# Fuzzy Sliding Mode Control for the Vehicle Height and Leveling Adjustment System of an Electronic Air Suspension

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## Abstract

The accurate control for the vehicle height and leveling adjustment system of an electronic air suspension (EAS) still is a challenging problem that has not been effectively solved in prior researches. This paper proposes a new adaptive controller to control the vehicle height and to adjust the roll and pitch angles of the vehicle body (leveling control) during the vehicle height adjustment procedures by an EAS system. A nonlinear mechanism model of the full-car vehicle height adjustment system is established to reflect the system dynamic behaviors and to derive the system optimal control law. To deal with the nonlinear characters in the vehicle height and leveling adjustment processes, the nonlinear system model is globally linearized through the state feedback method. On this basis, a fuzzy sliding mode controller (FSMC) is designed to improve the control accuracy of the vehicle height adjustment and to reduce the peak values of the roll and pitch angles of the vehicle body. To verify the effectiveness of the proposed control method more accurately, the full-car EAS system model programmed using AMESim is also given. Then, the co-simulation study of the FSMC performance can be conducted. Finally, actual vehicle tests are performed with a city bus, and the test results illustrate that the vehicle height adjustment performance is effectively guaranteed by the FSMC, and the peak values of the roll and pitch angles of the vehicle body during the vehicle height adjustment procedures are also reduced significantly. This research proposes an effective control methodology for the vehicle height and leveling adjustment system of an EAS, which provides a favorable control performance for the system.

## Keywords

Electronic air suspension Height adjustment Leveling control Fuzzy sliding mode control Vehicle tests## 1 Introduction

Since the first application of electronic technology to the vehicle air suspension system recorded in 1980s, electronic air suspension (EAS) system has gained wide attentions [1, 2, 3, 4]. The main advantage of EAS system is the improvement of the vehicle dynamic performance, including the driving comfort, the handling stability and the fuel economy [5, 6]. As a well-established vehicle suspension system, EAS system has been researched in depth and been used in a large number of vibration isolation occasions, such as the vehicle seat suspension and the railway vehicle suspension [7, 8, 9, 10]. However, the vehicle height and leveling adjustment control problem of an EAS system still poses difficulties for researchers, which are reflected in the previous academic reports on this theme. The progresses of the intelligent control theories together with the increasing compute capability of the relevant hardware allow solving the control problem from a new perspective [8, 9]. In this paper, a fuzzy sliding mode control (FSMC) approach for precise vehicle height adjustment and leveling the vehicle body at appropriate posture by an EAS system is presented.

The heights at four corners of the vehicle body are adjusted by controlling the air mass flow rates flowing into or flowing out of the air springs for an EAS system. During off-road driving condition, the EAS system can lift the vehicle body to prevent the suspension from hitting the position limiter [10], while for driving on an expressway with high speed, the air drag can be reduced and the driving safety can be enhanced by lowering the vehicle body. Meanwhile, in order to prevent the phenomenon of vehicle posture instability, i.e., the peak values of the roll and pitch angles of the vehicle body are too large, which is mainly caused by the different payloads at four corners and the system parameters difference between the front and rear air suspensions, the effective control of the vehicle leveling is also very essential for the EAS system dynamic performance [11].

The difficulties in controlling the vehicle height and leveling of an EAS system arise from several aspects. In general, the controller needs to adjust the air mass flow rates flowing into or flowing out of the four air springs synchronously with several on-off solenoid valves. Furthermore, the nonlinear dynamic behaviors of the air spring will also give some troubles to the derivation of the system control law. In particular, the vehicle height and leveling control problem is further complicated by the real-time variations of the payloads and the system parameters [12]. In that case, only robust control methods could be effective. In Ref. [13], Kim and Lee proposed a sliding mode control (SMC) strategy for solving the height and leveling control problem of a closed-loop EAS system. Although the control method is robust and the system performance is improved, the controller may be sensitive to external disturbances due to the fixed control parameters. Xu et al. [14] proposed a variable structure vehicle height control method for single-wheel air suspension system, and on this basis, by calculating the deviations of the pitch and roll angles of the vehicle body, a leveling controller is also designed based on the fuzzy logic control (FLC) algorithm. However, due to the separated design of the vehicle height and leveling controller, the overall control system design procedure is lack of systematic character and fault tolerance.

