Research Papers

Iterative Identification of Discrete-Time Systems With Bilinear Forms in the Presence of Colored Noises Based on the Hierarchical Principle

[+] Author and Article Information
Mengting Chen

Key Laboratory of Advanced Process Control for
Light Industry (Ministry of Education),
Jiangnan University,
Wuxi 214122, China
e-mail: mtchen11@126.com

Feng Ding

Key Laboratory of Advanced Process Control for
Light Industry (Ministry of Education),
Jiangnan University,
Wuxi 214122, China;
College of Automation and
Electronic Engineering,
Qingdao University of
Science and Technology,
Qingdao 266061, China;
Department of Mathematics,
King Abdulaziz University,
Jeddah 21589, Saudi Arabia
e-mail: fding@jiangnan.edu.cn

1Corresponding author.

Contributed by the Design Engineering Division of ASME for publication in the JOURNAL OF COMPUTATIONAL AND NONLINEAR DYNAMICS. Manuscript received December 11, 2018; final manuscript received May 30, 2019; published online July 15, 2019. Assoc. Editor: Bernard Brogliato.

J. Comput. Nonlinear Dynam 14(9), 091004 (Jul 15, 2019) (10 pages) Paper No: CND-18-1550; doi: 10.1115/1.4044013 History: Received December 11, 2018; Revised May 30, 2019

The paper focuses on the identification of discrete-time bilinear forms in the special case when the external noise (disturbance) is an autoregressive average moving process. The proposed estimation procedure is iterative where, at each iteration, two sets of parameter vectors are estimated interactively. Using the hierarchical technique, a hierarchical generalized extended least squares-based iterative (H-GELSI) algorithm is proposed for avoiding estimating the redundant parameters. In contrast to the hierarchical generalized extended gradient-based iterative (H-GEGI) algorithm, the proposed algorithm can give more accurate parameter estimates. The main results derived in this paper are verified by means of both the computational efficiency comparison and two numerical simulations.

Copyright © 2019 by ASME
Your Session has timed out. Please sign back in to continue.


