American Institute of Mathematical Sciences

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March  2021, 11(1): 1-12. doi: 10.3934/naco.2020011

Fault-tolerant control against actuator failures for uncertain singular fractional order systems

Xuefeng Zhang , and  Yingbo Zhang

School of Sciences, Northeastern University, Shenyang 110819, China

* Corresponding author: Xuefeng Zhang

Received  May 2019 Revised  August 2019 Published  February 2020

Fund Project: The first author is supported by NSFC under grant 61603055

A method of designing observer-based feedback controller against actuator failures for uncertain singular fractional order systems (SFOS) is presented in this paper. By establishing actuator fault model and state observer, an observer-based fault-tolerant state feedback controller is developed such that the closed-loop SFOS is admissible. The controller designed by the proposed method guarantees that the closed-loop system is regular, impulse-free and stable in the event of actuator failures. Finally, a numerical example is given to illustrate the effectiveness of the proposed design method.

Keywords: Dimension theory, Poincaré recurrences, multifractal analysis.
Mathematics Subject Classification: Primary: 58F15, 58F17; Secondary: 53C35.
Citation: Xuefeng Zhang, Yingbo Zhang. Fault-tolerant control against actuator failures for uncertain singular fractional order systems. Numerical Algebra, Control & Optimization, 2021, 11 (1) : 1-12. doi: 10.3934/naco.2020011
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.  Google Scholar

[2]

M. Blanke, M. Kinnaert, J. Lunze and M. Staroswiecki, Diagnosisi and Fault-Tolerant Control, Springer, Berlin, Germany, 2006. doi: 10.1007/978-3-540-35653-0.  Google Scholar

[3]

M. Blanke, R. Izadi-Zamanabadi, S. A. Bogh and C. P. Lunau, Fault-tolerant control systems$-$a holistic view, Contol Engineering Prac., 5 (1997), 693-702.  doi: 10.1016/s0967-0661(97)00051-8.  Google Scholar

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L. L. Fan and Y. D. Song, Neuro$-$adaptive model$-$referance fault$-$tolerant control with application to wind turbines, IET Control Theory & Appl., 6 (2012), 475-486.  doi: 10.1049/iet-cta.2011.0250.  Google Scholar

[5]

C. Farges, M. Moze and J. Sabatier, Pseudo-state feedback stabilization of commensurate fractional order systems, Automatica, 46 (2010), 1730-1734.  doi: 10.1016/j.automatica.2010.06.038.  Google Scholar

[6]

C. Farges, J. Sabatier and M. Moze, Robust stability analysis and stabilization of fractional order polytopic systems , Preprints of the 18th IFAC World Congress, (2011), Milano, Italy, 10800-10805. doi: 10.3182/20110828-6-IT-1002.00779.  Google Scholar

[7]

Bin Guo and Yong Chen, Adaptive fault tolerant control for time-varying delay system withactuator fault and mismatched disturbance, ISA Transcation, 89 (2019), 122-130.  doi: 10.1016/j.isatra.2018.12.024.  Google Scholar

[8]

Q. Hu, B. Xiao and M. I. Friswll, Robust fault$-$tolerant control for spacecraft attitude stabilisation subject to input saturation, IET Control Theory & Appl., 5 (2011), 271-282.  doi: 10.1049/iet-cta.2009.0628.  Google Scholar

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S. D. Huang, J. Lam, G. H. Yang and S. Y. Zhang, Fault tolerant decentralized $H_\infty$ control for symmetric composite systems, IEEE Trans. Autom. Control, 44 (1999), 2108-2114.  doi: 10.1109/9.802926.  Google Scholar

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R. Isermann, Fault-Diagnosis Systems: An Introduction from Fault Detection to Fault Tolerance, Springer, Berlin, Germany, 2006. doi: 10.1007/3-540-30368-5.  Google Scholar

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L. A. Jacyntho, M. C. M. Teixeira and E. Assunco, Identification of fractional-order transfer functions using a step excitation, IEEE Trans. Circuits Syst., 62 (2015), 896-900.  doi: 10.1109/tcsii.2015.2436052.  Google Scholar

