The Research On Quadratic System Decoupling

Quadratic differential systems arise extensively in many applications, such as control systems. In certain conditions, many high-order systems can be simplified into quadratic systems. Therefore, detailed research and analysis on the characteristics of quadratic systems have important practical significance, in which, the decoupling of quadratic systems are often required. The quadratic system decoupling means that based on the appropriate transformations or other methods, the multi-degree-of-freedom quadratic system can be linked with multiple totally independent single-degree-of-freedom quadratic subsystems.The research status of quadratic systems decoupled was outlined. Aiming at the poor decoupled method based on the Lancaster strcture, a solution is proposed.The decoupled system is identified by the isospectral of the systems, and the nonlinear problem to solve decoupling transformation is converted to the solution of similarity transformation matrix for two isospectral matrices. The Jordan canonical form which quadratic systems can be decoupled wad described in detail, the Jordan canonical forms of the real decoupled systems and complex decoupled systems are given respectively, a necessary and sufficient condition which general quadratic system can be decoupled is given.In order to describe the problem of quadratic system decoupled more clearly, we introduce the theory of Jordan triple. As to the general Jordan canonical form of real and complex decoupled system, a real Jordan triple and a complex Jordan triple are given, which can generate a real and a complex decoupled system. A proof which Jordan triple is equal to standard triple is given, and we get a method which can generate a decoupled system by the Jordan triple. Finally, we give a algorithm quadratic system can be decoupled, only need to compute the Jordan triple and the Similarity transformation matrix of original system, we can construct the decoupling matrix of the system, The numerical experiments are carried out respectively.This study in this paper is not only an improvement and development for the decoupling theory of quadratic system, but alao a starting point of the practical application.