Stable Model Order Reduction Method for Fractional-Order Systems Based on Unsymmetric Lanczos Algorithm
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摘要
This study explores a stable model order reduction method for fractional-order systems. Using the unsymmetric Lanczos algorithm, the reduced order system with a certain number of matched moments is generated. To obtain a stable reduced order system, the stable model order reduction procedure is discussed. By the revised operation on the tridiagonal matrix produced by the unsymmetric Lanczos algorithm, we propose a reduced order modeling method for a fractional-order system to achieve a satisfactory fitting effect with the original system by the matched moments in the frequency domain. Besides, the bound function of the order reduction error is offered. Two numerical examples are presented to illustrate the effectiveness of the proposed method.
This study explores a stable model order reduction method for fractional-order systems. Using the unsymmetric Lanczos algorithm, the reduced order system with a certain number of matched moments is generated. To obtain a stable reduced order system, the stable model order reduction procedure is discussed. By the revised operation on the tridiagonal matrix produced by the unsymmetric Lanczos algorithm, we propose a reduced order modeling method for a fractional-order system to achieve a satisfactory fitting effect with the original system by the matched moments in the frequency domain. Besides, the bound function of the order reduction error is offered. Two numerical examples are presented to illustrate the effectiveness of the proposed method.
This study explores a stable model order reduction method for fractional-order systems. Using the unsymmetric Lanczos algorithm, the reduced order system with a certain number of matched moments is generated. To obtain a stable reduced order system, the stable model order reduction procedure is discussed. By the revised operation on the tridiagonal matrix produced by the unsymmetric Lanczos algorithm, we propose a reduced order modeling method for a fractional-order system to achieve a satisfactory fitting effect with the original system by the matched moments in the frequency domain. Besides, the bound function of the order reduction error is offered. Two numerical examples are presented to illustrate the effectiveness of the proposed method.
引文
[1]A.Narang,S.L.Shah,and T.Chen,“Continuous-time model identification of fractional-order models with time delays,”IET Control Theory&Applications,vol.5,no.7,pp.900-912,2011.
[2]D.Idiou,A.Charef,and A.Djouambi,“Linear fractional order system identification using adjustable fractional order differentiator,”IET Signal Processing,vol.8,no.4,pp.398-409,2014.
[3]M.S.Tavazoei,“From traditional to fractional PI control:a key for generalization,”IEEE Industrial Electronics Magazine,vol.6,no.3,pp.41-51,2012.
[4]R.Magin,M.D.Ortigueira,I.Podlubny,and J.Trujillo,“On the fractional signals and systems,”Signal Processing,vol.91,no.3,pp.350-371,2015.
[5]C.A.Monje,Y.Q.Chen,B.M.Vinagre,D.Xue,and D.V.Feliu,Fractional-order Systems and Controls-Fundamentals and Applications,Springer-Verlag,London,2010.
[6]S.Victor,R.Malti,H.Garnier,and A.Oustaloup,”Parameter and differentiation order estimation in fractional models,”Automatica,vol49,no.4,pp.926-935,2013.
[7]L.Feng,“Parameter independent model order reduction,”Mathematics and Computers in Simulation,vol.68,no.3,pp.221-234,2005.
[8]R.W.Freund,“Krylov-subspace methods for reduced-order modeling in circuit simulation,”Journal of Computational and Applied Mathematics,vol.123,no.1-2,pp.395-421,2000.
[9]Z.Bai,“Krylov subspace techniques for reduced-order modeling of large-scale dynamical systems,”Applied Numerical Mathematics,vol.43,no.1-2,pp.9-44,2002.
[10]V.Druskina and V.Simoncini,“Adaptive rational Krylov subspaces for large-scale dynamical systems,”Systems&Control Letters,vol.60,no.8,pp.546-560,2011.
[11]Y.Lin,L.Bao,and Y.Wei,“Order reduction of bilinear MIMOdynamical systems using new block Krylov subspaces,”Computers&Mathematics with Applications,vol.58,no.6,pp.1093-1102,2009.
