Practical stability of a nonlinear system with delayed control input. (English) Zbl 1510.93234
Summary: The problem of delayed input control for a nonlinear system is discussed, where the nonlinearities of nonlinear systems are not assumed as Lipschitz continuous, they can be non-Lipschitz continuous or discontinuous in this paper. Notice that as a general nonlinear system, its sub-systems may have no common equilibrium or no equilibriums, but their trajectories may still be kept near equilibriums. Motivated by this, practical stability of nonlinear systems is considered by employing the Lyapuov method. Practical stability criteria in forms of linear matrix inequalities are obtained, where improved integral inequalities are given to reduce the conservatism of the obtained results. Finally, the obtained results are applied to analyze two problems of load frequency control of a one-area networked power system with sampled input and flight control of a two-degree-freedom helicopter system. The advantage and effectiveness of our approach are shown by a comparison with the literature.
MSC:
| 93D05 | Lyapunov and other classical stabilities (Lagrange, Poisson, \(L^p, l^p\), etc.) in control theory |
| 93C43 | Delay control/observation systems |
| 34K20 | Stability theory of functional-differential equations |
| 93C10 | Nonlinear systems in control theory |
Keywords:
nonlinear systems; delayed inputs; improved integral inequalities; practical stability; networked power system; helicopter systemReferences:
| [1] | Min, H.; Xu, S.; Zhang, B.; Ma, Q., Globally adaptive control for stochastic nonlinear time-delay systems with perturbations and its application, Automatica, 102, 105-110 (2019) · Zbl 1415.93146 |
| [2] | Liu, W.; Lu, J.; Xu, S.; Li, Y.; Zhang, Z., Sampled-data controller design and stability analysis for nonlinear systems with input saturation and disturbances, Appl. Math. Comput., 360, 14-27 (2019) · Zbl 1428.93071 |
| [3] | Min, H.; Xu, S.; Gu, J.; Zhang, Z., Adaptive finite-time stabilization of nonlinearly parameterized systems subject to mismatching disturbances, Int. J. Robust Nonlinear Control, 29, 11, 3469-3484 (2019) · Zbl 1426.93285 |
| [4] | Zhang, Z.; Xu, S.; Shen, H., Reduced-order observer-based output-feedback tracking control of nonlinear systems with state delay and disturbance, Int. J. Robust Nonlinear Control, 20, 15, 1723-1738 (2010) · Zbl 1204.93062 |
| [5] | Zhang, Z.; Xu, S.; Zhang, B., Asymptotic tracking control of uncertain nonlinear systems with unknown actuator nonlinearity, IEEE Trans. Autom. Control, 59, 5, 1336-1341 (2013) · Zbl 1360.93305 |
| [6] | Zhang, Z.; Xie, X. J., Asymptotic tracking control of uncertain nonlinear systems with unknown actuator nonlinearity and unknown gain signs, Int. J. Control, 87, 11, 2294-2311 (2014) · Zbl 1308.93118 |
| [7] | Zhang, Z.; Lu, J.; Gao, L.; Shao, H., Exponential synchronization of Genesio-Tesi chaotic systems with partially known uncertainties and completely unknown dead-zone nonlinearity, J. Franklin Inst., 350, 2, 347-357 (2013) · Zbl 1278.93119 |
| [8] | Zhang, Z.; Shen, H.; Li, Z.; Zhang, S., Zero-error tracking control of uncertain nonlinear systems in the presence of actuator hysteresis, Int. J. Syst. Sci., 46, 15, 2853-2864 (2015) · Zbl 1332.93189 |
| [9] | Zhang, Z.; Xu, S.; Zhang, B., Exact tracking control of nonlinear systems with time delays and dead-zone input, Automatica, 52, 272-276 (2015) · Zbl 1309.93089 |
| [10] | Zhang, Z.; Xu, S., Observer design for uncertain nonlinear systems with unmodeled dynamics, Automatica, 51, 80-84 (2015) · Zbl 1309.93032 |
| [11] | Jia, X.; Xu, S.; Jiao, T.; Chu, Y.; Zou, Y., Global state regulation by output feedback for feedforward systems with input and output dependent incremental rate, J. Franklin Inst., 352, 6, 2526-2538 (2015) · Zbl 1395.93480 |
| [12] | Jia, X.; Xu, S.; Cui, G.; Zhang, B.; Ma, Q., Global adaptive regulation of feedforward nonlinear time-delay systems by output feedback, Int. J. Robust Nonlinear Control, 27, 14, 2451-2472 (2017) · Zbl 1373.93177 |
