Iterative learning control for conformable stochastic impulsive differential systems with randomly varying trial lengths
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Wanzheng Qiu
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Dong Shen
Abstract
In this paper, we introduce a new kind of conformable stochastic impulsive differential systems (CSIDS) involving discrete distribution of Bernoulli. For random discontinuous trajectories, we modify the tracking error of piecewise continuous variables by a zero-order holder. First, the improved P-type and PD α -type learning laws of the random iterative learning control (ILC) scheme are designed through global and local averaging operators. Next, we establish sufficient conditions for convergence of the tracking error in the expectation sense and prove the main results by using the impulsive Gronwall inequality and mathematical analysis tools. Finally, the theoretical results are verified by two numerical examples, and the tracking performance is compared for different conformable order of α.
Funding source: National Natural Science Foundation of China
Award Identifier / Grant number: 12161015
Funding source: Guizhou Data Driven Modeling Learning and Optimization Innovation Team
Award Identifier / Grant number: [2020]5016
Funding source: Super Computing Algorithm and Application Laboratory of Guizhou University and Gui’an Scientific Innovation Company
Award Identifier / Grant number: K22-0116-003
Funding source: Major Project of Guizhou Postgraduate Education and Teaching Reform
Award Identifier / Grant number: YJSJGKT[2021]041
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Author contributions: All the authors have accepted responsibility for the entire content of this submitted manuscript and approved submission.
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Research funding: This work is partially supported by the National Natural Science Foundation of China (grant numbers 12161015; 62173333), Super Computing Algorithm and Application Laboratory of Guizhou University and Gui’an Scientific Innovation Company (K22-0116-003), Major Project of Guizhou Postgraduate Education and Teaching Reform (YJSJGKT[2021]041), the Slovak Research and Development Agency under the contract No. APVV-18-0308, and the Slovak Grant Agency VEGA No. 1/0358/20 and No. 2/0127/20.
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Conflict of interest statement: The authors declare no conflicts of interest regarding this article.
References
[1] Z. Bien and J. Xu, Iterative Learning Control: Analysis, Design, Integration and Applications, New York, NY, Springer Science & Business Media, 2012.Suche in Google Scholar
[2] Y. Chen and C. Wen, Iterative Learning Control: Convergence, Robustness and Applications, London, Springer London, 1999.10.1007/BFb0110114Suche in Google Scholar
[3] S. Arimoto, S. Kawamura, and F. Miyazaki, “Bettering operation of robots by learning,” J. Rob. Syst., vol. 1, pp. 123–140, 1984. https://doi.org/10.1002/rob.4620010203.Suche in Google Scholar
[4] S. Ibrir, “Iterative-learning procedures for nonlinear-model-order reduction in discrete time,” IMA J. Math. Control Inf., vol. 37, pp. 953–986, 2020. https://doi.org/10.1093/imamci/dnz028.Suche in Google Scholar
[5] I. Hussain, X. Ruan, and Y. Liu, “Convergence characteristics of iterative learning control for discrete-time singular systems,” Int. J. Syst. Sci., vol. 52, pp. 217–237, 2021. https://doi.org/10.1080/00207721.2020.1824030.Suche in Google Scholar
[6] D. Huang and J. Xu, “Steady-state iterative learning control for a class of nonlinear PDE processes,” J. Process Control, vol. 21, pp. 1155–1163, 2011. https://doi.org/10.1016/j.jprocont.2011.06.018.Suche in Google Scholar
