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January  2022, 15(1): 197-212. doi: 10.3934/dcdss.2021036

Frequency domain $ H_{\infty} $ control design for active suspension systems

1. 

University of Sidi Mohammed Ben Abdellah, Department of Physics, Faculty of Sciences Dhar El Mehraz, LISAC, BP 1796, Fes-Atlas, Morocco

2. 

University of Sidi Mohammed Ben Abdellah, Engineering, Systems and Applications, Laboratory National School of Applied Sciences (ENSA), Fez, Morocco

* Corresponding author: Jamal Mrazgua

Received  August 2020 Revised  February 2021 Published  January 2022 Early access  March 2021

A methodology for fault-tolerant-control(FTC) is proposed that compensates actuator failures in active suspension systems (ASS). This methodology is based on a Frequency Domain approach that represents failures using a scale factor to optimize the ASS and improve ride comfort. The controller design is carried out using off-the-shelf tools based on linear matrix inequalities (LMIs), guaranteeing asymptotic stability, compensating the effect of actuator faults, and ensuring certain $ H_{\infty} $ performance. In the context of ASS, the performance guarantees correspond to ride comfort in the presence of road disturbances. To validate the approach, controllers are developed and tested in simulation for a quarter-car model: the results illustrate the effectiveness of the proposed approach.

Citation: Jamal Mrazgua, El Houssaine Tissir, Mohamed Ouahi. Frequency domain $ H_{\infty} $ control design for active suspension systems. Discrete and Continuous Dynamical Systems - S, 2022, 15 (1) : 197-212. doi: 10.3934/dcdss.2021036
References:
[1]

M. Bahreini and J. Zarei, Robust fault-tolerant control for networked control systems subject to random delays via static-output feedback, ISA Transactions, 86 (2019), 153-162. 

[2]

Y. Bo and W. Fuzhong, LMI approach to reliable $ H_{\infty} $ control of linear systems, Journal of Systems Engineering and Electronics, 17 (2006), 381-386.  doi: 10.1016/S1004-4132(06)60065-0.

[3]

P. BucciJ. KirschenbaumL. A. ManganT. AldemirC. Smith and T. Wood, Construction of event-tree/fault-tree models from a Markov approach to dynamic system reliability, Reliability Engineering & System Safety, 93 (2008), 1616-1627.  doi: 10.1016/j.ress.2008.01.008.

[4]

D. CaoX. Song and M. Ahmadian, Editors perspectives: Road vehicle suspension design, dynamics, and control, Vehicle System Dynamics, 49 (2011), 3-28.  doi: 10.1080/00423114.2010.532223.

[5]

H. Chen and K.-H. Guo, Constrained $ H_{\infty} $ control of active suspensions: An LMI approach, IEEE Transactions on Control Systems Technology, 13 (2005), 412-421. doi: 10.1109/TCST.2004.841661.

[6]

M. Z. Q. ChenY. HuC. Li and G. Chen, Performance benefits of using inerter in semiactive suspensions, IEEE Transactions on Control Systems Technology, 23 (2015), 1571-1577.  doi: 10.1109/TCST.2014.2364954.

[7]

W. Chen and M. Saif, An iterative learning observer for fault detection and accommodation in nonlinear time-delay systems, Internat. J. Robust Nonlinear Control, 16 (2006), 1-19.  doi: 10.1002/rnc.1033.

[8]

H. DuW. Li and N. Zhang, Integrated seat and suspension control for a quarter car with driver model, IEEE Transactions on Vehicular Technology, 61 (2012), 3893-3908.  doi: 10.1109/TVT.2012.2212472.

[9]

H. Du and N. Zhang, $ H_{\infty} $ control of active vehicle suspensions with actuator time delay, J. Sound Vibration, 301 (2007), 236-252.  doi: 10.1016/j.jsv.2006.09.022.

[10]

F. EL HaoussiE. H. Tissir and F. Tadeo, Advances in the robust stabilization of neutral systems with saturating actuators, International Journal of Ecology and Development, 28 (2014), 49-62. 

[11]

C. El-Kasri, A. Hmamed, E. H. Tissir and F. Tadeo, Robust $ H_{\infty} $ filtering for uncertain two-dimensional continuous systems with time varying delays, Multidimens. Syst. Signal Process., 24 (2013), 685-706. doi: 10.1007/s11045-013-0242-7.