[12]R.S.Puri and D.A.Morrey,“Krylov-Arnoldi reduced order modelling framework for efficient,fully coupled,structural-acoustic optimization,”Structural and Multidisciplinary optimization,vol.43,no.4,pp.495-517,2011.
[13]J.H.Chen,S.M.Kang,J.Zou,and C.Liu,“Reduced-order modeling of weakly nonlinear MEMS devices with Taylor-series expansion and Arnoldi approach,”Journal of Microelectromechanical Systems,vol.13,no.3,pp.441-451,2004.
[14]H.J.Lee,C.C.Chu,and W.S.Feng,“An adaptive-order rational Arnoldi method for model-order reductions of linear time-invariant systems,”Linear Algebra and Its Applications,vol.415,no.2-3,pp.235-261,2006.
[15]Y.Yue and K.Meerbergen,“Parametric model order reduction of damped mechanical systems via the block Arnoldi process,”Applied Mathematics Letters,vol.26.no.6,pp.643-648,2013.
[16]M.Ahmadloo and A.Dounavis,“Parameterized model order reduction of electromagnetic systems using multiorder Arnoldi,”IEEE Transactions on Advanced Packaging,vol.33,no.4,pp.1012-1020,2010.
[17]Z.J.Bai,R.D.Slone,and W.T.Smith,“Error bound for reduced system model by Pade approximation via the Lanczos process,”IEEETransactions on Computer-Aided Design of Integrated Circuits and Systems,vol.18,no.2,133-141,1999.
[18]L.M.Silveira,M.Kamon,I.Elfadel,and J.White,“A coordinatetransformed Arnoldi algorithm for generating guaranteed stable reducedorder models of RLC circuits,”Computer Methods in Applied Mechanics and Engineering,vol.169,no.3-4,pp.377-389,1999.
[19]Z.Bai and R.W.Freund,“A partial pade-via-Lanczos method for reduced-order modeling,”Linear Algebra and Its Applications,vol.332,pp.139-164,2001.
[20]Y.L.Jiang and Z.H.Xiao,“Arnoldi-based model reduction for fractional order linear systems,”International Journal of Systems Science,vol.46,no.8,pp.1411-1420,2015.
[21]Z.Gao,“Reduced order modelling method for linear fractional-order systems based on ynsymmetric Lanczos algorithm,”Control and Decision,vol.31,no.8,pp.1499-1504,2016.
[22]P.Feldmann and R.W.Freund,“Efficient linear circuit analysis by Pade approximation via the Lanczos process,”IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems,vol.14,no.5,pp.639-649,1995.
[23]J.Shen and J.Lam,“H∞model reduction for positive fractional order systems,”Asian Journal of Control,vol.16,no.2,pp.441-450,2014.
[24]D.Matignon,“Stability properties for generalized fractional differential system,”in Proc.of Fractional Differential Systems,Models,Methods and Applications,Paris,1998,pp.145-158.
[2]D.Idiou,A.Charef,and A.Djouambi,“Linear fractional order system identification using adjustable fractional order differentiator,”IET Signal Processing,vol.8,no.4,pp.398-409,2014.
[3]M.S.Tavazoei,“From traditional to fractional PI control:a key for generalization,”IEEE Industrial Electronics Magazine,vol.6,no.3,pp.41-51,2012.
[4]R.Magin,M.D.Ortigueira,I.Podlubny,and J.Trujillo,“On the fractional signals and systems,”Signal Processing,vol.91,no.3,pp.350-371,2015.
[5]C.A.Monje,Y.Q.Chen,B.M.Vinagre,D.Xue,and D.V.Feliu,Fractional-order Systems and Controls-Fundamentals and Applications,Springer-Verlag,London,2010.
[6]S.Victor,R.Malti,H.Garnier,and A.Oustaloup,”Parameter and differentiation order estimation in fractional models,”Automatica,vol49,no.4,pp.926-935,2013.
