Maximum-Likelihood Parametric Identification Technique for Objects of Randomly Varying Structure

Abstract

Nonlinearities of many systems require descriptions, whose analytical forms depend on the actual state of the system or even on its past. Under random conditions both the coefficient functionals and the set of parameters have to be changed in random instances of time. Thus, not one model but many consecutive models have to be used in a random sequence. A maximum-likelihood (ML) method of online parametric identification of a wide class of such systems is concerned in this paper. The class consists of dynamical objects whose consecutive models take the form of a stochastic (ev. multidimensional) differential equations, which are affine in parameters but may be nonlinear with respect to the state (or state history). The answers to three major questions are given. Firstly, it is shown that, despite switching between models, the likelihood functional for each of the individual models can be defined similarly as in the case of one-model-systems and that the temporary change of a model does not affect the properties of ML estimator. Secondly, it is established that, if certain parameters appear in two (or more) models then the information gained while one of the models was being identified can be utilized during the identification of another model. This leads to the improvement of estimation in both models. The global likelihood function (for a group of models) is introduced to treat such cases. Thirdly, it is shown that also when the choice of the model, which should actually be considered depends on the (unknown) parameters being estimated then, in some cases, the consistent estimators can be established. The above is exemplified by results of estimation of parameters of two nonlinear oscillators. In one example, the restoring force is linear around the equilibrium and is strengthened apart from it. First, the identification is performed in each of these regions independently and next, simultaneous identification is applied. Another example concerns the hysteretic system with dry friction. Identification is performed in the state space (the data are the displacement and velocity only and the values of restoring force are not measured).