Robust stability and \(H_\infty\) control problem for 2-D nonlinear uncertain switched system with mixed time-varying delays under fuzzy rules. (English) Zbl 1538.93035
Summary: In this paper, the robust stability and \(H_\infty\) control problem of two-dimensional (2-D) nonlinear polytopic uncertain switched system with mixed time-varying delays is studied based on Roesser model, and the nonlinear system is cleverly broken down into linear forms under the Takagi-Sugeno (T-S) fuzzy rules. Assuming its uncertain parameters are given by convex bounded polyhedral domain. First of all, an improved multi-parameter Lyapunov-Krasovskii function (LKF) is proposed to work hard to get additional information related to time delays, so that it is less conservative. Second, using the aforementioned LKF coupled with Finsler’s lemma, finite-sum inequalities, and Jensen inequalities, a new sufficient condition in linear matrix inequalities (LMIs) is derived for robust \(H_\infty\) performance analysis of 2-D nonlinear polytopic uncertain switched system. Third, for this system with time delays, a memory-state feedback controller including information about the past of the system state is designed to remove the effect of time delays on the system, and the resulting closed-loop system is robustly asymptotically stable under the specified \(H_\infty\) disturbance attenuation level \(\gamma\). Finally, the superiority and effectiveness of the proposed results are illustrated by simulation examples.
MSC:
| 93D09 | Robust stability |
| 93B36 | \(H^\infty\)-control |
| 93C10 | Nonlinear systems in control theory |
| 93C41 | Control/observation systems with incomplete information |
| 93C30 | Control/observation systems governed by functional relations other than differential equations (such as hybrid and switching systems) |
| 93C43 | Delay control/observation systems |
| 93C42 | Fuzzy control/observation systems |
Keywords:
mixed time-varying delays; robust stability; \(H_\infty\) control; Takagi-Sugeno fuzzy rules; 2-D nonlinear switched system; polytopic uncertainReferences:
| [1] | Badie, K.; Alfidi, M.; Chalh, Z., Robust \(H_{\infty }\) control for 2-D discrete state delayed systems with polytopic uncertainties, Multidimens Syst Signal Process, 30, 3, 1327-1343 (2019) · Zbl 1416.93160 · doi:10.1007/s11045-018-0606-0 |
| [2] | Badie, K.; Alfidi, M.; Tadeo, F.; Chalh, Z., Robust state feedback for uncertain 2-D discrete switched systems in the Roesser model, J Control Decis, 8, 3, 331-342 (2021) · doi:10.1080/23307706.2020.1803774 |
| [3] | Badie K, Alfidi M, Chalh Z (2019) Delay-dependent exponential stability of discrete 2-D switched systems with delays. In: 2019 8th International Conference on Systems and Control (ICSC). IEEE, pp 513-518 · Zbl 1451.93319 |
| [4] | Benzaouia, A.; Hmamed, A.; Tadeo, F.; Hajjaji, AE, Stabilisation of discrete 2D time switching systems by state feedback control, Int J Syst Sci, 42, 3, 479-487 (2011) · Zbl 1209.93127 · doi:10.1080/00207720903576522 |
| [5] | Ding, X.; Liu, X., On stabilizability of switched positive linear systems under state-dependent switching, Appl Math Comput, 307, 92-101 (2017) · Zbl 1411.93130 |
| [6] | Duan, Z.; Xiang, Z., State feedback \({H_\infty }\) control for discrete 2D switched systems, J Franklin Inst, 350, 6, 1513-1530 (2013) · Zbl 1293.93608 · doi:10.1016/j.jfranklin.2013.04.001 |
| [7] | Duan, Z.; Xiang, Z.; Karimi, HR, Delay-dependent \({H_\infty }\) control for 2-D switched delay systems in the second FM model, J Franklin Inst, 350, 7, 1697-1718 (2013) · Zbl 1392.93009 · doi:10.1016/j.jfranklin.2013.04.019 |
| [8] | Duan, Z.; Ghous, I.; Shen, J., Fault detection observer design for discrete-time 2-D TS fuzzy systems with finite-frequency specifications, Fuzzy Sets Syst, 392, 24-45 (2020) · Zbl 1452.93015 · doi:10.1016/j.fss.2019.05.004 |
