Skip to main content
SpringerLink
Log in

Parameter Estimation Methods of Linear Continuous-Time Time-Delay Systems from Multi-frequency Response Data

  • Published:
Circuits, Systems, and Signal Processing Aims and scope Submit manuscript

Abstract

This paper considers the identification problem of the linear continuous time-delay systems. By using the multi-frequency responses, a stochastic gradient gradient-based iterative (SG-GI) algorithm is derived. The proposed algorithm can estimate the unknown parameters and the unknown time delays simultaneously. To improve the parameter estimation accuracy of the SG-GI algorithm, a multi-innovation stochastic gradient gradient-based iterative (MISG-GI) algorithm is derived by using the multi-innovation identification theory. In addition, a forgetting factor is introduced to increase the parameter estimation accuracy. The resulting algorithm is called the multi-innovation forgetting gradient gradient-based iterative (MIFG-GI) algorithm. The effectiveness of the proposed strategies is illustrated by a numerical example.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4

Similar content being viewed by others

Data Availability Statement

Data sharing is not applicable to this article as no new data were created or analyzed in this study.

References

  1. Y. Cao, Y. An, S. Su et al., A statistical study of railway safety in China and Japan 1990–2020. Accid. Anal. Prevent. 175, 106764 (2022)

    Article  Google Scholar 

  2. Y. Cao, Y. Ji, Y. Sun et al., The fault diagnosis of a switch machine based on deep random forest fusion. IEEE Intell. Transp. Syst. Mag. (2023). https://doi.org/10.1109/MITS.2022.3174238

    Article  Google Scholar 

  3. Y. Cao, L. Ma, S. Xiao et al., Standard analysis for transfer delay in CTCS-3. Chin. J. Electron. 26(5), 1057–1063 (2017)

    Article  Google Scholar 

  4. Y. Cao, Y. Sun, G. Xie et al., A sound-based fault diagnosis method for railway point machines based on two-stage feature selection strategy and ensemble classifier. IEEE Trans. Intell. Transp. Syst. (2022). https://doi.org/10.1109/TITS.2021.3109632

    Article  Google Scholar 

  5. Y. Cao, Y. Sun, G. Xie et al., Fault diagnosis of train plug door based on a hybrid criterion for IMFs selection and fractional wavelet package energy entropy. IEEE Trans. Veh. Technol. 68(8), 7544–7551 (2019)

    Article  Google Scholar 

  6. Y. Cao, Z. Wang, F. Liu et al., Bio-inspired speed curve optimization and sliding mode tracking control for subway trains. IEEE Trans. Veh. Technol. 68(7), 6331–6342 (2019)

    Article  Google Scholar 

  7. Y. Cao, J. Wen, A. Hobiny et al., Parameter-varying artificial potential field control of virtual coupling system with nonlinear dynamics. Fractals 30(2), 2240099 (2022)

    Article  MATH  Google Scholar 

  8. Y. Cao, J. Wen, L. Ma, Tracking and collision avoidance of virtual coupling train control system. Alex. Eng. J. 60(2), 2115–2125 (2021)

    Article  Google Scholar 

  9. Y. Cao, Y. Yang, L. Ma et al., Research on virtual coupled train control method based on GPC & VAPF. Chin. J. Electron. 31(5), 897–905 (2022)

    Article  Google Scholar 

  10. Y. Cao, Z.X. Zhang, F. Cheng et al., Trajectory optimization for high-speed trains via a mixed integer linear programming approach. IEEE Trans. Intell. Transp. Syst. (2023). https://doi.org/10.1109/TITS.2022.3155628

    Article  Google Scholar 

  11. K. Chen, W. Ai, B. Chen, Mechanical parameter identification of two-mass drive system based on variable forgetting factor recursive least squares method. Trans. Inst. Meas. Control. 41(2), 1–10 (2018)

    MathSciNet  Google Scholar 

  12. F. Chen, H. Garnier, M. Gilson, Robust identification of continuous-time models with arbitrary time-delay from irregularly sampled data. J. Process Control 25, 19–27 (2015)

