Improving Vehicle Dynamics: A Fractional-Order PIλDμ Control Approach to Active Suspension Systems
Abstract
:1. Introduction
2. Modeling
2.1. Active Suspension
- (1)
- The differential equation for the vertical motion of the unsprung mass in each suspension is shown below:
- (2)
- The vertical motion of the vehicle body is related to the change in contact between the wheels and the ground as the vehicle passes over an uneven road surface. The pitching motion of the vehicle body is most pronounced when the vehicle is accelerating and braking. The control of the pitching motion is critical to improving the dynamic stability and comfort of the vehicle. Roll motion occurs mainly when the vehicle is turning, as the vehicle’s center of gravity shifts to the outside, creating roll. Controlling the roll motion is important to maintain vehicle stability and improve cornering performance. The differential equations for the vertical, pitch, and roll motions of a vehicle are as follows [78,79]:
- (3)
- (4)
- The vertical, rolling, and pitching movements of the engine are related to the motion of the pistons. The rapid up-and-down movement of the pistons in the cylinders generates vibrations, which are transmitted to the vehicle body through the engine mounts, potentially causing vertical movement, rolling, and pitching of the engine. The differential equations for the vertical, rolling, and pitching movements of the engine are as follows [89,90]:
2.2. Road Excitation
2.2.1. Single-Wheel Road Input Analysis
2.2.2. Temporal Displacement of Axle Trajectories During Vehicle Motion
2.2.3. Coherence of Left and Right Wheel Tracks
2.2.4. Four-Wheels Road Excitation Model
3. Methods
3.1. Fractional-Order PIλDμ
3.1.1. Theoretical Framework of Fractional-Order Calculus
3.1.2. Architecture of PIλDμ Controller System
3.2. Optimization Algorithm
3.2.1. Gray Wolf Optimizer
- (1)
- Initialize the population:
- (2)
- Identification of α-wolf, β-wolf, and δ-wolf
- (3)
- Predation process
3.2.2. Penalty Optimization Process
4. Results and Analysis
4.1. Logical Architecture
4.2. Parameter Optimization Process
4.3. Dynamic Processes on C-Level Road Surfaces
4.3.1. Controller Input
4.3.2. Time-Domain Indicators of Output Response
4.3.3. Frequency Domain Comparison of Output Response
4.3.4. RMS Values on Five Different Road Surfaces
5. Comparison and Discussion
5.1. Previous Literature
5.2. Enhancing the Smoothness of Suspension Through the Utilization of Other Technologies
6. Conclusions
Supplementary Materials
Author Contributions
Funding
Data Availability Statement
Acknowledgments
