Dynamic monitoring feedback iteration control for damping-adjustable semi-active suspension systems

Abstract

In order to further improve vehicle ride performance, a dynamic monitoring feedback iteration control algorithm is proposed by combining the features of a variable-damping semi-active suspension system and applying them to the system. A seven-degree-of-freedom finished vehicle simulation model is built based on MATLAB/Simulink. The root-mean-square values of the acceleration of the sprung mass, the dynamic travel of the suspension and the dynamic tire load are taken as evaluation indicators of vehicle ride performance. An analytic hierarchy process (AHP) is used to determine the weighting coefficients of the evaluation indicators, and a genetic algorithm is utilized to determine the optimal damping of the suspension under various typical working conditions. Suspension damping is controlled with a dynamic monitoring feedback iteration algorithm. The correction coefficients of the control algorithm are determined according to the deviation between the obtained damping and the optimized damping so that the control parameters will agree with the optimal result under typical working conditions, and the control effect under other working conditions is verified. The simulation results indicate that the proposed dynamic monitoring feedback iteration control algorithm can effectively reduce the root-mean-square value of the acceleration of the sprung mass by 10.56% and the root-mean-square value of the acceleration of the dynamic travel of the suspension by 11.98% under mixed working conditions, thus improving vehicle ride performance. The study in this paper provides a new attempt for damping control of semi-active suspension and lays a theoretical foundation for its application in engineering.

Keywords

Semi-active suspension controlled damping dynamic monitoring feedback iteration analytic hierarchy process genetic algorithm ride performance

1 Introduction

Vehicle ride performance and handling stability are inherently contradictory. After the parameters of the traditional passive suspension are determined, the dynamic properties will also be determined and cannot be changed. However, people have different requirements for ride performance and handling stability under different actual running conditions. In order to improve vehicle adaptability for all kinds of working conditions, the variable-damping semi-active suspension system was created and has been extensively applied. The variable-damping semi-active suspension system can adjust the damping of the suspension shock absorber according to pavement characteristics and the vehicle load, with advantages such as low-frequency vibration and road impact resistance, so it can improve vehicle ride comfort under all kinds of working conditions [1 –3].

Implementing the effective control of the variable-damping semi-active suspension system and adjusting the suspension stiffness or damping according to running conditions can improve vehicle adaptability to the pavement, vehicle speed and load, thereby improving vehicle ride performance. Semi-active suspension control theory is an important research hotspot in the vehicle engineering field; mature control theories such as PID control, fuzzy control, and robust control have already been formed, all of which have improved suspension performance to a certain degree. A dynamic monitoring feedback iteration control algorithm that evolved from the traditional iteration algorithm is proposed in this paper. According to the real-time vibration of sprung mass (external environment), the control algorithm can obtain different damping iteration results, adjust the damping of the shock absorber in the suspension system based on the iteration result, and continue to track sprung mass vibration for the next damping iteration until the iteration result becomes stable. This control algorithm can provide a new idea for semi-active suspension control by virtue of discrete reaching and gradual improvement of the control effect. In this paper, the Dynamic Monitoring Feedback Iteration Control is proposed for the first time for the purpose of improving vehicle ride comfort, and it is applied to the damping control of semi-active suspension, which not only improves the control efficiency, but also lays a theoretical foundation for future engineering applications [4, 5].

2 System modeling

A seven-degrees-of-freedom finished semi-active pneumatic suspension model is established based on the mathematical model of a variable-damping shock absorption system. Related parameters of a light bus are shown in Table 1. The seven degrees of freedom are vertical displacement at the center of the sprung mass, vehicle body roll and vehicle body pitch, and four vertical displacements of the unsprung mass.

