# Suspension System Control Based on Type-2 Fuzzy Sliding Mode Technique

Authors: Zheng Wang; Longyi Ran; Bing Kong; Xuezeng Jia; Liya Liu; Jafar Tavoosi
Journal: Complexity
 (2022)

---

## Abstract

This paper presents a new method of intelligent control for vehicle suspension. First, the suspension system is modeled with all the details, and then based on the obtained model, the sliding mode control is designed for it. The controller parameters and coefficients are calculated and updated by a type-2 fuzzy system. The chattering phenomenon is eliminated with a unique technique. In order to evaluate the performance of the proposed control system, two-model uncertainty of the road is applied. Simulations have been performed for both active and passive modes. The simulation results show the high efficiency of the proposed control system.


---

## Body

## 1. Introduction

A compromise between passenger convenience and the ability to direct vehicle is required in the design of the suspension system [1, 2]. To achieve the convenience of passengers, a soft suspension system is required to minimize the displacement and acceleration of the vehicle body. But, the suspension system must be stiff to direct vehicles in different road conditions [3, 4]. In a passive suspension system, spring and damper parameters are constant, so the vehicle body will fluctuate in the face of road terrain [4]. An active suspension system could adapt itself to different conditions of the road and decreases displacement and acceleration of the vehicle body simultaneously [5, 6]. A lot of research has been conducted to design the active suspensions system by different approaches (see [7] and its references).One of the important advantages of sliding mode control (SMC) is invariance property to indeterminacy [8]. Considering this advantage, sliding mode control is a strong tool to confront parametric or nonparametric uncertainties, disturbances, and noise [9]. It is worth noting that invariance is a stronger property than robustness [10, 11]. Robustness means to achieve an optimal outcome in the worst conditions, and invariance means to achieve the optimal outcome without influencing the system with noise, disturbances, and indeterminacy. In the face of noise, disturbance, and uncertainty, a robust system may malfunction, while invariance means robustness with strong performance [12, 13]. Chattering is one of the bad properties of sliding phase; therefore, the arrival time to the sliding surface and arrival phase should be limited in a nonlinear system. Chattering is defined as high but limited frequencies fluctuates that cause heat loss in electrical equipment and depreciation of mechanical components [14]. Four methods have been proposed to decrease or eliminate chattering: boundary, adaptive boundary layer, higher-order sliding mode control (HOSMC), and dynamical SMC [15, 16]. Chattering was not eliminated but decreased in the boundary layer and adaptive boundary layer [17, 18]. In HOSMC, chattering is eliminated by transferring it to higher derivatives of the sliding surface [19, 20]. A lot of algorithms have been proposed to apply the second or higher order of sliding mode control [21]. Although model derivatives of the system are required for SMC calculations, for example, dynamical model derivatives of the system are required for two relative degree systems [22]. While in dynamic sliding mode control, an integrator (or low-pass filter) is placed before the system, which causes to eliminate chattering because this filter eliminates high-frequency fluctuations [23]. The integrator increases the system degree; therefore, the model and dynamic of the system should be clear to apply sliding mode control on the additive system [24]. In other words, in dynamic sliding mode control, it is the required system model, but in HOSM, the model derivatives of the system should be clear. This case indicates precedence of dynamic on high degree.In [25] fuzzy adaptive sliding mode, control has been used along with proportional-integral controller to apply in active suspension system. In the mentioned paper, the stability of the system is asymptotically proven and the convergence time of the system is long, so there is no sliding phase and chattering is not eliminated. Moreover, the sign function is emerged in the input signal directly, which makes chattering. Control methods such as PI and PID can also be used as appropriate solutions [26]. Applying nonlinear systems identification is also very important to have an accurate model of the system [27]. The proposed approach in [28] uses the proportional-integral sliding mode control for the quarter-car model, in which stability demonstration is asymptotical. Then, there is no sliding phase and so invariance is eliminated. In [29], Kalman filter applies on nonlinear model of Quarter-Car suspension system to estimate linear model states. Moreover, due to stability proof being asymptotical, the sliding phase does not exist, and chattering is eliminated. In [30], linearization of hydraulic actuator equation is used firstly. Then, a dynamic sliding mode controller is applied in the output control loop. The authors have not considered the servo valve equation to simplify applying feedback linearization, and the closed-loop system has no invariance property because feedback linearization is not robust. Fewer people today do not own a vehicle or use a car, so an active and intelligent suspension is essential for the well-being of the car occupants. This article deals with this issue in detail and the ultimate goal is to design an active suspension system with minimal oscillation and vibration.In this paper, for the first time, we use a recurrent type-2 fuzzy to update SMC coefficients for use in the suspension system. Another innovation of this article is the use of SMC with three terms (while others have used two-term SMC). As can be seen in (12), there are three λ, but in the other’s work, there are two λ, meaning that they set only two parameters. So, our SMC also has greater degree of freedom that can control the system more precisely. Also, in this paper, road roughness is applied by two models in transparent manners and it is observed that the active suspension system with fuzzy neural network has a better response.The main contribution of this paper is as follows:(1)
Propose a new recurrent type-2 fuzzy system to update the SMC coefficients for use in the suspension system. Figure1 (page 6) shows a new structure of the type-2 fuzzy neural network, which is carefully observed in (13) that weights are also type-2 fuzzy sets (in addition to membership functions.)(2)
Another innovation of this article is the use of SMC with three terms (while others have used two-term SMC). As can be seen in (15), there are three λ, but in the other’s work, there are two λ, meaning that they set only two parameters. So, our SMC also has greater degree of freedom that can control the system more precisely.(3)
Also, in this paper, road roughness is applied by two models in transparent manners (in similar articles, only one type of roughness has been investigated) and it is observed that the active suspension system with fuzzy neural network has a better response.Figure 1
The structure of the proposed type-2 fuzzy system.In Section2, dynamic components of the applied suspension system are explained. In Section 3, the proposed approach and question definition are presented. The controller design is presented in Section 4. Important notes in the proposed approach are stated in Section 5. Simulation and conclusion are considered in Sections 6 and 7, respectively.