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Y. Jiang, Q. L. Hu and G. F. Ma, Adaptive backsteping fault$-$tolerant control for flexible spacecraft with unknown bounded disturbances and actuator failures, ISA Transcations, 49 (2010), 57-69.  doi: 10.1016/j.isatra.2009.08.003.  Google Scholar

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E. Last, Linear matrix inequalities in system and control theory, Proceedings of the IEEE, 86 (1994), 2473-2474.  doi: 10.1109/JPROC.1998.735454.  Google Scholar

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X. J. Li and G. H. Yang, Robust adaptive fault$-$tolerant control for uncertain linear systems with actuator failures, IET Control Theory & Appl., 6 (2012), 1544-1551.  doi: 10.1049/iet-cta.2011.0599.  Google Scholar

[15]

Y. Li, Y. Q. Chen and I. Podlubny, Stability of fractional order nonlinear dynamic systems, Automatica, 45 (2009), 1965-1969.  doi: 10.1016/j.automatica.2009.04.003.  Google Scholar

[16]

C. Lin, B. Chen, P. 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.  Google Scholar

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J. G. Lu and G. R. Chen, Robust stability and stabilization of fractional order interval systems: An LMI approach, IEEE Transactions on Automatic Control, 54 (2009), 1294-1299.  doi: 10.1109/tac.2009.2013056.  Google Scholar

[18]

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

[19]

H. J. Ma and G. H. Yang, Detection and adaptive accommodation for actuator faults of a class of non$-$linear systems, IET Control Theory & Appl., 6 (2012), 2292-2307.  doi: 10.1049/iet-cta.2011.0265.  Google Scholar

[20]

S. Marir, M. Chadli and D. Bouagada, New admissibility conditions for singular linear continuous$-$time fractional$-$order systems, Jouranl of the Franklin Institute, 354 (2017), 752-766.  doi: 10.1016/j.jfranklin.2016.10.022.  Google Scholar

[21]

S. Marir, M. 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.  Google Scholar

[22]

D. Matignon, Stability results for fractional differential equations with applications to control processing , Proc. Computational Engineering in Systems and Applications Multiconferences (IMACS), (1996), 963–868. Google Scholar

[23]

C. Peng, T. C. Yang and E. G. Tian, Robust fault-tolerant control of networked control systems with stochastic actuator failure, IET Control Theory & Appl., 4 (2012), 3003-3011.  doi: 10.1049/iet-cta.2009.0427.  Google Scholar

[24]

I. Podlubny, Fractional$-$order systems and $PI^\lambda D^\mu-$controllers, IEEE Transactions on Automat. Control, 44 (1999), 208-214.  doi: 10.1109/9.739144.  Google Scholar

[25] I. Podlubny, Fractional Differertial Equations, Academic Press, New York, 1999.  doi: 10.1007/978-3-642-39765-33.  Google Scholar
[26]

H. Shen, X. N. Song and Z. Wang, Robust fault-tolerant control of uncertain fractional-order systems against actuator faults, IET Control Theory & Appl., 7 (2013), 1233-1241.  doi: 10.1049/iet-cta.2012.0822.  Google Scholar

[27]

J. Shen and J. D. Cao, Necessary and sufficient conditions for consensus of delayed fractional$-$order systems, Asian J. Control, 14 (2012), 1690-1697.  doi: 10.1002/asjc.492.  Google Scholar

[28]

X. N. Song, Y. Q. Chen and H. Shen, LMI fault tolerant control for interval fractional-order systems with sensor failures , Proc. Fourth IFAC Workshop Fractional Differentiation and its Applications, (2010), Article no. FDA10-126, Badajoz, Spain. Google Scholar

[29]

X. N. Song and H. Shen, Fault tolerant control for interval fractional-order systems with sensor failures, Advances in Mathematical Physics, 2013 (2013), 1-11.  doi: 10.1155/2013/836743.  Google Scholar

[30]

F. Tao and Q. Zhao, Synthesis of active fault$-$tolerant control based on Markovian jump system models, IET Control Theory & Appl., 1 (2007), 1160-1168.  doi: 10.1049/iet-cta:20050492.  Google Scholar

[31]

M. Tavakoli-Kakhki and M. S. Tavazoei, Estimation of the order and parameters of a fractional order model from a noisy step response data, Journal of Dynamic Systems, Measurement Control, 136 (2014), 1-7.  doi: 10.1115/1.4026345.  Google Scholar

