基于随机逼近算法的无模型直接自适应控制
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摘要
为了解决对受控对象数学模型结构的依赖和未建模动态的问题,自适应控制领域提出了无模型自适应控制的概念,即不需要建立被控对象的数学模型,直接利用系统的输入输出数据来设计控制器对系统进行控制。无模型直接自适应控制从方法上讲更加贴近真实意义上的控制思想,而非模型论。实践中,因为避免了建立数学模型的过程,使其应用更为广泛。
由于随机逼近算法具有算式结构简单,不需要对象的具体数学模型以及对含有噪声的数据有较好的处理能力等特点,使其非常适用于无模型直接自适应控制。因此,本文在对随机逼近算法进行研究的基础之上,主要侧重于研究基于随机逼近算法的无模型直接自适应控制。通过各种控制方案进行分析研究,发现存在的问题,并对方案进行进一步的改进。
本文在研究直接自适应神经网络控制(DA-NNC)中发现,系统的跟踪响应存在一定程度的偏差,针对这一问题提出了加入PID补偿器混合控制的方案,并通过仿真验证了混合控制可以有效地减小系统跟踪响应偏差。另外,在针对虚拟参考概念的研究中,深入讨论了滤波器在系统设计中的重要作用,给出了其具体的计算形式。并结合虚拟参考的概念,提出了基于随机逼近算法的虚拟参考无模型直接自适应控制,最后,通过仿真例子验证了控制方法的正确性和实用性。
本文还对基于随机逼近算法的无模型直接自适应控制,提出了一些需要深入研究和探讨的问题。
In order to solve the problems related to the dependence with the model of the system and the unmodeling system dynamics, model-free adaptive control method was proposed. Model-free adaptive control can implement the control only by the input data and output data without knowing the system model. Mode-free direct adaptive control is not model based theory; it is theoretically more close to the true meaning of control concept than other methods. Because of the avoidance of setting up the mathematical model, it is widely used in practice.
Stochastic approximation method is suited to be used in model-free direct adaptive control due to its simple algorithm, free of the specific mathematic model of plant and the excellent ability to handle data with noise. Therefore, this paper pay much attention to the study of model-free direct adaptive control based on stochastic approximation algorithm. Through analyzing various control methods, the paper finds some exist problems and make some further improvements.
In the process of studying the DA-NNC method, we find the bias of certain degrees exist in system tracking response. The proposed hybrid direct adaptive control with PID compensator can effectively reduced the existing error, and simulation result shows that this improvement is efficient. In addition, a further discussion on the importance of prefilter in the design is made while researching the virtual reference concept. We give the specific form of the filter, and proposed virtual reference model-free direct control method, and finally, the correctness and usefulness is proved through simulations.
This paper also gives out some questions, which need further research and discussion, about the Model-free direct adaptive control based on stochastic algorithms.
由于随机逼近算法具有算式结构简单,不需要对象的具体数学模型以及对含有噪声的数据有较好的处理能力等特点,使其非常适用于无模型直接自适应控制。因此,本文在对随机逼近算法进行研究的基础之上,主要侧重于研究基于随机逼近算法的无模型直接自适应控制。通过各种控制方案进行分析研究,发现存在的问题,并对方案进行进一步的改进。
本文在研究直接自适应神经网络控制(DA-NNC)中发现,系统的跟踪响应存在一定程度的偏差,针对这一问题提出了加入PID补偿器混合控制的方案,并通过仿真验证了混合控制可以有效地减小系统跟踪响应偏差。另外,在针对虚拟参考概念的研究中,深入讨论了滤波器在系统设计中的重要作用,给出了其具体的计算形式。并结合虚拟参考的概念,提出了基于随机逼近算法的虚拟参考无模型直接自适应控制,最后,通过仿真例子验证了控制方法的正确性和实用性。
本文还对基于随机逼近算法的无模型直接自适应控制,提出了一些需要深入研究和探讨的问题。
In order to solve the problems related to the dependence with the model of the system and the unmodeling system dynamics, model-free adaptive control method was proposed. Model-free adaptive control can implement the control only by the input data and output data without knowing the system model. Mode-free direct adaptive control is not model based theory; it is theoretically more close to the true meaning of control concept than other methods. Because of the avoidance of setting up the mathematical model, it is widely used in practice.
Stochastic approximation method is suited to be used in model-free direct adaptive control due to its simple algorithm, free of the specific mathematic model of plant and the excellent ability to handle data with noise. Therefore, this paper pay much attention to the study of model-free direct adaptive control based on stochastic approximation algorithm. Through analyzing various control methods, the paper finds some exist problems and make some further improvements.
