Modelling and Analysis of Automobile Vibration System Based on Fuzzy Theory under Different Road Excitation Information

A fuzzy increment controller is designed aimed at the vibration system of automobile active suspension with seven degrees of freedom (DOF). For decreasing vibration, an active control force is acquired by created Proportion-Integration-Differentiation (PID) controller.The controller’s parameters are adjusted by a fuzzy increment controller with self-modifying parameters functions, which adopts the deviation and its rate of change of the body’s vertical vibration velocity and the desired value in the position of the front and rear suspension as the input variables based on 49 fuzzy control rules. Adopting Simulink, the fuzzy increment controller is validated under different road excitation, such as thewhite noise input with four-wheel correlation in time-domain, the sinusoidal input, and the pulse input of C-grade road surface. The simulation results show that the proposed controller can reduce obviously the vehicle vibration compared to other independent control types in performance indexes, such as, the root mean square value of the body’s vertical vibration acceleration, pitching, and rolling angular acceleration.


Introduction
It is generally known that automobile active suspension system can produce and adjust the active control force in time to restrain the vehicle body's vibration for improving the ride comfort according to the road surface excitation.Recently, there have been large research achievements in the automobile suspension system, such as the optimization control theory, the PID theory, the theory of variable structure control, and the skyhook damping theory which have been applied generally to the vibration control system.Zhang et al. proposed a semiactive suspension controller with magnetorheological dampers to realize independent control based on modified skyhook damping scheme of quartervehicle subsuspension system in the full vehicle [1].Ren et al. presented an adaptive hybrid control algorithm combined with the ground-hook and skyhook control strategies based on quarter car model of semiactive suspension and designed an unscented Kalman filter to estimate the suspension states [2].Singh and Aggarwal designed a hybrid Fuzzy-PID controller to evaluate the passenger ride comfort based on a semiactive quarter car model having MR shock absorber [3].Du et al. presented an approach in designing a robust controller for half-car model active suspensions considering changes in vehicle inertial properties, such as the suspension deflection limitation and the controller saturation problem [4].Owing to the randomness of road surface roughness and the nonlinearity and uncertainty of vehicle system, the abovementioned method has its disadvantages and adaptability [5][6][7].It has been known that the intelligent control theory has the ability of logical reasoning and decision-making, and it is best suited to solve the complexity and uncertainty system [8][9][10].Therefore, a variety of intelligent control methods has been widely used like the fuzzy control, neural network, genetic algorithm, and so on [11][12][13][14].Huang et al. designed a fuzzy controller for converting chaos into periodic motion of stable performance based on the neuralnet tyre model [10].Riaz et al. presented adaptive Neuro Fuzzy Takagi-Sugeno-Kang control strategies for the vehicle active suspension to improve the ride quality and vehicle stability [13].Zirkohi and Lin proposed an interval type-2 fuzzy-neural network approach incorporating the Lyapunov design approach and SMC method to improve controller robustness [14].Balamurugan   to generate the desired damping force and adjust the voltage level to MR damper based on the quarter car vehicle model of semiactive suspension [15].Zhang et al. designed a semiactive controller based on the inverse model and sliding mode control strategies for the quarter-vehicle suspension with the magnetorheological damper [16].Bououden et al. investigated the problem of time-varying delays and input constraints for active suspension systems with a quartervehicle model based on the Lyapunov-Krasovskii method [17].From the analysis results on automobile active suspension research, we learned that much of the relevant literature focus on a quarter-or a half-car model that cannot fully reflect or evaluate riding comfort of a whole vehicle [18][19][20][21][22]. So, it is our goal to research the time-domain response of the automobile active suspension with seven DOF for decreasing the body vibration considering the pitching and rolling motion under the condition of the white noise input with four wheels correlated effects and different road excitation.At last, the submodule of fuzzy increment controller and the random input model have been given, and the effectiveness of proposed controller has been validated adopting Simulink.