| [9] | Duan, Z.; Ghous, I.; Xia, Y.; Akhtar, J., \({H_\infty }\) control problem of discrete 2-D switched mixed delayed systems using the improved Lyapunov-Krasovskii functional, Int J Control Autom Syst, 18, 8, 2075-2087 (2020) · doi:10.1007/s12555-019-0331-y |
| [10] | Dzung, NT, Robust stabilization of non-stationary Markov jump 2-D systems with multiplicative noises, J Franklin Inst, 358, 15, 7413-7425 (2021) · Zbl 1472.93149 · doi:10.1016/j.jfranklin.2021.07.033 |
| [11] | Fornasini, E.; Marchesini, G., Doubly-indexed dynamical systems: state-space models and structural properties, Theory Comput Syst, 12, 1, 59-72 (1978) · Zbl 0392.93034 |
| [12] | Fridman, E.; Shaked, U., An improved stabilization method for linear time-delay systems, IEEE Trans Autom Control, 47, 11, 1931-1937 (2002) · Zbl 1364.93564 · doi:10.1109/TAC.2002.804462 |
| [13] | Ghous, I.; Xiang, Z.; Karimi, HR, State Feedback \(H_\infty\) control for 2-D switched delay systems with actuator saturation in the second FM model, Circ Syst Signal Process, 34, 7, 2167-2192 (2015) · Zbl 1341.93025 · doi:10.1007/s00034-014-9960-9 |
| [14] | Ghous, I.; Huang, S.; Xiang, Z., State Feedback \({L_1}\) gain control of positive 2-D continuous switched delayed systems via state-dependent switching, Circ Syst Signal Process, 35, 7, 2432-2449 (2016) · Zbl 1346.93201 · doi:10.1007/s00034-015-0161-y |
| [15] | Ghous, I.; Xiang, Z.; Karimi, HR, \({H_\infty }\) control of 2-D continuous Markovian jump delayed systems with partially unknown transition probabilities, Inf Sci, 382, 274-291 (2017) · Zbl 1432.93085 · doi:10.1016/j.ins.2016.12.018 |
| [16] | Hua, D.; Wang, W.; Yao, J.; Ren, Y., Non-weighted \({H_\infty }\) performance for 2-D FMLSS switched system with maximum and minimum dwell time, J Franklin Inst, 356, 11, 5729-5753 (2019) · Zbl 1415.93103 · doi:10.1016/j.jfranklin.2019.05.025 |
| [17] | Hua, D.; Wang, W.; Ren, Y.; Juan, Y., Finite-region stabilization and \({H_\infty }\) control for 2-D FMLSS asynchronously switched system, J Franklin Inst, 357, 14, 9127-9153 (2020) · Zbl 1448.93312 · doi:10.1016/j.jfranklin.2020.04.039 |
| [18] | Jin, SH; Park, JB, Robust \({H_\infty }\) filtering for polytopic uncertain systems via convex optimisation, IEE Proc-Control Theory Appl, 148, 1, 55-59 (2001) · doi:10.1049/ip-cta:20010237 |
| [19] | Kaczorek, T., Two dimensional linear systems, 1985 (1985), Berlin: Springer, Berlin · Zbl 0593.93031 |
| [20] | Kaddour, AA; Benjelloun, K.; Elalami, N., Static output-feedback controller design for a fish population system, Appl Soft Comput, 29, 280-287 (2015) · doi:10.1016/j.asoc.2014.12.033 |
| [21] | Koumir M, El-Amrani A, Boumhidi \(I (2020) {H_\infty }\) model reduction and finite frequency for 2-D fuzzy systems. In: 2020 Fourth International Conference On Intelligent Computing in Data Sciences. IEEE, pp 1-6 |
| [22] | Le, VH; Trinh, H., Stability of two-dimensional Roesser systems with time-varying delays via novel 2D finite-sum inequalities, IET Control Theory Appl, 10, 14, 1665-1674 (2016) · doi:10.1049/iet-cta.2016.0078 |
| [23] | Li, X.; Gao, H., Robust finite frequency \({H_\infty }\) filtering for uncertain 2-D Roesser systems, Automatica, 48, 6, 1163-1170 (2012) · Zbl 1244.93159 · doi:10.1016/j.automatica.2012.03.012 |
| [24] | Liang, J.; Huang, T.; Hayat, T.; Alsaadi, F., \({H_\infty }\) filtering for two-dimensional systems with mixed time delays, randomly occurring saturations and nonlinearities[J], Int J Gen Syst, 44, 2, 226-239 (2015) · Zbl 1309.93169 · doi:10.1080/03081079.2014.973733 |
| [25] | Mao, D.; Ma, Y., Robust \({{H}_{\infty }}\) control for uncertain Takagi-Sugeno fuzzy systems with state and input time-varying delays, Comput Appl Math, 41, 5, 1-23 (2022) · Zbl 1513.93016 · doi:10.1007/s40314-022-01879-2 |
