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 自动化学报  2018, Vol. 44 Issue (10): 1812-1823 PDF

1. 华南理工大学自动化科学与工程学院 广州 510640

Dynamic Feature Extraction of Nonlinear Systems With Deterministic Learning Theory and Spatio-temporal Lempel-Ziv Complexity
WANG Qian1, WANG Cong1
1. School of Automation Science and Engineering, South China University of Technology, Guangzhou 510640
Manuscript received : October 26, 2016, accepted: July 18, 2017.
Foundation Item: Supported by National Major Scientiflc Instruments Development Project (61527811) and National Science Fund for Distinguished Young Scholars (61225014)
Corresponding author. WANG Cong  Professor at the School of Automation Science and Engineering, South China University of Technology. His research interest covers adaptive neural network control/identication, deterministic learning theory, dynamical pattern recognition, pattern-based intelligent control, oscillation fault diagnosis and the application in aerospace and biomedical engineering. Corresponding author of this paper.
Abstract: Effective feature characterization of nonlinear and non-stationary signals is an important and challenging problem in feature extraction. This paper presents a new dynamic feature extraction method for nonlinear dynamical systems based on deterministic learning theory and Lempel-Ziv complexity (LZ complexity). The proposed method extracts features from the dynamics trajectory of nonlinear system. Through the deterministic learning theory, the unknown system dynamics is accurately identified in a local region along the recurrent trajectories of nonlinear system. Firstly, the LZ complexity is used to characterize the obtained dynamics trajectory. A temporal-LZ complexity (TLZC) index and a spatio-LZ complexity (SLZC) index are constructed to quantify the complexity of the system dynamics trajectory in the time-domain and space-domain. In addition, sensitivity analysis is conducted for the dynamics feature characterization, which evaluates the sensitivity of system dynamic indices with respect to parameter changes from period trajectory to chaotic trajectory. Finally, numerical simulation and experiments are carried out to demonstrate the effectiveness of the proposed method. Compared with the state features, the advantage of using the proposed dynamic features is a better representation of the original system by inclusion of internal dynamics information.
Key words: Dynamic feature extraction     deterministic learning     spatio-temporal Lempel-Ziv complexity     sensitivity analysis

1 预备知识 1.1 确定学习理论

 \begin{align} \dot{x}=F(x;p), \quad x(t_{0})=x_{0} \end{align} (1)

 \begin{align} \dot{\hat{x}}_{i}=-a_{i}(\hat{x}_{i}-x_{i})+\hat{W}_{i}^{{\rm T}}S_{i}(x) \end{align} (2)

 \begin{align} \dot{\hat{W}}_{i} = -\Gamma_{i}S_{i}(x)\tilde{x}_{i}-\sigma_{i}\Gamma_{i}\hat{W}_{i} \end{align} (3)

 \begin{align} f_{i}(\phi_{\zeta};p) = \hat{W}_{i}^{{\rm T}}S_{i}(\phi_{\zeta})+\epsilon_{\zeta i} = \bar{W}_{i}^{{\rm T}}S_{i}(\phi_{\zeta})+\epsilon_{\zeta 1i} \end{align} (4)

1.2 Lempel-Ziv复杂度算法

 \begin{align} u(i)= \begin{cases} 0, &y(i) < y_{\rm ave}\\ 1,&y(i) \geq y_{\rm ave} \end{cases} \end{align} (5)

 \begin{align} \lim\limits_{n\rightarrow\infty}c(n)=b(n)=\frac{n}{{\rm log}_{l}(n)} \end{align} (6)

 \begin{align} 0\leq C=\frac{c(n)}{b(n)}\leq 1 \end{align} (7)

 \begin{align} \eta=\dfrac{\Delta C }{\Delta p}{\dfrac{p }{ C}} \end{align} (16)

4 数值仿真

 \begin{align} & \dot{x}_{1}=-x_{2}-x_{3}\notag\\ & \dot{x}_{2}=x_{1}+p_{1}x_{3}\notag\\ & \dot{x}_{3}=p_{2}+x_{3}(x_{1}-p_{3}) \end{align} (17)

 图 1 Rossler系统状态$x_{1}$的倍周期分岔过程 Figure 1 The period-doubling bifurcation diagram of the state $x_{1}$ of the Rossler system