Gan, M. , Li, H. X. , and Peng, H. , 2015, “ A Variable Projection Approach for Efficient Estimation of RBF-ARX Model,” IEEE Trans. Cybern., 45(3), pp. 476–485. [CrossRef] [PubMed]
Zarei, A. , and Tavakoli, S. , 2018, “ Synchronization of Quadratic Chaotic Systems Based on Simultaneous Estimation of Nonlinear Dynamics,” ASME J. Comput. Nonlinear Dyn., 13(8), p. 081001. [CrossRef]
Kim, P. , Rogers, J. , Sun, J. , and Bollt, E. , 2016, “ Causation Entropy Identifies Sparsity Structure for Parameter Estimation of Dynamic Systems,” ASME J. Comput. Nonlinear Dyn., 12(1), p. 011008. [CrossRef]
Xu, L. , 2014, “ A Proportional Differential Control Method for a Time-Delay System Using the Taylor Expansion Approximation,” Appl. Math. Comput., 236, pp. 391–399.
Gan, M. , Chen, C. L. P. , Chen, G. Y. , and Chen, L. , 2018, “ On Some Separated Algorithms for Separable Nonlinear Least Squares Problems,” IEEE Trans. Cybern., 48(10), pp. 2866–2874. [CrossRef] [PubMed]
Chen, G. Y. , Gan, M. , and Chen, G. L. , 2018, “ Generalized Exponential Autoregressive Models for Nonlinear Time Series: Stationarity, Estimation and Applications,” Inf. Sci., 438, pp. 46–57. [CrossRef]
Kolansky, J. , and Sandu, C. , 2017, “ Enhanced Polynomial Chaos-Based Extended Kalman Filter Technique for Parameter Estimation,” ASME J. Comput. Nonlinear Dyn., 13(2), p. 021012. [CrossRef]
Zhang, X. , Ding, F. , Xu, L. , and Yang, E. F. , 2019, “ Highly Computationally Efficient State Filter Based on the Delta Operator,” Int. J. Adapt. Control Signal Process., 33(6), pp. 875–889. [CrossRef]
Na, J. , Herrmann, G. , and Zhang, K. Q. , 2017, “ Improving Transient Performance of Adaptive Control Via a Modified Reference Model and Novel Adaptation,” Int. J. Robust Nonlinear Control., 27(8), pp. 1351–1372. [CrossRef]
Ding, F. , 2013, “ Two-Stage Least Squares Based Iterative Estimation Algorithm for CARARMA System Modeling,” Appl. Math. Model., 37(7), pp. 4798–4808. [CrossRef]
Ding, F. , 2013, “ Decomposition Based Fast Least Squares Algorithm for Output Error Systems,” Signal Process., 93(5), pp. 1235–1242. [CrossRef]
Zhang, X. , Xu, L. , Ding, F. , and Hayat, T. , 2018, “ Combined State and Parameter Estimation for a Bilinear State Space System With Moving Average Noise,” J. Franklin Inst., 355(6), pp. 3079–3103. [CrossRef]
Yu, C. , and Verhaegen, M. , 2018, “ Data-Driven Fault Estimation of Non-Minimum Phase LTI Systems,” Automatica, 92, pp. 181–187. [CrossRef]
Xu, L. , Xiong, W. L. , Alsaedi, A. , and Hayat, T. , 2018, “ Hierarchical Parameter Estimation for the Frequency Response Based on the Dynamical Window Data,” Int. J. Control Autom. Syst., 16(4), pp. 1756–1764. [CrossRef]
Xu, L. , 2017, “ The Parameter Estimation Algorithms Based on the Dynamical Response Measurement Data,” Adv. Mech. Eng., 9(11), pp. 1–12.
Liu, C. Y. , Gong, Z. H. , and Teo, K. L. , 2018, “ Robust Parameter Estimation for Nonlinear Multistage Time-Delay Systems With Noisy Measurement Data,” Appl. Math. Model., 53, pp. 353–368. [CrossRef]
Zhang, X. , Ding, F. , Xu, L. , and Yang, E. F. , 2018, “ State Filtering-Based Least Squares Parameter Estimation for Bilinear Systems Using the Hierarchical Identification Principle,” IET Control Theory Appl., 12(12), pp. 1704–1713. [CrossRef]
Zhang, X. , Ding, F. , Alsaadi, F. E. , and Hayat, T. , 2017, “ Parameter Identification of the Dynamical Models for Bilinear State Space Systems,” Nonlinear Dyn., 89(4), pp. 2415–2429. [CrossRef]
Zhang, X. , Ding, F. , Xu, L. , Alsaedi, A. , and Hayat, T. , 2019, “ A Hierarchical Approach for Joint Parameter and State Estimation of a Bilinear System With Autoregressive Noise,” Mathematics, 7(4), pp. 1–17. [CrossRef]
Allafi, W. , Zajic, I. , Uddin, K. , and Burnham, K. J. , 2017, “ Parameter Estimation of the Fractional-Order Hammerstein-Wiener Model Using Simplified Refined Instrumental Variable Fractional-Order Continuous Time,” IET Control Theory Appl., 11(15), pp. 2591–2598. [CrossRef]
