August  2019, 2(3): 267-277. doi: 10.3934/mfc.2019017

Alternative criteria for admissibility and stabilization of singular fractional order systems

School of Sciences, Northeastern University, No. 3 Wenhua Road Heping District, Shengyang 110819, Liaoning, China

* Corresponding author: Xuefeng Zhang

Received  June 2019 Revised  August 2019 Published  September 2019

Fund Project: The first author is supported by NSFC 61603055.

This paper discusses admissibility problem of singular fractional order systems with order $ 1<\alpha<2 $. The alternative necessary and sufficient admissibility conditions are proposed, in which include linear matrix inequalities (LMIs) with equality constraints and LMIs without equality constraints. Moreover, these criteria are brand-new and different from the existing results. The state feedback control to stabilize singular fractional order systems is derived. Two numerical examples are presented to shown the effectiveness of our results.

Citation: Xuefeng Zhang, Zhe Wang. Alternative criteria for admissibility and stabilization of singular fractional order systems. Mathematical Foundations of Computing, 2019, 2 (3) : 267-277. doi: 10.3934/mfc.2019017
References:
[1]

H. S. Ahn and Y. Q. Chen, Necessary and sufficient stability condition of fractional order interval linear systems, Automatica, 44 (2008), 2985-2988.  doi: 10.1016/j.automatica.2008.07.003.

[2]

L. Dai, Singular Control Systems, Berlin: Springer-Verlag, 1989. doi: 10.1007/BFb0002475.

[3]

Y. D. JiL. Q. Su and J. Q. Qiu, Design of fuzzy output feedback stabilization of uncertain fractional-order systems, Neurocomputing, 173 (2016), 1683-1693.  doi: 10.1016/j.neucom.2015.09.041.

[4]

B. K. Lenka, Fractional comparison method and asymptotic stability results for multivariable fractional order systems, Commun Nonlinear Sci. Numer. Simulat., 69 (2019), 398-415.  doi: 10.1016/j.cnsns.2018.09.016.

[5]

Y. LiY. Q. Chen and I. Podlubny, Mittag-Leffler stability of fractional order nonlinear dynamic systems, Automatica, 45 (2009), 1965-1969.  doi: 10.1016/j.automatica.2009.04.003.

[6]

B. X. Li and X. F. Zhang, Observer-based robust control of $0 < \alpha < 1$ fractional-order linear uncertain control systems, IET Control Theory & Applications, 10 (2016), 1724-1731.  doi: 10.1049/iet-cta.2015.0453.

[7]

C. LinB. ChenP. Shi and J. P. Yu, Necessary and sufficient conditions of observer-based stabilization for a class of fractional-order descriptor systems, Systems & Control Letters, 112 (2018), 31-35.  doi: 10.1016/j.sysconle.2017.12.004.

[8]

C. LinQ. G. Wang and T. Lee, Robust normalization and stabilization of uncertain descriptor systems with norm-bounded perturbations, IEEE Transactions on Automatic Control, 50 (2005), 515-520.  doi: 10.1109/TAC.2005.844908.

[9]

D. Y. LiuG. ZhengD. Boutat and H. R. Liu, Non-asymptotic fractional order differentiator for a class of fractional order linear systems, Automatica, 8 (2017), 61-71.  doi: 10.1016/j.automatica.2016.12.017.

[10]

J. G. Lu and Y. Q. Chen, Robust stability and syabilization of fractional order systems with order $\alpha$: the $0 < \alpha < 1$ case, IEEE Transactions on Automatic Control, 55 (2010), 152-158.  doi: 10.1109/TAC.2009.2033738.

[11]

S. Marir and M. Chadli, New admissibility conditions for singular linear continuous-time fractional-order systems, Journal of The Franklin Institute, 354 (2017), 752-766.  doi: 10.1016/j.jfranklin.2016.10.022.

[12]

S. MarirM. Chadli and D. Bouagada, A novel approach of admissibility for singular linear continuous-time fractional-order systems, International Journal of Control Automation and Systems, 15 (2017), 959-964.  doi: 10.1007/s12555-016-0003-0.

