Springer Nature is making SARS-CoV-2 and COVID-19 research free. View research | View latest news | Sign up for updates

# Modeling of nonlinear dynamical systems based on deterministic learning and structural stability

## Abstract

Recently, a deterministic learning (DL) theory was proposed for accurate identification of system dynamics for nonlinear dynamical systems. In this paper, we further investigate the problem of modeling or identification of the partial derivative of dynamics for dynamical systems. Firstly, based on the locally accurate identification of the unknown system dynamics via deterministic learning, the modeling of its partial derivative of dynamics along the periodic or periodic-like trajectory is obtained by using the mathematical concept of directional derivative. Then, with accurately identified system dynamics and the partial derivative of dynamics, a C 1-norm modeling approach is proposed from the perspective of structural stability, which can be used for quantitatively measuring the topological similarities between different dynamical systems. This provides more incentives for further applications in the classification of dynamical systems and patterns, as well as the prediction of bifurcation and chaos. Simulation studies are included to demonstrate the effectiveness of this modeling approach.

## 摘要

### 创新点

1. 1)

基于结构稳定和确定学习理论, 提出了 C1 -范数下的动力学建模方法。 在已有研究中, 结构稳定概念主要限于定性分析系统在参数微小扰动下, 系统结构的改变; 而本文给出的 C1 范数模型为定量计算非线性系统之间的动力学差异提供了工具。 该模型可进一步用于非线性系统及动态模式的分类, 分岔和混沌的预测等实际问题。

2. 2)

通过引入沿轨迹的方向导数这一数学概念, 实现了对系统偏导部分动力学的准确建模。 偏导部分的动力学信息, 是判断受扰系统结构稳定与否的必要部分。 确定学习算法获得了对系统动力学的准确辨识和建模, 在此基础上, 通过方向导数完成了偏导部分动力学的准确建模, 继而实现了 C1 范数意义下的系统动力学度量。

This is a preview of subscription content, log in to check access.

## References

1. 1

Narendra K S, Parthasarathy K. Identification and control of dynamical systems using neural networks. IEEE Trans Neural Netw, 1990, 1: 4–27

2. 2

Li H B, Sun Z Q, Min H B, et al. Fuzzy dynamic characteristic modeling and adaptive control of nonlinear systems and its application to hypersonic vehicles. Sci China Inf Sci, 2011, 54: 460–468

3. 3

Zhao W X, Chen H F. Markov chain approach to identifying winner systems. Sci China Inf Sci, 2012, 55: 1201–1217

4. 4

Hassani V, Tjahjowidodo T, Thanh N D. A survey on hysteresis modeling, identification and control. Mech Syst Signal Process, 2014, 49: 209–233

5. 5

Fekih A, Xu H, Chowdhury F N. Neural networks based system identification techniques for model based fault detection of nonlinear systems. Int J Comp Info Contr, 2007, 3: 1073–1085

6. 6

Ko C N. Identification of nonlinear systems with outliers using wavelet neural networks based on annealing dynamical learning algorithm. Eng Appl Artif Intell, 2012, 25: 533–543

7. 7

Han H G, Wu X L, Qiao J F. Nonlinear systems modeling based on self-organizing fuzzy-neural network with adaptive computation algorithm. IEEE Trans Cyber, 2014, 44: 554–564

8. 8

Polycarpou M M, Ioannou P A. Modelling, identification and stable adaptive control of continuous-time nonlinear dynamical systems using neural networks. In: Proceedings of the IEEE American Control Conference, Boston, 1992. 36–40

9. 9

Sanner R M, Slotine J E. Gaussian networks for direct adaptive control. IEEE Trans Neural Netw, 1992, 3: 837–863

10. 10

Willems J C, Rapisarda P, Markovsky I. A note on persistency of excitation. Syst Control Lett, 2005, 54: 325–329

11. 11

Guo J, Zhao Y L. Identification of the gain system with quantized observations and bounded persistent excitations. Sci China Inf Sci, 2014, 57: 012205

12. 12

Lu S, Basar T. Robust nonlinear system identification using neural network models. IEEE Trans Neural Netw, 1998, 9: 407–429

13. 13

Kurdila A J, Narcowich F J, Ward J D. Persistancy of excitation in identification using radial basis function approximants. SIAM J Contr Optim, 1995, 33: 625–642

14. 14

Gorinevsky D. On the persistancy of excitation in radial basis function network identification of nonlinear systems. IEEE Trans Neural Netw, 1995, 6: 1237–1244

15. 15

Chen W S, Wen C Y, Hua S Y, et al. Distributed cooperative adaptive identification and control for a group of continuous-time systems with a cooperative PE condition via consensus. IEEE Trans Autom Control, 2014, 59: 91–106

16. 16

Wang C, Hill D J. Learning from neural control. IEEE Trans Neural Netw, 2006, 17: 130–146

