A high-order fully actuated system approach to stabilization of discrete-time periodic affine nonlinear systems

Lingling Lv; Ruijie Fu; Xingyu Xue; Liyan Kang; Sunjie Chen; Lingling Lv; Ruijie Fu; Xingyu Xue; Liyan Kang; Sunjie Chen

doi:10.3934/jimo.2026200

Journal of Industrial and Management Optimization

2026, Volume 22, Issue 11: 5944-5960. doi: 10.3934/jimo.2026200

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Research article

A high-order fully actuated system approach to stabilization of discrete-time periodic affine nonlinear systems

1.
School of Electrical Engineering, North China University of Water Resources and Electric Power, Zhengzhou, People’s Republic of China
2.
Hangzhou Turui Intelligent Equipment Co., Ltd., Hangzhou, People’s Republic of China
3.
Department of Electrical and Computer Engineering, University of California at Los Angeles, California, USA

Received: 14 September 2026 Revised: 14 October 2026 Accepted: 19 October 2026 Published: 23 October 2026
93D15, 93C55

This study addresses stabilization for a class of discrete-time periodic affine nonlinear systems within the high-order fully actuated (HOFA) system framework. The nonlinear affine contribution is first handled through an input transformation, while the periodic high-order linear part is retained for controller synthesis. The resulting dynamics are then represented in lifted coordinates, so that the closed-loop behavior can be characterized by its transition over one period. Instead of computing the feedback matrices directly from the monodromy product, a prescribed periodic target model is introduced and related to the closed-loop dynamics through a periodic Sylvester matrix equation. This construction allows the assigned spectrum to be transferred to the actual closed-loop system, and asymptotic stability follows when the prescribed poles lie inside the unit disk. A third-order numerical study, together with a comparison against discrete-time sliding mode control, illustrates the resulting closed-loop behavior.
- high-order fully actuated system approach,
- discrete-time periodic systems,
- pole placement,
- periodic sylvester matrix equation
Citation: Lingling Lv, Ruijie Fu, Xingyu Xue, Liyan Kang, Sunjie Chen. A high-order fully actuated system approach to stabilization of discrete-time periodic affine nonlinear systems[J]. Journal of Industrial and Management Optimization, 2026, 22(11): 5944-5960. doi: 10.3934/jimo.2026200

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Abstract

This study addresses stabilization for a class of discrete-time periodic affine nonlinear systems within the high-order fully actuated (HOFA) system framework. The nonlinear affine contribution is first handled through an input transformation, while the periodic high-order linear part is retained for controller synthesis. The resulting dynamics are then represented in lifted coordinates, so that the closed-loop behavior can be characterized by its transition over one period. Instead of computing the feedback matrices directly from the monodromy product, a prescribed periodic target model is introduced and related to the closed-loop dynamics through a periodic Sylvester matrix equation. This construction allows the assigned spectrum to be transferred to the actual closed-loop system, and asymptotic stability follows when the prescribed poles lie inside the unit disk. A third-order numerical study, together with a comparison against discrete-time sliding mode control, illustrates the resulting closed-loop behavior.

