A new nonlinear filter for parameters identification in dynamic systems and application to a transmission channel

In this paper we propose general nonlinear models for off-line and on-line parameters identification in dynamic systems. These numerical filters can be applied to any nonlinear system represented by a state equation and an observation equation both nonlinear. The theory of hidden Markov models is used to derive these algorithms starting from a of Baum & Welch type method. The proposed identification algorithm has two levels: the first level is an iterative and global algorithm (IGA), it estimates iteratively the parameters from a block of data. The second level is an ergodic recursive algorithm (ERA), it estimates the parameters in an adaptive manner. The estimators defined by these algorithms converge almost surely to the true values of the model parameters studied in [A. Khoukhi, T. Aliziane, M. Souilah, Un Algorithme Multi-Niveau d’Identification d’un Canal en Communication Numérique, JESA 36(4) (2002) 519–537; M. Souilah, A. Khoukhi, T. Aliziane, A new multi-level algorithm for identification and stochastic adaptive control of industrial manipulators, Eng. Simulation 26(4) (2004) 83– 98; M. Souilah, A new strategy for identification and control of mobile robots, Eng. Simulation 28(3) (2006) 35–48]. The advantage of the proposed nonlinear filters in relation to classical autoregressive models is the fact that the nonlinearity of the model is taken into account as it is and no linearizations are made around a nominal position. The variances of the added noises are also estimated. The mathematical convergence of the algorithms IGA and ERA is an open problem in the general case. We propose to this end an interesting conjecture based on ergodic theory. These algorithms are applied to identify the parameters of a transmission channel in data communication. Some simulation results showing the convergence of these algorithms are given.