Table 1
Parameters of the finished vehicle

Parameter name Parameter value

Finished vehicle mass (full load) /kg 3900

Finished vehicle mass (no load) /kg 2800

Wheel mass /kg 50

Horizontal distance from front axis to center of mass /mm 2800

Horizontal distance from rear axis to center of mass /mm 1500

Wheelbase /mm 2000

Parameter name	Parameter value
Finished vehicle mass (full load) /kg	3900
Finished vehicle mass (no load) /kg	2800
Wheel mass /kg	50
Horizontal distance from front axis to center of mass /mm	2800
Horizontal distance from rear axis to center of mass /mm	1500
Wheelbase /mm	2000

Vertical displacements at the connecting positions between the four suspensions and the sprung mass are denoted Z_fL0, Z_fR0, Z_rL0 and Z_rR0; thus ${\begin{matrix} Z_{fL 0} = Z_{cb} + \frac{d}{2} θ - l_{f} ϕ \\ Z_{fR 0} = Z_{cb} - \frac{d}{2} θ - l_{f} ϕ \\ Z_{rL 0} = Z_{cb} + \frac{d}{2} θ + l_{r} ϕ \\ Z_{rR 0} = Z_{cb} - \frac{d}{2} θ + l_{r} ϕ \end{matrix}$ (1) where Z_cb is the vertical displacement of the center of the sprung mass; θ is the roll angle of the sprung mass; ϕ is the pitch angle of the sprung mass; lf and lr are the horizontal distances from the center of the sprung mass to its front and rear axles, respectively; and d is the wheelbase.

The forces of the four suspensions are ${\begin{matrix} F_{fL} = k_{f} \times Z_{fL} + C_{f} ({\dot{Z}}_{fL} - {\dot{Z}}_{fL 0}) \\ F_{rL} = k_{r} \times Z_{rL} + C_{r} ({\dot{Z}}_{rL} - {\dot{Z}}_{rL 0}) \\ F_{fR} = k_{f} \times Z_{fR} + C_{f} ({\dot{Z}}_{fR} - {\dot{Z}}_{fR 0}) \\ F_{rR} = k_{r} \times Z_{rR} + C_{r} ({\dot{Z}}_{rR} - {\dot{Z}}_{rR 0}) \end{matrix}$ (2) where F_fL, F_rL, F_fR and F_rR are acting forces at the connecting positions between the four suspensions and the sprung mass; k_f and k_r are the stiffness values of front and rear suspensions, respectively; C_f and C_r are the absorber damping coefficients of the front and rear suspensions; and Z_fL, Z_rL, Z_fR and Z_rR are the vertical displacements of the left front wheel, right front wheel, left rear wheel and right rear wheel, respectively.

The seven-degrees-of-freedom finished semi-active pneumatic suspension model is built according to Newton’s second law: ${\begin{matrix} m_{wf} {\ddot{Z}}_{fL} - k_{f} (q_{fL} - Z_{fL}) + F_{fL} = 0 \\ m_{wr} {\ddot{Z}}_{rL} - k_{r} (q_{rL} - Z_{rL}) + F_{rL} = 0 \\ m_{wf} {\ddot{Z}}_{fR} - k_{f} (q_{fR} - Z_{fR}) + F_{fR} = 0 \\ m_{wr} Z_{rR} - k_{r} (q_{rR} - Z_{rR}) + F_{rR} = 0 \\ m_{cb} {\ddot{Z}}_{cb} - F_{fL} - F_{rL} - F_{fR} - F_{rR} = 0 \\ J_{x} \ddot{θ} - \frac{d}{2} (F_{fL} + F_{rL}) + \frac{d}{2} (F_{fR} + F_{rR}) = 0 \\ J_{y} \ddot{ϕ} - l_{r} (F_{rL} + F_{rR}) + l_{f} (F_{fL} + F_{fR}) = 0 \end{matrix}$ (3) where qfL, qrL, qfL and qrR are the vertical displacement excitations borne by the left front wheel, right front wheel, left rear wheel and right rear wheel from the pavement, respectively; mwf and mwr are the masses of front and rear tires, respectively; mcb is the sprung mass; Jx is the rotational inertia of the sprung mass around the longitudinal axis of its center; and Jy is the rotational inertia of the sprung mass around the transverse axis of its center.

