Jinwei Sun, Mingming Dong, Zhiguo Wang, Baoyu Li and Liang Gu,

(1.School of Mechanical Engineering, Beijing Institute of Technology, Beijing 100081, China; 2.Beijing North Vehicle Group Corporation, Beijing 100072, China)

Abstract: Vehicle riding comfort and handling stability are directly affected by suspension performance. A novel dual chamber hydro-pneumatic (DCHP) suspension system is developed in this paper. Based on the structural analysis of the DCHP suspension, an equivalent suspension model is proposed for the control purpose. A cuckoo search (CS) based fuzzy PID controller is proposed for the control of the DCHP suspension system. The proposed controller combines the advantage of fuzzy logic and PID controller, and CS algorithm is used to regulate the membership functions and PID parameters. Compared with tradition LQR controller and passive suspension system, the CSFPID controller can reduce the sprung mass acceleration, and at the same time with no deterioration of tire deflection.

Key words: vehicle dynamics; dual chamber hydro-pneumatic (DCHP) suspension; cuckoo search (CS); fuzzy PID

Well-designed suspension systems can provide better riding comfort and at the same time keep the tire in contact with road surface[1-2]. Hydro-pneumatic suspension systems use an inert gas as an elastic medium. Compared with the traditional suspension system, the nonlinear stiffness characteristic of the hydro-pneumatic system can enhance the vehicle ride comfort and handling stability[3]. The dual chamber hydro-pneumatic (DCHP) suspension is an improvement of the single chamber suspension, and its equivalent stiffness and damping are the functions of the excitation frequency[4]. Eltawwab[5]described the structure of the double chamber hydro-pneumatic suspension system and established a simplified linear model. The influence of parameters on the performance of the suspension is analyzed by using a series of stiffness and damping parameters, and the root mean square (RMS) value of the sprung mass acceleration. Suspension deflection and tire deflection are taken as the indexes to evaluate the suspension performance. The results show that the performance of the double chamber hydro-pneumatic suspension improved compared to the single chamber hydro-pneumatic suspension. Erin[6]established a nonlinear DCHP suspension system, and the results of the time domain simulation and frequency domain simulation are compared with the experimental data to validate the accuracy of the model. Other researchers have also studied the characteristics of DHCP suspension[7]. Consider the hydro-pneumatic suspension control: some researcher use the semi-active control model while others use the active control method for the system. Wang et al.[8]proposed a variable universe fuzzy control method for the hydro-pneumatic suspension to suppress the sprung mass acceleration. A hybrid reference model combining skyhook and ground-hook control is used in the semi-active hydro-pneumatic suspension[9]to improve both riding comfort and handling stability. Other control techniques such as sliding mode control, PID etc. are also implemented for the hydro-pneumatic suspension control. Although many researchers have worked on the hydro-pneumatic suspension modeling and control, seldom have they considered the DCHP control problem. After the analysis and model of the DCHP suspension system, this paper proposed a novel cuckoo search (CS) based fuzzy PID (FPID) controller for the system. The CS is utilized to get the best membership functions and PID parameters to improve the riding comfort, and at the same time not deteriorate the handling stability.

1 Nonlinear Double Chamber Hydro-Pneumatic Suspension

1.1 Structure of the DCHP suspension

The structure of the DCHP suspension is shown in Fig.1. The system consists of four parts[3]: action cylinder, main damper, main accumulator and external components. The action cylinder is composed of a piston cylinder and an outer cylinder, and the external components include dampers and two accumulators connected via a variable orifice i.e. an adjustable throttle valve. The cylinder is mounted between the body and the lower arm to provide support force, and the piston rod moves along the outer cylinder.

As shown in Fig.1, the main piston divides the cylinder interior space into two parts, the lower chamber and upper chamber, which are the compression and recovery chamber, respectively. When the main piston moves down, the pressure of the compression chamber increases and the oil flows into the recovery chamber through the valve block. When the pressure on both sides of the main piston increases to a certain extent, the oil flows to the recovery chamber through the compression valve, resulting in compression damping force. On the contrary, when the main piston moves upward, it results in recovery damping force.

Fig.1 Structure of the dual-chamber hydro-pneumatic suspension

1.2 Modeling of DCHP suspension

The pressure equations of the main and additional accumulator can be derived from the polytropic equation of state[10]as

(1)

wherep10,p20,V10andV20are the initial pressure and volume of the main and additional accumulator, respectively, andp10=p20.p1is the pressure of the main accumulator, andp2is the pressure of the additional accumulator.V12is the oil volume flow into the additional accumulator,Arepresents the piston rod section area andris the polytrophic exponent. When the excitation frequency is very low, the compression and expansion of the gas can be regarded as isothermal,r=1; when the excitation frequency is high, the cooling conditions of the gas are poor,r=1.4.