Wang, Y. J. , and Ding, F. , 2017, “ A Filtering Based Multi-Innovation Gradient Estimation Algorithm and Performance Analysis for Nonlinear Dynamical Systems,” IMA J. Appl. Math., 82(6), pp. 1171–1191. [CrossRef]
Ayala, V. , Silva, A. D. , and Ferreira, M. , 2018, “ Affine and Bilinear Systems on Lie Groups,” Syst. Control Lett., 117, pp. 23–39. [CrossRef]
Bai, E. W. , 1998, “ An Optimal Two-Stage Identification Algorithm for Hammerstein-Wiener Nonlinear Systems,” Automatica, 34(3), pp. 333–338. [CrossRef]
Yu, F. , Mao, Z. Z. , Jia, M. X. , and Yuan, P. , 2014, “ Recursive Parameter Identification of Hammerstein-Wiener Systems With Measurement Noise,” Signal Process., 105, pp. 137–147. [CrossRef]
Benesty, J. , Paleologu, C. , and Ciochina, S. , 2017, “ On the Identification of Bilinear Forms With the Wiener Filter,” IEEE Signal Process. Lett., 24(5), pp. 653–657. [CrossRef]
Paleologu, C. , Benesty, J. , and Ciochină, S. , 2018, “ Adaptive Filtering for the Identification of Bilinear Forms,” Digit. Signal Process., 75, pp. 153–167. [CrossRef]
Bai, E. W. , and Liu, Y. , 2006, “ Least Squares Solutions of Bilinear Equations,” Syst. Control Lett., 55(6), pp. 466–472. [CrossRef]
Wang, J. D. , Zhang, Q. H. , and Ljung, L. , 2009, “ Revisiting Hammerstein System Identification Through the Two-Stage Algorithm for Bilinear Parameter Estimation,” Automatica, 45(11), pp. 2627–2633. [CrossRef]
Filipovic, V. Z. , 2015, “ Consistency of the Robust Recursive Hammerstein Model Identification Algorithm,” J. Franklin Inst., 352(5), pp. 1932–1945. [CrossRef]
Xu, L. , Chen, L. , and Xiong, W. L. , 2015, “ Parameter Estimation and Controller Design for Dynamic Systems From the Step Responses Based on the Newton Iteration,” Nonlinear Dyn., 79(3), pp. 2155–2163. [CrossRef]
Ding, F. , Pan, J. , Alsaedi, A. , and Hayat, T. , 2019, “ Gradient-Based Iterative Parameter Estimation Algorithms for Dynamical Systems From Observation Data,” Mathematics, 7(5), pp. 1–15. [CrossRef]
Pan, J. , Ma, H. , and Jiang, X. , 2018, “ Adaptive Gradient-Based Iterative Algorithm for Multivariate Controlled Autoregressive Moving Average Systems Using the Data Filtering Technique,” Complexity, 2018, p. 9598307.
Xu, L. , 2015, “ Application of the Newton Iteration Algorithm to the Parameter Estimation for Dynamical Systems,” J. Comput. Appl. Math., 288, pp. 33–43. [CrossRef]
Xu, L. , 2016, “ The Damping Iterative Parameter Identification Method for Dynamical Systems Based on the Sine Signal Measurement,” Signal Process., 120, pp. 660–667. [CrossRef]
Zhang, Z. N. , Ding, F. , and Liu, X. G. , 2011, “ Hierarchical Gradient Based Iterative Parameter Estimation Algorithm for Multivariable Output Error Moving Average Systems,” Comput. Math. Appl., 61(3), pp. 672–682. [CrossRef]
Ding, F. , Liu, X. G. , and Chu, J. , 2013, “ Gradient-Based and Least-Squares-Based Iterative Algorithms for Hammerstein Systems Using the Hierarchical Identification Principle,” IET Control Theory Appl., 7(2), pp. 176–184. [CrossRef]
Chen, M. T. , Ding, F. , Xu, L. , Hayat, T. , and Alsaedi, A. , 2017, “ Iterative Identification Algorithms for Bilinear-in-Parameter Systems With Autoregressive Moving Average Noise,” J. Franklin Inst., 354(17), pp. 7885–7898. [CrossRef]
Chen, M. T. , Ding, F. , Alsaed, A. , and Hayat, T. , 2018, “ Iterative Identification Algorithms for Bilinear-in-Parameter Systems by Using the Over-Parameterization Model and the Decomposition,” Int. J. Control Autom. Syst., 16(6), pp. 2634–2643. [CrossRef]
Farina, M. , Zhang, X. L. , and Scattolini, R. , 2018, “ A Hierarchical Multi-Rate MPC Scheme for Interconnected Systems,” Automatica, 90, pp. 38–46. [CrossRef]
Wang, D. Q. , Li, L. W. , Ji, Y. , and Yan, Y. R. , 2018, “ Model Recovery for Hammerstein Systems Using the Auxiliary Model Based Orthogonal Matching Pursuit Method,” Appl. Math. Model., 54, pp. 537–550. [CrossRef]