Since the system uncertainties can be dealt with directly under the control of the SMC method, many SMC applications in engineering have been reported. Nevertheless, the SMC methodology without adaptive capability may suffer from chattering problem in control [15]. Several approaches have been proposed to eliminate the chattering phenomenon. However, these approaches, which are based on the use of a boundary layer in the sliding mode, are known to degrade the robustness [16]. To conquer the vulnerability of single SMC method and guarantee the system stability, the combination of SMC and FLC could be useful [17]. Many publications have reported effective schemes to apply fuzzy sliding mode control (FSMC) method for the controller design of several high-order nonlinear systems [18, 19, 20, 21].

The main contribution of this paper is that a novel control approach is proposed by combining the merits of SMC and FLC to address the vehicle height and leveling control problem of a full-car EAS system. The FSMC is designed by defining a switching sliding surface for the system, whose slope is modified by a FLC system. On this basis, the switching surface is also used as an input to FLC, thus the continuous term of SMC can be computed by using equivalent method. One advantage of the proposed approach is that the chattering control phenomenon is attenuated. Another advantage of the method is that the system controller has higher adaptive ability for the external disturbances. As far as we know, this approach is efficiently applied for the vehicle height and leveling control problem of an EAS system for the first time.

The organization of this paper is as follows: The EAS system dynamics, leveling control problem and the system nonlinear mathematical model are presented in Section 2. In Section 3, the design method for the FSMC of the EAS system is described. Section 4 illustrates various aspects of the controller behaviors with co-simulation conducted based on Matlab and AMESim. The actually potential advantages of the control method are verified in Section 5 by actual vehicle tests. The conclusions of this paper are finally summarized in Section 6.

## 2 System Description and Modeling

### 2.1 System Outline

- (1)
Height and leveling adjustment during the lifting procedure: In the lifting process, the charging solenoid valve or the compressor is opened, thus the compressed air can flow into the four air springs. If the vehicle body does not maintain at an appropriate posture, the control signals for the four air spring solenoid valves would be reassigned to reduce the air mass flow rates flowing into the air springs whose height lifting speed are relatively fast.

- (2)
Height and leveling adjustment during the lowering procedure: To lower the vehicle body, the ECU generates control signals to open the discharging solenoid valve and the air spring solenoid valves, thus the compressed air in the air springs can flow into the environment directly. Similarly, if the vehicle body does not maintain at an appropriate posture in the height lowering procedure, the air mass flow rates flowing out of the air springs whose height lowering speed are relatively fast would be reduced.

### 2.2 Leveling Control Problem

In the figures, \(z_{a - i}^{fr} ,\;z_{a - i}^{fl} ,\;z_{a - i}^{rl} \;{\text{and}}\;z_{a - i}^{rr}\) are the initial vehicle heights at four corners of the vehicle body, \(z_{a - u}^{f} ,\;z_{a - u}^{f} ,\;z_{a - u}^{rl} \;{\text{and}}\;z_{a - u}^{rr}\) are the vehicle heights at four corners without coordination control, \(z_{a - c}^{fr} ,\;z_{a - c}^{f} ,\;z_{a - c}^{rl} \;{\text{and}}\;z_{a - c}^{rr}\) are the vehicle heights at four corners under ideal control, *θ* and *φ* denote the roll and pitch angles of the vehicle body. In order to prevent the phenomenon of vehicle leveling instability, the control system needs to adjust the air mass flow rates flowing into or flowing out of the air springs effectively during the vehicle height lifting or lowering procedures.

### 2.3 System Modeling

The mechanism model describing the nonlinear dynamic behaviors of the vehicle height and leveling adjustment system of EAS consists of the model describing the thermodynamic behaviors of the air springs, the model describing the airflow properties of the solenoid valves and the full-car dynamic model.

