# American Institute of Mathematical Sciences

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

 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.

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 and 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. [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. [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. [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. [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. [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. [7] M. W. Iruthayarajan and S. Baskar, Evolutionary algorithms based design of multivariate PID controller, Expert systems and Applications, 36 (2009), 9159-9167. [8] L. S. Jennings and M. E. Fisher, Miser3: Optimal Control Toolbox: User Manual, , Matlab Beta Version 2.0, Nedlands, WA 6907, Australia, 2002. [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 [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. [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. [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. [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. [14] H. W. J. Lee and K. L. Teo, Acta Astronautic, 74 (2012), 131–140., doi: 10.1023/A:1024735407694. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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.

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. [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. [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. [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. [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. [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. [7] M. W. Iruthayarajan and S. Baskar, Evolutionary algorithms based design of multivariate PID controller, Expert systems and Applications, 36 (2009), 9159-9167. [8] L. S. Jennings and M. E. Fisher, Miser3: Optimal Control Toolbox: User Manual, , Matlab Beta Version 2.0, Nedlands, WA 6907, Australia, 2002. [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 [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. [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. [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. [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. [14] H. W. J. Lee and K. L. Teo, Acta Astronautic, 74 (2012), 131–140., doi: 10.1023/A:1024735407694. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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.
Graphs of $\operatorname{sign}_{\delta}(x)$ versus $x$
A schematic diagram for the vehicle suspension model
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)
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)
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)
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)
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 5.308.75 -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
 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 5.308.75 -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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