Some Model Order Reduction Methods Via Optimization On Matrix Manifolds

Some engineering application problems and physical phenomena are often modeled as differential equations.Generically,the more accurate the problem description is,the higher dimension the differential equation systems have.Directly simulating these dif-ferential equation systems require a lot of storage and time.Model order reduction uses a lower-dimensional system to approximate the original large system.It can efficiently reduce the storage and shorten the simulation time,and improve the efficiency of simu-lation and analysis by simulating the lower-dimensional system.Recently,model order reduction methods receive more and more attention,and are extensively applied to the areas of control,integrated circuits,solving partial differential equations and so on.Associated with optimization algorithms on matrix manifolds,this dissertation in-vestigates some model order reduction methods,including model order reduction of linear systems via the computation of Hankel singular values,H₂optimal model order reduction of coupled systems based on the gradient descent method on the Grassmann manifold,H₂optimal model order reduction of bilinear systems via the trust-region method on the Stiefel manifold and structure-preserving model order reduction of K-power systems on the Grassmann manifold.Specifically,this dissertation consists of the following several parts.By converting the computation of Hankel singular values of linear systems to lin-ear eigenproblem,the Riemannian Rayleigh quotient iteration and the Jacobi-Davidson method are individually employed to solve the linear eigenproblem,and the correspond-ing model order reduction algorithms are established.Then,the computation of Hankel singular values is converted into a generalized eigenproblem,and the Jacobi-Davidson method is used to solve this eigenproblem.The corresponding model order reduction algorithm is also developed.Regarding both the linear eigenproblem and the generalized eigenprobem,it is verified that the Jacobi-Davidson method can be seen as an acceler-ation of the Riemannian Rayleigh quotient iteration.Furthermore,it is demonstrated by theoretical analysis that these model order reduction methods are equivalent to the balanced truncation method to a degree.Concerning the coupled system with differential algebraic equation subsystems,theε-embedding technique and the stable representation are introduced.Then,this dis-sertation studies the H₂optimal model order reduction method for ordinary differential equation systems on the Grassmann manifold.Apply theε-embedding technique and the stable representation to the coupled system such that the closed-loop system is a stable ordinary equation system or the coupled system consists of stable ordinary differential equation subsystems.Then,the model order reduction method for ordinary differential equation systems can be applicable for the coupled system.It leads to a H₂optimal model order reduction method for the closed-loop system and a structure-preserving model order reduction method for the coupled system.Moreover,the H₂optimality for the reduced systems resulting from these two methods are discussed.In view of the trust-region method on the Stiefel manifold,H₂optimal model order reduction of bilinear systems is explored.According to the Euclidean gradient of the H₂error and the inner product associated with the Riemannian Hessian,the formula of the Riemannian trust-region subproblem is simplified,the corresponding model or-der reduction method is established,and the convergence of the proposed algorithm is analyzed.Compared with bilinear systems,K-power systems have their own special structure.This dissertation considers the structure-preserving H₂optimal model order reduction method for K-power systems on the Grassmann manifold.On the basis of the Euclidean gradient of the H₂error for bilinear systems,the Euclidean gradient of the H₂error for K-power systems is of the block-diagonal form.By restricting the projection matrix,the gradient of the H₂error on the Grassmann manifold is also block-diagonal.Then,the H₂optimal model order reduction method for K-power systems is accordingly proposed.The reduction process can refer to the block-diagonal structure,which can reduce the operation complexity.Numerical experiments demonstrate these proposed model order reduction methods.It is indicated that the reduced systems can be efficiently generated and the dynamic behavior of original systems can be well retained.Further,the reduced systems can also preserve the structure of coupled systems and K-power systems.