A multi-observer approach for the state estimation of nonlinear systems

Abstract

Abstract-We present a prescriptive approach for the state estimation of nonlinear systems. We first assume that we know a local observer, i.e. a dynamical system whose state converges to the plant state when it is initialized nearby the plant initial condition. We sample the set where the plant initial condition is assumed to lie with a finite number of points; noting that this set can be arbitrarily large. A local observer is initialized at each of these sampled points to form a bank of observers called multi-observer. A supervisor is then constructed to select one of these observers at any time instant. The selected state estimate is guaranteed to converge to the state of the plant, provided the number of samples is sufficiently large and a detectability property holds, which is expressed in terms of Lyapunov-based conditions. An explicit lower bound on the required number of observers is given when an estimate of the basin of convergence of the local observer is available. We explain how to apply the approach to nonlinear systems with globally Lipschitz and differentiable nonlinearities. Simulations results are presented for a Wilson-Cowan oscillator.

Uniting Observers

“…The temporal regularization τ imposes to use global observer for at least a T units of (continuous) time. This is only done to avoid unnecessary multiple consecutive jumps in the scheme (10) and to always enforce a flow after a jump when T > 0. Note that since T can be chosen arbitrarily small, as we will see, we do not lose in generality in designing the proposed hybrid scheme from a practical point of view.…”

“…The main message of this work, namely to combine different observers in order to improve the overall performance, can be adapted to take into account also different scenarios which have not been explicitly addressed here. For instance, in order to relax the matching conditions of Assumption 5, it may be possible to substitute the local observer by a bank of local observers, in the same spirit of [10], as mentioned in Remark 1. Compared to [10], uniting a single global observer with a bank of local oscillators could be more efficient from the computation point of view.…”

“…This may occur when the basin of attraction of local observer (2) is too small, namely ε 0 is too small; the global observer (3) is not enough accurate, namely ε 1 is too large, or the effect of the measurement noise w is too large, namely the level set ε 1 of V 1 that can be estimated via z 1 is too large. In these cases, one can overcome this problem by enlarging the domain of attraction of the local observer by replacing local observer (2) with a bank of local observers and by using the multi-observer design technique in [10].…”

“…We consider in this section the case in which we only know how to design a global asymptotic observer (in the absence of 10 Recall that the existence of the Lyapunov function V 1 in Assumption 9 implies the existence of another Lyapunov function with arbitrary λ 1 . See [12], [35].…”

“…A natural idea to overcome these limitations is therefore that of combining different observers or gains to inherit the "good" properties of each of them. This approach, often used by practitioners, have been studied by means of switching, adaptive or gain-scheduling strategies, see for instance [6]- [10].…”