A decoupled approach to filter design for stochastic systems

ABSTRACT This paper presents a new theorem to guarantee the almost sure exponential stability for a class of stochastic triangular systems by studying only the stability of each diagonal subsystems. This result allows to solve the filtering problem of the stochastic systems with multiplicative noises by using the almost sure exponential stability concept. Two kinds of observers are treated: the full-order and reduced-order cases.

[1]  E. Yaz,et al.  Observer design for discrete and continuous non-linear stochastic systems , 1993 .

[2]  H. Trinh,et al.  On the Existence and Design of Functional Observers for Linear Systems , 2007, 2007 International Conference on Mechatronics and Automation.

[3]  Xuerong Mao,et al.  Stochastic stabilization and destabilization , 1994 .

[4]  Zhengzhi Han,et al.  A note on observers for Lipschitz nonlinear systems , 2002, IEEE Trans. Autom. Control..

[5]  Urszula Ledzewicz,et al.  OPTIMAL CONTROL FOR A CLASS OF COMPARTMENTAL MODELS IN CANCER CHEMOTHERAPY , 2003 .

[6]  G. Besancon,et al.  Reduced order observer for a class of non-uniformly observable systems , 1995, Proceedings of 1995 34th IEEE Conference on Decision and Control.

[7]  Ian R. Petersen,et al.  Absolute Stabilization and Minimax Optimal Control of Uncertain Systems with Stochastic Uncertainty , 1997 .

[8]  Alfredo Germani,et al.  A new approach to nonlinear filtering via a mixed state observer and polynomial scheme , 2011 .

[9]  Exponential stability for stochastic differential equations with respect to semimartingales , 1990 .

[10]  Alfredo Germani,et al.  Linear filtering for bilinear stochastic differential systems with unknown inputs , 2002, IEEE Trans. Autom. Control..

[11]  O. Grasselli,et al.  An existence theorem for observers of bilinear systems , 1981 .

[12]  Krishan Nagpal,et al.  Linearized Reduced Order Filtering , 1987, 1987 American Control Conference.

[13]  X. Mao,et al.  Mathematical modelling of internal HIV dynamics , 2009 .

[14]  Shengyuan Xu,et al.  Robust H∞ filtering for uncertain impulsive stochastic systems under sampled measurements , 2003, Autom..

[15]  Urszula Ledzewicz,et al.  Optimal Bang-Bang Controls for a Two-Compartment Model in Cancer Chemotherapy , 2002 .

[16]  Alfredo Germani,et al.  A New Suboptimal Approach to the Filtering Problem for Bilinear Stochastic Differential Systems , 2000, SIAM J. Control. Optim..

[17]  Hong-ke Wang,et al.  On the Exponential Stability of Stochastic Differential Equations , 2009, ICFIE.

[18]  M. Vidyasagar,et al.  Nonlinear systems analysis (2nd ed.) , 1993 .

[19]  Victor M. Becerra,et al.  Optimal control , 2008, Scholarpedia.

[20]  Driss Boutat,et al.  Synchronisation of chaotic systems via reduced observers , 2011 .

[21]  I. Petersen.  A stabilization algorithm for a class of uncertain linear systems , 1987 .

[22]  K. S. Banerjee.  Generalized Inverse of Matrices and Its Applications , 1973 .

[23]  T. Damm.  Rational Matrix Equations in Stochastic Control , 2004 .

[24]  Y. Funahashi.  Stable state estimator for bilinear systems , 1979 .

[25]  Shengyuan Xu,et al.  Reduced-order H∞ filtering for stochastic systems , 2002, IEEE Trans. Signal Process..

[26]  Theodore F. Elbert,et al.  Estimation and control of systems , 1984 .

[27]  Bor-Sen Chen,et al.  Robust H∞ filtering for nonlinear stochastic systems , 2005 .

[28]  Xuerong Mao,et al.  Exponential stability of non-linear stochastic evolution equations , 1998 .

[29]  E. Feron,et al.  History of linear matrix inequalities in control theory , 1994, Proceedings of 1994 American Control Conference - ACC '94.

[30]  Xuerong Mao,et al.  Exponential stability of large-scale stochastic differential equations , 1992 .

[31]  Y. Phillis.  Estimation and control of systems with unknown covariance and multiplicative noise , 1989, 26th IEEE Conference on Decision and Control.

[32]  Frank Kozin,et al.  A survey of stability of stochastic systems , 1969, Autom..

