Structure-preserving Isospectral Transformation For Total And Partial Decoupling Of Self-adjoint Quadratic Systems

There are many systems with complex structure in practical life application and social production.The system is coupled when there are many inter-related subsystems.Coupled system is widely used in many fields,including astronautical technology,shipbuilding development,economical development,industrial manufacture,agricultural production and so on.However,the associated effects of a coupled system are difficult to analyze and control.In order to control the coupled system more effectively,the research of decoupling is particularly important.The decoupling research of the second-order linear system has great practical value and theoretical significance,and it is widely used in many disciplines such as applied mechanics,acoustic systems,electrical circuit simulation,fluid mechanics,microelectronic design and so on.At present,scientists and scholars focus on total decoupling of the second-order linear system,but there is a lack of research on partial decoupling for the coupled systems which only parts need to analyze and control.Besides,it often needs to rebuild the links of subsystems in many practical problems.To solve the above problems in the field of decoupling of secondorder linear systems,this paper establishes an algorithm of decoupling for the self-adjoint second-order linear system.This algorithm is capable of solving the problems of both total and partial decoupling,and reconstruction of the inter-related of subsystems.The emphasis of this paper is on researching the following four parts.Structure-preserving isospectral flow(SPIF)is an algorithm that based on the theory of maintaining the Lancaster structure of the quadratic eigenvalue problem and thus keep the spectrum invariant,which solves the corresponding second-order linear system decoupling problem.If the spectrum of the quadratic eigenvalue problem is invariant,the dynamical behavior of the corresponding second-order linear system is invariant.By introducing the spectrum preserving theory of Lancaster structure and SPIF algorithm,the supplementary analysis of SPIF is given,which provides a theoretical basis for the follow-up research of both total and partial decoupling of self-adjoint second-order linear system,and reconstruction of the inter-related of subsystems.To solve the self-adjoint second-order linear system decoupling problem,an algorithm called Lancaster structure-preserving congruence transformation is proposed by combining SPIF and the spectrum preserving theory of Lancaster structure.This algorithm can totally and partially decouple the self-adjoint second-order linear system,and reconstruct the inter-related of subsystems without solving the corresponding quadratic eigenvalue problem.Meanwhile,it can find the transformation that does the decoupling.Besides,the computational overhead is reduced by introducing the skew-symmetric parameter matrices.Numerical experiment results show that the given algorithm can realize the system decoupling problem.The nonlinear equation of motion for a class of second-order nonlinear system with triangular structure and the linearized equation are given.By proving the decrease of the total energy,the nonlinear system will reach an equilibrium position when the motion calms down.By comparing the coordinate changes in the motion of the nonlinear and its linearized subsystems,the equation of the linearized system can describe the motion of the original nonlinear system.The numerical experiment results show that the given algorithm can realize the linearized system decoupling problem.The theoretical proof of the Lancaster structure invariance of the quadratic eigenvalue problem and the error analysis of the structure in the numerical experiments are given.The error analysis reveals that the deviation of the structure due to numerical integration error of the transformation,and the deviation is of the same order of numerical integration error.In view of the change of spectrum caused by the deviation of the Lancaster structure,the proof that the change of spectrum has an upper bound and which is of the same order of numerical integration error of the transformation is given.Finally,convergence analysis of the transformation is propsed,and three improved tactics of the numerical experiment are given for the case that the decoupling is not ideal caused by the unideal of the convergence.