References

[1]	M. Ghodrat, H. J. Marquez, A multirate approach to event-triggered control of periodic sampled-data systems, IEEE Trans. Control Netw. Syst., 12 (2025), 648–660. https://doi.org/10.1109/TCNS.2024.3395848 doi: 10.1109/TCNS.2024.3395848
[2]	T. Choe, K. Ri, C. Yun, K. Kim, K. Kim, Nonlinear vibration analysis and stability analysis of rotor systems with multiple localized nonlinearities, AIP Adv., 12 (2022), 125004. https://doi.org/10.1063/5.0128600 doi: 10.1063/5.0128600
[3]	Y. Chen, Q. Zong, X. Zhang, J. Li, Data-driven active vibration control for helicopter with trailing-edge flaps using adaptive dynamic programming, Chin. J. Aeronaut., 37 (2024), 151–166. https://doi.org/10.1016/j.cja.2024.02.019 doi: 10.1016/j.cja.2024.02.019
[4]	N. Athanasopoulos, M. Lazar, C. Böhm, F. Allgöwer, On stability and stabilization of periodic discrete-time systems with an application to satellite attitude control, Automatica, 50 (2014), 3190–3196. https://doi.org/10.1016/j.automatica.2014.10.028 doi: 10.1016/j.automatica.2014.10.028
[5]	A. Czornik, E. Makarov, M. Niezabitowski, S. Popova, V. Zaitsev, Uniform global asymptotic stabilization of semilinear periodic discrete-time systems, IEEE Trans. Autom. Control, 67 (2022), 3598–3605. https://doi.org/10.1109/TAC.2021.3105648 doi: 10.1109/TAC.2021.3105648
[6]	L. Lv, G. Duan, B. Zhou, Parametric pole assignment and robust pole assignment for discrete-time linear periodic systems, SIAM J. Control Optim., 48 (2010), 3975–3996. https://doi.org/10.1137/080730469 doi: 10.1137/080730469
[7]	J. Gao, C. Han, Y. Zhao, L. Lv, Output feedback based poles configuration for LDP systems with varying state and input dimensions, IET Control Theory Appl., 17 (2023), 318–330. https://doi.org/10.1049/cth2.12316 doi: 10.1049/cth2.12316
[8]	Z. Chen, Y. Guo, Z. Zou, Z. Zhang, An adaptive Levenberg–Marquardt identification algorithm for time-delay linear discrete periodic systems, J. Ind. Manag. Optim., 22 (2026), 3023–3049. https://doi.org/10.3934/jimo.2026111 doi: 10.3934/jimo.2026111
[9]	S. Sinha, S. P. Nandanoori, J. Drgona, D. Vrabie, Data-driven stabilization of discrete-time control-affine nonlinear systems: A Koopman operator approach, in 2022 European Control Conference, (2022), 552–559. https://doi.org/10.23919/ECC55457.2022.9837986
[10]	I. Karafyllis, S. Kotsios, Conditions for global asymptotic stabilization of discrete-time systems, IFAC Proc. Vol., 37 (2004), 719–724. https://doi.org/10.1016/S1474-6670(17)31390-3 doi: 10.1016/S1474-6670(17)31390-3
[11]	G. Duan, High-order fully actuated system approaches: Part I. Models and basic procedure, Int. J. Syst. Sci., 52 (2021a), 422–435. https://doi.org/10.1080/00207721.2020.1829167 doi: 10.1080/00207721.2020.1829167
[12]	G. Duan, High-order fully actuated system approaches: Part VII. Controllability, stabilisability and parametric designs, Int. J. Syst. Sci., 52 (2021b), 3091–3114. https://doi.org/10.1080/00207721.2021.1921307 doi: 10.1080/00207721.2021.1921307
[13]	G. Duan, High-order fully actuated system approaches: Part X. Basics of discrete-time systems, Int. J. Syst. Sci., 53 (2022), 810–832. https://doi.org/10.1080/00207721.2021.1975848 doi: 10.1080/00207721.2021.1975848
[14]	G. Duan, B. Zhou, Fully actuated system approach for linear systems control: A frequency-domain solution, J. Syst. Sci. Complex., 35 (2022), 2046–2061. https://doi.org/10.1007/s11424-022-1361-8 doi: 10.1007/s11424-022-1361-8
[15]	D. Gu, S. Wang, A high-order fully actuated system approach for a class of nonlinear systems, J. Syst. Sci. Complex., 35 (2022), 714–730. https://doi.org/10.1007/s11424-022-2041-4 doi: 10.1007/s11424-022-2041-4
[16]	G. Duan, Discrete-time delay systems: Part 1. Global fully actuated case, Sci. China Inf. Sci., 65 (2022), 182201. https://doi.org/10.1007/s11432-021-3417-3 doi: 10.1007/s11432-021-3417-3