 图 2 Rossler系统的状态轨迹和动力学轨迹图 Figure 2 The state trajectory and dynamics trajectory of the Rossler system

 图 3 系统的状态轨迹和动力学轨迹时间复杂度指标图 Figure 3 The TLZC indices of state trajectory and dynamics trajectory of the Rossler system
 图 4 系统的状态轨迹和动力学轨迹空间复杂度指标图 Figure 4 The SLZC indices of state trajectory and dynamics trajectory of the Rossler system

 图 5 Rossler系统2倍周期模态分解图 Figure 5 The EMD of period-2 of Rossler system
 图 6 Rossler系统2倍周期Hilbert谱图 Figure 6 The Hilbert spectrum of period-2 of Rossler system
 图 7 Rossler系统2倍周期Hilbert边际谱图 Figure 7 The Hilbert marginal spectrum of period-2 of Rossler system

5 实验分析

 \begin{align} \begin{cases} E \dot{\phi} = -A \phi + \Psi_{c}(\phi) - T \bar{\psi}\\[2mm] \dot{\bar{\psi}} = \dfrac{1}{4l_{c}B^{2}}(S \phi - \Phi_{T}(\bar{\psi})) \end{cases} \end{align} (18)

 图 8 系统从失速前进入到旋转失速初始扰动阶段的过程 Figure 8 Time evolution of the first flow state before rotating stall

 图 9 失速前到旋转失速初始扰动阶段状态、动力学轨迹的时间复杂度TLZC Figure 9 The TLZC index of the system state and dynamics trajectory before rotating stall
 图 10 失速前到旋转失速初始扰动阶段状态、动力学轨迹的空间复杂度SLZC Figure 10 The SLZC index of the system state and dynamics trajectory before rotating stall

6 结论

 图 A1 随机数据序列的归一化复杂度 Figure A1 The normalized Lempel-Ziv complexity of random sequence

