辨识方法的计算效率(2):迭代算法
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摘要
讨论了最小二乘迭代辨识算法及其计算效率问题.最小二乘迭代算法由于涉及矩阵求逆运算,为减小计算量,提出了基于块矩阵求逆的最小二乘迭代辨识算法.基于块矩阵求逆的最小二乘迭代辨识算法不是一种新算法,只是从辨识算法的实现方式上降低计算负担,它与最小二乘迭代算法产生相同的参数估计,但计算量小.文中研究了伪线性回归系统、多元伪线性回归系统、多变量伪线性回归系统的最小二乘迭代辨识算法及其基于块矩阵求逆的最小二乘迭代算法.
This paper focuses on the computational efficiency of the least squares based iterative algorithms. The computational burdens of the least squares based iterative ( LSI) algorithms are heavy due to computing large-size matrix inversion. In order to reduce the computational burdens,the block matrix inversion based LSI algorithms are presented. The proposed methods can reduce the computational cost through simplifying the implementation of the least squares based iterative algorithms,thus the estimation accuracies remain unchanged. The least squares based iterative algorithms and the block matrix inversion based LSI methods are studied for pseudo-linear regression sys- tems,multivariate pseudo-linear regression systems and multivariable pseudo-linear systems.
This paper focuses on the computational efficiency of the least squares based iterative algorithms. The computational burdens of the least squares based iterative ( LSI) algorithms are heavy due to computing large-size matrix inversion. In order to reduce the computational burdens,the block matrix inversion based LSI algorithms are presented. The proposed methods can reduce the computational cost through simplifying the implementation of the least squares based iterative algorithms,thus the estimation accuracies remain unchanged. The least squares based iterative algorithms and the block matrix inversion based LSI methods are studied for pseudo-linear regression sys- tems,multivariate pseudo-linear regression systems and multivariable pseudo-linear systems.
引文
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[5]丁锋.系统辨识(4):辅助模型辨识思想与方法[J].南京信息工程大学学报:自然科学版,2011,3(4):289-318DING Feng.System identification.Part D:Auxiliary model identification idea and methods[J].Journal of Nanjing University of Information Science&Technolo-gy:Natural Science Edition,2011,3(4):289-318
[6]丁锋.系统辨识(5):迭代搜索原理与辨识方法[J].南京信息工程大学学报:自然科学版,2011,3(6):481-510DING Feng.System identification.Part E:Iterative search principle and identification methods[J].Journal of Nanjing University of Information Science&Technol-ogy:Natural Science Edition,2011,3(6):481-510
[7]丁锋.系统辨识(6):多新息辨识理论与方法[J].南京信息工程大学学报:自然科学版,2012,4(1):1-28DING Feng.System identification.Part F:Multi-innova-tion identification theory and methods[J].Journal of Nanjing University of Information Science&Technolo-gy:Natural Science Edition,2012,4(1):1-28
[8]丁锋.系统辨识(7):递阶辨识原理与方法[J].南京信息工程大学学报:自然科学版,2012,4(2):97-124DING Feng.System identification.Part G:Hierarchical i-dentification principle and methods[J].Journal of Nan-jing University of Information Science&Technology:Natural Science Edition,2012,4(2):97-124
[9]丁锋.系统辨识(8):耦合辨识概念与方法[J].南京信息工程大学学报:自然科学版,2012,4(3):193-212DING F.System identification.Part H:Coupling identifi-cation concept and methods[J].Journal of Nanjing Uni-versity of Information Science and Technology:Natural Science Edition,2011,4(3):193-212
[10]Ding F,Chen T.Identification of dual-rate systems based on finite impulse response models[J].International Journal of Adaptive Control and Signal Processing,2004,18(7):589-598
[11]Ding F,Chen T.Combined parameter and output estima-tion of dual-rate systems using an auxiliary model[J].Automatica,2004,40(10):1739-1748
[12]Ding F,Chen T.Parameter estimation of dual-rate sto-chastic systems by using an output error method[J].IEEE Transactions on Automatic Control,2005,50(9):1436-1441
[13]Ding F,Ding J.Least squares parameter estimation with irregularly missing data[J].International Journal of A-daptive Control and Signal Processing,2010,24(7):540-553
[14]Ding F,Liu G,Liu X P.Parameter estimation with scarce measurements[J].Automatica,2011,47(8):1646-1655