[17]	D. W. Zhang, G. P. Liu, L. Cao, Predictive control of discrete-time high-order fully actuated systems with application to air-bearing spacecraft simulator, J. Franklin Inst., 360 (2023), 5910–5927. https://doi.org/10.1016/j.jfranklin.2023.04.003 doi: 10.1016/j.jfranklin.2023.04.003
[18]	S. Zhang, Y. Qiu, M. Hou, X. Wu, H. Zhang, J. Wang, Design of a fast model predictive controller based on the fully actuated system approach, Control Eng. Pract., 169 (2026), 106762. https://doi.org/10.1016/j.conengprac.2026.106762 doi: 10.1016/j.conengprac.2026.106762
[19]	S. Zhang, Y. Qiu, M. Hou, X. Wu, H. Zhang, J. Wang, Multiple filtered high-gain observers-based robust control for mixed-order fully actuated systems, J. Franklin Inst., 363 (2026), 108599. https://doi.org/10.1016/j.jfranklin.2026.108599 doi: 10.1016/j.jfranklin.2026.108599
[20]	D. Jin, J. Qiu, W. Ji, T. Wang, Attitude–orbit integrated modeling and control for flexible spacecraft based on twistors and fully actuated system approach, IEEE Trans. Syst. Man Cybern. Syst., 56 (2026), 1369–1378. https://doi.org/10.1109/TSMC.2025.3647684 doi: 10.1109/TSMC.2025.3647684
[21]	S. Jia, J. Qiu, T. Wang, High-order fully actuated system approach-based attitude stabilization for underactuated rigid and flexible spacecraft, IEEE Trans. Autom. Sci. Eng., 22 (2025), 15094–15105. https://doi.org/10.1109/TASE.2025.3567541 doi: 10.1109/TASE.2025.3567541
[22]	G. Liu, K. Zhang, B. Li, Fully-actuated system approach based optimal attitude tracking control of rigid spacecraft with actuator saturation, J. Syst. Sci. Complex., 35 (2022), 688–702. https://doi.org/10.1007/s11424-022-1515-8 doi: 10.1007/s11424-022-1515-8
[23]	G. Tian, J. Tan, B. Li, G. Duan, Optimal fully actuated system approach-based trajectory tracking control for robot manipulators, IEEE Trans. Cybern., 54 (2024), 7469–7478. https://doi.org/10.1109/TCYB.2024.3467386 doi: 10.1109/TCYB.2024.3467386
[24]	L. Lv, Z. Zhang, L. Zhang, W. Wang, An iterative algorithm for periodic Sylvester matrix equations, J. Ind. Manag. Optim., 14 (2018), 413–425. https://doi.org/10.3934/jimo.2017053 doi: 10.3934/jimo.2017053
[25]	C. Ma, T. Yan, A finite iterative algorithm for the general discrete-time periodic Sylvester matrix equations, J. Franklin Inst., 359 (2022), 4410–4432. https://doi.org/10.1016/j.jfranklin.2022.03.047 doi: 10.1016/j.jfranklin.2022.03.047
[26]	L. Lv, J. Chen, L. Zhang, F. Zhang, Gradient-based neural networks for solving periodic Sylvester matrix equations, J. Franklin Inst., 359 (2022), 10849–10866. https://doi.org/10.1016/j.jfranklin.2022.05.023 doi: 10.1016/j.jfranklin.2022.05.023
[27]	S. Li, C. Ma, An improved gradient neural network for solving periodic Sylvester matrix equations, J. Franklin Inst., 360 (2023), 4056–4070. https://doi.org/10.1016/j.jfranklin.2023.02.019 doi: 10.1016/j.jfranklin.2023.02.019
[28]	W. Wang, C. Song, Iterative algorithms for discrete-time periodic Sylvester matrix equations and its application in antilinear periodic system, Appl. Numer. Math., 168 (2021), 251–273. https://doi.org/10.1016/j.apnum.2021.06.006 doi: 10.1016/j.apnum.2021.06.006
[29]	R. Qi, C. Song, Iterative algorithms for the generalized discrete-time periodic Sylvester transpose matrix equations with application in the periodic state observer design of linear systems, J. Appl. Anal. Comput., 15 (2025), 286–315. https://doi.org/10.11948/20240085 doi: 10.11948/20240085
[30]	L. Lv, W. Lai, X. Liu, A parametric poles assignment algorithm for high-order linear discrete periodic systems, J. Franklin Inst., 360 (2023), 11855–11870. https://doi.org/10.1016/j.jfranklin.2023.09.010 doi: 10.1016/j.jfranklin.2023.09.010

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