## 2. Dynamic Model of the Suspension System

The dynamics of the intended active suspension system of a quarter-car, which is described by the following equations, is shown in Figure2(1)mszs¨=−bsżs−żu−kszs−zu+fa,(2)muz¨u=+bsz˙s−z˙u+kszs−zu−fa+btz˙r−z˙u+ktzr−zu,where zs, zu, and zr are the vertical displacement of the body and wheels and height of land terrains to baseline, respectively. Also, the first and second derivatives zs and zu indicate the vertical velocity and acceleration of vehicle body and wheels. Note that we assume velocity and acceleration in the system are perpendicular to the ground. In addition, ms, mu, ks, kt, bs, and bt are the mass, stiffness, vehicle body, damping, and wheels, zs−zu is the suspension deflection, and zu−zr is the wheel deflection. Variable fae is the output force of the hydraulic actuator, which is placed between the body and wheels. If fa=0, the intended suspension system will be inactive.Figure 2
Quarter-car suspension system.To design an active suspension system, the desiredfa value should be determined so that zs, żs, and zs¨ have the minimum value. We assume that all parameters are defined, and all variables are measurable. This assumption is logical because the variables can measure with different sensors in a mechanical system. In practical applications, sensors are installed on the vehicle and are used then. zr and its derivative are the only variables that are not measurable practically that is considered as indeterminacy. This structure is shown in Figure 2. The servo valve equations activate the hydraulic actuator as follows. It is worth noting that the input of the valve is current too.(3)x˙sp=−xsp+isvτ,where isv is the input current of the valve and τ is the mechanical time constant of the servo valve, and xsp is the spool valve displacement (or input) which is attached to the hydraulic actuator. Dynamic equations of the hydraulic actuator are as follows:(4)f˙a=−αAp2z˙s−z˙u−βfa+γApxspPs−BklxspfaAp,where Bkl represents the backlash in the output of the solenoid valve depicted in Figure 3. For this figure, we have a=b=0.1. It should be noted that the derivative of the output with respect to the input is always either 0 or 1.Figure 3
Backlash of hydraulic actuator.As seen in (3) and (4), isv is the only input and the only parameter required to be calculated to reach the optimal fa. This structure as well is demonstrated in Figure 4 where Ps and Pr are the direct and return pressures of the actuator, respectively, and Pu and Pl are the pressures in the upper and lower of the actuator cylinder chambers, respectively. Plus, Ap is the piston area and β=αCtp,⋅α=4βe/Vt,⋅γ=αCdw1/ρ where βe is the bulk modulus of hydraulic fluid, Vt is the volume of the hydraulic cylinder, Ctp is the leakage coefficient, Cd is the discharge coefficient, w is the spool valve area gradient, and ρ is the hydraulic oil density.Figure 4
Hydraulic actuator system with double active.