[13]

U. Michael and B. Thierry, Fractional splines and wavelets, SIAM Review, 42 (2000), 43-67.  doi: 10.1137/S0036144598349435.

[14]

I. N'DoyeM. DarouachM. Zasadzinski and N. Radhy, Robust stabilization of uncertain descriptor fractional-order systems, Automatica, 49 (2013), 1907-1913.  doi: 10.1016/j.automatica.2013.02.066.

[15]

I. N' Doye, M. Darouach, M. Zasadzinski and N. Radhy, Stabilization of singular fractional-order systems: An LMI approach, In Control and Automation (MED), (2010), 209–213.

[16]

M. D. OrtigueiraD. Valério and J. T. Machado, Variable order fractional systems, Commun Nonlinear Sci. Numer. Simulat., 71 (2019), 231-243.  doi: 10.1016/j.cnsns.2018.12.003.

[17]

I. Podlubny, Fractional-order systems and $\text{P}{{\text{I}}^{\lambda }}{{\text{D}}^{\mu }}$-controllers, IEEE Trans. Autom. Control, 44 (1999), 208-214.  doi: 10.1109/9.739144.

[18]

Y. A. Rossikhin and M. V. Shitikova, Application of fractional derivatives to the analysis of damped vibrations of viscoelastic single mass systems, Acta Mechanica, 120 (1997), 109-125.  doi: 10.1007/BF01174319.

[19]

Y. A. Rossikhin and M. V. Shitikova, Applications of fractional calculus to dynamic problems of linear and nonlinear hereditary mechanics of solids, Appl. Mech. Rev., 50 (1997), 15-67.  doi: 10.1115/1.3101682.

[20]

J. SabatierM. Moze and C. Farges, LMI stability conditions for fractional order systems, Computers and Mathematics with Applications, 59 (2010), 1594-1609.  doi: 10.1016/j.camwa.2009.08.003.

[21]

X. N. Song, L. P. Liu and Z. Wang, Stabilization of Singular Fractional-Order Systems: A Linear Matrix Inequality Approach, IEEE International Conference on Automation and Logistics, Zhengzhou, China, 2012.

[22]

Y. H. WeiT. PeterZ. Yao and Y. Wang, The output feedback control synthesis for a class of singular fractional order systems, ISA Transactions, 69 (2017), 1-9.  doi: 10.1016/j.isatra.2017.04.020.

[23]

Y. H. WeiJ. C. WangT. Y. Liu and Y. Wang, Sufficient and necessary conditions for stabilizing singular fractional order systems with partially measurable state, Journal of The Franklin Institute, 356 (2019), 1975-1990.  doi: 10.1016/j.jfranklin.2019.01.022.

[24]

S. Y. Xu and J. Lam, Robust Control and Filtering of Singular Systemms, Lecture Notes in Control and Information Sciences, 332. Springer-Verlag, Berlin, 2006.

[25]

S. Y. XuC. W. YangY. G. Niu and J. Lam, Robust stabilization for uncertain discrete singular systems, Automatica, 37 (2001), 769-774.  doi: 10.1016/S0005-1098(01)00013-9.

[26]

Z. H. Yang and Y. M. Ying, Donline optimization for residential pv-ess energy system scheduling, Mathematical Foundations of Computing, 2 (2019), 55-71. 

[27]

Y. YuZ. Jiao and C. Y. Sun, Sufficient and necessary condition of admissibility for fractional-order singular system, Acta Automatica Sinica, 39 (2013), 2160-2164.  doi: 10.1016/S1874-1029(14)60003-3.

[28]

X. F. Zhang, Relationship between integer order systems and fractional order systems and its two applications, IEEE/CAA Journal of Autmatica Sinica, 5 (2018), 639-643.  doi: 10.1109/JAS.2016.7510205.

[29]

X. F. Zhang and Y. Q. Chen, Admissibility and robust stabilization of continuous linear singular fractional order systems with the fractional order $\alpha$: The $0 < \alpha < 1$ case, ISA Transactions, 82 (2018), 42-50. 

