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  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 & Management Optimization, 2018, 14 (1) : 19-33. doi: 10.3934/jimo.2017035
References:
[1]

A. BalluchiM. D. BenedettoC. PinelloC. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[4]

J. CaiC. WenH. 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.  Google Scholar

[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.  Google Scholar

[6]

D. DuB. JiangP. 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.  Google Scholar

[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.  Google Scholar

[8]

J. P. HespanhaD. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[11]

D. LeithR. ShortenW. 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.  Google Scholar

[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.  Google Scholar

[13]

R. LuB. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[16]

P. ShiM. 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.  Google Scholar

[17]

Y. TangH. GaoW. 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.  Google Scholar

[18]

W. XiangJ. 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.  Google Scholar

[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.  Google Scholar

[20]

W. XiangJ. 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.  Google Scholar

[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.  Google Scholar

[22]

Z. XiangC. Liang and M. S. Mahmoud, Robust L2L 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.  Google Scholar

[23]

Z. XiangC. 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.  Google Scholar

[24]

Z. XiangC. Liang and Q. Chen, Robust L2L 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.  Google Scholar

[25]

D. XieL. 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.   Google Scholar

[26]

S. XuJ. Lam and Y. Zou, H filtering for singular systems, IEEE Transactions on Automatic Control, 48 (2003), 2217-2222.  doi: 10.1109/TAC.2003.820149.  Google Scholar

[27]

G. S. ZhaiB. HuK. 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.  Google Scholar

[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.  Google Scholar

[29]

W. ZhangM. S. Branicky and S. M. Phillips, Stability of networked control systems, IEEE Control Systems Magazine, 21 (2001), 84-99.  doi: 10.1109/37.898794.  Google Scholar

show all references

References:
[1]

A. BalluchiM. D. BenedettoC. PinelloC. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[4]

J. CaiC. WenH. 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.  Google Scholar

[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.  Google Scholar

[6]

D. DuB. JiangP. 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.  Google Scholar

[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.  Google Scholar

[8]

J. P. HespanhaD. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[11]

D. LeithR. ShortenW. 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.  Google Scholar

[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.  Google Scholar

[13]

R. LuB. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[16]

P. ShiM. 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.  Google Scholar

[17]

Y. TangH. GaoW. 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.  Google Scholar

[18]

W. XiangJ. 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.  Google Scholar

[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.  Google Scholar

[20]

W. XiangJ. 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.  Google Scholar

[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.  Google Scholar

[22]

Z. XiangC. Liang and M. S. Mahmoud, Robust L2L 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.  Google Scholar

[23]

Z. XiangC. 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.  Google Scholar

[24]

Z. XiangC. Liang and Q. Chen, Robust L2L 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.  Google Scholar

[25]

D. XieL. 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.   Google Scholar

[26]

S. XuJ. Lam and Y. Zou, H filtering for singular systems, IEEE Transactions on Automatic Control, 48 (2003), 2217-2222.  doi: 10.1109/TAC.2003.820149.  Google Scholar

[27]

G. S. ZhaiB. HuK. 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.  Google Scholar

[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.  Google Scholar

[29]

W. ZhangM. S. Branicky and S. M. Phillips, Stability of networked control systems, IEEE Control Systems Magazine, 21 (2001), 84-99.  doi: 10.1109/37.898794.  Google Scholar

Figure 1.  Illustration of state projection approach
Figure 2.  Illustration of projection of filter state
Figure 3.  State response of $x_1 (-)$ and $x_2 (\cdots)$
Figure 4.  State response of $\hat x_1 (-)$ and $\hat x_2 (\cdots)$
Figure 5.  Response of $z (-)$ and $\hat z (\cdots)$
Figure 6.  Response of $\tilde z = z-\hat z$
[1]

Joackim Bernier. Bounds on the growth of high discrete Sobolev norms for the cubic discrete nonlinear Schrödinger equations on $ h\mathbb{Z} $. Discrete & Continuous Dynamical Systems - A, 2019, 39 (6) : 3179-3195. doi: 10.3934/dcds.2019131

[2]

Xinliang An, Avy Soffer. Fermi's golden rule and $ H^1 $ scattering for nonlinear Klein-Gordon equations with metastable states. Discrete & Continuous Dynamical Systems - A, 2020, 40 (1) : 331-373. doi: 10.3934/dcds.2020013

[3]

Monica Motta, Caterina Sartori. On ${\mathcal L}^1$ limit solutions in impulsive control. Discrete & Continuous Dynamical Systems - S, 2018, 11 (6) : 1201-1218. doi: 10.3934/dcdss.2018068

[4]

Yu-Zhao Wang. $ \mathcal{W}$-Entropy formulae and differential Harnack estimates for porous medium equations on Riemannian manifolds. Communications on Pure & Applied Analysis, 2018, 17 (6) : 2441-2454. doi: 10.3934/cpaa.2018116