[32]

E. Uezato and M. Ikeda, Strict LMI conditions for stability, robust stabilization, and $H_\infty$ control of descriptor systems , IEEE Conference on Decision & Contol. IEEE, (1994), 4092–4097. doi: 10.1109/CDC.1999.828001.  Google Scholar

[33]

Y. H. Wei, P. W. Tse, Y. Zhao 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.  Google Scholar

[34]

Y. H. Wei, J. C. Wang, T. Y. Liu and Y. Wang, Sufficient and necessary conditions for stabilizing singular fractional order systems with partially measurable state, Jouranl of the Franklin Institute, 356 (2019), 1975-1990.  doi: 10.1016/j.jfranklin.2019.01.022.  Google Scholar

[35]

Z. G. Wu, P. Shi, H. Y. Su and J. Chu, Reliable $H_\infty$ control for control for discrete$-$time fuzzy systems with infinite$-$distributed delay, IEEE Trans. on Fuzzy Syst., 20 (2012), 22-31.  doi: 10.1109/TFUZZ.2011.2162850.  Google Scholar

[36]

L. H. Xie, Output feedback $H_\infty$ control of systems with parameter uncertainty, International Journal of Control, 63 (1996), 741-750.  doi: 10.1080/00207179608921866.  Google Scholar

[37]

S. Y. Xu and J. Lam, Robust Control and Filtering of Singular Systems, Springer, Berlin, 2006. doi: 10.1080/00207721.2014.998751.  Google Scholar

[38]

G. H. Yang, J. L. Wang and Y. C. Soh, Reliable $H_\infty$ controller design for linear systems, Automatica, 37 (2001), 717-725.  doi: 10.1016/S0005-1098(01)00007-3.  Google Scholar

[39]

H. Yang, V. Cocquempot and B. Jiang, Robust fault tolerant tracking control with application to hybrid nonlinear systems, Control Theory & Appl., 3 (2009), 211-224.  doi: 10.1049/iet-cta:20080015.  Google Scholar

[40]

T. Zhan and S. P. Ma, The controller design for singular fractional-order systems with fractional order $0<\alpha<1$, The ANZIAM Journal, 60 (2018), 230-248.  doi: 10.1017/S1446181118000202.  Google Scholar

[41]

X. F. Zhang and Y. Q. Chen, D$-$stability based LMI criteria of stability and stabilization for fractional order systems , ASME 2015 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference, DETC2015-46692, (2015), 1–6. Google Scholar

[42]

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 Transcations, 82 (2018), 42-50.  doi: 10.1016/j.isatra.2017.03.008.  Google Scholar

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.  Google Scholar

[2]

M. Blanke, M. Kinnaert, J. Lunze and M. Staroswiecki, Diagnosisi and Fault-Tolerant Control, Springer, Berlin, Germany, 2006. doi: 10.1007/978-3-540-35653-0.  Google Scholar

[3]

M. Blanke, R. Izadi-Zamanabadi, S. A. Bogh and C. P. Lunau, Fault-tolerant control systems$-$a holistic view, Contol Engineering Prac., 5 (1997), 693-702.  doi: 10.1016/s0967-0661(97)00051-8.  Google Scholar

[4]

L. L. Fan and Y. D. Song, Neuro$-$adaptive model$-$referance fault$-$tolerant control with application to wind turbines, IET Control Theory & Appl., 6 (2012), 475-486.  doi: 10.1049/iet-cta.2011.0250.  Google Scholar

[5]

C. Farges, M. Moze and J. Sabatier, Pseudo-state feedback stabilization of commensurate fractional order systems, Automatica, 46 (2010), 1730-1734.  doi: 10.1016/j.automatica.2010.06.038.  Google Scholar

[6]

C. Farges, J. Sabatier and M. Moze, Robust stability analysis and stabilization of fractional order polytopic systems , Preprints of the 18th IFAC World Congress, (2011), Milano, Italy, 10800-10805. doi: 10.3182/20110828-6-IT-1002.00779.  Google Scholar

[7]

Bin Guo and Yong Chen, Adaptive fault tolerant control for time-varying delay system withactuator fault and mismatched disturbance, ISA Transcation, 89 (2019), 122-130.  doi: 10.1016/j.isatra.2018.12.024.  Google Scholar

