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1. 北京航空航天大学 电子信息工程学院, 北京 100083;
2. 西安卫星测控中心 宇航动力学国家重点实验室, 西安 710043

An SVR based hybrid modeling method
SUN Zebin1, ZHAO Qi1, ZHAO Hongbo1, FENG Wenquan1, ZHANG Wenfeng1, YANG Tianshe2
1. School of Electronic and Information Engineering, Beijing University of Aeronautics and Astronautics, Beijing 100083, China;
2. State Key Laboratory of Astronautic Dynamics, Xi'an Satellite Control Center, Xi'an 710043, China
Received: 2016-04-19; Accepted: 2016-04-29; Published online: 2016-05-05
Foundation item: National Basic Research Program of China
Corresponding author. ZHAO Hongbo, E-mail:bhzhb@buaa.edu.cn
Abstract: As computing power increases in recent years, data-driven modeling method receives much attention. Modeling methods to analyze quantitative behavior of systems with single mode have been researched much. However, most systems have multiple modes which own different continuous behavior and are influenced by continuous state when switching. This paper proposes the empirical probabilistic hybrid automata model and the qualitative and quantitative hybrid modeling method based on support vector regression (SVR).First, switching points between modes are recognized via wavelet and then the SVR sub-models are constructed for each mode. Finally, all sub-models are integrated within D-Markov machine. The example verification results demonstrate that the proposed method is as stable as traditional SVR model, and much more accurate than it.
Key words: hybrid modeling     support vector regression (SVR)     D-Markov machine     wavelet     data-driven modeling

1 理论基础

1.1 支持向量回归

 (1)

 (2)

 (3)

 (4)

1.2 基于小波的奇异点分析

 (5)

 (6)

 (7)

 (8)

ψ(x) 满足一定条件时[7]，有区间小波奇异点定理：如果j=jn满足j=jn→∞(n→∞) 且，那么当n→∞时：

1) 对于所有的kI(tl, 2-jn)

 (9)

2) 对于所有

 (10)

3)kI(sl, 2-jn) 时，则存在一个与kn无关常数C0>0，使得

 (11)

4)

 (12)

1.3 D-Markov重构

Pennsylvania州立大学机械系的Ray等[8-10]提出一种基于D-Markov机的模型重构方法，在该方法中，他将系统在“慢时间尺度”和“快时间尺度”上分析, 见图 1。对应于混合系统中模式间切换与模式内部行为。在快慢尺度上的分析能够有效地挖掘动态系统的特征[11]

 图 1 慢时间尺度与快时间尺度 Fig. 1 Slow time scale and fast time scale
 (13)

 (14)
 (15)

 (16)

2 定性定量混合建模 2.1 混合建模方法

 图 2 混合建模框图 Fig. 2 Block diagram of hybrid modeling

EPHA重构算法如下:

EPHA-Re-Construct

//使用奇异点识别技术，探测历史数据中模式切

switching_points

SwitchingPointsDetection (data)

//根据模式切换点得到分段数据

data_section=SplitData (data, switching_points)

//分段拟合

//归类与符号映射

//当svri与映射表中对应的svr输出结果一致时，认为已经为svri分配了符号

if AssignedSymbol (svri, map_set)

//已经分配了符号时，什么都不用做

else

//否则为svri分配新的符号，并将映射关系添加到map_set中

//符号化切换点附近的数据

sym_set=Symbol (data, D, switching_points)

//将模式符号与数据符号有序排列

sym_seq=Sequence (symbol in map_set+sym_set)

//D-Markov重构

markov_machine=MarkovReConstruction (D, sym_seq)

//得到EPHA模型

EPHA=map_set+markov_machine

2.2 方法分析

3 实验验证

 (17)

1) 对该系统按0.01 s等间隔采样，前4 s的系统行为如图 3所示。其中信噪比为10 dB。

 图 3 混合系统行为 Fig. 3 Behavior of hybrid system

2) 使用小波探测奇异点，结果如图 4所示，在0~4 s之中，真实模式切换点为1、2与3 s，探测到的模式切换点分别为0.97、2.05与2.96 s。

 图 4 模式切换点识别 Fig. 4 Recognition of mode switching points

3) 对历史数据在各个模式下使用SVR单独建模，实验中使用高斯核函数，并使用粒子群优化算法确定参数。

4) 将数据等间隔量化，选取D=1，在模式切换点附近建立D-Markov机。

 图 5 EPHA模型与SVR模型系统行为曲线 Fig. 5 System behavior curves of EPHA model and SVR model
 (18)

 序号 均方误差 SVR EPHA 1 0.089 6 0.071 6 2 0.094 5 0.067 9 3 0.094 4 0.072 8 4 0.091 6 0.073 3 5 0.093 5 0.071 8 6 0.089 6 0.072 2 7 0.091 2 0.070 1 8 0.094 2 0.071 6 9 0.093 5 0.068 7 10 0.094 5 0.072 2 方差 3.95×10-6 3.11×10-6 均值 0.092 7 0.071 2

4 结论

1) 本文提出的EPHA混合建模方法能够有效解决混合系统中传统定量建模方法精度差的问题，特别是在模式切换点附近，精确度远高于传统SVR模型，实验表明，建模误差减少约23%。

2) EPHA模型与传统SVR模型的稳定性接近，是一种可靠的数学模型。

3) EPHA混合建模方法能够准确识别混合系统模式切换关系，对通过数据分析系统行为有着极大的帮助。

4) EPHA模型具备良好的适应能力，当放宽假设2与假设3时，仍然能够非常容易地扩展，从而得到准确的混合模型。

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#### 文章信息

SUN Zebin, ZHAO Qi, ZHAO Hongbo, FENG Wenquan, ZHANG Wenfeng, YANG Tianshe

An SVR based hybrid modeling method

Journal of Beijing University of Aeronautics and Astronsutics, 2017, 43(2): 352-359
http://dx.doi.org/10.13700/j.bh.1001-5965.2016.0319