Dimension estimation for autonomous nonlinear systems Alberto Padoan and Alessandro Astolfi Abstract— The problem of estimating the dimension of the state-space of an autonomous nonlinear systems is considered. Assuming that sampled measurements of the output and finitely many of its time derivatives are available, an iterative algorithm able to retrieve the dimen- sion of the minimal state-space realization is derived. The performance of the proposed algorithm are evaluated on benchmark nonlinear systems. I. Introduction Determining the dimension of the state space from experimental observations is a crucial step in the math- ematical modelling of dynamical systems. Among the infinitely many state space descriptions of a dynamical system, it is desirable to find the dimension of the smallest one. Conceptually, the dimension of the state space can be thought as a measure of the complexity of a dynamical system. The difficulty of the problem of estimating the dimension of the state space of a dy- namical system is related to the properties of the class of dynamical systems considered. For example, the dimension of the state space space of a linear system can be recovered by means of subspace identification methods [1–4], provided the measured input-output data are not too noisy. Understanding the properties of nonlinear dynamical systems from experimental ob- servations is more difficult, even under simplifying assumptions [5]. A popular approach to state space reconstruction for autonomous nonlinear systems is based on the “method of delays” [6]. This method hinges upon the fact that finitely many functions of the output of an autonomous nonlinear system can be used to build vectors which lie generically on an embedded manifold of the original state space, provided that the selected functions are more than twice the dimension of the state space of the system. Due to their practical acces- sibility, time-delayed versions of the output are often chosen as such functions (hence the name method of delays). This method of state space reconstruction has led several authors to disregard the problem of deter- mining the dimension of the state space of the system, but to estimate the minimum embedding dimension A. Padoan is with the Department of Electrical and Electronic En- gineering, Imperial College London, London SW7 2AZ, UK (e-mail: alberto.padoan13@imperial.ac.uk). A. Astolfi is with the Department of Electrical and Electronic Engineering, Imperial College London, London SW7 2AZ, UK and with the Dipartimento di Ingegneria Civile e Ingegneria Informatica, Universit` a di Roma “Tor Vergata”, Via del Politecnico 1, Rome 00133, Italy (e-mail: a.astolfi@imperial.ac.uk). instead, i.e. the minimum number of time-delayed ver- sions of the output needed to describe a state space in a higher-dimensional manifold. The computation of the minimum embedding dimension is usually carried out with the method of false nearest neighbours [7]. The reader is referred to [5], and references therein, for further detail. In some special situations the dimension of the (por- tion of the) state space can be reconstructed by visual inspection (see, for an example, [8]). The dimension of the state space of the system, however, may not coincide with the dimension of the geometric objects obtained with this method. In addition, the determination of the dimension by visual inspection is not possible for systems of dimension greater than three. In general, the lack of quantitative arguments makes visual inspection extremely subjective and thus not suitable to provide reliable estimates of the dimension of the state space. The main contribution of this work is an iterative algorithm which estimates the dimension of the state space of an autonomous systems from measurements of the output and finitely many of its time derivatives. The proposed approach hinges upon a local observabil- ity assumption and is partly inspired by the subspace approach to linear system identification [1–4]. Under the assumption that the sampling operation is suffi- ciently fast, an approximate linear relationship between the measured data can be derived. This relationship is used as a test condition and to derive an iterative algorithm to estimate the dimension of the state space of the system. Simulation results show that, if the sampling period is sufficiently small, the dimension of the state space of the system can be correctly estimated. The remainder of the paper is organized as follows. Section II defines the problem and introduces basic assumptions. Section III illustrates the main results, including an iterative algorithm which estimates the di- mension of the state space from measured output data. In Section IV simulation results show the effectiveness of the proposed algorithm on benchmark nonlinear systems. Practical considerations and future directions of research are discussed in Section V. Notation: Standard notation is used. R, R n and R p⇥m denote the set of real numbers, of n-dimensional vec- tors with real entries, and of p ⇥ m-dimensional matri- ces with real entries, respectively. The Schur comple- 1