Stability Control for Vehicle Dynamic Management with Multi-Objective Fuzzy Continuous Damping Control

applied sciences

Article Stability Control for Vehicle Dynamic Management with Multi-Objective Fuzzy Continuous Damping Control

Xu Zhang 1, Chuanxue Song 1, Shixin Song 2, Jingwei Cao 1 , Silun Peng 1, Chunyang Qi 1, Feng Xiao 1,3 and Da Wang 1,* 1 State Key Laboratory of Automotive Simulation and Control, Jilin University, Changchun 130022, China; [email protected] (X.Z.); [email protected] (C.S.); [email protected] (J.C.); [email protected] (S.P.); [email protected] (C.Q.); [email protected] (F.X.) 2 College of Mechanical and Aerospace Engineering, Jilin University, Changchun 130022, China; [email protected] 3 Taizhou Automobile Power Transmission Research Institute, Jilin University, Taizhou 225322, China * Correspondence: [email protected]

Received: 29 September 2020; Accepted: 6 November 2020; Published: 8 November 2020

Abstract: Vehicle dynamic management (VDM) is a vehicle chassis integrated control system based on electronic stability program (ESP) and continuous damping control (CDC) that has been developed in recent years. In this work, the ideal yaw angle rate and sideslip angle of the mass center are calculated deriving an ideal monorail model with two degrees of freedom. Then, a direct yaw moment proportional-integral-differential control strategy for ESP is proposed as the foundation of VDM. In addition, a multi-objective fuzzy continuous damping control (MFCDC) is proposed to achieve comfort, handling stability, and rollover prevention. The effect of the MFCDC strategy is analyzed and verified through a sine wave steer input test, double line change test, and fishhook test. The results indicate that MFCDC-ESP has a significant advantage in preventing rollover. MFCDC-ESP can maintain the optimized distribution of damping force through its own compensation under possible instability and predict the critical stable state to some extent. MFCDC-ESP exhibits strong real-time sensitivity to the control state of the damping force of each wheel. Hence, it can ensure the comfort of passengers under good driving conditions and exert strong adaptability and control effects under extreme working conditions.

Keywords: continuous damping control; multi-objective fuzzy control; electronic stability program; sky-hook control; rollover prevention

1. Introduction The automobile is a typical complex system. Many parts of a vehicle are involved in the research of vehicle dynamics, and the performance of each part affects the performance of the whole vehicle in varying degrees [1]. An automobile system can be divided into several small systems. Overall vehicle performance can be improved by enhancing each small system from the perspective of reductionism. For cars, they should be regarded as a large system, and all their parts should be analyzed in a unified way. Moreover, a unified and global implementation scheme should be proposed and subsequently implemented to each subsystem holistically. The chassis system is an important assembly in automobiles that involves suspension, steering, braking, driving, and other subsystems with coupling relationship. The extensive use of underlying controllers may lead to functional conflicts between different chassis controllers and result in unnecessary expenses and the reduction of the control effect in the

Appl. Sci. 2020, 10, 7923; doi:10.3390/app10217923 www.mdpi.com/journal/applsci Appl. Sci. 2020, 10, 7923 2 of 22 wide use of control systems and control technologies. Active control technology can greatly improve the control performance of all parts of the chassis system [2]. In 2006, a nonlinear constrained optimal allocation control method was proposed on the basis of the tire force distribution method and concept of friction circle [3]. Wang et al. studied a hierarchical integrated control method for vehicle dynamics in which a sliding mode controller is used to obtain the total longitudinal force, total lateral force, and total yaw moment of a vehicle in the upper control layer; this approach distributes the force and torque of the body to the longitudinal slip ratio and side slip angle of the four tires [4]. Roshanbin A and Naraghi M studied and designed a vehicle integrated control framework that comprehensively considers the control actuators of each system and proposed the concept of vehicle dynamics integrated control. The concept skips the traditional idea of electronic control system design and involves function based on control structure [5]. A logic control of active front wheel steering (AFS) and electronic stability system was studied by Hwang et al. in 2008 [6]. Tin Lun Lam et al. established a coordinated control method for four-wheel driving force that is mainly used in four-wheel independent driving and four-wheel independent steering structures to minimize the energy loss of vehicle systems while improving their overall stability [7]. Tin Lun Lam et al. also studied a new four-wheel angle assignment method for four-wheel independent steering electric vehicles by minimizing tire slip [8]. Tjonnas Johannes and Johansen Tor A used a hierarchical control system in which the maximum friction coefficient between the wheel and the road surface is adaptively controlled to minimize the output force of the controller [9]. A nonlinear vehicle dynamics model integrated with AFS, electronic stability control (ESC), and variable torque distribution was established by Elmarakbi Ahmed et al. [10]. A chassis control algorithm based on differential braking, front/rear traction torque, and active yaw moment control was proposed by Her Hyundong et al. The results indicate that the control effect of the chassis control system is better than that of a sole bottom plate control system and that it can maximize speed and vehicle steering lateral stability [11,12]. Through coordinated control with AFS, ESC, and active rear-wheel steering, Yim Seongjin observed the interaction among systems by using a variety of combinations and distributed wheel tire control force on the basis of inverse weighted pseudo control allocation. The Plackett–Burman method was used to analyze the sensitivity and check the influence of variable weights [13]. In the field of chassis control, suspension has been used to improve comfort. However, with the emergence of semiactive suspension and active suspension, the need to establish the status of suspension control with consideration of ride comfort and stability has increased. Passive suspension control cannot be adjusted in real time, hence the availability of active and semiactive suspension control in the field of suspension control. Semiactive suspension can achieve ride comfort control similar to that of active suspension but with minimal power consumption [14,15]; it is thus the most widely used and applicable controllable suspension system [16]. The concept of sky-hook damping “on-off” control with semiactive suspension was first proposed by Karnopp in 1974. Sky-hook control can improve ride comfort but handling stability inevitably deteriorates [17]. In 1983, Toyota developed a semiactive suspension vehicle (Toyota Soarer 280GT) equipped with three adjustable working conditions. In 1989, semiactive control was combined with active control in Ford’s Thunderbird, and the combination surpassed passive suspension in terms of ride comfort and handling stability [18]. In 1997, VALÁŠEK M et al. proposed a ground-hook control algorithm to improve handling stability further [19]. Continuous damping control (CDC) technology of ZF Sachs is based on proportional valve control. Damping control is achieved by changing the damper aperture at a high corresponding frequency with CDC [20]. The new Audi A6 allroad Quattro technology combines CDC technology with air suspension technology to realize adaptive control. A similar adaptive control technology is Hydractive, which was proposed by Citroen [21]. With the maturity of semiactive suspension technology, semiactive control has been integrated into chassis systems. The linear quadratic Gaussian (LQG) method for integrated control of suspension and steering system was adopted by Harada m et al. to analyze lateral stability of vehicle [22]. Yoshimura and Emoto developed a hierarchical integrated control of steering and active suspension systems; Appl. Sci. 2020, 10, 7923 3 of 22 the strategy effectively solves the problem of steering effect on the active suspension actuator force and significantly improves the ride comfort and handling performance of vehicles [23]. Gaspar P et al. designed a robust controller for active suspension by using the hybrid µ synthesis method [24]. Fan Yu added a central differential brake to the integrated control of active yaw and active traction distribution to study the effects of front and rear axle load transfer on lateral stability [25]. In 2008, an experiment on a hybrid vehicle console frame was performed on dSPACE [26]. Yoon et al. designed an index called RI to prevent rollover, characterize rollover risk, and reduce the use of ESC [27]. On the basis of this RI, a model-based state estimator was designed and used to suppress roll motion caused by maneuvers and road disturbances [28]. In 2009, Kangwon Lee presented a method involving the use of the derivative of yaw rate to control CDC and thereby integrate ESC and CDC [29]. A double-layer control strategy between suspension, steering, and braking systems was proposed to improve the performance of vehicles under different driving conditions. On the basis of integral sliding mode control theory, the control modes under different working conditions were switched according to the monitoring and identification of states [30,31]. Zhang Xinjie et al. regarded the ride comfort and handling stability of vehicles as functions of vehicle speed and road driving conditions and proposed the control principle of hybrid suspension based on vehicle speed and road conditions [32]. Her Hyundong et al. proposed a three-step integrated chassis control algorithm with braking force, active rolling moment, and damping coefficient distributed [33]. Mazlan Saiful Amri et al. used linear quadratic regulator and LQG optimal control theory to design an active lateral stabilizer bar to reduce vehicle roll angle and roll angle rate [34]. Vehicle dynamics management (VDM) offers great advantages, including the reduction of sensors and system complexity. It can also eliminate interference conflicts between various systems while achieving global optimal performance. Given the complexity of centralized control models, design difficulties, and low universality, the coordinated control method is mainly used for integrated design in VDM. The control of a chassis system should be based on system theory, and the design of the subsystem controller should be combined with the design of the whole chassis controller to obtain the best control performance for the chassis system. Considering the complexity of the vehicle chassis system, the current work takes CDC suspension as the research object on the basis of VDM to improve and optimize the overall control performance for vehicle chassis systems. This paper comprises five parts: introduction, basic handling stability control strategy of electronic stability program (ESP), design of multi-objective fuzzy continuous damping controller, simulation and result analysis, and conclusion. In Section1, the research background and research status are introduced, and a brief overview of the overall layout of the paper is presented. In Section2, the state space expression of lateral stability control is given by studying the control mechanism of the ESP. The ideal yaw angle rate and sideslip angle of the mass center are calculated by deriving the ideal monorail model with two degrees freedom. Finally, a direct yaw moment proportional–integral–differential (PID) control strategy based on yaw rate and centroid sideslip angle is proposed. In Section3, the multi-objective fuzzy continuous damping controller based on ESP is proposed to achieve sky-hook control and thereby ensure passenger comfort while preventing rollover. The effect of the ESP can be improved by redistributing the vertical force at the same time. In Sections4 and5, the e ffect of the multi-objective fuzzy continuous damping control strategy is analyzed and verified through a sine wave steer input test, double line change (DLC) test, and fishhook test. The simulation results show that the multi-objective fuzzy control strategy is successful. The strategy can ensure the comfort of passengers under good driving conditions while ensuring strong adaptability and control effect under dangerous and extreme working conditions. Appl. Sci. 2020, 10, 7923 4 of 22

