American Institute of Mathematical Sciences

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September  2020, 16(5): 2305-2330. doi: 10.3934/jimo.2019055

Differential equation approximation and enhancing control method for finding the PID gain of a quarter-car suspension model with state-dependent ODE

H. W. J. Lee 1, , Y. C. E. Lee 1, and  Kar Hung Wong 2,,

1. 

Department of Applied Mathematics, The Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong, China
2. 

Department of Computational and Applied Mathematics, University of the Witwatersrand, Johannesburg, South Africa

* Corresponding author: Kar Hung Wong

Received  June 2018 Revised  December 2018 Published  September 2020 Early access  May 2019

Active suspension control strategy design in vehicle suspension systems has been a popular issue in road vehicle applications. In this paper, we consider a quarter-car suspension problem. A nonlinear objective function together with a system of state-dependent ODEs is involved in the model. A differential equation approximation method, together with the control parametrization enhancing transform (CPET), is used to find the optimal proportional-integral-derivative (PID) feedback gains of the above model. Hence, an approximated optimal control problem is obtained. Proofs of convergences of the state and the optimal control of the approximated problem to those of the original optimal control problem are provided. A numerical example is solved to illustrate the efficiency of our method.

Keywords: Car suspension, control parametrization enhancing technique, differential equation approximations, optimal control, PID controllers, time-varying feedback gains.
Mathematics Subject Classification: Primary: 49M15, 65M60; Secondary: 35Q92.
Citation: H. W. J. Lee, Y. C. E. Lee, Kar Hung Wong. Differential equation approximation and enhancing control method for finding the PID gain of a quarter-car suspension model with state-dependent ODE. Journal of Industrial & Management Optimization, 2020, 16 (5) : 2305-2330. doi: 10.3934/jimo.2019055
References:
[1]

P. Brezas, M. C. Smith and W. Hoult, A clipped-optimal control algorithm for semi-active vehicle, suspensions: theory and experimental evaluation, Automatica, 53 (2015), 188-194.  doi: 10.1016/j.automatica.2014.12.026.  Google Scholar

[2]

M. Z. Q. Chen, Y. Hu, C. Li and G. Chen, Application of semi-active inerter in semi-active suspensions via force tracking, Journal of Vibration and Acoustics, 138 (2016), 041014–1– 041014–11. doi: 10.1115/1.4033357.  Google Scholar

[3]

M. Čorič, J. Deur, L. Xu, H. E. Tseng and D. Horvat, Optimization of active suspension control inputs for improved vehicle ride performance, Vehicle System Dynamics, 54 (2016), 1004-1030.  doi: 10.1080/00423114.2016.1177655.  Google Scholar

[4]

F. Fruhauf, R. Kasper and J. L. Luckel, Design of an active suspension for a passenger vehicle model using input processes with time delays, Vehicle System Dynamic, 14 (1985), 115-120.  doi: 10.1080/00423118508968811.  Google Scholar

[5]

T. J. Gordan, C. Marsh and M. G. Milsted, A comparison of adaptive LQG and nonlinear controllers for vehicle suspension systems, Vehicle System Dynamic, 20 (1991), 321-340.  doi: 10.1080/00423119108968993.  Google Scholar

[6]

A. Hac, Suspension optimization of a 2-DOF vehicle model using a stochastic optimal control technique, Journal of Sound and Vibration, 100 (1985), 343-357.   Google Scholar

[7]

M. W. Iruthayarajan and S. Baskar, Evolutionary algorithms based design of multivariate PID controller, Expert systems and Applications, 36 (2009), 9159-9167.   Google Scholar

[8]

L. S. Jennings and M. E. Fisher, Miser3: Optimal Control Toolbox: User Manual, , Matlab Beta Version 2.0, Nedlands, WA 6907, Australia, 2002. Google Scholar

[9]

H. W. J. Lee, K. L. Teo, V. Rehbock and L. S. Jennings, Control parametrization enhancing technique for time optimal control problems, Dynamic Systems and Applications, 6 (1997), 243-262. Control parametrization enhancing technique for solving a special ODE class with state dependent switch, Journal of optimization Theory and Applications, 118 (2003), 55-66  Google Scholar

[10]

