Joint state and parameter estimation for a class of cascade systems: Application to a hemodynamic model

In this paper, we address a special case of state and parameter estimation, where the system can be put on a cascade form allowing to estimate the state components and the set of unknown parameters separately. Inspired by the nonlinear Balloon hemodynamic model for functional Magnetic Resonance Imaging problem, we propose a hierarchical approach. The system is divided into two subsystems in cascade. The state and input are first estimated from a noisy measured signal using an adaptive observer. The obtained input is then used to estimate the parameters of a linear system using the modulating functions method. Some numerical results are presented to illustrate the efficiency of the proposed method.

[1]  Michel Fliess,et al.  Analyse non standard du bruit , 2006, ArXiv.

[2]  Karl J. Friston,et al.  DEM: A variational treatment of dynamic systems , 2008, NeuroImage.

[3]  Heinz A. Preisig,et al.  Theory and application of the modulating function method—I. Review and theory of the method and theory of the spline-type modulating functions , 1993 .

[4]  R. Marino,et al.  Global adaptive output-feedback control of nonlinear systems , 1991, [1991] Proceedings of the 30th IEEE Conference on Decision and Control.

[5]  Olivier Gibaru,et al.  Identification of fractional order systems using modulating functions method , 2013, 2013 American Control Conference.

[6]  Qinghua Zhang,et al.  Adaptive observer for multiple-input-multiple-output (MIMO) linear time-varying systems , 2002, IEEE Trans. Autom. Control..

[7]  Naoki Miura,et al.  A state-space model of the hemodynamic approach: nonlinear filtering of BOLD signals , 2004, NeuroImage.

[8]  Olivier Faugeras,et al.  Using nonlinear models in fMRI data analysis: Model selection and activation detection , 2006, NeuroImage.

[9]  Chadia Zayane-Aissa,et al.  Estimation of the neuronal activation using fMRI data: An observer-based approach , 2013, 2013 American Control Conference.

[10]  Hieu Minh Trinh,et al.  State and input simultaneous estimation for a class of nonlinear systems , 2004, Autom..

[11]  Olivier Gibaru,et al.  Parameters estimation of a noisy sinusoidal signal with time-varying amplitude , 2011, 2011 19th Mediterranean Conference on Control & Automation (MED).

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

[13]  M. Gevers,et al.  Stable adaptive observers for nonlinear time-varying systems , 1987 .

[14]  Karl J. Friston,et al.  Nonlinear Responses in fMRI: The Balloon Model, Volterra Kernels, and Other Hemodynamics , 2000, NeuroImage.

[15]  Yeng Chai Soh,et al.  High-Gain Observers With Sliding Mode for State and Unknown Input Estimations , 2009, IEEE Transactions on Industrial Electronics.

[16]  Vince D. Calhoun,et al.  Dynamic modeling of neuronal responses in fMRI using cubature Kalman filtering , 2011, NeuroImage.

[17]  Marvin Shinbrot.  On the analysis of linear and nonlinear dynamical systems from transient-response data , 1954 .

[18]  T. Westerlund,et al.  Remarks on "Asymptotic behavior of the extended Kalman filter as a parameter estimator for linear systems" , 1980 .

[19]  R. Marino,et al.  Adaptive observers with arbitrary exponential rate of convergence for nonlinear systems , 1995, IEEE Trans. Autom. Control..

[20]  Arthur J. Krener,et al.  Linearization by output injection and nonlinear observers , 1983 .

[21]  H. Unbehauen,et al.  Identification of continuous-time systems , 1991 .

[22]  G. Kreisselmeier.  Adaptive observers with exponential rate of convergence , 1977 .

[23]  Peter C. Young,et al.  Direct Identification of Continuous-time Models from Sampled Data: Issues, Basic Solutions and Relevance , 2008 .

[24]  R. Buxton,et al.  Dynamics of blood flow and oxygenation changes during brain activation: The balloon model , 1998, Magnetic resonance in medicine.