## 3. Problem Description

The recommended method has two steps. In the first step, we assumefa is the input of system (1). In this case, by controlling the dynamic sliding mode, the desired smooth and free of chattering, that is, fd will be calculated such that the dislocation, velocity, and acceleration of the car body (zs, żs, and zs¨, respectively) converge to 0, leading to more comfort of the passengers.In the second step, using another dynamic sliding mode control, the smooth and free of chattering current of the solenoid valveisv will be determined such that in (4), fa follows the optimal fd value obtained in the first step.To apply the suggested method, we define the following state variables of (1):(5)x1=zS,x2=z˙S,Hence,(6)x˙1=x2,x˙2=−bsmsx2−ksmsx1+bsmsz˙u+ksmszu+fams.On the other hand, (2) can be rewritten as(7)z¨u=bsmux2+ksmux1+−bsmu−btmuz˙u+−ksmu−ktmuzu−famu+fdis.Also, as stated,fdis that represents the terrains on the ground is considered as the uncertainty,(8)fdis=btmuz˙r+ktmuzr.Also, (4) can be reformulated as well.(9)f˙a=−βfa−αAp2x2+αAp2z˙u+γApxspgxsp,fa,where(10)g=Ps−BklxspfaAp.


## 4. Type-2 Fuzzy System

In this section, the proposed type-2 fuzzy system is discussed to predict the parameters of SMC. Numerous studies have shown that type-2 fuzzy system performs better than type-1 fuzzy system. Figure3 shows the structure of the proposed type-2 fuzzy system.The type-2 fuzzy neural network presented is completely new (both membership functions and weights are type-2 fuzzy sets) and is used for the first time in this article. This type-2 fuzzy neural network is used to update sliding mode control parameters (equation (15), λ_1, λ_2, λ_3) according to the amount of error. Figure 5 shows that the fuzzy system considers the system parameters online and tunes the λ parameters. In the work of others, these parameters are determined in trial and error way, but in our work, the parameters are calculated online based on changes in system’s dynamic. In the proposed method, data are first collected to train the neural network. At this stage, rich data are extracted both for the error-free system and by applying some deliberate errors to the system. Naturally, the more the system dynamics can be changed and the more the errors applied, the more useful the data will be obtained for neural network training.Figure 5
Block diagram of the proposed control system.The internal phase of the first layer calculations is given as follows:(11)∅¯jiuj=uj−cji12,uj<cji1,1,cji1≤uj≤cji2,uj−cji22uj>cji2,∅¯jiuj=uj−cji22,uj≤cji1+cji22,uj−cji12uj>cji1+cji22,where ∅¯ji and ∅¯ji are the upper and lower of the jth input and ith neuron, respectively. Therefore, the outputs of the first layer are as follows:(12)∅¯iu=exp−∑j=1n+1∅˜jiujσi2,∅¯iu=exp−∑j=1n+1∅¯jiujσi2,where ∅¯i and ∅¯i are the upper and lower of the ith neuron i=1,2,…,m, respectively. u∈uj,j=1,…,n is the input vector, and cji∈cji1,cji2 is the center of all the RBF neurons. It should be noted that the left and right endpoints of the second layer are as follows:(13)y^l=∑i=1q∅¯iucwi2σwi+∑i=q+1m∅¯iucwi1σwi∑i=1q∅¯iuσwi+∑i=q+1m∅¯iuσwiy^r=∑i=1p∅¯iucwi1σwi+∑i=p+1m∅¯iucwi2σwi∑i=1p∅¯iuσwi+∑i=p+1m∅¯iuσwi,where p and q are the left and right switching points of the type-2 fuzzy system, which can be calculated using the trial and error method or the Karnik–Mendel (KM) algorithm. Also, m, wi, cwi, and σwi are the mean values of the first layer neurons, the weights, the center of weights, and the spread of weights, respectively. Lastly, the general output of the network can be derived as follows:(14)y^=y^l+y^r2.The gradient descent method is used to teach the network. See [31, 32], for more details.