[15]Ding F,Chen T.Performance analysis of multi-innovation gradient type identification methods[J].Automatica,2007,43(1):1-14
[16]Ding F,Liu X P,Liu G.Multi-innovation least squares i-dentification for system modeling[J].IEEE Transactions on Systems,Man,and Cybernetics,Part B:Cybernetics,2010,40(3):767-778
[17]Ding F,Liu X P,Liu G.Auxiliary model based multi-in-novation extended stochastic gradient parameter estima-tion with colored measurement noises[J].Signal Pro-cessing,2009,89(10):1883-1890
[18]Zhang J B,Ding F,Shi Y.Self-tuning control based on multi-innovation stochastic gradient parameter estimation[J].Systems&Control Letters,2009,58(1):69-75
[19]Ding F,Chen T.Hierarchical gradient-based identifica-tion of multivariable discrete-time systems[J].Automati-ca,2005,41(2):315-325
[20]Ding F,Chen T.Hierarchical least squares identification methods for multivariable systems[J].IEEE Transac-tions on Automatic Control,2005,50(3):397-402
[21]Ding F,Chen T.Hierarchical identification of lifted state-space models for general dual-rate systems[J].IEEE Transactions on Circuits and Systems—I:Regular Pa-pers,2005,52(6):1179-1187
[22]Han H Q,Xie L,Ding F,et al.Hierarchical least squares based iterative identification for multivariable systems with moving average noises[J].Mathematical and Com-puter Modelling,2010,51(9-10):1213-1220
[23]Zhang Z N,Ding F,Liu X G.Hierarchical gradient based iterative parameter estimation algorithm for multivariable output error moving average systems[J].Computers and Mathematics with Applications,2011,61(3):672-682
[24]Ding J,Ding F,Liu X P,et al.Hierarchical least squares identification for linear SISO systems with dual-rate sam-pled-data[J].IEEE Transactions on Automatic Control,2011,56(11):2677-2683
[25]Ding F,Liu G,Liu X P.Partially coupled stochastic gra-dient identification methods for non-uniformly sampled systems[J].IEEE Transactions on Automatic Control,2010,55(8):1976-1981
[26]Ding F.Coupled least squares identification for multiva-riable systems[J].IET Control Theory and Applica-tions,2013,Accepted
[27]Ding F,Qiu L,Chen T.Reconstruction of continuous-time systems from their non-uniformly sampled discrete-time systems[J].Automatica,2009,45(2):324-332
[28]丁锋.辨识方法的计算效率(1):递推算法[J].南京信息工程大学学报:自然科学版,2012,4(4):289-300DING Feng.Computational efficiency of the identification methods.Part A:Recursive algorithms[J].Journal of Nanjing University of Information Science and Technolo-gy:Natural Science Edition,2012,4(4):289-300
[29]Golub G H,Van Loan C F.Matrix Computations[M],3rd Ed.Baltimore,MD:Johns Hopkins University Press,1996
[30]Hu H Y,Ding F.An iterative least squares estimation al-gorithm for controlled moving average systems based on matrix decomposition[J].Applied Mathematics Letters,2012,25(12):2332-2338
[31]Ding F,Liu X P,Liu G.Gradient based and least-squares based iterative identification methods for OE and OEMA systems[J].Digital Signal Processing,2010,20(3):664-677
[32]丁锋,杨慧中,刘飞.弱条件下随机梯度算法性能分析[J].中国科学:E辑,2008,38(12):2173-2184
[33]Ding F,Yang H Z,Liu F.Performance analysis of sto-chastic gradient algorithms under weak conditions[J].Science in China Series F:Information Sciences,2008,51(9):1269-128
[34]Bao B,Xu Y Q,Sheng J,et al.Least squares based itera-tive parameter estimation algorithm for multivariable con-trolled ARMA systems modelling with finite measurement data[J].Mathematical and Computer Modelling,2011,53(9/10):1664-1669
[35]Wang D Q,Ding F.Input-output data filtering based re-cursive least squares parameter estimation for CARAR-MA systems[J].Digital Signal Processing,2010,20(4):991-999
[36]Ding F,Liu Y J,Bao B.Gradient based and least squares based iterative estimation algorithms for multi-input multi-output systems[J].Journal of Systems and Control Engineering,2012,226(1):43-55
[37]丁锋,杨家本.大系统的递阶辨识[J].自动化学报,1999,25(5):647-654DING Feng,YANG Jiaben.Hierarchical identification of large scale systems[J].Acta Automatica Sinica,1999,25(5):647-654
