Numerical design of Luenberger observers for nonlinear systems

Ramos, Louise da Costa; Meglio, Florent Di; Morgenthaler, Valery; Silva, Luı́s Fernando Figueira da; Bernard, Pauline

doi:10.1109/cdc42340.2020.9304163

Cited by 21 publications

(24 citation statements)

References 17 publications

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“…Implementing a KKL observer thus follows the steps: 1) Choose matrices D and F 2) Compute the corresponding transformation T * 3) Simulate (3) from an arbitrary z(0) and compute the estimate x(t) = T * (z(t)). In [12], a method to complete step 2 by performing nonlinear regression on trajectories of (1) and ( 3) is proposed. In this paper, we propose an approach to assist the user in completing step 1 by learning the dependency of T * with respect to a parameter defining D. We then define a criterion to be optimized by this parameter.…”

Section: Kkl Observersmentioning

confidence: 99%

“…In [12], a method is proposed to approximate the mappings by performing nonlinear regression on datasets generated from trajectories of the system and the observer. Given fixed observer parameters, a neural network interpolates between dataset points, and the resulting mapping is used to compute state estimates from observer values.…”

Section: Introductionmentioning

confidence: 99%

“…In this paper, we build on the approach of [12] and propose a first step towards calibration of the observer. Our main contribution is a procedure to learn the dependency of the mappings involved in KKL observer design with respect to some of the observer parameters.…”

Section: Introductionmentioning

confidence: 99%

“…We then approximate the KKL mappings as functions of this frequency using neural networks. This approximation is enabled by appropriately sampling the state-space, improved upon [12]. We then propose an easily computable performance criterion to choose the frequency that, in some sense, trades off the transient performance against the sensitivity to noise of the state estimate.…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Towards gain tuning for numerical KKL observers

Buisson-Fenet¹,

Bahr²,

Morgenthaler³

et al. 2022

Preprint

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This paper presents a first step towards tuning observers for nonlinear systems. Relying on recent results around Kazantzis-Kravaris/Luenberger (KKL) observers, we propose to design a family of observers parametrized by the cutoff frequency of a linear filter. We use neural networks to learn the mapping between the observer and the nonlinear system as a function of this frequency, and present a novel method to sample the state-space efficiently for nonlinear regression. We then propose a criterion related to noise sensitivity, which can be used to tune the observer by choosing the most appropriate frequency. We illustrate the merits of this approach in numerical simulations.

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Section: Kkl Observersmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Towards gain tuning for numerical KKL observers

Buisson-Fenet¹,

Bahr²,

Morgenthaler³

et al. 2022

Preprint

Self Cite

View full text Add to dashboard Cite

show abstract

“…After proving the existence of a generative model under this particular form, we verify that it also respects the hypothesis required for our upper bound. To demonstrate the feasibility of this solution, and inspired by [25], we design a learning algorithm to discover such KKL models. In our experiments, the KKL-based predictor exhibits remarkable forecasting capabilities, excellent generalization and robustness to noise.…”

Section: B the Output Prediction Problemmentioning

confidence: 99%

Deep KKL: Data-driven Output Prediction for Non-Linear Systems

Janny,

Andrieu,

Nadri

et al. 2021

Preprint

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We address the problem of output prediction, ie. designing a model for autonomous nonlinear systems capable of forecasting their future observations. We first define a general framework bringing together the necessary properties for the development of such an output predictor. In particular, we look at this problem from two different viewpoints, control theory and data-driven techniques (machine learning), and try to formulate it in a consistent way, reducing the gap between the two fields. Building on this formulation and problem definition, we propose a predictor structure based on the Kazantzis-Kravaris/Luenberger (KKL) observer and we show that KKL fits well into our general framework. Finally, we propose a constructive solution for this predictor that solely relies on a small set of trajectories measured from the system. Our experiments show that our solution allows to obtain an efficient predictor over a subset of the observation space.

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Data‐driven state observation for nonlinear systems based on online learning

Tang

2023

AIChE Journal

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For controlling nonlinear processes represented by state‐space models, a state observer is needed to estimate the states from the trajectories of measured variables. While model‐based observer synthesis is traditionally challenging due to the difficulty of solving pertinent partial differential equations, this article proposes an efficient model‐free, data‐driven approach for state observation, which is suitable for data‐driven nonlinear control without accurate nonlinear models. Specifically, by using a Chen–Fliess series representation of the observer dynamics, state observation is endowed with an online least squares regression formulation that can be solved by gradient flow with performance guarantees. When the target state trajectories for regression are unavailable, by exploiting the Kazantzis–Kravaris/Luenberger observer structure, state observation is reduced to a dimensionality reduction problem amenable to an online implementation of kernel principal component analysis. The proposed approach is demonstrated by a limit cycle dynamics and a chaotic system.

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Numerical design of Luenberger observers for nonlinear systems

Cited by 21 publications

References 17 publications

Towards gain tuning for numerical KKL observers

Towards gain tuning for numerical KKL observers

Deep KKL: Data-driven Output Prediction for Non-Linear Systems

Data‐driven state observation for nonlinear systems based on online learning

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