Model Order Reduction Method For The Discrete Linear Periodically Time-varying Systems

Model order reduction is an effective approximate method to deal with the highorder or complex systems in the engineering field.It’s purpose is to find a low-order system which can retain some important properties of the original system to approximate the original complex high-order systems.Replacing the original system with the low-order system in the related process of computer-aided engineering,the analysis difficulty and the amount of calculation can be reduced,the efficiency of simulation can be improved.At present,the model order reduction method has been used widely in the engineering fields,such as control systems,communication systems and circuit systems.In this paper,the singular value decomposition(SVD)truncation model order reduction method based on cross Gramian of the discrete linear periodically time-varying systems is studied firstly.The cross Gramian of the discrete linear periodically timevarying systems is defined,and two important properties of the cross Gramian of the discrete linear time-invariant systems are generalized to the discrete linear periodically time-varying systems.On the basis of the definition and properties of the cross Gramian,the SVD truncation model order reduction method which retains the time-varying structure is proposed.The output sample polyphase(OSP)transfer function is employed to prove the equivalence of the proposed method and the traditional balanced truncation method.Secondly,the model order reduction method based on the Fourier lifting system of the discrete linear periodically time-varying systems is discussed in this paper.By using exponentially modulated periodic(EMP)-Fourier series to expand the coefficient matrices of the original system,the Fourier lifting system of the original system is established.The Krylov subspace is constructed by expanding the transfer function of the Fourier lifting system on the discrete Laguerre functions,and the basis matrix of the Krylov subspace obtained by block Arnoldi algorithm is utilized as the projection matrix to construct the reduced order system.Furthermore,the moment matching property of the transfer function of the resulting reduced order system and the harmonic transfer function(HTF)of the original system is discussed,and the corresponding error bound is also derived.Finally,this paper explore the model order reduction method based on the standard lifting system of the discrete linear periodically time-varying systems.By expanding the transfer function of the standard lifting system on a set of discrete Laguerre functions,the Krylov subspace is constructed to obtain the projection matrix,and the corresponding reduced order system is established.Similarly,the moment matching property and the error estimation are also discussed.In addition,the numerical examples are employed to verify the feasibility and the effectiveness of the three model order reduction methods,respectively.