# American Institute of Mathematical Sciences

January  2018, 14(1): 19-33. doi: 10.3934/jimo.2017035

## $\mathcal{H}_∞$ filtering for switched nonlinear systems: A state projection method

 1 School of Environment and Resource, Southwest University of Science and Technology, Mianyang 621000, China 2 Mianyang Polytechnic, Mianyang 621000, China

* Corresponding author: Lin Du

Received  April 2015 Revised  February 2017 Published  January 2018 Early access  April 2017

Fund Project: This work was supported by the National Natural Science Foundation of China under Grant No. 61603312.

In this paper, the $\mathcal{H}_∞$ filtering problem of switched nonlinear system with linear hyper plane switching surface is investigated. A state projection method is introduced to ensure the stability of error system and guarantee a prescribed disturbance attenuation level in the $\mathcal{H}_∞$ sense, by designing filter gains for each subsystem via solving a set of LMIs and formulating a state projection relation for filter state at switching instant. It is worthwhile to note that the state projection relation is deduced by both Lyapunov functions and the switching surface, which implies the state projection method is suitable for switched system with linear hyper plane switching surface. Finally, a numerical example is provided to illustrate our theoretic findings in this paper.

Citation: Lin Du, Yun Zhang. $\mathcal{H}_∞$ filtering for switched nonlinear systems: A state projection method. Journal of Industrial and Management Optimization, 2018, 14 (1) : 19-33. doi: 10.3934/jimo.2017035
##### References:
 [1] A. Balluchi, M. D. Benedetto, C. Pinello, C. Rossi and A. Sangiovanni-Vincentelli, Cut-off in engine control: A hybrid system approach, Proceedings of the 36th IEEE Conference on Decision and Control, 5 (1997), 4720-4725.  doi: 10.1109/CDC.1997.649753. [2] B. E. Bishop and M. W. Spong, Control of redundant manipulators using logic-based switching, Proceedings of the 36th IEEE Conference on Decision and Control, 2 (1998), 16-18.  doi: 10.1109/CDC.1998.758498. [3] M. S. Branicky, Multiple Lyapunov functions and other analysis tools for switched and hybrid systems, IEEE Transactions on Automatic Control, 43 (1998), 475-482.  doi: 10.1109/9.664150. [4] J. Cai, C. Wen, H. Su and Z. Liu, Robust adaptive failure compensation of hysteretic actuators for a class of uncertain nonlinear systems, IEEE Transactions on Automatic Control, 58 (2013), 2388-2394.  doi: 10.1109/TAC.2013.2251795. [5] Y. Chen and W. X. Zheng, Stochastic state estimation for neural networks with distributed delays and Markovian jump, Neural Networks, 25 (2012), 14-20.  doi: 10.1016/j.neunet.2011.08.002. [6] D. Du, B. Jiang, P. Shi and S. Zhou, H∞ filtering of discrete-time switched systems with state delays via switched Lyapunov function approach, IEEE Transactions on Automatic Control, 52 (2007), 1520-1524.  doi: 10.1109/TAC.2007.902777. [7] A. Elsayed and M. Grimble, A new approach to design for optimal digital linear filters, IMA J. Math. Control Inf, 6 (1989), 233-251.  doi: 10.1093/imamci/6.2.233. [8] J. P. Hespanha, D. Liberzon and A. S. Morse, Stability of switched systems with average dwell time, Proceedings of 38th Conference on Decision and Control, (1999), 2655-2660.  doi: 10.1109/CDC.1999.831330. [9] K. Hu and J. Yuan, Improved robust H∞ filtering for uncertain discrete-time switched systems, IET Control Theory Applications, 3 (2009), 315-324.  doi: 10.1049/iet-cta:20070253. [10] D. Koenig and B. Marx, H∞ filtering and state feedback control for discrete-time switched descriptor systems, IET Control Theory Applications, 3 (2009), 661-670.  doi: 10.1049/iet-cta.2008.0132. [11] D. Leith, R. Shorten, W. Leithead and O. Mason, Issue in the design of switched linear control systems: A benchmark study, International Journal of Adaptive Control, 17 (2003), 103-118.  