Identification of Stochastic MIMO Systems: Statistical Linearization and Anisotropic Norm Based on Hellinger-Tsallis Divergence

The paper analyzes problems of the stochastic system identification, which relate to the application of nonlinear measures of dependence of random values and processes. The authors considered the approaches using a consistent (as A.N. Kolmogorov defines it) measure of dependence based on the Hellinger-Tsallis divergence, viewed in the paper as a direct corollary of the Tsallis divergence of order ½ and the Hellinger distance. A constructive procedure of deriving a linear input/output model is proposed, which is a statistical equivalent of a non-linear multi-input/multi-output dynamic stochastic system driven by a Gaussian white-noise process. The keystone of such a procedure is the statistical linearization criterion that is the condition of the component-wise coincidence of the Hellinger-Tsallis mutual information (as a partial case of a corresponding divergence measure) of the system input and output processes, on the one hand, and the input and output processes of the linearized model, on the another hand. This approach enables one to obtain explicit analytical expressions determining matrix valued coefficients of the weight function of the linearized model. Meanwhile, the proposed technique does not depend on distribution and does not require applying restricting assumptions on the explicit analytical form of the joint probability distribution for the system input and output processes, which may degenerate the problem statement entity at all. As a feature, applying the Hellinger-Tsallis divergence is dual: on the one hand, within the statistical linearization problem statement it is a basis to construct the statistical linearization criterion; on the another hand, it is a basis of constructing the anisotropic norm of random vectors, which is used as an index of suitability of the input system process within the given statistical linearization problem statement.