
-
Previous Article
Projection methods for solving split equilibrium problems
- JIMO Home
- This Issue
-
Next Article
Two nonparametric approaches to mean absolute deviation portfolio selection model
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 |
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.
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. 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 |
[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. 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. |
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. 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 |
[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. 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. |





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 |
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 |
[1] |
Fabio Camilli, Serikbolsyn Duisembay, Qing Tang. Approximation of an optimal control problem for the time-fractional Fokker-Planck equation. Journal of Dynamics & Games, 2021 doi: 10.3934/jdg.2021013 |
[2] |
Diana Keller. Optimal control of a linear stochastic Schrödinger equation. Conference Publications, 2013, 2013 (special) : 437-446. doi: 10.3934/proc.2013.2013.437 |
[3] |
Paula A. González-Parra, Sunmi Lee, Leticia Velázquez, Carlos Castillo-Chavez. A note on the use of optimal control on a discrete time model of influenza dynamics. Mathematical Biosciences & Engineering, 2011, 8 (1) : 183-197. doi: 10.3934/mbe.2011.8.183 |
[4] |
Changjun Yu, Lei Yuan, Shuxuan Su. A new gradient computational formula for optimal control problems with time-delay. Journal of Industrial & Management Optimization, 2021 doi: 10.3934/jimo.2021076 |
[5] |
Jamal Mrazgua, El Houssaine Tissir, Mohamed Ouahi. Frequency domain $ H_{\infty} $ control design for active suspension systems. Discrete & Continuous Dynamical Systems - S, 2021 doi: 10.3934/dcdss.2021036 |
[6] |
Wensheng Yin, Jinde Cao, Guoqiang Zheng. Further results on stabilization of stochastic differential equations with delayed feedback control under $ G $-expectation framework. Discrete & Continuous Dynamical Systems - B, 2021 doi: 10.3934/dcdsb.2021072 |
[7] |
Elimhan N. Mahmudov. Second order discrete time-varying and time-invariant linear continuous systems and Kalman type conditions. Numerical Algebra, Control & Optimization, 2021 doi: 10.3934/naco.2021010 |
[8] |
Bingru Zhang, Chuanye Gu, Jueyou Li. Distributed convex optimization with coupling constraints over time-varying directed graphs†. Journal of Industrial & Management Optimization, 2021, 17 (4) : 2119-2138. doi: 10.3934/jimo.2020061 |
[9] |
Tobias Geiger, Daniel Wachsmuth, Gerd Wachsmuth. Optimal control of ODEs with state suprema. Mathematical Control & Related Fields, 2021 doi: 10.3934/mcrf.2021012 |
[10] |
Andrea Signori. Penalisation of long treatment time and optimal control of a tumour growth model of Cahn–Hilliard type with singular potential. Discrete & Continuous Dynamical Systems, 2021, 41 (6) : 2519-2542. doi: 10.3934/dcds.2020373 |
[11] |
Guirong Jiang, Qishao Lu. The dynamics of a Prey-Predator model with impulsive state feedback control. Discrete & Continuous Dynamical Systems - B, 2006, 6 (6) : 1301-1320. doi: 10.3934/dcdsb.2006.6.1301 |
[12] |
Quan Hai, Shutang Liu. Mean-square delay-distribution-dependent exponential synchronization of chaotic neural networks with mixed random time-varying delays and restricted disturbances. Discrete & Continuous Dynamical Systems - B, 2021, 26 (6) : 3097-3118. doi: 10.3934/dcdsb.2020221 |
[13] |
Marita Holtmannspötter, Arnd Rösch, Boris Vexler. A priori error estimates for the space-time finite element discretization of an optimal control problem governed by a coupled linear PDE-ODE system. Mathematical Control & Related Fields, 2021 doi: 10.3934/mcrf.2021014 |
[14] |
Lorenzo Freddi. Optimal control of the transmission rate in compartmental epidemics. Mathematical Control & Related Fields, 2021 doi: 10.3934/mcrf.2021007 |
[15] |
Marzia Bisi, Maria Groppi, Giorgio Martalò, Romina Travaglini. Optimal control of leachate recirculation for anaerobic processes in landfills. Discrete & Continuous Dynamical Systems - B, 2021, 26 (6) : 2957-2976. doi: 10.3934/dcdsb.2020215 |
[16] |
Qi Lü, Xu Zhang. A concise introduction to control theory for stochastic partial differential equations. Mathematical Control & Related Fields, 2021 doi: 10.3934/mcrf.2021020 |
[17] |
Luke Finlay, Vladimir Gaitsgory, Ivan Lebedev. Linear programming solutions of periodic optimization problems: approximation of the optimal control. Journal of Industrial & Management Optimization, 2007, 3 (2) : 399-413. doi: 10.3934/jimo.2007.3.399 |
[18] |
Xiaohong Li, Mingxin Sun, Zhaohua Gong, Enmin Feng. Multistage optimal control for microbial fed-batch fermentation process. Journal of Industrial & Management Optimization, 2021 doi: 10.3934/jimo.2021040 |
[19] |
John T. Betts, Stephen Campbell, Claire Digirolamo. Examination of solving optimal control problems with delays using GPOPS-Ⅱ. Numerical Algebra, Control & Optimization, 2021, 11 (2) : 283-305. doi: 10.3934/naco.2020026 |
[20] |
Livia Betz, Irwin Yousept. Optimal control of elliptic variational inequalities with bounded and unbounded operators. Mathematical Control & Related Fields, 2021 doi: 10.3934/mcrf.2021009 |
2019 Impact Factor: 1.366
Tools
Metrics
Other articles
by authors
[Back to Top]