2. Basic Handling Stability Control Strategy of ESP VDM is a vehicle chassis integrated control system based on the ESP and CDC and other chassis control subsystems, such as the antilock braking system, traction control system, and even four-wheel steering system [35,36]. Direct yaw moment control (DYC) is a common way to handle stability control.

2.1. Construction of Direct Yaw Moment Controller

2.1.1. Linear Monorail Standard Equation with DYC The stability of a vehicle when turning can be significantly improved by DYC, which is designed to boost vehicle stability. The basic idea of DYC is to change the ratio of the driving or braking force between the inner and outer wheels of a vehicle so as to make the vehicle produce an additional restoring yaw moment, which prevents the instability of the vehicle as much as possible. The research and development of handling and stability systems is mainly completed by tracking the ideal vehicle model on the basis of the actual motion state of vehicles. However, the definition of an ideal vehicle model remains unclear. In the whole process of vehicle motion, numerous uncertain parameters affect the modeling accuracy of complex vehicle systems. Although the design of vehicle controllers is considerably difficult, the variations of bounded uncertain parameters make the whole vehicle system controllable. The linear monorail model of vehicle motion with two degrees of freedom includes several important parameters, such as body mass, yaw rate, sideslip angle, wheelbase, and lateral stiffness of the front and rear axles. Collectively, these parameters can reflect the lateral motion of vehicles. Therefore, such linear monorail model with two degrees of freedom is usually used as an ideal model for the design of stability control strategies, and its steering characteristics are considered as vehicle stability control objectives. When the vehicle is regarded as a monorail model with two degrees of freedom (Figure1), the following equilibrium relationships of the system dynamic force of the vehicle can be established as follows: . mt v + ur = Fy1 + Fy2 (1) . Izr = L F L Fy (2) 1 y1 − 2 2 F = K α (3) yj − αj j v = v + L r = u tan(δ α ) (4) 1 1 1 − 1 v = v L r = u tan( α ) (5) 2 − 2 − 2 v = tan β (6) u where mt is the total mass of the vehicle; Fyj is the lateral force of each vehicle axle; u, v are the longitudinal and lateral speeds of the vehicle, respectively; r is the yaw angle rate of the vehicle; Kαj is the lateral slip stiffness of each axle; αj is the slip angle of each axle; vj is the longitudinal speed of each axle; Lj is the longitudinal distance from each axle to the mass center; δ1 is the front wheel angle; β is the sideslip angle of mass center; and subscript j indicates front axle (subscript 1) and rear axle (subscript 2). When an angle such as αj, δ1, and β is considerably small, the tangent value is approximately equal to the angle radian value. Therefore, Equations (4) and (5) can be respectively simplified into the following: v + L r α = δ 1 (7) 1 1 − u v L r α = − 2 (8) 2 − u Appl. Sci. 2020, 10, 7923 5 of 22

Appl. Sci. 2020, 10, x FOR PEER REVIEW 5 of 22

FigureFigure 1.1. MonorailMonorail modelmodel withwith twotwo degreesdegrees ofof freedom.freedom. TheThe twotwo wheelswheels onon thethe axleaxle indicatedindicated byby thethe blueblue dotteddotted lineline areare replacedreplaced byby aa monorailmonorail equivalentequivalent toto thethe blackblack solidsolid lineline inin thisthis model.model.

ByWhen substituting an angle Equationssuch as αj, (3), δ1, (6)–(8)and β is into considerably Equations (1)small, and the (2) andtangent with value the application is approximately of the DYCequal moment to the angleMzd-ESP radian, the value. following Therefore, are obtained: Equations (4) and (5) can be respectively simplified into the following: . 1 umt β + r = (Kα1 + Kα2)β + 푣(+L1퐿K1α푟1 L2Kα2)r Kα1δ1 (9) 훼 = 훿 − u − − (7) 1 1 푢 . 1 I r = (L K L K )β + L2K + L2K r L K δ + M (10) z 1 α1 2 α2 1 α1푣 − 퐿22푟α2 1 α1 1 zd ESP − u훼 = − − − (8) 2 푢 Equations (9) and (10) can be transformed into the following standard state space equation:

By substituting Equations (3), (6)–(8) into. Equations (1) and (2) and with the application of the DYC moment Mzd-ESP, the following are obtainedX = AX: + BU (11) 1 푢푚 (훽̇ + 푟) = (퐾 + 퐾 )훽 + (퐿 퐾 − 퐿 퐾 )푟- where state vector, input vector, and relevant푡 matrix are: 훼1 훼2 푢 1 훼1 2 훼2 (9) 퐾훼1훿1 T X = β r 1 T 퐼 푟̇ = (퐿 퐾 − 퐿 퐾 )훽 + (퐿2퐾 + 퐿2 퐾 )푟 − 퐿 퐾 훿 + 푀 (10) 푧 1 훼1 2 훼2U = 푢 δ11 훼M1zd ESP2 훼2 1 훼1 1 푧푑−퐸푆푃  − 2   Kα1 + Kα2 L1Kα1 L2Kα2 u mt  Equations (9) and (10) can be transformed into the following− − standard state space equation:  um u2m  A =  t t   푿̇ = 푨푿 +L푩푼2K + L2K   L1Kα1 L2Kα2 1 α 1 2 α2  (11)  −  I uI where state vector, input vector, and relevantz  matrix are:  z Kα1  0   um 푇  B = 푿 =−(훽 t 푟)   L1Kα1 1    − Iz Iz 푇 푼 = (훿1 푀푧푑−퐸푆푃)

2.1.2. Control Target: Ideal Yaw Rate퐾 and+ Sideslip퐾 Angle퐿 퐾 − of퐿 Mass퐾 − Center푢2푚 훼1 훼2 1 훼1 2 훼2 푡 When no DYC control is added, 푢 the푚 ideal vehicle should푢2푚 always be in a steady driving state. 푨 = 푡 푡 ( 2 2 ) When the vehicle is in a steady state, 퐿1퐾훼 the1 − lateral퐿2퐾훼2 acceleration퐿1퐾훼1 + and퐿2퐾 yaw훼2 angular acceleration are zero, [ 퐼푧 푢퐼푧 ] Appl. Sci. 2020, 10, 7923 6 of 22 and the diﬀerential term on the left side of Equations (9) and (10) disappears and becomes a solvable equation. The steady-state yaw rate gain and centroid sideslip angle gain can be obtained as ! r∗ u/(L + L2) u/(L + L2) d = 1 = 1 (12) ! 2 δ1 m L L 1 + Du s 1 + t 1 2 u2 2 K K (L1 + L2) α2 − α1 ! ( + ) + 2 2 + 2 β∗ 1 L1Kα1 L1 L2 1 Du L1Kα1 L2Kα2 d = − (13) 2 δ1 (L + L2) (1 + Du )(L K L K ) s 1 1 α1 − 2 α2 where coeﬃcient D is a stability factor, which is only related to the structure of the vehicle but is independent of the vehicle motion state. It is an important parameter in the study of vehicle steady-state response, and it can be expressed as ! m L L D = t 1 2 (14) 2 K K (L1 + L2) α2 − α1