In the process of studying the DA-NNC method, we find the bias of certain degrees exist in system tracking response. The proposed hybrid direct adaptive control with PID compensator can effectively reduced the existing error, and simulation result shows that this improvement is efficient. In addition, a further discussion on the importance of prefilter in the design is made while researching the virtual reference concept. We give the specific form of the filter, and proposed virtual reference model-free direct control method, and finally, the correctness and usefulness is proved through simulations.
This paper also gives out some questions, which need further research and discussion, about the Model-free direct adaptive control based on stochastic algorithms.
引文
[1] 韩曾晋,自适应控制系统. 清华大学出版社,1995
[2] K.J.奥斯特隆姆 B.威顿马克,自适应控制. 北京:科学出版社,1992
[3] Hakan Hjalmarsson, Svante Gunnarsson and Michel Gevers, A convergent iterative Restricted Complexity Control Design Scheme, Proceedings of the 33rd Conference on Descision and control Lake Buena Vista, December 1994
[4] J.C.Spall and J.A.Cristion. Model-free control of nonlinear stochastic systems approximation. Automatica., Vol.33,pp.109-112,1998
[5] 夏长亮、祁温雅,基于 RBF 神经网络的超声波奠基参数辨识与模型参考自适应控制,中国科技论文在线
[6] Robbins and S.Monro, A stochastic approximation method , Ann.math. Statist, Vol. 22 pp.400-407 1951
[7] E.Kiefer and J.Wolfwitz, Stochastic estimation of the Maximum of a regression function, Ann. Math. Statist., Vol.23 pp.462-466 1952
[8] J.R.Blum, Multidimensional stochastic approximation methods, Ann.Math. Statist, Vol.25 pp.737-744 1954
[9] J.C.Spall,. A Stochastic Approximation Technique for Generating Maximum Likelihood Parameter Estimates, Proceedings of the American Control Conference, Minneapolis, MN, pp. 1161-1167,10-12 June 1987
[10]J.C.Spall, Multivariate Stochastic Approximation Using a Simultaneous Pertur- bation Gradient Approximation, IEEE Transactions on Automatic Control, vol. 37, pp. 332-341 1992
[11]Sadegh, P. and Spall, J.C, Optimal Random Perturbations for Stochastic Approximation Using a Simultaneous Perturbation Gradient Approximation, Proceedings of the American Control Conference, Albuquerque, NM, pp. 3582-3586 ,June 1997
[12]Spall, J.C. and Cristion, J.A. , Nonlinear Adaptive Control Using Neural Networks: Estimation With a Smoothed Form of Simultaneous Perturbation Gradient Approximation Statistica Sinica, vol. 4, pp. 1- 27,1994
[13]Spall, J.C. Adaptive Stochastic Approximation by the Simultaneous Perturbation Method, IEEE Transactions on Automatic Control, vol. 45, pp. 1839-1853,2000
[14]Zhu, X. and Spall, J.C.A Modifed Second-Order SPSA Optimization Algorithm for Finite Samples International Journal of Adaptive Control and Signal Processing, vol. 16, pp. 397-409, 2002
[15]狄昂照 ,随机逼近,科学出版社 1994
[16]朱玉,王先来,对未建模型的工业过程对象进行控制的新方法,自动化与仪器仪表, pp35-39,2002 年第 3 期(总第 101 期)
[17]聂赞坎, 徐宗本,随机逼近理论与自适应算法,科学出版社 2003
[18]L.jung, Analysis of recursive stochastic algorithms, IEEE Trans AC-22(4), pp551-575,1977
[19]Kushner and Clark, Stochastic Approximation Methods for Constrained and Unconstrained Systems, Springer, 1978
[20]陈翰馥,朱允民,变界截尾随机逼近算法,中国科学(A 辑) 1986
[21]Chin, D.C., Comparative Study of Stochastic Algorithms for System Optimization Based on Gradient Approximations, IEEE Transactions on Systems, Man, and Cybernetics, vol. 27, pp. 244-249. 1997
[22]J.C.Spall, Introduction to stochastic search and optimization :Estimazation, simulation and control, pp.176- 204, Willey 2003
[23]Dippon, J. and Renz, J. Weighted Means In Stochastic ApproximationOf Minima, SIAM Journal Of Control and Optimization, vol. 35, pp. 1811-1827.1997