Conflicts of Interest
References
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Parameter | Value | Parameter | Value |
---|---|---|---|
mu1 | 50 kg | ku1 | 217,751 N/m |
mu2 | 50 kg | ku2 | 217,751 N/m |
mu3 | 68 kg | ku3 | 217,751 N/m |
mu4 | 68 kg | ku4 | 217,751 N/m |
l1 | 1.55 m | lf | 0.30 m |
l2 | 1.45 m | lr | 0.15 m |
l3 | 0.75 m | l0 | 0.58 m |
l4 | 0.75 m | kαf | 29.4 N·m/° |
εu | 0.1 | kαr | 12.8 N·m/° |
Parameter | Value | Parameter | Value |
---|---|---|---|
Cfb | 1290 N·s/m | βfb | 0.3 |
ηkb | 1.4 | vkb | 0.3 m/s |
ηbb | 1.6 | vbb | −0.3 m/s |
Parameter | Value | Parameter | Value |
---|---|---|---|
mb | 1350 kg | l11 | 0.05 m |
Jβ | 5368 kg∙m2 | l12 | 0.15 m |
Jγ | 529 kg∙m2 | l13 | 0.27 m |
l5 | 0.15 m | l14 | 0.10 m |
l6 | 0.53 m | kt1 | 24,606 N/m |
l7 | 0.35 m | kt2 | 24,606 N/m |
l8 | 0.35 m | kt3 | 26,115 N/m |
l9 | 0.50 m | kt4 | 26,115 N/m |
l10 | 0.50 m | εt | 0.12 |
ks1 | 22,000 N/m | ks2 | 22,000 N/m |
ks3 | 17,000 N/m | ks4 | 17,000 N/m |
εs | 0.11 | km3 | 250,000 N/m |
km1 | 250,000 N/m | εm | 0.1 |
km2 | 250,000 N/m |
Parameter | Value | Parameter | Value |
---|---|---|---|
kd,2 | −68.2 N/m | fy,2 | −51.7 N |
kd,1 | −1156.5 N/m | fy,2 | 439.7 N |
kd,0 | 10,236.7 N/m | fy,2 | 134.8 N |
cpo,2 | 162.1 N·s/m | λ2,2 | −11.6907 s/m |
cpo,1 | 711.0 N·s/m | λ2,1 | 11.1987 s/m |
cpo,0 | 871.9 N·s/m | λ2,0 | 137.7681 s/m |
mf,2 | −0.207,1 kg | f0 | 80.0 N |
mf,1 | 0.4785 kg | λ1 | 0.000016 s/m |
mf,0 | 0.6638 kg |
Parameter | Value | Parameter | Value |
---|---|---|---|
ms1 | 80 kg | ms2 | 80 kg |
ms3 | 85 kg | ms4 | 85 kg |
Parameter | Value | Parameter | Value |
---|---|---|---|
me | 230 kg | θσ | 6.54 kg∙m2 |
θω | 11.84 kg∙m2 | h1 | 0.48 m |
h2 | 0.45 m | h3 | 0.15 m |
h4 | 0.06 m | h5 | 0.04 m |
h6 | 0.35 m |
Parameter | Value | Parameter | Value |
---|---|---|---|
mc | 0.82 kg | L | 0.015 m |
λ | 0.33 | r | 0.06 m |
Category | Gq(n0)/(10−6 m3) (n0 = 0.1 m−1) | σq/(10−3 m) |
---|---|---|
A | 16 | 3.81 |
B | 64 | 7.61 |
C | 256 | 15.23 |
D | 1024 | 30.45 |
E | 4096 | 60.90 |
F | 16384 | 121.89 |
G | 65536 | 243.61 |
H | 262144 | 487.22 |
Engine Mountings | Seats | Actuators | ||||||
---|---|---|---|---|---|---|---|---|
PID | I | F | PID | I | F | PID | I | F |
kp1 | 0.0001 | 0.0198 | kp2 | 0.5066 | 1.3749 | kp3 | 3269.5862 | 238.4866 |
ki1 | 1.1224 | 0.1334 | k2 | 2.4706 | 0.0123 | ki3 | 3.1035 | 4613.0458 |
kd1 | 0.0020 | 0.0003 | kd2 | 0.0047 | 0.4002 | kd3 | 0.0976 | 2.8533 |
λ1 | / | 0.9837 | λ2 | / | 0.2986 | λ3 | / | 0.0458 |
μ1 | / | 0.0126 | μ2 | / | 0.6701 | μ3 | / | 0.0628 |
