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; Y. C. E. Lee; Kar Hung Wong

doi:10.3934/jimo.2019055

Article Contents

2020, Volume 16, Issue 5: 2305-2330. Doi: 10.3934/jimo.2019055

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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
^* Corresponding author: Kar Hung Wong

Received: May 31, 2018

Revised: November 30, 2018

Early access: May 2019

Published: August 2020

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Abstract

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:

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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 K_P(t)	Values of K_I(t)	Values of K_D(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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