| [26] | Oliveira MC, Skelton RE (2001) Stability tests for constrained linear systems. In: Perspectives in robust control. Springer, London 241-257 · Zbl 0997.93086 |
| [27] | Peaucelle, D.; Arzelier, D.; Bachelier, O.; Jacques, B., A new robust D-stability condition for real convex polytopic uncertainty, Syst Control Lett, 40, 1, 21-30 (2000) · Zbl 0977.93067 · doi:10.1016/S0167-6911(99)00119-X |
| [28] | Peng, D.; Hua, C., Improved approach to delay-dependent stability and stabilisation of two-dimensional discrete-time systems with interval time-varying delays, IET Control Theory Appl, 9, 12, 1839-1845 (2015) · doi:10.1049/iet-cta.2014.0886 |
| [29] | Peng, D.; Nie, H., Stabilisation for 2-D discrete-time switched nonlinear systems with mixed time-varying delays under all modes unstable, Int J Control Autom Syst, 53, 4, 757-777 (2022) · Zbl 1498.93662 |
| [30] | Peng, D.; Xu, H., Quantized feedback control for 2D uncertain nonlinear systems with time-varying delays in a networked environment, Comput Appl Math, 41, 3, 1-28 (2022) · Zbl 1499.93034 · doi:10.1007/s40314-022-01803-8 |
| [31] | Ren, W.; Xiong, J., Robust filtering for 2-D discrete-time switched systems, IEEE Trans Autom Control, 66, 10, 4747-4760 (2021) · Zbl 1536.93935 · doi:10.1109/TAC.2020.3037056 |
| [32] | Roesser, R., A discrete state-space model for linear image processing, IEEE Trans Autom Control, 20, 1, 1-10 (1975) · Zbl 0304.68099 · doi:10.1109/TAC.1975.1100844 |
| [33] | Takagi, T.; Sugeno, M., Fuzzy identification of systems and its applications to modeling and control, IEEE Trans Syst Man Cybern-Syst, 15, 1, 116-132 (1985) · Zbl 0576.93021 · doi:10.1109/TSMC.1985.6313399 |
| [34] | Wu, L.; Yang, R.; Shi, P.; Su, X., Stability analysis and stabilization of 2-D switched systems under arbitrary and restricted switchings, Automatica, 59, 206-215 (2015) · Zbl 1326.93110 · doi:10.1016/j.automatica.2015.06.008 |
| [35] | Xu, H.; Xu, S.; Lam, J., Positive real control for 2-D discrete delayed systems via output feedback controllers, J Comput Appl Math, 216, 1, 87-97 (2008) · Zbl 1330.93194 · doi:10.1016/j.cam.2007.04.014 |
| [36] | Yang, R.; Xie, L.; Zhang, C., \({H_2}\) and mixed \({H_2 }/{H_\infty }\) control of two-dimensional systems in Roesser model, Automatica, 42, 9, 1507-1514 (2006) · Zbl 1128.93327 · doi:10.1016/j.automatica.2006.04.002 |
| [37] | Yu, Q.; Yan, J., A novel average dwell time strategy for stability analysis of discrete-time switched systems by T-S fuzzy modeling, J Comput Appl Math, 391 (2021) · Zbl 1459.93138 · doi:10.1016/j.cam.2020.113306 |
| [38] | Zhang, L.; Shi, P.; Boukas, EK; Wang, C., \({H_\infty }\) control of switched linear discrete-time systems with polytopic uncertainties, Opt Control Appl Methods, 27, 5, 273-291 (2006) · doi:10.1002/oca.782 |
| [39] | Zhang, J.; Liu, D.; Ma, Y., Finite-time dissipative control of uncertain singular T-S fuzzy time-varying delay systems subject to actuator saturation, Comput Appl Math, 39, 3, 1-22 (2020) · Zbl 1448.93288 · doi:10.1007/s40314-020-01183-x |
| [40] | Zhang L, Wang Y, Yang H (2020) Observer-based memory state feedback control for switched fuzzy systems. In: 2020 Chinese Control And Decision Conference. IEEE, pp 1647-1652 |
| [41] | Zheng, W.; Wang, H.; Wang, H.; Wen, S., Stability analysis and dynamic output feedback controller design of T-S fuzzy systems with time-varying delays and external disturbances, J Comput Appl Math, 358, 111-135 (2019) · Zbl 1416.93175 · doi:10.1016/j.cam.2019.01.022 |
| [42] | Zhou, Z.; Li, L., Conservative domain decomposition schemes for solving two-dimensional heat equations, Comput Appl Math, 38, 1, 1-26 (2019) · Zbl 1438.65204 · doi:10.1007/s40314-019-0767-y |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