PID | ut1 | ut2 | ut3 | ut4 | ||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|
I | RMS | 0.0059 | 0.0035 | 0.0036 | 0.0151 | 0.0105 | 0.0146 | 0.0105 | 133.9700 | 144.8095 | 128.7949 | 146.7290 |
AVE | 0.0033 | 0.0018 | 0.0018 | 0.0085 | 0.0057 | 0.0057 | 0.0055 | 9.8852 | 9.8852 | 7.4277 | 7.42767 | |
F | RMS | 0.0009 | 0.0005 | 0.0005 | 0.0482 | 0.0484 | 0.0488 | 0.0498 | 163.2996 | 180.5513 | 156.2748 | 185.4619 |
AVE | 0.0005 | 0.0002 | 0.0003 | 0.0235 | 0.0259 | 0.0242 | 0.0262 | 11.76018 | −5.0096 | 8.3197 | −8.4500 |
Road Types | − lg(RMS(Yi)) | |||||||||
RMS(Y1) | RMS(Y2) | RMS(Y3) | RMS(Y4) | RMS(Y5) | RMS(Y6) | RMS(Y7) | RMS(Y8) | RMS(Y9) | ||
A | P | 2.6965 | 2.9008 | 2.6405 | 0.4514 | 0.4687 | 0.4656 | 0.4693 | 0.4553 | 0.8725 |
I | 2.6973 | 2.9026 | 2.6408 | 0.6403 | 0.7358 | 0.6895 | 0.7680 | 0.5948 | 0.9109 | |
F | 2.6964 | 2.9017 | 2.6405 | 0.6546 | 0.7559 | 0.7074 | 0.7921 | 0.6030 | 0.9112 | |
B | P | 2.6938 | 2.8937 | 2.6398 | 0.1866 | 0.1751 | 0.1805 | 0.1624 | 0.2271 | 0.6986 |
I | 2.6964 | 2.9004 | 2.6401 | 0.4560 | 0.5041 | 0.4680 | 0.5100 | 0.4784 | 0.7461 | |
F | 2.6963 | 2.9003 | 2.6402 | 0.4805 | 0.5334 | 0.4944 | 0.5296 | 0.4995 | 0.7462 | |
C | P | 2.6821 | 2.8665 | 2.6370 | −0.1211 | −0.1453 | −0.1383 | −0.1667 | −0.5545 | 0.4454 |
I | 2.6929 | 2.8913 | 2.6380 | 0.2043 | 0.2303 | 0.2006 | 0.2155 | 0.2694 | 0.5074 | |
F | 2.6927 | 2.8912 | 2.6378 | 0.2360 | 0.2655 | 0.2314 | 0.2485 | 0.3037 | 0.5076 | |
D | P | 2.6440 | 2.7822 | 2.6268 | −0.4488 | −0.4741 | −0.4714 | −0.5002 | −0.3563 | 0.1561 |
I | 2.6797 | 2.8601 | 2.6305 | −0.0718 | −0.0536 | −0.0753 | −0.0647 | −0.0031 | 0.2295 | |
F | 2.6807 | 2.8629 | 2.6306 | −0.0347 | −0.0169 | −0.0422 | −0.0301 | 0.0375 | 0.2296 | |
E | P | 2.5292 | 2.5894 | 2.5783 | −0.7719 | −0.7964 | −0.7972 | −0.8251 | −0.6648 | −0.1490 |
I | 2.6317 | 2.7563 | 2.5955 | −0.3597 | −0.3432 | −0.3593 | −0.3487 | −0.2995 | −0.0677 | |
F | 2.6351 | 2.7607 | 2.5911 | −0.3194 | −0.2998 | −0.3156 | −0.3049 | −0.2579 | −0.0681 | |
Road Types | − lg(RMS(Yi)) | |||||||||
RMS(Y10) | RMS(Y11) | RMS(Y12) | RMS(Y13) | RMS(Y14) | RMS(Y15) | RMS(Y16) | RMS(Y17) | RMS(Y18) | ||
A | P | 0.2128 | 2.1047 | 2.1373 | 2.2162 | 2.1976 | 2.6890 | 2.6950 | 2.6949 | 2.6921 |
I | 0.3398 | 2.2291 | 2.3219 | 2.3967 | 2.4293 | 2.7150 | 2.7255 | 2.7067 | 2.7099 | |
F | 0.3446 | 2.3335 | 2.3239 | 2.3898 | 2.4205 | 2.7176 | 2.7283 | 2.7058 | 2.7092 | |
B | P | 0.0237 | 1.8416 | 1.8458 | 1.9006 | 1.8787 | 2.3777 | 2.3819 | 2.3786 | 2.3787 |
I | 0.2000 | 1.9828 | 2.0330 | 2.0828 | 2.1181 | 2.4053 | 2.4136 | 2.3908 | 2.3981 | |
