Abstract
The aim of this study is to offer a way for dealing with the stabilization of discrete delayed systems with missing measurements using an anti-windup compensator. Linear matrix inequalities are used to establish a set of sufficient conditions based on the Lyapunov–Krasovskii function that ensure the asymptotic stability of the system through the anti-windup controller and the performance of the \(H_\infty \) norm. A characterization of the stability domain is also given. Numerical examples are given to contrast with previous studies, and show the efficiency of our methodology in obtaining the largest estimated domain of attraction, upper bounds of the delay as well as the minimum value of performance attenuation \(H_{\infty }\), which means that our approach gives less conservative results than the other methods.
Similar content being viewed by others
Data Availability
Data sharing is not applicable to this article as no datasets were generated or analyzed during the current study.
References
Benzaouia, A., Mesquine, F., Hmamed, A., & Aoufoussi, H. (2006). Stability and control synthesis for discrete-time linear systems subject to actuator saturation by output feedback. Mathematical Problems in Engineering. https://doi.org/10.1155/MPE/2006/43970
Chandra Pal, V., & Negi, R. (2017). Robust output feedback control of 2D discrete systems with actuator saturation and time-varying delay. Transactions of the Institute of Measurement and Control, 39(11), 1673–1695. https://doi.org/10.1177/0142331216644045
Chandra Pal, V., & Negi, R. (2018). \({H_\infty }\) Based anti-windup controller for two-dimensional discrete delayed systems in presence of actuator saturation. IMA Journal of Mathematical Control and Information, 35(2), 627–660. https://doi.org/10.1093/imamci/dnw067
Chen, Y., Fu, Z., Fei, S., & Song, S. (2020). Delayed anti-windup strategy for input-delay systems with actuator saturations. Journal of the Franklin Institute, 357(8), 4680–4696. https://doi.org/10.1016/j.jfranklin.2020.02.008
Chen, Y., Li, Y., & Fei, S. (2017). Anti-windup design for time-delay systems via generalised delay-dependent sector conditions. IET Control Theory and Applications, 11(10), 1634–1641. https://doi.org/10.1049/iet-cta.2016.0785
Chen, J., Lu, J., & Xu, S. (2016). Summation inequality and its application to stability analysis for time-delay systems. IET Control Theory and Applications, 10(4), 391–395. https://doi.org/10.1049/iet-cta.2015.0576
Chen, J., Park, J. H., Xu, S., & Zhang, X.-M. (2020). Stability of discrete-time systems with time-varying delay via a novel Lyapunov–Krasovskii functional. International Journal of Robust and Nonlinear Control, 30(12), 4779–4788. https://doi.org/10.1002/rnc.5009
Chen, J., Xu, S., Jia, X., & Zhang, B. (2016). Novel summation inequalities and their applications to stability analysis for systems with time-varying delay. IEEE Transactions on Automatic Control, 62(5), 2470–2475. https://doi.org/10.1109/TAC.2016.2606902
Chen, J., Xu, S., Ma, Q., Li, Y., Chu, Y., & Zhang, Z. (2017). Two novel general summation inequalities to discrete-time systems with time-varying delay. Journal of the Franklin Institute, 354(13), 5537–5558. https://doi.org/10.1016/j.jfranklin.2017.06.008
Darouiche, F. Z., & Tissir, E. H. (2022). Design of robust \({H_\infty }\) filtering controller for discrete-time-varying delay systems with missing measurements. International Journal of Dynamics and Control, 10, 1–12. https://doi.org/10.1007/s40435-022-01084-3
Darouiche, F. Z., & Tissir, E. H. (2023). \({H_\infty }\) filtering controller for discrete time-varying delay system with missing measurements. Journal of Circuits, Systems and Computers, 32(9), 1673–1695. https://doi.org/10.1142/S0218126623501463
El Fezazi, N., El Haoussi, F., Tissir, E. H., & Tadeo, F. (2015). Delay dependent anti-windup synthesis for time-varying delay systems with saturating actuators. International Journal of Computer Applications, 111(1), 1–6. https://doi.org/10.5120/19499-1107
El Fezazi, N., Tissir, E. H., & El Haoussi, F. (2018). Anti-windup design for state delayed discrete-time systems with input saturation. International Journal of Ecology and Development, 33(2), 145–155.
