Solutions To Periodic Sylvester Bimatrix Equations And Its Applications

Periodic system is a relatively simple time-varying system,and people’s interest in this kind of system comes from many aspects.Firstly,many physical dynamic systems with cyclic characteristics can be modeled as periodic systems,and periodic systems are widely used in many theoretical analysis and engineering practice.In addition,periodic feedback can improve the performance of time-invariant systems in some cases,and the closed-loop system thus obtained is also a periodic system.In the study of periodic system,the solutions of periodic matrix equations are often encountered.When we study complex-valued linear periodic systems,we will encounter the periodic Sylvester bimatrix equations.At present,some research achievements have been made on the solutions of the bimatrix equations,but the solutions of the periodic Sylvester bimatrix equations are still very few,which require further analysis and research.This paper studies the solutions of periodic Sylvester bimatrix equations and its applications of the subject,the subject includes iterative algorithm based on conjugate gradient design periodic Sylvester bimatrix equations and using parameter method to solve generalized periodic Sylvester bimatrix equation,and based on this,this research will be theory applied to the pole assignment of maglev system,the system has strong robustness.The main contents of this paper include the following points:First,the parametric solutions of the generalized periodic Sylvester bimatrix equations are studied.Using the bimatrix mapping tool and some algebraic techniques,the generalized periodic Sylvester bimatrix equations are sorted out,and a parametric algorithm is proposed by using the right mutual prime decomposition method.The exact solution can be obtained by selecting different free parameters,thus providing sufficient degrees of freedom.Finally,a numerical example is given to verify the correctness and effectiveness of the method.Secondly,the finite iterative solutions of periodic Sylvester bimatrix equations are studied.By setting up the iteration step size,using the least square rule and the principle of conjugate gradient,the finite iteration algorithm is given.It is proved that the algorithm can solve the target equations by finite iteration under any initial condition.Finally,a simple example is given to verify the correctness of the proposed method.Thirdly,parametric pole assignment for complex-valued linear periodic systems is studied.By using the research theory in this paper,a period controller for complex-valued linear periodic system is designed,through which the system can have strong stability.When the bimatrices in a complex-valued linear periodic system are all matrices,the system becomes a linear periodic system.Next,for the linear periodic system,a periodic controller can also be used to make the system have good stability,and the method is applied to the magnetic suspension system,through theoretical derivation,the design problem of the periodic controller is transformed into the solutions of the corresponding matrix equations.On the magnetic levitation system mathematical model is set up in the first place,the internal fixed constant parameters after get a simplified mathematical model,through a series of transformations,we get the state space expression of magnetic levitation system,then we will discretization,adopt periodic control law,the design of the controller are ideal,finally through Matlab simulation,the simulation results show that the controller can make the system stable goals are met.This method can not only stabilize the system,but also meet other performance requirements of the system due to its design freedom.