Dong, S. J. , Liu, T. , and Wang, Q. G. , 2018, “ Identification of Hammerstein Systems With Time Delay Under Load Disturbance,” IET Control Theory Appl., 12(7), pp. 942–952.
Abrahamsson, R. , Kay, S. M. , and Stoica, P. , 2007, “ Estimation of the Parameters of a Bilinear Model With Applications to Submarine Detection and System Identification,” Digit. Signal Process., 17(4), pp. 756–773. [CrossRef]
Bai, E. W. , 2002, “ A Blind Approach to the Hammerstein-Wiener Model Identification,” Automatica, 38(6), pp. 967–979. [CrossRef]
Chen, M. T. , Ding, F. , and Yang, E. F. , 2018, “ Gradient-Based Iterative Parameter Estimation for Bilinear-in-Parameter Systems Using the Model Decomposition Technique,” IET Control Theory Appl., 12(17), pp. 2380–2389. [CrossRef]
Xu, L. , and Ding, F. , 2017, “ The Parameter Estimation Algorithms for Dynamical Response Signals Based on the Multi-Innovation Theory and the Hierarchical Principle,” IET Signal Process., 11(2), pp. 228–237. [CrossRef]
Xu, L. , and Ding, F. , 2017, “ Recursive Least Squares and Multi-Innovation Stochastic Gradient Parameter Estimation Methods for Signal Modeling,” Circuits Syst. Signal Process., 36(4), pp. 1735–1753. [CrossRef]
Xu, L. , Ding, F. , Gu, Y. , Alsaedi, A. , and Hayat, T. , 2017, “ A Multi-Innovation State and Parameter Estimation Algorithm for a State Space System With D-Step State-Delay,” Signal Process., 140, pp. 97–103. [CrossRef]
Xu, L. , and Ding, F. , 2017, “ Parameter Estimation for Control Systems Based on Impulse Responses,” Int. J. Control Autom. Syst., 15(6), pp. 2471–2479. [CrossRef]
Xu, L. , and Ding, F. , 2018, “ Iterative Parameter Estimation for Signal Models Based on Measured Data,” Circuits Syst. Signal Process., 37(7), pp. 3046–3069. [CrossRef]
Xu, L. , Ding, F. , and Zhu, Q. M. , 2019, “ Hierarchical Newton and Least Squares Iterative Estimation Algorithm for Dynamic Systems by Transfer Functions Based on the Impulse Responses,” Int. J. Syst. Sci., 50(1), pp. 141–151. [CrossRef]
Sun, Z. Y. , Zhang, D. , Meng, Q. , and Chen, C. C. , 2019, “ Feedback Stabilization of Time-Delay Nonlinear Systems With Continuous Time-Varying Output Function,” Int. J. Syst. Sci., 50(2), pp. 244–255. [CrossRef]
Wang, T. Z. , Dong, J. J. , Xie, T. , Diallo, D. , and Benbouzid, M. , 2019, “ A Self-Learning Fault Diagnosis Strategy Based on Multi-Model Fusion,” Information, 10(3), p. 116. [CrossRef]
Zhan, X. S. , Guan, Z. H. , Zhang, X. H. , and Yuan, F. S. , 2013, “ Optimal Tracking Performance and Design of Networked Control Systems With Packet Dropout,” J. Frankl. Inst., 350(10), pp. 3205–3216. [CrossRef]
Zhan, X. S. , Cheng, L. L. , Wu, J. , Yang, Q. S. , and Han, T. , 2019, “ Optimal Modified Performance of MIMO Networked Control Systems With Multi-Parameter Constraints,” ISA Trans., 84(1), pp. 111–117. [CrossRef] [PubMed]
Pan, J. , Li, W. , and Zhang, H. , P., 2018, “ Control Algorithms of Magnetic Suspension Systems Based on the Improved Double Exponential Reaching Law of Sliding Mode Control,” Int. J. Control Autom. Syst., 16(6), pp. 2878–2887. [CrossRef]
Ma, F. Y. , Yin, Y. K. , and Li, M. , 2019, “ Start-Up Process Modelling of Sediment Microbial Fuel Cells Based on Data Driven,” Math. Probl. Eng., 2019, p. 7403732.
Tian, X. P. , and Niu, H. M. , 2019, “ A Bi-Objective Model With Sequential Search Algorithm for Optimizing Network-Wide Train Timetables,” Comput. Ind. Eng., 127, pp. 1259–1272. [CrossRef]
Yang, F. , Zhang, P. , and Li, X. X. , 2019, “ The Truncation Method for the Cauchy Problem of the Inhomogeneous Helmholtz Equation,” Appl. Anal., 98(5), pp. 991–1004. [CrossRef]
Jiang, C. M. , Zhang, F. F. , and Li, T. X. , 2018, “ Synchronization and Antisynchronization of N-Coupled Fractional-Order Complex Chaotic Systems With Ring Connection,” Math. Meth. Appl. Sci., 41(7), pp. 2625–2638. [CrossRef]
