A new $ H_{\infty} $ control method of switched nonlinear systems with persistent dwell time: $ H_{\infty} $ fuzzy control criterion with convergence rate constraints

Han Geng; Huasheng Zhang; Han Geng; Huasheng Zhang

doi:10.3934/math.20241275

AIMS Mathematics

2024, Volume 9, Issue 9: 26092-26113. doi: 10.3934/math.20241275

Previous Article Next Article

Research article

A new $ H_{\infty} $ control method of switched nonlinear systems with persistent dwell time: $ H_{\infty} $ fuzzy control criterion with convergence rate constraints

Han Geng ,
Huasheng Zhang ^,

School of Mathematical Sciences, Liaocheng University, Liaocheng 252059, China

Received: 14 July 2024 Revised: 11 August 2024 Accepted: 19 August 2024 Published: 10 September 2024
MSC : 34D20, 93B36, 93C42, 93D09, 93D20

This study aims to explore the problem of $ H_{\infty} $ fuzzy control with an adjustable convergence rate for switched nonlinear systems with time-varying delays under the persistent dwell time (PDT) switching. Compared to the widely studied dwell time (DT) switching or average dwell time (ADT) switching in existing literature, PDT switching provides a more comprehensive consideration of the switching frequency and has a broader range of applicability. Subsequently, by combining the interval stability definition, T-S fuzzy model, PDT technique, and Lyapunov-Krasovskii (L-K) functional, a new $ H_{\infty} $ fuzzy control criterion for adjusting the convergence rate of switched nonlinear systems with time-varying delays is proposed. This criterion enables the development of a novel method for constructing $ H_{\infty} $ fuzzy controllers, which can regulate the system's convergence rate and achieve the specified $ H_{\infty} $ performance. Combining the above methods, an algorithm is introduced to precisely control the convergence rate of the target system. Finally, the effectiveness of this method is validated through a control example of a single-link robot arm.
- switched T-S fuzzy systems,
- $ H_{\infty} $ control,
- convergence rate,
- persistent dwell time,
- time-varying delay
Citation: Han Geng, Huasheng Zhang. A new $ H_{\infty} $ control method of switched nonlinear systems with persistent dwell time: $ H_{\infty} $ fuzzy control criterion with convergence rate constraints[J]. AIMS Mathematics, 2024, 9(9): 26092-26113. doi: 10.3934/math.20241275

Related Papers:

Abstract

This study aims to explore the problem of $ H_{\infty} $ fuzzy control with an adjustable convergence rate for switched nonlinear systems with time-varying delays under the persistent dwell time (PDT) switching. Compared to the widely studied dwell time (DT) switching or average dwell time (ADT) switching in existing literature, PDT switching provides a more comprehensive consideration of the switching frequency and has a broader range of applicability. Subsequently, by combining the interval stability definition, T-S fuzzy model, PDT technique, and Lyapunov-Krasovskii (L-K) functional, a new $ H_{\infty} $ fuzzy control criterion for adjusting the convergence rate of switched nonlinear systems with time-varying delays is proposed. This criterion enables the development of a novel method for constructing $ H_{\infty} $ fuzzy controllers, which can regulate the system's convergence rate and achieve the specified $ H_{\infty} $ performance. Combining the above methods, an algorithm is introduced to precisely control the convergence rate of the target system. Finally, the effectiveness of this method is validated through a control example of a single-link robot arm.

