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Low Integer-Order Approximation of Fractional-Order Systems Using Grey Wolf Optimizer-Based Cuckoo Search Algorithm

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Abstract

This paper presents a stable reduced order approximation method of fractional-order systems based on the grey wolf optimizer hybridized with cuckoo search algorithm (GWO-CS). The proposed method is applied to factional order transfer functions (FOTF) in order to obtain a low-order model exhibiting good fits to the Bode’s ideal transfer function (TF) of the original system, taking into account the stability criteria of the reduced model. In the first stage, the Oustaloup approximation of a given FOTF is applied to obtain a high integer-order TF that matches with the original FOTF. In the second stage, The GWO-CS method is compared with the frequency-limited balanced truncation method resulting from Oustaloup approximation of the original FOTF in a specified frequency range and other existing meta-heuristic based model order reduction approaches. Simulation of different numerical examples confirms and validates the effectiveness of the proposed approach.

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Data availability

The datasets generated and analyzed during the current study are available from the corresponding author on reasonable request.

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  1. EEA&TI Laboratory, Faculty of Sciences and Techniques Mohammedia, Hassan 2 University of Casablanca, BP 146, 28806, Mohammedia, Casablanca, Morocco

    Abdelaziz Mouhou & Abdelmajid Badri

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  1. Abdelaziz Mouhou
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Appendix A: Singular Values Decomposition of the Studied Examples

Appendix A: Singular Values Decomposition of the Studied Examples

Table 15 HSVD of the studied examples
Full size table

Taking into account the HSVD depicted in Table 15, the reduced order can be calculated as follows:

  1. (1)

    Example 1: \(\sum _{i>2}^{n} \sigma _{i}= 0.0396< \sigma _{2}=0.7695\)

  2. (2)

    Example 2: \(\sum _{i>2}^{n} \sigma _{i}= 0.0772< \sigma _{2}=2.1219\)

  3. (3)

    Example 3: \(\sum _{i>2}^{n} \sigma _{i}= 0.01< \sigma _{2}=0.2420\)

  4. (4)

    Example 4: \(\sum _{i>2}^{n} \sigma _{i}= 0.0375< \sigma _{2}=0.0687\)

Therefore, according to (16) the suitable reduced order for the studied examples is r = 3.

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Mouhou, A., Badri, A. Low Integer-Order Approximation of Fractional-Order Systems Using Grey Wolf Optimizer-Based Cuckoo Search Algorithm. Circuits Syst Signal Process 41, 1869–1894 (2022). https://doi.org/10.1007/s00034-021-01872-w

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