#### 2.3.1 Thermodynamic Behaviors of the Air Spring

During the vehicle height and leveling adjustment procedures, the dynamic behaviors of the air springs are similar to a variable mass gas charging/discharging thermodynamic system. Based on the derivation of the relevant thermodynamic theories, the thermodynamic behaviors of the air spring are given as [25, 26]

*P*

_{ as }is the air pressure inside the air spring,

*V*

_{ as }is the volume of the air spring,

*κ*refers to the polytropic constant,

*R*denotes the universal gas constant,

*T*is the temperature of the air inside the air spring,

*q*

_{ in }and

*q*

_{ out }are the air mass flow rates flowing into and flowing out of the air spring respectively.

The pipe can also be considered as a variable mass gas charging/discharging thermodynamic system, but its volume remains constant in the vehicle height and leveling adjustment procedures, thus the mechanism model of the pipe can be given as

*P*

_{ pi }is the air pressure inside the pipe,

*V*

_{ pi }is the volume of the pipe,

*q*

_{pi-in}and

*q*

_{pi-out}are the air mass flow rates flowing into and flowing out of the pipe.

#### 2.3.2 Solenoid Valve Airflow Properties

The air mass flow rate through the solenoid valve is mainly decided by the upstream and downstream air pressures of the solenoid valve. Based on the modeling assumptions, the on-off solenoid valve is abstracted to a thin wall orifice, thus the nonlinear air mass flow characteristics through the solenoid valve is expressed mathematically as [27, 28]

*q*(

*P*

_{ u },

*P*

_{ d }) refers to the air mass flow rate through the solenoid valve,

*s*denotes the solenoid valve cross-sectional area,

*P*

_{ u }refers to the upstream air pressure,

*P*

_{ d }refers to the downstream air pressure, and

*b*refers to the critical pressure ratio. It is noted that the airflow characteristics of the six on-off solenoid valves are assumed to be identical to each other.

#### 2.3.3 Full-car Dynamic Model

The motion equations of the full-car dynamic model can be presented mathematically as [29]

*m*

_{ s }refers to the sprung mass,

*z*

_{ s }refers to the vertical displacement of the vehicle body,

*F*

_{ si }(

*i*= 1, 2, 3, 4) denote the applied forces exerted on the vehicle body by the four air suspensions,

*P*

_{ asi }(

*i*= 1, 2, 3, 4) denote the pressures of the air in the air springs,

*c*

_{ di }(

*i*= 1, 2, 3, 4) refer to the damping coefficients of the four dampers,

*A*

_{ asi }(

*i*= 1, 2, 3, 4) refer to the effective areas of the air springs,

*z*

_{ fr },

*z*

_{ fl },

*z*

_{ rr }and

*z*

_{ rl }are the vertical displacements of the four unsprung masses,

*z*

_{ ai }(

*i*= 1, 2, 3, 4) refer to the displacements of the vehicle body at four corners,

*I*

_{ θ }and

*I*

_{ φ }refer to the roll and pitch moments of inertia respectively,

*l*

_{ a }is half of the wheel-track,

*l*

_{ f }and

*l*

_{ r }refer to the distances between the vehicle centroid to the front and the rear axis,

*m*

_{ fr },

*m*

_{ fl },

*m*

_{ rr }and

*m*

_{ rl }are the four unsprung masses,

*k*

_{ tf }and

*k*

_{ tr }are the tyre stiffnesses of the front and rear wheels respectively,

*z*

_{ ri }(

*i*= 1, 2, 3, 4) are the road roughness inputs.

