Research On The Realization Of Multidimensional Singular Systems Based On Elementary Transformation

The system realization problem refers to the problem of the internal description of the whole state space by the external relation between the input and output of the system.For example,for a linear system,the external description can be the form of a transfer function,or sufficient input and output data,and its state space description is attributed to the determination of sufficient state variables and a complete coefficient matrix of differential or differential equations.At present,the realization of the regular system has been gradually improved,but the realization of the singular system has not been effectively resolved.The transfer function of singular system is non-true but rational,which adds essential difficulty to the realization of singular system.Furthermore,the existing one-dimensional system realization can get the minimum realization of the sys-tem,that is,the resulting coefficient matrix order is the smallest.However,for a given multidimensional system,so far there are no conditions to determine whether it is the minimum realization.Therefore,the realization of multidimensional singular system not only contains the difficult points of singular system realization,but also considers the realization of the minimum order problem of multidimensional system,which has not been perfected and solved.The results show that in the multidimensional singular system,the determinant value of the parameter matrix E on one side of the state variable is 0,and the 0 ma-trix on one side of the state variable can not be realized by using the regular realization method.Aiming at the characteristics of multidimensional singular system,this paper proposed a realization method based on elementary transformation,and applies it to the problem of establishing multi-dimensional singular system state space model of three-DOF helicopter.In this paper,a two-dimensional singular system realization method based on elementary transformation is proposed,the concrete realization steps are given,the equivalent relationship between the corresponding tectonic matrix and the transfer function,the applicable conditions are analyzed and mathematically proved,and the validity of the method is verified by the example of the transfer function and the advan-tages of the proposed method are analyzed.On this basis,the algorithm proposed in this paper is extended from the realization problem of two-dimensional singular system to the realization of multi-dimensional(more than three-dimensional)singular system.It is worth noting that the state space model of multidimensional singular system and two-dimensional singular system is obviously different,so the corresponding realization algorithm and steps are not the same.The calculation results of the transfer function ex-ample show that the method proposed in this paper can usually get a multi-dimensional singular system model with lower order.In addition,in order to verify the application ef-fect of the proposed method in the actual system,we target the multi-parameter complex characteristics of the three-DOF helicopter dynamic model,use the physics knowledge to analyze the relationship between the variables to obtain the transfer function,and use the realization method proposed in this paper to obtain the multi-dimensional singular system dynamics model of the helicopter,it provides strong support for the design of subsequent system analysis optimization,stability analysis,observer and compensator.