[7]L.Feng,“Parameter independent model order reduction,”Mathematics and Computers in Simulation,vol.68,no.3,pp.221-234,2005.
[8]R.W.Freund,“Krylov-subspace methods for reduced-order modeling in circuit simulation,”Journal of Computational and Applied Mathematics,vol.123,no.1-2,pp.395-421,2000.
[9]Z.Bai,“Krylov subspace techniques for reduced-order modeling of large-scale dynamical systems,”Applied Numerical Mathematics,vol.43,no.1-2,pp.9-44,2002.
[10]V.Druskina and V.Simoncini,“Adaptive rational Krylov subspaces for large-scale dynamical systems,”Systems&Control Letters,vol.60,no.8,pp.546-560,2011.
[11]Y.Lin,L.Bao,and Y.Wei,“Order reduction of bilinear MIMOdynamical systems using new block Krylov subspaces,”Computers&Mathematics with Applications,vol.58,no.6,pp.1093-1102,2009.
[12]R.S.Puri and D.A.Morrey,“Krylov-Arnoldi reduced order modelling framework for efficient,fully coupled,structural-acoustic optimization,”Structural and Multidisciplinary optimization,vol.43,no.4,pp.495-517,2011.
[13]J.H.Chen,S.M.Kang,J.Zou,and C.Liu,“Reduced-order modeling of weakly nonlinear MEMS devices with Taylor-series expansion and Arnoldi approach,”Journal of Microelectromechanical Systems,vol.13,no.3,pp.441-451,2004.
[14]H.J.Lee,C.C.Chu,and W.S.Feng,“An adaptive-order rational Arnoldi method for model-order reductions of linear time-invariant systems,”Linear Algebra and Its Applications,vol.415,no.2-3,pp.235-261,2006.
[15]Y.Yue and K.Meerbergen,“Parametric model order reduction of damped mechanical systems via the block Arnoldi process,”Applied Mathematics Letters,vol.26.no.6,pp.643-648,2013.
[16]M.Ahmadloo and A.Dounavis,“Parameterized model order reduction of electromagnetic systems using multiorder Arnoldi,”IEEE Transactions on Advanced Packaging,vol.33,no.4,pp.1012-1020,2010.
[17]Z.J.Bai,R.D.Slone,and W.T.Smith,“Error bound for reduced system model by Pade approximation via the Lanczos process,”IEEETransactions on Computer-Aided Design of Integrated Circuits and Systems,vol.18,no.2,133-141,1999.
[18]L.M.Silveira,M.Kamon,I.Elfadel,and J.White,“A coordinatetransformed Arnoldi algorithm for generating guaranteed stable reducedorder models of RLC circuits,”Computer Methods in Applied Mechanics and Engineering,vol.169,no.3-4,pp.377-389,1999.
[19]Z.Bai and R.W.Freund,“A partial pade-via-Lanczos method for reduced-order modeling,”Linear Algebra and Its Applications,vol.332,pp.139-164,2001.
[20]Y.L.Jiang and Z.H.Xiao,“Arnoldi-based model reduction for fractional order linear systems,”International Journal of Systems Science,vol.46,no.8,pp.1411-1420,2015.
[21]Z.Gao,“Reduced order modelling method for linear fractional-order systems based on ynsymmetric Lanczos algorithm,”Control and Decision,vol.31,no.8,pp.1499-1504,2016.
[22]P.Feldmann and R.W.Freund,“Efficient linear circuit analysis by Pade approximation via the Lanczos process,”IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems,vol.14,no.5,pp.639-649,1995.
[23]J.Shen and J.Lam,“H∞model reduction for positive fractional order systems,”Asian Journal of Control,vol.16,no.2,pp.441-450,2014.
[24]D.Matignon,“Stability properties for generalized fractional differential system,”in Proc.of Fractional Differential Systems,Models,Methods and Applications,Paris,1998,pp.145-158.