A backstepping-based fault compensation scheme for a class of Euler-Bernoulli beam-ODE cascade systems. (English) Zbl 1480.93109

Summary: This paper develops a fault compensation scheme for a class of Euler-Bernoulli beam-ODE cascade systems to deal with certain boundary input faults, using a model transform-based backstepping control design. A model transformation is introduced to convert the Euler-Bernoulli beam-ODE cascade system to a Schrödinger-ODE cascade system. The FTC law is designed to compensate boundary input faults for the Schrödinger-ODE cascade system by constructing a normal intermediate system and a new exponentially stable target system. The purposes of the two-step backstepping are to bring in a state feedback and to improve the system performance. The performance of the target system, intermediate system, and Schrödinger-ODE cascade system is analysed in the Riesz basis frame. The stability analysis of the original Euler-Bernoulli beam-ODE cascade system is further provided. Simulation results illustrate the effectiveness of the proposed fault compensation scheme.


93B35 Sensitivity (robustness)
93C20 Control/observation systems governed by partial differential equations
93C15 Control/observation systems governed by ordinary differential equations
93B52 Feedback control
Full Text: DOI


[1] Blanke, M.; Kinnaert, M.; Lunze, J.; Staroswiecki, M., Diagnosis and fault-tolerant control (2015), Berlin: Springer Verlag, Berlin
[2] Cheng, M. B.; Radisavljevic, V.; Su, W. C., Sliding mode boundary control of a parabolic PDE system with parameter variations and bouondary uncertainties, Automatica, 47, 2, 381-387 (2011) · Zbl 1213.35239
[3] Franze, G.; Tedesco, F.; Famularo, D., Actuator fault tolerant control: A receding horizon set-theoretic approach, IEEE Transactions on Automatic Control, 60, 8, 2225-2230 (2015) · Zbl 1360.93450
[4] Gao, S. Q.; Liu, J. K., Adaptive fault-tolerant vibration control of a wind turbine blade with actuator stuck, International Journal of Control, 1-12 (2018)
[5] Gao, Y.; Wu, H. N.; Wang, J. W.; Guo, L., Feedback control design with vibration suppression for flexible air-breathing hypersonic vehicles, Sciece China Information Sciences, 57, 3, 1-14 (2014) · Zbl 1336.93069
[6] Ge, S. S.; Zhang, S.; He, W., Vibration control of an Euler-Bernoulli beam under unknown spatiotemporally varying disturbance, International Journal of Control, 84, 5, 947-960 (2011) · Zbl 1245.93065
[7] Ghantasala, S.; El-Farra, N. H., Robust actuator fault isolation and management in constrained uncertain parabolic PDE systems, Automatica, 45, 10, 2368-2373 (2009) · Zbl 1179.93103
[8] Guo, B. Z.; Jin, F. F., The active disturbance rejection and sliding mode control approach to the stabilization of Euler-Bernoulli beam equation with boundary input disturbance, Automatica, 49, 9, 2911-2918 (2013) · Zbl 1364.93637
[9] Guo, B. Z.; Liu, J. J.; Al-Fhaid, A. S.; Younas, A. M. M.; Asiri, A., The active disturbance rejection control approach to stabilisation of coupled heat and ODE system subject to boundary control matched disturbance, International Journal of Control, 88, 8, 1554-1564 (2015) · Zbl 1337.93078
[10] Hasan, A.; Aamo, O. M.; Krstic, M., Boundary observer design for hyperbolic PDE-ODE cascade systems, Automatica, 68, 75-86 (2016) · Zbl 1334.93037
[11] He, W.; Ge, S. S.; How, B. V. E.; Choo, Y. S.; Hong, K. S., Robust adaptive boundary control of a flexible marine riser with vessel dynamics, Automatica, 47, 4, 722-732 (2011) · Zbl 1215.93073
[12] He, W.; Zhang, S.; Ge, S. S., Adaptive boundary control of a nonlinear flexible string system, IEEE Transactions on Control Systems Technology, 22, 3, 1088-1093 (2014)
[13] Jiang, B.; Staroswiecki, M.; Cocquempot, V., Fault accommodation for nonlinear dynamic systems, IEEE Transactions on Automatic Control, 51, 9, 1578-1583 (2006) · Zbl 1366.93694
[14] Kang, W.; Fridman, E., Sliding mode control of Schrödinger-ODE in the presence of unmatched disturbance, Systems & Control Letters, 98, 65-73 (2016) · Zbl 1351.93036
[15] Kang, W.; Fridman, E., Boundary constrained control of delayed nonlinear Schrödinger equation, IEEE Transactions on Automatic Control, 63, 11, 3873-3880 (2018) · Zbl 1423.93303
[16] Krstic, M., Delay compensation for nonlinear, adaptive, and PDE systems (2009), Birkhauser: Springer, Birkhauser · Zbl 1181.93003
[17] Krstic, M.; Kanellakopoulos, I.; Kokotovic, P. V., Nonlinear and adaptive control design (1995), New York, NY: Wiley, New York, NY
[18] Krstic, M.; Smyshlyaev, A., Boundary control of PDEs: A course on backstepping designs (2008), Philadelphia: Society for Industrial and Applied Mathematics, Philadelphia · Zbl 1149.93004
[19] Liu, Z. J.; Liu, J. K.; He, W., Robust adaptive fault tolerant control for a linear cascaded ODE-beam system, Automatica, 98, 42-50 (2018) · Zbl 1406.93094