    Article  Google Scholar 

  13. F. Chen, H. Garnier, A. Padilla et al., Recursive IV identification of continuous-time models with time delay from sampled data. IEEE Trans. Control Syst. Technol. 28(3), 1074–1082 (2020)

    Article  Google Scholar 

  14. Y. Chen, C. Zhang, C. Liu et al., Atrial fibrillation detection using feedforward neural network. J. Med. Biol. Eng. 42(1), 63–73 (2022)

    Article  Google Scholar 

  15. T. Cui, F. Ding, T. Hayat, Moving data window-based partially-coupled estimation approach for modeling a dynamical system involving unmeasurable states. ISA Trans. Part B 128, 437–452 (2022)

    Article  Google Scholar 

  16. F. Ding, L. Lv, J. Pan et al., Two-stage gradient-based iterative estimation methods for controlled autoregressive systems using the measurement data. Int. J. Control Autom. Syst. 18(4), 886–896 (2020)

    Article  Google Scholar 

  17. F. Ding, H. Ma, J. Pan et al., Hierarchical gradient- and least squares-based iterative algorithms for input nonlinear output-error systems using the key term separation. J. Franklin Inst. 358(9), 5113–5135 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  18. F. Ding, F. Wang, L. Xu et al., Decomposition based least squares iterative identification algorithm for multivariate pseudo-linear ARMA systems using the data filtering. J. Franklin Inst. 354(3), 1321–1339 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  19. J.L. Ding, W. Zhang, Finite-time adaptive control for nonlinear systems with uncertain parameters based on the command filters. Int. J. Adapt. Control Signal Process. 35(9), 1754–1767 (2021)

    Article  MathSciNet  Google Scholar 

  20. S. Dong, T. Liu, W. Wang et al., Identification of discrete-time output error model for industrial processes with time delay subject to load disturbance. J. Process Control 50, 40–55 (2017)

    Article  Google Scholar 

  21. Y. Fan, X. Liu, Auxiliary model-based multi-innovation recursive identification algorithms for an input nonlinear controlled autoregressive moving average system with variable-gain nonlinearity. Int. J. Adapt. Control Signal Process. 36(3), 690–707 (2022)

    Article  MathSciNet  Google Scholar 

  22. F.Z. Geng, X.Y. Wu, A novel kernel functions algorithm for solving impulsive boundary value problems. Appl. Math. Lett. 134, 108318 (2022)

    Article  MathSciNet  MATH  Google Scholar 

  23. Y. Gu, Q.M. Zhu, H. Nouri, Identification and U-control of a state-space system with time-delay. Int. J. Adapt. Control Signal Process. 36(1), 138–154 (2022)

    Article  MathSciNet  Google Scholar 

  24. H. Ha, J.S. Welsh, M. Alamir, Useful redundancy in parameter and time delay estimation for continuous-time models. Automatica 95, 455–462 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  25. J. Hou, F. Chen, P. Li et al., Gray-box parsimonious subspace identification of Hammerstein-type systems. IEEE Trans. Ind. Electron. 68(10), 9941–9951 (2021)

    Article  Google Scholar 

  26. J. Hou, H. Su, C. Yu et al., Bias-correction errors-in-variables Hammerstein model identification. IEEE Trans. Ind. Electron. (2022). https://doi.org/10.1109/TIE.2022.3199931

    Article  Google Scholar 

  27. J. Hou, H. Su, C. Yu et al., Consistent subspace identification of errors-in-variables Hammerstein systems. IEEE Trans. Syst. Man Cybern. Syst. (2022). https://doi.org/10.1109/TSMC.2022.3213809

    Article  Google Scholar 

  28. J.K. Hwang, Y. Liu, Identification of interarea modes from an effectual impulse response of ringdown frequency data. Electr. Power Syst. Res. 144, 96–106 (2017)

    Article  Google Scholar 

  29. Y. Ji, X.K. Jiang, L.J. Wan, Hierarchical least squares parameter estimation algorithm for two-input Hammerstein finite impulse response systems. J. Franklin Inst. 357(8), 5019–5032 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  30. Y. Ji, Z. Kang, Three-stage forgetting factor stochastic gradient parameter estimation methods for a class of nonlinear systems. Int. J. Robust Nonlinear Control 31(3), 971–987 (2021)

