A New Orthogonal Least Squares Identification Method for a Class of Fractional Hammerstein Models
Abstract
:1. Introduction
2. Model and Problem Formulation
2.1. Mathematical Background of Fractional Differentiation
2.2. MISO Fractional Hammerstein System
3. Identification Algorithm
3.1. Orthogonal Least Squares Algorithm Based on Householder Transformation
3.2. Construction of the Permutation Matrix
3.3. Determination of Sparsity Level
3.4. Identification of Orders and Separation of Parameters
Algorithm 1 H-GOLS algorithm for MISO fractional Hammerstein system |
Input: , p, l and . Output: , ⋯, , , ⋯, , , ⋯, , and .
|
4. Experimental Results and Discussions
5. Conclusions
Author Contributions
Funding
Data Availability Statement
Conflicts of Interest
References
- Huang, K.; Tang, Y.; Liu, X.; Wu, D.; Yang, C.; Gui, W. Knowledge-informed neural network for nonlinear model predictive control with industrial applications. IEEE Trans. Syst. Man-Cybern. Syst. 2023, 4, 2241–2253. [Google Scholar] [CrossRef]
- Dragan, P.; Novak, N.; Vojislav, F.; Ljubisa, D. Multilinear model of heat exchanger with Hammerstein structure. J. Control Sci. Eng. 2016, 1, 1–7. [Google Scholar] [CrossRef]
- Wu, W.; Jhao, D. Control of a direct internal reforming molten carbonate fuel cell system using wavelet network-based Hammerstein models. J. Process Control 2012, 22, 653–658. [Google Scholar] [CrossRef]
- Van der Veen, G.; Wingerden, J.W.; Verhaegen, M. Global identification of wind turbines using a Hammerstein identification method. IEEE Trans. Control Syst. Technol. 2013, 4, 1471–1478. [Google Scholar] [CrossRef]
- Guha, D.; Roy, P.K.; Banerjee, S. Adaptive fractional-order sliding-mode disturbance observer-based robust theoretical frequency controller applied to hybrid wind-diesel power system. ISA Trans. 2023, 133, 160–183. [Google Scholar] [CrossRef]
- Qureshi, S.; Yusuf, A.; Shaikh, A.A.; Inc, M.; Baleanu, D. Fractional modeling of blood ethanol concentration system with real data application. Chaos 2019, 29, 013143. [Google Scholar] [CrossRef] [PubMed]
- Jajarmi, A.; Baleanu, D. On the fractional optimal control problems with a general derivative operator. Asian J. Control 2021, 23, 1062–1071. [Google Scholar] [CrossRef]
- Dai, Y.; Wei, Y.; Hu, Y.; Wang, Y. Modulating function based identification for fractional order systems. Neurocomputing 2016, 173, 1959–1966. [Google Scholar] [CrossRef]
- Cui, R.; Wei, Y.; Chen, Y.; Chen, S.; Wang, Y. An innovative parameter estimation for fractional-order systems in the presence of outliers. Nonlinear Dyn. 2017, 89, 453–463. [Google Scholar] [CrossRef]
- Djamah, T.; Bettayeb, M.; Djennoune, S. Identification of multivariable fractional order systems. Asian J. Control 2013, 15, 741–750. [Google Scholar] [CrossRef]
- Ding, F.; Liu, X.; Lin, G. Identification methods for Hammerstein nonlinear systems. Digit. Signal Process. 2011, 21, 215–238. [Google Scholar] [CrossRef]
- Ding, F.; Liu, X.; Chu, J. Gradient-based and least-squares-based iterative algorithms for Hammerstein systems using the hierarchical identification principle. IET Control Theory Appl. 2013, 7, 176–184. [Google Scholar] [CrossRef]
- Piao, H.; Cheng, D.; Chen, C.; Wang, Y.; Wang, P.; Pan, X. A high-accuracy CO2 carbon isotope sensing system using subspace identification of Hammerstein model for geochemical application. IEEE Trans. Instrum. Meas. 2021, 71, 1–9. [Google Scholar] [CrossRef]
- Bai, E.; Fu, M. A blind approach to Hammerstein model identification. IEEE Trans. Signal Process. 2002, 50, 1610–1619. [Google Scholar] [CrossRef]
- Chen, X.; Chai, Y.; Liu, Q.; Huang, P.; Fan, L. Identification of MISO Hammerstein system using sparse multiple kernel-based hierarchical mixture prior and variational Bayesian inference. ISA Trans. 2023, 137, 323–338. [Google Scholar] [CrossRef]
- Victor, S.; Malti, R.; Garnier, H.; Oustaloup, A. Parameter and differentiation order estimation in fractional models. Automatica 2013, 49, 926–935. [Google Scholar] [CrossRef]
- Jin, Q.; Wang, B.; Wang, Z. Recursive identification for MIMO fractional-order Hammerstein model based on AIAGS. Mathmatics 2022, 10, 212. [Google Scholar] [CrossRef]
- Rahmani, M.R.; Farrokhi, M. Identification of neuro-fractional Hammerstein systems: A hybrid frequency-/time-domain approach. Soft Comput. 2018, 22, 8097–8106. [Google Scholar] [CrossRef]
