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October  2015, 11(4): 1343-1354. doi: 10.3934/jimo.2015.11.1343

## Optimal tracking control for networked control systems with random time delays and packet dropouts

 1 School of Computer Science, Xi'an Shiyou University, Xi'an, 710065, China 2 Department of Automation, Northwestern Polytechnical University, Xi'an, 710072, China, China

Received  October 2012 Revised  February 2015 Published  March 2015

This paper studies the problem of optimal output tracking control for networked control system with uncertain time delays and packet dropouts. Active time-varying sampling period strategy is proposed to ensure the random variable time delays always shorter than one sampling period. Hybrid driven modes are adopted by sensor to solve the issues of long time delay and packet dropout. By using augmentation approach, the tracking problem of this formulated within-one-step delayed discrete-time system is transformed into a general problem of non-delayed state linear quadratic regulator. A “gridding” approach is introduced to guarantee the realization of optimal output feedback control law by the solution of a series of Riccati matrix equations from an offline database that is constructed by different combination of time delays and packet dropouts. Simulation results demonstrate the effectiveness of the optimal tracking control law.
Citation: Ying Wu, Zhaohui Yuan, Yanpeng Wu. Optimal tracking control for networked control systems with random time delays and packet dropouts. Journal of Industrial and Management Optimization, 2015, 11 (4) : 1343-1354. doi: 10.3934/jimo.2015.11.1343
##### References:
 [1] N. Duan and X. J. Xie, Further results on output-feedback stabilization for a class of stochastic nonlinear systems, IEEE Transactions on Automatic Control, 56 (2011), 1208-1213. doi: 10.1109/TAC.2011.2107112. [2] D. M. Dawson, et al., Tracking control of rigid-link electrically-driven robot manipulators, International Journal of Control, 56 (1992), 991-1006. doi: 10.1080/00207179208934354. [3] H. J. Gao and T. W. Chen, Network-based $H_\infty$ output tracking control, IEEE Transactions on Automatic Control, 53 (2008), 655-667. doi: 10.1109/TAC.2008.919850. [4] L. Gollmann and H. Maurer, Theory and applications of optimal control problems with multiple time-delays, Journal of Industrial and Management Optimization, 10 (2014), 413-441. [5] S. Y. Han, et al., Near-Optimal Tracking Control for Discrete-time Systems with Delayed Input, International Journal of Control Automation and Systems, 8 (2010), 1330-1335. doi: 10.1007/s12555-010-0619-4. [6] C. J. Hou, Y. P. Chen and Z. L. Lu, Superconvergence property of finite element methods for parabolic optimal control problems, Journal of Industrial and Management Optimization, 7 (2011), 927-945. doi: 10.3934/jimo.2011.7.927. [7] K. Ji and W. J. Kim, Stabilization of networked control system with time delays and data-packet losses, European Journal of Control, 13 (2007), 343-350. doi: 10.3166/ejc.13.343-350. [8] B. Jayawardhana and G. Weiss, Tracking and disturbance rejection for fully actuated mechanical systems, Automatica, 44 (2008), 2863-2868. doi: 10.1016/j.automatica.2008.03.030. [9] S. Liu, L. Xie and H. S. Zhang, Linear quadratic tracking problem for discrete-time systems with multiple delays in single input channel, International Journal of Robust and Nonlinear Control, 20 (2010), 1379-1394. doi: 10.1002/rnc.1520. [10] M. Mauder, Robust tracking control of nonholonomic dynamic systems with application to the bi-steerable mobile robot, Automatica, 44 (2008), 2588-2592. doi: 10.1016/j.automatica.2008.02.012. [11] A. Naskali and A. Onat, Random network delay in model based predictive networked control systems, in Proc. $6^{nd}$ WSEAS Int. Conf. Appl. Comput. Sci., 1 (2006), 199-206. [12] A. Onat, et al., Control over imperfect networks: Model-based predictive networked control