Model Order Reduction Of Discrete Linear Systems Based On The Approximate Cross Gramian

The simulation,optimization and control of large-scale dynamic systems have been widely involved in engineering applications.In general,the dimensions of the differential or difference equations describing these dynamic systems are large,which results in the tremendous amount of data storage and computation if these systems are directly simulated on the computer.Therefore,many researchers in the field of engineering and mathematics have been devoting to the research of reducing the amount of data storage and computation in simulating large-scale systems.Model order reduction can well solve the above problem,and its basic idea is to convert a large-scale system into an approximate small system,which retains some properties of the original one,such as stability and passivity.For discrete linear systems,this dissertation studies the balanced proper orthogonal decomposition model order reduction method based on the approximate cross Gramian.Specifically,it consists of the following contents:The model order reduction method based on the approximate cross Gramian for symmetric discrete linear systems is studied.First,the approximate cross Gramian of symmetric systems is constructed via the snapshot method.Then,on the basis of the approximate cross Gramian,the projection matrix is constructed via the SVD(singular value decomposition)truncated reduction method to obtain the reduced system.What’s more,the corresponding model order reduction algorithm is established.After that,it is proved that the proposed method is equivalent to the balanced proper orthogonal decom-position method based on the approximate controllability Gramian and the approximate observability Gramian.Finally,the feasibility and effectiveness of the proposed method is verified via numerical examples.The model order reduction method based on the approximate cross Gramian is studied for general multiple input and multiple output discrete linear systems.First,the general multiple input and multiple output system is decomposed into several single input and single output subsystems.Then,the model order reduction method based on the approximate cross Gramian for symmetric discrete linear systems is utilized to reduce these subsystems.Then,according to the relationship between these subsystems and the original system,the reduced order subsystems are combined to construct the reduced order system of the original multiple input and multiple output system.In addition,the correlation between the approximate controllability Gramian and the approximate obs-ervability Gramian of the original system and the approximate cross Gramian of the single input and single output subsystem is also investigated.Finally,numerical examples illustrate that the proposed method is effective.