[8]

Q. Hu, B. Xiao and M. I. Friswll, Robust fault$-$tolerant control for spacecraft attitude stabilisation subject to input saturation, IET Control Theory & Appl., 5 (2011), 271-282.  doi: 10.1049/iet-cta.2009.0628.  Google Scholar

[9]

S. D. Huang, J. Lam, G. H. Yang and S. Y. Zhang, Fault tolerant decentralized $H_\infty$ control for symmetric composite systems, IEEE Trans. Autom. Control, 44 (1999), 2108-2114.  doi: 10.1109/9.802926.  Google Scholar

[10]

R. Isermann, Fault-Diagnosis Systems: An Introduction from Fault Detection to Fault Tolerance, Springer, Berlin, Germany, 2006. doi: 10.1007/3-540-30368-5.  Google Scholar

[11]

L. A. Jacyntho, M. C. M. Teixeira and E. Assunco, Identification of fractional-order transfer functions using a step excitation, IEEE Trans. Circuits Syst., 62 (2015), 896-900.  doi: 10.1109/tcsii.2015.2436052.  Google Scholar

[12]

Y. Jiang, Q. L. Hu and G. F. Ma, Adaptive backsteping fault$-$tolerant control for flexible spacecraft with unknown bounded disturbances and actuator failures, ISA Transcations, 49 (2010), 57-69.  doi: 10.1016/j.isatra.2009.08.003.  Google Scholar

[13]

E. Last, Linear matrix inequalities in system and control theory, Proceedings of the IEEE, 86 (1994), 2473-2474.  doi: 10.1109/JPROC.1998.735454.  Google Scholar

[14]

X. J. Li and G. H. Yang, Robust adaptive fault$-$tolerant control for uncertain linear systems with actuator failures, IET Control Theory & Appl., 6 (2012), 1544-1551.  doi: 10.1049/iet-cta.2011.0599.  Google Scholar

[15]

Y. Li, Y. Q. Chen and I. Podlubny, Stability of fractional order nonlinear dynamic systems, Automatica, 45 (2009), 1965-1969.  doi: 10.1016/j.automatica.2009.04.003.  Google Scholar

[16]

C. Lin, B. Chen, P. 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.  Google Scholar

[17]

J. G. Lu and G. R. Chen, Robust stability and stabilization of fractional order interval systems: An LMI approach, IEEE Transactions on Automatic Control, 54 (2009), 1294-1299.  doi: 10.1109/tac.2009.2013056.  Google Scholar

[18]

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

[19]

H. J. Ma and G. H. Yang, Detection and adaptive accommodation for actuator faults of a class of non$-$linear systems, IET Control Theory & Appl., 6 (2012), 2292-2307.  doi: 10.1049/iet-cta.2011.0265.  Google Scholar

[20]

S. Marir, M. Chadli and D. Bouagada, New admissibility conditions for singular linear continuous$-$time fractional$-$order systems, Jouranl of the Franklin Institute, 354 (2017), 752-766.  doi: 10.1016/j.jfranklin.2016.10.022.  Google Scholar

[21]

S. Marir, M. 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.  Google Scholar

[22]

D. Matignon, Stability results for fractional differential equations with applications to control processing , Proc. Computational Engineering in Systems and Applications Multiconferences (IMACS), (1996), 963–868. Google Scholar

[23]

C. Peng, T. C. Yang and E. G. Tian, Robust fault-tolerant control of networked control systems with stochastic actuator failure, IET Control Theory & Appl., 4 (2012), 3003-3011.  doi: 10.1049/iet-cta.2009.0427.  Google Scholar

[24]

I. Podlubny, Fractional$-$order systems and $PI^\lambda D^\mu-$controllers, IEEE Transactions on Automat. Control, 44 (1999), 208-214.  doi: 10.1109/9.739144.  Google Scholar

[25] I. Podlubny, Fractional Differertial Equations, Academic Press, New York, 1999.  doi: 10.1007/978-3-642-39765-33.  Google Scholar
[26]

H. Shen, X. N. Song and Z. Wang, Robust fault-tolerant control of uncertain fractional-order systems against actuator faults, IET Control Theory & Appl., 7 (2013), 1233-1241.  doi: 10.1049/iet-cta.2012.0822.  Google Scholar