## 5. The Designing of the Active Controller

As stated,fa is considered as the dynamic input control signal (6) and is determined such that zs, żs, and zs¨ (x1,x2,x3=x2˙) converge to 0 in a limited time. The calculated value is the desirable force that needs to be generated by the hydraulic actuator. We name it fd and determine it using the dynamic sliding mode control. To execute this controller, the sliding surface s is defined as(15)s=λ1x1+λ2x2+λ3x3.The optimal values of the control parametersλ1,λ2,λ3 are calculated at any time by the type-2 fuzzy system. We determine the constant coefficients of the surface so that the following equation is Hurwitz:(16)λ3p2+λ2p+λ1=0.The derivative of this surface is(17)s˙=λ1x˙1+λ2x˙2+λ3x˙3=λ1x2+λ2x3+λ3x˙3.Now, using (6), we have(18)x˙3=x¨2=−bsmsx3−ksmsx2+bsmsz¨u+ksmsz˙u+f˙dms.Using this equation, the derivative of the sliding surface is(19)s˙=λ1x2+λ2−bsmsx2−ksmsx1+bsmsz˙u+ksmszu+fdms+λ3−bsmsx3−ksmsx2+bsmsz¨u+ksmsz˙u+f˙dms.This equation can be reformulated as(20)s˙=λ1−λ2bsms−λ3ksmsx2+−λ2ksmsx1+λ2bsms+λ3ksmsz˙u+λ2ksmszu+λ2fdms+λ3f˙dms+−λ3bsmsx3+λ3bsmsz¨uNow, we replace thez¨u from (7) into (16)(21)s˙|=λ1−λ2bsms−λ3ksms+λ3bsmsbsmux2+−λ2ksms+λ3bsmsksmux1+λ2bsms+λ3ksms−λ3bsmsbsmu−λ3bsmsbtmuz˙u+λ2ksms+λ3bsmsksmu−λ3bsmsktmuzu+λ3msf˙d+λ2ms−λ3bsms1mufd+−λ3bsms+λ3bsmsfdis.Again, considering thatx3=x˙2, we have(22)s˙|=λ1−λ2bsms−λ3ksms+λ3bsmsbsmu+λ3bsmsbsmux2+−λ2ksms+λ3bsmsksmu+λ3bsmsksmux1+λ3msf˙d+λ2ms+λ3bsms1mu−λ3bsms1mufd+λ3bsmsfdis+λ2bsms+λ3ksms−λ3bsmsbsmu−λ3bsmsbtmu−λ3bsmsbsmsz˙u+λ3bsms+λ3bsmsksmu−λ3bsmsktmu−λ3bsmsksmuzu.For the purpose of simplicity, (22) can be rewritten using the corresponding coefficients:(23)s˙=k1x1+k2x2+k3zu+k4z˙u+k5fd+k6f˙d+k7fdis.Theorem 1.
The following desirable input for the system (1) or (6) converges the states x1,x2,x3 to 0 in a limited time.(24)f˙d=−k1x1+k2x2+k3zu+k4z˙u+k5fd+ηfsk6,η=k7fdis+ε,ε>0,where fs=sats/φ is the saturation function and ϕ is the thickness of the border layer.Proof.
Considering theV=0.5s2 Lyapunov function, we have(25)V˙=ss˙=sk1x1+k2x2+k3zu+k4z˙u+k5fd+k6f˙d+k7fdis=sk7fdis−ηfs=sk7fdis−ηfs≤k7fdiss−ηfs=k7fdis−ηs=−εs≤0.
This concludests≤s0/ε where ts ,the limited time of the states, is reaching the marginal layer around the sliding surface. The suggested controller is pictured in Figure 1.