[30]

X. F. Zhang and Y. Q. Chen, D-stability based LMI criteria of stability and stabilization for fractional order systems, Proceedings of the ASME 2015 International Design Engineering Technical Conference and Computers and Information in Engineering Conference Boston, (2016), 1–6. doi: 10.1115/DETC2015-46692.

[31]

X. F. Zhang and Z. L. Zhao, Normalization and stabilization for rectangular singular fractional order T-S fuzzy systems, Fuzzy Sets and Systems, 2019. doi: 10.1016/j.fss.2019.06.013.

show all references

References:
[1]

H. S. Ahn and Y. Q. Chen, Necessary and sufficient stability condition of fractional order interval linear systems, Automatica, 44 (2008), 2985-2988.  doi: 10.1016/j.automatica.2008.07.003.

[2]

L. Dai, Singular Control Systems, Berlin: Springer-Verlag, 1989. doi: 10.1007/BFb0002475.

[3]

Y. D. JiL. Q. Su and J. Q. Qiu, Design of fuzzy output feedback stabilization of uncertain fractional-order systems, Neurocomputing, 173 (2016), 1683-1693.  doi: 10.1016/j.neucom.2015.09.041.

[4]

B. K. Lenka, Fractional comparison method and asymptotic stability results for multivariable fractional order systems, Commun Nonlinear Sci. Numer. Simulat., 69 (2019), 398-415.  doi: 10.1016/j.cnsns.2018.09.016.

[5]

Y. LiY. Q. Chen and I. Podlubny, Mittag-Leffler stability of fractional order nonlinear dynamic systems, Automatica, 45 (2009), 1965-1969.  doi: 10.1016/j.automatica.2009.04.003.

[6]

B. X. Li and X. F. Zhang, Observer-based robust control of $0 < \alpha < 1$ fractional-order linear uncertain control systems, IET Control Theory & Applications, 10 (2016), 1724-1731.  doi: 10.1049/iet-cta.2015.0453.

[7]

C. LinB. ChenP. Shi and J. P. Yu, Necessary and sufficient conditions of observer-based stabilization for a class of fractional-order descriptor systems, Systems & Control Letters, 112 (2018), 31-35.  doi: 10.1016/j.sysconle.2017.12.004.

[8]

C. LinQ. G. Wang and T. Lee, Robust normalization and stabilization of uncertain descriptor systems with norm-bounded perturbations, IEEE Transactions on Automatic Control, 50 (2005), 515-520.  doi: 10.1109/TAC.2005.844908.

[9]

D. Y. LiuG. ZhengD. Boutat and H. R. Liu, Non-asymptotic fractional order differentiator for a class of fractional order linear systems, Automatica, 8 (2017), 61-71.  doi: 10.1016/j.automatica.2016.12.017.

[10]

J. G. Lu and Y. Q. Chen, Robust stability and syabilization of fractional order systems with order $\alpha$: the $0 < \alpha < 1$ case, IEEE Transactions on Automatic Control, 55 (2010), 152-158.  doi: 10.1109/TAC.2009.2033738.

[11]

S. Marir and M. Chadli, New admissibility conditions for singular linear continuous-time fractional-order systems, Journal of The Franklin Institute, 354 (2017), 752-766.  doi: 10.1016/j.jfranklin.2016.10.022.

[12]

S. MarirM. Chadli and D. Bouagada, A novel approach of admissibility for singular linear continuous-time fractional-order systems, International Journal of Control Automation and Systems, 15 (2017), 959-964.  doi: 10.1007/s12555-016-0003-0.

[13]

U. Michael and B. Thierry, Fractional splines and wavelets, SIAM Review, 42 (2000), 43-67.  doi: 10.1137/S0036144598349435.

[14]

I. N'DoyeM. DarouachM. Zasadzinski and N. Radhy, Robust stabilization of uncertain descriptor fractional-order systems, Automatica, 49 (2013), 1907-1913.  doi: 10.1016/j.automatica.2013.02.066.

[15]

I. N' Doye, M. Darouach, M. Zasadzinski and N. Radhy, Stabilization of singular fractional-order systems: An LMI approach, In Control and Automation (MED), (2010), 209–213.

