# RBF-Based Fractional-Order SMC Fault-Tolerant Controller for a Nonlinear Active Suspension

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Problem Formulation

#### 2.1. Road Disturbance

$q\left(t\right)$ | road profile; |

$w\left(t\right)$ | random white noise signal; |

${f}_{0}=0.01$ | low cut-off frequency; |

${G}_{0}=64\times {10}^{-6}$ | road roughness; |

v = 60 km/h | vehicle speed. |

#### 2.2. Nonlinear Active Suspension System Model

${m}_{s}$ | car body mass; |

${m}_{1}\&{m}_{2}$ | unsprung mass of the front and rear; |

I | pitch moment of inertia; |

$\phi $ | pitch angle; |

${x}_{s1}\&{x}_{s2}$ | front and rear body displacement; |

${x}_{u1}\&{x}_{u2}$ | front and rear unsprung mass displacement; |

${x}_{g1}\&{x}_{g2}$ | front and rear road inputs; |

$a\&b$ | distance of suspension from the center mass of the vehicle body; |

${F}_{s1}\&{F}_{s2}$ | front and rear nonlinear spring force; |

${F}_{d1}\&{F}_{d2}$ | front and rear nonlinear damper force; |

${u}_{1}\&{u}_{2}$ | front and rear active force; |

${F}_{t1}\&{F}_{t2}$ | front and rear elastic force of the tire. |

${k}_{s}\&{k}_{sn}$ | linear and nonlinear spring stiffness; |

${c}_{s}\&{c}_{sn}$ | linear and nonlinear damping coefficients; |

${k}_{t}\&{c}_{t}$ | spring stiffness coefficient and damping coefficient of the tire. |

- Wheels have ideal terrain signals;
- The pitch angle $\phi $ is small.

#### 2.3. Nonlinear Active Suspension System Fault Model

${u}_{f}$ | active force produced by the fault actuator; |

j | the jth actuator; |

$\delta $ | the effectiveness factor; |

$\delta =0$ | the actuator cannot work anymore; |

$\delta =1$ | the actuator can work normally; |

$0<\delta <1$ | partial loss in the actuator. |

## 3. Fault-Tolerant Controller Design

#### 3.1. Fractional-Order Sliding-Mode Control Method

#### 3.2. RBF-Based Fractional-Order SMC Fault-Tolerant Controller

- Optimizing the equivalent control of SMC;
- Optimizing the toggle switch of SMC.