When the sideslip angle is very small, the ground adhesion constraint condition can be expressed as follows:

ur0 ay µg (15) d|≈| ≤ 2 2 + 2 L1Kα1u δ1 L1Kα1 L2Kα2 β0 µg − (16) d ≤ u2(L K L K ) 1 α1 − 2 α2 where ay is the lateral acceleration of the vehicle; µ is the friction coeﬃcient of the road; g is the gravity acceleration. The ideal yaw rate and sideslip angle control targets with consideration of steady-state driving and adhesion conditions can be written as: ( ) n o u/(L1 + L2) µg rd = min r∗ , r0 = min δ1 , (17) d d 1 + Du2 u

n o βd = min β∗ , β0 d d  ( + ) + 2 2 + 2 2 2 + 2   1 L1Kα1 L1 L2 1 Du L1Kα1 L2Kα2 L1Kα1u δ1 L1Kα1 L2Kα2  (18) = min − δ , µg −   2 1 2   (L + L2) (1 + Du )(L K L K ) u (L K L K )  1 1 α1 − 2 α2 1 α1 − 2 α2 2.1.3. PID Control Strategy for DYC Proportional–integral–differential (PID) control first appeared in all control strategies and thus was widely studied. It is also the most commonly used control law and is the embodiment of feedback theory. Given its simple algorithm, high reliability, and few setting parameters (i.e., proportional parameter P, integral parameter I, and differential parameter D), PID control is widely used in control systems. Appl. Sci.The 2020 PID, 10, control x FOR PEER principle REVIEW is shown in Figure2 and the ESP controller model is shown in Figure7 of 223.

Figure 2. Proportional–integral–diﬀerential (PID) control principle. Figure 2. Proportional–integral–differential (PID) control principle.

Figure 3. Electronic stability program (ESP) controller model. According to the monorail model with two degrees of freedom, the ideal values of yaw rate and mass sideslip angle of the ESP control system are generated, and the difference is calculated with the actual measured yaw rate and the center of the mass sideslip angle. PID control is carried out to output the difference of the left and right wheel moments.

3. Design of Multi-Objective Fuzzy Continuous Damping Controller CDC suspension can guarantee the comfort of passengers under good level road conditions. The damping forces in the suspension can be changed through the control function of CDC suspension. This step can reduce the vehicle roll degree and change the vertical load distribution of the vehicle so as to aid the ESP. The vertical force on the suspension of each wheel can be expressed by the following equations, and the suspension model is shown in Figure 4. 퐵 퐵 퐹 = 퐾 (푍 − 푍 + 퐿 sin 휃 + 1 sin 휑) + 퐶 (푍 ̇ − 푤 + 퐿 푞 cos 휃 + 1 푝 cos 휑) 푣푙1 푠푙1 푢푙1 푠 1 2 푠푙1 푢푙1 1 2 (19)

퐵 퐵 퐹 = 퐾 (푍 − 푍 − 퐿 sin 휃 + 2 sin 휑) + 퐶 (푍 ̇ − 푤 − 퐿 푞 cos 휃 + 2 푝 cos 휑) 푣푙2 푠푙2 푢푙2 푠 2 2 푠푙2 푢푙2 2 2 (20) Appl. Sci. 2020, 10, x FOR PEER REVIEW 7 of 22

Appl. Sci. 2020, 10, 7923 7 of 22 Figure 2. Proportional–integral–differential (PID) control principle.

FigureFigure 3. EElectroniclectronic stability stability program program (ESP (ESP)) controller controller model. model. According According to the to monorail the monorail model model with withtwo degrees two degrees of freedom, of freedom, the ideal the idealvalues values of yaw of rate yaw and rate mass and sideslip mass sideslip angle of angle the ESP of the control ESP controlsystem systemare generated, are generated, and the and difference the diﬀerence is calculated is calculated with withthe actual the actual measured measured yaw yaw rate rate and and the the center center of ofthe the mass mass sideslip sideslip angle. angle. PID PIDcontrol control is carried is carried out to out output to output the difference the diﬀerence of the of left the and left right and wheel right wheelmoments moments..

3. Design of Multi-Objective Fuzzy Continuous Damping Controller 3. Design of Multi-Objective Fuzzy Continuous Damping Controller CDC suspension can guarantee the comfort of passengers under good level road conditions. CDC suspension can guarantee the comfort of passengers under good level road conditions. The The damping forces in the suspension can be changed through the control function of CDC suspension. damping forces in the suspension can be changed through the control function of CDC suspension. This step can reduce the vehicle roll degree and change the vertical load distribution of the vehicle so This step can reduce the vehicle roll degree and change the vertical load distribution of the vehicle so as to aid the ESP. The vertical force on the suspension of each wheel can be expressed by the following as to aid the ESP. The vertical force on the suspension of each wheel can be expressed by the following equations, and the suspension model is shown in Figure4. equations, and the suspension model is shown in Figure 4. 퐵 . 퐵 퐹 = 퐾 (푍 − 푍 + 퐿 sin 휃 +B11 sin 휑) + 퐶 (푍 ̇ − 푤 + 퐿 푞 cos 휃 + B11푝 cos 휑 F 푣푙1= K 푠푙1Z 푢푙1 Zs +푠 L 1sin θ + sin ϕ + C푠푙1(Z푢푙1 w + L 1q cos θ + p cos ϕ) (19(19)) vl1 sl1 ul1 − 1 22 sl1 ul1 − 1 2 퐵2 ̇ 퐵2 퐹푣푙2 = 퐾푠푙2(푍푢푙2 − 푍푠 − 퐿2 sin 휃 +B2 sin 휑) + 퐶푠푙2(푍푢푙. 2 − 푤 − 퐿2푞 cos 휃 +B2 푝 cos 휑) (20) F = K Z Zs L sin θ + 2 sin ϕ + C (Z w L q cos θ + 2 p cos ϕ (20) vl2 sl2 ul2 − − 2 2 sl2 ul2 − − 2 2 . B1 B1 F = K Z Zs + L sin θ sin ϕ + C (Z w + L q cos θ p cos ϕ (21) vr1 sr1 ur1 − 1 − 2 sr1 ur1 − 1 − 2 B2 . B2 Fvr = Ksr Zur Zs L sin θ sin ϕ + Csr (Zur w L q cos θ p cos ϕ (22) 2 2 2 − − 2 − 2 2 2 − − 2 − 2 where Fvij is the vertical force of each wheel; p, q are the roll angle rate and pitch angle rate of the vehicle, respectively; w is the vertical speed of the vehicle; ϕ, θ are the roll angle and pitch angle of the vehicle; Ksij,Kuij are the stiffness coefficients of spring and wheel of each wheel, respectively; Csij,Cuij are the damping coefficients of the shock absorber and wheel, respectively; Zuij is the unsprung mass Appl. Sci. 2020, 10, x FOR PEER REVIEW 8 of 22

퐵 퐵 퐹 = 퐾 (푍 − 푍 + 퐿 sin 휃 − 1 sin 휑) + 퐶 (푍 ̇ − 푤 + 퐿 푞 cos 휃 − 1 푝 cos 휑) 푣푟1 푠푟1 푢푟1 푠 1 2 푠푟1 푢푟1 1 2 (21)

퐵 퐵 퐹 = 퐾 (푍 − 푍 − 퐿 sin 휃 − 2 sin 휑) + 퐶 (푍 ̇ − 푤 − 퐿 푞 cos 휃 − 2 푝 cos 휑) 푣푟2 푠푟2 푢푟2 푠 2 2 푠푟2 푢푟2 2 2 (22) where Fvij is the vertical force of each wheel; p, q are the roll angle rate and pitch angle rate of the vehicle, respectively; w is the vertical speed of the vehicle; φ, θ are the roll angle and pitch angle of Appl. Sci. 2020, 10, 7923 8 of 22 the vehicle; Ksij, Kuij are the stiffness coefficients of spring and wheel of each wheel, respectively; Csij, Cuij are the damping coefficients of the shock absorber and wheel, respectively; Zuij is the unsprung displacementmass displacement of each of wheel; each wheelBj is the; Bj wheelis the wheel distance distance of each of axle; each subscript axle; subscripti (l or r) i (indicatesl or r) indicates the left the or rightleft or side. right side.