[7] W. Chen, R. Li, and J. Li, “Observer-based adaptive iterative learning control for nonlinear systems with time-varying delays,” Int. J. Autom. Comput., vol. 7, pp. 438–446, 2010. https://doi.org/10.1007/s11633-010-0525-5.Suche in Google Scholar
[8] S. Liu, J. Wang, and W. Wei, “A study on iterative learning control for impulsive differential equations,” Commun. Nonlinear Sci. Numer. Simulat., vol. 24, pp. 4–10, 2015. https://doi.org/10.1016/j.cnsns.2014.12.002.Suche in Google Scholar
[9] J. Wang, M. Fečkan, and S. Liu, “Convergence characteristics of PD-type and PDD-type iterative learning control for impulsive differential systems with unknown initial states,” J. Vib. Control, vol. 24, pp. 3726–3743, 2018. https://doi.org/10.1177/1077546317710159.Suche in Google Scholar
[10] W. Chen and L. Zhang, “Adaptive iterative learning control for nonlinearly parameterized systems with unknown time-varying delays,” Int. J. Control Autom. Syst., vol. 8, pp. 177–186, 2010. https://doi.org/10.1007/s12555-010-0201-0.Suche in Google Scholar
[11] D. Shen, G. Qu, and X. Yu, “Averaging techniques for balancing learning and tracking abilities over fading channels,” IEEE Trans. Automat. Control, vol. 66, pp. 2636–2651, 2020. https://doi.org/10.1109/tac.2020.3011329.Suche in Google Scholar
[12] T. Seel, T. Schauer, and J. Raisch, “Iterative learning control for variable pass length systems,” IFAC Proc. Vol., vol. 44, pp. 4880–4885, 2011. https://doi.org/10.3182/20110828-6-it-1002.02180.Suche in Google Scholar
[13] R. W. Longman and K. D. Mombaur, “Investigating the use of iterative learning control and repetitive control to implement periodic Gaits,” Lect. Notes Control Inf. Sci., vol. 340, pp. 189–218, 2006.10.1007/978-3-540-36119-0_9Suche in Google Scholar
[14] X. Li, J. Xu, and D. Huang, “An iterative learning control approach for linear systems with randomly varying trial lengths,” IEEE Trans. Automat. Control, vol. 59, pp. 1954–1960, 2014. https://doi.org/10.1109/tac.2013.2294827.Suche in Google Scholar
[15] X. Li, J. Xu, and D. Huang, “Iterative learning control for nonlinear dynamic systems with randomly varying trial lengths,” Int. J. Adapt. Control Signal Process., vol. 29, pp. 1341–1353, 2015. https://doi.org/10.1002/acs.2543.Suche in Google Scholar
[16] S. Liu, A. Debbouche, and J. Wang, “On the iterative learning control for stochastic impulsive differential equations with randomly varying trial lengths,” J. Comput. Appl. Math., vol. 312, pp. 47–57, 2017. https://doi.org/10.1016/j.cam.2015.10.028.Suche in Google Scholar
[17] C. Zeng, D. Shen, and J. Wang, “Adaptive learning tracking for uncertain systems with partial structure information and varying trial lengths,” J. Franklin Inst., vol. 355, pp. 7027–7055, 2018. https://doi.org/10.1016/j.jfranklin.2018.07.031.Suche in Google Scholar
[18] X. Li and D. Shen, “Two novel iterative learning control schemes for systems with randomly varying trial lengths,” Syst. Control Lett., vol. 107, pp. 9–16, 2017. https://doi.org/10.1016/j.sysconle.2017.07.003.Suche in Google Scholar
[19] T. Abdeljawad, “On conformable fractional calculus,” J. Comput. Appl. Math., vol. 279, pp. 57–66, 2015. https://doi.org/10.1016/j.cam.2014.10.016.Suche in Google Scholar
[20] R. Khalil, M. A. Horani, A. Yousef, and M. Sababheh, “A new definition of fractional derivative,” J. Comput. Appl. Math., vol. 264, pp. 65–70, 2014. https://doi.org/10.1016/j.cam.2014.01.002.Suche in Google Scholar
[21] W. S. Chung, “Fractional Newton mechanics with conformable fractional derivative,” J. Comput. Appl. Math., vol. 290, pp. 150–158, 2015. https://doi.org/10.1016/j.cam.2015.04.049.Suche in Google Scholar
[22] T. Abdeljawad, Q. M. Al-Mdallal, and F. Jarad, “Fractional logistic models in the frame of fractional operators generated by conformable derivatives,” Chaos Solit. Fractals, vol. 119, pp. 94–101, 2019.10.1016/j.chaos.2018.12.015Suche in Google Scholar
