Stable Takens' Embeddings for Linear Dynamical Systems

Takens' Embedding Theorem remarkably established that concatenating M previous outputs of a dynamical system into a vector (called a delay coordinate map) can be a one-to-one mapping of a lowdimensional 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 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 hav...

• Type: Journal
• Year: 2010
• Where: CORR
• Authors: Han Lun Yap, Christopher J. Rozell
• Keywords: CORR 2010, Education, Embedding, Linear Dynamical Systems, Stable Embedding