Figure 4. Suspension model. The unsprung mass of the suspension components and wheels is replaced Figure 4. Suspension model. The unsprung mass of the suspension components and wheels is by the equivalent unsprung mass in this model. replaced by the equivalent unsprung mass in this model. The DYC moment can be written as: The DYC moment can be written as:

Mzd ESP = (Fbl1 Fbr1)B1 + (Fbl2 Fbr2)B2 (23) 푀푧푑−퐸푆푃 = (퐹푏푙1 −− 퐹푏푟1)퐵1 + (퐹푏푙2 −− 퐹푏푟2)퐵2 (23) where Fbijbij isis the the braking braking force force of of each wheel (direction back is positive). The braking force is constrained byby the adhesionadhesion ofof thethe ground:ground:

r 2 2 퐹푏푖푗 = √(휇퐹푣푖푗)2 − 퐹푦푖푗 (24) F = µF F 2 (24) bij vij − yij where Fyij is the lateral force of each wheel. whereTheFyij previousis the lateral equations force of indicate each wheel. that CDC can change the adhesion by changing the vertical forceThe of the previous ground, equations thereby indicateincreasing that the CDC threshold can change range the of adhesionthe braking by force changing. the vertical force of the ground, thereby increasing the threshold range of the braking force. 3.1. Control Objectives and Strategies 3.1. Control Objectives and Strategies 3.1.1. Sky-Hook Control 3.1.1. Sky-Hook Control Sky-hook control assumes that two ends of the damper are not fixed between two masses; Sky-hook control assumes that two ends of the damper are not fixed between two masses; instead, instead, it assumes that one end is fixed on one mass and that the other end is fixed in the sky. In this it assumes that one end is fixed on one mass and that the other end is fixed in the sky. In this way, way, the sky-hook shock absorber does not have the opposite effect on both masses at the same time the sky-hook shock absorber does not have the opposite effect on both masses at the same time (Figure 5). (Figure5). Appl. Sci. 2020, 10, 7923 9 of 22 Appl. Sci. 2020, 10, x FOR PEER REVIEW 9 of 22

Figure 5. Abridged general view of sky-hook control. Figure 5. Abridged general view of sky-hook control. CDC is a semiactive suspension with real-time control of damping force. The power of the CDC shockCDC absorber is a semiactive must be positive suspension under with the theoryreal-time of semiactivecontrol of damping suspension. force. Given The thepower absence of the of C anDC externalshock absorber energy injection must be inpositive semiactive under suspension, the theory theof semiactive form of CDC suspension. energy transfer Given can the onlyabsence be from of an mechanicalexternal energy energy injection into thermal in semiactive energy; suspension, it can be written the form as of CDC energy transfer can only be from mechanical energy into thermal energy; it can be written as  . . . 푃  FDij CDC w Zuij = CDij CDCw w Zuij , w w Zuij > 0 퐷푖푗−퐶퐷퐶P =  − − − − − . (25) Dij CDC ̇ ̇ ̇ 퐹퐷푖푗−퐶퐷퐶−(푤 − 푍푢푖푗) = 퐶퐷푖푗−퐶퐷퐶푤(푤 − 푍푢푖푗), 푤(푤 − 푍푢푖푗) > 00, w w Zuij 0 (25) = { − ≤ 0, 푤(푤 − 푍 ̇ ) ≤ 0 . 푢푖푗 F Sign Z 0 (26) Dij Dij ≥ 퐹퐷푖푗Sign(푍퐷푖푗̇ ) ≥ 0 (26) where the function Sign(x) is described as: where the function Sign(x) is described as:   1, 1x, 푥> >0 0  SignSign(x()푥)=={ 0, 0x,=푥 =0 0 (27)(27)   −1,1x, 푥<<0 0 − . ̇ . ̇ Z푍퐷푖푗==w푤 −Z푍푢푖푗 (28)(28) Dij − uij where PDij-CDC is the power of the CDC shock absorber of each wheel, FDij-CDC is the force of the CDC where PDij-CDC is the power of the CDC shock absorber of each wheel, FDij-CDC is the force of the CDC shock absorber of each wheel, CDij-roll is the damping coefficient of the CDC shock absorber of each shock absorber of each wheel, CDij-roll is the damping coeﬃcient of the CDC shock absorber of each wheel, ZDij is the displacement of the shock absorber of each wheel, and x is a variable. wheel, ZDij is the displacement of the shock absorber of each wheel, and x is a variable. Sky-hook damping control provides a damping force opposite to the sprung mass speed in real time. Hence, it can produce minimal spring mass vibration and ensure the comfort of passengers to the maximum extent. Appl. Sci. 2020, 10, 7923 10 of 22

Sky-hook damping control provides a damping force opposite to the sprung mass speed in real time.Appl. Sci. Hence, 2020, 10 it, x can FOR produce PEER REVIEW minimal spring mass vibration and ensure the comfort of passengers10 of to22 the maximum extent. In addition addition to to the the fact fact that that CDC CD hasC has no external no external energy energy constraint, constraint, it is a ﬀ itected is affected by its own by externalits own characteristics,external characteristics, and it has and maximum it has maximum and minimum and minimum damping damping force limits force (Figure limits6). (Figure 6).

Figure 6. Range of variation of CDC damping force. 3.1.2. Antirollover Control 3.1.2. Antirollover Control When a vehicle is rolling, its lateral load transfer ratio (LTR) can be expressed by the vertical force; When a vehicle is rolling, its lateral load transfer ratio (LTR) can be expressed by the vertical it is given by [37] force; it is given by [37] F Fvr LTR = vl − (29) F퐹푣푙 +− F퐹푣푟 퐿푇푅 = vl vr (29) ( 퐹 + 퐹 F = F 푣푙 + F푣푟 vl vl1 vl2 (30) F퐹vr == F퐹vr1 + F퐹vr2 { 푣푙 푣푙1 푣푙2 퐹 = 퐹 + 퐹 (30) LTR is an indicator of vehicle rollover푣푟 risk and푣푟1 is푣푟 defined2 as the proportion of the difference betweenLTR the is an vertical indicator load of thevehicle left androllover right risk wheels and in is the defined total vehicle as the mass.proportion LTR varies of the from difference1 to 1 − regardlessbetween the of vertical the working load conditionof the left ofand the right vehicle. wheels When in the the total vehicle vehicle runs mass. in a straight LTR varies line, thefrom vertical −1 to loads1 regardless of the leftof the and working right wheels condition are the of same,the vehicle. and the Wh LTRen the is equal vehicle to 0.runs When in a thestraight vehicle line, turns, the thevertical vertical loads force of the of oneleft and wheel right transfers wheels to are the the other same, side, and and the the LTR LTR is equal value to gradually 0. When increases the vehicle or decreases.turns, the vertical When force one side of one of thewheel wheel transfers is off tothe the road, other the side, LTR and value the LTR reaches value 1 or gradually1. This increases rollover − evaluationor decreases. method When canone achieveside of the accurate wheel rolloveris off the predictionroad, the LTR when value one reaches side of 1 the or wheel−1. This leaves rollover the groundevaluation at themethod same can time. achieve accurate rollover prediction when one side of the wheel leaves the ground at the same time. 3.2. Multi-Objective Fuzzy Continuous Damping Controller 3.2. MuFuzzylti-Objective control hasFuzzy a weak Continuo dependenceus Damping on mathematicalController model, so it is widely used in uncertain or nonlinearFuzzy control control systems has a weak [38–40 dependence]. Compared on with mathematical some traditional model, control so it is techniques, widely used it hasin uncertain a certain degreeor nonlinear of intelligence control systems and has strong[38–40] robustness. Compared to with parameter some changes.traditional In control recent years, techniques, there are it manyhas a researchescertain degree on fuzzy of intelligence control in and the fieldhas stro of dampingng robustness control to [41 parameter–43]. A multi-objective changes. In recent fuzzy years, continuous there dampingare many researches controller ison constructed fuzzy control to in realize the field vehicle of damping stability control control [41 under-43]. constraintA multi-objective control fuzzy with stabilitycontinuous and damping antirollover controller control. is constructed to realize vehicle stability control under constraint controlThe with damping stability forces and of antirollover the sky-hook control. damping control are calculated under any working condition. WhenThe other damping conditions forces requiring of the control sky-hook arise, damping the force control generated are by calculated fuzzy control under is any added working to the dampingcondition. force When of sky-hookother conditions damping requiring control. control arise, the force generated by fuzzy control is addedFuzzy to the inference damping controller force of sky is mainly-hook damping divided control. into three parts: fuzzification process, fuzzy rule reasoningFuzzy process,inference and controller clarity processis mainly [44 divided]. The fuzzyinto three control parts: structure fuzzification of this process, work is fuzzy shown rule in Figurereasoning7. process, and clarity process [44]. The fuzzy control structure of this work is shown in Figure 7. Appl. Sci. 2020, 10, 7923 11 of 22 Appl. Sci. 2020, 10, x FOR PEER REVIEW 11 of 22