El Fezazi, N., Tissir, E. H., El Haoussi, F., Alvarez, T., & Tadeo, F. (2018). Control based on saturated time-delay systems theory of Mach number in wind tunnels. Circuits, Systems, and Signal Processing, 37(4), 1505–1522.
El Haoussi, F., & Tissir, E. H. (2007). An LMI-based approach for robust stabilization of time delay systems containing saturating actuators. IMA Journal of Mathematical Control and Information, 24(3), 347–356. https://doi.org/10.1093/imamci/dnl030
El Haoussi, F., & Tissir, E. H. (2010). Delay and its time-derivative dependent robust stability of uncertain neutral systems with saturating actuators. International Journal of Automation and Computing, 7(4), 455–462. https://doi.org/10.1007/s11633-010-0527-3
El Haoussi, F., Tissir, E. H., Tadeo, F., & Hmamed, A. (2013). Robust stabilization with saturating actuators of neutral and state delayed systems. International Journal of sciences and techniques of Automatic control and computer engineering, 7(1), 1878–1889.
Flores Jeferson, V., Da Silva, J. M. G., & Seuret, A. (2011). Static anti-windup synthesis for linear systems with time-varying input delays. IFAC Proceedings Volumes, 44(1), 14483–14488. https://doi.org/10.3182/20110828-6-IT-1002.01648
Ge, X., Hoi, K., & Vong, S. (2018). A delay-variation-dependent stability criterion for discrete-time systems via a bivariate quadratic function negative-determination lemma. Journal of the Franklin Institute, 359(10), 4976–4996. https://doi.org/10.1016/j.jfranklin.2022.04.023
He, Y., Liu, G.-P., Rees, D., & Wu, M. (2009). \(H_\infty \) filtering for discrete-time systems with time-varying delay. Signal Processing, 89(3), 275–282. https://doi.org/10.1016/j.sigpro.2008.08.008
He, X., & Zhou, D. (2007). Robust \(H_\infty \) filtering for time-delay systems with missing measurements: A parameter-dependent approach. Journal of Control Theory and Applications, 5(4), 336–344. https://doi.org/10.1007/s11768-006-6095-y
Huaicheng, Y., Zhenzhen, S., Hao, Z., & Hongbo, S. (2012). Quantized \(H_\infty \) filtering for discrete-time networked systems with mixed delays and missing measurements. In: Proceedings of the 31st Chinese control conference, pp. 5966–5971.
Hu, T., Lin, Z., & Chen, B. M. (2002). An analysis and design method for linear systems subject to actuator saturation and disturbance. Automatica, 38(2), 351–359. https://doi.org/10.1016/S0005-1098(01)00209-6
Khallouk, H., & Mesquine, F. (2019). Output feedback control for discrete-time systems with actuators saturation. 2019 8th International conference on systems and control (ICSC), pp. 223–228. https://doi.org/10.1109/ICSC47195.2019.8950542.
Kundu, J., & Negi, R. (2012). Stability analysis of discrete time delay systems with actuator saturation. 2nd International conference on power, control and embedded systems, pp. 1–5. https://doi.org/10.1109/ICPCES.2012.6508117.
Lamrabet, O., Naamane, K., Tissir, E. H., El Haoussi, F., & Tadeo, F. (2020). An input-output approach to anti-windup design for sampled-data systems with time-varying delay. Circuits Systems and Signal Processing, 39(10), 4868–4889. https://doi.org/10.1007/s00034-020-01414-w
Lamrabet, O., Tissir, E. H., & El Haoussi, F. (2019). Anti-windup compensator synthesis for sampled-data delay systems. Circuits Systems and Signal Processing, 38(5), 2055–2071. https://doi.org/10.1007/s00034-018-0971-9
Liu, P.-L. (2011). Delay-dependent stabilization for linear time-delay uncertain systems with saturating actuators. International Journal of General Systems, 40(3), 301–312. https://doi.org/10.1080/03081079.2010.542414
Liu, Y., Alsaadi, F. E., Yin, X., & Wang, Y. (2015). Robust \(H_\infty \) filtering for discrete nonlinear delayed stochastic systems with missing measurements and randomly occurring nonlinearities. International Journal of General Systems, 44(2), 169–181.