[27]

J. Shen and J. D. Cao, Necessary and sufficient conditions for consensus of delayed fractional$-$order systems, Asian J. Control, 14 (2012), 1690-1697.  doi: 10.1002/asjc.492.  Google Scholar

[28]

X. N. Song, Y. Q. Chen and H. Shen, LMI fault tolerant control for interval fractional-order systems with sensor failures , Proc. Fourth IFAC Workshop Fractional Differentiation and its Applications, (2010), Article no. FDA10-126, Badajoz, Spain. Google Scholar

[29]

X. N. Song and H. Shen, Fault tolerant control for interval fractional-order systems with sensor failures, Advances in Mathematical Physics, 2013 (2013), 1-11.  doi: 10.1155/2013/836743.  Google Scholar

[30]

F. Tao and Q. Zhao, Synthesis of active fault$-$tolerant control based on Markovian jump system models, IET Control Theory & Appl., 1 (2007), 1160-1168.  doi: 10.1049/iet-cta:20050492.  Google Scholar

[31]

M. Tavakoli-Kakhki and M. S. Tavazoei, Estimation of the order and parameters of a fractional order model from a noisy step response data, Journal of Dynamic Systems, Measurement Control, 136 (2014), 1-7.  doi: 10.1115/1.4026345.  Google Scholar

[32]

E. Uezato and M. Ikeda, Strict LMI conditions for stability, robust stabilization, and $H_\infty$ control of descriptor systems , IEEE Conference on Decision & Contol. IEEE, (1994), 4092–4097. doi: 10.1109/CDC.1999.828001.  Google Scholar

[33]

Y. H. Wei, P. W. Tse, Y. Zhao 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.  Google Scholar

[34]

Y. H. Wei, J. C. Wang, T. Y. Liu and Y. Wang, Sufficient and necessary conditions for stabilizing singular fractional order systems with partially measurable state, Jouranl of the Franklin Institute, 356 (2019), 1975-1990.  doi: 10.1016/j.jfranklin.2019.01.022.  Google Scholar

[35]

Z. G. Wu, P. Shi, H. Y. Su and J. Chu, Reliable $H_\infty$ control for control for discrete$-$time fuzzy systems with infinite$-$distributed delay, IEEE Trans. on Fuzzy Syst., 20 (2012), 22-31.  doi: 10.1109/TFUZZ.2011.2162850.  Google Scholar

[36]

L. H. Xie, Output feedback $H_\infty$ control of systems with parameter uncertainty, International Journal of Control, 63 (1996), 741-750.  doi: 10.1080/00207179608921866.  Google Scholar

[37]

S. Y. Xu and J. Lam, Robust Control and Filtering of Singular Systems, Springer, Berlin, 2006. doi: 10.1080/00207721.2014.998751.  Google Scholar

[38]

G. H. Yang, J. L. Wang and Y. C. Soh, Reliable $H_\infty$ controller design for linear systems, Automatica, 37 (2001), 717-725.  doi: 10.1016/S0005-1098(01)00007-3.  Google Scholar

[39]

H. Yang, V. Cocquempot and B. Jiang, Robust fault tolerant tracking control with application to hybrid nonlinear systems, Control Theory & Appl., 3 (2009), 211-224.  doi: 10.1049/iet-cta:20080015.  Google Scholar

[40]

T. Zhan and S. P. Ma, The controller design for singular fractional-order systems with fractional order $0<\alpha<1$, The ANZIAM Journal, 60 (2018), 230-248.  doi: 10.1017/S1446181118000202.  Google Scholar

[41]

X. F. Zhang and Y. Q. Chen, D$-$stability based LMI criteria of stability and stabilization for fractional order systems , ASME 2015 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference, DETC2015-46692, (2015), 1–6. Google Scholar

[42]

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 Transcations, 82 (2018), 42-50.  doi: 10.1016/j.isatra.2017.03.008.  Google Scholar

Figure 1.  State responses of the closed-loop system in Example 1 with $ u(t) = 0. $
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Figure 2.  Observation errors of the selected system in Example 1 with $ u(t) = Kx(t). $
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Figure 3.  State responses of the closed-loop system in Example 1 with $ u(t) = Kx(t). $
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