# American Institute of Mathematical Sciences

April  2021, 14(4): 1447-1464. doi: 10.3934/dcdss.2020375

## Output feedback based sliding mode control for fuel quantity actuator system using a reduced-order GPIO

 1 School of Automation, Southeast University, Nanjing 210096, China 2 Key Laboratory of Measurement and Control of Complex Systems of Engineering, Ministry of Education, China 3 Shenzhen Research Institute of Southeast University, Shenzhen 518057, China

* Corresponding author: Shihua Li

Received  October 2019 Revised  February 2020 Published  April 2021 Early access  May 2020

In an electronically controlled VE distributive pump, the fuel quantity actuator is a significant component. It is responsible for governing the quantity of fuel being injected into diesel-type engines. The FQA system has nonlinearities and always confronts disturbances caused by the external torque and the input voltage variation in the real working condition, which can be regarded as a lumped disturbance. However, most existing results only focus on dealing with the so called constant disturbance in the FQA system which fail to remove the influence of time-varying disturbances. Therefore, to deal with the nonlinearities and reject the lumped disturbance, a reduced-order generalized proportional integral observer (GPIO) based sliding mode control approach is presented. By using a reduced-order GPIO, time-varying disturbances can be estimated accurately. In addition, a theoretical analysis of the closed-loop system is given. The proposed control scheme exhibits a satisfactory performance in terms of transient behavior and disturbance rejection. Finally, a set of experimental tests are carried out to validate the feasibility as well as efficiency of the proposed control framework.

Citation: Hao Sun, Shihua Li, Xuming Wang. Output feedback based sliding mode control for fuel quantity actuator system using a reduced-order GPIO. Discrete and Continuous Dynamical Systems - S, 2021, 14 (4) : 1447-1464. doi: 10.3934/dcdss.2020375
##### References:
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[7] Y. Li, W. Gao and X. Zhou, The adaptive fuzzy control of electromagnetic actuator in diesel fuel injection system, in Proceedings of the IEEE International Vehicle Electronics Conference (IVEC'99)(Cat. No. 99EX257), IEEE, 1999,149-152. [8] Y. Li, G. Liu and X. Zhou, Fuel-injection control system design and experiments of a diesel engine, IEEE Transactions on Control Systems Technology, 11 (2003), 565-570. [9] R. Madoński and P. Herman, Survey on methods of increasing the efficiency of extended state disturbance observers, ISA Transactions, 56 (2015), 18-27. [10] J. Mao, J. Yang, S. Li, Y. Yan and Q. Li, Output feedback-based sliding mode control for disturbed motion control systems via a higher-order ESO approach, IET Control Theory & Applications, 12 (2018), 2118-2126.  doi: 10.1049/iet-cta.2018.5197. [11] K. Mollenhauer, H. Tschöke and K. G. Johnson, Handbook of Diesel Engines, vol. 1, Springer, 2010. [12] K. Reif, Diesel Engine Management, Springer, 2014. doi: 10.1007/978-3-658-03981-3. [13] C. Ren and S. Ma, Generalized proportional integral observer based control of an omnidirectional mobile robot, Mechatronics, 26 (2015), 36-44.  doi: 10.1016/j.mechatronics.2015.01.001. [14] K. Shi, Z. Wang, C. Wu and S. Li, GPIO based backstepping control for electronic throttle,, in IECON 2017-43rd Annual Conference of the IEEE Industrial Electronics Society, IEEE, 2017, 6087-6092. doi: 10.1109/IECON.2017.8217057. [15] G. Stumpp and H. Kull, Strategy for a Fail-Safe Electronic Diesel Control System for Passenger Cars, Technical report, SAE Technical Paper, 1983. doi: 10.4271/830527. [16] H. Sun, C. Dai and S. Li, Modelling and composite control of fuel quantity actuator system for diesel engines, IFAC-PapersOnLine, 51 (2018), 807-812.  doi: 10.1016/j.ifacol.2018.10.124. [17] H. Sun and S. Li, Sliding mode control method for diesel engine fuel quantity actuator with disturbance estimation, Control Theory and Applications, 35 (2018), 1568-1576. [18] Z. Sun, T. Guo, Y. Yan, X. Wang and S. Li, A composite current-constrained control for permanent magnet synchronous motor with time-varying disturbance, Advances in Mechanical Engineering, 9 (2017), 1687814017728691. doi: 10.1177/1687814017728691. [19] M. Trenne and A. Ives, Closed loop design for electronic diesel injection systems, SAE Transactions, 2017, 1834-1840. doi: 10.4271/820447. [20] J. Wang, S. Li, J. Yang, B. Wu and Q. Li, Extended state observer-based sliding mode control for PWM-based DC-DC buck power converter systems with mismatched disturbances, IET Control Theory & Applications, 9 (2015), 579-586.  doi: 10.1049/iet-cta.2014.0220. [21] J. Wang, F. Wang, G. Wang, S. Li and L. Yu, Generalized proportional integral observer based robust finite control set predictive current control for induction motor systems with time-varying disturbances, IEEE Transactions on Industrial Informatics, 14 (2018), 4159-4168.  doi: 10.1109/TII.2018.2818153. [22] Z. Wang, S. Li, J. Wang and Q. Li, Robust control for disturbed buck converters based on two GPI observers, Control Engineering Practice, 66 (2017), 13-22.  doi: 10.1016/j.conengprac.2017.06.001. [23] Y. Xia, Z. Zhu and M. Fu, Back-stepping sliding mode control for missile systems based on an extended state observer, IET Control Theory & Applications, 5 (2011), 93-102.  doi: 10.1049/iet-cta.2009.0341. [24] J. Yang, H. Cui, S. Li and A. Zolotas, Optimized active disturbance rejection control for DC-DC buck converters with uncertainties using a reduced-order GPI observer, IEEE Transactions on Circuits and Systems Ⅰ: Regular Papers, 65 (2018), 832-841.  doi: 10.1109/TCSI.2017.2725386. [25] M. Yang and S. C. Sorenson, Survey of the Electronic Injection and Control of Diesel Engines, Technical report, SAE Technical Paper, 1994. doi: 10.4271/940378. [26] M. Yawei and X. Feiyun, Application of discrete tracking differentiator to electronic control system of diesel engine, in 2011 International Conference on Electric Information and Control Engineering, IEEE, 2011, 3005-3008.