[38]Duan H H,Jia J,Ding R F.Two-stage recursive least squares parameter estimation algorithm for output error models[J].Mathematical and Computer Modelling,2012,55(3/4):1151-1159
[39]Yao G Y,Ding R F.Two-stage least squares based itera-tive identification algorithm for controlled autoregressive moving average(CARMA)systems[J].Computers and Mathematics with Applications,2012,63(5):975-984
[40]Xiao Y S,Zhang Y,Ding J,et al.The residual based in-teractive least squares algorithms and simulation studies[J].Computers and Mathematics with Applications,2009,58(6):1190-1197
[2]丁锋.系统辨识(1):辨识导引[J].南京信息工程大学学报:自然科学版,2011,3(1):1-22DING Feng.System identification.Part A:Introduction to the identification[J].Journal of Nanjing University of Information Science&Technology:Natural Science Edi-tion,2011,3(1):1-22
[3]丁锋.系统辨识(2):系统描述的基本模型[J].南京信息工程大学学报:自然科学版,2011,3(2):97-117DING Feng.System identification.Part B:Basic models for system description[J].Journal of Nanjing University of Information Science&Technology:Natural Science E-dition,2011,3(2):97-117
[4]丁锋.系统辨识(3):辨识精度与辨识基本问题[J].南京信息工程大学学报:自然科学版,2011,3(3):193-226DING Feng.System identification.Part C:Identification accuracy and basic problems[J].Journal of Nanjing U-niversity of Information Science&Technology:Natural Science Edition,2011,3(3):193-226
[5]丁锋.系统辨识(4):辅助模型辨识思想与方法[J].南京信息工程大学学报:自然科学版,2011,3(4):289-318DING Feng.System identification.Part D:Auxiliary model identification idea and methods[J].Journal of Nanjing University of Information Science&Technolo-gy:Natural Science Edition,2011,3(4):289-318
[6]丁锋.系统辨识(5):迭代搜索原理与辨识方法[J].南京信息工程大学学报:自然科学版,2011,3(6):481-510DING Feng.System identification.Part E:Iterative search principle and identification methods[J].Journal of Nanjing University of Information Science&Technol-ogy:Natural Science Edition,2011,3(6):481-510
[7]丁锋.系统辨识(6):多新息辨识理论与方法[J].南京信息工程大学学报:自然科学版,2012,4(1):1-28DING Feng.System identification.Part F:Multi-innova-tion identification theory and methods[J].Journal of Nanjing University of Information Science&Technolo-gy:Natural Science Edition,2012,4(1):1-28
[8]丁锋.系统辨识(7):递阶辨识原理与方法[J].南京信息工程大学学报:自然科学版,2012,4(2):97-124DING Feng.System identification.Part G:Hierarchical i-dentification principle and methods[J].Journal of Nan-jing University of Information Science&Technology:Natural Science Edition,2012,4(2):97-124
[9]丁锋.系统辨识(8):耦合辨识概念与方法[J].南京信息工程大学学报:自然科学版,2012,4(3):193-212DING F.System identification.Part H:Coupling identifi-cation concept and methods[J].Journal of Nanjing Uni-versity of Information Science and Technology:Natural Science Edition,2011,4(3):193-212
[10]Ding F,Chen T.Identification of dual-rate systems based on finite impulse response models[J].International Journal of Adaptive Control and Signal Processing,2004,18(7):589-598
[11]Ding F,Chen T.Combined parameter and output estima-tion of dual-rate systems using an auxiliary model[J].Automatica,2004,40(10):1739-1748
[12]Ding F,Chen T.Parameter estimation of dual-rate sto-chastic systems by using an output error method[J].IEEE Transactions on Automatic Control,2005,50(9):1436-1441
[13]Ding F,Ding J.Least squares parameter estimation with irregularly missing data[J].International Journal of A-daptive Control and Signal Processing,2010,24(7):540-553
[14]Ding F,Liu G,Liu X P.Parameter estimation with scarce measurements[J].Automatica,2011,47(8):1646-1655
[15]Ding F,Chen T.Performance analysis of multi-innovation gradient type identification methods[J].Automatica,2007,43(1):1-14
[16]Ding F,Liu X P,Liu G.Multi-innovation least squares i-dentification for system modeling[J].IEEE Transactions on Systems,Man,and Cybernetics,Part B:Cybernetics,2010,40(3):767-778
[17]Ding F,Liu X P,Liu G.Auxiliary model based multi-in-novation extended stochastic gradient parameter estima-tion with colored measurement noises[J].Signal Pro-cessing,2009,89(10):1883-1890
[18]Zhang J B,Ding F,Shi Y.Self-tuning control based on multi-innovation stochastic gradient parameter estimation[J].Systems&Control Letters,2009,58(1):69-75
