Observability of multivariate differential embeddings
نویسندگان
چکیده
The present paper extends some results recently developed for the analysis of observability in nonlinear dynamical systems. The aim of the paper is to address the problem of embedding an attractor using more than one observable. A multivariate nonlinear observability matrix is proposed which includes the monovariable nonlinear and linear observability matrices as particular cases. Using the developed framework and a number of worked examples, it is shown that the choice of embedding coordinates is critical. Moreover, in some cases, to reconstruct the dynamics using more than one observable could be worse than to reconstruct using a scalar measurement. Finally, using the developed framework it is shown that increasing the embedding dimension, observability problems diminish and can even be eliminated. This seems to be a physically meaningful interpretation of the Takens embedding theorem. PACS number: 05.45.−a
منابع مشابه
Relation between observability and differential embeddings for nonlinear dynamics.
In the analysis of a scalar time series, which lies on an m-dimensional object, a great number of techniques will start by embedding such a time series in a d-dimensional space, with d>m. Therefore there is a coordinate transformation phi(s) from the original phase space to the embedded one. The embedding space depends on the observable s(t). In theory, the main results reached are valid regard...
متن کاملManifold Embeddings for Model-Based Reinforcement Learning under Partial Observability
Interesting real-world datasets often exhibit nonlinear, noisy, continuous-valued states that are unexplorable, are poorly described by first principles, and are only partially observable. If partial observability can be overcome, these constraints suggest the use of model-based reinforcement learning. We experiment with manifold embeddings to reconstruct the observable state-space in the conte...
متن کاملGenericity of Algebraically Observable Polynomial Systems
E. Sontag has introduced the concept of algebraic observability for n-dimensional polynomial systems. It is a stronger notion than the usual concept of observability and implies the existence of a polynomial expression of the state variables in terms of a finite number of derivatives of the output function. We prove that algebraic observability is a generic property for polynomial systems of bo...
متن کاملDelay Differential Analysis of Time Series
Nonlinear dynamical system analysis based on embedding theory has been used for modeling and prediction, but it also has applications to signal detection and classification of time series. An embedding creates a multidimensional geometrical object from a single time series. Traditionally either delay or derivative embeddings have been used. The delay embedding is composed of delayed versions of...
متن کاملObservability and the eigenstructure of multivariate canonical dynamic linear models
Multivariate canonical state space dynamic models are developed by studying the eigenstructure of their transition matrices. Observability is introduced for time-varying model components defining locally observable dynamic models. Single component models that have a simple transition matrix are first discussed and categorized according to their forecast function. Then more complicated models, d...
متن کامل