 1 Huang N E, Shen Z, Long S R, Wu M C, Shih H H, Zheng Q N, Yen N C, Tung C C, Liu H H. The empirical mode decomposition and the Hilbert spectrum for nonlinear and non-stationary time series analysis. Proceedings of the Royal Society A:Mathematical, Physical and Engineering Sciences, 1998, 454(1971): 903-995. DOI:10.1098/rspa.1998.0193 2 Zhang H J, Bi G A, Razul S G, See C M S. Robust time-varying filtering and separation of some nonstationary signals in low SNR environmentsx. Signal Processing, 2015, 106: 141-158. 3 Brigham E O. The Fast Fourier Transform and Its Applications. Englewood Cliffs, New Jersey, USA: Prentice Hall, 1988. 4 Rai V K, Mohanty A R. Bearing fault diagnosis using FFT of intrinsic mode functions in Hilbert-Huang transform. Mechanical Systems and Signal Processing, 2007, 21(6): 2607-2615. 5 Feng Z P, Liang M, Chu F L. Recent advances in time-frequency analysis methods for machinery fault diagnosis:a review with application examples. Mechanical Systems and Signal Processing, 2013, 38(1): 165-205. DOI:10.1016/j.ymssp.2013.01.017 6 Akhtar M T, Mitsuhashi W, James C J. Employing spatially constrained ICA and wavelet denoising, for automatic removal of artifacts from multichannel EEG data. Signal Processing, 2012, 92(2): 401-416. DOI:10.1016/j.sigpro.2011.08.005 7 Yan J H, Lu L. Improved Hilbert-Huang transform based weak signal detection methodology and its application on incipient fault diagnosis and ECG signal analysis. Signal Processing, 2014, 98: 74-87. DOI:10.1016/j.sigpro.2013.11.012 8 Boffetta G, Cencini M, Falcioni M, Vulpiani A. Predictability:a way to characterize complexity. Physics Reports, 2002, 356(6): 367-474. DOI:10.1016/S0370-1573(01)00025-4 9 Hu J, Gao J B, Tung W W. Characterizing heart rate variability by scale-dependent Lyapunov exponent. Chaos, 2009, 19(2): Article No. 028506. 10 Aboy M, Hornero R, Abásolo D, Álvarez D. Interpretation of the Lempel-Ziv complexity measure in the context of biomedical signal analysis. IEEE Transactions on Biomedical Engineering, 2006, 53(11): 2282-2288. DOI:10.1109/TBME.2006.883696 11 Tabor M. Chaos and Integrability in Nonlinear Dynamics:An Introduction. New York, USA: Wiley, 1989. 12 Lee J, Wu F J, Zhao W Y, Ghaffari M, Liao L X, Siegel D. Prognostics and health management design for rotary machinery systems——reviews, methodology and applications. Mechanical Systems and Signal Processing, 2014, 42(1-2): 314-334. 13 Shilnikov L P, Shilnikov A L, Turaev D V, Chua L O. Methods of Qualitative Theory in Nonlinear Dynamics, Part Ⅰ. Singapore: World Scientific, 2001. 14 Shilnikov L P, Shilnikov A L, Turaev D V, Chua L O. Methods of Qualitative Theory in Nonlinear Dynamics, Part Ⅱ. Singapore: World Scientific, 2001. 15 Wang C, Hill D J. Learning from neural control. IEEE Transactions on Neural Networks, 2006, 17(1): 130-146. 16 Wang C, Hill D J. Deterministic Learning Theory for Identification, Recognition, and Control. Boca Raton, FL, USA: CRC Press, 2009. 17 Wang Cong, Chen Tian-Rui, Liu Teng-Fei. Deterministic learning and data-based modeling and control. Acta Automatica Sinica, 2009, 35(6): 693-706.( 王聪, 陈填锐, 刘腾飞. 确定学习与基于数据的建模及控制. 自动化学报, 2009, 35(6): 693-706.) 18 Wang C, Chen T R. Rapid detection of small oscillation faults via deterministic learning. IEEE Transactions on Neural Networks, 2011, 22(8): 1284-1296. 19 Wang Cong, Wen Bin-He, Si Wen-Jie, Peng Tao, Yuan Cheng-Zhi, Chen Tian-Rui, Lin Wen-Yu, Wang Yong, Hou An-Ping. Modeling and detection of rotating stall in axial flow compressors, Part Ⅰ:investigation on high-order M-G models via deterministic learning. Acta Automatica Sinica, 2014, 40(7): 1265-1277.( 王聪, 文彬鹤, 司文杰, 彭滔, 袁成志, 陈填锐, 林文愉, 王勇, 侯安平. 轴流压气机旋转失速建模与检测, Ⅰ:基于确定学习理论与高阶Moore-Greitzer模型的研究. 自动化学报, 2014, 40(7): 1265-1277.) 20 Wang C, Dong X D, Ou S X, Wang W, Hu J M, Yang F F. A new method for early detection of myocardial ischemia:cardiodynamicsgram (CDG). Science China Information Sciences, 2016, 59(1): 1-11. 21 Lempel A, Ziv J. On the complexity of finite sequences. IEEE Transactions on Infromation Theory, 1976, 22(1): 75-81. DOI:10.1109/TIT.1976.1055501 22 Kaspar F, Schuster H G. Easily calculable measure for the complexity of spatiotemporal patterns. Physical Review A, 1987, 36(2): 842-848. DOI:10.1103/PhysRevA.36.842 23 Rapp P E, Cellucci C J, Korslund K E, Watanabe T A, Jiménez-Moñtano M A. Effective normalization of complexity measurements for epoch length and sampling frequency. Physical Review E, 2001, 64(1): Article No. 016209. 