Solutions To Periodic Sylvester-conjugate Matrix Equations And Its Applications

In modern control theory,the theoretical development of engineering has always been inseparable from the linear periodic system.The linear periodic systems can be modified by modeling and designed to be suitable for linear time-varying systems or state-space equations for linear time-invariant systems.Therefore,linear periodic systems have a wide range of applications in life.In the actual process,we often encounter many situations in which the plural is involved.As a linear periodic system that fits the real life,we need to understand the practicability it has in each case,ensure that the system have their consistency characteristics in the field of real numbers or the complex number field.To this end,the convergence and consistency of linear periodic system applications can be investigated and verified by complex conjugate periodic matrix equations,so as to avoid such situations as communication system error or power system equipment damage and so on.The main contents of this paper can be summarized as follows:First,the finite iterative solution of the complex conjugate forward / backward periodic Sylvester matrix equation is studied.By setting up a new iteration step and applying the principle of conjugate gradient,the algorithm of time-invariant equation is extended to the domain of time-varying,and a new finite iteration method is given.Through theoretical derivation and simulation,it is proved that the algorithm can solve the objective equation without error by finite step iteration under arbitrary initial value condition.Second,the parametric solution of the complex conjugate period Sylvester matrix equation is studied.Flexibly use the superposition of the linear periodic system,obtain the matrix equation with the same solution as the original matrix equation and the input vector as the initial value,and use the right mutual decomposition principle to complete the parameterized operation of the system.Finally,the solution of the matrix equation satisfies periodicity,which verifies the consistency of the parameterized algorithm in the real and complex fields.Third,the parameterized pole assignment of the linear periodic matrix equation is studied.For the closed-loop system of linear discrete-cycle systems,the set of controllers is established,which is the so-called parameterization problem of control law.Then the system poles are arranged in the unit circle to make the system stable.In addition,the feedback gain of periodic system is obtained by using the parametric solution method of Sylvester matrix equation,that is,the so-called control law.Let the system meet certain robust performance to maintain the stability of the system.On this basis,the desired robust controller can be obtained by integrating optimal control law parameterization.Then,the parametric robust controller is put into the excitation system for application.Based on the parametric control principle,the optimal damping pole of the excitation system is introduced into the parameterized solution of the time-invariant Sylvester matrix equation,which makes the system have strong robustness.In order to reduce system function loss and enhance the degrees of freedom,the system can also be adjusted by periodic parametric control law.