Abstract
This paper puts forward an augmented Lyapunov–Krasovskii functional (LKF) method to investigate the robust \({{H}_{\infty }}\) control problem of uncertain Takagi–Sugeno fuzzy systems (TSFSs) with delays which includes delay-product-type terms. On the basis of the new augmented LKF and improved boundary techniques, the sufficient asymptotic stability condition is derived in this paper. And the robust \({{H}_{\infty }}\) controller is further designed, which is able to ensure that the fuzzy time delay system (TDS) has the specified \({{H}_{\infty }}\) performance index. Subsequently, four numerical examples are presented to illustrate the effectiveness and advantage of the put forward methodology.
Similar content being viewed by others
References
Cai X, Zhong S, Wang J, Shi K (2020) Robust \({{H}_{\infty }}\) control for uncertain delayed T-S fuzzy systems with stochastic packet dropouts. Appl Math Comput 385:125432
Datta R, Dey R, Bhattacharya B, Saravanakumar R, Kwon OM (2020) Stability and stabilization of T-S fuzzy systems with variable delays via new Bessel-Legendre polynomial based relaxed integral inequality. Inf Sci 522:99–123
Elias LJ, Faria FA, Araujo R, Oliveira VA (2021) Stability analysis of Takagi-Sugeno systems using a switched fuzzy Lyapunov function. Inf Sci 543:43–57
Feng Z, Zheng W (2017) Improved stability condition for Takagi-Sugeno fuzzy systems with time-varying delay. IEEE Trans Cybern 47(3):661–670
Ge C, Shi Y, Park JH, Hua C (2019) Robust \({{H}_{\infty }}\) stabilization for T-S fuzzy systems with time-varying delays and memory sampled-data control. Appl Math Comput 346:500–512
Jia T, Pan Y, Liang H, Lam HK (2021) Event-Based adaptive fixed-time fuzzy control for active vehicle suspension systems with time-varying displacement constraint. IEEE Trans Fuzzy Syst. https://doi.org/10.1109/TFUZZ.2021.3075490
Kwon OM, Park MJ, Lee SM, Park JH (2012) Augmented Lyapunov-Krasovskii functional approaches to robust stability criteria for uncertain Takagi-Sugeno fuzzy systems with time-varying delays. Fuzzy Sets Syst 201:1–19
Kwon OM, Park MJ, Park JH, Lee SM (2016) Stability and stabilization of T-S fuzzy systems with time-varying delays via augmented Lyapunov-Krasovskii functionals. Inf Sci 372:1–15
Lee SH, Selvaraj P, Park MJ, Kwon OM (2020) Improved results on \({{H}_{\infty }}\) stability analysis of sampled-data systems via looped-functionals and zero equalities. Appl Math Comput 373:125003
Li R, Zhang Q (2018) Robust \({{H}_{\infty }}\) sliding mode observer design for a class of Takagi-Sugeno fuzzy descriptor systems with time-varying delay. Appl Math Comput 337:158–178
Li M, Shu F, Liu D, Zhong S (2018) Robust \({{H}_{\infty }}\) control of T-S fuzzy systems with input time-varying delays: A delay partitioning method. Appl Math Comput 321:209–222
Li Y, Liu L, Feng G (2020) Finite-Time \(\cal{H}_{\infty }\) controller synthesis of T-S Fuzzy systems. IEEE Trans Syst Man Cybern 50(5):1956–1963
Lian Z, He Y, Zhang C, Wu M (2017) Further robust stability analysis for uncertain Takagi-Sugeno fuzzy systems with time-varying delay via relaxed integral inequality. Inf Sci 409–410:139–150
Lian Z, He Y, Wu M (2021) Stability and stabilization for delayed fuzzy systems via reciprocally convex matrix inequality. Fuzzy Sets Syst 402:124–141
Liang H, Liu G, Huang T, Lam HK, Wang B (2019) Cooperative fault-tolerant control for networks of stochastic nonlinear systems with nondifferential saturation nonlinearity. IEEE Trans Syste Man Cybern Syst. https://doi.org/10.1109/TSMC.2020.3020188
Lin H, Zeng H, Wang W (2021) New Lyapunov-Krasovskii functional for stability analysis of linear systems with time-varying delay. J Syst Sci Complex 34(2):632–641
