Nonlinear dynamic system identification using neuro-fractional-order Hammerstein model

Abstract

This paper presents a neuro-fractional-order Hammerstein model with a systematic identification algorithm for identifying unknown nonlinear dynamic systems. The proposed model consists of a Radial Basis Function Neural Network (RBF NN) followed by a Fractional-Order System (FOS). The proposed identification scheme is performed in two stages. First, the fractional-order and the number of state variables (or degree) of the state-space realization of the FOS are estimated in the frequency domain. Then, the parameters of the RBF NN (the weights, centers and widths of the Gaussian functions) and the state matrix of the FOS are determined using the time domain data via the Lyapunov stability theory. Simulating as well as experimental examples are provided to verify the effectiveness of the proposed method. The identification results show that the proposed neuro-fractional-order Hammerstein modeling is superior as compared with the existing Hammerstein modeling in literature.

Keywords

Hammerstein model fractional-order differential equations neural networks frequency domain identification

Get full access to this article

View all access options for this article.

References

Aguila-Camacho

Duarte-Mermoud

Gallegos

(2014) Lyapunov functions for fractional order systems. Communications in Nonlinear Science and Numerical Simulation 19(9): 2951–2957.

Aoun

Maltl

Cois

et al . (2002) System identification using fractional Hammerstein models. In: 15th IFAC World Congress, Barcelona, Spain.

Atitallah

Bedoui

Abderrahim

(2016) An optimal two stage identification algorithm for discrete hammerstein time delay systems. In: 13th IFAC Workshop on Time Delay Systems (TDS 2016), Istanbul, Turkey.

Atitallah

Bedoui

Abderrahim

(2017) Multistage for identification of Wiener time delay systems based on hierarchical gradient approach. Mathematical and Computer Modelling of Dynamical Systems 23(2): 222–239.

Chen

Ding

(2015) Hierarchical least squares identification for Hammerstein nonlinear controlled autoregressive systems. Circuits, Systems, and Signal Processing 34(1): 61–75.

Chen

Hong

Gao

et al . (2014) Complex-valued B-spline neural networks for modeling and inverting Hammerstein systems. IEEE Transactions on Neural Networks and Learning Systems 25(9): 1673–1685.

Cui

Liu

et al . (2014) Identification of Hammerstein model using functional link artificial neural network. Neurocomputing 142: 419–428.

De Moor

De Gersem

De Schutter

et al . (1997) DAISY: A database for identification of systems. Journal A 38: 4–5.

Deng

Ding

(2014) Newton iterative identification method for an input nonlinear finite impulse response system with moving average noise using the key variables separation technique. Nonlinear Dynamics 76(2): 1195–1202.

10.

Ding

Liu

(2016) Kalman state filtering based least squares iterative parameter estimation for observer canonical state space systems using decomposition. Journal of Computational and Applied Mathematics 301: 135–143.

11.

Djamah

Mansouri

Bettayeb

et al . (2009) State space realization of fractional order systems. In: 2nd Mediterranean Conference on Intelligent Systems and Automation (CISA), pp. 37–42. Zarzis, Tunisia: AIP Publishing.

12.

Dudul

(2007) Identification of a liquid saturated steam heat exchanger using focused time lagged recurrent neural network model. IETE Journal of Research 53(1): 69–82.

13.

Efe

MÖ

(2008) Fractional fuzzy adaptive sliding-mode control of a 2-DOF direct-drive robot arm. IEEE Transactions on Systems, Man, and Cybernetics, Part B: Cybernetics 38(6): 1561–1570.

14.

Z-J

Xie

W-F

Luo

W-D

(2013) Robust on-line nonlinear systems identification using multilayer dynamic neural networks with two-time scales. Neurocomputing 113: 16–26.

15.

Giri

Bai

E-W

(2010) Block-oriented Nonlinear System Identification: Springer.

16.

Goethals

Pelckmans

Suykens

et al . (2005) Identification of MIMO Hammerstein models using least squares support vector machines. Automatica 41(7): 1263–1272.

17.

Gotmare

Patidar

George

(2014) Nonlinear system identification using a cuckoo search optimized adaptive Hammerstein model. Expert Systems with Applications 42(5): 2538–2546.

18.

Hammar

Djamah

Bettayeb

(2015) Fractional Hammerstein system identification using particle swarm optimization. In: 7th International Conference on Modelling, Identification and Control (ICMIC), Sousse, Tunisia.

19.

