional dependance upon the system state and then, if an estimate of the state of the system is required, this can be readily computed from the obtained time derivatives of the output in a static way (see Diop and Fliess [3]). As a time signal, the output signal can be efficiently approximated in a manner that is independent of the initial values of its time derivatives using the “algebraic derivative” method and, moreover, a non-asymptotic, fast, approximation is entirely possible to obtain its var- ious derivatives over an open time interval. The result of our algebraic estimation approach is a piecewise con- tinuous approximation to the actual system output and its first few time derivatives, along with an updating mechanism that allows for the automatic resetting of the involved computations when the validity of the adopted approximation ceases to be valid. Incidentally, our for- mulae for the on line generation of the output signal time derivatives may be written, solely, in terms of in- tegrations and convolutions of the originally measured output signal. We test the robustness of our approach to non-structured perturbation inputs using computer simulations and computer generated noises. An initial report on the theoretical features of algebraic methods for the output feedback control of flat systems was given by the authors in [8]. Section 2 of this article presents a brief summary of the algebraic derivative method and illustrates it by means of an elementary example. Section 3 presents the model of the synchronous generator, establishes its flatness and proposes a feedback controller, requiring a