# zbMATH — the first resource for mathematics

Regulation of linear systems with both pointwise and distributed input delays by memoryless feedback. (English) Zbl 1415.93204
Summary: This paper is concerned with the stabilization of linear systems with both pointwise and distributed input delays, which can be arbitrarily large yet exactly known. The state vector used in the well-known Artstein transformation is firstly linked with the future state of the system. Pseudo-predictor feedback (PPF) approaches are then established to design memory stabilizing controllers. Necessary and sufficient conditions guaranteeing the stability of the closed-loop system are established in terms of the stability of some integral delay systems. Furthermore, since the PPF still is infinite-dimensional state feedback law and may cause difficulties in their practical implementation, truncated pseudo-predictor feedback (TPPF) approaches are established to design finite dimensional (memoryless) controllers. It is shown that the pointwise and distributed input delays can be compensated properly by the TPPF as long as the open-loop system is polynomially unstable. Finally, two numerical examples, one of which is the spacecraft rendezvous control system, are carried out to support the obtained theoretical results.

##### MSC:
 93D15 Stabilization of systems by feedback 93C05 Linear systems in control theory
DDE-BIFTOOL
Full Text:
##### References:
 [1] Artstein, Z., Linear systems with delayed controls: a reduction, IEEE Trans. Autom. Control, 27, 4, 869-879, (1982) · Zbl 0486.93011 [2] Bekiaris-Liberis, N.; Krstic, M., Lyapunov stability of linear predictor feedback for distributed input delays, IEEE Trans. Autom. Control, 56, 3, 655-660, (2011) · Zbl 1368.93527 [3] Bekiaris-Liberis, N.; Krstic, M., Predictor-feedback stabilization of multi-input nonlinear systems, IEEE Trans. Autom. Control, 62, 2, 516-531, (2017) · Zbl 1364.93620 [4] Cacace, F.; Germani, A., Output feedback control of linear systems with input, state and output delays by chains of predictors, Automatica, 85, 455-461, (2017) · Zbl 1375.93053 [5] Cacace, F.; Germani, A.; Manes, C., Exponential stabilization of linear systems with time-varying delayed state feedback via partial spectrum assignment, Syst. Control Lett., 69, 47-52, (2014) · Zbl 1288.93065 [6] Carter, T. E., State transition matrices for terminal rendezvous studies: brief survey and new example, AIAA J. Guid. Control Dyn., 21, 1, 148-155, (1998) · Zbl 0926.70031 [7] K. Engelborghs, T. Luzyanina, G. Samaey, DDE-BIFTOOL v.2.00: a matlab package for bifurcation analysis of delay differential equation, 2001, T.W. Reports, 330. Department of Computer Science, K. U. Leuven. Berlin: Springer. [8] Fridman, E.; Shaikhet, L., Stabilization by using artificial delays: an LMI approach, Automatica, 81, 429-437, (2017) · Zbl 1417.93252 [9] Goebel, G.; Münz, U.; Allgöwer, F., Stabilization of linear systems with distributed input delay, Proceedings of the American Control Conference (ACC), 5800-5805, (2010) [10] Gu, K.; Kharitonov, V. L.; Chen, J., Stability of Time-Delay Systems, (2003), Springer Science & Business Media · Zbl 1039.34067 [11] Hu, M. J.; Xiao, J. W.; Xiao, R. B.; Chen, W. H., Impulsive effects on the stability and stabilization of positive systems with delays, J. Frankl. Inst., 354, 10, 4034-4054, (2017) · Zbl 1367.93560 [12] Jankovic, M., Recursive predictor design for state and output feedback controllers for linear time delay systems, Automatica, 46, 3, 510-517, (2010) · Zbl 1194.93077 [13] Kharitonov, V. L., An extension of the prediction scheme to the case of systems with both input and state delay, Automatica, 50, 1, 211-217, (2014) · Zbl 1298.93196 [14] Kharitonov, V. L., Predictor based stabilization of neutral type systems with input delay, Automatica, 52, 125-134, (2015) · Zbl 1309.93129 [15] Krstic, M., Input delay compensation for forward complete and strict-feedforward nonlinear systems, IEEE Trans. Autom. Control, 55, 2, 287-303, (2010) · Zbl 1368.93546 [16] Kwon, W. H.; Pearson, A. E., Feedback stabilization of linear systems with delayed control, IEEE Trans. Autom. Control, 25, 2, 266-269, (1980) · Zbl 0438.93055 [17] Li, X.; Ding, Y., Razumikhin-type theorems for time-delay systems with persistent impulses, Syst. Control Lett., 107, 22-27, (2017) · Zbl 1376.93088 [18] Li, X.; Song, S., Stabilization of delay systems: delay-dependent impulsive control, IEEE