#### 3.3. Stability of the Control System

## 4. Simulation Results and Analysis

#### 4.1. The Case of No Actuator Fault

#### 4.2. Case of Actuator Fault

## 5. Discussion and Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

- Jin, P.; Xue, W.; Li, K. Actuator Fault Estimation for Vehicle Active Suspensions Based on Adaptive Observer and Genetic Algorithm. Shock Vib.
**2019**, 3, 1783850. [Google Scholar] [CrossRef] - Du, H.; Yim, S.K.; Lam, J. Semi-active H∞ control of vehicle suspension with magneto-rheological dampers. J. Sound Vib.
**2005**, 283, 981–996. [Google Scholar] [CrossRef] - Shao, X.; Naghdy, F.; Du, H. Reliable fuzzy H-infinity control for active suspension of in-wheel motor driven electric vehicles with dynamic damping. Mech. Syst. Signal Process.
**2017**, 87, 365–383. [Google Scholar] [CrossRef] - Chen, L.; Wang, S.; Sun, X.; Yuan, C.; Cai, Y. Model predictive control of an air suspension system with a damping multi-mode switching damper based on a hybrid model. Mech. Syst. Signal Process.
**2017**, 94, 94–110. [Google Scholar] - Chen, H.; Guo, K.H. Constrained H
_{∞}control of active suspensions: An LMI approach. IEEE Trans. Control. Syst. Technol.**2005**, 13, 412–421. [Google Scholar] [CrossRef] - Zhang, L.P.; Gong, D.L. Passive fault-tolerant control for vehicle active suspension system based on H
_{2}/H_{∞}approach. J. Vibroeng.**2018**, 20, 1828–1849. [Google Scholar] [CrossRef] - Hu, Y.; Hou, Z. Multiplexed model predictive control for active vehicle suspensions. Int. J. Control
**2014**, 88, 347–363. [Google Scholar] [CrossRef] - Liang, J.; Lu, Y.; Pi, D. A decentralized cooperative control framework for active steering and active suspension: Multi-agent approach. IEEE Trans. Transp. Electrif.
**2022**, 8, 1414–1429. [Google Scholar] [CrossRef] - Wang, Z.; Unbehauen, H. Robust Hinfinity Control for Systems with Time-Varying Parameter Uncertainty and Variance Constraints. Cybern. Syst.
**2000**, 31, 175–191. [Google Scholar] - Kim, C.; Ro, P.I. A sliding mode controller for vehicle active suspension systems with non-linearities. Proc. Inst. Mech. Eng. Part D J. Automob. Eng.
**2016**, 212, 79–92. [Google Scholar] [CrossRef] - Yang, G.; Zhang, S. Reliable Control Using Redundant Controllers. IEEE Trans. Autom. Control
**1998**, 43, 1588. [Google Scholar] [CrossRef] - Peng, S.; El Kebir, B.; Sing, K.N.; Guo, X. Robust disturbance attenuation for discrete-time active fault tolerant control systems with uncertainties. Optim. Control. Appl. Methods
**2003**, 24, 85–101. [Google Scholar] - Chamseddine, A.; Noura, H. Control and Sensor Fault Tolerance of Vehicle Active Suspension. IEEE Trans. Control. Syst. Technol.
**2008**, 16, 416–433. [Google Scholar] [CrossRef] - Li, H.; Liu, H.; Gao, H.; Shi, P. Reliable Fuzzy Control for Active Suspension Systems With Actuator Delay and Fault. IEEE Trans. Fuzzy Syst.
**2012**, 20, 342–357. [Google Scholar] [CrossRef] - Wang, R.; Jing, H.; Karimi, H.R.; Chen, N. Robust fault-tolerant H∞ control of active suspension systems with finite-frequency constraint-ScienceDirect. Mech. Syst. Signal Process.
**2015**, 62–63, 341–355. [Google Scholar] [CrossRef] - Feng, C.; Hao, S.; Li, Y.; Tong, S. Fuzzy Adaptive Fault-Tolerant Control for a Class of Active Suspension Systems with Time Delay. Int. J. Fuzzy Syst.
**2019**, 21, 2054–2065. [Google Scholar] - Soon, K.B.; Daejun, K.; Kyongsu, Y. Fault-tolerant control with state and disturbance observers for vehicle active suspension systems. Proc. Inst. Mech. Eng. Part D J. Automob. Eng.
**2020**, 234, 1912–1929. [Google Scholar] - Utkin Vadim, I.; Chang, H.C. Sliding mode control on electro-mechanical systems. Math. Probl. Eng.
**2002**, 8, 451–473. [Google Scholar] [CrossRef] - Morteza, M.; Afef, F. Adaptive PID-Sliding-Mode Fault-Tolerant Control Approach for Vehicle Suspension Systems Subject to Actuator Faults. IEEE Trans. Veh. Technol.
**2014**, 63, 1041–1054. [Google Scholar] - Li, M.; Zhang, Y.; Geng, Y. Fault-tolerant sliding mode control for uncertain active suspension systems against simultaneous actuator and sensor faults via a novel sliding mode observer. Optim. Control. Appl. Methods
**2018**, 39, 1728–1749. [Google Scholar] [CrossRef] - Slotine, J.J.E. Tracking Control of Nonlinear System Using Sliding Surface with Application to Robot Manipulator. Int. J. Control
**1983**, 38, 465–492. [Google Scholar] [CrossRef] - Slotine, J.J.E. Sliding Controller Design for Non-linear Systems. Int. J. Control
**1984**, 40, 421–434. [Google Scholar] [CrossRef] - Gao, W.; Hung, J.C. Variable Structure Control of Nonlinear Systems: A New Approach. IEEE Trans. Ind. Electron.
**1993**, 40, 45–55. [Google Scholar] - Zou, X.; Ding, H.; Li, J. Sensorless Control Strategy of Permanent Magnet Synchronous Motor Based on Fuzzy Sliding Mode Controller and Fuzzy Sliding Mode Observer. J. Electr. Eng. Technol.
**2023**, 18, 2355–2369. [Google Scholar] [CrossRef] - Ming, Y.; Cong, S.; Xu, J. Design of Nonlinear Motor Adaptive Fuzzy Sliding Mode Controller Based on GA. J. Syst. Simul.
**2008**, 20, 3141–3145. [Google Scholar] - Zhang, Q.; Zhang, Q.; Jiang, C.; Huang, H.; Zhang, S. Research on SMC Control of Permanent Magnet Synchronous Motor System Based on Particle Swarm Optimization. Equip. Manuf. Technol.
**2019**, 4, 35–39. [Google Scholar] - Guo, X.; Liu, X. Particle swarm optimization sliding mode control on interconnected power system. In Proceedings of the 33rd Chinese Control Conference, Nanjing, China, 28–30 July 2014. [Google Scholar]
- Saleh, T.M.; Mohammad, H. Synchronization of chaotic fractional-order systems via active sliding mode controller. Phys. Stat. Mech. Its Appl.
**2008**, 387, 57–70. [Google Scholar] - Zhang, Y. Time domain model of road irregularities simulated using the harmony superposition method. Trans. Chin. Soc. Agric. Eng.
**2003**, 19, 32–35. [Google Scholar] - Liu, X.; Pang, H.; Shang, Y. An Observer-Based Active Fault Tolerant Controller for Vehicle Suspension System. Appl. Sci.
**2018**, 8, 2568. [Google Scholar] [CrossRef] - Chen, W.; Zhao, L. Intelligent Vehicle Fault Tolerant Control Technology and Application; Science Press: Beijing, China, 2021. [Google Scholar]
- Podlubny, I. Fractional Differential Equations. In Mathematics in Science and Engineering; Academic Press: San Diego, CA, USA, 1999. [Google Scholar]
- Li, H.; Yu, J.; Hilton, C. Adaptive Sliding-Mode Control for Nonlinear Active Suspension Vehicle Systems Using T–S Fuzzy Approach. IEEE Trans. Ind. Electron.
**2013**, 60, 3328–3338. [Google Scholar] [CrossRef] - Duarte-Mermoud, M.; Aguila-Camacho, N. Using general quadratic Lyapunov functions to prove Lyapunov uniform stability for fractional order systems. Commun. Nonlinear Sci. Numer. Simul.
**2015**, 22, 650–659. [Google Scholar] [CrossRef]