doi: 10.1002/acs.741. [12] H. Lin and P. J. Antsaklis, Stability and stabilizability of switched linear systems: A survey of recent results, IEEE Transactions on Automatic Control, 54 (2009), 308-322.  doi: 10.1109/TAC.2008.2012009. [13] R. Lu, B. Lou and A.-K. Xue, Mode-dependent quantised $H_∞$ filtering for Markovian jump singular system, International Journal of Systems Science, 46 (2015), 1817-1824.  doi: 10.1080/00207721.2013.837539. [14] A. S. Morse, Supervisory control of families of linear set-point controllers, part 1: Exact matching, IEEE Transactions on Automatic Control, 41 (1996), 1413-1431.  doi: 10.1109/9.539424. [15] K. S. Narendra and J. A. Balakrishnan, Common Lyapunov function for stable LTI systems with commuting A-matrices, IEEE Transactions on Automatic Control, 39 (1994), 2469-2471.  doi: 10.1109/9.362846. [16] P. Shi, M. Mahmoud and S. Nguang, Robust filtering for jumping systems with modedependent delays, Signal Process, 86 (2006), 140-152.  doi: 10.1016/j.sigpro.2005.05.005. [17] Y. Tang, H. Gao, W. Zou and J. Kurths, Distributed synchronization in networks of agent systems with nonlinearities and random switchings, IEEE Transactions On Cybernetics, 43 (2013), 358-370.  doi: 10.1109/TSMCB.2012.2207718. [18] W. Xiang, J. Xiao and N. Iqbal, Robust observer design for nonlinear uncertain switched systems under asynchronous switching, Nonlinear Analysis: Hybrid Systems, 6 (2012), 754-773.  doi: 10.1016/j.nahs.2011.08.001. [19] W. Xiang and J. Xiao, H∞ filtering for switched nonlinear systems under asynchronous switching, International Journal of System Science, 42 (2011), 751-765.  doi: 10.1080/00207721.2010.488763. [20] W. Xiang, J. Xiao and M. N. Iqbal, Fault detection for switched nonlinear systems under asynchronous switching, International Journal of Control, 84 (2011), 1362-1376.  doi: 10.1080/00207179.2011.598191. [21] W. Xiang and J. Xiao, Stabilization of switched continuous-time system with all modes unstable via dwell time switching, Automatica, 50 (2014), 940-945.  doi: 10.1016/j.automatica.2013.12.028. [22] Z. Xiang, C. Liang and M. S. Mahmoud, Robust L2 − L∞ filtering for switched time-delay systems with missing measurements, Circuits, Systems, and Signal Processing, 31 (2012), 1677-1697.  doi: 10.1007/s00034-012-9396-z. [23] Z. Xiang, C. Qiao and S. Mahmoud, Robust H∞ filtering for switched stochastic systems under asynchronous switching, Journal of the Franklin Institute, 349 (2012), 1213-1230.  doi: 10.1016/j.jfranklin.2012.01.008. [24] Z. Xiang, C. Liang and Q. Chen, Robust L2 − L∞ filtering for switched systems under asynchronous switching, Communications in Nonlinear Science and Numerical Simulation, 16 (2011), 3303-3318.  doi: 10.1016/j.cnsns.2010.10.029. [25] D. Xie, L. Wang and F. Hao, Robust stability analysis and control synthesis for discrete-time uncertain switched systems, Proceedings of Conference on Decision and Control, (2003), 4812-4817. [26] S. Xu, J. Lam and Y. Zou, H∞ filtering for singular systems, IEEE Transactions on Automatic Control, 48 (2003), 2217-2222.  doi: 10.1109/TAC.2003.820149. [27] G. S. Zhai, B. Hu, K. Yasuda and A. N. Michel, Stability analysis of switched systems with stable and unstable subsystems: An average dwell time approach, Proceedings of the American Control Conference, (2000), 200-204.  doi: 10.1109/ACC.2000.878825. [28] B. Zhang and S. Xu, Robust $H_∞$ filtering for uncertain discrete piecewise time-delay systems, International Journal of Control, 80 (2007), 636-645.  doi: 10.1080/00207170601131982. [29] W. Zhang, M. S. Branicky and S. M. Phillips, Stability of networked control systems, IEEE Control Systems Magazine, 21 (2001), 84-99.  doi: 10.1109/37.898794.

