Quadratic System Decoupling Algorithm Researches Based On Lancaster Structure And Its Applications

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, it is often necessary to research system decouplings. The quadratic system decoupling is to seek the appropriate transformations or other methods so that the multi-degree-of-freedom quadratic system is linked with multiple totally independent single-degree-of-freedom quadratic subsystems. In this paper, the quadratic system decoupling methods based on Lancaster structure are lucubrated, and ship pitch-heave motion system examples are used to testify the applicability of proposed algorithm. The detail discussion and results may be summarized as follows.Firstly, a solution method of decoupling transformations is proposed based on Sylvester equation. As one of forefront researches in numerical algebra field, the Structure Preserving Isospectral Flows simplifies quadratic systems through the Lancaster structure preserving transformations and isospectral transformations. Although this method can output decoupling systems, it can not output the corresponding decoupling transformations. In this paper, a solution method of decoupling transformations is proposed based on Sylvester equation. The nonlinear problem to solve decoupling transforms is converted to homogeneous Sylvester equations solution, and is converted to homogeneous linear equations solution by matrix Kronecker product theory furthermore. In addition, the existence of non-singular decoupling transformations is discussed. Finally, for quadratic system without external force, a structure preserving flows method based on similarity transformation is proposed. The numerical experiments are carried out to verify the methods respectively.Secondly, a quadratic system decoupling method is proposed based on spectrality. Whether the Structure Preserving Isospectral Flows is feasible or not is judged by the system's isospectrality before and after decoupling, although the algorithms designed to take into account the system's isospectrality, but it is not easy at the algorithm implementation. For this problem, an isosperctral decouling method is presented for quadratic system. First of all, an equivalent decoupling system is formatted according to the spectral characteristics of the original system. Then, the corresponding decoupling transformations are output by make use of Sylvester equation solution. Finally, in view of quadratic system without external force, the similar decoupling transformation solution method is deduced. The numerical experiments are carried out respectively.Thirdly, the structural characteristics of quadratic system decoupling transformations are researched. Decoupling transformations structural depict is always a nodus for quadratic system decoupling. At present, although it has been proved in theory that the decoupling transformations exist for almost all the quadratic systems based on Lancaster structure, it is difficult to characterize the structure of these transformations. In this paper, a structural characteristic is proposed by block matrix operations. In particular, the existence of diagonal form and triangular form decoupling transform are researched, and the relative theorem, proof and solution methods are proposed.Fourthly, to testify the applicability of the algorithms by practical system decoupling, the numerical experiments are shown based on 40 group ship pitch-heave motion data from pool experiments. On one side, the Structure Preserving Isospectral Flows and decoupling transformation solution based on Sylvester equation are used to ship longitudinal motion decoupling. In numerical experiments,8 group ideal results are shown, and the problems in experiment are analyzed. On the other side, the quadratic system decoupling method based spectrum information is applied. In numerical experiments, the rest 32 group ideal results are shown. This part of researches not only testify the feasibility and applicability of the proposed algorithms, and show the comparative analysis of proposed algorithms, but also reduce ship pitch-heave motion system to two totally indepent single-degree-of-freedom quadratic systems, in other words, decouple the system.The study in this paper is not only an improvement and development for the decoupling researches of quatratic differential system, but also a new attempt for its application in practical systems.