### 5.1. Calculation of the Valve Current (Step 2)

The purpose of this step is the calculation ofisv from (3) such that fa in (4) or (9) follows the fd desirable value that is obtainable in the first step from equation (24). To this end and to obtain a smooth and free of chattering isv that is practical, the dynamic sliding mode control is employed. Now, the sliding surface is defined:(26)sf=λ4f˙a−f˙d+λ5fa−fd.Note that the optimal values of the control parametersλ4,λ5 are calculated at any time by the type-2 fuzzy system. Also, the constant coefficients will be determined so that the following equation is Hurwitz:(27)λ4p+λ5=0.The derivative of this surface is(28)s˙f=λ4f¨a−f¨d+λ5f˙a−f˙d.By replacingf¨a from (4) and f¨d from (20), we have(29)s˙f=λ4f¨a−f¨d+λ5f˙a−f˙d=λ4−βf˙a−αAp2x˙2+αAp2z¨u+γApx˙spg+γApxspg˙+λ4k1x2+k2x˙2+k3z˙u+k4z¨u+k5f˙d+ηf˙sk6+λ5−βfa−αAp2x2+αAp2z˙u+γApxspg+λ5k1x1+k2x2+k3zu+k4z˙u+k5fd+ηfsk6.This can be organized as(30)s˙f=−λ4αAp2+λ4k2k6x˙2+−λ5αAp2+λ4k1k6+λ5k2k6x2+λ5k1k6x1+λ4αAp2+λ4k4k6z¨u+λ4ηk6f˙s+λ5αAp2+λ5k4k6+λ4k3k6z˙u+λ5k3k6zu+−λ4βf˙a+−λ5βfa+λ4γApgx˙sp+λ4γApg˙+λ5γApgxsp+λ4k5k6f˙d+λ5k5k6fd+λ5ηk6fs.By replacingx˙2, z¨u, f˙a, f˙d, and x˙sp, we have(31)s˙f=+−k2k6λ4k5k6−αAp2−λ4β+bsmuλ4αAp2+λ4k4k6−bsmu−λ4αAp2+λ4k2k6−λ5αAp2+λ4k1k6+λ5k21k6x2+−k1k6λ4k5k6+ksmuλ4αAp2+λ4k4k6−ksmu−λ4αAp2+λ4k2k6+λ5k1k6x1+−k4k6λ4k5k6+αAp2−λ4βλ4k3k6−bsmu−btmuλ4αAp2+λ4k4k6+bsms−λ4αAp2+λ4k2k6+λ5αAp2+λ5k4k6z˙u+−k3k6λ4k5k6+−ksmu−ktmuλ4αAp2+λ4k4k6−ksms−λ4αAp2+λ4k2k6+λ5k3k6zu+−β−λ4β−1muλ4αAp2+λ4k4k6+1ms−λ4αAp2+λ4k2k6+λ5βfa+−1τλ4γApg+γApg−λ4β+1τλ4γApgisv++λ4γApg˙+λ5γApgxsp+−k5k6λ4k5k6+λ5k5k6fd+−ηk6λ4k5k6+λ5ηk6fs+λ4ηk6f˙s+λ4αAp2+λ4k4k6fdis.This can be simplified as(32)s˙f=k8x1+k9x2+k10zu+k11z˙u+k12fa+k13fd+k14xsp+k15isv+k16fs+k17f˙s+k18fdis.Theorem 28.
The following input for systems (3) and (4) or (3) and (9) causes fa−fd to converge to 0 in a limited time.(33)isv=−η2fsf+k16fs+k17f˙s+k8x1+k9x2k15−k10zu+k11z˙u+k12fa+k13fd+k14xspk15,η2=k18fdis+ε,ε>0,where fsf=satsf/φ is the saturation function and ϕ is the thickness of the marginal layer.Proof.
Considering the Lyapunov functionV=0.5sf2, we have(34)V˙=sfS˙f=sfk8x1+k9x2+k10zu+k11z˙u+k12fa+k13fd+k14xsp+k15isv+k16fs+k17f˙s+k18fdis=sfk18fdis−η2fsf=sfk18fdis−η2fsf≤k18fdissf−η2fsf=k18fdis−η2sf=−εsf≤0.
This concludests≤s0/ε where ts is the limited time of the states reaching the marginal layer around the sliding surface. The general structure of the proposed controller is shown in Figure 5. As can be seen in this structure, using the dynamic sliding mode control of the current without chattering of the solenoid, inlet isv is determined so that the output force of the hydraulic actuator follows our desired force. In other words, it is defined such that fa−fd converges to zero in a limited time. On the other hand, this optimum force is determined and calculated by controlling the dynamic sliding state and mechanical equations of the suspension system to have the desired conditions of the passenger and the vehicle. Note that in (24), there is a sign function or discontinuity in f˙d and is therefore without switching, because the integrator acts as a low-pass filter. The same is true for isv and xsp in (3) and (33), and as a result, they are smooth and without chattering.
According to the second part of (6), the dynamic slip mode control causes x1,x2,andx3=x˙2 to converge to zero and as a result,(35)bsz˙u+kszu=−fd,meaning that signals zu and z˙u are also under control indirectly.
In case of using the ordinary (instead of the dynamic) sliding mode control to obtainisv in (33), xsp is calculated instead of isv. This yields the emergence of a discontinuity in xsp. Plus, the differential (3) needs to be solved. The solution of (3) means using an integrator that will increase the noise effect plus pay attention, that in the ordinary sliding mode control, the surface is defined as(36)sf=λ5fa−fd.
In this case, a chattering will appear infa during the calculation.