F | 0.2088 | 1.9861 | 2.0336 | 2.0766 | 2.1103 | 2.4090 | 2.4172 | 2.3902 | 2.3976 | |
C | P | −0.3141 | 1.5464 | −0.3141 | 1.5465 | 1.5371 | 2.0679 | 2.0622 | 2.0574 | 2.0588 |
I | −0.0290 | 1.7034 | 1.7337 | 1.7720 | 1.7970 | 2.0863 | 2.0945 | 2.0698 | 2.0788 | |
F | −0.0093 | 1.7068 | 1.7344 | 1.7666 | 1.7905 | 2.0907 | 2.0985 | 2.0696 | 2.0786 | |
D | P | −0.6168 | 1.2431 | 1.2280 | 1.2689 | 1.2396 | 1.7436 | 1.7480 | 1.7423 | 1.7444 |
I | −0.3048 | 1.4111 | 1.4326 | 1.4657 | 1.4838 | 1.7724 | 1.7808 | 1.7549 | 1.7646 | |
F | −0.2816 | 1.4146 | 1.4336 | 1.4606 | 1.4781 | 1.7766 | 1.7849 | 1.7549 | 1.7647 | |
E | P | −0.9231 | 0.9382 | 0.9210 | 0.9566 | 0.9264 | 1.4349 | 1.4394 | 1.4331 | 1.4353 |
I | −0.3018 | −0.1141 | 1.1316 | 1.1629 | 1.1765 | 1.4643 | 1.4727 | 1.4460 | 1.4559 | |
F | −0.5772 | 1.1176 | 1.1328 | 1.1573 | 1.1712 | 1.4687 | 1.4768. | 1.4461 | 1.4561 |
Optimization | RMS(Y1) | RMS(Y2) | RMS(Y3) | RMS(Y4) | RMS(Y5) | RMS(Y6) | RMS(Y7) | RMS(Y8) | RMS(Y9) | |
Ratio (%) | I/P | 0.3 | 0.65 | 1.5 | 35.0 | 46.0 | 40.0 | 50.0 | 27.0 | 8.0 |
F/P | 0.2 | 0.6 | 0.5 | 36.0 | 48.0 | 42.0 | 52.0 | 30.0 | 9.0 | |
F/I | −0.03 | 0.1 | −0.03 | 3 | 4.0 | 4.0 | 5.0 | 1.0 | 1.0 | |
Optimization | RMS(Y10) | RMS(Y11) | RMS(Y12) | RMS(Y13) | RMS(Y14) | RMS(Y15) | RMS(Y16) | RMS(Y17) | RMS(Y18) | |
Ratio (%) | I/P | 25.0 | 25.0 | 34.0 | 33.2 | 40.0 | 5.0 | 5.0 | 32.0 | 4.5 |
F/P | 75.0 | 26.0 | 35.0 | 33.6 | 41.0 | 6.0 | 6.0 | 30.0 | 4.4 | |
F/I | 67.0 | 2.0 | 2.0 | 1.3 | 0.9 | 1.0 | 1.0 | −0.3 | −0.3 |
Studies | Control Algorithm | Optimization Amplitude (%) | ||||||
---|---|---|---|---|---|---|---|---|
RMS(Y1–3) | RMS(Y4–7) | RMS(Y8) | RMS(Y9) | RMS(Y10) | RMS(Y11–14) | RMS(Y15–18) | ||
Present | PIλDμ | 0.3% | 46% | 30% | 9% | 75% | 34% | 2.5% |
Dridi et al. [123] | LSTM | / | / | 27.9% | / | / | / | / |
Nagarkar et al. [124] | FLC | / | / | 46.0% | / | / | 3.6% | 18.7% |
Mrazgua et al. [125] FLC | T-S fuzzy | / | / | 57.16% | / | / | / | / |
Yin et al. [126] | fuzzy PID | / | 21.17% | 22.00% | 21.37% | 24.17% | 15% | 10% |
Lee et al. [127] | CDC | / | / | 14.53% | / | / | / | / |
Shen et al. [128] | sky-hook | / | / | 12.8% | / | / | 37.3% | 8.9% |
Nagarkar et al. [129] | NSGA-II algorithm | / | / | 47% | / | / | 9.2% | 35.8% |
Wei et al. [130] | fuzzy PID | / | / | 59.08% | 3.06% | 3.54% | 11.98% | 2.09% |
Shen et al. [131] | tructure-immittance approach | / | / | 18% | 15% | / | / | / |
Anandan et al. [132] | PID | / | 21% | 35% | 33% | / | 18% | / |
Theunissen et al. [133] | e-MPC | / | / | 10% | 8–21% | 8–21% | / | / |
Yang et al. [134] | ground-hook | / | / | 4.87% | / | / | / | 16.19% |
