by H.L. Yap and C.J. Rozell
Abstract:
Takens' Embedding Theorem remarkably established that concatenating $M$ previous outputs of a dynamical system into a vector (called a \emphdelay coordinate map) can be a one-to-one mapping of a low-dimensional attractor from the system state-space. However, Takens' theorem is fragile because even small imperfections can induce arbitrarily large errors in the attractor representation. We extend Takens' result to establish explicit, non-asymptotic sufficient conditions for a delay coordinate map to form a \emphstable embedding in the restricted case of linear dynamical systems and observation functions. Our work is inspired by the field of Compressive Sensing (CS), where results guarantee that low-dimensional signal families can be robustly reconstructed if they are stably embedded by a measurement operator. However, in contrast to typical CS results, i) our sufficient conditions are independent of the size of the ambient state space ($N$), and ii) some system and measurement pairs have fundamental limits on the conditioning of the embedding (i.e., how close it is to an isometry), meaning that further measurements beyond some point add no further significant value. We use several simple simulations to explore the conditions of the main results, including the tightness of the bounds and the convergence speed of the stable embedding.
Reference:
Stable Takens' Embeddings for Linear Dynamical SystemsH.L. Yap and C.J. Rozell. IEEE Transactions on Signal Processing, 59(10), pp. 4781–4794, October 2011.
Bibtex Entry:
@Article{yap.11e,
author = {Yap, H.L. and Rozell, C.J.},
title = {Stable {T}akens' Embeddings for Linear Dynamical Systems},
year = {2011},
journal = {IEEE Transactions on Signal Processing},
volume = {59},
number = {10},
pages = {4781--4794},
month = oct,
abstract = {Takens' Embedding Theorem remarkably established that concatenating $M$ previous outputs of a dynamical system into a vector (called a \emph{delay coordinate map}) can be a one-to-one mapping of a low-dimensional attractor from the system state-space.
However, Takens' theorem is fragile because even small imperfections can induce arbitrarily large errors in the attractor representation. We extend Takens' result to establish explicit, non-asymptotic sufficient conditions for a delay coordinate map to form a \emph{stable embedding} in the restricted case of linear dynamical systems and observation functions.
Our work is inspired by the field of Compressive Sensing (CS), where results guarantee that low-dimensional signal families can be robustly reconstructed if they are stably embedded by a measurement operator.
However, in contrast to typical CS results, i) our sufficient conditions are independent of the size of the ambient state space ($N$), and ii) some system and measurement pairs have fundamental limits on the conditioning of the embedding (i.e., how close it is to an isometry), meaning that further measurements beyond some point add no further significant value.
We use several simple simulations to explore the conditions of the main results, including the tightness of the bounds and the convergence speed of the stable embedding.
},
url = {http://siplab.gatech.edu/pubs/yapTSP2011.pdf}
}