An adaptive Levenberg-Marquardt identification algorithm for time-delay linear discrete periodic systems

Zhuo Chen; Yingying Guo; Zhuocheng Zou; Zhen Zhang; Zhuo Chen; Yingying Guo; Zhuocheng Zou; Zhen Zhang

doi:10.3934/jimo.2026111

Journal of Industrial and Management Optimization

2026, Volume 22, Issue 6: 3023-3049. doi: 10.3934/jimo.2026111

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Research article

An adaptive Levenberg-Marquardt identification algorithm for time-delay linear discrete periodic systems

1.
School of Management and Economics, North China University of Water Resources and Electric Power, Zhengzhou 450046, China
2.
School of Materials Science and Engineering, North China University of Water Resources and Electric Power, Zhengzhou 450046, China
3.
Zhejiang Shenjia Technology Co., Ltd. No. 133 Wuchang Road, Hangzhou, Zhejiang 310023, China
4.
Taizhou Yinxin Zhishu Technology Co., Ltd. No. 383, Shide Road, Fuxi Street, Tiantai, Zhejiang 317200, China

Received: 10 February 2026 Revised: 12 May 2026 Accepted: 15 May 2026 Published: 28 May 2026
93B30, 93C55, 65K10, 39Axx

This paper proposes an adaptive auxiliary-model-based Levenberg-Marquardt (AM-ILM) algorithm for parameter identification of linear discrete-time periodic systems with time delays. Unlike existing recursive least squares (AM-RLS) based methods that assume slowly varying parameters, the proposed LM framework dynamically balances gradient descent and Gauss-Newton steps via an adaptive regularization strategy, enabling accurate tracking of periodic variations. An auxiliary model reconstructs unmeasurable intermediate variables from input-output data, addressing the non-causal structure caused by time delays. The algorithm introduces two innovations: an adaptive regularization that adjusts the damping parameter, and an auxiliary model that reconstructs latent variables. By replacing unknown variables with auxiliary models and designing an adaptive updating strategy, an improved LM algorithm is developed. Numerical examples demonstrate its effectiveness, with comparisons to LM, AM-RLS, RLS, fully connected neural network, model-agnostic meta-learning, and Bayesian algorithms. Identification results under varying parameters and initial conditions, along with an application study, are provided. The results show that the proposed algorithm effectively handles the coupling between time delay and periodic variations, achieving superior accuracy and stability.
- periodic system identification,
- time-delay periodic systems,
- auxiliary models,
- adaptive LM algorithm
Citation: Zhuo Chen, Yingying Guo, Zhuocheng Zou, Zhen Zhang. An adaptive Levenberg-Marquardt identification algorithm for time-delay linear discrete periodic systems[J]. Journal of Industrial and Management Optimization, 2026, 22(6): 3023-3049. doi: 10.3934/jimo.2026111

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Abstract

This paper proposes an adaptive auxiliary-model-based Levenberg-Marquardt (AM-ILM) algorithm for parameter identification of linear discrete-time periodic systems with time delays. Unlike existing recursive least squares (AM-RLS) based methods that assume slowly varying parameters, the proposed LM framework dynamically balances gradient descent and Gauss-Newton steps via an adaptive regularization strategy, enabling accurate tracking of periodic variations. An auxiliary model reconstructs unmeasurable intermediate variables from input-output data, addressing the non-causal structure caused by time delays. The algorithm introduces two innovations: an adaptive regularization that adjusts the damping parameter, and an auxiliary model that reconstructs latent variables. By replacing unknown variables with auxiliary models and designing an adaptive updating strategy, an improved LM algorithm is developed. Numerical examples demonstrate its effectiveness, with comparisons to LM, AM-RLS, RLS, fully connected neural network, model-agnostic meta-learning, and Bayesian algorithms. Identification results under varying parameters and initial conditions, along with an application study, are provided. The results show that the proposed algorithm effectively handles the coupling between time delay and periodic variations, achieving superior accuracy and stability.

References

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