Abstract
This paper proposes an accelerated adaptive backstepping control scheme for a micro-electro-mechanical systems (MEMS) triaxial gyroscope, which has complicated nonlinear behaviors. The mathematical model of the gyroscope with output constraints is established. Its dynamical evolution laws are analyzed through phase diagrams, time histories and Lyapunov exponents. In designing the controller, a type-2 fuzzy wavelet neural network (T2FWNN) is utilized to approximate the nonlinear unknown functions of the system. Then, in backstepping, a speed function is employed to realize the accelerated convergence with less fluctuations and after that, a time-varying barrier Lyapunov function (BLF) is constructed to restrict state variables into the prescribed ranges. Meanwhile, a hyperbolic tangent tracking differentiator (HTTD) is employed to approximate the virtual control inputs with high precision thus reducing the computation complexity in the backstepping framework. The whole controller fuses the T2FWNN, the speed function, the time-varying BLF, the HTTD and the adaptive law into the backstepping scheme. Besides, stability analysis proves that all signals in the closed-loop system are ultimately bounded. Finally, the majority of the simulated results prove that the designed controller not only satisfies the constraints of state variables, but also suppresses any chaotic oscillations with good tracking performance.
Similar content being viewed by others
Data Availability
Enquiries about data availability should be directed to the authors.
References
Wang, J., Lou, W., Wang, D., Feng, H.: Design, analysis, and fabrication of silicon-based MEMS gyroscope for high-g shock platform. Microsyst. Technol. 25, 4577–4586 (2019). https://doi.org/10.1007/s00542-019-04596-9
Luo, S., Ma, H., Li, F., Ouakad, H.M.: Dynamical analysis and chaos control of MEMS resonators by using the analog circuit. Nonlinear Dyn. 108, 97–112 (2022). https://doi.org/10.1007/s11071-022-07227-7
Trivedi, S., Shen, T., Chang, C.-Y., Huang, P.-W., Li, S.-S.: Design of piezoelectric MEMS accelerometer module and its application in surface roughness prediction of fused silica substrate. IEEE Sens. J. 21, 21979–21988 (2021). https://doi.org/10.1109/JSEN.2021.3103059
Splith, T., Kaps, A., Stallmach, F.: Phase plot of a gravity pendulum acquired via the MEMS gyroscope and magnetic field sensors of a smartphone. Am. J. Phys. 90, 314–316 (2022). https://doi.org/10.1119/10.0009254
Mostafa, M.Z., Khater, H.A., Rizk, M.R., Bahasan, A.M.: A novel GPS/ RAVO/MEMS-INS smartphone-sensor-integrated method to enhance USV navigation systems during GPS outages. Meas. Sci. Technol. 30, 095103 (2019). https://doi.org/10.1088/1361-6501/ab161c
Lee, J.H., Lee, J.I., Kim, D.H., Nam, K.H., Jeon, T.J., Han, I.H.: Validation of a gyroscope-based wearable device for real-time position monitoring of patients in a hospital. Technol. Health Care. 29, 843–848 (2021). https://doi.org/10.3233/THC-202575
Frolov, S.V., Potlov, A.Y.: An Endoscopic optical coherence tomography system with improved precision of probe positioning. Biomed. Eng. 53, 6–10 (2019). https://doi.org/10.1007/s10527-019-09866-4
Bojja, J., Collin, J., Kirkko-Jaakkola, M., Payne, M., Griffiths, R., Takala, J.: Compact north finding system. IEEE Sens. J. 16, 2554–2563 (2016). https://doi.org/10.1109/JSEN.2016.2518860
Hoang, M.L., Pietrosanto, A.: Yaw/Heading optimization by drift elimination on MEMS gyroscope. Sensor Actuat. A-Phys. 325, 112691 (2021). https://doi.org/10.1016/j.sna.2021.112691