3 Optimization of system control parameters

In consideration of vehicle ride performance and ride comfort, the root-mean-square values of the sprung mass (fACC), dynamic load of the tire (fDTL) and dynamic travel of the suspension (fSWS) are used as the optimization objective parameters. The multiobjective optimization function f(x) is established via the linear weighted sum method. After weighting coefficients are assigned and the dimension and order of magnitude are unified, the optimization objective function is built as in Equation (4); its variables are the damping values of the front and rear suspension systems (cf and cr):

$f (x) = m i n [ω_{1} \frac{f_{A C C}}{m i n (f_{A C C})} + ω_{2} \frac{f_{S W S}}{m i n (f_{SWS})} + ω_{3} \frac{f_{D T L}}{\min (f_{D T L})}]$ (4) where ω1 is the weighting coefficient of the root-mean-square value of acceleration of the sprung mass; ω2 is the weighting coefficient of the root-mean-square value of the dynamic travel of the suspension; ω3 is the weighting coefficient of the root-mean-square value of the dynamic tire load; min(f_ACC) is the minimum root-mean-square value of the acceleration of the sprung mass within the damping variable range; min(f_SWS) is the minimum root-mean-square value of dynamic travel of the suspension within the damping variable range; and min(f_DTL) is the minimum root-mean-square value of the dynamic tire load within the damping variable range.

In the optimization design of suspension system parameters, in order to guarantee system practicability and safety based on the realization of system functions, the following constraint conditions on the system variables are proposed:

The root-mean-square value of the dynamic tire load has a direct bearing on vehicle handling stability and driving safety. When the ratio of the dynamic tire load to the finished vehicle mass (relative dynamic wheel load) is greater than 1, the adhesive force of the wheel to the ground is zero, causing a loss of driving force and braking force; therefore, the relative dynamic load between a wheel and the pavement should be controlled within a reasonable range. According to the literature, when the root-mean-square value of the relative dynamic load of a wheel is less than 1/3, the probability that the wheel will break away from the ground is less than 0.15%, so it can be reasonably assumed that the wheel will not be divorced from the ground.

When the ratio of the root-mean-square value of the dynamic travel of the suspension to the limit travel is less than 1/3, the probability that the suspension system will impact the limit block is less than 0.15%, so the impact may not occur.

An analytic hierarchy process (AHP) is a systematic method that takes a complicated multiobjective decision-making problem as a system, decomposes the objective into multiple subobjectives to form several multi-index layers, and solves the weight values through the fuzzy quantitative method of qualitative indicators for the purpose of multiobjective and multischeme optimized decision making. This method determines the weighting coefficients of evaluation indicators by comparing their importance degrees [6].

Let a_mn be the comparison value of the importance degree between index m and index n. As stipulated by this method, if m and n have the same importance degree, then a_mn = 1, and a high importance degree is indicated by the value a_mn = 9. The judgment matrix shown in Table 2 can be constructed according to the importance comparisons between the evaluation indicators [7].

Table 2

Judgment matrix

Comparison value	Index 1	Index 2	Index 3
Index 1	1	a₁₂	a₁₃
Index 2	1/a₁₂	1	a₂₃
Index 3	1/a₁₃	1/a₂₃	1

The root-mean-square values of the acceleration of the sprung mass, the dynamic travel of the suspension and the dynamic tire load are selected as the evaluation indicators of vehicle ride performance. As a primary evaluation index of vehicle ride performance, the root-mean-square value of the acceleration of the sprung mass has a higher importance degree than the other two indicators, which share the same importance. In summary, the judgment matrix of the evaluation indicators is obtained as shown in Table 3.

Table 3

Judgment matrix

Comparison value	Acceleration of sprung mass	Dynamic travel of suspension	Dynamic tire load
Acceleration of sprung mass	1	5	5
Dynamic travel of suspension	1/5	1	1
Dynamic tire load	1/5	1	1

According to the judgment matrix, the weighting coefficients of the evaluation indicators are calculated according to the following method:

Calculate the multiplication vector of the row elements in the judgment matrix: $X_{m} = \prod_{n = 1}^{3} a_{mn}$ (5)

X_m= [25,1/5,1/5]^T is obtained.