| [13] | Jia, X.; Chen, X.; Xu, S.; Zhang, B.; Zhang, Z., Adaptive output feedback control of nonlinear time-delay systems with application to chemical reactor systems, IEEE Trans. Ind. Electron., 64, 6, 4792-4799 (2017) |
| [14] | Jia, X.; Xu, S.; Qi, Z.; Zhang, Z.; Chu, Y., Adaptive output feedback tracking of nonlinear systems with uncertain nonsymmetric dead-zone input, ISA Trans., 35-44 (2019) |
| [15] | Jia, X.; Xu, S.; Ma, Q.; Qi, Z.; Zou, Y., Global practical tracking by output feedback for a class of non-linear time-delay systems, IMA J. Math. Control Inf., 33, 4, 1067-1080 (2016) · Zbl 1397.93081 |
| [16] | Jia, X.; Xu, S.; Chen, J.; Li, Z.; Zou, Y., Global output feedback practical tracking for time-delay systems with uncertain polynomial growth rate, J. Franklin Inst., 352, 12, 5551-5568 (2015) · Zbl 1395.93300 |
| [17] | Ji, H.; Cui, B.; Liu, X., Networked sampled-data control of distributed parameter systems via distributed sensor networks, Commun. Nonlinear Sci. Numer. Simul., 98, 1, 105773 (2021) · Zbl 1464.93046 |
| [18] | Zhang, H.; Liu, J.; Xu, S., H-infinity load frequency control of networked power systems via an event-triggered scheme, IEEE Trans. Ind.Electron. (2019) |
| [19] | Zhang, H.; Liu, J., Event-triggered fuzzy flight control of a two-degree-of-freedom helicopter system, IEEE Trans. Fuzzy Syst. (2020) |
| [20] | Xu, S.; Lam, J., Improved delay-dependent stability criteria for time-delay systems, IEEE Trans. on Autom. Control, 50, 3, 384-387 (2005) · Zbl 1365.93376 |
| [21] | Xu, S.; Lam, J.; Chen, T.; Zou, Y., A delay-dependent approach to robust \(h_\infty\) filtering for uncertain distributed delay systems, IEEE Trans. Signal Process., 53, 10, 3764-3772 (2005) · Zbl 1370.93109 |
| [22] | Xu, S.; Lam, J.; Zou, Y., New results on delay-dependent robust \(h_\infty\) control for systems with time-varying delays, Automatica, 42, 2, 343-348 (2006) · Zbl 1099.93010 |
| [23] | Xu, S.; Lam, J.; Mao, X., Delay-dependent \(h_\infty\) control and filtering for uncertain Markovian jump systems with time-varying delays, IEEE Trans. Circuits Syst. I Regul. Pap., 54, 9, 2070-2077 (2007) · Zbl 1374.93134 |
| [24] | Xu, S.; Lam, J., On equivalence and efficiency of certain stability criteria for time-delay systems, IEEE Trans. Automat. Control, 52, 1, 95-101 (2007) · Zbl 1366.93451 |
| [25] | Xu, S.; Lam, J., A survey of linear matrix inequality techniques in stability analysis of delay systems, Int. J. Syst. Sci., 39, 12, 1095-1113 (2008) · Zbl 1156.93382 |
| [26] | Xu, S.; Lam, J.; Zhang, B.; Zou, Y., New insight into delay-dependent stability of time-delay systems, Int. J. Robust Nonlinear Control, 25, 7, 961-970 (2015) · Zbl 1312.93077 |
| [27] | Chen, J.; Xu, S.; Zhang, B., Single/multiple integral inequalities with applications to stability analysis of time-delay systems, IEEE Trans. Autom. Control, 62, 7, 3488-3493 (2016) · Zbl 1370.34128 |
| [28] | Chen, J.; Xu, S.; Chen, W.; Zhang, B.; Ma, Q.; Zou, Y., Two general integral inequalities and their applications to stability analysis for systems with time-varying delay, Int. J. Robust Nonlinear Control, 26, 18, 4088-4103 (2016) · Zbl 1351.93103 |
| [29] | Chen, J.; Xu, S.; Zhang, B.; Liu, G., A note on relationship between two classes of integral inequalities, IEEE Trans. Autom. Control, 62, 8, 4044-4049 (2016) · Zbl 1373.26025 |
| [30] | Chen, J.; Park, J. H.; Xu, S., Stability analysis for neural networks with time-varying delay via improved techniques, IEEE Trans. Syst. Man Cybern., 49, 12, 4495-4500 (2019) |
| [31] | Wang, Y.; Karimi, H. R.; Lam, H. K.; Shen, H., An improved result on exponential stabilization of sampled-data fuzzy systems, IEEE Trans. Fuzzy Syst., 26, 6, 3875-3883 (2018) |
| [32] | Kim, J. H., Further improvement of Jensen inequality and application to stability of time-delayed systems, Automatica, 64, 121-125 (2016) · Zbl 1329.93123 |