[33]  Alessandro Astolfi,et al.  Invariant Manifold Based Reduced-Order Observer Design for Nonlinear Systems , 2008, IEEE Transactions on Automatic Control.

[34]  D. Luenberger.  An introduction to observers , 1971 .

[35]  F. Thau.  Observing the state of non-linear dynamic systems† , 1973 .

[36]  Alfredo Germani,et al.  Filtering of nonlinear stochastic feedback systems , 2000, Proceedings of the 39th IEEE Conference on Decision and Control (Cat. No.00CH37187).

[37]  Driss Boutat,et al.  A nonlinear canonical form for reduced order observer design , 2010 .

[38]  Michel Zasadzinski,et al.  Hinfinity functional filtering for stochastic bilinear systems with multiplicative noises , 2009, Autom..

[39]  H. Souley Ali,et al.  H ∞ functional filtering for stochastic bilinear systems with multiplicative noises , 2009 .

[40]  X. Mao,et al.  Environmental Brownian noise suppresses explosions in population dynamics , 2002 .

[41]  Liangjian Hu,et al.  Almost sure exponential stabilisation of stochastic systems by state-feedback control , 2008, Autom..

[42]  E. Kolárová,et al.  Vector Stochastic Differential Equations Used to Electrical Networks with Random Parameters , 2012 .

[43]  Alan J. Mayne,et al.  Generalized Inverse of Matrices and its Applications , 1972 .

[44]  K. Grigoriadis,et al.  Optimal unbiased filtering via linear matrix inequalities , 1997, Proceedings of the 1997 American Control Conference (Cat. No.97CH36041).

[45]  H. Schattler,et al.  Optimal control for a bilinear model with recruiting agent in cancer chemotherapy , 2003, 42nd IEEE International Conference on Decision and Control (IEEE Cat. No.03CH37475).

[46]  Anna Doubova,et al.  Extinction-time for stochastic population models , 2016, J. Comput. Appl. Math..

[47]  Erling B. Andersen,et al.  Asymptotic Properties of , 1970 .

[48]  U. Shaked,et al.  Robust $L_{\infty}$ -Induced Filtering and Control of Stochastic Systems With State-Multiplicative Noise , 2010, IEEE Transactions on Automatic Control.

[49]  F. Kozin,et al.  On almost sure sample stability of nonlinear stochastic dynamic systems , 1994, IEEE Trans. Autom. Control..

[50]  G. Besancon,et al.  On characterizing classes of observer forms for nonlinear systems , 1997, 1997 European Control Conference (ECC).

[51]  Xuerong Mao.  Stochastic Differential Equations in Applications , 2010 .

[52]  Xuerong Mao,et al.  A stochastic model for internal HIV dynamics , 2008 .

[53]  Harold J. Kushner,et al.  A partial history of the early development of continuous-time nonlinear stochastic systems theory , 2014, Autom..

[54]  T. Tarn,et al.  Observers for nonlinear stochastic systems , 1975, 1975 IEEE Conference on Decision and Control including the 14th Symposium on Adaptive Processes.

[55]  Stephen P. Boyd,et al.  Linear Matrix Inequalities in Systems and Control Theory , 1994 .

[56]  Feiqi Deng,et al.  Asymptotic properties of stochastic population dynamics , 2008 .

[57]  Isaac Yaesh,et al.  Robust H∞ filtering of stationary continuous-time linear systems with stochastic uncertainties , 2001, IEEE Trans. Autom. Control..

[58]  Mohamed Darouach,et al.  Robust I reduced order filtering for uncertain bilinear systems , 2006, Autom..

[59]  Alfredo Germani,et al.  A state observer approach to filter stochastic nonlinear differential systems , 2011, IEEE Conference on Decision and Control and European Control Conference.

[60]  Yanbin Zhao,et al.  Reduced-order observer design for one-sided Lipschitz non-linear systems , 2009, IMA J. Math. Control. Inf..

[61]  G. Besançon.  Remarks on nonlinear adaptive observer design , 2000 .

[62]  S. Marcus,et al.  Analysis of bilinear noise models in circuits and devices , 1976 .

[63]  B. Øksendal.  Stochastic differential equations : an introduction with applications , 1987 .

[64]  S. Hara,et al.  Minimal order state observers for bilinear systems , 1976 .

[65]  D. Luenberger.  Observers for multivariable systems , 1966 .

[66]  H. Hammouri,et al.  On uniform observation of nonuniformly observable systems , 1996 .

[67]  Alfredo Germani,et al.  The observer follower filter: A new approach to nonlinear suboptimal filtering , 2013, Autom..