[24]Wang, I.-J. and Chong, E.K.P. A Deterministic Analysis of Stochastic Approximation with Randomized Directions, IEEE Transactions on Automatic Control, vol. 43, pp. 1745- 1749,1998
[25]Gerencser.L, Rate of convergence of moments for a simultaneous perturbation stochastic approximation method for function minimization, IEEE Trans. Automat. Contr, 44:894-906, 1999
[26]L.Gerencser, Z Vago, A stochastic approximation method for nosie-free optimization, Proceedings of the European Control Conference, ECC, 2001
[27]Maeda, Y. and De Figueiredo, R.J.P. Learning Rules for Neuro-Controller via Simultaneous Perturbation, IEEE Transactions on Neural Networks, vol. 8, pp. 1119-1130,1997
[28]J.C.Spall, A one measurement form of simultaneous perturbation stochastic approximation, Automatica., Vol.33 pp,109-112, 1997
[29]Maryak, J.L., and Chin, D.C. Global Random Optimization by Simultaneous Perturbation Stochastic Approximation, Proceedings of the American ControlConference, 25-27 June 2001, Arlington, VA, pp. 756-762, 2001
[30]Sadegh, P. Constrained Optimization via Stochastic Approximation with a Simultaneous Perturbation Gradient Approximation Automatica, vol. 33, pp. 889-892, 1997
[31]Fu, M.C. and Hill, S.D. Optimization of Discrete Event Systems via Simultaneous Perturbation Stochastic Approximation, Transactions of the Institute of Industrial Engineers, vol. 29, pp. 233-243,1997
[32]Wang, I.-J. and Spall, J.C. Stochastic Optimization with Inequality Constraints Using Simultaneous Perturbations and Penalty Functions, Proceedings of the IEEE Conference on Decision and Control, 9-12 December 2003, Maui, Hawaii, pp. 3808-3813,2003
[33]R. Hildebrand, A. Lecchini, G. Solari and M. Gevers,Prefiltering in Iterative Feedback Tuning: optimization of the prefilter for accuracy,IEEE Transactions on Automatic Control, 49(10):1801-1805, 2004
[34] Stochastic Approximation, M.T.Wasan, Cambridge UniversityPress, Cambridge , Great Britain.
[35] H.Hjalmarsson. Control of nonlinear systems using iterative feedback tuning .In Proc 1998 American Control Conference pp 2083-2089
[36]J.sjoberg and F.De Bruyne. On a nonlinear controller tuning strategy. In 14th IFAC World Congress volume I pages 343-348, Beijing1999
[37] H.Hjalmarsson Efficient tuning of linear multivariable controllers using iterative feedback tuning, International Journal on Adaptive Control and Signal Processing 1999
[38] H.Hjalmarsson Model-free tuning of controllers :Experience with time-varying inear systems. In Proceedings of European Control Conference Rome Italy 1995
[39] Control of Nonlinear systems using iterative feedback tuning, Hakan Hjalmarsson Proceedings of the American Control Conference Philadelphia, Pennsylvania, 1999
[40]Sandor Veres Tuning for Robusitness and performance using iterative feedback tunning. Proceedings of the 41st IEEE conference on Decision and control .Las Vegas,Nevada USA 2003
[41]T.Poggio and F.Girosi ,”Networks for approximation an learning “ in Proc.IEEE,vol .78,pp.889-895,1996
[42]S.H.Lane,D.A.Handelman, and J.J.Gelford,Theory and development of higherorder CMAC neural networks, IEEE Contr.Syst.Mag..vol12 ,1992
[43]T.Chen an H.Chen “universal approximation to nonlinear operators by neural networks with arbitrary activation functions and its application to dynamical systems”IEEE Trans.Neural Networks, Vol.6, pp.911-917,1997
[44]P.Sadegh “Constrained optimization via stochastic approximation with a simultaneous perturbation gradient approximation “ Automatica, vol 33,pp 889-892, 1997