    Article  MathSciNet  Google Scholar 

  31. Y. Ji, Z. Kang, X.M. Liu, The data filtering based multiple-stage Levenberg-Marquardt algorithm for Hammerstein nonlinear systems. Int. J. Robust Nonlinear Control 31(15), 7007–7025 (2021)

    Article  MathSciNet  Google Scholar 

  32. Y. Ji, Z. Kang, C. Zhang, Two-stage gradient-based recursive estimation for nonlinear models by using the data filtering. Int. J. Control Autom. Syst. 19(8), 2706–2715 (2021)

    Article  Google Scholar 

  33. Y. Ji, C. Zhang, Z. Kang, T. Yu, Parameter estimation for block-oriented nonlinear systems using the key term separation. Int. J. Robust Nonlinear Control 30(9), 3727–3752 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  34. Z. Kang, Y. Ji, X.M. Liu, Hierarchical recursive least squares algorithms for Hammerstein nonlinear autoregressive output-error systems. Int. J. Adapt. Control Signal Process. 35(11), 2276–2295 (2021)

    Article  Google Scholar 

  35. J.M. Li, F. Ding, T. Hayat, A novel nonlinear optimization method for fitting a noisy Gaussian activation function. Int. J. Adapt. Control Signal Process. 36(3), 690–707 (2022)

    Article  Google Scholar 

  36. M.H. Li, X.M. Liu, Maximum likelihood hierarchical least squares-based iterative identification for dual-rate stochastic systems. Int. J. Adapt. Control Signal Process. 35(2), 240–261 (2021)

    Article  MathSciNet  Google Scholar 

  37. M.H. Li, X.M. Liu, Iterative identification methods for a class of bilinear systems by using the particle filtering technique. Int. J. Adapt. Control Signal Process. 35(10), 2056–2074 (2021)

    Article  MathSciNet  Google Scholar 

  38. M.H. Li, X.M. Liu, Particle filtering-based iterative identification methods for a class of nonlinear systems with interval-varying measurements. Int. J. Control Autom. Syst. 20(7), 2239–2248 (2022)

    Article  Google Scholar 

  39. M.H. Li, X.M. Liu, The filtering-based maximum likelihood iterative estimation algorithms for a special class of nonlinear systems with autoregressive moving average noise using the hierarchical identification principle. Int. J. Adapt. Control Signal Process. 33(7), 1189–1211 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  40. J. Li, D. Ma, Laplace transforms and valuations. J. Funct. Anal. 272(2), 739–758 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  41. M. Li, G. Xu, Q. Lai, J. Chen, A chaotic strategy-based quadratic opposition-based learning adaptive variable-speed whale optimization algorithm. Math. Comput. Simul. 193, 71–99 (2022)

    Article  MathSciNet  MATH  Google Scholar 

  42. M. Li, G. Xu, L. Zeng et al., Hybrid whale optimization algorithm based on symbiosis strategy for global optimization. Appl. Intell. (2022). https://doi.org/10.1007/s10489-022-04132-9

    Article  Google Scholar 

  43. J.H. Li, T.C. Zong, G.P. Lu, Parameter identification of Hammerstein-Wiener nonlinear systems with unknown time delay based on the linear variable weight particle swarm optimization. ISA Trans. 120, 89–98 (2022)

    Article  Google Scholar 

  44. M. Liu, H. Chen, H-infinity state estimation for discrete-time delayed systems of the neural network type with multiple missing measurements. IEEE Trans. Neural Netw. Learn. Syst. 26(12), 2987–2998 (2015)

    Article  MathSciNet  Google Scholar 

  45. Q. Liu, F. Ding, Gradient-based recursive parameter estimation for a periodically nonuniformly sampled-data Hammerstein-Wiener system based on the key-term separation. Int. J. Adapt. Control Signal Process. 35(10), 1970–1989 (2021)

    Article  MathSciNet  Google Scholar 

  46. Q. Liu, F. Chen, T. Hayat, Recursive least squares estimation methods for a class of nonlinear systems based on non-uniform sampling. Int. J. Adapt. Control Signal Process. 35(8), 1612–1632 (2021)

