A multi observers approach when observability index is higher than the state dimension - a case study -

Abstract

We address a family of observation problems that would classically require the construction of an embedding, by a different approach which consists in the design of several estimators in parallel. In principle, for a dynamics in dimension n with a scalar output y, each estimator uses the knowledge of only n − 1 derivatives of the output, and the further derivatives are used to discriminate at any time among the estimators. Estimators are built here by roots tracking technique. We illustrate our approach on the parameter estimation of a polynomial dynamics. The simulations show that the final estimation jumps from one estimator to another when passing through observability singularities, or when the parameter suddenly changes, preserving a good estimation error.

A Multiobservers Approach for a Class of Bidimensional Nonuniformly Observable Systems

“…a) The dynamics of ŝ cannot be explicitly obtained from (8), especially when multiple solutions of (4) appear. In order to define the dynamics of ŝ explicitly from (8), a preliminary solution has been proposed in [10] consists in saturating the norm of (∂ s F (z(t), ŝ(t))) −1 near the singularities.…”

“…By consequence, approaching a singularity two different estimators may merge into each other leading to loose a part of the solution. A preliminary solution based on an approximation dynamics coupling the relative vector fields corresponding to different initial conditions is proposed in [10].…”

“…Observe that, under Assumption 1, the polynomial F ε (z, s) given by equation (10) does not have multiple complex roots (this is a property of holomorphic functions). By consequence, for each z ∈ R 2 , there exist p-distinct complex roots of the polynomial F ε (z, s).…”

“…This approach does not require any coordinates transformation, and by consequence no problem of transformation inversion of the observability map nor its Jacobian is posed. The technical task with this approach lies in the roots tracking method which, as described in the preliminary result of this work (see [10]), needs to be robust near singularities (multiple roots); robust in the sense that it always distinguishes between different neighboring roots. Indeed, knowing that each estimator should have a subset of (dynamical) roots as a global attractor and knowing that such attractors intersect at multiple roots, then, close to singularities, two distinct estimators may exchange roles and even fuse together.…”

Illustration of the application of the multi-observers approach

“…The value of a is fixed to a = 2 and the initial condition of the original system (1)-( 2) is fixed at (0, 2). The parameter k, relative to (8) has been fixed to k = 150. An explicit Euler scheme with a discretization step equal to h = 10 −3 has been chosen for the simulation.…”