The Research On Solving Method Of Decoupling Transformation Based On Lancaster System

The system described by the second order differential equation is called the second-order system.The second-order system is widely used in the control system and other applications.In some cases,many higher-order systems can be converted into second-order systems.Therefore,it is of great practical significance to study the second-order system.For the research of second-order system people often need to decouple the system equations.Select the appropriate coordinate transformation to convert a multivariable coupled second-order system into a system with multiple independent single variable to study and remove the interaction between variables.The decoupling of the second order system involves the simultaneous diagonalization of three matrices,but the three matrices are generally impossible to achieve simultaneously diagonalization.In the numerical field,the decoupling of the second order system is studied by the simultaneous diagonalization of the Lancaster system,and it has been proved that almost all the second order systems can be realized.This article in view of based on Lancaster structure system decoupling transformation solution method research.First,The solution of the decoupling transformation of the second order system is transformed into the solution of the nonsingular solution of homogeneous Sylvester equation.Based on the Jordan decomposition theory and the homogeneous structure of the system,the non-singular solution of Sylvester equation is constructed.And the nonsingular complex solution is obtained by selecting appropriate parameters.On this basis,using MATLAB to do numerical experiments,the method to calculate the nonsingular solution of any homogeneous Sylvester equation is given,and the possibility of the method is proved.Secondly,according to the real matrix is similar to a similar instances,and the use of non-singular plural to construct a non-exotic real number solution.Regarding real number solution non opposite sex research,in order to enhance the real number solution non the opposite sex,as far as possible selects the matrix condition number small parameter.Since the parameter is not a fixed value,it is difficult to find the specific eigenvalue and the specific expression,so the application aspect is greatly restricted.By introducing the non-orthogonaldegree of the matrix,the non-orthogonality of the matrix is the reciprocal of the absolute value of the standard determinant of the matrix.The non-orthogonality can be used to find out the concrete expression,and the proof of the theory is given by using the non-orthogonal degree.Moreover,by using MATLAB to draw the image of the conditional number and the non-orthogonal degree with the parameter change,the feasibility of selecting the parameter with the non-orthogonal degree is proved.In this paper,the decoupling transformation method of the Lancaster system is given and some progress has been made in both theoretical and numerical fields.Not only has consummated regarding the second order system decoupling research,but also simple for the construction of nonsingular solutions.So it is of important practical significance.