    Article  MathSciNet  Google Scholar 

  47. S.Y. Liu, F. Ding, L. Xu et al., Hierarchical principle-based iterative parameter estimation algorithm for dual-frequency signals. Circuits Syst Signal Process. 38(7), 3251–3268 (2019)

    Article  Google Scholar 

  48. Q. Liu, C. Shang, D. Huang, Efficient low-order system identification from low-quality step response data with rank-constrained optimization. Control. Eng. Pract. 107, 104671 (2021)

    Article  Google Scholar 

  49. T. Liu, H. Tian, S. Rong et al., Heating-up control with delay-free output prediction for industrial jacketed reactors based on step response identification. ISA Trans. 83, 227–238 (2018)

    Article  Google Scholar 

  50. H. Liu, J. Wang, Y. Ji, Maximum likelihood recursive generalized extended least squares estimation methods for a bilinear-parameter systems with ARMA noise based on the over-parameterization model. Int. J. Control Autom. Syst. 20(8), 2606–2615 (2022)

    Article  Google Scholar 

  51. P. Ma, F. Ding, New gradient based identification methods for multivariate pseudo-linear systems using the multi-innovation and the data filtering. J. Franklin Inst. 354(3), 1568–1583 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  52. J. Ma, F. Ding, Filtering-based multistage recursive identification algorithm for an input nonlinear output-error autoregressive system by using the key term separation technique. Circuits Syst. Signal Process. 36(2), 577–599 (2017)

    Article  MATH  Google Scholar 

  53. H. Ma, J. Pan, W. Ding, Partially-coupled least squares based iterative parameter estimation for multi-variable output-error-like autoregressive moving average systems. IET Control Theory Appl. 13(18), 3040–3051 (2019)

    Article  MathSciNet  Google Scholar 

  54. P. Ma, L. Wang, Filtering-based recursive least squares estimation approaches for multivariate equation-error systems by using the multiinnovation theory. Int. J. Adapt. Control Signal Process. 35(9), 1898–1915 (2021)

    Article  MathSciNet  Google Scholar 

  55. J. Ma, W. Xiong, J. Chen et al., Hierarchical identification for multivariate Hammerstein systems by using the modified Kalman filter. IET Control Theory Appl. 11(6), 857–869 (2017)

    Article  MathSciNet  Google Scholar 

  56. X. Meng, Y. Ji, J. Wang, Iterative parameter estimation for photovoltaic cell models by using the hierarchical principle. Int. J. Control Autom. Syst. 20(8), 2583–2593 (2022)

    Article  Google Scholar 

  57. J.Y. Ni, L. Xu, T. Hayat, Parameter estimation for time-delay systems based on the frequency responses and harmonic balance methods. Int. J. Adapt. Control Signal Process. 34(12), 1779–1798 (2020)

    Article  MathSciNet  Google Scholar 

  58. J. Ni, Y. Zhang, T. Hayat, Parameter estimation algorithms of linear systems with time-delays based on the frequency responses and harmonic balances under the multi-frequency sinusoidal signal excitation. Signal Process. 181, 107904 (2021)

    Article  Google Scholar 

  59. J. Pan, Q. Chen, J. Xiong, G. Chen, A novel quadruple boost nine level switched capacitor inverter. J. Electr. Eng. Technol. (2022). https://doi.org/10.1007/s42835-022-01130-2

    Article  Google Scholar 

  60. J. Pan, X. Jiang, X. Wan et al., A filtering based multi-innovation extended stochastic gradient algorithm for multivariable control systems. Int. J. Control Autom. Syst. 15(3), 1189–1197 (2017)

    Article  Google Scholar 

  61. J. Pan, W. Li, H.P. Zhang, 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), 2878–2887 (2018)

    Article  Google Scholar 

  62. J. Pan, S. Liu, J. Shu et al., Hierarchical recursive least squares estimation algorithm for secondorder Volterra nonlinear systems. Int. J. Control Autom. Syst. 20(12), 3940–3950 (2022)

    Article  Google Scholar 

  63. J. Pan, H. Ma, X. Zhang et al., Recursive coupled projection algorithms for multivariable output-error-like systems with coloured noises. IET Signal Process. 14(7), 455–466 (2020)