24 Out H H, Sayood K. A new sequence distance measure for phylogenetic tree construction. Bioinformatics (Oxford, England), 2003, 19(16): 2122-2130. DOI:10.1093/bioinformatics/btg295 25 Yan R Q, Gao R X. Complexity as a measure for machine health evaluation. IEEE Transaction on Instrumentation and Measurement, 2004, 53(4): 1327-1334. DOI:10.1109/TIM.2004.831169 26 Savageau M A. Parameter sensitivity as a criterion for evaluating and comparing the performance of biochemical systems. Nature, 1971, 229(5286): 542-544. DOI:10.1038/229542a0 27 Wu W H, Wang F S, Chang M S. Dynamic sensitivity analysis of biological systems. BMC Bioinformatics, 2008, 9(Suppl 12): S17. DOI:10.1186/1471-2105-9-S12-S17 28 Wang P, Lv J H, Ogorzalek M J. Global relative parameter sensitivities of the feed-forward loops in genetic networks. Neurocomputing, 2012, 78(1): 155-165. DOI:10.1016/j.neucom.2011.05.034 29 Lu B Y, Yue H. Developing objective sensitivity analysis of periodic systems:case studies of biological oscillators. Acta Automatica Sinica, 2012, 38(7): 1065-1073. 30 Varma A, Morbidelli M, Wu H. Parametric Sensitivity in Chemical Systems. Cambridge: Cambridge University Press, 1999. 31 Kolmogorov A N. Three approaches to the quantitative definition of information. International Journal of Computer Mathematics, 1968, 2(1-4): 157-168. 32 Zhang X S, Roy R J, Jensen E W. EEG complexity as a measure of depth of anesthesia for patients. IEEE Transactions on Biomedical Engineering, 2001, 48(12): 1424-1433. 33 Sen A K. Complexity analysis of riverflow time series. Stochastic Environmental Research and Risk Assessment, 2009, 23(3): 361-366. DOI:10.1007/s00477-008-0222-x 34 Hong H, Liang M. Fault severity assessment for rolling element bearings using the Lempel-Ziv complexity and continuous wavelet transform. Journal of Sound and Vibration, 2009, 320(1-2): 452-468. DOI:10.1016/j.jsv.2008.07.011 35 Hu J, Gao J B, Principe J C. Analysis of biomedical signals by the Lempel-Ziv complexity:the effect of finite data size. IEEE Transactions on Biomedical Engineering, 2006, 53(12): 2606-2609. 36 Estevez-Rams E, Serrano R L, Fernández B A, Reyes I B. On the non-randomness of maximum Lempel-Ziv complexity sequences of finite size. Chaos, 2013, 23(2): 023118. DOI:10.1063/1.4808251 37 Amigo J M, Kennel M B. Variance estimators for the Lempel-Ziv entropy rate estimator. Chaos, 2006, 16(4): Article No. 043102. 38 Estevezrams E, Loraserrano R, Nunes C A J, Aragónfernán-dez B. Lempel-Ziv complexity analysis of one dimensional cellular automata. Chaos, 2015, 25(12): Article No. 123106. DOI:10.1063/1.4936876 39 Li J K, Song X R, Yin K. Discrete capability of the Lempel-Ziv complexity algorithm on a vibration sequence. Chinese Physics Letters, 2010, 27(6): Article No. 060502. 40 Radhakrishnan N, Wilson J D, Loizou P C. An alternate partitioning technique to quantify the regularity of complex time series. International Journal of Bifurcation and Chaos, 2000, 10(7): 1773-1779. DOI:10.1142/S0218127400001092 41 Sarlabous L, Torres A, Fiz J A, Morera J, Jané R. Index for estimation of muscle force from mechanomyography based on the Lempel-Ziv algorithm. Journal of Electromyography and Kinesiology, 2013, 23(3): 548-557. DOI:10.1016/j.jelekin.2012.12.007 42 Gray A, Abbena E, Salamon S. Modern Differential Geometry of Curves and Surfaces with Mathematica. Boca Raton: CRC Press, 1998. 43 Chen D F, Wang C, Dong X D. Modeling of nonlinear dynamical systems based on deterministic learning and structural stability. Science China Information Sciences, 2016, 59(9): Article No. 92202. 44 Rössler O E. An equation for continuous chaos. Physics Letters A, 1976, 57(5): 397-398. DOI:10.1016/0375-9601(76)90101-8 45 Chen G R, Dong X N. From Chaos to Order:Methodologies, Perspectives, and Applications. Singapore: World Scientific, 1998. 46 Paduano J D. Analysis of compression system dynamics. Active Control of Engine Dynamics, 2002, 8: 1-36. 47 Kamath C. A risk stratification system to discriminate congestive heart failure patients using multivalued coarse-graining Lempel-Ziv complexity. Journal of Engineering Science and Technology, 2015, 10(1): 12-24.