No abstract

Abstract. This paper gives the sufficient and necessary conditions which guarantee the existence of a diffeomorphism in order to transform a nonlinear system without inputs into a canonical normal form depending on its output. Moreover we extend our results to a class of systems with inputs.1. Introduction. Since Luenberger's work [9], the design of an observer for observable linear systems with linear outputs has been a well-known concept. In order to use the same observer for nonlinear systems, the so-called observability linearization problem for nonlinear systems was born. The sufficient and necessary conditions which guarantee the existence of a diffeomorphism and of an output injection to transform a single output nonlinear system without inputs into a linear one with an output injection were firstly addressed in [12]. Then, for a multi-output nonlinear system without inputs, the linearization problem was partially solved in [13]. The complete solution to the linearization problem was given in [16]. Another approach was introduced for the analytical systems in [11] by assuming that the spectrum of the linear part must lie in the Poincaré domain and it was generalized in [14] by assuming that the spectrum of the linear part must lie in the Siegel domain. These assumptions are not in generally fulfilled. Other approaches using quadratic normal forms were given in [1] and [3]. All these approaches enable us to design an observer for a larger class of nonlinear systems.Meanwhile, other researchers worked on designing nonlinear observers directly, such as high-gain observers [6], [4], [7]. Nevertheless, even if the conditions which guarantee the linearization method to design an observer were not generically fulfilled, this method still remained important for the nonlinear observer design first because it works well for non-analytic systems, and second because it could be used in the adaptive theory and also be useful for the observation of systems with unknown inputs. All these reasons explain why researchers carry on investigating this matter.In [10], the author gave the sufficient and necessary geometrical conditions to transform a nonlinear system into a so-called output-dependent time scaling linear canonical form. While [5] gave the dual geometrical conditions of [10].In this paper, as an extension of [17], we will propose a method to deduce the geometrical conditions which are sufficient and necessary to guarantee the existence of a local diffeomorphism z = φ (x) which transforms the locally observable dynamical system

No abstract

No abstract