[45]G..N.Saridis Self-Organizing Control of Stochastic Systems.NewYork Marcel Dekker
[46]P.E.Moden and T.Soderstrom “Stationary performance of linear stochstic systems under single step optimal control” IEEE Trans Automat .Contr volAC-27 1982
[47]W.Fleming, Functions of Several Variables ,New York; Springer-Verlag,1997
[48]J.C.Spall” Multivariate stochastic approximation using a simultaneous perturbation gradient approximation” IEEE contr.Syst. Mag. Vol12,1992
[49]J.C.Spall and J.A.Cristion “A neural network controller for systems with unmodeled dynamics with applications to wastewater treatment” IEEE Trans. Syst. Man Cybern Part B,vol 27,pp.369-375, 1997
[50]J.C.Spall and J.A.Cristion “Nonlinear dadptive control using neural networks: Estimation based on a smoothed form of simultaneous perturbation gradient approximation, Statistica Sinica, 1994
[51]洪立,王世渊,王先来 神经网络控制器在未建模型的非线性系统中的应用, 化工自动化及仪表 2000 年第二期
[52]Han-fu chen and hai-tao Fang “output Tracking for Nonlinear Stochastic Systems by Iterative Learning Control” IEEE Transaction on Automatic Control Vol.49 2004
[53]诚志峰,周有训, 一种简易的无模型控制算法—改进的同时扰动随机逼近控制 2004 年第一期广东自动化与信息工程
[54]J.C.Spall. Accelerated second-order stochastic optimization using only function measurements. Proc.IEEE Conf.Decision Control. pp1417-1424,1997
[55]D.C.Chin. A more efficient global optimization algorithm based on Neural Networks pp573-574, 1994
[56]S.Yako witz. A globally convergent stochastic approximation. SIAM J.Contr. Optimization. vo1. 31:pp.30-40,1993
[57]M .C.Nechyba and Y.Xu. Human-control strategy: Abstraction, verification andreplication. IEEE Contr .Sust.Magazine. v o1 .17 ,pp.48-61, Oct. 1997
[58]Guido O.Guardabassi and Sergio M.savaresi, Virtual reference direct design method: an off-line approach to databased control system design, IEEE Trans.Automat. Contr., Vol.45,pp.1480-1484, 2000
[59]M.C. Campi, A. Lecchini and S.M Savaresi. Virtual Refrence Feedback Tuning (VRFT): a new direct approach to the design of feedback controllers. In Proc. 39th Conf. on Decision and Control, Sydney, pages 623-628, 2002
[60]鲍志军,王先来,径向基神经网络在系统辨识中的结构与算法研究[硕士学位论文],天津大学,2002
[61]张宇橙,张玲华,曹雪红 数字信号处理与应用 南京 东南大学出版社 1997 年
[62]张玲华,郑宝玉 随机信号处理 P96-108 清华大学出版社 2003 年
[63]黄忠霖,控制系统 MATALAB 方针,北京:电子工业出版社,2003 年
[64]Hjalmarsson, H,. Gunnarson, S.,&Gevers,M. Model-free tuning of a robust regulator for a flexible transmission system. European Journal of Control 1995
[2] K.J.奥斯特隆姆 B.威顿马克,自适应控制. 北京:科学出版社,1992
[3] Hakan Hjalmarsson, Svante Gunnarsson and Michel Gevers, A convergent iterative Restricted Complexity Control Design Scheme, Proceedings of the 33rd Conference on Descision and control Lake Buena Vista, December 1994
[4] J.C.Spall and J.A.Cristion. Model-free control of nonlinear stochastic systems approximation. Automatica., Vol.33,pp.109-112,1998
[5] 夏长亮、祁温雅,基于 RBF 神经网络的超声波奠基参数辨识与模型参考自适应控制,中国科技论文在线
[6] Robbins and S.Monro, A stochastic approximation method , Ann.math. Statist, Vol. 22 pp.400-407 1951
[7] E.Kiefer and J.Wolfwitz, Stochastic estimation of the Maximum of a regression function, Ann. Math. Statist., Vol.23 pp.462-466 1952
[8] J.R.Blum, Multidimensional stochastic approximation methods, Ann.Math. Statist, Vol.25 pp.737-744 1954
[9] J.C.Spall,. A Stochastic Approximation Technique for Generating Maximum Likelihood Parameter Estimates, Proceedings of the American Control Conference, Minneapolis, MN, pp. 1161-1167,10-12 June 1987
[10]J.C.Spall, Multivariate Stochastic Approximation Using a Simultaneous Pertur- bation Gradient Approximation, IEEE Transactions on Automatic Control, vol. 37, pp. 332-341 1992