Ma Y, Chen M (2016) Finite time non-fragile dissipative control for uncertain TS fuzzy system with time-varying delay. Neurocomputing 177:509–514
Ma Y, Chen M (2017) Memory feedback \({{H}_{\infty }}\) control of uncertain singular T-S fuzzy time-delay system under actuator saturation. Comput Appl Math 36:493–511
Mai VT, Dinh CH (2019) Robust finite-time stability and stabilization of a class of fractional-order switched nonlinear systems. J Syst Sci Complexity 32(6):1479–1497
Mrquez R, Guerra TM, Kruszewski A, Bernal M (2013) Improvements on non-quadratic stabilization of Takagi-Sugeno models via line-integral Lyapunov functions. IFAC Proc Vol 46(20):473–478
Park MJ, Kwon OM, Park JH, Lee SM, Cha EJ (2015) Stability of time-delay systems via Wirtinger-based double integral inequality. Automatica 55:204–208
Park PG, Lee WI, Lee SY (2015) Auxiliary function-based integral inequalities for quadratic functions and their applications to time-delay systems. J Franklin Inst 352(4):1378–1396
Peng C, Yue D, Tian Y (2009) New approach on robust delay-dependent \({{H}_{\infty }}\) control for uncertain T-S fuzzy systems with interval time-varying delay. IEEE Trans Fuzzy Syst 17(4):890–900
Sakthivel R, Shi P, Arunkumar A, Mathiyalagan K (2016) Robust reliable \({{H}_{\infty }}\) control for fuzzy systems with random delays and linear fractional uncertainties. Fuzzy Sets Syst 302:65–81
Seuret A, Gouaisbaut F (2013) Wirtinger-based integral inequality: Application to time-delay systems. Automatica 49(9):2860–2866
Seuret A, Gouaisbaut F (2018) Stability of linear Systems with time-varying delays using Bessel-Legendre inequalities. IEEE Trans Autom Control 63(1):225–232
Sun J, Liu G, Chen J, Rees D (2010) Improved delay-range-dependent stability criteria for linear systems with time-varying delays. Automatica 46(2):157–166
Takagi T, Sugeno M (1985) Fuzzy identification of systems and its applications to modeling and control. IEEE Trans Syst Man Cybern 15(1):116–132
Tan J, Dian S, Zhao T, Chen L (2018) Stability and stabilization of T-S fuzzy systems with time delay via Wirtinger-based double integral inequality. Neurocomputing 275:1063–1071
Tang P, Ma Y (2021) Exponential stabilization and non-fragile sampled-date dissipative control for uncertain time-varying delay T-S fuzzy systems with state quantization. Inf Sci 545:513–536
Wang L, Lam HK (2018) A new approach to stability and stabilization analysis for continuous-time Takagi-Sugeno fuzzy systems with time delay. IEEE Trans Fuzzy Syst 26(4):2460–2465
Wang G, Liu J, Lu S (2016) Stability analysis and stabilization for fuzzy hyperbolic time-delay system based on delay partitioning approach. Neurocomputing 214:555–566
Wang G, Jia R, Song H, Liu J (2018) Stabilization of unknown nonlinear systems with T-S fuzzy model and dynamic delay partition. J Intell Fuzzy Syst 35(2):2079–2090
Wang J, Xia J, Shen H, Xing M, Park JH (2021) \(\cal{H}_{\infty }\) aynchronization for fuzzy markov jump chaotic systems with piecewise-constant transition probabilities subject to PDT switching rule. IEEE Trans Fuzzy Syst 29:3082–3092
Wang J, Yang C, Xia J, Wu Z, Shen H (2021) Observer-based sliding mode control for networked fuzzy singularly perturbed systems under weighted try-once-discard protocol. IEEE Trans Fuzzy Syst. https://doi.org/10.1109/TFUZZ.2021.3070125
Wu M, He Y, She JH, Liu G (2004) Delay-dependent criteria for robust stability of time-varying delay systems. Automatica 40(8):1435–1439
Wu B, Chen M, Zhang L (2019) Disturbance-observer-based sliding mode control for T-S fuzzy discrete-time systems with application to circuit system. Fuzzy Sets Syst 374:138–151
