Set-membership identifiability and guaranteed parameter estimation for nonlinear uncertain dynamical systems

Abstract

Definitions of identifiability and methods for checking this property for linear and nonlinear systems are now well established. Recently, some works (Jauberthie et al. [2011], Braems et al. [2001]) have provided identifiability definitions for set-membership models in a bounded-error context and established links with classical identifiability definitions. These works are summarized in the first part of the paper, recalling the two complementary definitions : set-membership identifiability that is conceptual and μ-set-membership identifiability that can be put in correspondence with existing set-membership parameter estimation methods (Jauberthie et al. [2011]). In the second part, two methods for checking set-membership identifiability and μ-set-membership identifiability are proposed. The first one is an extension of a method proposed by Pohjanpalo [1978] based on the power series expansion of the solution that accounts for the initial conditions of the system. It generalizes to nonlinear systems the initial extension provided in Jauberthie et al. [2011]. The second method is based on differential algebra and makes use of relations linking the observations, the inputs and the unknown parameters of the system. Classically, when using this method, initial conditions are not considered but it has been shown recently (Saccomani et al. [2004]) that they can change the identifiability results. In this paper, an extension using initial conditions is proposed. In the third part of the paper, a numerical parameter estimation method is deduced from the differential algebra method and an example is presented.