Symbol | Meaning | Value |
---|---|---|

m | car body mass | 600 Kg |

${m}_{u1}={m}_{u2}$ | unsprung mass of front and rear | 50 Kg |

I | pitch moment | 1700 Kg·m^{2} |

$a=b$ | distance of axles to the center mass | 1.2 m |

${k}_{s1}={k}_{s2}$ | linear spring stiffness | 22.5 kN·m^{−1} |

${k}_{sn1}={k}_{sn2}$ | nonlinear spring stiffness | 150 kN·m^{−1} |

${c}_{s1}={c}_{s2}$ | linear damping coefficients | 1200 N·s·m^{−1} |

${c}_{sn1}={c}_{sn2}$ | nonlinear damping coefficients | 520 N·s·m^{−1} |

${k}_{t1}={k}_{t2}$ | spring stiffness coefficient of the tire | 440,000 N·m^{−1} |

${c}_{t1}={c}_{t2}$ | damping coefficient of the tire | 190 N·s·m^{−1} |

Index | SMC | GASMC | FRSMC |
---|---|---|---|

Car body acceleration | 0.1856 | 0.1464 | 0.1318 |

Pitch angle acceleration | 0.3207 | 0.3031 | 0.2678 |

Front suspension deflection | 0.0134 | 0.0125 | 0.0109 |

Rear suspension deflection | 0.0127 | 0.0117 | 0.0099 |

Front-tire dynamic load | 329.5 | 314.9 | 279.1 |

Rear-tire dynamic load | 320.1 | 303.4 | 266.94 |

Index | NF(SMC) | Fault | PSFTC | FRFTC |
---|---|---|---|---|

Car body acceleration | 0.1856 | 0.5041 | 0.1595 | 0.1514 |

Pitch angle acceleration | 0.3207 | 0.3653 | 0.3272 | 0.3195 |

Front suspension deflection | 0.0134 | 0.0256 | 0.0142 | 0.0129 |

Rear suspension deflection | 0.0127 | 0.2461 | 0.0133 | 0.0116 |

Front-tire dynamic load | 329.5 | 676.7 | 344.4 | 320.1 |

Rear-tire dynamic load | 320.1 | 642.1 | 332.5 | 316.5 |

Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content. |

© 2024 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Zhao, W.; Gu, L.
RBF-Based Fractional-Order SMC Fault-Tolerant Controller for a Nonlinear Active Suspension. *Machines* **2024**, *12*, 270.
https://doi.org/10.3390/machines12040270

**AMA Style**

Zhao W, Gu L.
RBF-Based Fractional-Order SMC Fault-Tolerant Controller for a Nonlinear Active Suspension. *Machines*. 2024; 12(4):270.
https://doi.org/10.3390/machines12040270

**Chicago/Turabian Style**

Zhao, Weipeng, and Liang Gu.
2024. "RBF-Based Fractional-Order SMC Fault-Tolerant Controller for a Nonlinear Active Suspension" *Machines* 12, no. 4: 270.
https://doi.org/10.3390/machines12040270