## 5.1. Calculation of the Valve Current (Step 2)

The purpose of this step is the calculation ofisv from (3) such that fa in (4) or (9) follows the fd desirable value that is obtainable in the first step from equation (24). To this end and to obtain a smooth and free of chattering isv that is practical, the dynamic sliding mode control is employed. Now, the sliding surface is defined:(26)sf=λ4f˙a−f˙d+λ5fa−fd.Note that the optimal values of the control parametersλ4,λ5 are calculated at any time by the type-2 fuzzy system. Also, the constant coefficients will be determined so that the following equation is Hurwitz:(27)λ4p+λ5=0.The derivative of this surface is(28)s˙f=λ4f¨a−f¨d+λ5f˙a−f˙d.By replacingf¨a from (4) and f¨d from (20), we have(29)s˙f=λ4f¨a−f¨d+λ5f˙a−f˙d=λ4−βf˙a−αAp2x˙2+αAp2z¨u+γApx˙spg+γApxspg˙+λ4k1x2+k2x˙2+k3z˙u+k4z¨u+k5f˙d+ηf˙sk6+λ5−βfa−αAp2x2+αAp2z˙u+γApxspg+λ5k1x1+k2x2+k3zu+k4z˙u+k5fd+ηfsk6.This can be organized as(30)s˙f=−λ4αAp2+λ4k2k6x˙2+−λ5αAp2+λ4k1k6+λ5k2k6x2+λ5k1k6x1+λ4αAp2+λ4k4k6z¨u+λ4ηk6f˙s+λ5αAp2+λ5k4k6+λ4k3k6z˙u+λ5k3k6zu+−λ4βf˙a+−λ5βfa+λ4γApgx˙sp+λ4γApg˙+λ5γApgxsp+λ4k5k6f˙d+λ5k5k6fd+λ5ηk6fs.By replacingx˙2, z¨u, f˙a, f˙d, and x˙sp, we have(31)s˙f=+−k2k6λ4k5k6−αAp2−λ4β+bsmuλ4αAp2+λ4k4k6−bsmu−λ4αAp2+λ4k2k6−λ5αAp2+λ4k1k6+λ5k21k6x2+−k1k6λ4k5k6+ksmuλ4αAp2+λ4k4k6−ksmu−λ4αAp2+λ4k2k6+λ5k1k6x1+−k4k6λ4k5k6+αAp2−λ4βλ4k3k6−bsmu−btmuλ4αAp2+λ4k4k6+bsms−λ4αAp2+λ4k2k6+λ5αAp2+λ5k4k6z˙u+−k3k6λ4k5k6+−ksmu−ktmuλ4αAp2+λ4k4k6−ksms−λ4αAp2+λ4k2k6+λ5k3k6zu+−β−λ4β−1muλ4αAp2+λ4k4k6+1ms−λ4αAp2+λ4k2k6+λ5βfa+−1τλ4γApg+γApg−λ4β+1τλ4γApgisv++λ4γApg˙+λ5γApgxsp+−k5k6λ4k5k6+λ5k5k6fd+−ηk6λ4k5k6+λ5ηk6fs+λ4ηk6f˙s+λ4αAp2+λ4k4k6fdis.This can be simplified as(32)s˙f=k8x1+k9x2+k10zu+k11z˙u+k12fa+k13fd+k14xsp+k15isv+k16fs+k17f˙s+k18fdis.Theorem 28.
The following input for systems (3) and (4) or (3) and (9) causes fa−fd to converge to 0 in a limited time.(33)isv=−η2fsf+k16fs+k17f˙s+k8x1+k9x2k15−k10zu+k11z˙u+k12fa+k13fd+k14xspk15,η2=k18fdis+ε,ε>0,where fsf=satsf/φ is the saturation function and ϕ is the thickness of the marginal layer.Proof.
Considering the Lyapunov functionV=0.5sf2, we have(34)V˙=sfS˙f=sfk8x1+k9x2+k10zu+k11z˙u+k12fa+k13fd+k14xsp+k15isv+k16fs+k17f˙s+k18fdis=sfk18fdis−η2fsf=sfk18fdis−η2fsf≤k18fdissf−η2fsf=k18fdis−η2sf=−εsf≤0.