The transmission force of the DCHP suspension can be expressed as

(2)

whereFdis the main damper force andQ12andcaare the additional damper volumetric flow rate and damping coefficient, respectively. The values of the parameters are shown in Tab.1. As shown in Fig.2a, the dynamic model of the one quarter vehicle with DCHP suspension is expressed as

(3)

wherexbandxwrepresent the body and wheel displacement, respectively,xris the road roughness, andktis the tire stiffness.

Tab.1 Parameters of the DCHP suspension

Fig.2 Quarter model of DCHP suspension system

The equivalent stiffness and damping coefficients of the DCHP suspension are the function of the excitation frequency, the property is independent of the nonlinear stiffness of the elastic medium itself, but is determined by the structural characteristics of the DCHP system. So, the elastic and damping components are linearized to better study the feature of the system. Eq.(3) can be rewritten as

(4)

wherek1=p0A2r/V10,k2=p0A2r/V20,xa=V12/Aandp0is the static equilibrium position pressure of the main and additional accumulator.V10andV20are the initial gas volume of the main and additional accumulator. The equivalent model of the DCHP suspension system is shown in Fig.2b. The equivalent stiffness and damping values of the system areceqandkeq. By comparing the sprung mass and unsprung mass displacement transfer functionxb/xw, the equivalent stiffness and damping can be expressed as

(5)

whereωis the excitation frequency. From Eq.(5) we can see that despite the linear characteristic of the spring and damping component, the stiffness and damping coefficient have a frequency-dependent characteristic.

The control of the system is achieved through the control of the additional damper, and the DCHP suspension system can be equivalent to a slow active suspension system. The equivalent DCHP suspension with active force is shown in Fig.2c. DHCP suspension with active force is

(6)

1.3 Random road disturbance model

Random road excitation can be considered as a symmetrical and isotropic Gaussian random process with statistical characteristics[11-12]. The power spectral density (PSD) of the random road disturbance can be expressed as

(7)

wherenis the spatial frequency (m-1) andn0is the reference spatial frequency.Gq(n0) represents the road roughness coefficient of different road levels. The road excitation is regarded as the response of the unit white noise excitation, and then the system frequency response function is[13]

(8)

vis the vehicle speed (m/s),ωis the round frequency (rad/s) andn00represents road space cutoff frequency (m-1). The time domain model of the road surface excitation can be obtained by converting Eq.(8) into a differential equation as

(9)

Considering the D-classes road,Gq(n0)=1 024×10-6m3,W=2,n0=0.1, and vehicle speed is 60 km/h. The displacement of the road profile can be built by integrating white noise. Fig.3 shows the quarter vehicle random road profile. To verify the statistical characteristics of random road profile, the power spectral density (PSD) of the ISO class D pavement is used for the purpose of comparison, and the result is shown in Fig.4.

Fig.3 Random road profile

Fig.4 Comparison of PSD curves between the generated pavement and the ISO pavement

2 Cuckoo Search Based Fuzzy PID Control

2.1 Cuckoo search algorithm

CS is a new metaheuristic search algorithm developed by Yang Xinshe at the University of Cambridge in 2009[14]. The basic idea of CS is the reproductive behavior of cuckoo and the characteristics of Levy flight. The flow chart of the cuckoo search is shown in Fig.5. Preliminary studies show that it is superior to the existing algorithms, such as PSO and GA[15]. Compared with PSO, CS combines the local random walk strategy and the global search random walk strategy. They are switched through the probability parameterpa, thus improving the global searching efficiency and increasing the probability of getting the global optimal solutions. Local random walk can be expressed as

(10)

(11)

(12)

Fig.5 Flow chart of the cuckoo search algorithm

2.2 PID control

PID control is widely used in industry because it has a simple structure. The basic function of the PID control is to eliminate or reduce the steady state error and improve dynamic response of the system, and it is the combination of proportional, integral and derivative. The PID control law is

(13)

wheree(t) is the error of the model state and reference state,u(t) is the control law andKP,KIandKDare the proportional gain, integral gain and derivative gain, respectively. However，for its linear characteristics, classical PID is not suitable for nonlinear systems. So, the introduction of fuzzy logic control in this paper is not only to reduce the suspension vibration, but also to deal with system nonlinearity.

