基于重启Lanczos过程的模型降阶方法
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摘要
针对大规模的线性时不变系统,提出了基于重启Lanczos过程的模型降阶方法。首先,通过重启Lanczos过程分别得到原始系统的可控Gram矩阵的近似矩阵及可观Gram矩阵的近似矩阵。然后,根据原始系统的可控Gram矩阵及可观Gram矩阵所满足的Lyapunov方程构造映射Sylvester方程并求解,对解进行双正交化,得到降阶所需的变换矩阵,从而得到降阶系统。运用此方法对大规模线性时不变系统进行降阶,能够得到具有较高近似精度的稳定的降阶系统。最后,数值算例验证了此方法是行之有效的。
We propose a model reduction method based on the restarted Lanczos process for some large scale linear time-invariant systems.Firstly,we utilize the restarted Lanczos process to obtain the approximate matrices of both the reachability Gramian matrices and the observability Gramian matrices of the original system.Then,according to the Lyapunov equations satisfied by the reachability Gramian matrices and the observability Gramian matrices respectively,we can construct the projected Sylvester equations,solve them and carry out the biorthogonal process on the solution to get the required transformation matrices,thus obtaining the order-reduced system of the original system.Using the proposed algorithm to reduce the order of the large scale linear time-invariant system,we can obtain a stable reduced order system in higher precision.Numerical examples verify the effectiveness of the proposed algorithm.
We propose a model reduction method based on the restarted Lanczos process for some large scale linear time-invariant systems.Firstly,we utilize the restarted Lanczos process to obtain the approximate matrices of both the reachability Gramian matrices and the observability Gramian matrices of the original system.Then,according to the Lyapunov equations satisfied by the reachability Gramian matrices and the observability Gramian matrices respectively,we can construct the projected Sylvester equations,solve them and carry out the biorthogonal process on the solution to get the required transformation matrices,thus obtaining the order-reduced system of the original system.Using the proposed algorithm to reduce the order of the large scale linear time-invariant system,we can obtain a stable reduced order system in higher precision.Numerical examples verify the effectiveness of the proposed algorithm.
引文
[1]Jiang Yao-lin.Model order reduction methods[M].Beijing:Science Press,2010.(in Chinese)
[2]Megretski A.H-infinity model reduction with guaranteed suboptimality bound[C]∥Proc of the 2006 American Control Conference,2006:448-453.
[3]Xu Kang-li,Yang Zhi-xia,Jiang Yao-lin.Arnoldi model reduction method based on implicitly restarted Krylov-Schur technology[J].Computer Engineering and Applications,2015,52(12):251-255.doi:10.3778/j.issn.1002-8331.1408-0083.(in Chinese)
[4]Gugercin S,Antoulas A C,Beattie C A.H2 model reduction for large scale linear dynamical systems[J].SIAM Journal on Matrix Analysis and Applications,2008,30(2):609-638.
[5]Antoulas A C.Approximation of large scale dynamical systems[M].Philadelphia:Society for Industrial and Applied Mathematics,2005.
[6]Sun Dao-heng,Ma Hai-yang,Li Wen-wang.Model order reduction and simulation of MEMS based on Krylov subspace[J].Chinese Journal of Mechanical Engineering,2004,40(5):58-61.(in Chinese)
[7]Liu Zhi-zhen,Li Dong-xu.Model reduction method for elastic-viscoelastic composite structure based on Krylov subspace[J].Journal of Vibration and Shock,2007,26(6):96-99.(in Chinese)
[8]Grimme E J,Sorensen D C,Dooren P V.Model reduction of state space systems via an implicitly restarted Lanczos method[J].Numerical Algorithms,1996,12(1):1-31.
[9]Bai Z,Slone R D.Error bound for reduced system model by Pade approximation via the Lanczos process[J].IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems,1999,18(2):133-141.
[10]Chu C C,Lai M H,Feng W S.The multiple point global Lanczos method for multiple-inputs multiple-outputs interconnect order reduction[J].IEICE Transactions on Fundamentals of Electronics Communications and Computer Sciences,2006,E89-A(10):2706-2716.
[11]Gallivan K,Vandendorpe A,Dooren P V.Sylvester equations and projection-based model reduction[J].Journal of Computational and Applied Mathematics,2004,162(1):213-229.
[12]Xu Shu-fang,Qian Jiang.Six lectures of matrix computations[M].Beijing:Higher Education Press,2011.(in Chinese)
[13]Rosa C S,Boris L,Rudy E.Stability preservation in projection-based model order reduction of large scale system[J].European Journal of Control,2012,18(2):122-132.