[23] M. Bohner and V. F. Hatipoǧlu, “Dynamic cobweb models with conformable fractional derivatives,” Nonlinear Anal. Hybri. Syst., vol. 32, pp. 157–167, 2019. https://doi.org/10.1016/j.nahs.2018.09.004.Suche in Google Scholar
[24] X. Ma, W. Wu, B. Zeng, Y. Wang, and X. Wu, “The conformable fractional grey system model,” ISA Trans., vol. 96, pp. 255–271, 2020. https://doi.org/10.1016/j.isatra.2019.07.009.Suche in Google Scholar PubMed
[25] W. Wu, X. Ma, Y. Zhang, W. Li, and Y. Wang, “A novel conformable fractional non-homogeneous grey model for forecasting carbon dioxide emissions of BRICS countries,” Sci. Total Environ., vol. 707, p. 135447, 2020. https://doi.org/10.1016/j.scitotenv.2019.135447.Suche in Google Scholar PubMed
[26] A. Nazir, N. Ahmed, U. Khan, S. T. Mohyud-Din, K. S. Nisar, and I. Khan, “An advanced version of a conformable mathematical model of Ebola virus disease in Africa,” Alex. Eng. J., vol. 59, pp. 3261–3268, 2020. https://doi.org/10.1016/j.aej.2020.08.050.Suche in Google Scholar
[27] A. Zheng, Y. Feng, and W. Wang, “The Hyers-Ulam stability of the conformable fractional differential equation,” Math. Aeterna, vol. 5, pp. 485–492, 2015.Suche in Google Scholar
[28] J. Tariboon and S. K. Ntouyas, “Oscillation of impulsive conformable fractional differential equations,” Open Math., vol. 14, pp. 497–508, 2016. https://doi.org/10.1515/math-2016-0044.Suche in Google Scholar
[29] M. Li, J. Wang, and D. O’Regan, “Existence and Ulams stability for conformable fractional differential equations with constant coefficients,” Bull. Malays. Math. Sci. Soc., vol. 42, pp. 1791–1812, 2019. https://doi.org/10.1007/s40840-017-0576-7.Suche in Google Scholar
[30] Y. Qi and X. Wang, “Asymptotical stability analysis of conformable fractional systems,” J. Taibah Univ. Sci., vol. 14, pp. 44–49, 2020. https://doi.org/10.1080/16583655.2019.1701390.Suche in Google Scholar
[31] X. Wang, J. Wang, D. Shen, and Y. Zhou, “Convergence analysis for iterative learning control of conformable fractional differential equations,” Math. Methods Appl. Sci., vol. 41, pp. 8315–8328, 2018. https://doi.org/10.1002/mma.5291.Suche in Google Scholar
[32] W. Qiu, M. Fečkan, D. O’Regan, and J. Wang, “Convergence analysis for iterative learning control of conformable impulsive differential equations,” Bull. Iran. Math. Soc., vol. 48, pp. 193–212, 2022. https://doi.org/10.1007/s41980-020-00510-6.Suche in Google Scholar
[33] R. Durrentt, Probability: Theory and Examples, Cambridge, Cambridge University Press, 2019.Suche in Google Scholar
[34] W. Qiu, J. Wang, and D. O’Regan, “Existence and Ulam stability of solutions for conformable impulsive differential equations,” Bull. Iran. Math. Soc., vol. 46, pp. 1613–1637, 2020. https://doi.org/10.1007/s41980-019-00347-8.Suche in Google Scholar
[35] K. H. Park, “An average operator-based PD-type iterative learning control for variable initial state error,” IEEE Trans. Automat. Control, vol. 50, pp. 865–869, 2005.10.1109/TAC.2005.849249Suche in Google Scholar
[36] Y. Qin, Integral and Discrete Inequalities and Their Applications, Berlin, Germany, Birkhauser, 2016.10.1007/978-3-319-33304-5Suche in Google Scholar
© 2022 Walter de Gruyter GmbH, Berlin/Boston
Artikel in diesem Heft
- Frontmatter
- Original Research Article
- Hybrid solitary wave solutions of the Camassa–Holm equation
- Numerical simulations of wave propagation in a stochastic partial differential equation model for tumor–immune interactions
- A Chebyshev collocation method for solving the non-linear variable-order fractional Bagley–Torvik differential equation
- Higher order codimension bifurcations in a discrete-time toxic-phytoplankton–zooplankton model with Allee effect
- Numerical modeling of the dam-break flood over natural rivers on movable beds
- A class of piecewise fractional functional differential equations with impulsive
- Lie symmetry analysis for two-phase flow with mass transfer