- Liao, Z.; Zhu, Z.; Liang, S.; Peng, C.; Wang, Y. Subspace identification for fractional order Hammerstein systems based on instrumental variables. Int. J. Control Autom. Syst. 2012, 10, 947–953. [Google Scholar] [CrossRef]
- Aoun, M.; Malti, R.; Cois, O.; Oustaloup, A. System identification using fractional Hammerstein models. IFAC Proc. Vol. 2002, 35, 265–269. [Google Scholar] [CrossRef]
- Mohammad, M.J.; Hamed, M.; Mohammad, T. Recursive identification of multiple-input single-output fractional-order Hammerstein model with time delay. Appl. Soft Comput. 2018, 70, 486–500. [Google Scholar] [CrossRef]
- Qi, Z.; Sun, Q.; Ge, W.; He, Y. Nonlinear modeling of PEMFC based on fractional order subspace identification. Asian J. Control 2020, 22, 1892–1900. [Google Scholar] [CrossRef]
- Gao, Z.; Lin, X.; Zheng, Y. System identification with measurement noise compensation based on polynomial modulating function for fractional-order systems with a known time-delay. ISA Trans. 2018, 79, 62–72. [Google Scholar] [CrossRef] [PubMed]
- Wu, X.; Li, J.; Chen, G. Chaos in the fractional order unified system and its synchronization. J. Frankl. Inst. 2008, 345, 392–401. [Google Scholar] [CrossRef]
- Wang, D.; Li, L.; Ji, Y.; Yan, Y. Model recovery for Hammerstein systems using the auxiliary model based orthogonal matching pursuit method. Appl. Math. Model. 2018, 54, 537–550. [Google Scholar] [CrossRef]
- Liu, X.; Liu, Y.; Zhu, Q.; Ding, F. Joint parameter and time-delay estimation for a class of Wiener models based on a new orthogonal least squares algorithm. Nonlinear Dyn. 2024, 112, 12159–12170. [Google Scholar] [CrossRef]
- Gnanasekaran, A.; Darve, E. Hierarchical orthogonal factorization: Sparse least squares problems. J. Sci. Comput. 2022, 91, 50. [Google Scholar] [CrossRef]
- Kim, Y.H. QR factorization-based sampling set selection for bandlimited graph signals. Signal Process. 2021, 179, 107847. [Google Scholar] [CrossRef]
- Burnham, K.P.; Anderson, D.R. Multimodel inference: Understanding AIC and BIC in model selection. Sociol. Methods Res. 2004, 33, 261–304. [Google Scholar] [CrossRef]
- Efron, B.; Hastie, T.; Johnstone, I.; Tibshirani, R. Least angle regression. Ann. Stat. 2004, 32, 407–499. [Google Scholar] [CrossRef]
- Brunton, S.L.; Proctor, J.L.; Kutz, J.N. Discovering governing equations from data by sparse identification of nonlinear dynamical systems. Proc. Natl. Acad. Sci. USA 2016, 113, 3932–3937. [Google Scholar] [CrossRef] [PubMed]
- Chen, Y.; Liu, Y.; Chen, J.; Ma, J. A novel identification method for a class of closed-loop systems based on basis pursuit de-noising. IEEE Access 2020, 8, 99648–99654. [Google Scholar] [CrossRef]
- Li, Y.; Ling, B.; Xie, L.; Dai, Q. Using LASSO for formulating constraint of least-squares programming for solving one-norm equality constrained problem. Signal Image Video Process. 2017, 11, 179–786. [Google Scholar] [CrossRef]
- De Moor, B.L.R. DDaisy: Database for the Identification of Systems; Department of Electrical Engineering, Ed.; ESAT/STADIUS; KU Leuven: Leuven, Belgium, 2024. [Google Scholar]
Method | Addition Operations | Multiplication Operations |
---|---|---|
H-GOLS | ||
total: | ||
BPDN | ||
total: | ||
OMP | ||
total: | ||
LASSO | ||
total: |
Method | RMSE | MAE | ||||
---|---|---|---|---|---|---|
Training | Validation | Training | Validation | Training | Validation | |
H-GOLS-Fractional | ||||||
BPDN | 0.1701 | 0.1798 | 0.1343 | 0.1411 | 0.9202 | 0.9317 |
OMP | 0.2539 | 0.2432 | 0.1873 | 0.1998 | 0.8489 | 0.8074 |
LASSO | 0.1625 | 0.1600 | 0.1271 | 0.1286 | 0.9272 | 0.9459 |
H-GOLS-Integral | 0.1560 | 0.1614 | 0.1255 | 0.1289 | 0.9234 | 0.9477 |
Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content. |
© 2025 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).
Share and Cite
Yin, X.; Liu, Y. A New Orthogonal Least Squares Identification Method for a Class of Fractional Hammerstein Models. Algorithms 2025, 18, 201. https://doi.org/10.3390/a18040201
Yin X, Liu Y. A New Orthogonal Least Squares Identification Method for a Class of Fractional Hammerstein Models. Algorithms. 2025; 18(4):201. https://doi.org/10.3390/a18040201
Chicago/Turabian StyleYin, Xijian, and Yanjun Liu. 2025. "A New Orthogonal Least Squares Identification Method for a Class of Fractional Hammerstein Models" Algorithms 18, no. 4: 201. https://doi.org/10.3390/a18040201
APA StyleYin, X., & Liu, Y. (2025). A New Orthogonal Least Squares Identification Method for a Class of Fractional Hammerstein Models. Algorithms, 18(4), 201. https://doi.org/10.3390/a18040201