Liu, X.-G., Wang, F.-X., & Tang, M.-L. (2017). Auxiliary function-based summation inequalities and their applications to discrete-time systems. Automatica, 78, 211–215. https://doi.org/10.1016/j.automatica.2016.12.036
Manitius, A. Z. (1984). Feedback controllers for a wind tunnel model involving a delay: Analytical design and numerical simulation. IEEE Transactions on Automatic Control, 29(12), 1058–1068.
Naamane, K., Chaibi, R., Tissir, E. H., & Hmamed, A. (2017). Stabilization of discrete-time TS fuzzy systems with saturating actuators. In 2017 International conference on advanced technologies for signal and image processing, pp. 1–5. https://doi.org/10.1007/s00034-021-01849-9.
Naamane, K., & Tissir, E. H. (2019). Improved delay dependent stability of nonlinear quadratic T-S fuzzy systems. Journal of Circuits, Systems and Computers, 29(9), 2050134. https://doi.org/10.1142/S0218126620501340
Naamane, K., & Tissir, E. H. (2022). Robust anti-windup controller design for Takagi–Sugeno fuzzy systems with time-varying delays and actuator saturation. Circuits, Systems, and Signal Processing, 41(3), 1426–1452. https://doi.org/10.1007/s00034-021-01849-9
Nam, P. T., Trinh, H., & Pathirana, P. N. (2015). Discrete inequalities based on multiple auxiliary functions and their applications to stability analysis of time-delay systems. Journal of the Franklin Institute, 352(12), 5810–5831. https://doi.org/10.1016/j.jfranklin.2015.09.018
Negi, R., Purwar, S., & Kar, H. (2012). Delay-dependent stability analysis of discrete time delay systems with actuator saturation. Intelligent Control and Automation, 3(1), 34–43. https://doi.org/10.1016/j.jfranklin.2015.09.018
Nguyen, A., Dequidt, A., & Dambrine, M. (2015). Simultaneous LMI-based design of dynamic output feedback controller and anti-windup compensator for constrained Takagi-Sugeno fuzzy systems subject to persistent disturbances. 2015 IEEE international conference on fuzzy systems (FUZZ-IEEE), pp. 1–7. https://doi.org/10.1109/FUZZ-IEEE.2015.7337874.
Pal, V. C., Negi, R., & Zhu, Q. (2019). Stabilization of discrete-time delayed systems in presence of actuator saturation based on Wirtinger inequality. Mathematical Problems in Engineering. https://doi.org/10.1155/2019/5954642
Qiu, S.-B., Liu, X.-G., Wang, F.-X., & Chen, Q. (2019). Stability and passivity analysis of discrete-time linear systems with time-varying delay. Systems and Control Letters, 134, 104543. https://doi.org/10.1016/j.sysconle.2019.104543
Ran, M., Wang, Q., & Dong, C. (2016). Stabilization of a class of nonlinear systems with actuator saturation via active disturbance rejection control. Automatica, 63, 302–310. https://doi.org/10.1016/j.automatica.2015.10.010
Rotondo, D., & Buciakowski, M. (2021). Guaranteed cost estimation and control for a class of nonlinear systems subject to actuator saturation. European Journal of Control, 61, 119–132. https://doi.org/10.1016/j.ejcon.2021.07.002
Tarbouriech, S., Da Silva, J. M. G., & Garcia, G. (2004). Delay-dependent anti-windup strategy for linear systems with saturating inputs and delayed outputs. International Journal of Robust and Nonlinear Control, 14(7), 665–682. https://doi.org/10.1002/rnc.899
Tian, E., Yue, D., & Wei, G. (2013). Robust \(H_\infty \) filter for discrete-time linear system with uncertain missing measurements and non-linearity. IET Signal Processing, 7(3), 239–248. https://doi.org/10.1049/iet-spr.2012.0029
Ting, C.-S., & Chang, Y.-N. (2011). Robust anti-windup controller design of time-delay fuzzy systems with actuator saturations. Information Sciences, 181(15), 3225–3245. https://doi.org/10.1016/j.ins.2011.03.015
Tissir, E. H. (2007). Delay-dependent robust stability of linear systems with non commensurate time-varying delays. International Journal of Systems Sciences, 38(9), 749–757. https://doi.org/10.1080/00207720701597415
Wang, Z., Ho, D. W. C., Liu, Y., & Liu, X. (2009). Robust \(H_\infty \) control for a class of nonlinear discrete time-delay stochastic systems with missing measurements. Automatica, 45(3), 684–691.
