Uniting Observers

Abstract

We propose a framework for designing observers possessing global convergence properties and desired asymptotic behaviours for the state estimation of nonlinear systems. The proposed scheme consists in combining two given continuoustime observers: one, denoted as global, ensures (approximate) convergence of the estimation error for any initial condition ranging in some prescribed set, while the other, denoted as local, guarantees a desired local behaviour. We make assumptions on the properties of these two observers, and not on their structures, and then explain how to unite them as a single scheme using hybrid techniques. Two case studies are provided to demonstrate the applicability of the framework. Finally, a numerical example is presented.

Lyapunov Functions for State Observers of Dynamic Systems Using Hamilton–Jacobi Inequalities

“…In addition, we have addressed the problem of constructing observers with desired performances in terms of L 2 -gain, as shown in more detail with the case study on observers for a class of polynomial systems. Future work will regard the investigation of the proposed approach based on ISS for other kinds of observers such as those considered in [16,48]. Another direction of research will be focused on the study of ISS in a discrete-time setting for moving horizon estimation [49].…”

Performance Optimization via Sequential Processing for Nonlinear State Estimation of Noisy Systems

“…The functions δ, α P K 0 and λ P p0, 1q are obtained from a differential inequality of the form 9 vpx t , tq ď´λvpx t , tq`αp}y t }q`δp}d t }q (5) which must hold along the solutions of (2) for some continuously differentiable function v : R nˆR ě Ñ R and for all px 0 , d, tq P R nˆDˆR ě . On the other hand, the function β P K 1 8 , which is instrumental in (3) to determine an upper bound for the state norm }x t }, is related to vpx, tq as follows: there exists a time t ą 0 for which vpx, tq ě βp}x}q for all px, tq P R nˆr t,`8q. In section V we will discuss existence conditions (Proposition 5.1) and constructive methodologies (Propositions 5.2 and 5.3) for the functions v, β, δ and α with the above mentioned properties.…”

“…However, the error bounds depend on the state magnitude so that for large state initial conditions the state estimate has large excursions and significant deviations from the actual value of the state. More recently, the work [3] unites local observers, which have good error performances versus measurement noise like extended Kalman filters (EKF's) for instance, with semiglobal HGO's, which have bad error performances. Systems with bounded solutions are considered and the resulting observer has a switching structure which guarantees the compromise between bad (semiglobal) and good (local) error performances but its correct functioning depends on some local and semiglobal norm estimators together with the exact knowledge of the domains of attraction of the local and semiglobal observers.…”

“…In this paper we consider a quite general class of nonlinear systems with model uncertainties (or state noise) and measurement noise and design state observers with the primary objective to optimize the error performances in terms of robustness and noise sensitivity. Important contributions of our work over the existing literature are: 1) global convergence domains (in contrast with a priori given and bounded domains as in [19], [8] and [3]), 2) no specific system's structure (in contrast with feedback linearizable systems as in [19], [8] and [9]) taking into account state and measurement noise at the same time and 3) new results on global observers in the absence of noise (in contrast with more restrictive conditions as in [17] and [1]). We do not require boundedness of system's solutions: this has the beneficial effect of obtaining asymptotic error bounds and conditions for improving noise sensitivity not dependent on the convergence domain so that we may improve noise sensitivity without shrinking the estimation error convergence domain.…”

On Hamilton-Jacobi Approaches to State Reconstruction for Dynamic Systems

“…Unfortunately, the proof of stability of the estimation error is not easy for such estimators owing the need to have at disposal an analytic expression of vðt,xÞ. In practice, under general assumptions, one can find only approximate solutions to the HJ equation (11) and hence derive only local stability results [28], whereas, in general, global properties are preferable [29]. This motivates the investigation of the global stability properties that can be ensured by estimators having a structure like that in (18), as shown in the next section.…”