# zbMATH — the first resource for mathematics

Dynamical robust $$H_\infty$$ filtering for nonlinear uncertain systems: an LMI approach. (English) Zbl 1202.93157
Summary: A new approach to robust $$H_\infty$$ filtering for a class of nonlinear systems with time-varying uncertainties is proposed in the LMI framework based on a general dynamical observer structure. The nonlinearities under consideration are assumed to satisfy local Lipschitz conditions and appear in both state and measured output equations. The admissible Lipschitz constants of the nonlinear functions are maximized through LMI optimization. The resulting $$H_\infty$$ observer guarantees asymptotic stability of the estimation error dynamics with prespecified disturbance attenuation level and is robust against time-varying parametric uncertainties as well as Lipschitz nonlinear additive uncertainty.

##### MSC:
 93E11 Filtering in stochastic control theory 93C10 Nonlinear systems in control theory 93B36 $$H^\infty$$-control 93D20 Asymptotic stability in control theory
Full Text:
##### References:
 [1] de Souza, C.E.; Xie, L.; Wang, Y., $$H_\infty$$ filtering for a class of uncertain nonlinear systems, Systems and control letters, 20, 6, 419-426, (1993) · Zbl 0784.93095 [2] Wang, Y.; Xie, L.; de Souza, C.E., Robust control of a class of uncertain nonlinear systems, Systems and control letters, 19, 2, 139-149, (1992) · Zbl 0765.93015 [3] Chen, M.; Chen, C., Robust nonlinear observer for Lipschitz nonlinear systems subject to disturbances, IEEE transactions on automatic control, 52, 12, 2365-2369, (2007) · Zbl 1366.93459 [4] Zemouche, A.; Boutayeb, M.; Bara, G.I., Observers for a class of Lipschitz systems with extension to $$H_\infty$$ performance analysis, Systems and control letters, 57, 1, 18-27, (2008) · Zbl 1129.93006 [5] Fridman, E.; Shaked, U., On regional nonlinear $$H_\infty \text{-filtering}$$, Systems and control letters, 29, 4, 233-240, (1997) · Zbl 0877.93129 [6] Nguang, S.K.; Fu, M., Robust nonlinear $$H_\infty$$ filtering, Automatica, 32, 8, 1195-1199, (1996) · Zbl 0855.93086 [7] McEneaney, W.M., Robust $$H_\infty$$ filtering for nonlinear systems, Systems and control letters, 33, 5, 315-325, (1998) · Zbl 0902.93065 [8] Chen, X.; Fukuda, T.; Young, K.D., A new nonlinear robust disturbance observer, Systems and control letters, 41, 3, 189-199, (2000) · Zbl 0986.93045 [9] Yung, C.; Li, Y.; Sheu, H., $$H_\infty$$ filtering and solution bound for non-linear systems, International journal of control, 74, 6, 565-570, (2001) · Zbl 1031.93147 [10] Sundarapandian, V., Nonlinear observer design for a general class of nonlinear systems with real parametric uncertainty, Computers and mathematics with applications, 49, 7-8, 1195-1211, (2005) · Zbl 1210.93032 [11] Aguilar-Lopez, R.; Maya-Yescas, R., State estimation for nonlinear systems under model uncertainties: a class of sliding-mode observers, Journal of process control, 15, 3, 363-370, (2005) [12] Xu, J.X.; Xu, J., Observer based learning control for a class of nonlinear systems with time-varying parametric uncertainties, IEEE transactions on automatic control, 49, 2, 275-281, (2004) · Zbl 1365.93552 [13] Marquez, H.J., A frequency domain approach to state estimation, Journal of the franklin institute, 340, 2, 147-157, (2003) · Zbl 1042.93013 [14] Pertew, A.M.; Marquez, H.J.; Zhao, Q., $$H_\infty$$ observer design for Lipschitz nonlinear systems, IEEE transactions on automatic control, 51, 7, 1211-1216, (2006) · Zbl 1366.93162 [15] Xu, S., Robust $$H_\infty$$ filtering for a class of discrete-time uncertain nonlinear systems with state delay, IEEE transactions on circuits and systems I: fundamental theory and applications, 49, 12, 1853-1859, (2002) [16] Lu, G.; Ho, D.W.C., Robust $$H_\infty$$ observer for nonlinear discrete systems with time delay and parameter uncertainties, IEE Proceedings: control theory and applications, 151, 4, 439-444, (2004) [17] Gao, H.; Wang, C., Delay-dependent robust $$H_\infty$$ and $$L_2 - L_\infty$$ filtering for a class of uncertain nonlinear time-delay systems, IEEE transactions on automatic control, 48, 9, 1661-1665, (2003) · Zbl 1364.93210 [18] Abbaszadeh, M.; Marquez, H.J., Robust $$H_\infty$$ observer design for sampled-data Lipschitz nonlinear systems with exact and Euler approximate models, Automatica, 44, 3, 799-806, (2008) · Zbl 1283.93180 [19] Abbaszadeh, M.; Marquez, H.J., LMI optimization approach to robust $$H_\infty$$ observer design and static output feedback stabilization for discrete-time nonlinear uncertain systems, International journal of robust and nonlinear control, 19, 3, 313-340, (2009) · Zbl 1163.93027 [20] M. Abbaszadeh, H.J. Marquez, A robust observer design method for continuous-time Lipschitz nonlinear systems, in: Proceedings of the 45th IEEE Conference on Decision and Control, San Diego, USA, 2006, pp. 3895-3900. [21] M. Abbaszadeh, H.J. Marquez, Robust $$H_\infty$$ observer design for a class of nonlinear uncertain systems via convex optimization, in: Proceedings of the American Control Conference, New York, USA, 2007, pp. 1699-1704. [22] Boyd, S.; Ghaoui, L.E.; Feron, E.; Balakrishnan, V., Linear matrix inequalities in system and control theory, (1994), SIAM PA · Zbl 0816.93004 [23] Khargonekar, P.P.; Petersen, I.R.; Zhou, K., Robust stabilization of uncertain linear systems: quadratic stabilizability and $$H_\infty$$ control theory, IEEE transactions on automatic control, 35, 3, 356-361, (1990) · Zbl 0707.93060 [24] Marquez, H.J., Nonlinear control systems: analysis and design, (2003), Wiley NY · Zbl 1037.93003
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.