Dynamic Feature Extraction of Nonlinear Systems With Deterministic Learning Theory and Spatio-temporal Lempel-Ziv Complexity
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摘要: 对非线性系统产生的非线性非平稳信号进行有效的特征表达是特征提取领域重要且困难的问题.本文基于确定学习理论和Lempel-Ziv复杂度（LZ复杂度）提出一种新的非线性系统动态特征提取方法.新方法将从系统的动力学轨迹中提取特征.通过确定学习理论对产生回归轨迹的非线性动力学系统的未知系统动态进行局部准确建模/辨识，1）使用LZ复杂度对辨识得到的动力学轨迹进行特征表达，并提出时间复杂度和空间复杂度两个指标组成时空LZ复杂度，从时间域和空间域的角度刻画系统动力学轨迹的复杂程度.2）对提出的动态特征提取方法进行敏感度分析，定量评价系统的动态特征指标相对于系统从周期轨迹到混沌轨迹的参数变化敏感程度.3）通过数值仿真和实验分析以验证动态特征提取的有效性.与从系统状态轨迹中提取特征相比，本文提出的动态特征提取方法可以从系统内在动态的角度对原系统进行更好的表达.
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关键词:
- 动态特征提取 /
- 确定学习 /
- 时空Lempel-Ziv复杂度 /
- 敏感度分析
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. -
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图 1 Rossler系统状态$x_{1}$的倍周期分岔过程
Fig. 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
Fig. 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
Fig. 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
Fig. 4 The SLZC indices of state trajectory and dynamics trajectory of the Rossler system
图 7 Rossler系统2倍周期Hilbert边际谱图
Fig. 7 The Hilbert marginal spectrum of period-2 of Rossler system
图 8 系统从失速前进入到旋转失速初始扰动阶段的过程
Fig. 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
Fig. 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
Fig. 10 The SLZC index of the system state and dynamics trajectory before rotating stall
表 1 Rossler系统的敏感度系数
Table 1 The sensitivity coefficients of the Rossler system
系统状态变化 η1(TLZC) η2(TLZC) η1(SLZC) η2(SLZC) period (1 ~ 2) 0.0018 0.0229 0.0247 0.0268 period (2 ~ 4) 0.0906 0.1345 0.0577 0.0925 period (4 ~ 8) 0.3646 0.6401 0.2980 0.5098 period (8 ~ chaos) 1.4589 2.2767 0.9603 1.3192 
下载: 导出CSV
表 2 失速前到初始扰动过程的时空复杂度指标敏感度系数
Table 2 The sensitivity coefficients of the normal system to stall precursors
失速前到初始扰动状态变化 η1 η2 TLZC 0.022 0.027 SLZC 0.022 0.043
下载: 导出CSV
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