Although the explicit commutativitiy conditions for second-order linear time-varying systems have been appeared in some literature, these are all for initially relaxed systems. This paper presents explicit necessary and sufficient commutativity conditions for commutativity of second-order linear time-varying systems with non-zero initial conditions. It has appeared interesting that the second requirement for the commutativity of non-relaxed systems plays an important role on the commutativity conditions when non-zero initial conditions exist. Another highlight is that the commutativity of switched systems is considered and spoiling of commutativity at the switching instants is illustrated for the first time. The simulation results support the theory developed in the paper.
Together with the dynamic development of modern computer systems, the possibilities of applying refined methods of nonparametric estimation to control engineering tasks have grown just as fast. This broad and complex theme is presented in this paper for the case of estimation of density of a random variable distribution. Nonparametric methods allow here the useful characterization of probability distributions without arbitrary assumptions regarding their membership to a fixed class. Following an illustratory description of the fundamental procedures used to this end, results will be generalized and synthetically presented of research on the application of kernel estimators, dominant here, in problems of Bayes parameter estimation with asymmetrical polynomial loss function, as well as for fault detection in dynamical systems as objects of automatic control, in the scope of detection, diagnosis and prognosis of malfunctions. To this aim the basics of data analysis and exploration tasks - recognition of outliers, clustering and classification - solved using uniform mathematical apparatus based on the kernel estimators methodology were also investigated