2.3 Fuzzy logic control

Fuzzy logic control (FLC) algorithm is based on the expert’s knowledge or experience. Generally speaking, a fuzzy system consists of three parts: fuzzifier, fuzzy inference engine for fuzzy rules and defuzzifier. Fuzzifier maps input variables into fuzzy sets, and fuzzy sets are characterized by membership functions. According to the fuzzy rules, inference engine performs mappings from input fuzzy sets to output fuzzy sets and defuzzifier maps the fuzzy output into crisp output. There are many different methods for each part of the fuzzy system; so FLC can be a combination of different methods. The controller contains two inputs and one output. The errore(t) and its derivate are used to get the control force. The linguistic variables are classified as: NB (Negative Big), NM (Negative Medium), NS (Negative Small), ZE (Zero), PS (Positive Small), PM (Positive Medium), and PB (Positive Big). The general form of the FLC rules can be defined as

(14)

In this paper, the controller for DHCP suspension is implemented by the CS based FLC. The two input variables to FLC are

(15)

2.4 CS based fuzzy PID control

As discussed above, fuzzy PID controller can integrate the advantages of the fuzzy logic control (FLC) and PID.The controller proposed in this paper is combination of the classical fuzzy PID control and the CS optimal algorithm mentioned above. The structure of the CS based fuzzy PID controller for DCHP suspension is shown in Fig.6, where the CS is used to regulate membership functions and PID parameters for the controller. Differ from many optimization methods that require complete information of the plant parameters, CS operates without knowledge of the plant, and only needs the lower and upper bound of the optimal variables. Hence, CS is more suitable to handle the problem of lacking experience or knowledge. The design procedure is as follows.

Step1Suppose cuckoo birds layneggs each time, and put these eggs into randomly chosenmnests. The parameter of the fuzzy PID controller, such as membership functions and PID parameters (Kp,KI,KD) make up the host nests position parameters. The fitness value will be calculated and the best nest will be kept to the next generation.

Step2There are three rules to simulate the parasitic behavior of cuckoo birds to search nests to lay eggs:

①Each bird lays one eggs to a randomly chosen nest.

②Best nest will be kept to next generation.

③The eggs will be found with a probabilitypa∈[0,1]. The host birds may either throw the different eggs, or abandon the nests and build new ones.

Step3Choose the suspension deflection as the constraint, sprung mass acceleration and tire deflection as the optimal objects; the optimal problem can be expressed as

Constraint: 6rms(xb-xw)≤lim(xb-xw)

Step4The initial settings for the CS should be defined, such as the population and the range of the optimal variables.

Step5Applying the CS optimal procedure to the DCHP suspension, after several generations, we can obtain the best membership functions and PID parameters for the controller.

Fig.6 CS based fuzzy PID controller

3 Simulation and Results

For the quarter DCHP suspension system discussed in section 1, the equivalent model shown in Fig.2c is used to introduce the control techniques proposed in this paper. The parameters of the DCHP suspension[16]system are selected ask1k2/(k1+k2)=12 500 N/m,mb=317.5 kg,mw=45.4 kg,cs=1 250 N·s/m, andkt=170 000 N/m. The random disturbance road profile described in section 1.3 is used as the model input disturbance. The compared method is denoted by LQR. Considering the riding comfort as the main indicator, the pareto front of the weighing factors are chosen asq1=50 823.3,q2=9.956 3×107andq3=9.350 2×107[17], whereq1,q2,q3are the weighing factors of sprung mass acceleration, suspension deflection and tire deflection, respectively. Fig.7 shows the evolution process of the CS algorithm. It can be seen that the search converges toward the minimal fitness value after 10 generations. Fig.8 shows the final membership function regulated by the CS.

Fig.7 Evolution process

Fig.8 Membership function

The main function of the optimization algorithm used in this paper is to find the optimal control parameters. In order to achieve the purpose of optimal control, the offline search optimization algorithm is utilized to obtain the most appropriate parameters; so, the algorithm will not affect the real-time features of the system.

The suspension response with the passive suspension system, the classical LQR and the proposed CSFPID, are shown in Fig.9, including sprung mass acceleration, suspension deflection and tire deflection. Tab.2 is the root mean square value of the dynamic response of the DHCP suspension with random road profile. Tab.2 shows that compared to the passive suspension system, the RMS value of body acceleration and suspension deflection reduced by 48% and 53%, respectively, by using CSFPID; The above values reduced by 37.5% and 25%, respectively, by using the LQR algorithm. However, the tire deflection, when using the LQR algorithm, had a 41% increase compared with the passive system, and the CSFPID control was almost the same with the passive system.

Fig.9 Suspension response

Control strategyVertical acceleration/(m·s-2)Suspension deflection/mTire deflection/mCSFPID0.3820.0050.0033LQR0.3430.0060.0048Passive suspension0.7450.0080.0034

4 Conclusion

In this paper, parametric DCHP suspension model and an equivalent model are developed, and CS based fuzzy PID controller is used in an equivalent DHCP suspension model. By using the CS characteristics, the control gain and the initial value of the PID controller and the fuzzy membership function can be obtained by defining the object functions and constraint of the proposed suspension. The numerical results indicate that the CS based FPID controller not only reduces the sprung mass acceleration and suspension deflection of the suspension system, but also shows that there is no deterioration of tire deflection compared to that obtained when using a classical LQR control method and passive suspension.