B. Li, K. L. Teo, C. C. Lim and G. R. Duan, An optimal PID controller design for nonlinear optimal constrained control problems, Discrete and Continuous Dynamical System, Series B, 16 (2011), 1101-1117.  doi: 10.3934/dcdsb.2011.16.1101.  Google Scholar

[11]

B. Li, C. Xu, K. L. Teo and J. Chu, Time optimal Zermelo's navigation problem with moving and fixed obstacles, Applied Mathematics and Computation, 224 (2013), 866-875.  doi: 10.1016/j.amc.2013.08.092.  Google Scholar

[12]

B. Li, C. J. Yu, K. L. Teo and G. R. Duan, An exact penalty function method for continuous inequality constrained optimal control problem, Journal of Optimization Theory and Applications, 151 (2011), 260-291.  doi: 10.1007/s10957-011-9904-5.  Google Scholar

[13]

C. J. Li, B. Li, K. L. Teo and G. F. Ma, A constrained optimal PID-like controller design for spacecraft attitude stabilization, doi: 10.1016/j.actaastro.2011.12.021.  Google Scholar

[14]

H. W. J. Lee and K. L. Teo, Acta Astronautic, 74 (2012), 131–140., doi: 10.1023/A:1024735407694.  Google Scholar

[15]

R. Li, K. L. Teo, K. H. Wong and G. R. Duan, Control Parameterization enhancing transform for optimal control of switched wystems, Mathmatical and Computer Modelling, 43 (2006), 1393-1403.  doi: 10.1016/j.mcm.2005.08.012.  Google Scholar

[16]

Q. Lin, R. Loxton and K. L. Teo, The control parameterization method for nonlinear optimal control: A survey, Journal of Industrial and Management Optimization, 10 (2014), 275-309.  doi: 10.3934/jimo.2014.10.275.  Google Scholar

[17]

C. Y. Liu, Z. H. Gong, E. M. Feng and H. C. Yin, Optimal switching control for microbial fed-batch culture, Nonlinear Analysis: Hybrid Systems, 2 (2008), 1168-1174.  doi: 10.1016/j.nahs.2008.09.005.  Google Scholar

[18]

C. Y. Liu, Z. H. Gong, E. M. Feng and H. C. Yin, Optimal switching control of a fed-batch fermentation process, Journal of Global Optimization, 52 (2012), 265-280.  doi: 10.1007/s10898-011-9663-8.  Google Scholar

[19]

C. Y. Liu, Z. H. Gong, K. L. Teo, R. Loxton and E. M. Feng, Bi-objective dynamic optimization of a nonlinear time-delay system in microbial batch process, Optimization Letter, 12 (2018), 1249-1264.  doi: 10.1007/s11590-016-1105-6.  Google Scholar

[20]

C. Y. Liu, R. Loxton, Q. Lin and K. L. Teo, Dynamic optimization for switched time-delay systems with state-dependent switching conditions, SIAM Journal on Control and Optimization, 56 (2018), 3499-3523.  doi: 10.1137/16M1070530.  Google Scholar

[21]

C. Myburgh and K. H. Wong, Computational control of an HIV Model, Annals of Operations Research, 133 (2005), 277-283.  doi: 10.1007/s10479-004-5038-6.  Google Scholar

[22]

B. Nagaraj and A. Vijayakumar, A comparative study of PID controller tuning using GA, PSO, EP and ACO, Journal of Automation, Mobile Robotics and Intelligent Systems, 5 (2011), 305-313.  doi: 10.1109/ICCCCT.2010.5670571.  Google Scholar

[23]

J. O. Pedro and O. A. Dahunsi, Neural network based feedback linearization control of a servo-hydraulic vehicle suspension system, International Journal of Applied Mathematics and Computer Science, 21 (2011), 137-147.  doi: 10.2478/v10006-011-0010-5.  Google Scholar

[24]

J. O. Pedro, M. Dangor, O. A. Dahunsi and M. M. Ali, CRS and PS optimised PID controller for nonlinear, electrohydraulic suspension systems, 9th Asian Control Conference (ASCC), (2013), 1–6. doi: 10.1109/ASCC.2013.6606012.  Google Scholar

[25]

