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Research Papers

# State Space Reconstruction of Nonstationary Time-Series

[+] Author and Article Information
Hong-Guang Ma

Professor
Xi'an Research Institute of High Technology,
Xi'an 710025, China
e-mail: mhg_xian@163.com

Chun-Liang Zhang

Aviation School,
Beijing Institute of Technology,
Zhuhai 519088, China
e-mail: 1910334008@qq.com

Fu Li

Aviation School,
Beijing Institute of Technology,
Zhuhai 519088, China
e-mail: 1065078862@qq.com

1Present address: Aviation School, Beijing Institute of Technology, Zhuhai 519088, China.

Contributed by the Design Engineering Division of ASME for publication in the JOURNAL OF COMPUTATIONAL AND NONLINEAR DYNAMICS. Manuscript received April 18, 2016; final manuscript received October 8, 2016; published online December 5, 2016. Assoc. Editor: Mohammad Younis.

J. Comput. Nonlinear Dynam 12(3), 031009 (Dec 05, 2016) (9 pages) Paper No: CND-16-1199; doi: 10.1115/1.4034998 History: Received April 18, 2016; Revised October 08, 2016

## Abstract

In this paper, a new method of state space reconstruction is proposed for the nonstationary time-series. The nonstationary time-series is first converted into its analytical form via the Hilbert transform, which retains both the nonstationarity and the nonlinear dynamics of the original time-series. The instantaneous phase angle θ is then extracted from the time-series. The first- and second-order derivatives $θ˙$, $θ¨$ of phase angle θ are calculated. It is mathematically proved that the vector field $[θ,θ˙,θ¨]$ is the state space of the original time-series. The proposed method does not rely on the stationarity of the time-series, and it is available for both the stationary and nonstationary time-series. The simulation tests have been conducted on the stationary and nonstationary chaotic time-series, and a powerful tool, i.e., the scale-dependent Lyapunov exponent (SDLE), is introduced for the identification of nonstationarity and chaotic motion embedded in the time-series. The effectiveness of the proposed method is validated.

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Topics: Signals , Time series

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## Figures

Fig. 3

Projections of phase points on the planes [θ,θ˙] and [θ˙,θ¨]

Fig. 4

Λ(t) curves for Lorenz system

Fig. 5

SDLE curves for Lorenz system

Fig. 1

Phase space of Lorenz system

Fig. 2

Reconstructed state space (vector field)

Fig. 9

Λ(t) curves for chirp signal

Fig. 10

SDLE curves for chirp signal

Fig. 11

Vibration signals of four-stroke diesel engine: (a) normal state, (b) air leakage, (c) exhaust valve clearance is too large, and (d) exhaust valve clearance is too small

Fig. 12

Reconstructed state space of each vibration signal: (a) normal state, (b) air leakage, (c) exhaust valve clearance is too large, and (d) exhaust valve clearance is too small

Fig. 13

SDLE curves for vibration signal of four-stroke diesel engine in normal state

Fig. 14

SDLE curves for vibration signal of four-stroke diesel engine in air leakage state

Fig. 15

SDLE curves for vibration signal of four-stroke diesel engine when exhaust valve clearance is too large

Fig. 16

SDLE curves for vibration signal of four-stroke diesel engine when exhaust valve clearance is too small

Fig. 6

(a) FFT spectrum of chirp signal and (b) time-domain waveform of chirp signal

Fig. 7

Reconstruct the state space of chirp signal

Fig. 8

Projections of phase points on the planes [θ,θ˙] and [θ˙,θ¨]

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