Analysis And Control Of Complex System Based On Dynamic Graph’s Inclusion Principle

Complexity science is a burgeoning discipline in21st century, the exploration for it has become a most revolutionary direction in modern science. The representing carrier of complexity is complex system, which consists of multiple subsystems and many interconnections among them, just because of these interconnections, the whole system represents complexity, it brings us difficulty to coordinate and control the system. In the framework of the dynamic graphs and its inclusion principle, the dissertation takes a class of dynamic interconnected complex systems as research objects, detailedly discusses the problems of modeling, decomposing, connective stability, decentralized control and system coordination.The main contributions and creative works of this dissertation are listed as follows:(1) The inclusion principle of dynamic graphs is proposed and extented to the complex system. In view of the existing dynamic graph theoretical basis and practical demand, the paper sums up and develops the theory of dynamic graph systematically. It presents a complete definition of the dynamic graph and relevant basic concepts such as norm, distance, balance graph, stability, etc.. A mathematical model of dynamic graph is aslo established. For solving the problem of overlapping structure decomposition of dynamic graph, the inclusion principle and permuted inclusion principle of dynamic graphs are proposed first. By using these principles, dynamic graph can be decoupled into a group pair-wise subgraphs, and it is conducive to the realization of decentralized coordianted control. The method simplifies the structure of controller and reduces the control complexity. On this basis, according to the topology structure characteristics of dynamic graph and complex system, the dynamic inclusion principle for open-loop and closed-loop and permuted inclusion principle are proposed, these principles provides a theoretical framework for the decomposition and contract of complex system, the design of controllers and observers, and decentralized coordinated control of complex system.(2) Based on dynamic graph, dynamic inclusion principle and Lyapunov stability theory, relative stability conditions and determination methods are given after the stability of complex system with dynamic interconnections was discussed, Firstly, the connective stability of the pseudo-linear complex system with interconnections is considered. By using the dynamic inclusion principle, the system is decomposed as a group of pair-wise subsystems in the expanded space, after analyzing the stability of each pair-wise subsystems, the conective stability of the whole system is discussed, and the conditions of connective stability are obtained. Secondly, it studies the stability of Lotka-Volterra system for species, furthermore, it analyzes stability of complex system with L-V interconnected model. Finally, the stability of the nonlinear complex system with interconnections is discussed. By introduce the M-matrix theory, the asymptotically stable conditions are obtained.(3) A decentralized coordinated control method for complex system is proposed. By utilizing expanded transformation and permuted transformation, complex system is decomposed of a group of decoupled pair-wise subsystems in the expanded space. Then the fundmental coordinate controllers of each pair-wise subsystem and dynamic topology structure coordinater of the whole system are designed. The obtained decentralized coordinated controller can be contracted to the original space. The contracted controller has the same topology structure as the original system, it can follows the information structure change of orignal system, and can be used to control orignal system directly. Finally, the feasibility and effectiveness of the proposed methodology are demonstrated by an application of a four-area electric power system.