a controlled system regarded as an aggregation of interrelated subsystems united by one overall functional purpose.

Examples of complex systems are the energy system, including the natural sources of energy (rivers, chemical or nuclear fuel deposits, and solar and wind energy), electric power plants, transformer substations, maintenance and repair personnel, transmission lines, and consumers of energy; the productive enterprise system, which includes the sources of raw material and power supply, personnel, technical equipment, the means of repair and upkeep of equipment, technical documents and records, finances, disposal of output, and accounting and bookkeeping; the commercial chain system, including suppliers of goods, storage facilities, merchandising facilities, personnel, finances, and accounting and bookkeeping; or the living organism, with its nervous system and hormonal regulating system and systems for nutrition, respiration, and motion, renewal of elements (cells) that may be destroyed, and reproduction of organisms.

The concept of the complex system arose as an expression of the systems approach to the formulation and solution of control problems related to cybernetics. The concept was introduced not for the purpose of classifying systems—that is, of separating them into complex and simple—but in order to specify a way of viewing the functioning of controlled systems on a large scale, while taking into account the many forms of the phenomena at work within them. The characteristic features of a complex system are the existence of distinguishable parts (controllable subsystems), the participation in the system of human beings along with machines and the natural environment, and the presence of material, power, and informational connections between parts of the systems as well as between the systems being studied and other systems.

In the systems approach, only those methods are used for the purpose of studying and improving complex systems that do not ignore the presence of close interconnections between the large number of factors which determine the functioning of the system being studied. Allowance is made for a greater or lesser degree of indefiniteness in the functioning of the system as a whole and of its separate parts. This indefiniteness is a result of the operation of accidental factors and the participation of human beings in the system. The interacting influences of the system and its natural surroundings are also taken into consideration, as are changes with the passage of time in the components of the system and of its external environment. Such an approach is effective in investigating complex technological, economic, and biological systems which have not been successfully studied by traditional methods based on the study of separate features of the system or particular phenomena in the system one at a time, or based on oversimplification of the subject under investigation.

The theory of the complex system is being developed in the context of solving five problems: the problem of language, the problem of a model, the problem of decomposition, the problem of aggregation, and the problem of strategy.

The problem of language consists of establishing a system of concepts necessary and sufficient for discussing the problems that relate to the complex system and for describing the facts and laws disclosed. No scientific trend can exist or develop without a language in which its ideas and methods can be formulated.

The problem of a model includes all the problems of constructing idealized (simplified) models of real systems suitable for the theoretical and experimental study of their qualities. The basic tasks are reduced to finding substitutes for real systems that are impossible to study directly because of their great complexity; the substitutes are expressed in the form of systems that are simpler and more accessible to theoretical investigation. The chief difficulty is that the models to be constructed must be sufficiently complex so that their properties correspond to the necessary degree to the similar properties of the original and are at the same time simple enough that they may be described and the necessary tasks solved by using the descriptions so compiled. Finding a compromise between these contradictions is frequently a very difficult problem which has been solved to date only for a relatively limited range of systems.

The problem of decomposition consists of breaking down the initial system into relatively discrete elements because the control of a complex system can be essentially simplified if the system is conceived as a kind of multiplication of the task of controlling the system’s parts. In this connection, however, it is necessary to overcome the difficulties involved in the selection of a method for breaking the system down. A method must be chosen that will provide the required simplification for solution procedures but will not lead to still greater errors resulting from disregard of some of the connections between elements of the system while it is being broken down into parts.

The problem of aggregation, which refers to the merging of several indexes into a single, summary index, has the aim of simplifying the solution of control problems for a complex system. As with decomposition, the aim is to break through the so-called multidimensionality barrier. This can be solved if the choice of means for merging indexes essentially facilitates the solution of control problems without resulting in inadmissible errors arising from reducing detail in describing the system.

The problem of strategy is that of choosing a means for assessing the condition of the system and of the environment and for working out a program of controlling influences that will best assure achievement of the aims of control. The main difficulties in forming a strategy of control are connected with the need for forecasting changes in the system and the environment which, by their very nature, cannot be precise.

In addition to the fundamental problems indicated above, a number of functional and operational problems must be solved in order to construct and use a complex system. Among the functional problems are determining the measures that must be taken to ensure that the system will fulfill its purpose and that its ability to function will be maintained. Operational tasks are directed toward solving the problems of planning complex operations and of controlling resources, reserves, and development of the systems.

Control of a complex system is based on the combined participation in the process of people and technical means. The fundamental technical means are computers and means for gathering, transmitting, presenting, and storing information. Control personnel together with the technical means constitute an automated system of control which carries out certain functions: information and research, planning, accounting and bookkeeping, operational control, and control of resources and supplies. Formalized operations are carried out by computers, whereas the people in charge are responsible for making decisions on the basis of nonformal methods.

Control of a complex system is organized, as a rule, in the form of a hierarchical system, the highest body organ of which controls several subdivisions on a lower level, each of which in turn has subdivisions at a still lower level. This kind of control structure makes it possible to enjoy the advantages of centralized and decentralized systems while avoiding their shortcomings.

A characteristic feature of the present trend in the growth of control technology for complex systems is the merging of systems for controlling technological processes and systems of organizational control into unified control systems in which the most efficient and economic use of information and technical means is assured.

In spite of the limited experience in organizing control of complex systems on the basis of scientific methods, the theory and technology of complex systems is developing intensively and is finding application in many branches of the national economy and defense, as well as in the sphere of repair and maintenance and of administrative management, where many factors must be taken into account and a large volume of information must be processed.