Mathematical Formalization and Algorithmization of the Main Modules of Organizational and Technical Systems

Abstract

The purpose of the research is to develop a generalized structural scheme of organizational and technical systems based on the general theory of management, which contains the necessary and sufficient number of modules and formalize on this basis the main management tasks that act as goals of the behavior of the management object. The main modules that directly implement the management process are the status assessment module of organizational and technical systems and the management module. It is shown that in traditional organizational and technical systems, including the decision-maker, the key module is the state assessment module of organizational and technical systems. In this regard, the key aspect of the work is to study the optimal algorithms for evaluating the state of processes occurring in the organizational and technical systems and develop on this basis the principles of mathematical formalization and algorithmization of the status assessment module. The research method is the application of the principles of the theory of statistical estimates of random processes occurring in the organizational and technical systems against the background of interference and the synthesis of algorithms for the functioning of the status assessment module on this basis. It is shown that a characteristic feature of random processes occurring in organizational and technical systems is their essentially discrete nature and Poisson statistics. A mathematical description of the statistical characteristics of point random processes is formulated, which is suitable for solving the main problems of process evaluation and management in organizational and technical systems. The main results were the definition of state space of the organizational and technical systems, the development of a generalized structural scheme of the organizational and technical systems in state space that includes the modules forming the state variable of the module assessment and module management. This mathematical interpretation of the organizational and technical systems structure allowed us to formalize the main problems solved by typical organizational and technical systems and consider optimal algorithms for solving such problems. The assumption when considering the problems of synthesis of optimal algorithms is to optimize the status assessment module of organizational and technical systems and the control module separately, while the main attention is paid to the consideration of optimal estimation algorithms. The formalization and algorithmization of the organizational and technical systems behavior is undertaken mainly in terms of the Bayesian criterion of optimal statistical estimates. Various methods of overcoming a priori uncertainty typical for the development of real organizational and technical systems are indicated. Methods of adaptation are discussed, including Bayesian adaptation of the decision-making procedure under conditions of a priori uncertainty. Using a special case of the Central limit theorem, an asymptotic statistical relationship between the mentioned point processes and traditional Gaussian processes is established. As an example, a nontrivial problem of optimal detection of Poisson signal against a background of Poisson noise is considered; graphs of the potential noise immunity of this algorithm are calculated and presented. The corresponding references are given to the previously obtained results of estimates of Poisson processes. For automatic organizational and technical systems, the generally accepted criteria for the quality of management of such systems are specified. The result of the review is a classification of methods for formalization and algorithmization of problems describing the behavior of organizational and technical systems.