show all references

##### References:
 [1] A. Balluchi, M. D. Benedetto, C. Pinello, C. Rossi and A. Sangiovanni-Vincentelli, Cut-off in engine control: A hybrid system approach, Proceedings of the 36th IEEE Conference on Decision and Control, 5 (1997), 4720-4725.  doi: 10.1109/CDC.1997.649753. [2] B. E. Bishop and M. W. Spong, Control of redundant manipulators using logic-based switching, Proceedings of the 36th IEEE Conference on Decision and Control, 2 (1998), 16-18.  doi: 10.1109/CDC.1998.758498. [3] M. S. Branicky, Multiple Lyapunov functions and other analysis tools for switched and hybrid systems, IEEE Transactions on Automatic Control, 43 (1998), 475-482.  doi: 10.1109/9.664150. [4] J. Cai, C. Wen, H. Su and Z. Liu, Robust adaptive failure compensation of hysteretic actuators for a class of uncertain nonlinear systems, IEEE Transactions on Automatic Control, 58 (2013), 2388-2394.  doi: 10.1109/TAC.2013.2251795. [5] Y. Chen and W. X. Zheng, Stochastic state estimation for neural networks with distributed delays and Markovian jump, Neural Networks, 25 (2012), 14-20.  doi: 10.1016/j.neunet.2011.08.002. [6] D. Du, B. Jiang, P. Shi and S. Zhou, H∞ filtering of discrete-time switched systems with state delays via switched Lyapunov function approach, IEEE Transactions on Automatic Control, 52 (2007), 1520-1524.  doi: 10.1109/TAC.2007.902777. [7] A. Elsayed and M. Grimble, A new approach to design for optimal digital linear filters, IMA J. Math. Control Inf, 6 (1989), 233-251.  doi: 10.1093/imamci/6.2.233. [8] J. P. Hespanha, D. Liberzon and A. S. Morse, Stability of switched systems with average dwell time, Proceedings of 38th Conference on Decision and Control, (1999), 2655-2660.  doi: 10.1109/CDC.1999.831330. [9] K. Hu and J. Yuan, Improved robust H∞ filtering for uncertain discrete-time switched systems, IET Control Theory Applications, 3 (2009), 315-324.  doi: 10.1049/iet-cta:20070253. [10] D. Koenig and B. Marx, H∞ filtering and state feedback control for discrete-time switched descriptor systems, IET Control Theory Applications, 3 (2009), 661-670.  doi: 10.1049/iet-cta.2008.0132. [11] D. Leith, R. Shorten, W. Leithead and O. Mason, Issue in the design of switched linear control systems: A benchmark study, International Journal of Adaptive Control, 17 (2003), 103-118.  doi: 10.1002/acs.741. [12] H. Lin and P. J. Antsaklis, Stability and stabilizability of switched linear systems: A survey of recent results, IEEE Transactions on Automatic Control, 54 (2009), 308-322.  doi: 10.1109/TAC.2008.2012009. [13] R. Lu, B. Lou and A.-K. Xue, Mode-dependent quantised $H_∞$ filtering for Markovian jump singular system, International Journal of Systems Science, 46 (2015), 1817-1824.  doi: 10.1080/00207721.2013.837539. [14] A. S. Morse, Supervisory control of families of linear set-point controllers, part 1: Exact matching, IEEE Transactions on Automatic Control, 41 (1996), 1413-1431.  doi: 10.1109/9.539424. [15] K. S. Narendra and J. A. Balakrishnan, Common Lyapunov function for stable LTI systems with commuting A-matrices, IEEE Transactions on Automatic Control, 39 (1994), 2469-2471.  