[19]Ding F,Chen T.Hierarchical gradient-based identifica-tion of multivariable discrete-time systems[J].Automati-ca,2005,41(2):315-325
[20]Ding F,Chen T.Hierarchical least squares identification methods for multivariable systems[J].IEEE Transac-tions on Automatic Control,2005,50(3):397-402
[21]Ding F,Chen T.Hierarchical identification of lifted state-space models for general dual-rate systems[J].IEEE Transactions on Circuits and Systems—I:Regular Pa-pers,2005,52(6):1179-1187
[22]Han H Q,Xie L,Ding F,et al.Hierarchical least squares based iterative identification for multivariable systems with moving average noises[J].Mathematical and Com-puter Modelling,2010,51(9-10):1213-1220
[23]Zhang Z N,Ding F,Liu X G.Hierarchical gradient based iterative parameter estimation algorithm for multivariable output error moving average systems[J].Computers and Mathematics with Applications,2011,61(3):672-682
[24]Ding J,Ding F,Liu X P,et al.Hierarchical least squares identification for linear SISO systems with dual-rate sam-pled-data[J].IEEE Transactions on Automatic Control,2011,56(11):2677-2683
[25]Ding F,Liu G,Liu X P.Partially coupled stochastic gra-dient identification methods for non-uniformly sampled systems[J].IEEE Transactions on Automatic Control,2010,55(8):1976-1981
[26]Ding F.Coupled least squares identification for multiva-riable systems[J].IET Control Theory and Applica-tions,2013,Accepted
[27]Ding F,Qiu L,Chen T.Reconstruction of continuous-time systems from their non-uniformly sampled discrete-time systems[J].Automatica,2009,45(2):324-332
[28]丁锋.辨识方法的计算效率(1):递推算法[J].南京信息工程大学学报:自然科学版,2012,4(4):289-300DING Feng.Computational efficiency of the identification methods.Part A:Recursive algorithms[J].Journal of Nanjing University of Information Science and Technolo-gy:Natural Science Edition,2012,4(4):289-300
[29]Golub G H,Van Loan C F.Matrix Computations[M],3rd Ed.Baltimore,MD:Johns Hopkins University Press,1996
[30]Hu H Y,Ding F.An iterative least squares estimation al-gorithm for controlled moving average systems based on matrix decomposition[J].Applied Mathematics Letters,2012,25(12):2332-2338
[31]Ding F,Liu X P,Liu G.Gradient based and least-squares based iterative identification methods for OE and OEMA systems[J].Digital Signal Processing,2010,20(3):664-677
[32]丁锋,杨慧中,刘飞.弱条件下随机梯度算法性能分析[J].中国科学:E辑,2008,38(12):2173-2184
[33]Ding F,Yang H Z,Liu F.Performance analysis of sto-chastic gradient algorithms under weak conditions[J].Science in China Series F:Information Sciences,2008,51(9):1269-128
[34]Bao B,Xu Y Q,Sheng J,et al.Least squares based itera-tive parameter estimation algorithm for multivariable con-trolled ARMA systems modelling with finite measurement data[J].Mathematical and Computer Modelling,2011,53(9/10):1664-1669
[35]Wang D Q,Ding F.Input-output data filtering based re-cursive least squares parameter estimation for CARAR-MA systems[J].Digital Signal Processing,2010,20(4):991-999
[36]Ding F,Liu Y J,Bao B.Gradient based and least squares based iterative estimation algorithms for multi-input multi-output systems[J].Journal of Systems and Control Engineering,2012,226(1):43-55
[37]丁锋,杨家本.大系统的递阶辨识[J].自动化学报,1999,25(5):647-654DING Feng,YANG Jiaben.Hierarchical identification of large scale systems[J].Acta Automatica Sinica,1999,25(5):647-654
[38]Duan H H,Jia J,Ding R F.Two-stage recursive least squares parameter estimation algorithm for output error models[J].Mathematical and Computer Modelling,2012,55(3/4):1151-1159
[39]Yao G Y,Ding R F.Two-stage least squares based itera-tive identification algorithm for controlled autoregressive moving average(CARMA)systems[J].Computers and Mathematics with Applications,2012,63(5):975-984
[40]Xiao Y S,Zhang Y,Ding J,et al.The residual based in-teractive least squares algorithms and simulation studies[J].Computers and Mathematics with Applications,2009,58(6):1190-1197