*z*

_{ s },

*θ*and

*φ*can be obtained as [30]

*q*

_{ asi }(

*i*= 1, 2, 3, 4), the system nonlinear mechanism model can be further obtained as

*refers to the road roughness disturbance. The specific contents of*

**r***α*(

*),*

**x***β*(

*) and*

**x***k*(

*) can be obtained based on Eqs. (1), (2), (4) and (5), which are omitted here because of lack of space.*

**x**## 3 Controller Design

To conduct the derivation of FSMC control law for the vehicle height and leveling adjustment system of EAS in the linear field, the nonlinear mechanism model of the system needs to be further globally linearized through the state feedback method based on the differential geometry theory. The following are the related concepts of differential geometry theory [31].

*refers to the system state variables,*

**x***f*and

*g*are the smooth functions of the vector field

*h*,

*refers to the system inputs and*

**u***refers to the system outputs.*

**y**### **Definition 1: Lie derivative**

### **Definition 2: Relative degree**

**x**_{0}∈

*, if the neighborhood of*

**x**

**x**_{0}and a positive integer

*r*, which make the system (9) meet the following conditions, exist,

Then the relative degree of the system is defined as *r*.

Differential geometry theory is a controller design method for the nonlinear system. By using appropriate nonlinear state feedback and coordinate transformation method, a nonlinear system is transformed into a linear system partially or wholly, then the system controller can be designed based on the linear control method.

### 3.1 Linearization of the System Model

According to the specific contents of *α*(* x*) and

*β*(

*), the following calculated results can be obtained based on the differential geometry theory*

**x***i*=

*j*=6. Thus the relative degree of the nonlinear system {

*r*

_{ i }}(

*i*= 1, 2, 3, 4, 5, 6) = {6}, which satisfies the necessary and sufficient conditions for the linearization of the system. Therefore, by introducing the following non-singular coordinate transformation:

*L*

_{ f }

*h*(

*) is called the lie derivative of*

**x***h*with respect to

*f*[32], the nonlinear model of the vehicle height and leveling adjustment system of EAS is transformed into a new system model by the linearized coordinates. Then, the new system state equation in the linear space can be presented mathematically as

### 3.2 SMC Design

Generally, the vehicle height and leveling adjustment system has high nonlinearities because of the nonlinear dynamic behaviors of the air springs and the airflow characteristics through the solenoid valve. Furthermore, the real-time variations of the payload and the system parameters also give troubles to the controller design. Thus, an effective control method is essential to solve these problems and to guarantee the vehicle height and leveling are controlled in high performance. The SMC is a representative of the robust control method and has been used in many pneumatic control systems [33, 34, 35]. The SMC design procedure contains two parts. One part is to define the sliding surfaces and the other part is to formulate an effective control law.

*h*

_{ d }denotes the desired vehicle height,

*θ*

_{ d }and

*φ*

_{ d }refer to the desired roll and pitch angles of the vehicle body.

*c*

_{ ij }(

*i*= 1,…, 6,

*j*= 1, 2) are positive constants.

The control outputs of the SMC consist of two parts. The first part is the equivalent control section and the second part is the switching control section. By differentiating the sliding surfaces *s* for getting the equivalent control outputs of SMC, and the exponential approach law is used in the switching control section, then several new equations can be obtained as

*k*

_{ i }(

*i*= 1,…, 6) are the SMC parameters,

*ε*

_{ i }(

*i*= 1,…, 6) are the gain coefficients of the switching control.

### 3.3 FSMC Design

*s*and the fuzzy rules are defined by triangular form and trapezoidal form of membership functions. The membership function of the output, i.e., the gain coefficient of the switching control, is defined as a singleton function, whose fuzzy sets are defined on the normalized universe of discourse ± 1, as shown in Figure 3. The fuzzy variables defined to describe the output membership functions are {NB, N, M, P, PB}.

Then, the outputs of the FSMC can be obtained. The FSMC determines a mapping from the sliding surface to the system outputs. The control rules of the FSMC are contrary to the modified SMC on the separated semi-plane, where the magnitudes of the fuzzy control signals are proportional to the states away from the sliding surface *s *= 0. Therefore, the FSMC actions will help to ensure the system state vectors stay on the sliding surfaces, so as to realize the system optimal control performance.