show all references

##### References:
 [1] W.-H. Chen, J. Yang, L. Guo and S. Li, Disturbance-observer-based control and related methods-an overview, IEEE Transactions on Industrial Electronics, 63 (2016), 1083-1095.  doi: 10.1109/TIE.2015.2478397. [2] H. Eisele, Electronic Control of Diesel Passenger Cars, Technical report, SAE Technical Paper, 1980. doi: 10.4271/800167. [3] A. Gutiérrez-Giles and M. A. Arteaga-Pérez, GPIbased velocity/force observer design for robot manipulators, ISA Transactions, 53 (2014), 929-938. [4] W. He, S. Li, J. Yang and Z. Wang, Incremental passivity based control for {DC-DC} boost converters under time-varying disturbances via a generalized proportional integral observer, Journal of Power Electronics, 18 (2018), 147-159. [5] H. K. Khalil, Nonlinear systems, Upper Saddle River. [6] K.-S. Kim, K.-H. Rew and S. Kim, Disturbance observer for estimating higher order disturbances in time series expansion, IEEE Transactions on Automatic Control, 55 (2010), 1905-1911.  doi: 10.1109/TAC.2010.2049522. [7] Y. Li, W. Gao and X. Zhou, The adaptive fuzzy control of electromagnetic actuator in diesel fuel injection system, in Proceedings of the IEEE International Vehicle Electronics Conference (IVEC'99)(Cat. No. 99EX257), IEEE, 1999,149-152. [8] Y. Li, G. Liu and X. Zhou, Fuel-injection control system design and experiments of a diesel engine, IEEE Transactions on Control Systems Technology, 11 (2003), 565-570. [9] R. Madoński and P. Herman, Survey on methods of increasing the efficiency of extended state disturbance observers, ISA Transactions, 56 (2015), 18-27. [10] J. Mao, J. Yang, S. Li, Y. Yan and Q. Li, Output feedback-based sliding mode control for disturbed motion control systems via a higher-order ESO approach, IET Control Theory & Applications, 12 (2018), 2118-2126.  doi: 10.1049/iet-cta.2018.5197. [11] K. Mollenhauer, H. Tschöke and K. G. Johnson, Handbook of Diesel Engines, vol. 1, Springer, 2010. [12] K. Reif, Diesel Engine Management, Springer, 2014. doi: 10.1007/978-3-658-03981-3. [13] C. Ren and S. Ma, Generalized proportional integral observer based control of an omnidirectional mobile robot, Mechatronics, 26 (2015), 36-44.  doi: 10.1016/j.mechatronics.2015.01.001. [14] K. Shi, Z. Wang, C. Wu and S. Li, GPIO based backstepping control for electronic throttle,, in IECON 2017-43rd Annual Conference of the IEEE Industrial Electronics Society, IEEE, 2017, 6087-6092. doi: 10.1109/IECON.2017.8217057. [15] G. Stumpp and H. Kull, Strategy for a Fail-Safe Electronic Diesel Control System for Passenger Cars, Technical report, SAE Technical Paper, 1983. doi: 10.4271/830527. [16] H. Sun, C. Dai and S. Li, Modelling and composite control of fuel quantity actuator system for diesel engines, IFAC-PapersOnLine, 51 (2018), 807-812.  doi: 10.1016/j.ifacol.2018.10.124. [17] H. Sun and S. Li, Sliding mode control method for diesel engine fuel quantity actuator with disturbance estimation, Control Theory and Applications, 35 (2018), 1568-1576. [18] Z. Sun, T. Guo, Y. Yan, X. Wang and S. Li, A composite current-constrained control for permanent magnet synchronous motor with time-varying disturbance, Advances in Mechanical Engineering, 9 (2017), 1687814017728691. doi: 10.1177/1687814017728691. [19] M. Trenne and A. Ives, Closed loop design for electronic diesel injection systems, SAE Transactions, 2017, 1834-1840. doi: 10.4271/820447. [20] J. Wang, S. Li, J. Yang, B. Wu and Q. Li, Extended state observer-based sliding mode control for PWM-based DC-DC buck power converter systems with mismatched disturbances, IET Control Theory & Applications, 9 (2015), 579-586.  