|Refined Jensen-Based Multiple Integral Inequality and Its Application to Stability of Time-Delay Systems
|Jidong Wang1,2, Zhanshan Wang2, Sanbo Ding3, Huaguang Zhang4
|1. College of Information Science and Engineering, Northeastern University, Shenyang 110819, China, and also with the State Key Laboratory of Synthetical Automation for Process Industries, Shenyang 110819, China;
2. College of Electrical Engineering, North China University of Water Resources and Electric Power, Zhengzhou 450011, China
|  S. L. Niculescu, Delay Effects on Stability:A Robust Control Approach. London, UK:Springer-Verlag, 2001.
 K. Q. Gu, V. L. Kharitonov, and J. Chen, Stability of Time-Delay Systems. Cambridge, MA, USA:Birkhäuser, 2003.
 E. Fridman, "Tutorial on Lyapunov-based methods for time-delay systems," Eur. J. Control, vol. 20, no. 6, pp. 271-283, Nov. 2014.
 Z. S. Wang, Z. W. Liu, and C. D. Zheng, Qualitative Analysis and Control of Complex Neural Networks with Delays. Beijing, China:Science Press, 2015.
 M. M. S. Pasand and M. Montazeri, "Structural properties, LQG control and scheduling of a networked control system with bandwidth limitations and transmission delays," IEEE/CAA J. of Autom. Sinica, doi:10.1109/JAS.2017.7510373, 2017.
 Y. H. Sun, Y. X. Wang, Z. N. Wei, G. Q. Sun, and X. P. Wu, "Robust H∞ load frequency control of multi-area power system with time delay:A sliding mode control approach," IEEE/CAA J. of Autom. Sinica, vol. 5, no. 2, pp. 610-617, Mar. 2018.
 H. B. Zeng, Y. He, M. Wu, and J. H. She, "New results on stability analysis for systems with discrete distributed delay," Automatica, vol. 60, pp. 189-192, Oct. 2015.
 Z. S. Wang, L. Liu, Q. H. Shan, and H. G. Zhang, "Stability criteria for recurrent neural networks with time-varying delay based on secondary delay partitioning method," IEEE Trans. Neural Networks Learn. Syst., vol. 26, no. 10, pp. 2589-2595, Oct. 2015.
 Y. L. Jiang and C. D. Li, "Globally exponential stability of memristive neural networks with time-varying delays and synchronous switching," Acta Autom. Sinica, vol. 43, no. 8, pp. 1465-1469, Aug. 2017.
 X. D. Li and S. J. Song, "Stabilization of delay systems:delay-dependent impulsive control," IEEE Trans. Autom. Control, vol. 62, no. 1, pp. 406-411, Jan. 2017.
 X. D. Li and J. H. Wu, "Stability of nonlinear differential systems with state-dependent delayed impulses," Automatica, vol. 64, pp. 63-69, Feb. 2016.
 Q. L. Han, "A discrete delay decomposition approach to stability of linear retarded and neutral systems," Automatica, vol. 45, no. 2, pp. 517-524, Feb. 2009.
 Y. J. Liu, S. M. Lee, O. M. Kwon, and J. H. Park, "A study on H∞ state estimation of static neural networks with time-varying delays," Appl. Math. Comput., vol. 226, pp. 589-597, Jan. 2014.
 V. B. Kolmanovskii, "On the Liapunov-Krasovskii functionals for stability analysis of linear delay systems," Int. J. Control, vol. 72, no. 4, pp. 374-384, Mar. 1999.
 Y. Zhang, K. Lou, and Y. Ge, "New result on delay-dependent stability for Markovian jump time-delay systems with partial information on transition probabilities," IEEE/CAA J. of Autom. Sinica, doi:10.1109/JAS.2016.7510229, 2017.
 H. G. Zhang and Z. W. Liu, "Stability analysis for linear delayed systems via an optimally dividing delay interval approach," Automatica, vol. 47, no. 9, pp. 2126-2129, Sep. 2011.
 C. K. Zhang, Y. He, L. Jiang, M. Wu, and H. B. Zeng, "Stability analysis of systems with time-varying delay via relaxed integral inequalities," Syst. Control Lett., vol. 92, pp. 52-61, Jun. 2016.