References

[1]	E. Skafidas, R. J. Evans, A. V. Savkin, I. R. Petersen, Stability results for switched controller systems, Automatica, 35 (1999), 553–564. https://doi.org/10.1016/S0005-1098(98)00167-8 doi: 10.1016/S0005-1098(98)00167-8
[2]	L. Zhang, K. Xu, J. Yang, M. Han, S. Yuan, Transition-dependent bumpless transfer control synthesis of switched linear systems, IEEE Trans. Automat. Control, 68 (2023), 1678–1684. https://doi.org/10.1109/TAC.2022.3152721 doi: 10.1109/TAC.2022.3152721
[3]	H. Li, Input-to-state stability for discrete-time switched systems by using Lyapunov functions with relaxed constraints, AIMS Math., 8 (2023), 30827–30845. http://dx.doi.org/10.3934/math.20231576 doi: 10.3934/math.20231576
[4]	T. C. Lee, Z. P. Jiang, Uniform asymptotic stability of nonlinear switched systems with an application to mobile robots, IEEE Trans. Automat. Control, 53 (2008), 1235–1252. https://doi.org/10.1109/TAC.2008.923688 doi: 10.1109/TAC.2008.923688
[5]	B. Niu, X. Zhao, X. Fan, Y. Cheng, A new control method for state-constrained nonlinear switched systems with application to chemical process, Int. J. Control, 88 (2015), 1693–1701. https://doi.org/10.1080/00207179.2015.1013062 doi: 10.1080/00207179.2015.1013062
[6]	U. Ali, M. Egerstedt, Hybrid optimal control under mode switching constraints with applications to pesticide scheduling, ACM Trans. Cyber-Phys. Syst., 2 (2018), 1–17. https://doi.org/10.1145/3047411 doi: 10.1145/3047411
[7]	G. Zhang, D. Tong, Q. Chen, W. Zhou, Sliding mode control against false data injection attacks in DC microgrid systems, IEEE Syst. J., 17 (2023), 6159–6168. https://doi.org/10.1109/JSYST.2023.3280185 doi: 10.1109/JSYST.2023.3280185
[8]	T. Takagi, M. Sugeno, Fuzzy identification of systems and its applications to modeling and control, IEEE Trans. Syst. Man Cybern., 15 (1985), 116–132. https://doi.org/10.1109/TSMC.1985.6313399 doi: 10.1109/TSMC.1985.6313399
[9]	X. Xie, L. Wan, Z. Gu, D. Yue, J. Sun, Enhanced resilient fuzzy stabilization of discrete-time Takagi-Sugeno systems based on augmented time-variant matrix approach, IEEE Trans. Cybern., 54 (2022), 929–934. https://doi.org/10.1109/TCYB.2022.3179048 doi: 10.1109/TCYB.2022.3179048
[10]	X. Xie, Z. Zhang, D. Yue, J. Xia, Relaxed observer design of discrete-time Takagi-Sugeno fuzzy systems based on a lightweight gain-scheduling law, IEEE Trans. Fuzzy Syst., 30 (2022), 5544–5550. https://doi.org/10.1109/TFUZZ.2022.3179598 doi: 10.1109/TFUZZ.2022.3179598
[11]	W. Xiang, J. Xiao, M. N. Iqbal, $H_{\infty}$ control for switched fuzzy systems via dynamic output feedback: hybrid and switched approaches, Commun. Nonli. Sci., 18 (2013), 1499–1514. https://doi.org/10.1016/j.cnsns.2012.10.003 doi: 10.1016/j.cnsns.2012.10.003
[12]	X. Zhao, Y. Yin, L. Zhang, H. Yang, Control of switched nonlinear systems via T-S fuzzy modeling, IEEE Trans. Fuzzy Syst., 24 (2016), 235–241. https://doi.org/10.1109/TFUZZ.2015.2450834 doi: 10.1109/TFUZZ.2015.2450834
[13]	S. Sun, Y. Wang, H. Zhang, J. Sun, Multiple intermittent fault estimation and tolerant control for switched TS fuzzy stochastic systems with multiple time-varying delays, Appl. Math. Comput., 377 (2020), 125114. https://doi.org/10.1016/j.amc.2020.125114 doi: 10.1016/j.amc.2020.125114
[14]	L. Cao, Y. Pan, H. Liang, T. Huang, Observer-based dynamic event-triggered control for multiagent systems with time-varying delay, IEEE Trans. Cybern., 53 (2023), 3376–3387. https://doi.org/10.1109/TCYB.2022.3226873 doi: 10.1109/TCYB.2022.3226873
[15]	M. Shi, D. Tong, Q. Chen, W. Zhou, $P$th moment exponential synchronization for delayed multi-agent systems with Livy noise and Markov switching, IEEE Trans. Circuits Syst. II, Exp Briefs, 71 (2024), 697–701. https://doi.org/10.1109/TCSII.2023.3304635 doi: 10.1109/TCSII.2023.3304635
[16]	D. Tong, B. Ma, Q. Chen, Y. Wei, P. Shi, Finite-time synchronization and energy consumption prediction for multilayer fractional-order networks, IEEE Trans. Circuits Syst. II, Exp Briefs, 70 (2023), 2176–2180. https://doi.org/10.1109/TCSII.2022.3233420 doi: 10.1109/TCSII.2022.3233420
[17]	J. Chen, J. H. Park, S. Xu, Stability analysis for neural networks with time-varying delay via improved techniques, IEEE Trans. Cybern., 49 (2019), 4495–4500. https://doi.org/10.1109/TCYB.2018.2868136 doi: 10.1109/TCYB.2018.2868136
[18]	X. Wang, J. Xia, J. H. Park, X. Xie, G. Chen, Event-triggered adaptive tracking with guaranteed transient performance for switched nonlinear systems under asynchronous switching, IEEE Trans. Cybern., 54 (2024), 496–505. https://doi.org/10.1109/TCYB.2022.3223983 doi: 10.1109/TCYB.2022.3223983