*q*

_{ o _ i _ con }refer to the air mass flow rates through the air spring solenoid valves, which are controlled by the proposed FSMC.

*q*

_{ o _ max }refers to the maximum air mass flow rate through the air spring solenoid valve when the solenoid valve is fully opened.

- (1)
When the absolute values of the desired air mass flow rates through the air spring solenoid valves are larger than the maximum air mass flow rates, the duty ratios of the on-off solenoid valves are defined as 1.

- (2)
When the absolute values of the desired air mass flow rates through the air spring solenoid valves are less than the maximum air mass flow rates, the duty ratios of the on-off solenoid valves are defined as the desired air mass flow rates divided by the maximum air mass flow rates.

- (3)
When the desired air mass flow rates through the air spring solenoid valves are zero, the solenoid valves should be closed.

*δ*

_{ c }and

*δ*

_{ d }are the on-off statuses of the charging solenoid valve and the discharging solenoid valve respectively,

*δ*

_{ a }

^{ fr },

*δ*

_{ a }

^{ fl },

*δ*

_{ a }

^{ rl }and

*δ*

_{ a }

^{ rr }refer to the on-off statuses of the four air spring solenoid valves.

## 4 Simulation Study

System parameters

Parameter | Value |
---|---|

Mass of the vehicle body | 11560 |

Roll moment of inertia | 16750 |

Pitch moment of inertia | 85036 |

Unsprung mass of front wheel | 435 |

Unsprung mass of rear wheel | 845 |

Distance from the centre of gravity to front axis | 3.32 |

Distance from the centre of gravity to front axis | 2.38 |

Half of the wheel-track | 1.21 |

Front wheel damping coefficients | 11085 |

Rear wheel damping coefficients | 13900 |

Tyre stiffness | 650 |

Cross-sectional area of the air spring | 0.042 |

Cross-sectional area of the solenoid valve | 7 × 10 |

Volume of the pipeline | 1.7 × 10 |

Air pressure of the air reservoir | 1.8 |

Air pressure of the environment | 1.01 × 10 |

Air temperature | 293.15 |

Universal gas constant | 287.1 |

Polytropic index | 1.4 |

Critical pressure ratio | 0.528 |

The control objectives of the proposed method are to ensure the vehicle height can be controlled in high precision and the peak values of the roll and pitch angles of the vehicle body can be reduced significantly despite payload variations and parameters difference. To obtain the simulation calculation results, the road roughness input is given mathematically as [41, 42]

*u*denotes the vehicle speed,

*G*

_{ q }(

*n*

_{0}) refers to the road random roughness coefficient and

*w*(

*t*) refers to the Gaussian white noise. The road roughness used in simulation and vehicle tests corresponds to the road of class B (

*G*

_{ q }(

*n*

_{0}) = 64 × 10

^{−6}m

^{3}) [43, 44].

Analysis of the co-simulation results

Control performance | Simulation scenarios | ||||||
---|---|---|---|---|---|---|---|

During standstill | During driving | ||||||

Vehicle height | Desired value | Actual value | Precision (%) | Desired value | Actual value | Precision (%) | |

Height lifting |
| 0.270 | 0.2698 | 99.92 | 0.270 | 0.2693 | 99.74 |

| 0.270 | 0.2697 | 99.89 | 0.270 | 0.2691 | 99.67 | |

| 0.270 | 0.2696 | 99.85 | 0.270 | 0.2685 | 99.44 | |

| 0.270 | 0.2696 | 99.85 | 0.270 | 0.2690 | 99.63 | |

Height lowering |
| 0.250 | 0.2502 | 99.92 | 0.250 | 0.2493 | 99.72 |

| 0.250 | 0.2503 | 99.88 | 0.250 | 0.2505 | 99.80 | |

| 0.250 | 0.2502 | 99.92 | 0.250 | 0.2491 | 99.64 | |

| 0.250 | 0.2503 | 99.88 | 0.250 | 0.2506 | 99.76 |

Posture | Without control | With control | Decrease (%) | Without control | With control | Decrease (%) | |
---|---|---|---|---|---|---|---|