Figure 7. Multi-objective fuzzy continuous damping control structure. Figure 7. Multi-objective fuzzy continuous damping control structure. 3.2.1. Fuzzification Process 3.2.1. Fuzzification Process The main function of fuzzification is to process accurate variables and transform them into correspondingThe main universe function to of form fuzzification fuzzy variables. is to process accurate variables and transform them into correspondingThe inputs universe of fuzzy controlto form arefuzzy rollover variables index. (RI; i.e., the value of the LTR mentioned previously) and yawThe angle inputs rate. of The fuzzy outputs control are are the rollover damping index reference (RI; coe i.e.,ffi cients the value of each of wheel the LTR that mentioned generates apreviously) corresponding and yaw damping angle forcerate. throughThe outputs PID are control. the damping The objective reference is tocoefficients reduce the of vibration each wheel of passengersthat generates under a corresponding normal driving damping conditions force and through to achieve PID stability control. and The rollover objective control is to in reduce the case the of rollovervibration or of instability. passengers under normal driving conditions and to achieve stability and rollover control in theThe case process of rollover of fuzzification or instability. needs to complete the operation of accurate clear quantity into fuzzyThe quantity. process Theof fuzzification fuzzy universe needs of to each complete variable the shouldoperation be of defined accurate at clear first, quantity which can into realize fuzzy thequantity. transformation The fuzzy from universe physical of clarity each variable to fuzziness should by quantization be defined at factors. first, which The fuzzy can universe realize the of RItransformation is defined as from [ 1, +physical1], the fuzzyclarity universe to fuzziness of yaw by quantization angle rate is factors. defined The as [fuzzy8, +8] universe and the of fuzzy RI is − − universesdefined as of[−1, the +1], damping the fuzzy reference universe coe of ffiyawcients angle of rate each is wheeldefined are as defined[−8, +8] and as [0, the+ fuzzy1]. In universe order tos of the damping reference coefficients of each wheel are defined as [0, +1]. In order to correspond the correspond the actual variables to the normalized universe, the quantization factors are defined as NRI actual variables to the normalized universe, the quantization factors are defined as NRI = 1, Nyaw = = 1, Nyaw = 8/|rmax|,Nu = 1. 8/|rmaxTaking|, Nu = RI 1. as an example, the fuzzy language variable is used to describe the input variable RI [45]. AfterTaking fuzzifying, RI as the an fuzzyexample, set T(RI) the fuzzyof the language RI can be variable expressed is used as to describe the input variable RI [45]. After fuzzifying, the fuzzy set T(RI) of the RI can be expressed as n o ( ) 푇T(푅퐼RI) =={푁푒푔푎푡푖푣푒Negative 퐵푖푔 Big푁푒푔푎푡푖푣푒 Negative Small푆푚푎푙푙 Zero푍푒푟표 Positive푃표푠푖푡푖푣푒 Small 푆푚푎푙푙 Positive푃표푠푖푡푖푣푒 Big 퐵푖푔}

ItIt cancan bebe abbreviatedabbreviated asas 푇(푅퐼)n = {푁퐵 푁푆 푍푂 푃푆 푃퐵} o T(RI) = NB NS ZO PS PB Similarly, fuzzy sets of the yaw angle rate and the damping reference coefficient can be obtained. Similarly,The fuzzy fuzzy set T( setsr) of ofthe the yaw yaw angle angle rate rate can and be the exp dampingressed as reference coeﬃcient can be obtained. The fuzzy set T(r) of the yaw angle rate can be expressed as 푇(푟) = {푁퐵 푁푀(푁푒푔푎푡푖푣푒 푀푖푑푑푙푒) 푁푆 푍푂 푃푆 푃푀(푃표푠푖푡푖푣푒 푀푖푑푑푙푒) 푃퐵} n o TheT(r) fuzzy= NBNM set U(dc) of(Negative the damping Middle reference) NS coefficient ZO PS of PMeach(Positive wheel can Middle be )expressedPB as 푈(푑푐) = {푆푆(푉푒푟푦 푆푚푎푙푙) 푆(푆푚푎푙푙) 푀(푀푖푑푑푙푒) 퐵(퐵푖푔) 퐵퐵(푉푒푟푦 퐵푖푔)}

Appl. Sci. 2020, 10, 7923 12 of 22

The fuzzy set U(dc) of the damping reference coeﬃcient of each wheel can be expressed as n o U(dc) = SS(Very Small) S(Small) M(Middle) B(Big) BB(Very Big)

Appl. Sci. 2020, 10, x FOR PEER REVIEW 12 of 22 3.2.2. Fuzzy Rule Reasoning Process 3.2.2.The Fuzzy triangular Rule Reasoning membership Process function is easy to operate on the premise of meeting the accuracy requirementsThe triangular and it membership is more conducive function to simulation; is easy to operate the language on the variablespremise of are meeting almost allthe triangular accuracy membershiprequirements functions and it is more (NB andconducive PB of the to yawsimulation angle; rate the arelanguage trapezoidal variables membership are almost functions). all triangular membershipThe expression functions of triangular(NB and PB membership of the yaw functionangle rate is are trapezoidal membership functions). The expression of triangular membership function is  0, x a1  x a0, ≤푥 ≤ 푎1  − 1 , a < x a 푥 − 푎1 1 2 f (x, a , a , a ) =  a2 a1, 푎 <≤푥 ≤ 푎 (31) 1 2 3 푎 a−3−푎x 1 2  2 −1 , a2 < x < a3 푓(푥, 푎1, 푎2, 푎3) = 푎a − 푥a (31)  33 3  − 0,, x 푎a2 < 푥 < 푎3 푎3 − 푎3 ≥ 3 { 0, 푥 ≥ 푎3 where a2 is the vertex coordinate value of the triangle; a1 and a3 are the intersection coordinates of the leftwhere and a2 right is the sides vertex of thecoordinate triangle value and the of the bottom triangle edge,; a1 respectively. and a3 are the intersection coordinates of the left andThe right input sides and of output the triangle variables and and the the bottom rules edge of fuzzy, respectively control are. shown in Figure8. The input and output variables and the rules of fuzzy control are shown in Figure 8.

(a) (b) (c)

(d) (e) (f) (g)

Figure 8. MultiMulti-objective-objective fuzzy fuzzy continuous continuous damping damping control control model. model. (a), (b (a),– cand) are (c) diagrams are diagrams of the of membershipthe membership functions functions of theof the inputs inputs RI RI and and yaw yaw rate rate and and the the output output dampingdamping reference coefficient, coeﬃcient, respectively.respectively. ( d),–g (e)), are (f), the and diagrams (g) are the of diagrams the rule surface of the rule of each surface wheel of each (CL1 wheel/CR1 /(CL1/CR1/CL2/CR2:CL2/CR2: damping referencedamping coereferenceﬃcient coefficient of front left of/front front rightleft/front/rear leftright/rear/rear right left/rear wheel, right respectively). wheel, respectively).

