Study On The Realization Of Multidimensional Fornasini-Marchesini State-space Model

In the field of control,one of the basic problem is the realization of system statespace models.The realization of the model is mainly aimed at the given system,according to the external relationship between input and output,to find a internal state-space description process in the system,so as to analyze and monitor the system better.However,compared with the one-dimensional system,the model structure of multidimensional system is very complex due to it contains many independent variables,and the process of model realization is also very challenging.In order to reduce the complexity of model analysis and the cost of hardware implementation in the subsequent,it is very important to realize a low-order state-space model result.This thesis mainly studies the realization of multidimensional Fornasini-Marchesini(F-M)state-space model,and puts forward three methods,namely,a novel constructive procedure to low-order F-M model realization,elementary operation approach to F-M model realization,and an improved elementary operation approach to F-M model realization.A novel constructive procedure is firstly proposed for the realization of a low-order F-M state-space model.For a given multidimensional system,essential properties for a backward shift space and a resolvent invariant space associated with the Gleason's problem from a given transfer function matrix are investigated,and some new sufficient conditions for low-order space construction are developed.Then,based on these conditions,a systematic constructive procedure is proposed for the multidimensional F-M model realization.It turns out that the new constructive procedure can generate an F-M model realization with much lower orders than the existing methods.After that,this thesis proposes a realization approach to generate an F-M model for multidimensional systems based on elementary operation approach,which can deal with the left matrix fraction description(MFD)and right MFD of transfer function at the same time,overcoming the weakness that only the left(or right)MFD of transfer function can be dealt within existing methods.First,a matrix property relation is revealed,and for the left MFD and the right MFD of a given transfer matrix,a corresponding initial polynomial matrix is established respectively.Then,reduce the F-M model realization problem to obtain a standard desired matrix form by applying elementary operations on the matrices.At last,a general construction procedure and two corresponding techniques to compute the F-M models are established.It turns out that the proposed approach can determine a lower-order F-M model easily compared with the existing methods.Finally,in order to solve the two problems of the high realization order w.r.t the right MFD and the choice of variable sequence is sensitive to the order of realization in elementary operation approach,an improved F-M model realization method for elementary operation approach is proposed.On the one hand,by exploring the relationship between the internal blocks of the initial polynomial matrix,a new technique is given,so that the realization can be performed between two matrix blocks at the same time,thus a lower order state-space model can be obtained compared with the previous realization techniques.On the other hand,the Gr(?)bner Bases algorithm is introduced,so as to obtain the Gr(?)bner basis of the polynomial set,such that the variables in the polynomial are extracted in a targeted manner during the process of the model realization,and it is even possible to extract the polynomials,greatly reducing the realization order of the final model.