分数阶广义线性系统观测器设计

数学物理学报 ›› 2021, Vol. 41 ›› Issue (5): 1529-1544.

分数阶广义线性系统观测器设计

冯再勇^1,²(),陈宁^3,*(),台永鹏⁴,向峥嵘²()

¹ 南京铁道职业技术学院基础部南京 210031
² 南京理工大学自动化学院南京 210094
³ 南京林业大学机械电子工程学院南京 210037
⁴ 南京林业大学汽车与交通工程学院南京 210037

收稿日期:2020-09-24 出版日期:2021-10-26 发布日期:2021-10-08
通讯作者: 陈宁 E-mail:zyfeng-97040@163.com;chenning@njfu.edu.cn
作者简介:冯再勇, E-mail: zyfeng-97040@163.com|向峥嵘, E-mail: zyfeng-97040@163.com
基金资助:
国家自然科学基金(61873128);江苏省高等学校自然科学研究面上项目(17KJB120008);江苏省轨道交通控制工程技术研究开发中心项目(KFJ2010);江苏省高校"青蓝工程"项目;南京铁道职业技术学院"青蓝工程"项目

On the Observer Design for Fractional Singular Linear Systems

Zaiyong Feng^1,²(),Ning Chen^3,*(),Yongpeng Tai⁴,Zhengrong Xiang²()

¹ Department of Basic Sciences, Nanjing Institute of Railway Technology, Nanjing 210031
² School of Automation, Nanjing University of Science and Technology, Nanjing 210094
³ College of Mechanical and Electronic Engineering, Nanjing Forestry University, Nanjing 210037
⁴ College of Automobile and Traffic Engineering, Nanjing Forestry University, Nanjing 210037

Received:2020-09-24 Online:2021-10-26 Published:2021-10-08
Contact: Ning Chen E-mail:zyfeng-97040@163.com;chenning@njfu.edu.cn
Supported by:
the NSFC(61873128);the NSF of the Jiangsu Higher Education Institution of China(17KJB120008);the Jiangsu Rail Transit Control Engineering Technology Research and Development Center(KFJ2010);the Qing Lan Project of the Jiangsu Higher Education Institutions;Qing Lan Project of Nanjing Institute of Railway Technology

摘要/Abstract

摘要：

研究了脉冲函数$\delta(t)$的Caputo导数${^C_0}{D}_t^\alpha\delta(t) $及其Laplace变换, 由此得到含脉冲项的分数阶广义线性系统的分布解. 阐明了分布解如何反映分数阶系统的"记忆特性". 以系统的分布解为基础, 证明了含脉冲项的分数阶广义线性系统的渐近稳定性定理. 据此, 给出了具有分数阶广义线性系统形式的状态观测器存在定理. 研究并总结了仅针对慢子系统进行极点配置, 得到系统全维状态观测器的简单设计方法. 最后, 针对具体实例, 设计了状态观测器, 验证了所给方法的有效性.

关键词: 分数阶广义线性系统, Caputo导数, 分布解, 渐近稳定性, 状态观测器

Abstract:

The paper discussed the Caputo fractional derivatives of impulse function $\delta(t) $, i.e., ${^C_0}{D}_t^\alpha\delta(t) $ and its Laplace transform, the distributional solution of Fractional Singular Linear Systems (FSLS) was consequently obtained. Based on the distributional solution of the system, the asymptotic stability theorem for FSLS was given, and the existence theorem of state observer for FSLS was proved. A simplified design method of full state observer for FSLS was investigated and summarized by pole assignment only for the slow subsystem. Finally, a state observer as an example was designed to verify the effectiveness of the proposed method.

Key words: Fractional singular linear system, Caputo fractional derivatives, Distributional solution, Asymptotic stability, State observer

中图分类号:

TP13

冯再勇,陈宁,台永鹏,向峥嵘. 分数阶广义线性系统观测器设计[J]. 数学物理学报, 2021, 41(5): 1529-1544.

Zaiyong Feng,Ning Chen,Yongpeng Tai,Zhengrong Xiang. On the Observer Design for Fractional Singular Linear Systems[J]. Acta mathematica scientia,Series A, 2021, 41(5): 1529-1544.