The basic idea of multi-objectivemulti-objective fuzzy continuous damping control is described as follows. Under good road conditions (RI and yaw rate valuesvalues areare extremelyextremely small),small), thethe additionaladditional dampingdamping force generatedgenerated by fuzzy control should be as small as possible to preventprevent any impact on sky-hooksky-hook control. When thethe RIRI reachesreaches a a relatively relatively large large degree, degree, the the compressed compressed and and damping damping reaction reaction should should be increasedbe increased to prevent to prevent excessive excessive roll moment roll moment and thereby and produce thereby a dampingproduce forcea damping with a compensation force with a ecompensationffect to resist rollover. effect to Whenresist therollover. RI value When is relatively the RI value small is andrelatively the yaw small rate and is large, the yaw the prevention rate is large, of instabilitythe prevention should of beinstability the top consideration, should be the and top the consideration, ground adhesion and the should ground be increased adhesion by should boosting be theincreased damping by force boosting of relevant the damping wheels so force as to of aid relevant the ESP wheels in producing so as to satisfactory aid the ESP effects. in producing When the RIsatisfactory and yaw rateeffects. are largeWhen but the show RI and opposite yaw rate effects are (this large state butis show rare andopposite is generally effects caused(this state by tripped is rare rollover),and is generally a moderate caused damping by tripped force rollover), can be considered a moderate to damping build a certain force complementarycan be considered relationship to build a betweencertain complementary stability and rollover. relationship between stability and rollover. The fuzzyfuzzy implicationimplication relationrelation that that consists consists of of a a series series of of “if “...if…then then…... ”” is is composed composed of linguistic fuzzy rules, whichwhich cancan be converted into tabular fuzzy rules [4646].]. The The tabular tabular fuzzy fuzzy rules rules are are intuitive, intuitive, simple,simple, and easy to check, which are oftenoften usedused inin thethe designdesign ofof fuzzyfuzzy controllers.controllers. The fuzzy rule tables can be established as shown in the following Tables 1–4 (CL1/CR1/CL2/CR2: damping reference coefficient of front left/front right/rear left/rear right wheel, respectively).

Appl. Sci. 2020, 10, 7923 13 of 22 tables can be established as shown in the following Tables1–4 (CL1 /CR1/CL2/CR2: damping reference coeﬃcient of front left/front right/rear left/rear right wheel, respectively).

Table 1. Fuzzy rule: RI, yaw angle rate and damping reference coeﬃcient of front left wheel.

CL1 Yaw Rate NB NM NS ZO PS PM PB RI NB SS SS M BB BB BB BB NS S S B B BB BB BB ZO S S S S M B BB PS BBBBBSS PB BB BB BB BB M S S

Table 2. Fuzzy rule: RI, yaw angle rate and damping reference coeﬃcient of front right wheel.

CL1 Yaw Rate NB NM NS ZO PS PM PB RI NB SS SS M BB BB BB BB NS SSBBBBB ZO BB B M S S S S PS BB BB BB B B S S PB BB BB BB BB M S S

Table 3. Fuzzy rule: RI, yaw angle rate and damping reference coeﬃcient of rear left wheel.

CL2 Yaw Rate NB NM NS ZO PS PM PB RI NB S S M BB BB BB BB NS S S B B BB BB BB ZO SS SS SS SS M B BB PS BBBBBSS PB BB BB BB BB M S S

Table 4. Fuzzy rule: RI, yaw angle rate and damping reference coeﬃcient of rear right wheel.

CL2 Yaw Rate NB NM NS ZO PS PM PB RI NB S S M BB BB BB BB NS SSBBBBB ZO BB B M SS SS SS SS PS BB BB BB B B S S PB BB BB BB BB M S S

3.2.3. Clarity Process After the fuzzy logic reasoning, the output conclusion is a fuzzy variable which cannot directly drive the actuator to control and needs to be changed into a clear variable. The process of transforming fuzzy variable into clear variable is called clarity. The method of calculating the curve of membership Appl. Sci. 2020, 10, 7923 14 of 22 function of fuzzy set and the center of area surrounded by abscissa is centroid method [42]. The solution x0 and the mathematical expression of centroid method is as follows: R x f (x, a1, a2, a3)dx x0 = R (32) f (x, a1, a2, a3)dx

The quantization factors are deﬁned as Nu = 1. The actual damping forces of suspension are calculated by the range of CDC and the damping reference coeﬃcients of fuzzy control.

4. Simulation and Result Analysis A real vehicle model is constructed in the software CarSim, and a control model is constructed in MATLAB/Simulink. The simulation analysis is performed through cosimulation in MATLAB/Simulink and CarSim. Passive suspension without any control (abbreviated as NC) and passive suspension with ESP (abbreviated as P-ESP) are compared with the multi-objective fuzzy continuous damping control strategy with ESP (abbreviated as MFCDC-ESP) to illustrate the eﬀect. The main parameters of vehicle model and control model are shown in Tables5 and6, respectively.

Table 5. Main parameters of vehicle model.

Vehicle Parameters Value Unit

mt 900 kg 2 Iz 708 Kg m L1 970 mm L2 903 mm g 9.8 m/s2 Kα1 144,000 N/rad Kα2 312,000 N/rad

Table 6. Main parameters of PID control model of ESP.

Vehicle Parameters Value P-yaw angle rate 150 I-yaw angle rate 0.05 D-yaw angle rate 2 P-sideslip angle 10 I-sideslip angle 0.1 D-sideslip angle 2

The real vehicle selected for the funding project is an A0 class high centroid vehicle. Although it has a high centroid, its mass is not large, and it is more prone to sideslip than to rollover in the absence of an ESC system when driving. Therefore, the selection of its parameters is the critical condition obtained via repeated simulations, and it can show the effect of vehicle control under the limited state. To verify the MFCDC-ESP effect of stability and antirollover, this work compares and simulates three test conditions, namely, sine wave steer input test, DLC test, and fishhook test.

4.1. Sine Wave Steer Input Test The sine wave steer input test is the most frequently used condition for verifying driver control stability. The basic parameters of the simulation analysis of the sine wave steer input test are shown in Table7. Appl. Sci. 2020, 10, 7923 15 of 22

Table 7. Main parameters of sine wave steer input test.

Test Parameters Value Unit Longitude target speed 50 km/h Friction coeﬃcient 0.16 - Steer input angle peak value 90 deg Simulation time 20 second Appl. Sci. 2020, 10, x FOR PEER REVIEW 15 of 22

The results of simulation are shown in FigureFigure9 9..

(a) (b)

Figure 9. Steering angle, vehicle longitudinal speed, vehicle sideslip angle, and yaw angle rate of sine Figure 9. Steering angle, vehicle longitudinal speed, vehicle sideslip angle, and yaw angle rate of sine wave steer input test. The result of NC is expressed with a black dotted line; the result of P-ESP is wave steer input test. The result of NC is expressed with a black dotted line; the result of P-ESP is expressedexpressed with a blue chain line; the result of MFCDC MFCDC-ESP-ESP i iss expressed with a red solid line.line. ( a) is is the the steersteer angle inputinput ofof thethe simulation. simulation (.b (–bd),) ( arec), theand comparisons (d) are the comparisons of the vehicle of longitudinal the vehicle speed, longitudinal vehicle speed,sideslip vehicle angle, sideslip and yaw angle, angle and rate, yaw respectively. angle rate, respectively.

As shown inin FigureFigure9 ,9 the, the longitudinal longitudinal speed speed of NCof NC decreases decreases rapidly rapidly at 2 s,at and 2 s, the and sideslip the sideslip angle angleof the of mass the center mass andcenter the and yaw the angle yaw rate angle increase rate atincrease the same at time.the same Then, time. violent Then, vibration violent occurs, vibration and occurs,the frequency and the of frequency the oscillation of the cycle oscillation increases. cycle In increases. terms of vehicle In terms speed, of vehicle the speed speed, of the the speed vehicle of with the vehicleP-ESP and with MFCDC-ESP P-ESP and MFCDC is only slightly-ESP is reducedonly slightly at the reduced beginning at duethe beginning to the respective due to e fftheects respective of lateral effectsforce and of lateral braking force force; and then braking the vehicleforce; then maintains the vehicle a stable maintains target speed.a stable With target regard speed. to With the sideslip regard toangle the sideslip of the mass angle center of the and mass yaw center rate, and the rangeyaw rate, of variations the range isof relatively variations small. is relatively These comparisonssmall. These comparisonsshow that the show vehicle that without the vehicle control without loses stability control and loses that stability the vehicles and that with the P-ESP vehicles and MFCDC-ESP with P-ESP andpresent MFCDC a good-ESP control present eff ecta good on vehicle control stability. effect on vehicle stability. InIn illustrating illustrating the the control control effect effect of MFCDC MFCDC-ESP,-ESP, this work shows the vertical force and damping damping forceforce in in Figure 10 and the simulation data results in Table8 8. .