Calculate the cube root vector H_m of multiplication: $H_{m} = \sqrt[3]{X_{m}}$ (6)

H_m= [2.924,0.5848,0.5848]^T is obtained.

Calculate regular vector β of the vector H_m: $β = H_{m} / \sum_{i = 1}^{n} H_{i}$ (7)

β= [0.7142,0.1429,0.1429] is obtained. The vector β is the weighting coefficient corresponding to each evaluation index; the weighting coefficients of the root-mean-square values of the acceleration of the sprung mass, the dynamic travel of the suspension and the dynamic tire load are ω1 = 0.7142, ω2 = 0.1429 and ω3 = 0.1429, respectively.

Check the consistency of the judgment matrix:

The consistency ratio is given by: $CR = \frac{λ_{\max} - n}{RI (n - 1)} = \frac{\sum_{i = 1}^{n} \frac{(EH)_{i}}{n H_{i}} - n}{RI (n - 1)}$ (8)

According to the above equation, when n = 3, RI = 0.58 can be obtained from the related table, so CR =&thinsp–2.05<0.1, and the consistency check is passed.

Based on the seven-degrees-of-freedom variable-damping semi-active suspension system Simulink simulation model built in section 1, Equation (4) is taken as the optimization objective function. The genetic algorithm optimization program is compiled in MATLAB, and its control parameters are selected as follows: population size 50, mutation probability 0.065 and crossover probability 0.73. The algorithm termination conditions are: evolution algebra 500, value range of front suspension stiffness (k_f) 10000 N/m–50000 N/m; value range of rear suspension stiffness (k_r) 20000 N/m100000 N/m; value of range of front suspension damping (c_f) 2000 N·s/m20000 N·s/m; and value range of rear suspension damping (c_r) 5000 N·s/m30000 N·s/m. Under typical working conditions (i.e., pavement conditions are selected as Class A, B or C pavements; vehicle speed is 90–120 km/h for Class A pavement, 60–90 km/h for Class B pavement and 40–70 km/h for Class C pavement, with an interval of 10 km/h; and load condition is divided into no load and full load), optimal stiffness values (k_f and k_r) of the front and rear variable-damping semi-active suspensions are sought [8].

Tables 4 and 5 show the optimization results for the damping values (k_f, k_r, c_f and c_r) of the front and rear variable-damping semi-active suspensions of Class A pavement under full-load and no-load conditions, respectively.

Table 4

Optimization results of Class a pavement under full-load conditions

v(km/h)	90	100	110	120
k_f(N/m)	29176	29200	29213	29410
k_r(N/m)	55590	56223	56120	56854
c_f (N·s/m)	8767	8579	8565	8417
c_r (N·s/m)	8333	8244	8085	7996

Table 5

Optimization results of Class A pavement under no-load conditions

v(km/h)	90	100	110	120
k_f(N/m)	20553	20714	20850	21012
k_r(N/m)	40538	41120	42190	42638
c_f (N ⋯/m)	6578	6578	6578	6578
c_f (N ⋯/m)	6352	6352	6352	6352

Tables 6 and 7 show the optimization results for the damping values (c_f and c_r) of variable-damping semi-active suspensions on Class B pavement under full-load and no-load conditions, respectively.

Table 6

Optimization results of Class B pavement under full-load conditions

v(km/h)	60	70	80	90
k_f(N/m)	29208	29282	29301	29323
k_r(N/m)	57590	58635	58966	58590
c_f (N ⋯/m)	8874	8874	8874	8874
c_f (N ⋯/m)	8659	8659	8659	8659

Table 7

Optimization results of Class B pavement under no-load conditions

v(km/h)	60	70	80	90
k_f(N/m)	15953	16180	16283	16465
k_r(N/m)	30538	32120	33352	33538
c_f (N ⋯/m)	6574	6574	6574	6574
c_f (N ⋯/m)	5381	5381	5381	5381

Tables 8 and 9 show the optimization results for the damping values (c_f and c_r) of variable-damping semi-active suspensions on Class C pavement under full-load and no-load conditions, respectively.