Wu, M. H. , Li, X. , Liu, C. , Liu, M. , Zhao, N. , Wang, J. , Wan, X. K. , Rao, Z. H. , and Zhu, L. , 2019, “ Robust Global Motion Estimation for Video Security Based on Improved k-Means Clustering,” J. Ambient Intell. Humanized Comput., 10(2), pp. 439–448. [CrossRef]
Wan, X. K. , Wu, H. , Qiao, F. , Li, F. , Li, Y. , Yan, Y. , and Wei, J. , 2019, “ Electrocardiogram Baseline Wander Suppression Based on the Combination of Morphological and Wavelet Transformation Based Filtering,” Comput. Math. Method Med., 2019, p. 7196156. [CrossRef]
Feng, L. , Li, Q. X. , and Li, Y. F. , 2019, “ Imaging With 3-D Aperture Synthesis Radiometers,” IEEE Trans. Geosci. Remote Sensing, 57(4), pp. 2395–2406. [CrossRef]
Shi, W. X. , Liu, N. , Zhou, Y. M. , and Cao, X. A. , 2019, “ Effects of Postannealing on the Characteristics and Reliability of Polyfluorene Organic Light-Emitting Diodes,” IEEE Trans. Electron Devices., 66(2), pp. 1057–1062. [CrossRef]
Fu, B. , Ouyang, C. X. , Li, C. S. , Wang, J. W. , and Gul, E. , 2019, “ An Improved Mixed Integer Linear Programming Approach Based on Symmetry Diminishing for Unit Commitment of Hybrid Power System,” Energies, 12(5), p. 833. [CrossRef]
Wu, T. Z. , Shi, X. , Liao, L. , Zhou, C. J. , Zhou, H. , and Su, Y. H. , 2019, “ A Capacity Configuration Control Strategy to Alleviate Power Fluctuation of Hybrid Energy Storage System Based on Improved Particle Swarm Optimization,” Energies, 12(4), p. 642. [CrossRef]
Wen, Y. Z. , and Yin, C. C. , 2019, “ Solution of Hamilton-Jacobi-Bellman Equation in Optimal Reinsurance Strategy Under Dynamic VaR Constraint,” J. Funct. space., 2019, p. 6750892.
Rao, Z. H. , Zeng, C. Y. , Wu, M. H. , Wang, Z. F. , Zhao, N. , Liu, M. , and Wan, X. K. , 2018, “ Research on a Handwritten Character Recognition Algorithm Based on an Extended Nonlinear Kernel Residual Network,” KSII Trans. Internet Inf. Syst., 12(1), pp. 413–435.
Zhao, N. , Liu, R. , Chen, Y. , Wu, M. , Jiang, Y. H. , Xiong, W. , and Liu, C. , 2018, “ Contract Design for Relay Incentive Mechanism Under Dual Asymmetric Information in Cooperative Networks,” Wireless Networks, 24(8), pp. 3029–3044. [CrossRef]
Ding, J. , Chen, J. Z. , Lin, J. X. , and Wan, L. J. , 2019, “ Particle Filtering Based Parameter Estimation for Systems With Output-Error Type Model Structures,” J. Franklin Inst., 356(10), pp. 5521–5540. [CrossRef]
Sha, X. Y. , Xu, Z. S. , and Yin, C. C. , 2019, “ Elliptical Distribution-Based Weight-Determining Method for Ordered Weighted Averaging Operators,” Int. J. Intell. Syst., 34(5), pp. 858–877. [CrossRef]
Li, X. Y. , Li, H. X. , and Wu, B. Y. , 2019, “ Piecewise Reproducing Kernel Method for Linear Impulsive Delay Differential Equations With Piecewise Constant Arguments,” Appl. Math. Comput., 349, pp. 304–313. [CrossRef]


Grahic Jump Location
Fig. 1

The Hammerstein nonlinear system

Grahic Jump Location
Fig. 2

The Hammerstein–Wiener nonlinear system

Grahic Jump Location
Fig. 3

The H-GEGI estimation errors δk versus k with different σe2

Grahic Jump Location
Fig. 4

The H-GELSI estimation errors δk versus k with different σe2

Grahic Jump Location
Fig. 5

The O-RLS estimation errors δt versus t with σe12=0.102 and σe22=0.202

Grahic Jump Location
Fig. 6

The H-GELSI estimation errors δk versus k with σe12=0.102 and σe22=0.202

Grahic Jump Location
Fig. 7

The true output and the predicted outputs



Some tools below are only available to our subscribers or users with an online account.

Related Content

Customize your page view by dragging and repositioning the boxes below.

Related Journal Articles
Related eBook Content
Topic Collections

Sorry! You do not have access to this content. For assistance or to subscribe, please contact us:

  • TELEPHONE: 1-800-843-2763 (Toll-free in the USA)
  • EMAIL: asmedigitalcollection@asme.org
Sign In