doi: 10.1109/9.362846. [16] P. Shi, M. Mahmoud and S. Nguang, Robust filtering for jumping systems with modedependent delays, Signal Process, 86 (2006), 140-152.  doi: 10.1016/j.sigpro.2005.05.005. [17] Y. Tang, H. Gao, W. Zou and J. Kurths, Distributed synchronization in networks of agent systems with nonlinearities and random switchings, IEEE Transactions On Cybernetics, 43 (2013), 358-370.  doi: 10.1109/TSMCB.2012.2207718. [18] W. Xiang, J. Xiao and N. Iqbal, Robust observer design for nonlinear uncertain switched systems under asynchronous switching, Nonlinear Analysis: Hybrid Systems, 6 (2012), 754-773.  doi: 10.1016/j.nahs.2011.08.001. [19] W. Xiang and J. Xiao, H∞ filtering for switched nonlinear systems under asynchronous switching, International Journal of System Science, 42 (2011), 751-765.  doi: 10.1080/00207721.2010.488763. [20] W. Xiang, J. Xiao and M. N. Iqbal, Fault detection for switched nonlinear systems under asynchronous switching, International Journal of Control, 84 (2011), 1362-1376.  doi: 10.1080/00207179.2011.598191. [21] W. Xiang and J. Xiao, Stabilization of switched continuous-time system with all modes unstable via dwell time switching, Automatica, 50 (2014), 940-945.  doi: 10.1016/j.automatica.2013.12.028. [22] Z. Xiang, C. Liang and M. S. Mahmoud, Robust L2 − L∞ filtering for switched time-delay systems with missing measurements, Circuits, Systems, and Signal Processing, 31 (2012), 1677-1697.  doi: 10.1007/s00034-012-9396-z. [23] Z. Xiang, C. Qiao and S. Mahmoud, Robust H∞ filtering for switched stochastic systems under asynchronous switching, Journal of the Franklin Institute, 349 (2012), 1213-1230.  doi: 10.1016/j.jfranklin.2012.01.008. [24] Z. Xiang, C. Liang and Q. Chen, Robust L2 − L∞ filtering for switched systems under asynchronous switching, Communications in Nonlinear Science and Numerical Simulation, 16 (2011), 3303-3318.  doi: 10.1016/j.cnsns.2010.10.029. [25] D. Xie, L. Wang and F. Hao, Robust stability analysis and control synthesis for discrete-time uncertain switched systems, Proceedings of Conference on Decision and Control, (2003), 4812-4817. [26] S. Xu, J. Lam and Y. Zou, H∞ filtering for singular systems, IEEE Transactions on Automatic Control, 48 (2003), 2217-2222.  doi: 10.1109/TAC.2003.820149. [27] G. S. Zhai, B. Hu, K. Yasuda and A. N. Michel, Stability analysis of switched systems with stable and unstable subsystems: An average dwell time approach, Proceedings of the American Control Conference, (2000), 200-204.  doi: 10.1109/ACC.2000.878825. [28] B. Zhang and S. Xu, Robust $H_∞$ filtering for uncertain discrete piecewise time-delay systems, International Journal of Control, 80 (2007), 636-645.  doi: 10.1080/00207170601131982. [29] W. Zhang, M. S. Branicky and S. M. Phillips, Stability of networked control systems, IEEE Control Systems Magazine, 21 (2001), 84-99.  doi: 10.1109/37.898794.
Illustration of state projection approach
Illustration of projection of filter state
State response of $x_1 (-)$ and $x_2 (\cdots)$
State response of $\hat x_1 (-)$ and $\hat x_2 (\cdots)$
Response of $z (-)$ and $\hat z (\cdots)$
Response of $\tilde z = z-\hat z$
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