Recursive identification of multivariable ARX models in the presence of a priori information: Robustness and regularization

• A priori information about stochastic disturbance.
• A priori information about unknown parameters.
• A priori information about disturbance introduces robustness (in the sense of Huber).
• A priori information about parameters introduces regularization (in the sense of Tikhonov).
• Improvement convergence rate of recursive algorithm on initial iterations.

This paper considers the nonlinear recursive algorithm for the identification of multivariable ARX models (autoregressive with exogenous input). It is assumed that the following a priori information is available:(a)a distribution class to which the stochastic disturbance belongs(b)a distribution class to which unknown parameters of ARX model belongA priori information (a) allows a description of outliers (large realizations of the stochastic process which are inconsistent with the largest part of population of observations) and introduces the nonlinear transformation of the prediction error in the recursive algorithm. A priori information (b) defines the initial conditions of the algorithm (for the vector of unknown parameters and the matrix gain) which increases the convergence speed in initial iterations. This intervention in the algorithm represents its regularization. Regularization problems are recently actualized thanks to statistical learning theory. Thanks to a priori information, the considered recursive algorithm is robust with respect to the uncertainty of statistical characteristics of disturbances and has the increased convergence speed in initial iterations. It is formulated theorem for convergence of estimated parameters with probability one. Simulation results illustrate the practical behavior of the algorithm.