    Article  Google Scholar 

  64. M. Pouliquen, E. Pigeon, O. Gehan et al., Impulse response identification from input/output binary measurements. Automatica 123, 109307 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  65. X. Ren, A. Rad, P. Chan, W. Lo, Online identification of continuous-time systems with unknown time delay. IEEE Trans. Autom. Control 50(9), 1418–1422 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  66. H.C. So, Noisy input-output system identification approach for time delay estimation. Signal Process. 82(10), 1471–1475 (2002)

    Article  MATH  Google Scholar 

  67. S. Su, J. She, K. Li et al., A nonlinear safety equilibrium spacing based model predictive control for virtually coupled train set over gradient terrains. IEEE Trans. Transp. Electrif. 8(2), 2810–2824 (2022)

    Article  Google Scholar 

  68. S. Su, T. Tang, J. Xun et al., Design of running grades for energy-efficient train regulation: a case study for Beijing Yizhuang line. IEEE Intell. Transp. Syst. Mag. 13(2), 189–200 (2021)

    Article  Google Scholar 

  69. S. Su, X. Wang, Y. Cao, J.T. Yin, An energy-efficient train operation approach by integrating the metro timetabling and eco-driving. IEEE Trans. Intell. Transp. Syst. 21(10), 4252–4268 (2020)

    Article  Google Scholar 

  70. S. Su, X. Wang, T. Tang et al., Energy-efficient operation by cooperative control among trains: A multi-agent reinforcement learning approach. Control Eng. Pract. 116, 104901 (2021)

    Article  Google Scholar 

  71. S. Su, Q. Zhu, J. Liu et al., Eco-driving of trains with a data-driven iterative learning approach. IEEE Trans. Ind. Inf. (2023). https://doi.org/10.1109/TII.2022.3195888

    Article  Google Scholar 

  72. Y.K. Sun, Y. Cao, P. Li, Contactless fault diagnosis for railway point machines based on multi-scale fractional wavelet packet energy entropy and synchronous optimization strategy. IEEE Trans. Veh. Technol. 71(6), 5906–5914 (2022)

    Article  Google Scholar 

  73. Y.K. Sun, Y. Cao, L.C. Ma, A fault diagnosis method for train plug doors via sound signals. IEEE Intell. Transp. Syst. Mag. 13(3), 107–117 (2021)

    Article  Google Scholar 

  74. Y.K. Sun, Y. Cao, G. Xie, T. Wen, Sound based fault diagnosis for RPMs based on multi-scale fractional permutation entropy and two-scale algorithm. IEEE Trans. Veh. Technol. 70(11), 11184–11192 (2021)

    Article  Google Scholar 

  75. J. Voros, Identification of Hammerstein systems with time-varying piecewise-linear characteristics. IEEE Trans. Circuits Syst. II Express Briefs 52(12), 865–869 (2005)

    Article  Google Scholar 

  76. L.J. Wan, F. Ding, Decomposition- and gradient-based iterative identification algorithms for multivariable systems using the multi-innovation theory. Circuits Syst. Signal Process. 38(7), 2971–2991 (2019)

    Article  Google Scholar 

  77. X. Wang, F. Ding, Modified particle filtering-based robust estimation for a networked control system corrupted by impulsive noise. Int. J. Robust Nonlinear Control 32(2), 830–850 (2022)

    Article  MathSciNet  Google Scholar 

  78. Y. Wang, F. Ding, M.H. Wu, Recursive parameter estimation algorithm for multivariate output-error systems. J. Franklin Inst. 355(12), 5163–5181 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  79. J. Wang, C. Ding, M. Wu et al., Lightweight multiple scale-patch dehazing network for real-world hazy image. KSII Trans. Int. Inf. Syst. 15(12), 4420–4438 (2022)

    Google Scholar 

  80. H. Wang, H. Fan, J. Pan, A true three-scroll chaotic attractor coined. Discrete Contin. Dyn. Syst. Ser. B 27(5), 2891–2915 (2022)

    Article  MathSciNet  MATH  Google Scholar 

  81. J. Wang, Y. Ji, C. Zhang, Iterative parameter and order identification for fractional-order nonlinear finite impulse response systems using the key term separation. Int. J. Adapt. Control Signal Process. 35(8), 1562–1577 (2021)

