Structure Classification And It's Application Research On Singular Linear Control Systems

In practical engineering applications,singular system had its broader meaning and a broader form of system representation which compared with the normal control system.They had been extensively penetrated into modern science and technology fields such as marine,nuclear fuel reactor engineering,solid vibration,intelligent robot control and depth learning.And research significance in the modern control theory methods more and more attentioned by scientific researchers.Generally,there were great difference between singular system and normal system in general solution,dynamic order,transfer fuction matrix,hierarchy,pole and stability,which had brought many difficulties to the study of singular system,and thus the singular system became a very difficult task on system control research.In this paper,the structure of the linear singular system matrix was decomposed based on the theory of meticulous restricted equivalent transformation,the new dynamic standard form of the decomposition of the restricted equivalent transformation was obtained and the classification of states of homogeneous and non-homogeneous linear singular systems under equivalent transformation was given.The decomposition algorithm of preserving regularly,pulse,rank and other algebra properties ware proved by using range and kernel space theory,and also the uniqueness of the decomposition algorithm for the dynamic standard form.Therefor the slow subspace,the fast subspace and the independent state subspace of the state of the general linear singular systems ware given,and the numerical simulation results are verified by an example of a dynamic system model.Several conclusions and applications were obtained using these 3 sub spaces of the singular system state.In view of the previous studies on the regularity of linear singular systems,new criterion condition of the regular system were given from the independent state subspace of the state of the general linear singular systems,and relation between the regular system and the system state was clarified.Therefor the dependence of the former regularity was overcomed from the new dynamic standard form of the system.Researching from the slow subspace,the eigenvalues and eigenvectors of linear singular systems were defined by the elementary factors of the system characteristic matrix,thebottleneck of the control study of non square singular systems were solved.The Jordan canonical form calculation of linear singular systems using elementary factor feature matrix of the slow subspace,and the eigenvalue system matrix standard form was given.Researching from the fast subspace,the analysis of the singular system may exist on the basis of pulse using the decomposition algorithm for dynamic standard form of fast subspace,the characteristics and conditions of non pulse in singular systems were obtained.Therefor the effect of control pulse were achieved using the system input.The condition of controllability criterion for singular systems was given by using the dynamic standard form.The main innovations of this paper are as follows:(1)For a linearly singular system,a method and procedure for constructing a finite equivalent transformation were proposed,which made the generalized linear singular system be reduced to a simpler and more precise standard form by using the equivalent equivalent transformation.The research on the type and state classification of linear singular systems was studied.The results showed that the state space of general linear singular systems could be composed of slow subspace,fast subspace and unrelated state subspace.(2)By means of matrix theory,the regularity,impulsivity and the dimension of the subsystems under the system of constrained equivalent transformation were proved to be the invariants of the system.The general forms of the invertible matrices P and Q in the constrained equivalent transformations which were normalized by singular systems were given by using the range of the matrix and the kernel space method.The dimension of the subsystems(independent zero-pulse fast subspace,associated 0-1 order impulse fast subspace,etc.)were uniquely determined by the system matrix(7)E,A(8).(3)The necessary and sufficient conditions for the linear singular system to be a regular system were obtained,which the state space of the singular system had no extrinsic state subspaces and the singular systems were linearly independent based on the standard form obtained by the system decomposition.(4)The generalized eigenvalues and generalized eigenvectors of linear singular systems were redefined by elementary factors of eigenmatrix of linear singular systems.The previous form of pole values was extended to overcome the difficulties and bottlenecks encountered in the study of nonsquare singular systems.The base algorithm of slow subsystem space and thespace dimension of slow sub-system were given,and the equivalent Jordan decomposition of slow sub-system space was obtained by cyclic equivalent decomposition algorithm.(5)The criterion of impulsive criteria for general linear singular systems was obtained based on the standard form obtained by the system decomposition,a gain matrix design method of state feedback impulse cancellation and design method of input impulse using system are presented by this kind of impulse criterion theorem,as well as the new criterion of C-controllable,R-controllable and I-controllable.of linear singular systems.Finally,the overall research work was summarized on the basis of the existing research,and the future research was prospected.