H₂ Optimal Model Order Reduction Methods For Discrete-time Linear Systems

In natural science,many physical phenomena can be described as mathematical models.With the increase of the complexity for research problems,the dimension-s of mathematical models are accordingly raised,which brings great challenges to engineers in designing and simulating these systems.Therefore,it is necessary to simplify these systems to reduce the difficulty of theoretical analysis and shorten the simulation time.Model order reduction uses a low-order system to approxi-mate the high-order system,which facilitates the theoretical analysis difficulty,and improves the simulation efficiency.After briefly introducing the backgrounds,sig-nificances and current research status,this dissertation studies H₂ optimal model order reduction methods for discrete-time linear systems.This paper first investigates a trust-region method for H₂ optimal model order reduction of discrete-time dynamical systems.According to the H₂ error norm given by residues and poles,the H₂ error norms of single-input single-output（SISO）sys-tems and multiple-input multiple-output（MIMO）systems are achieved respectively in the single poles case.Then,gradients and Hessians are derived concerning the residues and poles of the reduced systems.The gradients and Hessians are used to establish the trust-region algorithm,which can lead to a monotonically decreasing H₂ error norm sequence.The rational Krylov model order reduction method is an effective method,which is suitable for reducing large-scale systems.This paper studies the H₂ optimal iterative rational Krylov model order reduction method for SISO discrete-time linear systems.According to the H₂ norm of the error system,the interpolation-based first-order necessary conditions are given and the corresponding iterative rational Krylov algorithm for SISO discrete-time linear system is proposed.Meanwhile,we prove that the interpolation-based first-order necessary conditions are equivalent to Wilson conditions and Hyland-Bernstein conditions for discrete-time linear systems.Based on the cross-Gramian,the H₂ optimal model order reduction method is explored for SISO discrete-time systems.With the H₂ error norm expressed by the cross-Gramian,the gradients of the H₂ error norm are derived with respect to the coefficient matrices of the reduced-order system.Therefore,the H₂ optimal first-order necessary conditions are obtained for SISO discrete-time systems.