Trans. Autom. Control, 62, 1, 406-411, (2017) · Zbl 1359.34089 [19] Liu, Q.; Zhou, B., Delay compensation of discrete-time linear systems by nested prediction, IET Control Theory Appl., 10, 15, 1824-1834, (2016) [20] Liu, Q.; Zhou, B., Extended observer based feedback control of linear systems with both state and input delays, J. Frankl. Inst., 354, 18, 8232-8255, (2017) · Zbl 1380.93218 [21] Léchappé, V.; Moulay, E.; Plestan, F.; Glumineau, A.; Chriette, A., New predictive scheme for the control of LTI systems with input delay and unknown disturbances, Automatica, 52, 179-184, (2015) · Zbl 1309.93059 [22] Manitius, A. Z.; Olbrot, A. W., Finite spectrum assignment problem for systems with delays, IEEE Trans. Autom. Control, 24, 4, 541-553, (1979) · Zbl 0425.93029 [23] Mazenc, F.; Malisoff, M., Stabilization and robustness analysis for time-varying systems with time-varying delays using a sequential subpredictors approach, Automatica, 82, 118-127, (2017) · Zbl 1376.93091 [24] Michiels, W.; Vyhlíal, T., An eigenvalue based approach for the stabilization of linear time-delay systems of neutral type, Automatica, 41, 6, 991-998, (2005) · Zbl 1091.93026 [25] Mondié, S.; Ochoa, G.; Ochoa, B., Instability conditions for linear time delay systems: a Lyapunov matrix function approach, Int. J. Control, 84, 10, 1601-1611, (2011) · Zbl 1236.93136 [26] Najafi, M.; Hosseinnia, S.; Sheikholeslam, F.; Karimadini, M., Closed-loop control of dead time systems via sequential sub-predictors, Int. J. Control, 86, 4, 599-609, (2013) · Zbl 1278.93122 [27] Ponomarev, A., Nonlinear predictor feedback for input-affine systems with distributed input delays, IEEE Trans. Autom. Control, 61, 9, 2591-2596, (2016) · Zbl 1359.93208 [28] Sun, X. M.; Liu, G. P.; Rees, D.; Wang, W., Stability of systems with controller failure and time-varying delay, IEEE Trans. Autom. Control, 53, 10, 2391-2396, (2008) · Zbl 1367.93573 [29] Van Assche, V.; Dambrine, M.; Lafay, J. F.; Richard, J. P., Some problems arising in the implementation of distributed-delay control laws, Proceedings of the 38th IEEE Conference on Decision and Control, 4668-4672, (1999) [30] Verriest, E. I., Linear systems with rational distributed delay: reduction and stability, Proceedings of the European Control Conference (ECC), 3637-3642, (1999) [31] Park, M. J.; Kwon, O. M.; Ryu, J. H., Advanced stability criteria for linear systems with time-varying delays, J. Frankl. Inst., 355, 1, 520-543, (2018) · Zbl 1380.93189 [32] Qian, W.; Yuan, M.; Wang, L.; Chen, Y.; Yang, J., Robust stability criteria for uncertain systems with interval time-varying delay based on multi-integral functional approach, J. Frankl. Inst., 355, 849-861, (2018) · Zbl 1384.93114 [33] Xu, S.; Chen, T., An LMI approach to the h$$_{∞}$$ filter design for uncertain systems with distributed delays, IEEE Trans. Circuits Syst.-II: Express Briefs, 51, 4, 195-201, (2004) [34] Yang, X.; Li, X.; Cao, J., Robust finite-time stability of singular nonlinear systems with interval time-varying delay, J. Frankl. Inst., 355, 1241-1258, (2018) · Zbl 1393.93101 [35] Zhang, X. M.; Wu, M.; She, J. H.; He, Y., Delay-dependent stabilization of linear systems with time-varying state and input delays, Automatica, 41, 8, 1405-1412, (2005) · Zbl 1093.93024 [36] Zhang, B.; Lam, J.; Xu, S., Relaxed results on reachable set estimation of time-delay systems with bounded peak inputs, Int. J. Robust Nonlinear Control, 26, 9, 1994-2007, (2016) · Zbl 1342.93021 [37] Zhao, N.; Zhang, X.; Xue, Y.; Shi, P., Necessary conditions for exponential stability of linear neutral type systems with multiple time delays, J. Frankl. Inst., 355, 1, 458-473, (2018) · Zbl 1380.93215 [38] Zhou, B., Pseudo-predictor feedback stabilization of linear systems with time-varying input delays, Automatica, 50, 11, 2861-2871, (2014) · Zbl 1300.93141 [39] Zhou, B., Input delay compensation of linear systems with both state and input delays by nested prediction, Automatica,, 50, 5, 1434-1443, (2014) · Zbl 1296.93168 [40] Zhou, B.; Cong, S., Stabilisation and consensus of linear systems with multiple input delays by truncated pseudo-predictor feedback, Int. J. Syst. Sci., 47, 2, 328-342, (2016) · Zbl 1333.93203 [41] Zhou, B.; Lin, Z.; Duan, G. R., Truncated predictor feedback for linear systems with long time-varying input delays, Automatica, 48, 10, 2387-2399, (2012) · Zbl 1271.93123
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.