Mathematical and computer tools of discrete dynamic modeling and analysis of complex systems in control loop

equilibrium thermodynamics, etc. Among the computer simulation techniques are cellular automata, multi-agent techniques, evolutionary programming, Monte Carlo methods, etc. Since analytical treatments alone do not allow us to understand a complex system, computer simulations play a key role in our understanding of how these systems function and work. This is also true and possibly in a more degree for complex control systems. The main characteristic of modern complex control systems is that it is impossible to uniquely and adequately describe these systems, using classical mathematical methods. Classical mathematical models are suitable just for a few problem domains, which are static and comprehensible, and have most general properties. And there still remains a wide range of complex problems that can not be described by the existing formal methods.

Today we can distinguish several basic forms of complexity: structural (geometrical, topological), dynamical, hierarchical, algorithmic, and large scale. Taking into account the interplay between intellectualized mathematical and information technologies of control and decision support play an important role in modeling of processes of evolution and functioning of complex (large-scale) systems.

Complex systems are usually difficult to model, design, and control. In studying complex systems, the behavior of which depends on the elements interactions, an integrative system-theoretic (top-down) approach is more preferable, as compared to a reductionist (bottom-up) one. However, a compromise between both approaches should be found.

A central goal of this work is to propose models and modeling technique that are useful when applied to the complex systems, which can with a sufficient accuracy be described by models of development of hierarchical systems. Aiming at this, we develop method for constructing discrete models of complex hierarchical dynamic systems subject to external hierarchical dynamic control mechanism and their problem-oriented interpretation. The method includes:

1. Creating multivariate multilevel hierarchical structural model based on system analysis; 2. General mathematical formalization; 3. Constructing hierarchical dynamic graph model to solve a system development control problems and Mathematical and computer tools of discrete dynamic modeling and analysis of complex systems in control loop ARMEN G. BAGDASARYAN C to analyze system dynamic characteristics related to the attainability of desirable states and goals; 4. Specializing the model to the scenario-type schemes of control of complex systems.

Modeling and analysis of control and dynamic processes in complex multi-component large-scale systems make it necessary to operate with multiple state coordinates. This is caused by: (1) the fact that complex large-scale system behavior is influenced by a number of factors of various nature which leads to large amount of system parameters, indicators, and variables; (2) lack of sufficient information (incompleteness, uncertainty) on the state and processes that influence system development, especially, for systems and objects belonging to weakly-formalizable ones. Therefore, a peculiar approach which will allow taking into account all essential diverse factors that determine system activity and behavior under the influence of external control actions is needed. The modeling technique developed allows one to cope with the above mentioned problems. The control and problem domains have the following features:

1. Multilevel dynamical systems consisting of a set of autonomous elements (subsystems) with local (individual) and global (corporate, general) problems and goals is considered as a canonical model of control object; 2. The external dynamic control mechanism in a system is considered as a set of control actions initiating multilevel state dynamics of control object; 3. The long-term databases and monitoring that characterize the changing of parameters and indicators can be used as the main source of information about system behavior, development and control problems. Databases, monitoring data and other statistical material facilitate observing for the changes in parameters at different time intervals, which has an extreme promise for understanding the global regularities in system dynamics. Monitoring includes observation of the current situation around the system. The processes under monitoring are interpreted in the form of state dynamics and estimations, and tendencies of system development as well. The system goals are formulated as consistent dynamics of these processes. Monitoring of the present situation enables (1) discovering new factors and parameter estimations influencing the system development, (b) establishing possible new or desirable states and goals. In this case the model is updated; 4. Due to the hierarchical structure of a parameter set, a multilevel (hierarchical) control loop based on a set of independent closed control

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