State Estimation for Stochastic Nonlinear Systems with Applications to Viral Infections

State estimation of biological systems is a difficult task due to their complexity and stochasticity. In particular, bilinear and Michaelis-Menten terms are the base for many biological models such as in infectious diseases, cancer, diabetes, and many others. In this paper, mentioned non-linear terms are formulated into a polynomial form with state-dependent matrices driven by additive white Gaussian noises over linear observations. To show the effectiveness of the approach, two different models widely used for modeling viral infectious diseases are considered and compared with the extended Kalman filter (EKF) algorithm. Numerical results show the applicability of the polynomial approach.

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

[2]  Hugh F. Durrant-Whyte,et al.  A new method for the nonlinear transformation of means and covariances in filters and estimators , 2000, IEEE Trans. Autom. Control..

[3]  S. Andrew Gadsden,et al.  Gaussian filters for parameter and state estimation: A general review of theory and recent trends , 2017, Signal Process..

[4]  Corentin Briat,et al.  Simple interval observers for linear impulsive systems with applications to sampled-data and switched systems , 2017, ArXiv.

[5]  D. Simon.  Optimal State Estimation: Kalman, H Infinity, and Nonlinear Approaches , 2006 .

[6]  M. Basin.  New Trends in Optimal Filtering and Control for Polynomial and Time-Delay Systems , 2008 .

[7]  Alfredo Germani,et al.  Polynomial extended Kalman filter , 2005, IEEE Transactions on Automatic Control.

[8]  Kin Hong Wong,et al.  A computer vision based sea search method using Kalman filter and CAMSHIFT , 2013, 2013 The International Conference on Technological Advances in Electrical, Electronics and Computer Engineering (TAEECE).

[9]  Gilles Burel,et al.  On exact Kalman filtering of polynomial systems , 2006, IEEE Transactions on Circuits and Systems I: Regular Papers.

[10]  M. Basin,et al.  Optimal filtering for polynomial system states with polynomial multiplicative noise , 2006, 2006 American Control Conference.

[11]  Sirish L. Shah,et al.  Nonlinear Bayesian state estimation: A review of recent developments , 2012 .

[12]  R. E. Kalman,et al.  New Results in Linear Filtering and Prediction Theory , 1961 .

[13]  M. Basin,et al.  Optimal filtering for polynomial system states with polynomial multiplicative noise , 2006, 2006 American Control Conference.

[14]  Xuemin Shen,et al.  A dynamic system approach to speech enhancement using the H∞ filtering algorithm , 1999, IEEE Trans. Speech Audio Process..

[15]  Michael Basin,et al.  Optimal Filtering for Incompletely Measured Polynomial States over Linear Observations , 2007 .

[16]  Alma Y. Alanis,et al.  Observers for biological systems , 2014, Appl. Soft Comput..

[17]  Alberto L. Sangiovanni-Vincentelli,et al.  The design of dynamical observers for hybrid systems: Theory and application to an automotive control problem , 2012, Autom..

[18]  T. Başar,et al.  A New Approach to Linear Filtering and Prediction Problems , 2001 .

[19]  Ryan Zurakowski,et al.  Nonlinear observer output-feedback MPC treatment scheduling for HIV , 2011, Biomedical engineering online.

[20]  L. Szilagyi.  Application of the Kalman filter in cardiac arrhythmia detection , 1998, Proceedings of the 20th Annual International Conference of the IEEE Engineering in Medicine and Biology Society. Vol.20 Biomedical Engineering Towards the Year 2000 and Beyond (Cat. No.98CH36286).

[21]  R. E. Kalman,et al.  A New Approach to Linear Filtering and Prediction Problems , 2002 .

[22]  Michael V. Basin,et al.  Discrete-time high order neural network identifier trained with cubature Kalman filter , 2018, Neurocomputing.

[23]  Murat Barut,et al.  Braided extended Kalman filters for sensorless estimation in induction motors at high-low/zero speed , 2007 .

[24]  A. Perelson,et al.  Kinetics of Influenza A Virus Infection in Humans , 2006, Journal of Virology.

[25]  Alma Y. Alanis,et al.  Passivity-Based Inverse Optimal Impulsive Control for Influenza Treatment in the Host , 2020, IEEE Transactions on Control Systems Technology.

[26]  Alma Y. Alanis,et al.  Neural inverse optimal control for discrete-time impulsive systems , 2018, Neurocomputing.