systems, IEEE Transactions on Industrial Electronics, 58 (2011), 905-913. doi: 10.1109/TIE.2010.2051932. [13] A. Onat and E. Parlakay, Implementation of networked predictive control system, in Proc. 9th Real-Time Linux Workshop, 1 (2007), 85-93. [14] T. Suzuki, et al., Controllability and stabilizability of a networked control system with periodic communication constraints, Systems & Control Letters, 60 (2011), 977-984. doi: 10.1016/j.sysconle.2011.08.004. [15] Y. Shi and B. Yu, Output feedback stabilization of networked control systems with random delays modeled by Markov chains, IEEE Transactions on Automatic Control, 54 (2009), 1668-1674. doi: 10.1109/TAC.2009.2020638. [16] A. Sala, Computer control under time-varying sampling period: An LMI gridding approach, Automatica, 41 (2005), 2077-2082. doi: 10.1016/j.automatica.2005.05.017. [17] B. O. S. Teixeira, et al., Spacecraft tracking using sampled-data Kalman filters - An illustrative application of extended and unscented estimators, IEEE Control Systems Magazine, 28 (2008), 78-94. doi: 10.1109/MCS.2008.923231. [18] Y. H. Wang, Z. M. Wang and Y. F. Zheng, On the model-based networked control for singularly perturbed systems, International Journal of Robust and Nonlinear Control, 6 (2008), 153-162. doi: 10.1007/s11768-008-6152-9. [19] J. Wu and T. W. Chen, Design of networked control systems with packet dropouts, IEEE Transactions on Automatic Control, 52 (2007), 1314-1319. doi: 10.1109/TAC.2007.900839. [20] H. H. Wang, Optimal tracking for Discrete-Time Systems with Input Delays, Proceedings of the 2008 Chinese Control and Decision Conference, (2008), 4033-4037. [21] Y. L. Wang and G. H. Yang, Output tracking control for networked control systems with time delay and packet dropout, International Journal of Control, 81 (2008), 1709-1719. doi: 10.1080/00207170701836944. [22] Y. Wang and G. Yang, Output tracking control for discrete-time networked control systems, IEEE American Control Conference, (2009), 5109-5114. doi: 10.1109/ACC.2009.5159974. [23] D. Wu, J. Wu and S. Chen, Robust $H_\infty$ control for networked control systems with uncertainties and multiple-packet transmission, IET Control Theory Appl., 4 (2010), 701-709. doi: 10.1049/iet-cta.2009.0090. [24] Y. L. Wang and G. H. Yang, Output tracking control for continuous-time networked control systems with communication constraints, Proc. of the American Control Conference, (2009), 531-536. doi: 10.1109/ACC.2009.5159975. [25] H. H. Wang and G. Y. Tang, Observer-based optimal output tracking for discrete-time systems with multiple state and input delays, International Journal of Control Automation and Systems, 7 (2009), 57-66. doi: 10.1007/s12555-009-0108-9. [26] H. W. Yu, X. M. Zhang, G. P. Lu and Y. F. Zheng, On model based networked control for singularly perturbed systems with nonlinear uncertainties, Joint 48th IEEE Conference on Decision and Control and 28th Chinese Control Conference, Shanghai, (2009), 684-689. doi: 10.1109/CDC.2009.5400103. [27] D. Yue, Q. L. Han and J. Lam, Network-based robust $H_\infty$ control of systems with uncertainty, Automatica, 41 (2005), 999-1007. doi: 10.1016/j.automatica.2004.12.011. [28] X. M. Zhang, et al., Stochastic stability of networked control systems with network-induced delay and data dropout, in Proceedings of the 45th Ieee Conference on Decision and Control, 14 (2006), 5006-5011. doi: 10.1109/CDC.2006.376970. [29] H. G. Zhang, J. Yang and C. Y. Su, T-S fuzzy-model-based robust $H_\infty$ design for networked control systems with uncertainties, IEEE Trans. Ind. Inf., 3 (2007), 289-301. [30] Q. Zhang, et al., An Enhanced LMI Approach for Mixed $H_2/H_\infty$ Flight Tracking Control, Chinese Journal of Aeronautics, 24 (2011), 324-328. [31] H. B. Zeng, et al., Absolute stability and stabilization for Lurie networked control systems, International Journal of Robust and Nonlinear Control, 21 (2011), 1667-1676. doi: 10.1002/rnc.1658.