Wang, Z., Yang, F., Ho, D. W. C., & Liu, X. (2006). Robust \(H_\infty \) filtering for stochastic time-delay systems with missing measurements. IEEE Transactions on Signal Processing, 54(7), 2579–2587.
Xiao-Na, S., Zhu-Mu, F., & Lei-Po, L. (2012). Robust stabilization of state delayed discrete-time Takagi–Sugeno fuzzy systems with input saturation via an anti-windup fuzzy design. Chinese Physics B, 21(11), 118–701. https://doi.org/10.1088/1674-1056/21/11/118701
Xiao, S., Xu, L., Zeng, H.-B., & Teo, K. L. (2018). Improved stability criteria for discrete-time delay systems via novel summation inequalities. International Journal of Control, Automation and Systems, 16(4), 1592–1602. https://doi.org/10.1007/s12555-017-0279-8
You, J., Gao, H., & Basin, M. V. (2013). Further improved results on \(H_\infty \) filtering for discrete time-delay systems. Signal Processing , 93(7), 1845–1852. https://doi.org/10.1016/j.sigpro.2013.01.021
Zabari, A., Tissir, E. H., & Kririm, S. (2016). Delay dependent robust \(H_\infty \) Filter design for discrete time-delay systems with missing measurements via homogeneous polynomial matrices. International Journal of Automation and Smart Technology, 6(3), 163–175. https://doi.org/10.5875/ausmt.v6i3.1112
Zabari, A., Tissir, E. H., & Tadeo, F. (2017). Delay-dependent robust \(H_\infty \) filtering with lossy measurements for discrete-time systems. Arabian Journal for Science and Engineering, 42(12), 5263–5273. https://doi.org/10.1007/s13369-017-2608-x
Zhang, C.-K., He, Y., Jiang, L., & Wu, M. (2016). An improved summation inequality to discrete-time systems with time-varying delay. Automatica, 74, 10–15. https://doi.org/10.1016/j.automatica.2016.07.040
Zhang, X.-M., Wu, M., She, J.-H., & He, Y. (2005). Delay-dependent stabilization of linear systems with time-varying state and input delays. Automatica, 41(8), 1405–1412. https://doi.org/10.1016/j.automatica.2005.03.009
Zhang, B., Xu, S., & Zou, Y. (2008). Improved stability criterion and its applications in delayed controller design for discrete-time systems. Automatica, 44(11), 2963–2967. https://doi.org/10.1016/j.automatica.2008.04.017
Zhou, B. (2013). Analysis and design of discrete-time linear systems with nested actuator saturations. Systems and Control Letters, 62(10), 871–879. https://doi.org/10.1016/j.sysconle.2013.06.012
Zhou, B., Zheng, W. X., & Duan, G.-R. (2011). Stability and stabilization of discrete-time periodic linear systems with actuator saturation. Automatica, 47(8), 1813–1820. https://doi.org/10.1016/j.automatica.2011.04.015
Zong, G., & Hou, L. (2010). New delay-dependent stability result and its application to robust performance analysis for discrete-time systems with delay. IMA Journal of Mathematical Control and Information, 27(3), 373–386. https://doi.org/10.1093/imamci/dnq016
Zuo, Z., Li, Y., Wang, Y., & Li, H. (2018). Event-triggered control for switched systems in the presence of actuator saturation. International Journal of Systems Science, 49(7), 1478–1490. https://doi.org/10.1080/00207721.2018.1454538
Funding
This research received no external funding.
Author information
Authors and Affiliations
Contributions
T.E.H. was involved in supervision, reviewing and editing; D.F.Z. wrote the original draft; and N.K. contributed to reviewing and editing.
Corresponding author
Ethics declarations
Conflict of interest
The authors declare that they have no conflict of interest.
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
Darouiche, F.Z., Tissir, E.H. & Naamane, K. Anti-windup Controller Design for State-Delayed Discrete-Time Systems with Missing Measurements. J Control Autom Electr Syst 34, 1109–1122 (2023). https://doi.org/10.1007/s40313-023-01031-3
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s40313-023-01031-3