    Article  MathSciNet  Google Scholar 

  82. J. Wang, Y. Ji, X. Zhang et al., Two-stage gradient-based iterative algorithms for the fractional-order nonlinear systems by using the hierarchical identification principle. Int. J. Adapt. Control Signal Process. 36(7), 1778–1796 (2022)

    Article  MathSciNet  Google Scholar 

  83. X. Wang, S. Su, Y. Cao et al., Robust control for dynamic train regulation in fully automatic operation system under uncertain wireless transmissions. IEEE Trans. Intell. Transp. Syst. (2022). https://doi.org/10.1109/TITS.2022.3170950

    Article  Google Scholar 

  84. Y. Wang, S. Tang, M. Deng, Modeling nonlinear systems using the tensor network B-spline and the multi-innovation identification theory. Int. J. Robust Nonlinear Control 32(13), 7304–7318 (2022)

    Article  MathSciNet  Google Scholar 

  85. Y. Wang, S. Tang, X. Gu, Parameter estimation for nonlinear Volterra systems by using the multi-innovation identification theory and tensor decomposition. J. Franklin Inst. 359(2), 1782–1802 (2022)

    Article  MathSciNet  MATH  Google Scholar 

  86. Y. Wang, L. Yang, An efficient recursive identification algorithm for multilinear systems based on tensor decomposition. Int. J. Robust Nonlinear Control 31(16), 7920–7936 (2021)

    Article  MathSciNet  Google Scholar 

  87. Y. Wang, G. Yang, Arrhythmia classification algorithm based on multi-head self-attention mechanism. Biomed. Signal Process. Control 79, 104206 (2023)

    Article  Google Scholar 

  88. C. Wei, X. Zhang, L. Xu et al., Overall recursive least squares and overall stochastic gradient algorithms and their convergence for feedback nonlinear controlled autoregressive systems. Int. J. Robust Nonlinear Control 32(9), 5534–5554 (2022)

    Article  MathSciNet  Google Scholar 

  89. J. Xiong, J. Pan, G. Chen et al., Sliding mode dual-channel disturbance rejection attitude control for a quadrotor. IEEE Trans. Ind. Electron. 69(10), 10489–10499 (2022)

    Article  Google Scholar 

  90. L. Xu, Separable multi-innovation Newton iterative modeling algorithm for multi-frequency signals based on the sliding measurement window. Circuits Syst. Signal Process. 41(2), 805–830 (2022)

    Article  MathSciNet  MATH  Google Scholar 

  91. L. Xu, Separable Newton recursive estimation method through system responses based on dynamically discrete measurements with increasing data length. Int. J. Control Autom. Syst. 20(2), 432–443 (2022)

    Article  MathSciNet  Google Scholar 

  92. L. Xu, F.Y. Chen, T. Hayat, Hierarchical recursive signal modeling for multi-frequency signals based on discrete measured data. Int. J. Adapt. Control Signal Process. 35(5), 676–693 (2021)

    Article  Google Scholar 

  93. L. Xu, F. Ding, L. Wan, J. Sheng, Separable multi-innovation stochastic gradient estimation algorithm for the nonlinear dynamic responses of systems. Int. J. Adapt. Control Signal Process. 34(7), 937–954 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  94. L. Xu, F. Ding, E. Yang, Auxiliary model multiinnovation stochastic gradient parameter estimation methods for nonlinear sandwich systems. Int. J. Robust Nonlinear Control 31(1), 148–165 (2021)

    Article  MathSciNet  Google Scholar 

  95. L. Xu, F. Ding, Q. Zhu, Separable synchronous multi-innovation gradient-based iterative signal modeling from on-line measurements. IEEE Trans. Instrum. Meas. 71, 6501313 (2022)

    Google Scholar 

  96. L. Xu, F. Ding, Q. Zhu, Decomposition strategy-based hierarchical least mean square algorithm for control systems from the impulse responses. Int. J. Syst. Sci. 52(9), 1806–1821 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  97. L. Xu, G.L. Song, A recursive parameter estimation algorithm for modeling signals with multi-frequencies. Circuits Syst. Signal Process. 39(8), 4198–4224 (2020)