[12]

P. Finsler, Über das Vorkommen definiter semidefiniter Formen is Scharen quadratischer Formen, Comment. Math. Helv., 9 (1936), 188-192.

[13]

P. Gahinet, A. Nemirovskii, A. J. Laud and M. Chilali, LMI Control Toolbox User's Guide, Natick, MA: The Math. Works Inc, 1995.

[14]

H. Gao, W. Sun and O. Kaynak, Vibration suspension of vehicle active suspension systems in finite frequency domain, Joint 48th IEEE Conf. Decision and Control and 28th Chinese Control Conf., Shanghai, P.R. China, (2009), 5170-5175. doi: 10.1109/CDC.2009.5400753.

[15]

H. Gao, W. Sun and P. Shi, Robust sampled-data $ H_ {\infty} $ control for vehicle active suspension systems, IEEE Transactions on Control Systems Technology, 18 (2009), 238-245.

[16]

Y. HuM. Z. Q. Chen and Z. Shu, Passive vehicle suspensions employing inerters with multiple performance requirements, Journal of Sound and Vibration, 333 (2014), 2212-2225.  doi: 10.1016/j.jsv.2013.12.016.

[17]

T. Hu and Z. Lin, Control Systems With Actuator Saturation: Analysis and Design, Springer Science & Business Media, 2001.

[18]

T. Iwasaki and S. Hara, Generalized KYP Lemma: Unified frequency domain inequalities with design applications, IEEE Trans. Autom. Control, 50 (2005), 41-59. doi: 10.1109/TAC.2004.840475.

[19]

H. Li, J. Yu, C. Hilton and H. Liu, Adaptive sliding mode control for nonlinear active suspension vehicle systems using T-S fuzzy approach, IEEE Trans. Ind. Electron., 60 (2013), 3328-3338. doi: 10.1109/TIE.2012.2202354.

[20]

Y. LiuJ. WangG.-H. Yang and C. B. Soh, Reliable nonlinear control system design using duplicated control elements, Internat. J. Robust Nonlinear Control, 7 (1997), 1103-1122.  doi: 10.1002/(SICI)1099-1239(199712)7:12<1103::AID-RNC343>3.0.CO;2-R.

[21]

M.-M. Ma and H. Chen, Disturbance attenuation control of active suspension with non-linear actuator dynamics, IET Control Theory Appl., 5 (2011), 112-122. doi: 10.1049/iet-cta.2009.0457.

[22]

X. Ma, P. K. Wong and J. Zhao, Practical multi-objective control for automotive semi-active suspension system with nonlinear hydraulic adjustable damper, Mechanical Systems and Signal Processing, 117 (2019), 667-688. doi: 10.1016/j.ymssp.2018.08.022.

[23]

J. Madhavanpillai and B. B. Das, Detection of faults in flight control actuators with hard saturation and friction non-linearity, International Journal of Reliability and Safety, 7 (2013), 108-127. doi: 10.1504/IJRS.2013.056376.

[24]

J. Mrazgua, E. H. Tissir and M. Ouahi, Fault-tolerant $H_{\infty} $ control approach, application to active half-vehicle suspension systems with actuators failure accounts, 2019 8th International Conference on Systems and Control (ICSC), Marrakesh, Morocco, (2019), 271-276. doi: 10.1109/ICSC47195.2019.8950668.

[25]

J. Mrazgua, E. H. Tissir and M. Ouahi, Fuzzy fault-tolerant $ H_{\infty} $ control approach for nonlinear active suspension systems with actuator failure, Procedia Computer Science, 148 (2019), 465-474. doi: 10.1016/j.procs.2019.01.059.

[26]

P. Mukhija, I. Narayan Kar and R. K. P. Bhatt, Robust absolute stability criteria for uncertain lurie system with interval time-varying delay, ASME J. Dyn. Syst. Meas. Control, 136 (2014), 041020, 7 pp. doi: 10.1115/1.4026872.

[27]

N. P. Nguyen, T. T. Huynh, X. P. Do, N. Xuan Mung and S. K. Hong, Robust fault estimation using the intermediate observer: Application to the quadcopter, Sensors, 20 (2020), 4917. doi: 10.3390/s20174917.

[28]

P. Panagi and M. M. Polycarpou, Decentralized fault tolerant control of a class of interconnected nonlinear systems, IEEE Trans. Automat. Control, 56 (2011), 178-184. doi: 10.1109/TAC.2010.2089650.