Vehicle Vibration Model of the Active Suspension with Seven DOF
2.1.Dynamic Model of Active Suspension. Figure 1 describes a vehicle vibration model of the active suspension with seven DOF, including the body's vertical motion, pitching and rolling motion, and four vertical motions of the unsprung mass.
In Figure 1, M s represents the sprung mass; M 1 , M 2 , M 4 , and M 3 represent the mass between the front and rear wheels on left and right sides, respectively.z s denotes the vertical displacement of the body center of mass (7);  and  denote the pitch and roll angle displacement.z s1 , z s2 , z s4 , and z s3 represent the body vertical displacement in position of the front and rear suspension on left and right sides, respectively.z 1 , z 2 , z 4 , and z 3 represent the vertical displacement in position of the front and rear wheel on left and right sides, respectively.z q1 , z q2 , z q4 , and z q3 represent the road irregularity excitation, respectively.k 1 , k 2 , k 4 , and k 3 represent the elastic stiffness of the front and the rear suspension.k t1 , k t2 , k t4 , and k t3 represent the elastic stiffness of the front and rear wheel on left and right sides, respectively.c 1 , c 2 , c 4 , and c 3 represent the damping of the front and rear suspension, respectively.f 1 , f 2 , f 4 , and f 3 represent the control force from the active controller in position of the front and rear suspension on left and right sides, respectively.l a and l b denote the distance of front and rear axle to the body center of mass.  ,   denote the distance between the left and right wheel.V represents the vehicle's driving direction..According to the vehicle vibration model of the active suspension, the dynamic differential equations with seven DOF are derived as follows.

Dynamic Differential Equation
The dynamic differential equation of the body's vertical motion is listed as follows: The dynamic differential equation of the body's pitch motion is listed as follows: The dynamic differential equation of the body's roll motion is listed as follows: The dynamic differential equation of the vertical motion of four unsprung mass is listed as shown in equation (3).
where   from (1) to ( 4) is expressed as Here, ż  is listed as follows: In the above expression, z  , θ , and φ represent the vertical vibration and the pitch and roll angle acceleration of the body center of mass, respectively.z 1 , z 2 , z 4 , and z 3 represent the

Design of the Fuzzy Increment Controller of Automobile Active Suspension
3.1.Fuzzy Controller Design.The fuzzy increment controller includes two parts, the parameter self-adapting fuzzy controller and the PID controller, where the fuzzy controller adopts the deviation and its rate of change of the body vertical vibration velocity ( ż  ) and the desired value ( ż 0 ) in the position of the front and rear suspension is the input variables; the increments (ΔK p , ΔK i , and ΔK d ) of the PID controller parameters (K p , K i , and K d ) are derived according to a fuzzy control rule.The working principle of fuzzy increment controller is as shown in Figure 2.
where k e and k ec represent the conversion factor from the exact value to the fuzzy variable. p ,  i , and  d represent the modified coefficient of the increments value of the PID controller parameters.functions of all the variables are similar and as shown in Figure 3.

Construct Rule Base. Table 1 describes the relation between the input and output variable
, where the fuzzy control rules are characterized by If-Then statements involving fuzzy linguistic variables.For example, the generic form of the fuzzy rules in the case of MIMO is as follows: For Table 1, if the deviation of the body vertical vibration velocity ( ż  ) is NB, and the change rate of the velocity deviation ( z  ) is NB, then the output variable ΔK p is PB, ΔK i is NB, and ΔK d is PS.

Defuzzification.
The defuzzifier is utilized to yield a nonfuzzy decision or control action from an inferred fuzzy control action by the fuzzy reasoner.The method on the center of Gravity is used to calculate the defuzzified output as follows:  = ∑  =1     / ∑  =1   , where N is the number of fuzzy rules, u i is the output of fuzzy rule base, and   is the weight of u i .

PID Controller Design.
Adopting the deviation of vertical vibration velocity ( ż  ) and its desired value ( ż 0 ) in the position of the front and rear suspension as the input variables of the PID controller in the Figure 2, an active control force (  ( = 1, . . ., 4)) is obtained as shown in

Modelling Road Input Excitation with Four-Wheel Correlation in Time-Domain.
In order to validate the effectiveness of fuzzy increment controller, the white noise random input with four wheels correlated in time-domain, the sinusoidal input, and the pulse input of C-grade road surface are adopted, respectively.Here, the white noise random input on the left front wheel is shown as where  0 denotes a referenced spatial frequency and the value equals 0.1 (m −1 ).  ( 0 ) equals 256 × 10 −6 (m 2 /m −1 ) and it denotes a road roughness coefficient of C-grade road surface.() represents a white noise with mean zero.V denotes the vehicle speed (V = 20 m/s) and  0 equals 0.01.Considering the delay effect, the random input on the left rear wheel is denoted as Ż4 () = Ż1 ( − ).Here,  = /V and L represents wheelbase.So, the random input on the left rear wheel is expressed as According to the early literatures, the random input on the right front wheel is derived as Also, the random input on the right rear wheel is denoted as Ż3 () = Ż2 ( − ).Therefore, the random input on the right rear wheel is calculated as where In formula (10) and formula (11), B denotes the lateral distance between the left and right wheel,  represents the longitudinal distance between the front and rear wheel (i.e., the wheelbase), and V is car speed at the moment.