Kokuyama, W., Watanabe, T., Nozato, H., Ota, A.: Angular velocity calibration system with a self-calibratable rotary encoder. Measurement 82, 246–253 (2016). https://doi.org/10.1016/j.measurement.2016.01.011
Solouk, M.R., Shojaeefard, M.H., Dahmardeh, M.: Parametric topology optimization of a MEMS gyroscope for automotive applica-tions. Mech. Syst. Signal Pr. 128, 389–404 (2019). https://doi.org/10.1016/j.ymssp.2019.03.049
Fei, J., Zhou, J.: Robust Adaptive control of MEMS triaxial gyroscope using fuzzy compensator. IEEE Trans. Syst. Man Cybern. B. 42, 1599–1607 (2012). https://doi.org/10.1109/TSMCB.2012.2196039.5
Lestev, A.M.: Combination resonances in mems gyro dynamics. Gyroscopy Navig. 6, 41–44 (2015). https://doi.org/10.1134/S2075108715010083
Hamed, Y.S., El-Sayed, A.T., El-Zahar, E.R.: On controlling the vibrations and energy transfer in MEMS gyroscope system with simul-taneous resonance. Nonlinear Dyn. 83, 1687–1704 (2016). https://doi.org/10.1007/s11071-015-2440-3
Larkin, K., Ghommem, M., Hunter, A., Abdelkefi, A.: Nonlinear modeling and performance analysis of cracked beam microgyroscopes. Int. J. Mech. Sci. 188, 105965 (2020). https://doi.org/10.1016/j.ijmecsci.2020.105965
Ouakad, H.M.: Nonlinear structural behavior of a size-dependent MEMS gyroscope assuming a non-trivial shaped proof mass. Mi-crosyst. Technol. 26, 573–582 (2020). https://doi.org/10.1007/s00542-019-04530-z
Wei, Y., Dong, Y., Huang, X., Zhang, Z.: Nonlinearity measurement for low-pressure encapsulated MEMS gyroscopes by transient response. Mech. Syst. Signal Pr. 100, 534–549 (2018). https://doi.org/10.1016/j.ymssp.2017.07.034
Zhou, J., Wen, C., Wang, W.: Adaptive control of uncertain nonlinear systems with quantized input signal. Automatica 95, 152–162 (2018). https://doi.org/10.1016/j.automatica.2018.05.014
Zhao, L., Luo, S., Yang, G., Dong, R.: Chaos analysis and stability control of the MEMS resonator via the type-2 sequential FNN. Mi-crosyst. Technol. 27, 173–182 (2021). https://doi.org/10.1007/s00542-020-04935-1
Sui, S., Chen, C.L.P., Tong, S.: Event-trigger-based finite-time fuzzy adaptive control for stochastic nonlinear system with unmodeled dynamics. IEEE Trans. Fuzzy Syst. 29, 1914–1926 (2021). https://doi.org/10.1109/TFUZZ.2020.2988849
Luo, S., Lewis, F.L., Song, Y., Ouakad, H.M.: Optimal synchronization of unidirectionally coupled fo chaotic electromechanical devices with the hierarchical neural network. IEEE Trans. Neural Netw. Learn. Syst. 33, 1192–1202 (2022). https://doi.org/10.1109/TNNLS.2020.3041350
Li, W., Xiao, D., Wu, X., Su, J., Chen, Z., Hou, Z., Wang, X.: Enhanced temperature stability of sensitivity for MEMS gyroscope based on frequency mismatch control. Microsyst. Technol. 23, 3311–3317 (2017). https://doi.org/10.1007/s00542-016-3114-x
Fei, J., Yan, W., Yang, Y.: Adaptive nonsingular terminal sliding mode control of MEMS gyroscope based on backstepping design: adaptive nonsingular terminal sliding mode control. Int. J. Adapt. Control Signal Process. 29, 1099–1115 (2015). https://doi.org/10.1002/acs.2523
Guo, Y., Xu, B., Zhang, R.: Terminal sliding mode control of MEMS gyroscopes with finite-time learning. IEEE Trans. Neural Netw. Learn. Syst. 32, 4490–4498 (2021). https://doi.org/10.1109/TNNLS.2020.3018107
Rahmani, M.: MEMS gyroscope control using a novel compound robust control. ISA Trans. 72, 37–43 (2018). https://doi.org/10.1016/j.isatra.2017.11.009
Fei, J., Feng, Z.: Adaptive super-twisting sliding mode control for micro gyroscope based on double loop fuzzy neural network struc-ture. Int. J. Mach. Learn. & Cyber. 12, 611–624 (2021). https://doi.org/10.1007/s13042-020-01191-7