[5]

Mohan Mallick, R. Shivaji, Byungjae Son, S. Sundar. Bifurcation and multiplicity results for a class of $n\times n$ $p$-Laplacian system. Communications on Pure & Applied Analysis, 2018, 17 (3) : 1295-1304. doi: 10.3934/cpaa.2018062

[6]

Florin Diacu, Shuqiang Zhu. Almost all 3-body relative equilibria on $ \mathbb S^2 $ and $ \mathbb H^2 $ are inclined. Discrete & Continuous Dynamical Systems - S, 2018, 0 (0) : 1-13. doi: 10.3934/dcdss.2020067

[7]

Shengbing Deng. Construction solutions for Neumann problem with Hénon term in $ \mathbb{R}^2 $. Discrete & Continuous Dynamical Systems - A, 2019, 39 (4) : 2233-2253. doi: 10.3934/dcds.2019094

[8]

Peter Benner, Ryan Lowe, Matthias Voigt. $\mathcal{L}_{∞}$-norm computation for large-scale descriptor systems using structured iterative eigensolvers. Numerical Algebra, Control & Optimization, 2018, 8 (1) : 119-133. doi: 10.3934/naco.2018007

[9]

Jiao Du, Longjiang Qu, Chao Li, Xin Liao. Constructing 1-resilient rotation symmetric functions over $ {\mathbb F}_{p} $ with $ {q} $ variables through special orthogonal arrays. Advances in Mathematics of Communications, 2019, 0 (0) : 0-0. doi: 10.3934/amc.2020018

[10]

Liangming Chen, Ming Cao, Chuanjiang Li. Bearing rigidity and formation stabilization for multiple rigid bodies in $ SE(3) $. Numerical Algebra, Control & Optimization, 2019, 9 (3) : 257-267. doi: 10.3934/naco.2019017

[11]

Yongkuan Cheng, Yaotian Shen. Generalized quasilinear Schrödinger equations with concave functions $ l(s^2) $. Discrete & Continuous Dynamical Systems - A, 2019, 39 (3) : 1311-1343. doi: 10.3934/dcds.2019056

[12]

Yong-Yong Li, Yan-Fang Xue, Chun-Lei Tang. Ground state solutions for asymptotically periodic modified Schr$ \ddot{\mbox{o}} $dinger-Poisson system involving critical exponent. Communications on Pure & Applied Analysis, 2019, 18 (5) : 2299-2324. doi: 10.3934/cpaa.2019104

[13]

Pak Tung Ho. Prescribing $ Q $-curvature on $ S^n $ in the presence of symmetry. Communications on Pure & Applied Analysis, 2020, 19 (2) : 715-722. doi: 10.3934/cpaa.2020033

[14]

Vladimir Chepyzhov, Alexei Ilyin, Sergey Zelik. Strong trajectory and global $\mathbf{W^{1,p}}$-attractors for the damped-driven Euler system in $\mathbb R^2$. Discrete & Continuous Dynamical Systems - B, 2017, 22 (5) : 1835-1855. doi: 10.3934/dcdsb.2017109

[15]

Imed Bachar, Habib Mâagli. Singular solutions of a nonlinear equation in a punctured domain of $\mathbb{R}^{2}$. Discrete & Continuous Dynamical Systems - S, 2019, 12 (2) : 171-188. doi: 10.3934/dcdss.2019012

[16]

VicenŢiu D. RǍdulescu, Somayeh Saiedinezhad. A nonlinear eigenvalue problem with $ p(x) $-growth and generalized Robin boundary value condition. Communications on Pure & Applied Analysis, 2018, 17 (1) : 39-52. doi: 10.3934/cpaa.2018003

[17]

Gyu Eun Lee. Local wellposedness for the critical nonlinear Schrödinger equation on $ \mathbb{T}^3 $. Discrete & Continuous Dynamical Systems - A, 2019, 39 (5) : 2763-2783. doi: 10.3934/dcds.2019116

[18]

Silvia Frassu. Nonlinear Dirichlet problem for the nonlocal anisotropic operator $ L_K $. Communications on Pure & Applied Analysis, 2019, 18 (4) : 1847-1867. doi: 10.3934/cpaa.2019086

[19]

Yong Ren, Wensheng Yin. Quasi sure exponential stabilization of nonlinear systems via intermittent $ G $-Brownian motion. Discrete & Continuous Dynamical Systems - B, 2019, 24 (11) : 5871-5883. doi: 10.3934/dcdsb.2019110

[20]

Juntao Sun, Tsung-Fang Wu, Zhaosheng Feng. Non-autonomous Schrödinger-Poisson system in $\mathbb{R}^{3}$. Discrete & Continuous Dynamical Systems - A, 2018, 38 (4) : 1889-1933. doi: 10.3934/dcds.2018077

2018 Impact Factor: 1.025

Article outline

Figures and Tables

[Back to Top]