Delay Embeddings for Forced Systems. II. Stochastic Forcing

Stark, Jaroslav; Broomhead, D. S.; Davies, Michael; Huke, J. P.

doi:10.1007/s00332-003-0534-4

Cited by 161 publications

(160 citation statements)

References 20 publications

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“…In the case of controlled discrete-time systems, in article [9], the authors investigate controlled discrete-time systems and obtain some results which are similar (but not identical) to the one presented here.…”

Section: Introductionsupporting

confidence: 56%

On the genericity of the observability of controlled discrete-time systems

Ammar

Vivalda

2005

ESAIM: COCV

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Section: Introductionsupporting

confidence: 56%

On the genericity of the observability of controlled discrete-time systems

Ammar

Vivalda

2005

ESAIM: COCV

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“…However, in the limit of large concatenation parameters, the manifold geometry and, hence, the EOFs are biased towards the most stable component of the dynamics 7 , as supported by the reduction in the number of eigenvalues above a spectral gap from five (in the manifold of raw data) to one (after concatenation); see Supplementary Information section 13. One may therefore regard the timing jitter as a form of stochastic forcing, which has been extensively studied 14 . In Supplementary Information section 3, we describe how reliable dynamical information can be obtained on timescales substantially shorter than the timing jitter.…”

mentioning

confidence: 99%

Dynamics from noisy data with extreme timing uncertainty

Fung

Hanna

Vendrell

et al. 2016

Nature

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Imperfect knowledge of the times at which 'snapshots' of a system are recorded degrades our ability to recover dynamical information, and can scramble the sequence of events. In X-ray free-electron lasers, for example, the uncertainty-the so-called timing jitterbetween the arrival of an optical trigger ('pump') pulse and a probing X-ray pulse can exceed the length of the X-ray pulse by up to two orders of magnitude 1 , marring the otherwise precise time-resolution capabilities of this class of instruments. The widespread notion that little dynamical information is available on timescales shorter than the timing uncertainty has led to various hardware schemes to reduce timing uncertainty [2][3][4] . These schemes are expensive, tend to be specific to one experimental approach and cannot be used when the record was created under ill-defined or uncontrolled conditions such as during geological events. Here we present a data-analytical approach, based on singular-value decomposition and nonlinear Laplacian spectral analysis [5][6][7] , that can recover the history and dynamics of a system from a dense collection of noisy snapshots spanning a sufficiently large multiple of the timing uncertainty. The power of the algorithm is demonstrated by extracting the underlying dynamics on the few-femtosecond timescale from noisy experimental X-ray free-electron laser data recorded with 300-femtosecond timing uncertainty 1 . Using a noisy dataset from a pump-probe experiment on the Coulomb explosion of nitrogen molecules, our analysis reveals vibrational wave-packets consisting of components with periods as short as 15 femtoseconds, as well as more rapid changes, which have yet to be fully explored. Our approach can potentially be applied whenever dynamical or historical information is tainted by timing uncertainty.The fundamental premise of our approach is simple. A series of snapshots concatenated in the order of their inaccurate time stamps will contain some time-evolutionary information ('a weak arrow of time'), provided that the concatenation window spans a period comparable with, or longer than, the timing uncertainty associated with each individual snapshot. This realization leads one to consider a series of c-fold concatenated snapshots, formed by moving a c-frame-wide window over the raw dataset ordered according to the inaccurate time stamps. The dynamical history can then be extracted from the series of concatenated snapshots using techniques developed to extract signal from noise, such as singular-value decomposition (SVD) 8 . SVD determines a series of statistically significant modes, each consisting of a characteristic pattern (topogram) and its time evolution (chronogram). A topogram can be a characteristic image or spectrum, with the corresponding chronogram showing its change with time. For each mode, a singular value specifies the power contained in that mode 8 .Consider snapshots, such as images or spectra, that can be represented as vectors by using the pixel values of each snapshot as the components of a ve...

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“…Although the theory discussed above applies to autonomous systems, the embedding theorem has been extended to a general class of nonautonomous systems with deterministic forcing [15], state-dependant forcing [16], and stochastic forcing [17]. For nonautonomous stationary systems, it can be shown that the delay vector needs to also include the system inputs u [17]:…”

Section: Self-organizing Rulesmentioning

confidence: 99%

“…In other words, there exists a set of inputs, from limited observations, that can represent the system dynamics. The embedding theorem has been extended to a general class of nonautonomous systems with deterministic forcing [15], state-dependant forcing [16], and stochastic forcing [17], and counts numerous applications to analysis and identification of time series [18][19][20].…”

Section: Introductionmentioning

confidence: 99%

Adaptive structural control using dynamic hyperspace

Laflamme¹

2015

Int. J. CMEM

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The design of closed-loop structural control systems necessitates a certain level of robustness to cope with system uncertainties. Neurocontrollers, a type of adaptive control system, have been proposed to cope with those uncertainties. However, the performance of neural networks can be substantially influenced by the choice of the input space, or the hyperspace in which the representation lies. For instance, input selection may influence computation time, adaptation speed, effects of the curse of dimensionality, understanding of the representation, and model complexity. Input space selection is often overlooked in literature, and inputs are traditionally determined offline for an optimized performance of the neurocontroller. Such offline input selection is often unrealistic to conduct in the case of civil structures. In this paper, a novel method for automating the input selection process for neural networks is presented. The method is purposefully designed for online input selection during adaptive identification and control of nonlinear systems. Input selection is conducted online and sequentially, while the excitation is occurring. The algorithm designed for the adaptive input space assumes local quasi-stationarity of the time series, and embeds local maps sequentially in a delay vector using the embedding theorem. The input space of the representation is subsequently updated. The performance of the proposed dynamic input selection method is demonstrated through simulating semi-active control of an existing structure located in Boston, MA, U.S.A. Simulation results show the substantial performance of the proposed algorithm over traditional fixed-inputs strategies.

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Delay Embeddings for Forced Systems. II. Stochastic Forcing

Cited by 161 publications

References 20 publications

On the genericity of the observability of controlled discrete-time systems

On the genericity of the observability of controlled discrete-time systems

Dynamics from noisy data with extreme timing uncertainty

Adaptive structural control using dynamic hyperspace

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