Functional observability and subspace reconstruction in nonlinear systems

Abstract

Time-series analysis is fundamental for modeling and predicting dynamical behaviors from timeordered data, with applications in many disciplines such as physics, biology, finance, and engineering. Measured time-series data, however, are often low dimensional or even univariate, thus requiring embedding methods to reconstruct the original system's state space. The observability of a system establishes fundamental conditions under which such reconstruction is possible. However, complete observability is too restrictive in applications where reconstructing the entire state space is not necessary and only a specific subspace is relevant. Here, we establish the theoretic condition to reconstruct a nonlinear functional of state variables from measurement processes, generalizing the concept of functional observability to nonlinear systems. When the functional observability condition holds, we show how to construct a map from the embedding space to the desired functional of state variables, characterizing the quality of such reconstruction. The theoretical results are then illustrated numerically using chaotic systems with contrasting observability properties. By exploring the presence of functionally unobservable regions in embedded attractors, we also apply our theory for the early warning of seizure-like events in simulated and empirical data. The studies demonstrate that the proposed functional observability condition can be assessed a priori to guide time-series analysis and experimental design for the dynamical characterization of complex systems.

On functional observers for polynomial systems with nonlinear oscillations

“…We want to employ algebraic methods to obtain a functional observer. For the high gain observer (10) we need the map 𝛼 of the observability canonical form (6). The coordinates 𝑧 1 , … , 𝑧 𝑁 of the canonical form are defined by Lie derivatives of order 0, … , 𝑁 − 1, see (4) and (5).…”

“…This work was inspired by the literature [6], where the functional observability was investigated for the Lorenz system, among others, and a functional observer was designed. The observation was based on a partial inversion of the observability map, where the functional was expressed in terms of the output and its time derivatives.…”

Efficient Sensor Node Selection for Observability Gramian Optimization

“…For example, a previous study reported a tremendous increase in the computation time of the greedy method and the SDP relaxation method for a power grid system of such a size [66]. There are likely to be computational issues when the optimization objective is further extended to the observability for the nonlinear state space models [44,68,69], the error covariance of the Kalman filter [70][71][72][73][74], and the H 2 norm of an LTI system, which can be reduced through balanced truncation [75][76][77]. Accordingly, the main interest in this study is the applicability of the Gramian-based methods for the larger scale system constructed by the data-driven modeling methods.…”