The Structure On Nonsingular Solution Of The Sylvester Equation Based On Quadratic System Decoupling

Quadratic differential systems arise extensively in many applications, such as applied mechanics, electronic vibration, and hydromechanics. In certain conditions, many high-order systems can be simplified into quadratic systems, so it has an important practical significance to study and analyze the characteristics of quadratic systems deeply. The quadratic system decoupling means that based on the appropriate transformations or other methods, the multi-degree-of-freedom quadratic system can be transferred into the multiple totally independent single-degree-of-freedom quadratic subsystems. Quadratic system has three matrix parameters, therefore, the quadratic system decoupling is equivalent to the diagonalization of three matrix parameters simultaneously, which generally cannot be achieved by using the method of coordinate transformation directly. In the field of numerical algebraic, we achieve the decoupling of quadratic system through the diagonalization of Lancaster structure extended system block matrix of coefficient matrix simultaneously, and it has been proven to almost all of the quadratic differential systems.In this paper, we made a research of the solution of the decoupling transformations which was based on the Lancaster structure. Decoupling transformations was the key to system decoupling, and to determine whether the decoupling effectively or not. Therefore, it was important to study the solution method of decoupling transformations. But there was few research on the decoupling transformations at present, and the existing methods couldn't evaluate and control the degree of nonsingular transformation. The article focused on the issue for further research. Firstly, the problem was converted to the solution of the nonsingular solution of homogeneous Sylvester equations, and built the general solution of equations by the theory of matrix Jordan decomposition and isospectral flows, then obtained the nonsingular complex solution through selecting the parameters. Secondly, for the practical need of engineering, we construct the nonsingular real solution by means of the given nonsingular complex solution. Through the theory that the similarity of real matrix meaned the similarity of complex matrix, we got the nonsingular real solution by selcting the parameters. Finally, we reasearched the non-singularity of the real solutions. To improve the the non-singularity of the real solutions, relative knowledge of matrix condition number was used to determine the parameters. Through the deduction of the condition number bound, we drew the graph of of the condition number and its lower bound which varied from the parameters, and indicated the feasibility of parameters selection method.In the article, a practical method about the solutions of decoupling transformations for the quadratic system decoupling was proposed. Both in theory and practical engineering applications, we had made certain progress. And it made a contribution to the study of the quadratic system decoupling based on the Lancaster structure.