[29]

R. Sakthivel, L. Susana Ramya, Y.-K. Ma, M. Malik and A. Leelamani, Stabilization of uncertain switched discrete-time systems against actuator faults and input saturation, Nonlinear Anal. Hybrid Syst., 35 (2020), 100827, 12 pp. doi: 10.1016/j.nahs.2019.100827.

[30]

W. Sun, H. Gao and O. Kaynak, Finite frequency $ H_ {\infty} $ control for vehicle active suspension systems, IEEE Transactions on Control Systems Technology, 19 (2010), 416-422.

[31]

K. Telbissi and A. Benzaouia, Robust fault-tolerant control for discrete-time switched systems with time delays, Internat. J. Adapt. Control Signal Process., 34 (2020), 389-406. doi: 10.1002/acs.3091.

[32]

L. Xing, G. Levitin and C. Wang, Dynamic System Reliability: Modeling and Analysis of Dynamic and Dependent Behaviors, John Wiley & Sons, 2019.

[33]

M. Yamashita, K. Fujimori, K. Hayakawa and H. Kimura, Application of $ H_{\infty} $ control to active suspension systems, \textitAutomatica, 30 (1994), 1717-1729. doi: 10.1016/0005-1098(94)90074-4.

[34]

H. Zhang, Y. Shi and J. Wang, Observer-based tracking controller design for networked predictive control systems with uncertain Markov delays, Internat. J. Control, 86 (2013), 1824-1836. doi: 10.1080/00207179.2013.797107.

[35]

Q. Zhao and J. Jiang, Reliable state feedback control systemdesign against actuator failures, Automatica, 34 (1998), 1267-1272. doi: 10.1016/S0005-1098(98)00072-7.

[36]

Z. Zhu, Y. Xia and M. Fu, Adaptive sliding mode control for attitude stabilization with actuator saturation, IEEE Transactions on Industrial Electronics, 58 (2011), 4898-4907. doi: 10.1109/TIE.2011.2107719.

[37]

International Standards Organisation, Evaluation of Human Exposure to Whole-Body Vibration, Part 1-General requirements. ISO Geneva, Switzerland, 1997.

show all references

References:
[1]

M. Bahreini and J. Zarei, Robust fault-tolerant control for networked control systems subject to random delays via static-output feedback, ISA Transactions, 86 (2019), 153-162. 

[2]

Y. Bo and W. Fuzhong, LMI approach to reliable $ H_{\infty} $ control of linear systems, Journal of Systems Engineering and Electronics, 17 (2006), 381-386.  doi: 10.1016/S1004-4132(06)60065-0.

[3]

P. BucciJ. KirschenbaumL. A. ManganT. AldemirC. Smith and T. Wood, Construction of event-tree/fault-tree models from a Markov approach to dynamic system reliability, Reliability Engineering & System Safety, 93 (2008), 1616-1627.  doi: 10.1016/j.ress.2008.01.008.

[4]

D. CaoX. Song and M. Ahmadian, Editors perspectives: Road vehicle suspension design, dynamics, and control, Vehicle System Dynamics, 49 (2011), 3-28.  doi: 10.1080/00423114.2010.532223.

[5]

H. Chen and K.-H. Guo, Constrained $ H_{\infty} $ control of active suspensions: An LMI approach, IEEE Transactions on Control Systems Technology, 13 (2005), 412-421. doi: 10.1109/TCST.2004.841661.

[6]

M. Z. Q. ChenY. HuC. Li and G. Chen, Performance benefits of using inerter in semiactive suspensions, IEEE Transactions on Control Systems Technology, 23 (2015), 1571-1577.  doi: 10.1109/TCST.2014.2364954.

[7]

W. Chen and M. Saif, An iterative learning observer for fault detection and accommodation in nonlinear time-delay systems, Internat. J. Robust Nonlinear Control, 16 (2006), 1-19.  doi: 10.1002/rnc.1033.

[8]

H. DuW. Li and N. Zhang, Integrated seat and suspension control for a quarter car with driver model, IEEE Transactions on Vehicular Technology, 61 (2012), 3893-3908.  doi: 10.1109/TVT.2012.2212472.

[9]

H. Du and N. Zhang, $ H_{\infty} $ control of active vehicle suspensions with actuator time delay, J. Sound Vibration, 301 (2007), 236-252.  doi: 10.1016/j.jsv.2006.09.022.