[19]	A. S. Morse, Supervisory control of families of linear set-point controllers-Part Ⅰ. Exact matching, IEEE Trans. Automat. Control, 41 (1996), 1413–1431. https://doi.org/10.1109/9.539424 doi: 10.1109/9.539424
[20]	Q. Yu, Y. Feng, Stability analysis of switching systems with all modes unstable based on a $\Phi$-dependent max-minimum dwell time method, AIMS Math., 9 (2024), 4863–4881. http://dx.doi.org/10.3934/math.2024236 doi: 10.3934/math.2024236
[21]	J. P. Hespanha, Uniform stability of switched linear systems: Extensions of LaSalle's invariance principle, IEEE Trans. Automat. Control, 49 (2004), 470–482. https://doi.org/10.1109/TAC.2004.825641 doi: 10.1109/TAC.2004.825641
[22]	C. Edwards, T. Lombaerts, H. Smaili, Fault tolerant flight control, LNCIS, 399 (2010), 1–560. https://doi.org/10.1007/978-3-642-11690-2 doi: 10.1007/978-3-642-11690-2
[23]	J. Zhou, X. Ma, Z. Yan, S. Arik, Non-fragile output-feedback control for time-delay neural networks with persistent dwell time switching: a system mode and time scheduler dual-dependent design, Neural Networks, 169 (2024), 733–743. https://doi.org/10.1016/j.neunet.2023.11.007 doi: 10.1016/j.neunet.2023.11.007
[24]	H. Shen, M. Xing, Z. Wu, S. Xu, J. Cao, Multiobjective fault-tolerant control for fuzzy switched systems with persistent dwell time and its application in electric circuits, IEEE Trans. Fuzzy Syst., 28 (2020), 2335–2347. https://doi.org/10.1109/TFUZZ.2019.2935685 doi: 10.1109/TFUZZ.2019.2935685
[25]	J. Dong, X. Ma, L. He, S. Arik, Energy-to-peak control for switched systems with PDT switching, Elect. Res. Arch., 31 (2023), 5267–5285. http://doi.org/10.3934/era.2023268 doi: 10.3934/era.2023268
[26]	W. Zhang, L. Xie, Interval stability and stabilization of linear stochastic systems, IEEE Trans. Automat. Control, 54 (2009), 810–815. https://doi.org/10.1109/TAC.2008.2009613 doi: 10.1109/TAC.2008.2009613
[27]	H. Zhang, J. Xia, J. H. Park, W. Sun, G. Zhuang, Interval stability and interval stabilization of linear stochastic systems with time-varying delay, Int. J. Robust. Nonli. Control., 31 (2021), 2334–2347. https://doi.org/10.1002/rnc.5408 doi: 10.1002/rnc.5408
[28]	X. Wang, H. Zhang, J. Xia, W. Sun, G. Zhuang, Interval stability/stabilization of impulsive positive systems, Sci. China Inform. Sci., 66 (2023), 112203. https://doi.org/10.1007/s11432-021-3426-1 doi: 10.1007/s11432-021-3426-1
[29]	Y. Deng, H. Zhang, J. Xia, $H_{\infty}$ control With convergence rate constraint for time-varying delay switched systems, IEEE Trans. Syst. Man. Cybern. Syst., 53 (2023), 7354–7363. https://doi.org/10.1109/TSMC.2023.3298813 doi: 10.1109/TSMC.2023.3298813
[30]	H. Zhang, Y. Dai, C. Zhu, Region stability analysis and precise tracking control of linear stochastic systems, Appl. Math. Comput., 465 (2024), 128402. https://doi.org/10.1016/j.amc.2023.128402 doi: 10.1016/j.amc.2023.128402
[31]	L. Zhang, S. Zhuang, P. Shi, Non-weighted quasi-time-dependent $H_{\infty}$ filtering for switched linear systems with persistent dwell-time, Automatica, 54 (2015), 201–209. https://doi.org/10.1016/j.automatica.2015.02.010 doi: 10.1016/j.automatica.2015.02.010
[32]	D. Liberzon, Switching in systems and control, Boston: Birkhauser, 2003.
[33]	J. P. Hespanha, Root-mean-square gains of switched linear systems, IEEE Trans. Automat. Control, 48 (2003), 2040–2045. https://doi.org/10.1109/TAC.2003.819300 doi: 10.1109/TAC.2003.819300
[34]	Y. Chen, Z. Wang, B. Shen, Q. L. Han, Local stabilization for multiple input-delay systems subject to saturating actuators: The continuous-time case, IEEE Trans. Automat. Control, 67 (2022), 3090–3097. https://doi.org/10.1109/TAC.2021.3092556 doi: 10.1109/TAC.2021.3092556
[35]	Y. Mao, H. Zhang, Exponential stability and robust $H_{\infty}$ control of a class of discrete-time switched non-linear systems with time-varying delays via TS fuzzy model, Int. J. Syst. Sci., 45 (2014), 1112–1127. https://doi.org/10.1080/00207721.2012.745025 doi: 10.1080/00207721.2012.745025
[36]	H. N. Wu, K. Y. Cai, Mode-independent robust stabilization for uncertain Markovian jump nonlinear systems via fuzzy control, IEEE Trans. Syst. Man. Cybern., 36 (2006), 509–519. https://doi.org/10.1109/TSMCB.2005.862486 doi: 10.1109/TSMCB.2005.862486
[37]	B. Wang, J. Cheng, J. Zhan, A sojourn probability approach to fuzzy-model-based reliable control for switched systems with mode-dependent time-varying delays, Nonlinear Anal. Hybri. Syst., 26 (2017), 239–253. https://doi.org/10.1016/j.nahs.2017.05.006 doi: 10.1016/j.nahs.2017.05.006

Reader Comments

Your name:*

Email:*
© 2024 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)