Yan, W., Hou, S., Fang, Y., Fei, J.: Robust adaptive nonsingular terminal sliding mode control of MEMS gyroscope using fuzzy-neural-network compensator. Int. J. Mach. Learn. & Cyber. 8, 1287–1299 (2017). https://doi.org/10.1007/s13042-016-0501-7
Shao, X., Shi, Y.: Neural adaptive control for MEMS gyroscope with full-state constraints and quantized input. IEEE Trans. Ind. In-form. 16, 6444–6454 (2020). https://doi.org/10.1109/TII.2020.2968345
Asad, Y.P., Shamsi, A., Tavoosi, J.: Backstepping-based recurrent type-2 fuzzy sliding mode control for MIMO systems (MEMS triaxi-al gyroscope case study). Int. J. Unc. Fuzz. Knowl. Based Syst. 25, 213–233 (2017). https://doi.org/10.1142/S0218488517500088
Vafaie, R.H., Mohammadzadeh, A., Piran, M.J.: A new type-3 fuzzy predictive controller for MEMS gyroscopes. Nonlinear Dyn. 106, 381–403 (2021). https://doi.org/10.1007/s11071-021-06830-4
Zhou, Q., Zhao, S., Li, H., Lu, R., Wu, C.: Adaptive neural network tracking control for robotic manipulators with dead zone. IEEE Trans. Neural Netw. Learn. Syst. 30, 3611–3620 (2019). https://doi.org/10.1109/TNNLS.2018.2869375
Tognetti, E.S., de Oliveira, G.A.: Robust state feedback-based design of PID controllers for high-order systems with time-delay and parametric uncertainties. J. Control Autom. Electr. Syst. 33, 382–392 (2022). https://doi.org/10.1007/s40313-021-00846-2
Wang, C., Ji, X., Zhang, Z., Zhao, B., Quan, L., Plummer, A.R.: Tracking differentiator based back-stepping control for valve-controlled hydraulic actuator system. ISA Trans. 119, 208–220 (2022). https://doi.org/10.1016/j.isatra.2021.02.028
Zhu, G., Nie, L., Lv, Z., Sun, L., Zhang, X., Wang, C.: Adaptive fuzzy dynamic surface sliding mode control of large-scale power sys-tems with prescribe output tracking performance. ISA Trans. 99, 305–321 (2020). https://doi.org/10.1016/j.isatra.2019.08.063
Luo, S., Song, Y., Lewis, F.L., Garrappa, R.: Neuroadaptive optimal fixed-time synchronization and its circuit realization for unidirec-tionally coupled FO self-sustained electromechanical seismograph systems. IEEE Trans. Cybern. 53, 2454–2466 (2023). https://doi.org/10.1109/TCYB.2021.3121069
Luo, S., Lewis, F.L., Song, Y., Garrappa, R.: Dynamical analysis and accelerated optimal stabilization of the fractional-order self-sustained electromechanical seismograph system with fuzzy wavelet neural network. Nonlinear Dyn. 104, 1389–1404 (2021). https://doi.org/10.1007/s11071-021-06330-5
Guo, X., Xu, W., Wang, J., Park, J.H., Yan, H.: BLF-based neuroadaptive fault-tolerant control for nonlinear vehicular platoon with time-varying fault directions and distance restrictions. IEEE Trans. Intell. Transp. Syst. 23, 12388–12398 (2022). https://doi.org/10.1109/TITS.2021.3113928
Zirkohi, M.M.: Adaptive backstepping control design for MEMS gyroscope based on function approximation techniques with input saturation and output constraints. Comput. Electr. Eng. 97, 107547 (2022). https://doi.org/10.1016/j.compeleceng.2021.107547
Zhang, R., Xu, B., Zhao, W.: Finite-time prescribed performance control of MEMS gyroscopes. Nonlinear Dyn. 101, 2223–2234 (2020). https://doi.org/10.1007/s11071-020-05959-y
Sun, J., Yi, J., Pu, Z.: Fixed-time adaptive fuzzy control for uncertain nonstrict-feedback systems with time-varying constraints and input saturations. IEEE Trans. Fuzzy Syst. 30, 1114–1128 (2022). https://doi.org/10.1109/TFUZZ.2021.3052610
Huang, X., Song, Y., Lai, J.: Neuro-adaptive control with given performance specifications for strict feedback systems under full-state constraints. IEEE Trans. Neural Netw. Learn. Syst. 30, 25–34 (2019). https://doi.org/10.1109/TNNLS.2018.2821668