Xu Y, Xie Z, Zhao J, Li W, Li P, Wong PK (2021) Robust non-fragile finite frequency \({{H}_{\infty }}\) control for uncertain active suspension systems with time-delay using T-S fuzzy approach. J Franklin Inst 358:4209–4238
Yang Z, Yang YP (2010) New delay-dependent stability analysis and synthesis of T-S fuzzy systems with time-varying delay. Int J Robust Nonlinear Control 20(3):313–322
Yang F, Guan S, Wang D (2014) Quadratically convex combination approach to stability of T-S fuzzy systems with time-varying delay. J Franklin Inst 351(7):3752–3765
Zeng H, Park JH, Xia J, Xiao S (2014) Improved delay-dependent stability criteria for T-S fuzzy systems with time-varying delay. Appl Math Comput 235:492–501
Zeng H, He Y, Wu M, She J (2015) Free-matrix-based integral inequality for stability analysis of systems with time-varying delay. IEEE Trans Autom Control 60(10):2768–2772
Zhang Z, Lin C, Chen B (2015) New stability and stabilization conditions for T-S fuzzy systems with time delay. Fuzzy Sets Syst 263:82–91
Zhang C, He Y, Jiang L, Wu M, Zeng H (2016) Delay-variation-dependent stability of delayed discrete-time systems. IEEE Trans Autom Control 61(9):2663–2669
Zhang C, He Y, Jiang L, Wu M (2016) Stability analysis for delayed neural networks considering both conservativeness and complexity. IEEE Trans Neural Netw Learn Syst 27(7):1486–1501
Zhang C, He Y, Jiang L, Wang Q, Min Wu (2017a) Stability analysis of discrete-time neural networks with time-varying delay via an extended reciprocally convex matrix inequality. IEEE Trans Cybern 47(10):3040–3049
Zhang C, He Y, Jiang L, Wu M, Wang Q (2017b) An extended reciprocally convex matrix inequality for stability analysis of systems with time-varying delay. Automatica 85:481–485
Zhang C, He Y, Jiang L, Wu M (2017c) Notes on stability of time-delay systems: bounding inequalities and augmented Lyapunov-Krasovskii functionals. IEEE Trans Autom Control 62(10):5331–5336
Zhang Y, Ma Y, Fu L, Zhao W, Huang X (2020) Finite-time non-fragile \({{H}_{\infty }}\) sampled-data control for uncertain T-S fuzzy system with time-varying delay and nonlinear perturbation subject to Markovian jump. ISA Trans 99:59–73
Zhang J, Liu D, Ma Y (2020) Finite-time dissipative control of uncertain singular T-S fuzzy time-varying delay systems subject to actuator saturation. Comput Appl Math. https://doi.org/10.1007/s40314-020-01183-x
Zhao X, Lin C, Chen B, Wang Q (2018) A novel Lyapunov-Krasovskii functional approach to stability and stabilization for T-S fuzzy systems with time delay. Neurocomputing 313:288–294
Zhou K, Huang T, Zhao T, Gao F (2018) Membership-function-dependent stability and stabilization conditions for T-C S fuzzy time-delay systems. IETE J Res 1–14
Acknowledgements
This work was supported by the National Natural Science Foundation of China No. 61273004, and the Natural Science Foundation of Hebei province No. F2021203061. The authors also gratefully acknowledge the helpful comments and suggestions of the reviewers, which have improved the presentation.
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Marcos Eduardo Valle.
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
This work is supported by National Natural Science Foundation of China (No. 61273004) and the Natural Science Foundation of Hebei province (No. F2021203061).
Rights and permissions
About this article
Cite this article
Mao, D., Ma, Y. Robust \({{H}_{\infty }}\) control for uncertain Takagi–Sugeno fuzzy systems with state and input time-varying delays. Comp. Appl. Math. 41, 195 (2022). https://doi.org/10.1007/s40314-022-01879-2
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/s40314-022-01879-2
Keywords
- State and input delays
- Robust \({{H}_{\infty }}\) control
- Takagi–Sugeno fuzzy systems (TSFSs)
- Time delay system (TDS)