- Asymptotic behavior for stochastic plate equations with memory in unbounded domains
- Discussion on controllability of non-densely defined Hilfer fractional neutral differential equations with finite delay
- A linearized finite difference scheme for time–space fractional nonlinear diffusion-wave equations with initial singularity
- Ground state solutions of Schrödinger system with fractional p-Laplacian
- Bifurcation analysis of a new stochastic traffic flow model
- An uncertainty measure based on Pearson correlation as well as a multiscale generalized Shannon-based entropy with financial market applications
- Hilfer fractional stochastic evolution equations on infinite interval
- Iterative learning control for conformable stochastic impulsive differential systems with randomly varying trial lengths
- Chebyshev wavelet-Picard technique for solving fractional nonlinear differential equations
- Theoretical assessment of the impact of awareness programs on cholera transmission dynamic
- Theoretical and numerical analysis of a prey–predator model (3-species) in the frame of generalized Mittag-Leffler law
- Controllability discussion for fractional stochastic Volterra–Fredholm integro-differential systems of order 1 < r < 2
- Shehu transform on time-fractional Schrödinger equations – an analytical approach
- A (2 + 1)-dimensional variable-coefficients extension of the Date–Jimbo–Kashiwara–Miwa equation: Lie symmetry analysis, optimal system and exact solutions
- Mathematical model of fluid flow in a double constricted tapered tube with permeable boundary
Artikel in diesem Heft
- Frontmatter
- Original Research Article
- Hybrid solitary wave solutions of the Camassa–Holm equation
- Numerical simulations of wave propagation in a stochastic partial differential equation model for tumor–immune interactions
- A Chebyshev collocation method for solving the non-linear variable-order fractional Bagley–Torvik differential equation
- Higher order codimension bifurcations in a discrete-time toxic-phytoplankton–zooplankton model with Allee effect
- Numerical modeling of the dam-break flood over natural rivers on movable beds
- A class of piecewise fractional functional differential equations with impulsive
- Lie symmetry analysis for two-phase flow with mass transfer
- Asymptotic behavior for stochastic plate equations with memory in unbounded domains
- Discussion on controllability of non-densely defined Hilfer fractional neutral differential equations with finite delay
- A linearized finite difference scheme for time–space fractional nonlinear diffusion-wave equations with initial singularity
- Ground state solutions of Schrödinger system with fractional p-Laplacian
- Bifurcation analysis of a new stochastic traffic flow model
- An uncertainty measure based on Pearson correlation as well as a multiscale generalized Shannon-based entropy with financial market applications
- Hilfer fractional stochastic evolution equations on infinite interval
- Iterative learning control for conformable stochastic impulsive differential systems with randomly varying trial lengths
- Chebyshev wavelet-Picard technique for solving fractional nonlinear differential equations
- Theoretical assessment of the impact of awareness programs on cholera transmission dynamic
- Theoretical and numerical analysis of a prey–predator model (3-species) in the frame of generalized Mittag-Leffler law
- Controllability discussion for fractional stochastic Volterra–Fredholm integro-differential systems of order 1 < r < 2
- Shehu transform on time-fractional Schrödinger equations – an analytical approach
- A (2 + 1)-dimensional variable-coefficients extension of the Date–Jimbo–Kashiwara–Miwa equation: Lie symmetry analysis, optimal system and exact solutions
- Mathematical model of fluid flow in a double constricted tapered tube with permeable boundary