[10]

F. EL HaoussiE. H. Tissir and F. Tadeo, Advances in the robust stabilization of neutral systems with saturating actuators, International Journal of Ecology and Development, 28 (2014), 49-62. 

[11]

C. El-Kasri, A. Hmamed, E. H. Tissir and F. Tadeo, Robust $ H_{\infty} $ filtering for uncertain two-dimensional continuous systems with time varying delays, Multidimens. Syst. Signal Process., 24 (2013), 685-706. doi: 10.1007/s11045-013-0242-7.

[12]

P. Finsler, Über das Vorkommen definiter semidefiniter Formen is Scharen quadratischer Formen, Comment. Math. Helv., 9 (1936), 188-192.

[13]

P. Gahinet, A. Nemirovskii, A. J. Laud and M. Chilali, LMI Control Toolbox User's Guide, Natick, MA: The Math. Works Inc, 1995.

[14]

H. Gao, W. Sun and O. Kaynak, Vibration suspension of vehicle active suspension systems in finite frequency domain, Joint 48th IEEE Conf. Decision and Control and 28th Chinese Control Conf., Shanghai, P.R. China, (2009), 5170-5175. doi: 10.1109/CDC.2009.5400753.

[15]

H. Gao, W. Sun and P. Shi, Robust sampled-data $ H_ {\infty} $ control for vehicle active suspension systems, IEEE Transactions on Control Systems Technology, 18 (2009), 238-245.

[16]

Y. HuM. Z. Q. Chen and Z. Shu, Passive vehicle suspensions employing inerters with multiple performance requirements, Journal of Sound and Vibration, 333 (2014), 2212-2225.  doi: 10.1016/j.jsv.2013.12.016.

[17]

T. Hu and Z. Lin, Control Systems With Actuator Saturation: Analysis and Design, Springer Science & Business Media, 2001.

[18]

T. Iwasaki and S. Hara, Generalized KYP Lemma: Unified frequency domain inequalities with design applications, IEEE Trans. Autom. Control, 50 (2005), 41-59. doi: 10.1109/TAC.2004.840475.

[19]

H. Li, J. Yu, C. Hilton and H. Liu, Adaptive sliding mode control for nonlinear active suspension vehicle systems using T-S fuzzy approach, IEEE Trans. Ind. Electron., 60 (2013), 3328-3338. doi: 10.1109/TIE.2012.2202354.

[20]

Y. LiuJ. WangG.-H. Yang and C. B. Soh, Reliable nonlinear control system design using duplicated control elements, Internat. J. Robust Nonlinear Control, 7 (1997), 1103-1122.  doi: 10.1002/(SICI)1099-1239(199712)7:12<1103::AID-RNC343>3.0.CO;2-R.

[21]

M.-M. Ma and H. Chen, Disturbance attenuation control of active suspension with non-linear actuator dynamics, IET Control Theory Appl., 5 (2011), 112-122. doi: 10.1049/iet-cta.2009.0457.

[22]

X. Ma, P. K. Wong and J. Zhao, Practical multi-objective control for automotive semi-active suspension system with nonlinear hydraulic adjustable damper, Mechanical Systems and Signal Processing, 117 (2019), 667-688. doi: 10.1016/j.ymssp.2018.08.022.

[23]

J. Madhavanpillai and B. B. Das, Detection of faults in flight control actuators with hard saturation and friction non-linearity, International Journal of Reliability and Safety, 7 (2013), 108-127. doi: 10.1504/IJRS.2013.056376.

[24]

J. Mrazgua, E. H. Tissir and M. Ouahi, Fault-tolerant $H_{\infty} $ control approach, application to active half-vehicle suspension systems with actuators failure accounts, 2019 8th International Conference on Systems and Control (ICSC), Marrakesh, Morocco, (2019), 271-276. doi: 10.1109/ICSC47195.2019.8950668.

[25]

J. Mrazgua, E. H. Tissir and M. Ouahi, Fuzzy fault-tolerant $ H_{\infty} $ control approach for nonlinear active suspension systems with actuator failure, Procedia Computer Science, 148 (2019), 465-474. doi: 10.1016/j.procs.2019.01.059.

[26]

P. Mukhija, I. Narayan Kar and R. K. P. Bhatt, Robust absolute stability criteria for uncertain lurie system with interval time-varying delay, ASME J. Dyn. Syst. Meas. Control, 136 (2014), 041020, 7 pp. doi: 10.1115/1.4026872.