[16]

M. D. OrtigueiraD. Valério and J. T. Machado, Variable order fractional systems, Commun Nonlinear Sci. Numer. Simulat., 71 (2019), 231-243.  doi: 10.1016/j.cnsns.2018.12.003.

[17]

I. Podlubny, Fractional-order systems and $\text{P}{{\text{I}}^{\lambda }}{{\text{D}}^{\mu }}$-controllers, IEEE Trans. Autom. Control, 44 (1999), 208-214.  doi: 10.1109/9.739144.

[18]

Y. A. Rossikhin and M. V. Shitikova, Application of fractional derivatives to the analysis of damped vibrations of viscoelastic single mass systems, Acta Mechanica, 120 (1997), 109-125.  doi: 10.1007/BF01174319.

[19]

Y. A. Rossikhin and M. V. Shitikova, Applications of fractional calculus to dynamic problems of linear and nonlinear hereditary mechanics of solids, Appl. Mech. Rev., 50 (1997), 15-67.  doi: 10.1115/1.3101682.

[20]

J. SabatierM. Moze and C. Farges, LMI stability conditions for fractional order systems, Computers and Mathematics with Applications, 59 (2010), 1594-1609.  doi: 10.1016/j.camwa.2009.08.003.

[21]

X. N. Song, L. P. Liu and Z. Wang, Stabilization of Singular Fractional-Order Systems: A Linear Matrix Inequality Approach, IEEE International Conference on Automation and Logistics, Zhengzhou, China, 2012.

[22]

Y. H. WeiT. PeterZ. Yao and Y. Wang, The output feedback control synthesis for a class of singular fractional order systems, ISA Transactions, 69 (2017), 1-9.  doi: 10.1016/j.isatra.2017.04.020.

[23]

Y. H. WeiJ. C. WangT. Y. Liu and Y. Wang, Sufficient and necessary conditions for stabilizing singular fractional order systems with partially measurable state, Journal of The Franklin Institute, 356 (2019), 1975-1990.  doi: 10.1016/j.jfranklin.2019.01.022.

[24]

S. Y. Xu and J. Lam, Robust Control and Filtering of Singular Systemms, Lecture Notes in Control and Information Sciences, 332. Springer-Verlag, Berlin, 2006.

[25]

S. Y. XuC. W. YangY. G. Niu and J. Lam, Robust stabilization for uncertain discrete singular systems, Automatica, 37 (2001), 769-774.  doi: 10.1016/S0005-1098(01)00013-9.

[26]

Z. H. Yang and Y. M. Ying, Donline optimization for residential pv-ess energy system scheduling, Mathematical Foundations of Computing, 2 (2019), 55-71. 

[27]

Y. YuZ. Jiao and C. Y. Sun, Sufficient and necessary condition of admissibility for fractional-order singular system, Acta Automatica Sinica, 39 (2013), 2160-2164.  doi: 10.1016/S1874-1029(14)60003-3.

[28]

X. F. Zhang, Relationship between integer order systems and fractional order systems and its two applications, IEEE/CAA Journal of Autmatica Sinica, 5 (2018), 639-643.  doi: 10.1109/JAS.2016.7510205.

[29]

X. F. Zhang and Y. Q. Chen, Admissibility and robust stabilization of continuous linear singular fractional order systems with the fractional order $\alpha$: The $0 < \alpha < 1$ case, ISA Transactions, 82 (2018), 42-50. 

[30]

X. F. Zhang and Y. Q. Chen, D-stability based LMI criteria of stability and stabilization for fractional order systems, Proceedings of the ASME 2015 International Design Engineering Technical Conference and Computers and Information in Engineering Conference Boston, (2016), 1–6. doi: 10.1115/DETC2015-46692.

[31]

X. F. Zhang and Z. L. Zhao, Normalization and stabilization for rectangular singular fractional order T-S fuzzy systems, Fuzzy Sets and Systems, 2019. doi: 10.1016/j.fss.2019.06.013.

Figure 1.  The closed-loop fractional order system in Example 2
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