(a) (b) Appl. Sci. 2020, 10, x FOR PEER REVIEW 15 of 22

The results of simulation are shown in Figure 9.

(a) (b)

Figure 9. Steering angle, vehicle longitudinal speed, vehicle sideslip angle, and yaw angle rate of sine wave steer input test. The result of NC is expressed with a black dotted line; the result of P-ESP is expressed with a blue chain line; the result of MFCDC-ESP is expressed with a red solid line. (a) is the steer angle input of the simulation. (b), (c), and (d) are the comparisons of the vehicle longitudinal speed, vehicle sideslip angle, and yaw angle rate, respectively.

As shown in Figure 9, the longitudinal speed of NC decreases rapidly at 2 s, and the sideslip angle of the mass center and the yaw angle rate increase at the same time. Then, violent vibration occurs, and the frequency of the oscillation cycle increases. In terms of vehicle speed, the speed of the vehicle with P-ESP and MFCDC-ESP is only slightly reduced at the beginning due to the respective effects of lateral force and braking force; then the vehicle maintains a stable target speed. With regard to the sideslip angle of the mass center and yaw rate, the range of variations is relatively small. These comparisons show that the vehicle without control loses stability and that the vehicles with P-ESP and MFCDC-ESP present a good control effect on vehicle stability. Appl. Sci.In illustrating2020, 10, 7923 the control effect of MFCDC-ESP, this work shows the vertical force and damping16 of 22 force in Figure 10 and the simulation data results in Table 8.

Appl. Sci. 2020, 10, x FOR PEER(a )REVIEW (b) 16 of 22

Figure 10.10. Damping forcesforces andand vertical vertical forces forces of of each each wheel wheel of of the the sine sine wave wave steer steer input input test test with with P-ESP P- andESP MFCDC-ESP.and MFCDC-ESP. The detailsThe details are available are available in the in legend. the legend. L denotes L denotes left, left, R denotes R denotes right, right 1 denotes, 1 denotes the frontthe front axle, axle, and 2and denotes 2 denotes the rear the axle.rear (axle.a) is the(a) is comparison the comparison of the dampingof the damping forces offorces the front of the wheels. front (wheels.b) is the (b comparison) is the comparison of the damping of the damping forces of forc thees rear of wheels.the rear (wheels.c) is the (c comparison) is the comparison of the vertical of the forcesvertical of forces the front of the wheels. front wheels. (d) is the (d comparison) is the comparison of the vertical of the forcesvertical of forces the rear of wheels.the rear wheels.

Table 8. MMainain comparison results for the sine wave steer input test (P-ESP,(P-ESP, MFCDC-ESP).MFCDC-ESP). 1 Test ResultsTest (Unit)Results (Unit) P-ESPP-ESP MFCDC MFCDC-ESP-ESP PercentagePercentage Point 1 Point Yaw angleYaw rate angle (rad/ s)rate (rad/s) averageaverage 0.001353−0.001353 −0.000086050.00008605 −93.64% 93.64% − − − maximummaximum 26942694 27492749 2.04% 2.04% Vertical forceVertical L1 (N) force L1 (N) minimumminimum 19801980 19801980 0% 0% maximum 2758 2782 0.87% Vertical force R1 (N) maximum 2758 2782 0.87% Vertical force R1 (N) minimum 1981 1966 −0.76% minimum 1981 1966 0.76% maximum 2103 2170 3.19% − Vertical force L2 (N) maximum 2103 2170 3.19% Vertical force L2 (N) minimum 1268 1239 −2.29% minimum 1268 1239 2.29% maximum 2091 2150 2.82% − Vertical force R2 (N) maximum 2091 2150 2.82% Vertical force R2 (N) minimum 1357 1332 −1.84% minimum 1357 1332 1.84% maximum 245.7 176.2 −28.29% − maximum 245.7 176.2 28.29% Damping force L1 (N) minimum −299.3 −230.3 −23.05% − Damping force L1 (N) minimum 299.3 230.3 23.05% standard deviation − 93.77 65.56− −30.08% − standard deviation 93.77 65.56 30.08% maximum 253.7 223.8 −11.79% − maximum 253.7 223.8 11.79% Damping force R1 (N) minimum −315.3 −407.8 29.34% − Damping force R1 (N) minimum 315.3 407.8 29.34% standard deviation − 98.65 70.42− −28.62% standard deviationmaximum 98.65239.3 188.8 70.42 −21.10% 28.62% − Damping force L2 (N)maximum minimum 239.3−289.8 −219.2 188.8 −24.36% 21.10% − Damping force L2 (N) minimumstandard deviation 289.893.87 65.12 219.2 −30.63% 24.36% − − − standard deviationmaximum 93.87229.5 192.2 65.12 −16.25% 30.63% − Damping force R2 (N)maximum minimum 229.5−273.4 −495.1 192.2 81.09% 16.25% − Damping force R2 (N) minimumstandard deviation 273.491.31 65.02 495.1−28.79% 81.09% − − 1 standard deviation 91.31 65.02 28.79% Percentage point = [(MFCDC-ESP’s data)-(P-ESP’s data)]/(P-ESP’s data) × 100%.− 1 Percentage point = [(MFCDC-ESP’s data)-(P-ESP’s data)]/(P-ESP’s data) 100%. × The average value of MFCDC-ESP’s yaw angle rate is far less than that of P-ESP in the whole controlThe action average (Figure value 10 of MFCDC-ESP’sand Table 8). Hence, yaw angle the rate whole is far vehicle less than is in that a ofrelatively P-ESP in stable the whole state control in the actionwhole (Figure process 10 of and motion. Table8 ). As Hence, for the the vertical whole vehicle force, isthe in maximum a relatively vertical stable state force in the of eachwhole wheel process of ofMFCDC motion.-ESP As is for greater the vertical than that force, of the P-ESP maximum while the vertical minimum force of value each of wheel the former of MFCDC-ESP is less than is greaterthat of the latter; the difference for the rear axle wheels is even greater. These results show that MFCDC-ESP can maximize the rear axle, which lacks lateral force, for stability control. From the point of view of the damping force of each wheel, the damping forces of the front and rear axles on the right side of MFCDC-ESP increase suddenly at about 2 s. As shown in Figure 8, the moment at 2 s is the critical moment of lateral stability control (the vehicle with NC suddenly becomes unstable around this moment). The compensation effect of MFCDC-ESP also makes the vehicle establish good dynamic control. The standard deviation of the damping force of MFCDC-ESP is much smaller than that of P- ESP because of the sky-hook control effect after the vehicle enters the stationary period. This result also shows that the force on the vehicle body is reduced and that the passengers experience enhanced comfort. In practical application, MFCDC-ESP can automatically identify and switch the state between stability and comfort in real time due to its fuzzy control characteristics, so as to achieve the best driving effect. Appl. Sci. 2020, 10, 7923 17 of 22 than that of P-ESP while the minimum value of the former is less than that of the latter; the difference for the rear axle wheels is even greater. These results show that MFCDC-ESP can maximize the rear axle, which lacks lateral force, for stability control. From the point of view of the damping force of each wheel, the damping forces of the front and rear axles on the right side of MFCDC-ESP increase suddenly at about 2 s. As shown in Figure8, the moment at 2 s is the critical moment of lateral stability control (the vehicle with NC suddenly becomes unstable around this moment). The compensation effect of MFCDC-ESP also makes the vehicle establish good dynamic control. The standard deviation of the damping force of MFCDC-ESP is much smaller than that of P-ESP because of the sky-hook control effect after the vehicle enters the stationary period. This result also shows that the force on the vehicle body is reduced and that the passengers experience enhanced comfort. In practical application, MFCDC-ESP can automatically identify and switch the state between stability and comfort in real time due to its fuzzy control characteristics, so as to achieve the best driving effect.

4.2. DLC Test The DLC test is often used to measure the function eﬀect of ESP. The DLC test in this work only compares P-ESP with MFCDC-ESP to show the superiority of the latter, especially given the result of the sine wave input test emphasizing that a vehicle without control can easily lose stability. The basic parameters of the DLC test simulation analysis are shown in Table9.

Table 9. Main parameters of DLC test.