    Article  MATH  Google Scholar 

  98. C. Xu, H. Xu, Z. Guan et al., Observer-based dynamic event-triggered semi-global bipartite consensus of linear multi-agent systems with input saturation. IEEE Trans. Cybern. (2023). https://doi.org/10.1109/TCYB.2022.3164048

    Article  Google Scholar 

  99. X. Yang, W.L. Xiong, J.X. Ma et al., Robust identification of Wiener time-delay system with expectation-maximization algorithm. J. Franklin Inst. 354(13), 5678–5693 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  100. W. Yao, S. Marques, Prediction of transonic limit-cycle oscillations using an aeroelastic harmonic balance method. AIAA J. 53(7), 2040–2051 (2015)

    Article  Google Scholar 

  101. C.C. Yin, Y.Z. Wen, An extension of Paulsen-Gjessing’s risk model with stochastic return on investments. Insur. Math. Econom. 52(3), 469–476 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  102. C.C. Yin, J.S. Zhao, Nonexponential asymptotics for the solutions of renewal equations, with applications. J. Appl. Probab. 43(3), 815–824 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  103. C.C. Yin, K.C. Yuen, Optimality of the threshold dividend strategy for the compound Poisson model. Stat. Probab. Lett. 81(12), 1841–1846 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  104. C.C. Yin, K.C. Yuen, Optimal dividend problems for a jump-diffusion model with capital injections and proportional transaction costs. J. Ind. Manag. Optim. 11(4), 1247–1262 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  105. J. You, C. Yu, J. Sun et al., Generalized maximum entropy based identification of graphical ARMA models. Automatica 141, 110319 (2022)

    Article  MathSciNet  MATH  Google Scholar 

  106. C. Yu, Y. Li, H. Fang et al., System identification approach for inverse optimal control of finite-horizon linear quadratic regulators. Automatica 129, 109636 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  107. X. Zhang, F. Ding, Hierarchical parameter and state estimation for bilinear systems. Int. J. Syst. Sci. 51(2), 275–290 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  108. X.X. Zhang, J.C. Ji, J. Xu, Parameter identification of time-delayed nonlinear systems: an integrated method with adaptive noise correction. J. Franklin Inst. 356(11), 5858–5880 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  109. C. Zhang, H.B. Liu, Y. Ji, Gradient parameter estimation of a class of nonlinear systems based on the maximum likelihood principle. Int. J. Control Autom. Syst. 20(5), 1393–1404 (2022)

    Article  Google Scholar 

  110. X.X. Zhang, J. Xu, Identification of time delay in nonlinear systems with delayed feedback control. J. Franklin Inst. 352(8), 298–2998 (2015)

    Article  MathSciNet  Google Scholar 

  111. N. Zhao, A. Wu, Y. Pei et al., Spatial-temporal aggregation graph convolution network for efficient mobile cellular traffic prediction. IEEE Commun. Lett. 26(3), 587–591 (2022)

    Article  Google Scholar 

  112. W. Zhu, S. Wu, X. Wang et al., Harmonic balance method implementation of onlinear dynamic characteristics for compound planetary gear sets. Nonlinear Dyn. 81(3), 1511–1522 (2015)

    Article  MATH  Google Scholar 

Download references

Acknowledgements

This work was supported by the National Natural Science Foundation of China (No. 61873111).

Author information

Authors and Affiliations

  1. School of Internet of Things Engineering, Jiangnan University, Wuxi, 214122, People’s Republic of China

    Shunyuan Sun, Ling Xu & Feng Ding

Authors
  1. Shunyuan Sun
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Ling Xu
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Feng Ding
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Feng Ding.

Ethics declarations

Conflict of interest

The authors declare that there is no conflict of interests regarding the publication of this paper.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Sun, S., Xu, L. & Ding, F. Parameter Estimation Methods of Linear Continuous-Time Time-Delay Systems from Multi-frequency Response Data. Circuits Syst Signal Process 42, 3360–3384 (2023). https://doi.org/10.1007/s00034-022-02285-z

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00034-022-02285-z

Keywords

Search

Navigation