Table 8

Optimization results of Class C pavement under full-load conditions

v(km/h)	40	50	60	70
k_f(N/m)	36240	37266	37298	37328
k_r(N/m)	66451	68635	69966	69966
c_f (N ⋯/m)	7335	7335	7335	7335
c_f (N ⋯/m)	7124	7124	7124	7124

Table 9

Optimization results of Class C pavement under no-load conditions

v(km/h)	40	50	60	70
k_f(N/m)	20043	20233	20383	20450
k_r(N/m)	36590	39679	39352	39352
c_f (N ⋯/m)	5730	5730	5730	5730
c_f (N ⋯/m)	5715	5715	5715	5715

According to the optimization results from Table 4 to 9, as pavement conditions worsen, vehicle speed is accelerated and load capacity is reduced, and stiffness and damping are increased; when pavement conditions and vehicle speed remain unchanged, the maximum and minimum variable quantities of damping are 80.9% and 37.6%, respectively; when load capacity and vehicle speed are unchanged, the maximum and minimum variable quantities of damping are 60.0% and 11.0%, respectively; and when load capacity and pavement conditions remain the same, the maximum and minimum variable quantities of damping are 14.0% and 3.6%, respectively.

4 System control based on the dynamic monitoring feedback iteration algorithm

Dynamic monitoring feedback iteration control is proposed based on traditional iteration control and applied to the control of the variable-damping semi-active suspension system. This control method can realize real-time monitoring of the vibration signal of the sprung mass, input the signal into the iterative equation of the controller and calculate the damping of the shock absorber. The actuator will switch the damping of the shock absorber to the calculated value; the vibration condition of the sprung mass is continuously monitored, and the above steps are repeated until the calculated damping stabilizes (convergence). This control method can provide a new idea for the damping control of the shock absorber by virtue of discrete reaching and gradual improvement of the control effect. Its advantage over other control methods is that the vehicle corrects the damping value of the shock absorber to the target value according to different dynamic driving conditions in order to improve the performance of the shock absorber and improve vehicle ride performance.

4.1 Determination of the iterative equation

Taking the front suspension of the vehicle as an example, the initial iterative equation is based on the finished vehicle model and Newton’s second law, and the following equation holds at the front suspension of the vehicle: $\frac{c_{f}}{m_{f}} h_{f}^{.} + \frac{k_{f}}{m_{f}} h_{f} = h_{f}^{. .}$ (9) where c_f is the damping value of the front; k_f is the stiffness value of the front suspensions; m_f is the sprung mass components corresponding to the front suspensions; h_f, h_f and h_f are the vertical displacement, speed and acceleration of the sprung mass at the front suspension.

Within unit time, the root-mean-square values of the two sides of Equation (9) are simultaneously solved, and correction coefficients are added to obtain the relations of stiffness and damping with the root-mean-square value of the vertical acceleration of the sprung mass within unit time: $λ_{1} \frac{c_{f}}{m_{f}} RMS ({\dot{h}}_{f}) + λ_{2} \frac{k_{f}}{m_{f}} RMS (h_{f}) = RMS (h_{f}^{. .})$ (10) where λ₁ and λ₂ are correction coefficients.

In order to verify the convergence and divergence of Equation (10) at a vehicle speed of 60 km/h on Class B pavement under full-load conditions, the stiffness of the front suspension is set as 29280 N/m (the no-load damping is 14640 N/m, and that of the rear suspension is twice that of the front one); the root-mean-square value of the vibration signal (where the vertical displacement, speed and acceleration of the sprung mass are denoted h, $\dot{h}$ and $\overset{. .}{h}$ , respectively) of the sprung mass monitored for 10 s is substituted into iterative Equation (10) (λ₁ and λ₂ are first set equal to 1 in Equation (10) and subsequently corrected based on the result); the new damping is calculated; and after the damping of the shock absorber is adjusted, the new vibration signal of the sprung mass is monitored until the damping value is less than 50 N·s/m (or the number of iterations is less than 10). According to the above conditions, the iteration results under different damping values are shown in Tables 10 and 11. Equation (10) is strictly convergent and is the polynomial of each evaluation index and that the final convergence results are basically identical under different initial damping values (8000 N·s/m and 14000 N·s/m), with a difference rate of 0.007%. Similarly, the iteration results of different pavement classes, loads and vehicle speeds indicate that Equation (10) is convergent.