[11]Sadegh, P. and Spall, J.C, Optimal Random Perturbations for Stochastic Approximation Using a Simultaneous Perturbation Gradient Approximation, Proceedings of the American Control Conference, Albuquerque, NM, pp. 3582-3586 ,June 1997
[12]Spall, J.C. and Cristion, J.A. , Nonlinear Adaptive Control Using Neural Networks: Estimation With a Smoothed Form of Simultaneous Perturbation Gradient Approximation Statistica Sinica, vol. 4, pp. 1- 27,1994
[13]Spall, J.C. Adaptive Stochastic Approximation by the Simultaneous Perturbation Method, IEEE Transactions on Automatic Control, vol. 45, pp. 1839-1853,2000
[14]Zhu, X. and Spall, J.C.A Modifed Second-Order SPSA Optimization Algorithm for Finite Samples International Journal of Adaptive Control and Signal Processing, vol. 16, pp. 397-409, 2002
[15]狄昂照 ,随机逼近,科学出版社 1994
[16]朱玉,王先来,对未建模型的工业过程对象进行控制的新方法,自动化与仪器仪表, pp35-39,2002 年第 3 期(总第 101 期)
[17]聂赞坎, 徐宗本,随机逼近理论与自适应算法,科学出版社 2003
[18]L.jung, Analysis of recursive stochastic algorithms, IEEE Trans AC-22(4), pp551-575,1977
[19]Kushner and Clark, Stochastic Approximation Methods for Constrained and Unconstrained Systems, Springer, 1978
[20]陈翰馥,朱允民,变界截尾随机逼近算法,中国科学(A 辑) 1986
[21]Chin, D.C., Comparative Study of Stochastic Algorithms for System Optimization Based on Gradient Approximations, IEEE Transactions on Systems, Man, and Cybernetics, vol. 27, pp. 244-249. 1997
[22]J.C.Spall, Introduction to stochastic search and optimization :Estimazation, simulation and control, pp.176- 204, Willey 2003
[23]Dippon, J. and Renz, J. Weighted Means In Stochastic ApproximationOf Minima, SIAM Journal Of Control and Optimization, vol. 35, pp. 1811-1827.1997
[24]Wang, I.-J. and Chong, E.K.P. A Deterministic Analysis of Stochastic Approximation with Randomized Directions, IEEE Transactions on Automatic Control, vol. 43, pp. 1745- 1749,1998
[25]Gerencser.L, Rate of convergence of moments for a simultaneous perturbation stochastic approximation method for function minimization, IEEE Trans. Automat. Contr, 44:894-906, 1999
[26]L.Gerencser, Z Vago, A stochastic approximation method for nosie-free optimization, Proceedings of the European Control Conference, ECC, 2001
[27]Maeda, Y. and De Figueiredo, R.J.P. Learning Rules for Neuro-Controller via Simultaneous Perturbation, IEEE Transactions on Neural Networks, vol. 8, pp. 1119-1130,1997
[28]J.C.Spall, A one measurement form of simultaneous perturbation stochastic approximation, Automatica., Vol.33 pp,109-112, 1997
[29]Maryak, J.L., and Chin, D.C. Global Random Optimization by Simultaneous Perturbation Stochastic Approximation, Proceedings of the American ControlConference, 25-27 June 2001, Arlington, VA, pp. 756-762, 2001
[30]Sadegh, P. Constrained Optimization via Stochastic Approximation with a Simultaneous Perturbation Gradient Approximation Automatica, vol. 33, pp. 889-892, 1997
[31]Fu, M.C. and Hill, S.D. Optimization of Discrete Event Systems via Simultaneous Perturbation Stochastic Approximation, Transactions of the Institute of Industrial Engineers, vol. 29, pp. 233-243,1997
[32]Wang, I.-J. and Spall, J.C. Stochastic Optimization with Inequality Constraints Using Simultaneous Perturbations and Penalty Functions, Proceedings of the IEEE Conference on Decision and Control, 9-12 December 2003, Maui, Hawaii, pp. 3808-3813,2003
[33]R. Hildebrand, A. Lecchini, G. Solari and M. Gevers,Prefiltering in Iterative Feedback Tuning: optimization of the prefilter for accuracy,IEEE Transactions on Automatic Control, 49(10):1801-1805, 2004
[34] Stochastic Approximation, M.T.Wasan, Cambridge UniversityPress, Cambridge , Great Britain.