Height lifting |
| 0.022 | 0.013 | 40.91 | 0.040 | 0.029 | 27.50 |

| 0.035 | 0.022 | 37.14 | 0.053 | 0.042 | 20.75 | |

Height lowering |
| 0.019 | 0.011 | 42.11 | 0.028 | 0.017 | 39.29 |

| 0.023 | 0.016 | 30.43 | 0.054 | 0.036 | 33.33 |

As shown in Figure 6 and Table 2, the controller shows high precision in the vehicle height adjustment, and the peak values of the roll and pitch angles of the vehicle body are also reduced significantly. These results show the effectiveness of the proposed controller.

## 5 Vehicle Tests

The system controller designed based on FSMC is implemented using a D2P (development to production) rapid control prototyping platform, which can generate the control code directly based on the compilation of project files established in Simulink. The initial static vehicle height is assumed to be zero in the vehicle test.

## 6 Conclusions

- (1)
A FSMC controller with adaptive ability has been developed in this paper for solving the vehicle height and leveling control problem of an EAS system. Based on the analysis of the system working principle, a nonlinear mechanism model is established to describe the dynamic behaviors of the system and to derive the control law.

- (2)
As a robust control method, the FSMC technique can be used to ensure the vehicle height adjustment precision and stabilize the posture of the vehicle body during the vehicle height adjustment procedure of EAS system, i.e., the peak values of the roll and pitch angles can be decreased significantly.

- (3)
The resulting FSMC control algorithm can be implemented by the D2P rapid control prototyping platform, and on this basis, the actual vehicle tests of the control performance is conducted. The simulation and test results illustrate that the proposed FSMC method can control the vehicle height with high precision and regulate the roll and pitch angles of the vehicle body effectively. The proposed approach can be used to improve the performance of EAS system.

## Notes

### Authors’ contributions

LC was in charge of the whole trial; X-QS wrote the manuscript; Y-FC, C-CY and S-HW assisted with sampling and laboratory analyses. All authors read and approved the final manuscript.

### Authors’ Information

Xiao-Qiang Sun, born in 1989, is currently a lecturer at Automotive Engineering Research Institute, Jiangsu University, China. He received his PhD degree from Jiangsu University, China, in 2016. His research interests include electronically controlled air suspension and engineering application of advanced control theory. E-mail: sunxqujs@126.com

Ying-Feng Cai, born in 1985, is currently an associate professor and a master candidate supervisor at Automotive Engineering Research Institute, Jiangsu University, China. She received her PhD degree from Southeast University, China, in 2013. Her main research interests include vehicle system dynamics and intelligent automobile. E-mail: caicaixiao0304@126.com

Chao-Chun Yuan, born in 1978, is currently an associate professor and a master candidate supervisor at Automotive Engineering Research Institute, Jiangsu University, China. He received his PhD degree from Jiangsu University, China, in 2007. His main research interests include vehicle system dynamics and intelligent automobile. E-mail: yuancc_78@163.com

Shao-Hua Wang, born in 1978, is currently an associate professor and a master candidate supervisor at School of Automotive and Traffic Engineering, Jiangsu University, China. His main research interests include electronically controlled air suspension and hybrid electric vehicle. E-mail: wsh@ujs.edu.cn

Long Chen, born in 1958, is currently a professor and a PHD candidate supervisor at Automotive Engineering Research Institute, Jiangsu University, China. His main research interests include modeling and control of vehicle dynamic systems. E-mail: chenlong@ujs.edu.cn

### Competing interests

The authors declare no competing financial interests.

### Ethics approval and consent to participate

Not applicable.

### Funding

Supported by National Natural Science Foundation of China (Grant Nos. 51375212, 61601203), Priority Academic Program Development (PAPD) of Jiangsu Higher Education Institutions of China, Key Research and Development Program of Jiangsu Province (BE2016149), and Jiangsu Provincial Natural Science Foundation of China (BK20140555).

### Publisher’s Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

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