Table 10

Iteration results under initial damping of 8000 N·s/m

Damping / (N·s/m)	Root-mean-square value of vertical displacement of sprung mass /(m)	Root-mean-square value of speed of sprung mass /(m/s)	Root-mean-square value of acceleration of sprung mass /(m/s²)
8000	0.0084	0.0403	0.5103
6329	0.0086	0.0406	0.5113
5800	0.0087	0.0407	0.5145
5563	0.0090	0.0408	0.5148
5563	0.0090	0.0408	0.5148

Table 11

Iteration results under initial damping of 14000 N·s/m

Damping / (N·s/m)	Root-mean-square value of vertical displacement of sprung mass /(m)	Root-mean-square value of speed of sprung mass /(m/s)	Root-mean-square value of acceleration of sprung mass /(m/s²)
14000	0.0108	0.0798	0.8009
10046	0.0096	0.0481	0.5434
8015	0.0091	0.0431	0.5227
5563	0.0090	0.0408	0.5147
5563	0.0090	0.0408	0.5147

However, the above iteration results are greatly different from the parameter optimization results in section 2, so it is necessary to correct the coefficients of the initial iterative equation so that the difference value between the iteration result and the optimization result is smaller than 1%. In order to determine the effects of the changes in stiffness and damping coefficients on the final iteration result, the damping change after the iteration is compared through a simulation analysis, and it is shown that when the stiffness coefficient is increased, the iteration result decreases, and when the damping coefficient is increased, the iteration result also increases.

According to the above qualitative analysis, as the uncorrected iteration result (5563 N·s/m) is greater than the optimization result (8874 N·s/m), the correction coefficient λ₂ of the damping term is modified while the correction coefficient of the stiffness term is unchanged so that the iteration result will be approximately equal to the optimization result. The least square fitting method is used to obtain the relation between different correction coefficients and iteration results on Class A pavement under full load at a vehicle speed of 100 km/h and on Class C pavement under no load at a vehicle speed of 60 km/h, as shown in Figs. 1 and 2.

Fig. 1

Correction coefficient and iterative fitting curve for class A pavement under full load at a speed of 100 km/h.

Fig. 2

Correction coefficient and iterative fitting curve for class C pavement under full load at a speed of 60 km/h.

According to the curves shown in these figures, the optimization results for the corresponding working conditions are compared to obtain the final iterative Equation (11): $c_{f} = \frac{m_{f} RMS (h_{f}^{. .}) - 1.42 k_{f} RMS (h_{f})}{RMS (\dot{h_{f}})}$ (11)

Under typical working conditions on Class A and C pavements, the iteration results of iterative Equation (11) are shown in Tables 10 and 11.

Based on a comparison of the optimization results in Tables 12 and 13 with those in Table 4 and 9, the errors are 0.16% and the errors of the iteration results within other speed intervals and under other pavement classes are within 2%, so Equation (11) can be determined as the iterative equation.

Table 12

Iteration results for class A pavement under full load at a speed of 100 km/h

Damping / (N·s/m)	Root-mean-square value of vertical displacement of sprung mass /(m)	Root-mean-square value of speed of sprung mass /(m/s)	Root-mean-square value of acceleration of sprung mass /(m/s²)
8000	0.0121	0.0326	0.5022
7347	0.0118	0.0320	0.5021
6851	0.0117	0.0318	0.5021
6563	0.0113	0.0317	0.5021
6563	0.0113	0.0317	0.5021

Table 13

Iteration results for class C pavement under no load at a speed of 60 km/h

Damping / (N·s/m)	Root-mean-square value of vertical displacement of sprung mass /(m)	Root-mean-square value of speed of sprung mass /(m/s)	Root-mean-square value of acceleration of sprung mass /(m/s²)
8000	0.0115	0.0466	0.2710
7418	0.0113	0.0408	0.3760
6132	0.0111	0.0388	0.4535
5885	0.0110	0.0375	0.5788
5725	0.0110	0.0356	0.5816
5725	0.0110	0.0396	0.5816

4.2 Control flow

The dynamic monitoring feedback iteration control flow of the variable-damping semi-active suspension system is shown in Fig. 3.