K. L. Teo, C. J. Goh and K. H. Wong, A Unified Computational Approach to Optimal Control Problems, Longman Scientific and Technical, London, 1991.  Google Scholar

[26]

K. L. Teo, L. S. Jennings, H. W. J. Lee and V. Rehbock, Control parametrization enhancing transform for constrained optimal control problems, Journal of Australian Mathematical Society, Series B, 40 (1999), 314-335.  doi: 10.1017/S0334270000010936.  Google Scholar

[27]

H. E., Ts eng and D. Hrovat, State of the art survey: Active and semi-active suspension control, Vehicle System Dynamics, 53 (2015), 1034-1062.   Google Scholar

[28]

R. J. Wai, J. D. Lee and K. L. Chuang, Real-time PID control strategy for Maglev transportation system vis particle swarm optimization, IEEE Transactions on Industrial Electronics, 58 (2011), 629-646.   Google Scholar

[29]

F. Yang, K. L. Teo, R. Loxton, V. Rehbock, B. Li, C. Yu and L. S. Jennings, Visual Miser: An efficient user-friendly visual problem for solving optimal control problems, Journal of Industrial and Management Optimization, 12 (2016), 781-810.  doi: 10.3934/jimo.2016.12.781.  Google Scholar

show all references

References:
[1]

P. Brezas, M. C. Smith and W. Hoult, A clipped-optimal control algorithm for semi-active vehicle, suspensions: theory and experimental evaluation, Automatica, 53 (2015), 188-194.  doi: 10.1016/j.automatica.2014.12.026.  Google Scholar

[2]

M. Z. Q. Chen, Y. Hu, C. Li and G. Chen, Application of semi-active inerter in semi-active suspensions via force tracking, Journal of Vibration and Acoustics, 138 (2016), 041014–1– 041014–11. doi: 10.1115/1.4033357.  Google Scholar

[3]

M. Čorič, J. Deur, L. Xu, H. E. Tseng and D. Horvat, Optimization of active suspension control inputs for improved vehicle ride performance, Vehicle System Dynamics, 54 (2016), 1004-1030.  doi: 10.1080/00423114.2016.1177655.  Google Scholar

[4]

F. Fruhauf, R. Kasper and J. L. Luckel, Design of an active suspension for a passenger vehicle model using input processes with time delays, Vehicle System Dynamic, 14 (1985), 115-120.  doi: 10.1080/00423118508968811.  Google Scholar

[5]

T. J. Gordan, C. Marsh and M. G. Milsted, A comparison of adaptive LQG and nonlinear controllers for vehicle suspension systems, Vehicle System Dynamic, 20 (1991), 321-340.  doi: 10.1080/00423119108968993.  Google Scholar

[6]

A. Hac, Suspension optimization of a 2-DOF vehicle model using a stochastic optimal control technique, Journal of Sound and Vibration, 100 (1985), 343-357.   Google Scholar

[7]

M. W. Iruthayarajan and S. Baskar, Evolutionary algorithms based design of multivariate PID controller, Expert systems and Applications, 36 (2009), 9159-9167.   Google Scholar

[8]

L. S. Jennings and M. E. Fisher, Miser3: Optimal Control Toolbox: User Manual, , Matlab Beta Version 2.0, Nedlands, WA 6907, Australia, 2002. Google Scholar

[9]

H. W. J. Lee, K. L. Teo, V. Rehbock and L. S. Jennings, Control parametrization enhancing technique for time optimal control problems, Dynamic Systems and Applications, 6 (1997), 243-262. Control parametrization enhancing technique for solving a special ODE class with state dependent switch, Journal of optimization Theory and Applications, 118 (2003), 55-66  Google Scholar

[10]

B. Li, K. L. Teo, C. C. Lim and G. R. Duan, An optimal PID controller design for nonlinear optimal constrained control problems, Discrete and Continuous Dynamical System, Series B, 16 (2011), 1101-1117.  doi: 10.3934/dcdsb.2011.16.1101.  Google Scholar

[11]

B. Li, C. Xu, K. L. Teo and J. Chu, Time optimal Zermelo's navigation problem with moving and fixed obstacles, Applied Mathematics and Computation, 224 (2013), 866-875.  doi: 10.1016/j.amc.2013.08.092.  Google Scholar