17. 17

Wang C, Hill D J. Deterninistic Learning Theory for Identification, Recognition and Control. Boca Raton: CRC Press, 2009

18. 18

Wang C, Hill D J. Deterministic learning and rapid dynamical pattern recognition. IEEE Trans Neural Netw, 2007, 18: 617–630

19. 19

Pai M A, Sauer P W, Lesieutre B C. Structural stability in power systems-effect of load models. IEEE Trans Power Syst, 1995, 10: 609–615

20. 20

Pacifico M J. Structural stability of vector fields on 3-manifolds with boundary. J Differ Equations, 1984, 54: 346–372

21. 21

Wu J R, Yang C W. Structural stability in discrete singular systems. Chinese Phys, 2002, 11: 1221–1227

22. 22

Palmer K J, Pilyugin S Y, Tikhomirov S B. Lipschitz shadowing and structural stability of flows. J Differ Equations, 2012, 252: 1723–1747

23. 23

Fetea R, Petroianu A. The implications of structural stability in power systems. In: Proceedings of International Conference on Power System Technology, Beijing, 1998. 1336–1340

24. 24

John A T, Ivan Arango. Topological classification of limit cycles of piecewise smooth dynamical systems and its associated nonstandard bifurcations. Entropy, 2014, 16: 1949–1968

25. 25

Ma R, Liu H P, Sun F C. Linear dynamic system method for tactile object classification. Sci China Inf Sci, 2014, 57: 120205

26. 26

Ma S S, Lu M, Ding J F, et al. Weak signal detection method based on duffing oscillator with adjustable frequency. Sci China Inf Sci, 2015, 58: 102401

27. 27

Park J H. Adaptive synchronization of Rossler system with uncertain parameters. Chaos Solitons Fractals, 2005, 25: 333–338

28. 28

Powell M J D. The theory of radial basis function approximation in 1990. In: Advances in Numerical Analysis II: Wavelets, Subdivision, Algorithms, and Radial Basis Functions. Oxford: Oxford Univercity Press, 1992. 105–210

29. 29

Bitmead R R. Persistence of excitation conditions and the convergence of adaptive schemes. IEEE Trans Inf Theory, 1984, 30: 183–191

30. 30

Wang C, Hill D J. Persistence of excitation, RBF approximation and periodic orbits. In: Proceedings of the IEEE International Conference on Control and Automation, Budapest, 2005. 547–552

31. 31

Wang C, Hill D J, Chen G G. Deterministic learning of nonlinear dynamical systems. In: Proceedings of the IEEE International Symposium on Intelligent Control, Houston, 2003. 87–92

32. 32

Shilnikov L P, Shilnikov A L, Turaev D V, et al. Methods of Qualitative Theory in Nonlinear Dynamics. Singapore: World Scientific, 2001

33. 33

Gaull A, Kreuzer E. Exploring the qualitive behavour of uncertain dynamical systems a computational approach. Nonlinear Dynam, 2011, 63: 285–310

34. 34

Luo Q, Liao X X, Zeng Z G. Sufficient and necessary conditions for Lyapunov stability of Lorenz system and their application. Sci China Inf Sci, 2010, 53: 1574–1583

35. 35

Kuznetsov Y A. Elements of Applied Bifurcation Theory. 2nd ed. Berlin: Springer, 1998

36. 36

Gray A, Abbena E, Salamon S. Modern Differential Geometry of Curves and Surfaces With Mathematica. Boca Raton: CRC Press, 1998

37. 37

Yuan C Z, Wang C. Persistency of excitation and performance of deterministic learning. Syst Control Lett, 2011, 60: 952–959

38. 38

Yuan C Z, Wang C. Design and performance analysis of deterministic learning of sampled-data nonlinear systems. Sci China Inf Sci, 2014, 57: 032201

39. 39

Peixoto M M. Structural stability on two-dimensional manifolds. Topology, 1962, 1: 101–120

40. 40

Wang C, Wen B H, Si WJ, et al. Modeling and detection of rotating stall in axial flow compressors: part I-investigation on high-order M-G models via deterministic learning. Acta Autom Sin, 2014, 40: 1265–1277

41. 41

Divshali P H, Hosseinian S H, Nasr E. Reliable prediction of Hopf bifurcation in power systems. Electr Eng, 2009, 91: 61–68

## Author information

Correspondence to Cong Wang.

## Rights and permissions

Reprints and Permissions

Chen, D., Wang, C. & Dong, X. Modeling of nonlinear dynamical systems based on deterministic learning and structural stability. Sci. China Inf. Sci. 59, 92202 (2016). https://doi.org/10.1007/s11432-015-5498-0

• Accepted:

• Published:

### Keywords

• system modeling
• system identification
• deterministic learning
• nonlinear dynamics
• structural stability
• topological equivalence

• 系统建模
• 系统辨识
• 确定学习
• 非线性动力学
• 结构稳定
• 拓扑等价