[20] Ma, H. J.; Yang, G. H., Adaptive fault tolerant control of cooperative heterogeneous systems with actuator faults and unreliable interconnections, IEEE Transactions on Automatic Control, 61, 11, 3240-3255 (2016) · Zbl 1359.93228
[21] Mahmoud, M. S.; Memon, A. M.; Shi, P., Observer-based fault-tolerant control for a class of nonlinear network control systems, International Journal of Control, 87, 8, 1707-1715 (2014) · Zbl 1317.93089
[22] Mallavalli, S.; Fekih, A., A fault tolerant tracking control for a quadrotor UAV subject to simultaneous actuator faults and exogenous disturbances, International Journal of Control, 1-14 (2018) · Zbl 1440.93066
[23] Meglio, F.; Argomedo, F.; Hu, L.; Krstic, M., Stabilization of coupled linear heterodirectional hyperbolic PDE-ODE systems, Automatica, 87, 281-289 (2018) · Zbl 1378.93102
[24] Nwokah, O.; Hurmuzlu, Y., The mechanical systems design handbook: Modeling measurement and control (2002), Boca, Raton: CRC Press, Boca, Raton
[25] Patton, R. J.; Frank, P.; Clark, R., Issues of fault diagnosis for dynamic systems (2013), London: Springer Science & Business Media, London
[26] Ren, B. B.; Wang, J. M.; Krstic, M., Stabilization of an ODE-Schrödinger cascade, Systems & Control Letters, 62, 6, 503-510 (2013) · Zbl 1279.93085
[27] Smyshlyaev, A.; Krstic, M., Closed form boundary state feedbacks for a class of 1-D partial integro-differential equations, IEEE Transactions on Automatic Control, 49, 12, 2185-2202 (2004) · Zbl 1365.93193
[28] Su, X.; Shi, P.; Wu, L. G.; Song, Y. D., Fault detection filtering for nonlinear switched stochatic systems, IEEE Transactions on Automatic Control, 61, 5, 1310-1315 (2016) · Zbl 1359.93497
[29] Susto, G. A.; Krstic, M., Control of PDE-ODE cascades with Neumann interconnections, Journal of the Franklin Institute, 347, 1, 284-314 (2010) · Zbl 1298.93279
[30] Tao, G.; Chen, S.; Tang, X. D.; Joshi, S. M., Adaptive control of systems with actuator failures (2004), New York, NY: Springer-Verlag, New York, NY · Zbl 1063.93001
[31] Wang, J. M.; Liu, J. J.; Ren, B. B.; Chen, J. H., Sliding mode control to stabilization of cascaded heat PDE-ODE systems subject to boundary control matched disturbance, Automatica, 52, 23-34 (2015) · Zbl 1309.93124
[32] Wang, J. W.; Liu, Y. J.; Sun, C. Y., Pointwise exponential stabilization of a linear parabolic PDE system using non-collocated pointwise observation, Automatica, 93, 197-210 (2018) · Zbl 1400.93280
[33] Wang, Y.; Song, Y. D.; Krstic, M.; Wen, C. Y., Fault-tolerant finite time consensus for multiple uncertain nonlinear mechanical systems under single-way directly communication interactions and actuation failures, Automatica, 63, 374-383 (2016) · Zbl 1329.93016
[34] Wang, J. W.; Wu, H. N., Exponential pointwise stabilization of semilinear parabolic distributed parameter systems via the Takagi-Sugeno fuzzy PDE model, IEEE Transactions on Fuzzy Systems, 26, 1, 155-173 (2018)
[35] Wang, J. W.; Wu, H. N., Exponentially stabilizing fuzzy controller design for a nonlinear ODE-beam cascaded system and its application to flexible air-breathing hypersonic vehicle, Fuzzy Sets and Systems, 1-21 (2019)
[36] Wang, J. W.; Wu, H. N.; Li, H. X., Fuzzy control design for nonlinear ODE-hyperbolic PDE cascaded systems: A fuzzy and entropy-like lyapunov function approach, IEEE Transactions on Fuzzy Systems, 22, 5, 1313-1324 (2014)
[37] Wang, Z. P.; Wu, H. N.; Li, H. X., Sampled-data fuzzy control for nonlinear coupled parabolic PDE-ODE systems, IEEE Transactions on Cybernetics, 47, 9, 2603-2615 (2017) · Zbl 1386.93249
[38] Wu, H. N.; Feng, S., Mixed fuzzy/boundary control design for nonlinear coupled systems of ODE and boundary-disturbed uncertain beam, IEEE Transactions on Fuzzy Systems, 26, 6, 3379-3390 (2018)
[39] Wu, H. N.; Wang, J. W., Static output feedback control via PDE boundary and ODE measurements in linear cascaded ODE-beam systems, Automatica, 50, 2787-2798 (2014) · Zbl 1300.93081
[40] Yang, H.; Jiang, B.; Staroswiecki, M.; Zhang, Y. M., Fault recoverability and fault tolerant control for a class of interconnected nonlinear systems, Automatica, 54, 49-55 (2015) · Zbl 1318.93032
[41] Zhang, Y. M.; Jiang, J., Bibliographical review on reconfigurable fault-tolerant control systems, Annual Reviews in Control, 32, 2, 229-252 (2008)
[42] Zhang, X. D.; Polycarpou, M. M.; Parisini, T., Adaptive fault diagnosis and fault-tolerant control of MIMO nonlinear uncertain systems, International Journal of Control, 83, 5, 1054-1080 (2010) · Zbl 1197.93091
[43] Zhao, D.; Jiang, B.; Yang, H.; Tao, G., Fault-tolerant control of flexible air-breathing hypersonic vehicles in linear ODE-beam systems, International Journal of Control, 1-12 (2018)
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