Simulation Module Design.
In order to validate the feasibility of the fuzzy increment controller, car suspension parameters are as given in Table 2.
Using MATLAB/Simulink, the simulation modular of active suspension with seven DOF is designed.According to formula (9) to formula (12), a road excitation simulation module of white noise in time-domain is designed as shown in Figure 4 taking into account the delay and correlation characteristic between the wheels.Here, the submodule of fuzzy increment controller in a wheel side is described as in Figure 5.The simulation submodule of the other three controllers is similar to Figure 5. 6-8 show the acceleration's comparison results of the body vertical vibration, pitching, and rolling motion under the condition of white noise road surface excitation adopting different control modes.

Simulation Analysis. Figures
As you can see from Figures 6-8, these indexes values on the vertical vibration, pitching, and rolling angular acceleration of the vehicle body have been improved adopting the Fuzzy-PID code compared to other modes under the condition of white noise road surface excitation.In other words, the fuzzy increment controller can obviously reduce the body's vibration because the acceleration peak amplitude above indexes is far less than other modes.show the acceleration's comparison results of the body vertical vibration, pitching, and rolling motion under the condition of pulse excitation adopting different control mode.
From Figures 9-11 we can see that the vibration amplitude on vehicle body's acceleration is also far less than other modes adopting the Fuzzy-PID code under the condition of pulse excitation.
Figures 12-14 show the acceleration's comparison results of the body vertical vibration, pitching, and rolling motion under the condition of sinusoidal excitation adopting different control mode.As you can see from Figures 6-14, the fuzzy increment controller (i.e., the Fuzzy-PID code name) has good control effect in improving the body vertical vibration acceleration, pitching, and rolling angular acceleration comparing with the independent types of controls, such as the passive suspension, the PID control, and the fuzzy control.
In addition to comparison curves, the root mean square value of body's acceleration is counted regarding four kinds of control mode under the condition of different speed as shown in Table 3.
As seen from Table 3, the root mean square value of the body's acceleration, the pitching, and rolling angular acceleration adopting the Fuzzy-PID mode reduces 37.5 percent, 28.9 percent, and 26.2 percent compared with the passive mode under the condition of white noise road surface excitation.The root mean square values of indexes mentioned You can also see that the root mean square values of indexes mentioned above adopting the Fuzzy-PID mode have been improved obviously compared to other modes at 30 meters per second.Through analysis we can see that the designed controller can reduce obviously the vehicle vibration and has good control effect compared to the independent control type.

Conclusions
(1) The vibration model with seven degrees of freedom of automobile active suspension is derived and four fuzzy increment controllers are designed based on 49 rules for adapting the different real-time road input information.
(2) The white noise input model with four wheels correlated in characteristics of time-domain is designed that is closer to the actual road information input.
(3) Using MATLAB/Simulink, the simulation module of active suspension with seven DOF has been built and the fuzzy increment controller is validated taking the white noise excitation with four-wheel correlation in time-domain, the sinusoidal excitation, and the pulse excitation of C-grade road surface as the road input.The simulation results show

Figure 1 :
Figure 1: Vehicle model of the active suspension with seven DOF.

Figure 2 :
Figure 2: Principle diagram of the Fuzzy-PID controller.

Figure 4 :
Figure 4: Random input model with four-wheel correlation in time-domain.

Figure 13 :Figure 14 :
Figure 13: Comparison result of the pitching angular acceleration.

Table 1 :
Fuzzy control rule of the PID increment parameters.
p , K i , and K d represent the gain coefficient of the proportion, integral, and differential.The () denotes a deviation between the body vertical vibration velocity ( ż  ) and its desired value ( ż 0 is set to zero).Here,   = 3000,   = 20, and   = 30.

Table 2 :
Suspension structure parameters of a test car.