Test Parameters Value Unit Longitude target speed 120 km/h Friction coeﬃcient 0.6 - Steer input angle peak value 90 deg

The results of the simulation are shown in Figure 11. The simulation curve shows that the vehicle maintains good stability under the two control strategies because of the ESP. In comparison with those of P-ESP, the roll angle rate of MFCDC-ESP is smaller, and the comfort felt by passengers is better. Given the influence of the sky-hook control of MFCDC-ESP, the damping force of MFCDC-ESP is generally less than that of P-ESP when driving in a stable state. However, at 2.5 and 5.7 s, the damping force of MFCDC-ESP suddenly increases. Combined with the steering angle, these two positions are the critical moment for the change of the steering angle and are the most vulnerable periods for the vehicle to lose stability (this point of view can be also clearly reflected in the previous sine wave steer input test). Moreover, the vertical forces are redistributed. The rear axle with great braking potential undertakes heavy load distribution without lateral force; the same effect is desired for vehicle control. Therefore, the control strategy of MFCDC-ESP can change the distribution of vertical forces and quickly compensate for the occurrence of instability; it even exerts a certain predictive effect. In addition, CDC plays an important role in driving stability with auxiliary effects. The simulation data are reported in Table 10.

Table 10. Main comparison results of DLC test (P-ESP, MFCDC-ESP).

Test Results (Unit) P-ESP MFCDC-ESP Percentage Point absolute maximum 0.2266 0.2059 9.14% Yaw angle rate (rad/s) − standard deviation 0.1199 0.1158 3.42% − absolute maximum 0.3896 0.357 8.37% Roll angle rate (rad/s) − standard deviation 0.09699 0.09082 6.36% − Vertical force L1 (N) absolute maximum 3344 3368 0.72% Vertical force R1 (N) absolute maximum 3518 3508 0.28% − Vertical force L2 (N) absolute maximum 2960 3018 1.96% Vertical force R2 (N) absolute maximum 3135 3172 1.18% Appl. Sci. 2020, 10, x FOR PEER REVIEW 17 of 22

4.2. DLC Test The DLC test is often used to measure the function effect of ESP. The DLC test in this work only compares P-ESP with MFCDC-ESP to show the superiority of the latter, especially given the result of the sine wave input test emphasizing that a vehicle without control can easily lose stability. The basic parameters of the DLC test simulation analysis are shown in Table 9.

Table 9. Main parameters of DLC test.

Test Parameters Value Unit Longitude target speed 120 km/h Friction coefficient 0.6 - Steer input angle peak value 90 deg Appl. Sci. 2020, 10, 7923 18 of 22 The results of the simulation are shown in Figure 11.

(a) (b)

(e) (f)

(g) (h)

Figure 11. Simulation results of DLC test. (a–d) are the comparison results of the steering angle, vehicle longitudinal speed, yaw angle rate, and roll angle rate, respectively. The damping forces and vertical forces of each wheel of the DLC test with P-ESP and MFCDC-ESP. (e) is the comparison of the damping forces of the front wheels. (f) is the comparison of the damping forces of the rear wheels. (g) is the comparison of the vertical forces of the front wheels. (h) is the comparison of the vertical forces of the rear wheels. The details are available in the legend. L denotes left, R denotes right, 1 denotes the front axle, and 2 denotes the rear axle.

4.3. Fishhook Test The fishhook test is used to verify the effect of MFCDC-ESP on rollover. The basic parameters of the fishhook test simulation analysis are shown in Table 11. Appl. Sci. 2020, 10, 7923 19 of 22

Table 11. Main parameters of ﬁshhook test.

Test Parameters Value Unit Longitude target speed 80 km/h Friction coeﬃcient 0.85 - Steer input angle peak value 294 deg Simulation time 20 second Road condition straight -

The main parameters of the ﬁshhook test simulation are shown in Figure 12. Appl. Sci. 2020, 10, x FOR PEER REVIEW 19 of 22

(a) (b)

(e) (f)

Figure 12.12. MainMain resultsresults of of ﬁshhook fishhook test. test. The The result result of NCof NC is expressed is expressed with with a black a black dotted dotted line; the line; result the ofresult P-ESP of isP- expressedESP is expressed with a bluewith chain a blue line; chain the line; result the of result MFCDC-ESP of MFCDC is expressed-ESP is expressed with a red with solid a line.red (solida–f) areline. the (a comparison), (b), (c), (d for), (e LTR,), and vehicle (f) are sideslip the comparison angle, roll for angle, LTR, roll vehicle angle sideslip rate, yaw angle, angle roll rate, angle, and lateralroll angle acceleration, rate, yaw respectively.angle rate, and lateral acceleration, respectively.

The LTRLTR of of the the NC NC vehicle vehicle has has a step a step at 3 at s, with3 s, with the roll the angle roll angle and roll and angle roll rateangle increasing rate increasing sharply andsharply resulting and resulting in serious in rollover.serious rollover. Despite Despite the many the cases many in cases which in one which side one wheel side is wheel oﬀ the is ground off the (LTRground reaches (LTR 1 reaches or 1), no 1 or rollover −1), no occurs rollover with occurs P-ESP. with The main P-ESP. reason The is main that reason the lateral is that acceleration the lateral of − theacceleration vehicle is of reduced the vehicle due tois the reduced action due of the to ESP. the Fromaction the of yaw the ESP. angle From rate and the sideslipyaw angle angle rate of and the masssideslip center, angle the of wholethe mass vehicle center, vibrates the whole from vehicle 6 s to 8 vibrates s. Although from the 6 s yaw to 8 angles. Although rate is controlledthe yaw angle in a smallrate is rangecontrolled after in 8 s, a thesmall continuous range after oscillation 8 s, the continuous of the side oscillation slip angle of the side mass slip center angle indicates of the mass that thecenter vehicle indicates has entered that the an vehicle unstable has state. entered The vehiclean unstable with state. MFCDC-ESP The vehicle has nowith instability MFCDC phenomenon-ESP has no ininstability the whole phenomenon movement process, in the and whole all parameters movement are proc withiness, and the controllableall parameters range. are Because within the controllable range. Because the damping forces of CDC are controlled by electrical signal, the sudden changes of damping forces often lead to the sudden change of control current. The damping force current signal can be collected and identified by appropriate methods, and the driver can be given a corresponding real-time rollover warning, which improves the rollover prediction effect.

5. Conclusions The theoretical analysis and test results reveal that the vehicle with ESP can realize control over handling stability and that CDC can greatly improve the comfort of passengers under the action of sky-hook control. MFCDC-ESP combines these two functions organically and optimizes ride comfort. The control effect of the ESP can be improved, and extremely dangerous conditions can be effectively prevented when rollover occurs with MFCDC-ESP through the redistribution of the vertical force. MFCDC-ESP is superior to P-ESP in terms of ride comfort from the quantitative point of view; it also has an obviously different damping force and roll angle. In terms of stability control, MFCDC-ESP can effectively use the rear axle for load redistribution; the rear axle has a certain auxiliary lifting effect on the ESP. MFCDC-ESP has a significant advantage in preventing rollover (because P-ESP has no control strategy to inhibit rolling). Moreover, MFCDC-ESP has strong real-time sensitivity to the control state of the damping force of each wheel. MFCDC-ESP can maintain the optimized distribution of damping forces through its own compensation in the case of possible instability. it can even predict the critical stable state to some extent. In general, MFCDC-ESP vehicles can ensure the comfort of passengers under good driving conditions while having strong adaptability and control effect under various extreme working conditions. MFCDC-ESP perfectly meets the requirements of comfort, handling stability, and rollover prevention. In practical engineering simulation and application, fuzzy control can use human expert control experience to realize process automatic control for nonlinear and complex objects. Fuzzy control has a certain ability to judge and deal with uncertainty and it can realize the design of vehicle control systems which have a certain degree of intelligence.

Author Contributions: X.Z. conceived the control method, completed the modeling, analyzed the results, and finished the manuscript. C.S. provided overall guidance for the study and managed the project. S.S. and D.W. collected the data, reviewed, and revised the paper. J.C. and F.X. give crucial suggestions about fuzzy continuous damping force control strategy and perform simulation experiments. S.P. and C.Q. put forward the idea and debugged the model in MATLAB/Simulink. All authors have read and agreed to the published version of the manuscript. Funding: This research is supported by the Science and Technology Development Plan Program of Jilin Province (Grant No. 20200401112GX) and Industry Independent Innovation Ability Special Fund Project of Jilin Province (Grant No. 2020C021-3). Conflicts of Interest: The authors declare no conflict of interest.

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