show all references

##### References:
 [1] N. Duan and X. J. Xie, Further results on output-feedback stabilization for a class of stochastic nonlinear systems, IEEE Transactions on Automatic Control, 56 (2011), 1208-1213. doi: 10.1109/TAC.2011.2107112. [2] D. M. Dawson, et al., Tracking control of rigid-link electrically-driven robot manipulators, International Journal of Control, 56 (1992), 991-1006. doi: 10.1080/00207179208934354. [3] H. J. Gao and T. W. Chen, Network-based $H_\infty$ output tracking control, IEEE Transactions on Automatic Control, 53 (2008), 655-667. doi: 10.1109/TAC.2008.919850. [4] L. Gollmann and H. Maurer, Theory and applications of optimal control problems with multiple time-delays, Journal of Industrial and Management Optimization, 10 (2014), 413-441. [5] S. Y. Han, et al., Near-Optimal Tracking Control for Discrete-time Systems with Delayed Input, International Journal of Control Automation and Systems, 8 (2010), 1330-1335. doi: 10.1007/s12555-010-0619-4. [6] C. J. Hou, Y. P. Chen and Z. L. Lu, Superconvergence property of finite element methods for parabolic optimal control problems, Journal of Industrial and Management Optimization, 7 (2011), 927-945. doi: 10.3934/jimo.2011.7.927. [7] K. Ji and W. J. Kim, Stabilization of networked control system with time delays and data-packet losses, European Journal of Control, 13 (2007), 343-350. doi: 10.3166/ejc.13.343-350. [8] B. Jayawardhana and G. Weiss, Tracking and disturbance rejection for fully actuated mechanical systems, Automatica, 44 (2008), 2863-2868. doi: 10.1016/j.automatica.2008.03.030. [9] S. Liu, L. Xie and H. S. Zhang, Linear quadratic tracking problem for discrete-time systems with multiple delays in single input channel, International Journal of Robust and Nonlinear Control, 20 (2010), 1379-1394. doi: 10.1002/rnc.1520. [10] M. Mauder, Robust tracking control of nonholonomic dynamic systems with application to the bi-steerable mobile robot, Automatica, 44 (2008), 2588-2592. doi: 10.1016/j.automatica.2008.02.012. [11] A. Naskali and A. Onat, Random network delay in model based predictive networked control systems, in Proc. $6^{nd}$ WSEAS Int. Conf. Appl. Comput. Sci., 1 (2006), 199-206. [12] A. Onat, et al., Control over imperfect networks: Model-based predictive networked control systems, IEEE Transactions on Industrial Electronics, 58 (2011), 905-913. doi: 10.1109/TIE.2010.2051932. [13] A. Onat and E. Parlakay, Implementation of networked predictive control system, in Proc. 9th Real-Time Linux Workshop, 1 (2007), 85-93. [14] T. Suzuki, et al., Controllability and stabilizability of a networked control system with periodic communication constraints, Systems & Control Letters, 60 (2011), 977-984. doi: 10.1016/j.sysconle.2011.08.004. [15] Y. Shi and B. Yu, Output feedback stabilization of networked control systems with random delays modeled by Markov chains, IEEE Transactions on Automatic Control, 54 (2009), 1668-1674. doi: 10.1109/TAC.2009.2020638. [16] A. Sala, Computer control under time-varying sampling period: An LMI gridding approach, Automatica, 41 (2005), 2077-2082. doi: 10.1016/j.automatica.2005.05.017. [17] B. O. S. Teixeira, et al., Spacecraft tracking using sampled-data Kalman filters - An illustrative application of extended and unscented estimators, IEEE Control Systems Magazine, 28 (2008), 78-94. doi: 10.1109/MCS.2008.923231. [18] Y. H. Wang, Z. M. Wang and Y. F. Zheng, On the model-based networked control for singularly perturbed systems, International Journal of Robust and Nonlinear Control, 6 (2008), 153-162. doi: 10.1007/s11768-008-6152-9. [19] J. Wu and T. W. Chen, Design of networked control systems with packet dropouts, IEEE Transactions on Automatic Control, 52 (2007), 1314-1319. doi: 10.1109/TAC.2007.900839. [20] H. H. Wang, Optimal tracking for Discrete-Time Systems with Input Delays, Proceedings of the 2008 Chinese Control and Decision Conference, (2008), 4033-4037. [21] Y. L. Wang and G. H. Yang, Output tracking control for networked control systems with time delay and packet dropout, International Journal of Control, 81 (2008), 1709-1719. doi: 10.1080/00207170701836944. [22] Y. Wang and G. Yang, Output tracking control for discrete-time networked control systems, IEEE American Control Conference, (2009), 5109-5114. doi: 10.1109/ACC.2009.5159974. [23] D. Wu, J. Wu and S. Chen, Robust $H_\infty$ control for networked control systems with uncertainties and multiple-packet transmission, IET Control Theory Appl., 4 (2010), 701-709. doi: 10.1049/iet-cta.2009.0090. [24] Y. L. Wang and G. H. Yang, Output tracking control for continuous-time networked control systems with communication constraints, Proc. of the American Control Conference, (2009), 531-536. doi: 10.1109/ACC.2009.5159975. [25] H. H. Wang and G. Y. Tang, Observer-based optimal output tracking for discrete-time systems with multiple state and input delays, International Journal of Control Automation and Systems, 7 (2009), 57-66. doi: 10.1007/s12555-009-0108-9. [26] H. W. Yu, X. M. Zhang, G. P. Lu and Y. F. Zheng, On model based networked control for singularly perturbed systems with nonlinear uncertainties, Joint 48th IEEE Conference on Decision and Control and 28th Chinese Control Conference, Shanghai, (2009), 684-689. doi: 10.1109/CDC.2009.5400103. [27] D. Yue, Q. L. Han and J. Lam, Network-based robust $H_\infty$ control of systems with uncertainty, Automatica, 41 (2005), 999-1007. doi: 10.1016/j.automatica.2004.12.011. [28] X. M. Zhang, et al., Stochastic stability of networked control systems with network-induced delay and data dropout, in Proceedings of the 45th Ieee Conference on Decision and Control, 14 (2006), 5006-5011. doi: 10.1109/CDC.2006.376970. [29] H. G. Zhang, J. Yang and C. Y. Su, T-S fuzzy-model-based robust $H_\infty$ design for networked control systems with uncertainties, IEEE Trans. Ind. Inf., 3 (2007), 289-301. [30] Q. Zhang, et al., An Enhanced LMI Approach for Mixed $H_2/H_\infty$ Flight Tracking Control, Chinese Journal of Aeronautics, 24 (2011), 324-328. [31] H. B. Zeng, et al., Absolute stability and stabilization for Lurie networked control systems, International Journal of Robust and Nonlinear Control, 21 (2011), 1667-1676. doi: 10.1002/rnc.1658.
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