[12]

B. Li, C. J. Yu, K. L. Teo and G. R. Duan, An exact penalty function method for continuous inequality constrained optimal control problem, Journal of Optimization Theory and Applications, 151 (2011), 260-291.  doi: 10.1007/s10957-011-9904-5.  Google Scholar

[13]

C. J. Li, B. Li, K. L. Teo and G. F. Ma, A constrained optimal PID-like controller design for spacecraft attitude stabilization, doi: 10.1016/j.actaastro.2011.12.021.  Google Scholar

[14]

H. W. J. Lee and K. L. Teo, Acta Astronautic, 74 (2012), 131–140., doi: 10.1023/A:1024735407694.  Google Scholar

[15]

R. Li, K. L. Teo, K. H. Wong and G. R. Duan, Control Parameterization enhancing transform for optimal control of switched wystems, Mathmatical and Computer Modelling, 43 (2006), 1393-1403.  doi: 10.1016/j.mcm.2005.08.012.  Google Scholar

[16]

Q. Lin, R. Loxton and K. L. Teo, The control parameterization method for nonlinear optimal control: A survey, Journal of Industrial and Management Optimization, 10 (2014), 275-309.  doi: 10.3934/jimo.2014.10.275.  Google Scholar

[17]

C. Y. Liu, Z. H. Gong, E. M. Feng and H. C. Yin, Optimal switching control for microbial fed-batch culture, Nonlinear Analysis: Hybrid Systems, 2 (2008), 1168-1174.  doi: 10.1016/j.nahs.2008.09.005.  Google Scholar

[18]

C. Y. Liu, Z. H. Gong, E. M. Feng and H. C. Yin, Optimal switching control of a fed-batch fermentation process, Journal of Global Optimization, 52 (2012), 265-280.  doi: 10.1007/s10898-011-9663-8.  Google Scholar

[19]

C. Y. Liu, Z. H. Gong, K. L. Teo, R. Loxton and E. M. Feng, Bi-objective dynamic optimization of a nonlinear time-delay system in microbial batch process, Optimization Letter, 12 (2018), 1249-1264.  doi: 10.1007/s11590-016-1105-6.  Google Scholar

[20]

C. Y. Liu, R. Loxton, Q. Lin and K. L. Teo, Dynamic optimization for switched time-delay systems with state-dependent switching conditions, SIAM Journal on Control and Optimization, 56 (2018), 3499-3523.  doi: 10.1137/16M1070530.  Google Scholar

[21]

C. Myburgh and K. H. Wong, Computational control of an HIV Model, Annals of Operations Research, 133 (2005), 277-283.  doi: 10.1007/s10479-004-5038-6.  Google Scholar

[22]

B. Nagaraj and A. Vijayakumar, A comparative study of PID controller tuning using GA, PSO, EP and ACO, Journal of Automation, Mobile Robotics and Intelligent Systems, 5 (2011), 305-313.  doi: 10.1109/ICCCCT.2010.5670571.  Google Scholar

[23]

J. O. Pedro and O. A. Dahunsi, Neural network based feedback linearization control of a servo-hydraulic vehicle suspension system, International Journal of Applied Mathematics and Computer Science, 21 (2011), 137-147.  doi: 10.2478/v10006-011-0010-5.  Google Scholar

[24]

J. O. Pedro, M. Dangor, O. A. Dahunsi and M. M. Ali, CRS and PS optimised PID controller for nonlinear, electrohydraulic suspension systems, 9th Asian Control Conference (ASCC), (2013), 1–6. doi: 10.1109/ASCC.2013.6606012.  Google Scholar

[25]

K. L. Teo, C. J. Goh and K. H. Wong, A Unified Computational Approach to Optimal Control Problems, Longman Scientific and Technical, London, 1991.  Google Scholar

[26]

K. L. Teo, L. S. Jennings, H. W. J. Lee and V. Rehbock, Control parametrization enhancing transform for constrained optimal control problems, Journal of Australian Mathematical Society, Series B, 40 (1999), 314-335.  doi: 10.1017/S0334270000010936.  Google Scholar

[27]