Zhao, K., Song, Y., Chen, C.L.P., Chen, L.: Adaptive asymptotic tracking with global performance for nonlinear systems with unknown control directions. IEEE Trans. Automat. Contr. 67, 1566–1573 (2022). https://doi.org/10.1109/TAC.2021.3074899
de Souza, S.L.T., Caldas, I.L.: Calculation of Lyapunov exponents in systems with impacts. Chaos, Solitons Fractals 19, 569–579 (2004). https://doi.org/10.1016/S0960-0779(03)00130-9
Zhang, Z., Liu, Y., Sieber, J.: Calculating the Lyapunov exponents of a piecewise-smooth soft impacting system with a time-delayed feedback controller. Commun. Nonlinear Sci. Numer. Simul. 91, 105451 (2020). https://doi.org/10.1016/j.cnsns.2020.105451
Yang, Q., Osman, W.M., Chen, C.: A New 6D Hyperchaotic System with Four Positive Lyapunov Exponents Coined. Int. J. Bifurcation Chaos. 25, 1550060 (2015). https://doi.org/10.1142/S0218127415500601
Yue, Y., Xie, J.: Lyapunov exponents and coexistence of attractors in vibro-impact systems with symmetric two-sided rigid constraints. Phys. Lett. A. 373, 2041–2046 (2009). https://doi.org/10.1016/j.physleta.2009.04.009
Li, H., Wang, L., Du, H., Boulkroune, A.: Adaptive fuzzy backstepping tracking control for strict-feedback systems with input Delay. IEEE Trans. Fuzzy Syst. 25, 642–652 (2017). https://doi.org/10.1109/TFUZZ.2016.2567457
Abiyev, R., Abizada, S.: Type-2 fuzzy wavelet neural network for estimation energy performance of residential buildings. Soft Comput. 25, 11175–11190 (2021). https://doi.org/10.1007/s00500-021-05873-4
Mohammadzadeh, A., Zhang, W.: Dynamic programming strategy based on a type-2 fuzzy wavelet neural network. Nonlinear Dyn. 95, 1661–1672 (2019). https://doi.org/10.1007/s11071-018-4651-x
Mohammadzadeh, A., Castillo, O., Band, S.S., Mosavi, A.: A novel fractional-order multiple-model type-3 fuzzy control for nonlinear systems with unmodeled dynamics. Int. J. Fuzzy Syst. 23, 1633–1651 (2021). https://doi.org/10.1007/s40815-021-01058-1
Luo, S., Li, J., Li, S., Hu, J.: Dynamical analysis of the fractional-order centrifugal flywheel governor system and its accelerated adaptive stabilization with the optimality. Int. J. Elec. Power Energ. Syst. 118, 105792 (2020). https://doi.org/10.1016/j.ijepes.2019.105792
Li, F., Luo, S., Yang, G., Ouakad, H.M.: Dynamical analysis and accelerated adaptive backstepping funnel control for dual-mass MEMS gyroscope under event trigger. Chaos, Solitons Fractals 168, 113116 (2023). https://doi.org/10.1016/j.chaos.2023.113116
Acknowledgements
This project is supported by National Natural Science Foundation of China (Grant No. 52065008), Science and Technology Planning Project of Guizhou Province (No. [2021]5634), Innovation and Entrepreneurship Program for High-Level Talents of Guizhou Province (No. (2021)08) and International Influence Improvement Plan of Subject Double Promotion of Guizhou University (No. GDXKBZJH-YB-2023-24).
Funding
The authors have not disclosed any funding.
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflict of interest
The authors have not disclosed any competing interests.
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.
About this article
Cite this article
Li, F., Luo, S., He, S. et al. Dynamical analysis and accelerated adaptive backstepping control of MEMS triaxial gyroscope with output constraints. Nonlinear Dyn 111, 17123–17140 (2023). https://doi.org/10.1007/s11071-023-08741-y
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11071-023-08741-y