| [33] | Seuret, A.; Gouaisbaut, F., Stability of linear systems with time-varying delays using Bessel-Legendre inequalities, IEEE Trans. Autom. Control, 63, 1, 225-232 (2017) · Zbl 1390.34213 |
| [34] | Seuret, A.; Gouaisbaut, F., Wirtinger-based integral inequality: application to time-delay systems, Automatica, 49, 9, 2860-2866 (2013) · Zbl 1364.93740 |
| [35] | Liu, B.; Lu, W.; Chen, T., Generalized Halanay inequalities and their applications to neural networks with unbounded time-varying delays, IEEE Trans. Neural Netw., 22, 9, 1508-1513 (2011) |
| [36] | Park, P.; Ko, J. W.; Jeong, C., Reciprocally convex approach to stability of systems with time-varying delays, Automatica, 47, 1, 235-238 (2011) · Zbl 1209.93076 |
| [37] | Yu, G.; Chen, S. H., Nonlinear flight control of a two-degree-of-freedom helicopter using Takagi-Sugeno fuzzy model, IEEE International Conference on Systems Man and Cybernetics Conference Proceedings, 4354-4359 (2006) |
| [38] | Ma, Q.; Xu, S., Consensus switching of second-order multiagent systems with time delay, IEEE Trans. Cybern. (2020) |
| [39] | Ma, Q.; Xu, S., Exact delay bounds of second-order multi-agent systems with input and communication delays: from algebra and geometric prospective, IEEE Trans. Circuits Syst. II Express Briefs (2021) |
| [40] | Ma, Q.; Xu, S., Consensus ability of first-order multiagent systems under distributed PID controller with time delay, IEEE Trans. Neural Netw. Learn. Syst. (2021) |
| [41] | Cui, G.; Yu, J.; Shi, P., Observer-based finite-time adaptive fuzzy control with prescribed performance for nonstrict-feedback nonlinear systems, IEEE Trans. Fuzzy Syst. (2020) |
| [42] | Cui, G.; Yu, J.; Wang, Q., Finite-time adaptive fuzzy control for MIMO nonlinear systems with input saturation via improved command-filtered backstepping, IEEE Trans. Syst. ManCybern. (2020) |
| [43] | Xie, X.; Zhou, Q.; Yue, D.; Li, H., Relaxed control design of discrete-time Takagi-Sugeno fuzzy systems: an event-triggered real-time scheduling approach, IEEE Trans. Syst. ManCybern. Syst. (2017) |
| [44] | Zeng, H. B.; Liu, X. G.; Wang, W., A generalized free-matrix-based integral inequality for stability analysis of time-varying delay systems, Appl. Math. Comput., 354, 1-8 (2019) · Zbl 1428.34112 |
| [45] | Zeng, H. B.; Teo, K. L.; He, Y., A new looped-functional for stability analysis of sampled-data systems, Automatica, 82, 328-331 (2017) · Zbl 1372.93153 |
| [46] | Zhang, C. K.; Long, F.; He, Y.; Yao, W.; Wu, M., A relaxed quadratic function negative-determination lemma and its application to time-delay systems, Automatica, 113, 108764 (2020) · Zbl 1440.93144 |
| [47] | Lin, H.; Zeng, H.; Wang, W., New Lyapunov-Krasovskii functional for stability analysis of linear systems with time-varying delay, J. Syst. Sci. Complexity, 34, 2, 632-641 (2020) · Zbl 1460.93073 |
| [48] | Zeng, H. B.; Zhai, Z. L.; He, Y.; Teo, K. L.; Wang, W., New insights on stability of sampled-data systems with time-delay, Appl. Math. Comput., 374, 125041 (2020) · Zbl 1433.93110 |
| [49] | Zeng, H. B.; Lin, H. C.; He, Y.; Teo, K. L.; Wang, W., Hierarchical stability conditions for time-varying delay systems via an extended reciprocally convex quadratic inequality, J. Franklin Inst., 357, 14, 9930-9941 (2020) · Zbl 1448.93257 |
| [50] | Zhang, C. K.; He, Y.; Jiang, L.; Wu, M.; Zeng, H. B., Stability analysis of systems with time-varying delay via relaxed integral inequalities, Syst. Control Lett., 92, 52-61 (2016) · Zbl 1338.93290 |
| [51] | Zeng, H. B.; Zhai, Z. L.; Wang, W., Hierarchical stability conditions of systems with time-varying delay, Appl. Math. Comput., 404, 6, 126222 (2021) · Zbl 1510.34165 |
| [52] | Zeng, H. B.; Lin, H. C.; He, Y.; Zhang, C. K.; Teo, K. L., Improved negativity condition for a quadratic function and its application to systems with time-varying delay, IET Control Theory Appl., 14, 18, 2989-2993 (2020) · Zbl 1542.93298 |
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.