Fig. 3

Dynamic monitoring iteration control flow of the variable-damping semi-active suspension system.

Table 14

Performance comparison between passive (traditional) suspension and semi-active suspension

Performance index	Root-mean-square value of acceleration of sprung mass /(m/s²)	Root-mean-square value of dynamic travel of suspension /mm	Root-mean-square value of dynamic tire load /N
Traditional	0.9105	6.3	2273
Semi-active	0.8144	5.5	2266
Performance improvement	10.56%	11.98%	–0.31%

Based on the dynamic damping monitoring feedback iteration and the vibration condition of the variable-damping semi-active suspension system in the finished vehicle model, short-time tracking calculation of the damping values of the front and rear suspensions is performed, and the suspension damping is adjusted via the actuator; the vibration conditions of the sprung mass at each suspension under new suspension parameters are continuously tracked, followed by control, calculation and layer-by-layer iteration. When the vibration amplitude of the sprung mass caused by the control at a given time is smaller than a threshold value (here, the selected threshold value is that the difference between the damping value of the shock absorber and the last iteration result is less than 50 N·s/m), the control ends.

In order to verify the effectiveness of the dynamic monitoring feedback iteration control scheme, the performance changes of the traditional suspension and controlled semi-active suspension are compared under mixed working conditions: Class A pavement under full load at a speed of 80 km/h, Class B pavement under full load at a speed of 60 km/h, and Class C pavement under full load at a speed of 50 km/h. Figures 4 and 5 show the comparison charts between the traditional suspension and the controlled semi-active suspension in terms of the acceleration at the center of the sprung mass and the dynamic travel of the suspension, and Table 14 lists the performance comparison results for the traditional suspension and the controlled semi-active suspension.

Fig. 4

Comparison of acceleration of the vehicle body.

Fig. 5

Comparison of suspension displacement.

As shown in Figs. 4 and 5, after the dynamic monitoring feedback iteration tracking and control are implemented for the variable-damping semi-active suspension system, the root-mean-square value of the vertical acceleration of the vehicle body is reduced by 10.56%, and that of the dynamic travel of the suspension is reduced by 11.98%, so both are improved to a certain degree, and the shock resistance of the suspension is substantially improved. The root-mean-square value of the dynamic tire load is increased by a small amount but still falls within the allowed range, so it has no obvious effect on vehicle performance and satisfies the internationally stipulated limit value requirement.

5 Conclusions

An iterative equation used to control the variable-damping semi-active suspension system has been proposed in this paper. The least square method was used to fit correction coefficients and iterative damping curves, the correction coefficients of the iterative equation were determined, and the difference between the corrected iteration result and the optimization result was found to be less than 1%. On this basis, a dynamic monitoring iteration control algorithm was established.

The control effect was verified through a simulation analysis under mixed full-load working conditions (Class A pavement and vehicle speed of 80 km/h, Class B pavement and vehicle speed of 60 km/h, and Class C pavement and vehicle speed of 50 km/h). Based on this comparison, the root-mean-square value of the vertical acceleration of the sprung mass was reduced by 11.98%, and that of dynamic travel of the suspension was reduced by 11.98%, indicating that both vehicle ride performance and ride comfort are improved.

In this paper, the Dynamic Monitoring Feedback Iteration Control is proposed for the first time for the purpose of improving vehicle ride comfort, and it is applied to the damping control of semi-active suspension. The research shows that this control method can effectively reduce the acceleration and vibration of the vehicle body, which lays a foundation for its application in engineering practice.

Footnotes

Acknowledgments

This work was supported by The Natural Science Foundation of the Jiangsu Higher Education Institutions of China (20KJD460007).

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