H. E., Ts eng and D. Hrovat, State of the art survey: Active and semi-active suspension control, Vehicle System Dynamics, 53 (2015), 1034-1062.   Google Scholar

[28]

R. J. Wai, J. D. Lee and K. L. Chuang, Real-time PID control strategy for Maglev transportation system vis particle swarm optimization, IEEE Transactions on Industrial Electronics, 58 (2011), 629-646.   Google Scholar

[29]

F. Yang, K. L. Teo, R. Loxton, V. Rehbock, B. Li, C. Yu and L. S. Jennings, Visual Miser: An efficient user-friendly visual problem for solving optimal control problems, Journal of Industrial and Management Optimization, 12 (2016), 781-810.  doi: 10.3934/jimo.2016.12.781.  Google Scholar

Figure 1.  Graphs of $ \operatorname{sign}_{\delta}(x) $ versus $ x $
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Figure 2.  A schematic diagram for the vehicle suspension model
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Figure 3(a) and 3(b).  Comparison of the impact of the various controllers on the first two measures (i.e., the suspension travel and the sprung mass acceleration): (ⅰ) PI controller under the 10-control-switchings (optimal switching time) scenario (represented by dotted lines); (ⅱ) PD controller under the 10- control-switchings (optimal switching time) scenarios (represented by dash lines); (ⅲ) PID controller under the 10-control-switchings (optimal switching times) scenarios (represented by solid lines); (ⅳ) the static controller (represented by dash-dotted lines)
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Figure 3(c) and 3(d).  Comparison of the impact of the various controllers on the last two measures (i.e., the actuator force and the control voltage): (ⅰ) PI controller under the 10-control-switchings (optimal switching time) scenario (represented by dotted lines); (ⅱ) PD controller under the 10-controlswitchings (optimal switching time) scenarios (represented by dash lines); (ⅲ) PID controller under the 10-control-switchings (optimal switching times) scenarios (represented by solid lines); (ⅳ) the static controller (represented by dash-dotted lines)
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Figure 4(a) and 4(b).  Comparision of the impact of the different switching-time scenarios of the PID controllers on the first two measures (i.e., the suspension travel and the sprung mass acceleration) : (ⅰ) PID controller under the 1-control-switching (optimal switching time) scenario (represented by dotted lines); (ⅱ) PID controller under the 2-control-switchings (optimal switching time) scenario (represented by solid lines); (ⅲ) PID controller under the 2-control-switchings (fixed switching times) scenario (represented by dash lines)
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Figure 4(c) and 4(d).  Comparision of the impact of the different switching-time scenarios of the PID controller on the last two measures (i.e., the actuator force and the control voltage): (ⅰ) PID controller under the 1-control-switching (optimal switching time) scenario (represented by dotted lines); (ⅱ) PID controller under the 2-control-switchings (optimal switching time) scenario (represented by solid lines); (ⅲ) PID controller under the 2-control-switchings (fixed switching times) scenario (represented by dash lines)
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Table 1.  The optimal switching time (times) and the optimal values of the 3 gains under different switching scenarios of the PID controller
PID Setting Switching Time(s) Values of KP(t) Values of KI(t) Values of KD(t)
1 control-switching
(optimal switching time)		 1.516 5.30
8.75		 -11.71
-13.47		 -0.85
-34.10
2 control-switchings
(optimal switching times)		 0.781
1.608		 4.33
0.80
2.34		 1.436
0.299
0.446		 -0.71
-6.56
1.29
2control-switchings
(fixed switching times)		 1.000
2.000		 4.33
1.23
0.95		 1.889
0.038
-1.731		 -0.73
-8.59
-7.32
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PID Setting Switching Time(s) Values of KP(t) Values of KI(t) Values of KD(t)
1 control-switching
(optimal switching time)		 1.516 5.30
8.75		 -11.71
-13.47		 -0.85
-34.10
2 control-switchings
(optimal switching times)		 0.781
1.608		 4.33
0.80
2.34		 1.436
0.299
0.446		 -0.71
-6.56
1.29
2control-switchings
(fixed switching times)		 1.000
2.000		 4.33
1.23
0.95		 1.889
0.038
-1.731		 -0.73
-8.59
-7.32
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