About Evaluation of Complex Dynamical Systems

Methods are proposed for evaluation of complex dynamical systems, choice of their optimal operating modes, determination of optimal operating system out of given class of equivalent systems, system’s timeline behaviour analysis on the basis of versatile multicriteria, and multilevel analysis of behaviour of system's elements. 1. Introduction The study of different types of complex dynamical systems (technical, biological, social, economical) has attracted the attention of many researchers for a long time already [1–4]. Important direction of such a study is development of methods for evaluation of state, operating quality, and interaction between objects of those systems [5–10]. Among main problems arising in this case local and global evaluation of complex dynamical systems quality [10, 11] may be pointed out, as well as determination of their optimal operation modes [12], analysis of system behaviour during a certain period of time, and the choice of optimal operating system out of certain class of equivalent systems. We propose a unified approach to solving the above-listed problems. It is based on their comprehensive (which presupposes consideration of as many characteristics of system’s elements as possible), multicriteria, and multiparameter analysis. In order to provide operative processing of results the developed evaluation is multilevel, which means formulation of conclusions of different generalization degrees: from locally determining the behaviour of particular characteristics of system’s elements to finally determining quality of its operation in general. Problems considered in this paper and methods for their solution are illustrated with example of motion analysis of man’s musculoskeletal system (MMSS) with prosthetic lower limb [13–15]. 2. Problems Formulation Let us consider dynamical system consisting of elements able to operate under modes. In order to simplify the statement, let us assume that behaviour of every element of system under th mode is described by set of characteristics ,？？ ,？？ , where is the component of characteristic corresponding to th system’s element, , , where is the time for test research. Each of these characteristics results from experimental research or mathematical modelling for processes taking place in the system. To analyse the behaviour , let us use criteria. Let us denote with and the domains of reference and permissible values for characteristic according to , , , criterion. The term “equivalent” implies systems with the same content, type, and destination; the law of motion for which on the sequence of