Zhang et al. [135] | bridge network | / | / | 1.8% | / | / | 21.1% | 6.3% |
Zeng et al. [136] | Neuron PI | / | / | 37.2% | 45.2% | 38.6% | / | / |
Jiang et al. [137] | BP-PID | / | / | 27.58% | / | / | 4.48% | 4.17% |
Zhou et al. [138] | MPC | / | / | 22.38% | / | / | / | / |
Wang et al. [139] | pigeon-inspired optimization | / | / | 23.1 | / | / | 6.6% | / |
Fossati et al. [140] | NSGA-II | / | 21.14% | / | / | / | / | / |
Liu et al. [141] | SH-GH | / | / | 27.45% | / | / | / | 8.53% |
Xu et al. [142] | fuzzy | / | / | 14.6% | 9.6% | 5.3% | / | / |
Xu et al. [143] | multi-objective optimization | / | / | 43.88% | / | / | 24.38% | 46.46% |
Esmaeili et al. [144] | ANFIS | / | / | 62% | / | / | 5.83% | 56.8% |
Yang et al. [145] | hybrid-hook damping | 26.61% | -6.94% | 22.03% | ||||
Ho et al. [146] | SMC-NDOB | / | 41.5% | / | / | / | / | / |
Ning et al. [147] | T-S fuzzy | / | 45.5% | / | / | / | / | / |
Li et al. [148] | model reference adaptive | / | / | 8.70% | / | / | 28.26% | 18.21% |
Xu et al. [149] | LQG | / | / | 27.5% | / | / | 6.3% | 17.6% |
Liu et al. [150] | MPC-H∞ | / | / | 19.4% | / | / | / | 9.3% |
Wang et al. [151] | DSHIS | / | / | 48.17% | 56% | / | 17.39% | 4.35% |
Ghorbany et al. [152] | MOPSO | / | / | 71% | 57% | 33% | / | / |
Zhang et al. [153] | NESI | / | 23.59% | 23.97% | 27.48% | / | / | / |
Alfadhli et al. [154] | feedforward and feedback | / | 25% | / | / | / | / | / |
Wu et al. [155] | LQR | / | / | 58.25% | 55.41% | 31.39% | / | / |
Zhu et al. [156] | VUFC | / | / | 21.0% | / | / | / | / |
Cao et al. [157] | SOA-PID | / | / | 27.6% | 27.0% | 35.8% | / | / |
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Yin, Z.; Cui, C.; Wang, R.; Su, R.; Ma, X. Improving Vehicle Dynamics: A Fractional-Order PIλDμ Control Approach to Active Suspension Systems. Machines 2025, 13, 271. https://doi.org/10.3390/machines13040271
Yin Z, Cui C, Wang R, Su R, Ma X. Improving Vehicle Dynamics: A Fractional-Order PIλDμ Control Approach to Active Suspension Systems. Machines. 2025; 13(4):271. https://doi.org/10.3390/machines13040271
Chicago/Turabian StyleYin, Zongjun, Chenyang Cui, Ru Wang, Rong Su, and Xuegang Ma. 2025. "Improving Vehicle Dynamics: A Fractional-Order PIλDμ Control Approach to Active Suspension Systems" Machines 13, no. 4: 271. https://doi.org/10.3390/machines13040271
APA StyleYin, Z., Cui, C., Wang, R., Su, R., & Ma, X. (2025). Improving Vehicle Dynamics: A Fractional-Order PIλDμ Control Approach to Active Suspension Systems. Machines, 13(4), 271. https://doi.org/10.3390/machines13040271