一种改进的同伦算法与<i>H</i><sub>∞</sub>鲁棒控制器设计

刘斌; 孙久强; 翟志强; 李卓; 王常虹

doi:10.3724/SP.J.1004.2013.01374

一种改进的同伦算法与H_∞鲁棒控制器设计

doi: 10.3724/SP.J.1004.2013.01374

1.
东北石油大学电气信息工程学院大庆 163318;
2.
哈尔滨工业大学航天学院空间控制与惯性技术研究中心哈尔滨 150080;
3.
南京农业大学工学院南京 210095

基金项目:

国家自然科学基金(61004067);黑龙江省青年科学基金(QC2011C043);黑龙江省普通高校青年学术骨干支持计划;黑龙江省科学技术研究项目(12531058)资助

详细信息

作者简介:
孙久强东北石油大学硕士研究生. 主要研究方向为鲁棒控制.E-mail: 424392856@qq.com

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出版历程
- 收稿日期: 2011-12-08
- 修回日期: 2012-08-17
- 刊出日期: 2013-08-20

A Modified Homotopy Method and H_∞ Robust Controller Design

1.
School of Electric Information Engineering, Northeast Petroleum University, Daqing 163318;
2.
Space Control and Inertial Technology Research Center, Harbin Institute of Technology, Harbin 150080;
3.
College of Engineering, Nanjing Agriaultural University, Nanjing 210095

Funds:

Supported by National Natural Science Foundation of China (61004067), Youth Science Foundation of Heilongjiang Province (QC2011C043), Youth Academic Backbone Project of Heilongjiang Province, Science and Technology Research Project of Heilongjiang Province (12531058)

摘要

摘要: 提出了一种具有阶次限制的鲁棒控制器设计方法, 该算法将控制系统的性能指标转化为灵敏度函数问题, 并利用Nevanlinna-Pick插值算法进行求解. 提出了一种改进的同伦算法, 将其用于求解由灵敏度函数产生的非线性方程. 基于改进同伦算法设计的鲁棒控制器不仅避免了传统H∞控制中加权函数的选择问题, 而且克服了鲁棒控制器阶次较高的缺陷. 最后,文章以4阶系统为例, 设计了具有阶次限制的H∞鲁棒控制器, 通过与传统鲁棒控制器的比较可以看出, 基于本文方法设计的控制器不仅具有较低的阶次, 而且其控制性能也具有明显的优越性.
- H∞鲁棒控制 /
- 灵敏度 /
- Nevanlinna-Pick插值 /
- 同伦法
Abstract: A novel method of robust controller design with degree constraint is proposed for feedback control systems, where the performance indices are firstly transferred to the sensitivity function problem that will be solved by Nevanlinna-Pick interpolation. A modified homotopy method is presented to solve nonlinear equations induced by the sensitivity function problem. A new controller based on modified homotopy method is designed without using weighting functions, which can also overcome the defect of high order. At last, the 4th order plant is considered and the corresponding H∞ robust controller with degree constraint is designed. It is be shown from simulations that the robust controller has not only lower degree than traditional robust controller, but also superior quality obviously.
- H∞ robust control /
- sensitivity function /
- Nevanlinna-Pick interpolation /
- homotopy method

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参考文献(30)