 H. G. Zhang, Z. S. Wang, and D. R. Liu, "A comprehensive review of stability analysis of continuous-time recurrent neural networks," IEEE Trans. Neural Networks Learn. Syst., vol. 25, no. 7, pp. 1229-1262, Jul. 2014.
 L. Liu, Y. J. Liu, and C. L. P. Chen, "Adaptive neural network control for a DC motor system with dead-zone," Nonlinear Dyn., vol. 72, no. 1-2, pp. 141-147, Apr. 2013.
 Y. J. Liu and S. C. Tong, "Barrier Lyapunov functions for Nussbaum gain adaptive control of full state constrained nonlinear systems," Automatica, vol. 76, pp. 143-152, Feb. 2017.
 L. Liu, Z. S. Wang, and H. G. Zhang, "Adaptive fault-tolerant tracking control for MIMO discrete-time systems via reinforcement learning algorithm with less learning parameters," IEEE Trans. Autom. Sci. Eng., vol. 14, no. 1, pp. 299-313, Jan. 2017.
 Y. J. Liu, S. M. Lu, S. C. Tong, X. K. Chen, C. L. P. Chen, and D. J. Li, "Adaptive control-based Barrier Lyapunov Functions for a class of stochastic nonlinear systems with full state constraints," Automatica, vol. 87, pp. 83-93, Jan. 2018.
 A. Seuret and F. Gouaisbaut, "Wirtinger-based integral inequality:application to time-delay systems," Automatica, vol. 49, no. 9, pp. 2860-2866, Sep. 2013.
 H. B. Zeng, Y. He, M. Wu, and J. H. She, "Free-matrix-based integral inequality for stability analysis of systems with time-varying delay," IEEE Trans. Autom. Control, vol. 60, no. 10, pp. 2768-2772, Oct. 2015.
 L. Van Hien and H. Trinh, "Refined Jensen-based inequality approach to stability analysis of time-delay systems," IET Control Theory Appl., vol. 9, no. 14, pp. 2188-2194, Sep. 2015.
 A. Seuret, F. Gouaisbaut, and Y. Ariba, "Complete quadratic Lyapunov functionals for distributed delay systems," Automatica, vol. 62, pp. 168-176, Dec. 2015.
 M. Park, O. Kwon, J. H. Park, S. Lee, and E. Cha, "Stability of time-delay systems via Wirtinger-based double integral inequality," Automatica, vol. 55, pp. 204-208, May 2015.
 Z. S. Wang, S. B. Ding, Z. J. Huang, and H. G. Zhang, "Exponential stability and stabilization of delayed memristive neural networks based on quadratic convex combination method," IEEE Trans. Neural Network Learn. Syst., vol. 27, no. 11, pp. 2337-2350, Nov. 2016.
 Z. S. Wang, S. B. Ding, Q. H. Shan, and H. G. Zhang, "Stability of recurrent neural networks with time-varying delay via flexible terminal method," IEEE Trans. Neural Network Learn. Syst., vol. 28, no. 10, pp. 2456-2463, Oct. 2017.
 M. Fang and J. H. Park, "A multiple integral approach to stability of neutral time-delay systems," Appl. Math. Comput., vol. 224, pp. 714-718, Nov. 2013.
 S. B. Ding, Z. S. Wang, and H. G. Zhang, "Wirtinger-based multiple integral inequality for stability of time-delay systems," Int. J. Control, vol. 91, no. 1, pp. 12-18, Nov. 2018.
 Z. S. Wang, S. B. Ding, and H. G. Zhang, "Hierarchy of stability criterion for time-delay systems based on multiple integral approach," Appl. Math. Comput., vol. 314, pp. 422-428, Dec. 2017.
 J. Chen, S. Y. Xu, and B. Y. Zhang, "Single/Multiple integral inequalities with applications to stability analysis of time-delay systems," IEEE Trans. Autom. Control, vol. 62, no. 7, pp. 3488-3493, Jul. 2017.
 S. Boyd, L. El Ghaoui, E. Feron, and V. Balakrishnan, Linear Matrix Inequalities in System and Control Theory. Philadelphia, PA, USA:SIAM, 1994.