Redundance-free description of partitioned complex systems

Abstract

In this paper, we investigate a methodology of modelling a class of hierarchically structured complex systems. It is assumed that the subsystems of a system X are characterized as the equivalence classes of certain family of equivalence relations generated by a collection of classification criteria. The cardinalities of subsystems are assumed finite and are used as the system's quantitative description. The input data structure describing the system consists of a family of data matrices F containing the cardinalities of certain subsystems and may be redundant. We propose a formal approach to the operations on classification criteria and data matrices which allows us to investigate the properties of the above information structure in a systematic way. We define the base B of a family of data matrices F as its minimal subfamily, such that any coefficient from F can be derived from the coefficients of B. We give a constructive characterization of B and propose an algorithm to find the base. Further, we use the relations between the base matrices to select a subset of independent coefficients which will be called the reduced base. For time-varying systems, the base of the input data structures for each moment of time can be applied to construct a sequence of non-redundant state vectors. Finally, we formulate a discrete control system model of evolution of the system which may be used in Kalman filtering, prediction and optimization procedures.