Hong

Chen

(2012) The system identification and control of Hammerstein system using non-uniform rational B-spline neural network and particle swarm optimization. Neurocomputing 82: 216–223.

20.

Hong

Chen

Harris

et al . (2014) Single-carrier frequency domain equalisation for Hammerstein communication systems using complex-valued neural networks. IEEE Transactions on Signal Processing 62(17): 4467–4478.

21.

Ding

(2014) Least squares based iterative identification algorithms for input nonlinear controlled autoregressive systems based on the auxiliary model. Nonlinear Dynamics 76(1): 777–784.

22.

Ivanov

(2015) Identification discrete fractional order Hammerstein systems. In: 2015 International Siberian Conference on Control and Communications (SIBCON), Omsk, Russia.

23.

Janakiraman

Nguyen

Assanis

(2013) A lyapunov based stable online learning algorithm for nonlinear dynamical systems using extreme learning machines. In: The 2013 International Joint Conference on Neural Networks (IJCNN), Dallas, USA.

24.

Chen

Podlubny

(2009) Mittag–Leffler stability of fractional order nonlinear dynamic systems. Automatica 45(8): 1965–1969.

25.

Liao

Zhu

Liang

et al . (2012) Subspace identification for fractional order Hammerstein systems based on instrumental variables. International Journal of Control, Automation and Systems 10(5): 947–953.

26.

Liu

Wang

(2014) Modeling of a class of nonlinear dynamic system. Sensors & Transducers 169(4): 153–158.

27.

Mkadem

Boumaiza

(2009) Extended Hammerstein behavioral model using artificial neural networks. IEEE Transactions on Microwave Theory and Techniques 57(4): 745–751.

28.

B-Q

Chen

H-F

(2015) Recursive identification of multi-input multi-output errors-in-variables Hammerstein systems. IEEE Transactions on Automatic Control 60: 843–849.

29.

Narang

Shah

Chen

(2011) Continuous-time model identification of fractional-order models with time delays. IET Control Theory & Applications 5(7): 900–912.

30.

Oldham

Spanier

(1974) The Fractional Calculus: Integrations and Differentiations of Arbitrary Order. New York: Academic Press.

31.

Pintelon

Schoukens

Rolain

(2008) Frequency-domain approach to continuous-time system identification: Some practical aspects. Identification of Continuous-time Models from Sampled Data. London: Springer, pp. 215–248.

32.

Podlubny

(1999) Fractional differential equations. San Diego, USA: Academic Press.

33.

Poinot

Trigeassou

J-C

(2004) Identification of fractional systems using an output-error technique. Nonlinear Dynamics 38: 133–154.

34.

Schwartz

Sawin

Hatton

(1986) Frequency response analysis of nonsuperimposable systems using a pseudo-transfer function: Application to systems with coverage-dependent adsorption. Chemical Engineering Science 41(1–4): 495–510.

35.

Sun

Liu

(2013) A novel APSO-aided maximum likelihood identification method for Hammerstein systems. Nonlinear Dynamics 73(1–2): 449–462.

36.

Tang

Han

Wang

et al . (2016) A changing forgetting factor RLS for online identification of nonlinear systems based on ELM–Hammerstein model. Neural Computing and Applications: 1–15. doi: 10.1007/s00521-016-2394-5.

37.

Van Pelt

Bernstein

(2001) Non-linear system identification using Hammerstein and non-linear feedback models with piecewise linear static maps. International Journal of Control 74(18): 1807–1823.

38.

Wang

Liu

et al . (2015) Identification of chaotic system using Hammerstein-ELM model. Nonlinear Dynamics 81(3): 1081–1095.

39.

Wang

Ding

(2016) Modelling and multi-innovation parameter identification for Hammerstein nonlinear state space systems using the filtering technique. Mathematical and Computer Modelling of Dynamical Systems 22(2): 113–140.

40.

Wang

Ding

Alsaedi

et al . (2017) Filtering based parameter estimation for observer canonical state space systems with colored noise. Journal of the Franklin Institute 354(1): 593–609.

41.

Zhao

et al . (2014a) Hammerstein identification of supercharged boiler superheated steam pressure using Laguerre-Fuzzy model. International Journal of Heat and Mass Transfer 70: 33–39.

42.

Zhao

Chen

(2014b) Complete parametric identification of fractional order Hammerstein systems. In: IEEE International Conference on Fractional Differentiation and its Applications (ICFDA), Catania, Italy.