Generic Observability of Differentiable Systems

Aeyels, Dirk

doi:10.1137/0319037

Cited by 155 publications

(111 citation statements)

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“…Closely related is also the work of Takens [28], which shows that generically, a smooth dynamical system on an r-dimensional manifold can be embedded in R 2r+1 , as well as the control-theory work of Aeyels on generic observability, which shows in [2] that for generic vector fields and observation maps on an r-dimensional manifold, 2r+1 observations at randomly chosen times are enough for observability, and in [1] that this bound is best possible. Aeyels proofs, in particular, are based on transversality arguments of the general type that we use.…”

Section: Genericitymentioning

confidence: 99%

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For Differential Equations with r Parameters, 2r+1 Experiments Are Enough for Identification

Sontag¹

2003

J. Nonlinear Sci.

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Given a set of differential equations whose description involves unknown parameters, such as reaction constants in chemical kinetics, and supposing that one may at any time measure the values of some of the variables and possibly apply external inputs to help excite the system, how many experiments are sufficient in order to obtain all the information that is potentially available about the parameters? This paper shows that the best possible answer (assuming exact measurements) is 2r+1 experiments, where r is the number of parameters.

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Section: Genericitymentioning

confidence: 99%

“…Otherwise, X is an open subset of X (2) , and hence an open subset of X 2 , and is thus a manifold of dimension 2r. We let U (x 1 , x 2 ) be the set consisting of those u ∈ U such that…”

Section: Global Analytic Case: 2r+1 Experiments Are Enoughmentioning

confidence: 99%

For Differential Equations with r Parameters, 2r+1 Experiments Are Enough for Identification

Sontag¹

2003

J. Nonlinear Sci.

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“…Recall that Takens' Embedding Theorem ( [Takens, 1980]; see also [Aeyels, 1981] and [Sauer et al, 1991]) provides the theoretical foundation for the analysis of time series generated by nonlinear deterministic dynamical systems. Informally, it says that if we take a scalar observable ϕ of the state x of a deterministic dynamical system then typically we can reconstruct a copy of the original system by considering blocks (ϕ(x t ), ϕ(x t+τ ), ϕ(x t+2τ ), …, ϕ(x t+(d-1)τ )) of d successive observations of ϕ, for d sufficiently large.…”

Section: Introductionmentioning

confidence: 99%

Delay Embeddings for Forced Systems. II. Stochastic Forcing

Stark

Broomhead

Davies

et al. 2003

Journal of Nonlinear Science

160

154

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Takens' Embedding Theorem forms the basis of virtually all approaches to the analysis of time series generated by nonlinear deterministic dynamical systems. It typically allows us to reconstruct an unknown dynamical system which gave rise to a given observed scalar time series simply by constructing a new state space out of successive values of the time series. This provides the theoretical foundation for many popular techniques, including those for the measurement of fractal dimensions and Liapunov exponents, for the prediction of future behaviour, for noise reduction and signal separation, and most recently for control and targeting. Current versions of Takens' Theorem assume that the underlying system is autonomous (and noise free). Unfortunately this is not the case for many real systems. In a previous paper, one of us showed how to extend Takens' Theorem to deterministically forced systems. Here, we use similar techniques to prove a number of delay embedding theorems for arbitrarily and stochastically forced systems. As a special case, we obtain embedding results for Iterated Functions Systems, and we also briefly consider noisy observations.

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“…If at time i the system is in state x ∈ Λ, then at time i + 1 it is in state F x, and at time i − 1 it is in state F −1 x. That descriptions of this sort are, in a precise sense, generic follows from Takens's embedding theorem [34,4,33]. We do not assume that F (or F −1 ) is continuous.…”

Section: Introductionmentioning

confidence: 99%