[14]Geradin M,Chen S L.An exact model reduction technique for beam structures:Combination of transfer and dynamic stiffness matrices[J].Journal of Sound and Vibration,1995,185(3):431-440.
[15]Leibfritz F,Lipinski W.Description of the Benchmark examples in COMPLeib1.0[R].Berlin:Department of Mathematics,University of Trier,2003.
[1]蒋耀林.模型降阶方法[M].北京:科学出版社,2010.
[3]徐康丽,杨志霞,蒋耀林.基于Krylov-Schur重启技术的Arnoldi模型降阶方法[J].计算机工程与应用,2015,52(12):251-255.doi:10.3778/j.issn.1002-8331.1408-0083.
[6]孙道恒,马海阳,李文望.基于Krylov子空间法的MEMS降阶建模与仿真[J].机械工程学报,2004,40(5):58-61.
[7]刘志臻,李东旭.弹性-粘弹性复合结构的Krylov子空间模型降阶方法[J].振动与冲击,2007,26(6):96-99.
[12]徐树方,钱江.矩阵计算六讲[M].北京:高等教育出版社,2011.
[2]Megretski A.H-infinity model reduction with guaranteed suboptimality bound[C]∥Proc of the 2006 American Control Conference,2006:448-453.
[3]Xu Kang-li,Yang Zhi-xia,Jiang Yao-lin.Arnoldi model reduction method based on implicitly restarted Krylov-Schur technology[J].Computer Engineering and Applications,2015,52(12):251-255.doi:10.3778/j.issn.1002-8331.1408-0083.(in Chinese)
[4]Gugercin S,Antoulas A C,Beattie C A.H2 model reduction for large scale linear dynamical systems[J].SIAM Journal on Matrix Analysis and Applications,2008,30(2):609-638.
[5]Antoulas A C.Approximation of large scale dynamical systems[M].Philadelphia:Society for Industrial and Applied Mathematics,2005.
[6]Sun Dao-heng,Ma Hai-yang,Li Wen-wang.Model order reduction and simulation of MEMS based on Krylov subspace[J].Chinese Journal of Mechanical Engineering,2004,40(5):58-61.(in Chinese)
[7]Liu Zhi-zhen,Li Dong-xu.Model reduction method for elastic-viscoelastic composite structure based on Krylov subspace[J].Journal of Vibration and Shock,2007,26(6):96-99.(in Chinese)
[8]Grimme E J,Sorensen D C,Dooren P V.Model reduction of state space systems via an implicitly restarted Lanczos method[J].Numerical Algorithms,1996,12(1):1-31.
[9]Bai Z,Slone R D.Error bound for reduced system model by Pade approximation via the Lanczos process[J].IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems,1999,18(2):133-141.
[10]Chu C C,Lai M H,Feng W S.The multiple point global Lanczos method for multiple-inputs multiple-outputs interconnect order reduction[J].IEICE Transactions on Fundamentals of Electronics Communications and Computer Sciences,2006,E89-A(10):2706-2716.
[11]Gallivan K,Vandendorpe A,Dooren P V.Sylvester equations and projection-based model reduction[J].Journal of Computational and Applied Mathematics,2004,162(1):213-229.
[12]Xu Shu-fang,Qian Jiang.Six lectures of matrix computations[M].Beijing:Higher Education Press,2011.(in Chinese)
[13]Rosa C S,Boris L,Rudy E.Stability preservation in projection-based model order reduction of large scale system[J].European Journal of Control,2012,18(2):122-132.
[14]Geradin M,Chen S L.An exact model reduction technique for beam structures:Combination of transfer and dynamic stiffness matrices[J].Journal of Sound and Vibration,1995,185(3):431-440.
[15]Leibfritz F,Lipinski W.Description of the Benchmark examples in COMPLeib1.0[R].Berlin:Department of Mathematics,University of Trier,2003.
[1]蒋耀林.模型降阶方法[M].北京:科学出版社,2010.
[3]徐康丽,杨志霞,蒋耀林.基于Krylov-Schur重启技术的Arnoldi模型降阶方法[J].计算机工程与应用,2015,52(12):251-255.doi:10.3778/j.issn.1002-8331.1408-0083.
[6]孙道恒,马海阳,李文望.基于Krylov子空间法的MEMS降阶建模与仿真[J].机械工程学报,2004,40(5):58-61.
[7]刘志臻,李东旭.弹性-粘弹性复合结构的Krylov子空间模型降阶方法[J].振动与冲击,2007,26(6):96-99.
[12]徐树方,钱江.矩阵计算六讲[M].北京:高等教育出版社,2011.