区间分数阶系统的鲁棒稳定性判别准则：0 < α < 1

高哲; 廖晓钟

doi:10.3724/SP.J.1004.2012.00175

区间分数阶系统的鲁棒稳定性判别准则：0 < α < 1

doi: 10.3724/SP.J.1004.2012.00175

高哲^1,2,
廖晓钟^1,2, ,

1.
北京理工大学自动化学院北京 100081;
2.
复杂系统智能控制与决策教育部重点实验室 (北京理工大学) 北京 100081

详细信息

通讯作者:
廖晓钟, 北京理工大学自动化学院教授. 主要研究方向为运动控制,电力电子技术,绿色能源变换与控制技术. E-mail: liaoxiaozhong@bit.edu.cn

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- 文章访问数: 2895
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出版历程
- 收稿日期: 2011-07-18
- 修回日期: 2011-10-08
- 刊出日期: 2012-02-20

Robust Stability Criteria for Interval Fractional-order Systems: The 0 < α < 1 Case

GAO Zhe^1,2,
LIAO Xiao-Zhong^{1,2
, ,}

1.
School of Automation, Beijing Institute of Technology, Beijing 100081;
2.
Key Laboratory of Complex System Intelligent Control and Decision (Beijing Institute of Technology), Ministry of Education, Beijing 100081

摘要

摘要: 针对同元阶次在0和1之间的区间分数阶系统,提出了类似Kharitonov定理的鲁棒稳定性判别准则. 研究了区间分数阶系统分母的主分支函数值集不包含原点所需满足的条件.根据除零原理, 给出了区间分数阶系统鲁棒稳定的顶点和棱边条件. 定义了由分母函数系数构成的矩阵,通过检验矩阵是否在负实轴上存在特征值来检验棱边条件. 最后,通过对两个数值算例的分析说明了这种方法的有效性.
- 分数阶系统 /
- 区间不确定性 /
- 鲁棒稳定性 /
- 值集
Abstract: This paper presents a robust stability theorem like the Kharitonov theorem for interval fractional-order systems with commensurate order between 0 and 1. The condition that the origin is not contained in the value set of the principle branch function of denominator function in an interval fractional-order system is studied. The vertex and edge conditions for interval fractional-order systems are proposed based on the zero exclusion principle. Some matrices depending on parameters of the denominator function are defined and the edge conditions are tested by checking whether eigenvalues of each matrix lie on the negative real axis. Finally, two numerical examples are analyzed to illustrate the effectiveness of the proposed method.
- Fractional-order system /
- interval uncertainty /
- robust stability /
- value set

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参考文献(1)

[1]

Machado J T, Kiryakova V, Mainardi F. Recent history of fractional calculus. Communications in Nonlinear Science and Numerical Simulation, 2011, 16(3): 1140-1153[2] Jesus I S, Machado J A T. Fractional control of heat diffusion systems. Nonlinear Dynamics, 2008, 54(3): 263-282[3] Valerio D, Costa J S. Time-domain implementation of fractional order controllers. IEE Proceedings -- Control Theory and Applications, 2005, 152(5): 539-552[4] Zamani M, Karimi-Ghartemani M, Sadati N, Parniani M. Design of a fractional order PID controller for an AVR using particle swarm optimization. Control Engineering Practice, 2009, 17(12): 1380-1387[5] Merrikh-Bayat F, Karimi-Ghartemani M. Method for designing PI^{λ }D^{μ } stabilisers for minimum-phase fractional-order systems. IET Control Theory and Applications, 2010, 4(1): 61-70[6] Pisano A, Rapaic M R, Jelicic Z D, Usai E. Sliding mode control approaches to the robust regulation of linear multivariable fractional-order dynamics. International Journal of Robust Nonlinear Control, 2010, 20(18): 2045-2056[7] Jelicic Z D, Petrovacki N. Optimality conditions and a solution scheme for fractional optimal control problems. Structural and Multidisciplinary Optimization, 2009, 38(6): 571-581[8] Li Y, Chen Y Q, Ahn H S. Fractional-order iterative learning control for fractional-order linear systems. Asian Journal of Control, 2011, 13(1): 54-63[9] Tavazoei M S, Haeri M. A note on the stability of fractional order systems. Mathematics and Computers in Simulation, 2009, 79(5): 1566-1576[10] Lu J G, Chen Y Q. Robust stability and stabilization of fractional-order interval systems with the fractional order α : the 0α1 case. IEEE Transactions on Automatic Control, 2010, 55(1): 152-158[11] Lu J G, Chen G R. Robust stability and stabilization of fractional-order interval systems: an LMI approach. IEEE Transactions on Automatic Control, 54(6): 1294-1299[12] Liao Z, Peng C, Li W, Wang Y. Robust stability analysis for a class of fractional order systems with uncertain parameters. Journal of the Franklin Institute, 2011, 348(6): 1101-1113[13] Radwan A G, Soliman A M, Elwakil A S, Sedeek A. On the stability of linear systems with fractional-order elements. Chaos Solition and Fractals, 2009, 40(5): 2317-2328[14] Wang Zhen-Bin, Cao Guang-Yi, Zhu Xin-Jian. Stability conditions and criteria for fractional order linear time-invariant systems. Control Theory and Applications, 2004, 21(6): 922-926(王振滨, 曹广益, 朱新坚. 分数阶线性定常系统的稳定性及其判据. 控制理论与应用, 2004, 21(6): 922-926)[15] Auba T, Funahashi Y. A note on Kharitonov's theorem. IEEE Transactions on Automatic Control, 1993, 38(4): 663-664[16] Bhattacharyya S P, Chapellat H, Keel L H. Robust Control: the Parametric Approach. New Jersey: Prentice Hall, 1995. 37-39[17] Moornani K A, Haeri M. Robust stability testing function and Kharitonov-like theorem for fractional order interval systems. IET Control Theory and Applications, 2010, 4(10): 2097-2109

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区间分数阶系统的鲁棒稳定性判别准则：0 < α < 1

doi: 10.3724/SP.J.1004.2012.00175

通讯作者:
廖晓钟, 北京理工大学自动化学院教授. 主要研究方向为运动控制,电力电子技术,绿色能源变换与控制技术. E-mail: liaoxiaozhong@bit.edu.cn

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Robust Stability Criteria for Interval Fractional-order Systems: The 0 < α < 1 Case

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区间分数阶系统的鲁棒稳定性判别准则：0 < α < 1

doi: 10.3724/SP.J.1004.2012.00175

通讯作者: 廖晓钟, 北京理工大学自动化学院教授. 主要研究方向为运动控制,电力电子技术,绿色能源变换与控制技术. E-mail: liaoxiaozhong@bit.edu.cn

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出版历程

Robust Stability Criteria for Interval Fractional-order Systems: The 0 < α < 1 Case

计量

出版历程

目录

通讯作者:
廖晓钟, 北京理工大学自动化学院教授. 主要研究方向为运动控制,电力电子技术,绿色能源变换与控制技术. E-mail: liaoxiaozhong@bit.edu.cn