[35] H.Hjalmarsson. Control of nonlinear systems using iterative feedback tuning .In Proc 1998 American Control Conference pp 2083-2089
[36]J.sjoberg and F.De Bruyne. On a nonlinear controller tuning strategy. In 14th IFAC World Congress volume I pages 343-348, Beijing1999
[37] H.Hjalmarsson Efficient tuning of linear multivariable controllers using iterative feedback tuning, International Journal on Adaptive Control and Signal Processing 1999
[38] H.Hjalmarsson Model-free tuning of controllers :Experience with time-varying inear systems. In Proceedings of European Control Conference Rome Italy 1995
[39] Control of Nonlinear systems using iterative feedback tuning, Hakan Hjalmarsson Proceedings of the American Control Conference Philadelphia, Pennsylvania, 1999
[40]Sandor Veres Tuning for Robusitness and performance using iterative feedback tunning. Proceedings of the 41st IEEE conference on Decision and control .Las Vegas,Nevada USA 2003
[41]T.Poggio and F.Girosi ,”Networks for approximation an learning “ in Proc.IEEE,vol .78,pp.889-895,1996
[42]S.H.Lane,D.A.Handelman, and J.J.Gelford,Theory and development of higherorder CMAC neural networks, IEEE Contr.Syst.Mag..vol12 ,1992
[43]T.Chen an H.Chen “universal approximation to nonlinear operators by neural networks with arbitrary activation functions and its application to dynamical systems”IEEE Trans.Neural Networks, Vol.6, pp.911-917,1997
[44]P.Sadegh “Constrained optimization via stochastic approximation with a simultaneous perturbation gradient approximation “ Automatica, vol 33,pp 889-892, 1997
[45]G..N.Saridis Self-Organizing Control of Stochastic Systems.NewYork Marcel Dekker
[46]P.E.Moden and T.Soderstrom “Stationary performance of linear stochstic systems under single step optimal control” IEEE Trans Automat .Contr volAC-27 1982
[47]W.Fleming, Functions of Several Variables ,New York; Springer-Verlag,1997
[48]J.C.Spall” Multivariate stochastic approximation using a simultaneous perturbation gradient approximation” IEEE contr.Syst. Mag. Vol12,1992
[49]J.C.Spall and J.A.Cristion “A neural network controller for systems with unmodeled dynamics with applications to wastewater treatment” IEEE Trans. Syst. Man Cybern Part B,vol 27,pp.369-375, 1997
[50]J.C.Spall and J.A.Cristion “Nonlinear dadptive control using neural networks: Estimation based on a smoothed form of simultaneous perturbation gradient approximation, Statistica Sinica, 1994
[51]洪立,王世渊,王先来 神经网络控制器在未建模型的非线性系统中的应用, 化工自动化及仪表 2000 年第二期
[52]Han-fu chen and hai-tao Fang “output Tracking for Nonlinear Stochastic Systems by Iterative Learning Control” IEEE Transaction on Automatic Control Vol.49 2004
[53]诚志峰,周有训, 一种简易的无模型控制算法—改进的同时扰动随机逼近控制 2004 年第一期广东自动化与信息工程
[54]J.C.Spall. Accelerated second-order stochastic optimization using only function measurements. Proc.IEEE Conf.Decision Control. pp1417-1424,1997
[55]D.C.Chin. A more efficient global optimization algorithm based on Neural Networks pp573-574, 1994
[56]S.Yako witz. A globally convergent stochastic approximation. SIAM J.Contr. Optimization. vo1. 31:pp.30-40,1993
[57]M .C.Nechyba and Y.Xu. Human-control strategy: Abstraction, verification andreplication. IEEE Contr .Sust.Magazine. v o1 .17 ,pp.48-61, Oct. 1997
[58]Guido O.Guardabassi and Sergio M.savaresi, Virtual reference direct design method: an off-line approach to databased control system design, IEEE Trans.Automat. Contr., Vol.45,pp.1480-1484, 2000
[59]M.C. Campi, A. Lecchini and S.M Savaresi. Virtual Refrence Feedback Tuning (VRFT): a new direct approach to the design of feedback controllers. In Proc. 39th Conf. on Decision and Control, Sydney, pages 623-628, 2002
[60]鲍志军,王先来,径向基神经网络在系统辨识中的结构与算法研究[硕士学位论文],天津大学,2002
[61]张宇橙,张玲华,曹雪红 数字信号处理与应用 南京 东南大学出版社 1997 年
[62]张玲华,郑宝玉 随机信号处理 P96-108 清华大学出版社 2003 年
[63]黄忠霖,控制系统 MATALAB 方针,北京:电子工业出版社,2003 年
[64]Hjalmarsson, H,. Gunnarson, S.,&Gevers,M. Model-free tuning of a robust regulator for a flexible transmission system. European Journal of Control 1995
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