A Modified Riccati Approach To Analytic Interpolation With Applications To Systems And Control

Analytic interpolation is a classical mathematical problem.In engineering applications,many problems can be transformed into analytic interpolation problems,such as robust control problem and spectral estimation problem.However,the existing methods have many problems in solving analytic interpolation problems,especially in multivariable analytic interpolation problems,there is no effective method to solve.By combining this problem with the control theory,a modified Riccati equation is derived,and the relationship between the solution of this equation and the solution of the analytic interpolation problem is established.This paper firstly takes the rational covariance extension problem as an example to introduce the modified Riccati equation,and shows the derivation process of the equation,which provides a basis for solving analytic interpolation problem.Secondly,for the scalar case analytic interpolation problem,this paper lists the corresponding formulas according to the three constraints of the problem,and deduces the modified Riccati equation.The relation between the solution of the equation and the solution of the analytic interpolation problem is established.At the same time,the existence and uniqueness of the solution of this equation are proved and this paper introduces the homotopy continuation method to solve this problem.Again,for multivariable analytic interpolation problem,this paper introduces the related theory of multidimensional control system and uses observability indices to represent the observer canonical form of the system,and then introduces how to derive the modified Riccati equation for multivariable case.At the same time,the relevant theorems about the existence and uniqueness of the solution are provided.In order to solve this equation,this paper introduces the homotopy continuation method of multivariable case.Finally,examples are given to illustrate how to transform the problems of robust control and model reduction into analytic interpolation problem.The simulation results show that the proposed theory and algorithm can solve the related engineering problems well.