[1]	Zhou K M, Doyle J C. Essentials of Robust Control. Upper Saddle River: Prentice-Hall, 1998
[2]	Gahinet P, Apkarian P. A linear matrix inequality approach to H∞ control. International Journal of Robust and Nonlinear Control, 1994, 4(4): 421-448
[3]	Skelton R E, Iwasaki T, Grigoriadis D E. A Unified Algebraic Approach to Linear Control Design. New York: Taylor and Francis, 1997
[4]	Xin X, Guo L, Feng C B. Reduced-order controllers for continuous and discrete-time singular H∞ control problems based on LMI. Automatica, 1996, 32(11): 1581-1585
[5]	Zeng Jian-Ping, Cheng Peng. Design reduced-order controllers for a class of control problems. Acta Automatica Sinica, 2002, 28(2): 267-271(曾建平, 程鹏. 一类控制问题的降阶控制器设计. 自动化学报, 2002, 28(2): 267-271)
[6]	Watanabe T, Stoorvogel A A. Plant zero structure and further order reduction of a singular H∞ controller. International Journal of Robust and Nonlinear Control, 2002, 12(7): 591-619
[7]	Xin X. Reduced-order controllers for the H∞ control problem with unstable invariant zeros. Automatica, 2004, 40(2): 319-326
[8]	Zhong Rui-Lin, Cheng Peng. Constructing a reduced-order H-infinity controller using stable invariant zeros. Control Theory & Applications, 2007, 24(5): 707-710(钟瑞麟, 程鹏. 利用稳定零点构造降阶H∞控制器的方法. 控制理论与应用, 2007, 24(5): 707-710)
[9]	Byrnes C I, Georgiou T T, Lindquist A. Analytic interpolation with degree constraint: a constructive theory with applications to control and signal processing. In: Proceedings of the 38th IEEE Conference on Decision and Control. Phoenix, USA: IEEE, 1999. 982-988
[10]	Byrnes C I, Georgiou T T, Lindquist A. A generalized entropy criterion for Nevanlinna-Pick interpolation with degree constraint. IEEE Transactions on Automatic Control, 2001, 46(6): 822-839
[11]	Byrnes C I, Georgiou T T, Lindquist A, Megretski A. Generalized interpolation in H∞ with a complexity constraint. Transactions of American Mathematical Society, 2006, 358(3): 965-987
[12]	Georgiou T T. Realization of power spectra from partial covariance sequences. IEEE Transactions on Acoustics, Speech, and Signal Processing, 1987, 35(4): 438-449
[13]	Georgiou T T. A topological approach to Nevanlinna-Pick interpolation. SIAM Journal on Mathematical Analysis, 1987, 18(5): 1248-1260
[14]	Byrnes C I, Lindquist A, Gusev S V, Matveev A S. A complete parameterization of all positive rational extensions of a covariance sequence. IEEE Transactions on Automatic Control, 1995, 40(11): 1841-1857
[15]	Byrnes C I, Gusev S V, Lindquist A. A convex optimization approach to the rational covariance extension problem. SIAM Journal on Control and Optimization, 1998, 37(1): 211-229
[16]	Georgiou T T, Lindquist A. Kullback-Leibler approximation of spectral density functions. IEEE Transactions on Information Theory, 2003, 49(11): 2910-2917
[17]	Enqvist P. A homotopy approach to rational covariance extension with degree constraint. International Journal of Applied Mathematics Computer Science, 2001, 11(5): 1173-1201
[18]	Nagamune R. A robust solver using a continuation method for Nevanlinna-Pick interpolation with degree constraint. IEEE Transactions on Automatic Control, 2003, 48(1): 113-117
[19]	Liu C S, Atluri S N. A novel time integration method for solving a large system of non-linear algebraic equations. Computer Modeling in Engineering and Sciences, 2008, 31(2): 71-84
[20]	Sznaier M, Rotstein H, Bu J Y, Sideris A. An exact solution to continuous-time mixed H2/H∞ control problems. IEEE Transactions on Automatic Control, 2000, 45(11): 2095-2101
[21]	Arkowitz M. Introduction to Homotopy Theory. Berlin: Springer, 2011
[22]	Helton J W, Merino O. Classical Control Using H∞ Methods: Theory, Optimization, and Design. Philadelphia: Society for Industrial Mathematics, 1998
[23]	Walsh J L. Interpolation and Approximation by Rational Functions in The Complex Domain. New York: American Mathematical Society, 1956
[24]	Blomqvist A, Nagamune R. An extension of a Nevanlinna-Pick interpolation solver to cases including derivative constraints. In: Proceedings of the 41st IEEE Conference on Decision and Control. Las Vegas, USA: IEEE, 2002. 2552-2557
[25]	Delsarte P, Genin Y, Kamp Y. On the role of the Nevanlinna-Pick problem in circuit and system theory. International Journal of Circuit Theory and Applications, 1981, 9(2): 177-187
[26]	Youla D C, Saito M. Interpolation with positive real functions. Journal of the Franklin Institute, 1967, 284(2): 77-108
[27]	Byrnes C I, Gusev S V, Lindquist A. From finite covariance windows to modeling filters: a convex optimization approach. SIAM Review, 2001, 43(4): 645-675
[28]	Nagamune R. Closed-loop shaping based on Nevanlinna-Pick interpolation with a degree bound. IEEE Transactions on Automatic Control, 2004, 49(2): 300-305
[29]	Blomqvist A, Fanizza G, Nagamune R. Computation of bounded degree Nevanlinna-Pick interpolants by solving nonlinear equations. In: Proceedings of the 42nd IEEE Conference on Decision and Control. Hawaii, USA: IEEE, 2003. 4511-4516
[30]	Doyle J C, Francis B, Tannenbaum A. Feedback Control Theory. New York: Macmillan Publishing Company, 1990