[27]

N. P. Nguyen, T. T. Huynh, X. P. Do, N. Xuan Mung and S. K. Hong, Robust fault estimation using the intermediate observer: Application to the quadcopter, Sensors, 20 (2020), 4917. doi: 10.3390/s20174917.

[28]

P. Panagi and M. M. Polycarpou, Decentralized fault tolerant control of a class of interconnected nonlinear systems, IEEE Trans. Automat. Control, 56 (2011), 178-184. doi: 10.1109/TAC.2010.2089650.

[29]

R. Sakthivel, L. Susana Ramya, Y.-K. Ma, M. Malik and A. Leelamani, Stabilization of uncertain switched discrete-time systems against actuator faults and input saturation, Nonlinear Anal. Hybrid Syst., 35 (2020), 100827, 12 pp. doi: 10.1016/j.nahs.2019.100827.

[30]

W. Sun, H. Gao and O. Kaynak, Finite frequency $ H_ {\infty} $ control for vehicle active suspension systems, IEEE Transactions on Control Systems Technology, 19 (2010), 416-422.

[31]

K. Telbissi and A. Benzaouia, Robust fault-tolerant control for discrete-time switched systems with time delays, Internat. J. Adapt. Control Signal Process., 34 (2020), 389-406. doi: 10.1002/acs.3091.

[32]

L. Xing, G. Levitin and C. Wang, Dynamic System Reliability: Modeling and Analysis of Dynamic and Dependent Behaviors, John Wiley & Sons, 2019.

[33]

M. Yamashita, K. Fujimori, K. Hayakawa and H. Kimura, Application of $ H_{\infty} $ control to active suspension systems, \textitAutomatica, 30 (1994), 1717-1729. doi: 10.1016/0005-1098(94)90074-4.

[34]

H. Zhang, Y. Shi and J. Wang, Observer-based tracking controller design for networked predictive control systems with uncertain Markov delays, Internat. J. Control, 86 (2013), 1824-1836. doi: 10.1080/00207179.2013.797107.

[35]

Q. Zhao and J. Jiang, Reliable state feedback control systemdesign against actuator failures, Automatica, 34 (1998), 1267-1272. doi: 10.1016/S0005-1098(98)00072-7.

[36]

Z. Zhu, Y. Xia and M. Fu, Adaptive sliding mode control for attitude stabilization with actuator saturation, IEEE Transactions on Industrial Electronics, 58 (2011), 4898-4907. doi: 10.1109/TIE.2011.2107719.

[37]

International Standards Organisation, Evaluation of Human Exposure to Whole-Body Vibration, Part 1-General requirements. ISO Geneva, Switzerland, 1997.

Figure 1.  Quarter-car model with an active suspension
Figure 2.  Bump response of sprung mass acceleration
Figure 3.  Bump response of suspension deflection
Figure 4.  Bump response of tyre deflection
Figure 5.  Transfer function
Figure 6.  Force of the actuator
Table 1.  Quarter-Car Model Parameters
$ m_{s} $ Suspended mass
$ m_{u} $ Unsupported mass
$ c_{s} $ Damping of the suspension system
$ k_{s} $ Stiffness of the suspension system
$ c_{t} $ Damping of the pneumatic tyre
$ k_{t} $ Compressibility of the pneumatic tyre
$z_{s} $ Displacement of the sprung mass
$ z_{u} $ Displacement of unsprung mass
$ \delta^ {f} (t) $ Fault-tolerant-control
$ m_{s} $ Suspended mass
$ m_{u} $ Unsupported mass
$ c_{s} $ Damping of the suspension system
$ k_{s} $ Stiffness of the suspension system
$ c_{t} $ Damping of the pneumatic tyre
$ k_{t} $ Compressibility of the pneumatic tyre
$z_{s} $ Displacement of the sprung mass
$ z_{u} $ Displacement of unsprung mass
$ \delta^ {f} (t) $ Fault-tolerant-control
Table 2.  Quarter-Car Model Parameters
$ m_{s} $ $ m_{u} $ $ k_{s} $ $ k_{t} $ $ c_{s} $ $ c_{t} $
320 40 18 200 1 10
kg kg KN/m KN/m KNs/m Ns/m
$ m_{s} $ $ m_{u} $ $ k_{s} $ $ k_{t} $ $ c_{s} $ $ c_{t} $
320 40 18 200 1 10
kg kg KN/m KN/m KNs/m Ns/m
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