参考文献

1	Podlubny I . Fractional Differential Equations. New York: Academic Press, 1999
2	Diethelm K , Ford N J . Analysis of fractional differential equations. Journal of Mathematical Analysis and Applications, 2002, 265 (2): 229- 248
3	Daftardar-Gejji V , Babakhani A . Analysis of a system of fractional differential equations. Journal of Mathematical Analysis and Applications, 2004, 293 (2): 511- 522
4	Lakshmikantham V . Theory of fractional functional differential equations. Nonlinear Analysis Theory Methods and Applications, 2008, 69 (10): 3337- 3343
5	Kilbas A A, Srivastava H M, Trujillo J J. Theory and Applications of Fractional Differential Equations. New York: Elsevier, 2006
6	Du M, Wang Z, Hu H. Measuring memory with the order of fractional derivative. Scientific Reports, 2013. DOI: 10.1038/srep03431
7	Wang Z H . Fractiona calculus: A mathematical method of describing the memory characteristic and intermediate process. Scientific Chinese, 2011, 3, 76- 78
8	Koeller R . Applications of fractional calculus to the theory of viscoelasticity. Transactions of the Asme Journal of Applied Mechanics, 1984, 51 (2): 299- 307
9	Nadzharyan T A , Kostrov S A , Stepanov G V . Fractional rheological models of dynamic mechanical behavior of magnetoactive elastomers in magnetic fields. Polymer: The International Journal for the Science and Technology of Polymers, 2018, 142, 316- 329
10	Wang Z H , Hu H Y . Stability of a linear oscillator with damping force of fractional-order derivative. Sci China Ser G, 2009, 39 (10): 1495- 1502
11	蒲亦非. 分数阶微积分在现代信号分析与处理中应用的研究[D]. 成都: 四川大学, 2006
11	PU Y F. Research on Application of Fractional Calculus to Latest Signal Analysis and Processing[D]. Chengdu: Sichuan University, 2006
12	N'Doye I, Zasadzinski M, Darouach M. Stabilization of Singular Fractional-order Systems: an LMI Approach[C]//Raymond G, et al. 18th Mediterranean Conference on Control and Automation, MED'10. Marrakech, Morocco: IEEE, 2010: 209-213
13	N'Doye I , Darouach M , Zasadzinski M . Robust stabilization of uncertain descriptor fractional-order systems. Automatica, 2013, 49 (6): 1907- 1913
14	Marir S , Chadli M , Bouagada D . New admissibility conditions for singular linear continuous-time fractional-order systems. Journal of the Franklin Institute, 2016, 354 (2): 752- 766
15	Yao Y U , Zhuang J , Chang Y S . Sufficient and necessary condition of admissibility for fractional-order singular system. Acta Automatica Sinica, 2013, 39 (12): 2160- 2164
16	Lin C , Chen B , Shi P . Necessary and sufficient conditions of observer-based stabilization for a class of fractional-order descriptor systems. Systems and Control Letters, 2018, 112, 31- 35
17	Kaczorek T . Descriptor fractional linear systems with regular pencils. Asian Journal of Control, 2013, 15 (4): 1051- 1064
18	Feng Z Y , Chen N . On the existence and uniqueness of the solution of linear fractional differential-algebraic system. Mathematical Problems in Engineering, 2016, 1, 1- 9
19	Sajewski L. Descriptor Fractional Continuous-Time Linear System and Its Solution Comparison of Three Different Methods//Szewczyk R, Zieliński C, Kaliczyńska M, et al. International Conference Automation. Switzerland: Springer, 2017: 45-57
20	Kaczorek T . Singular fractional linear systems and electrical circuits. International Journal of Applied Mathematics and Computer Science, 2011, 21 (2): 379- 384
21	N'Doye I, Darouach M, Zasadzinski M. Observers Design for Singular Fractional-order Systems//2011 50th IEEE Conference on Decision and Control and European Control Conference. Orlando, FL: IEEE, 2011: 4017-4022
22	Zhan T , Ma S . Reduced-order observer design with unknown input for fractional order descriptor nonlinear systems. Transactions of the Institute of Measurement and Control, 2019, 41 (13): 3705- 3713
23	Campbell S L . Singular Systems of Differential Equations. London: Pitman Publishing, 1980
24	Debnath L , Bhatta D . Integral Transforms and Their Applications. New York: CRC Press, 2007
25	Dai L. Singular Control Systems. Berlin: Springer, 1989
26	Duan G R. Analysis and Design of Descriptor Linear Systems. New York: Springer, 2010

[1]	邓钧元, 张峻皓. 半空间中满足 Cattaneo 定律的双曲系统的稳定性[J]. 数学物理学报, 2025, 45(5): 1444-1462.
[2]	吕东霆. 两种群空间非均匀反应扩散竞争模型的全局渐近稳定性[J]. 数学物理学报, 2025, 45(4): 1171-1183.
[3]	徐斐, 张勇. 分数阶 Burgers 方程时间周期弱解的唯一性与渐近稳定性[J]. 数学物理学报, 2023, 43(6): 1710-1722.
[4]	李继泽,邱吉秀,周永辉. 受观察噪声影响的不可料非线性滤波方程及线性滤波稳定性[J]. 数学物理学报, 2023, 43(4): 1244-1254.
[5]	吴红星,袁邓彬,王胜华. 具结构化的细菌种群增生中迁移方程解的渐近稳定性分析[J]. 数学物理学报, 2022, 42(3): 807-817.
[6]	张丽红,杨泽栋,王国涛,BaleanuDumitru. 一类k-Hessian方程解的存在性和渐近稳定性[J]. 数学物理学报, 2021, 41(5): 1357-1371.
[7]	崔静,梁秋菊,毕娜娜. 分数布朗运动驱动的带脉冲的中立性随机泛函微分方程的渐近稳定性[J]. 数学物理学报, 2019, 39(3): 570-581.
[8]	张根根,王晚生,肖爱国. 一类变延迟中立型微分方程梯形方法的渐近估计[J]. 数学物理学报, 2019, 39(3): 560-569.
[9]	罗日才, 许弘雷, 王五生. 一类新的变时滞中立型神经网络的全局渐近稳定性条件[J]. 数学物理学报, 2015, 35(3): 634-640.
[10]	吴事良, 刘三阳. 反应扩散系统双稳波前解的全局渐近稳定性[J]. 数学物理学报, 2010, 30(2): 440-448.
[11]	陈福来;文贤章. 脉冲泛函微分方程的渐近性态[J]. 数学物理学报, 2006, 26(2): 287-296.
[12]	李小平, 蒋建初. 中立型时滞微分方程的渐近稳定性[J]. 数学物理学报, 2002, 22(2): 163-170.
[13]	滕志东. 一类无穷时滞周期Lotka-Volterra型系统的正周期解[J]. 数学物理学报, 2001, 21(1): 94-101.
[14]	滕志东陈兰荪. 具有时滞的周期Lotka-Volterra型系统的全局渐近稳定性[J]. 数学物理学报, 2000, 20(3): 296-303.
[15]	沈根成高国柱李力锋. 具有扰动和无界时滞的一维泛函微分方程的渐近稳定性[J]. 数学物理学报, 1999, 19(5): 493-500.