doi: 10.1049/iet-cta.2014.0220. [21] J. Wang, F. Wang, G. Wang, S. Li and L. Yu, Generalized proportional integral observer based robust finite control set predictive current control for induction motor systems with time-varying disturbances, IEEE Transactions on Industrial Informatics, 14 (2018), 4159-4168.  doi: 10.1109/TII.2018.2818153. [22] Z. Wang, S. Li, J. Wang and Q. Li, Robust control for disturbed buck converters based on two GPI observers, Control Engineering Practice, 66 (2017), 13-22.  doi: 10.1016/j.conengprac.2017.06.001. [23] Y. Xia, Z. Zhu and M. Fu, Back-stepping sliding mode control for missile systems based on an extended state observer, IET Control Theory & Applications, 5 (2011), 93-102.  doi: 10.1049/iet-cta.2009.0341. [24] J. Yang, H. Cui, S. Li and A. Zolotas, Optimized active disturbance rejection control for DC-DC buck converters with uncertainties using a reduced-order GPI observer, IEEE Transactions on Circuits and Systems Ⅰ: Regular Papers, 65 (2018), 832-841.  doi: 10.1109/TCSI.2017.2725386. [25] M. Yang and S. C. Sorenson, Survey of the Electronic Injection and Control of Diesel Engines, Technical report, SAE Technical Paper, 1994. doi: 10.4271/940378. [26] M. Yawei and X. Feiyun, Application of discrete tracking differentiator to electronic control system of diesel engine, in 2011 International Conference on Electric Information and Control Engineering, IEEE, 2011, 3005-3008.
Bosch electronically controlled VE distribution pump
Structure of fuel quantity actuator
Diagram of the fuel quantity actuator under the reduced-order GPIO based output feedback sliding mode control approach
Experimental test setup
Response curves in the presence of constant disturbance under SMC+ESO controller (34) (a) angular position; (b) duty ratio
Response curves in the presence of constant disturbance under SMC+GPIO controller (18) (a) angular position; (b) duty ratio
Response curves in the presence of time-varying disturbance under SMC+ESO controller (34) (a) angular position; (b) duty ratio
Response curves in the presence of time-varying disturbance under SMC+GPIO controller (18) (a) angular position; (b) duty ratio
Parameters of the fuel quantity actuator
 Parameter Symbol Value Nominal Input Voltage $V_{in0}$ 12 $V$ Reference Output Angle ${\theta}_{ref}$ 0.5 $rad$ Nominal Resistance $R$ 0.75 $\Omega$
 Parameter Symbol Value Nominal Input Voltage $V_{in0}$ 12 $V$ Reference Output Angle ${\theta}_{ref}$ 0.5 $rad$ Nominal Resistance $R$ 0.75 $\Omega$
Parameters values for simplified model (5)
 Parameter Names Parameter Values $a_{21}$ $-7.1121\times10^{3}$ $a_{22}$ $-41.6290$ $c_{2}$ $-36.1109$ $c$ $-2.3370\times10^{3}$ $b$ $3.7872\times10^{4}$
 Parameter Names Parameter Values $a_{21}$ $-7.1121\times10^{3}$ $a_{22}$ $-41.6290$ $c_{2}$ $-36.1109$ $c$ $-2.3370\times10^{3}$ $b$ $3.7872\times10^{4}$
Control parameters for fuel quantity actuator
 Controller Control Parameters $SMC+GPIO$ $k_1=1000, \lambda = 30, \beta=-200$ $SMC+ESO$ $k_2=1000, \lambda = 30, p=-200$
 Controller Control Parameters $SMC+GPIO$ $k_1=1000, \lambda = 30, \beta=-200$ $SMC+ESO$ $k_2=1000, \lambda = 30, p=-200$
Comparisons of disturbance rejection performance (Case Ⅰ: Constant disturbance)
 Controller MAPR RT IAE(3-6s) Case Ⅰ SMC+ESO 0.0294rad 520ms 8.7119 SMC+GPIO 0.0228rad 502ms 6.4476
 Controller MAPR RT IAE(3-6s) Case Ⅰ SMC+ESO 0.0294rad 520ms 8.7119 SMC+GPIO 0.0228rad 502ms 6.4476
Comparisons of disturbance rejection performance (Case Ⅱ: Time-varying disturbance)
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