Detectability Analysis and Observer Design for Linear Time Varying Systems

Tranninger, Markus; Seeber, Richard; Zhuk, Sergiy; Steinberger, Martin; Horn, Martin

doi:10.1109/lcsys.2019.2927549

Cited by 14 publications

(17 citation statements)

References 13 publications

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“…In the perfect-deterministic model setting, the asymptotic filter performance can likewise be understood principally in terms of the dynamical properties of the model. Particularly, the statistics are determined by the ability of the filter to correct for the dynamical instabilities of perturbations along the truth-twin trajectory with respect to the sensitivity of the filter to its observations Frank and Zhuk, 2018;Maclean and Van Vleck, 2019;Tranninger et al, 2019).…”

Section: Twin Experiments With Geophysical Modelsmentioning

confidence: 99%

On the numerical integration of the Lorenz-96 model, with scalar additive noise, for benchmark twin experiments

2020

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Relatively little attention has been given to the impact of discretization error on twin experiments in the stochastic form of the Lorenz-96 equations when the dynamics are fully resolved but random. We study a simple form of the stochastically forced Lorenz-96 equations that is amenable to higher-order time-discretization schemes in order to investigate these effects. We provide numerical benchmarks for the overall discretization error, in the strong and weak sense, for several commonly used integration schemes and compare these methods for biases introduced into ensemble-based statistics and filtering performance. The distinction between strong and weak convergence of the numerical schemes is focused on, highlighting which of the two concepts is relevant based on the problem at hand. Using the above analysis, we suggest a mathematically consistent framework for the treatment of these discretization errors in ensemble forecasting and data assimilation twin experiments for unbiased and computationally efficient benchmark studies. Pursuant to this, we provide a novel derivation of the order 2.0 strong Taylor scheme for numerically generating the truth twin in the stochastically perturbed Lorenz-96 equations.

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Section: Twin Experiments With Geophysical Modelsmentioning

confidence: 99%

On the numerical integration of the Lorenz-96 model, with scalar additive noise, for benchmark twin experiments

2020

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“…Finally we stress that our design is minimal in that one cannot make use of Lyapunov subspace reduction (e.g. [5], [13]) and restrict the observer's gain corrections just to the stable Lyapunov subspace due to the following fact: all the Lyapunov exponents of the considered dynamical system equal to zero. Nevertheless, unlike [5] we do not require to observe all "unstable directions" instead requiring the uniform complete observability along the observer's trajectory.…”

Section: Introductionmentioning

confidence: 99%

Homogeneous Observers for Projected Quadratic Partial Differential Equations

Zhuk

Polyakov

2020

2020 59th IEEE Conference on Decision and Control (CDC)

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The paper proposes a new homogeneous observer for finite-dimensional projections of quadratic homogeneous hyperbolic PDEs with compact state space. The design relies upon new sufficient conditions for fixed-time convergence of observer's gain, described as a solution of a non-linear homogeneous matrix differential equations, towards an ellipsoid in the space of symmetric non-negative matrices. Convergence of the observer is analyzed, and a numerical convergence test is proposed: numerical experiments confirm the test on ODEs obtained by finite-difference discretization of Burgers-Hopf equation.

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“…Introduction. In recent years, detectability analysis of complex dynamical systems gained a lot of research attention in different engineering fields [7,22,26,45,47]. Despite the fact that detectability is very well studied for linear time invariant systems [41], the underlying theory and existing detectability conditions are less mature for the time varying and/or nonlinear setting.…”

mentioning

confidence: 99%

Detectability Conditions and State Estimation for Linear Time-Varying and Nonlinear Systems

Tranninger,

Seeber,

Steinberger

et al. 2021

Preprint

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This work proposes a detectability condition for linear time-varying systems based on the exponential dichotomy spectrum. The condition guarantees the existence of an observer, whose gain is determined only by the unstable modes of the system. This allows for an observer design with low computational complexity compared to classical estimation approaches. An extension of this observer design to a class of nonlinear systems is proposed and local convergence of the corresponding estimation error dynamics is proven. Numerical results show the efficacy of the proposed observer design technique.

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Detectability Analysis and Observer Design for Linear Time Varying Systems

Cited by 14 publications

References 13 publications

On the numerical integration of the Lorenz-96 model, with scalar additive noise, for benchmark twin experiments

On the numerical integration of the Lorenz-96 model, with scalar additive noise, for benchmark twin experiments

Homogeneous Observers for Projected Quadratic Partial Differential Equations

Detectability Conditions and State Estimation for Linear Time-Varying and Nonlinear Systems

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