This concludests≤s0/ε where ts is the limited time of the states reaching the marginal layer around the sliding surface. The general structure of the proposed controller is shown in Figure 5. As can be seen in this structure, using the dynamic sliding mode control of the current without chattering of the solenoid, inlet isv is determined so that the output force of the hydraulic actuator follows our desired force. In other words, it is defined such that fa−fd converges to zero in a limited time. On the other hand, this optimum force is determined and calculated by controlling the dynamic sliding state and mechanical equations of the suspension system to have the desired conditions of the passenger and the vehicle. Note that in (24), there is a sign function or discontinuity in f˙d and is therefore without switching, because the integrator acts as a low-pass filter. The same is true for isv and xsp in (3) and (33), and as a result, they are smooth and without chattering.
According to the second part of (6), the dynamic slip mode control causes x1,x2,andx3=x˙2 to converge to zero and as a result,(35)bsz˙u+kszu=−fd,meaning that signals zu and z˙u are also under control indirectly.
In case of using the ordinary (instead of the dynamic) sliding mode control to obtainisv in (33), xsp is calculated instead of isv. This yields the emergence of a discontinuity in xsp. Plus, the differential (3) needs to be solved. The solution of (3) means using an integrator that will increase the noise effect plus pay attention, that in the ordinary sliding mode control, the surface is defined as(36)sf=λ5fa−fd.
In this case, a chattering will appear infa during the calculation.


## 6. Simulation

In this article, MATLAB software version 2019a has been used to model the suspension system and implement the proposed control system. The values of the parameters used for the suspension system are listed in Table1 [7]. Plus, the initial values of the controller parameters are chosen as(37)λ1=1,λ2=6,λ3=6,λ4=10,λ5=0.2,ε=20,φ=0.3.Table 1
Parameters of suspension system.
Unit (dimension)ValueVariableKg240msN/(m/s)375bsN/m14600ksKg45muN/(m/s)415btN/m124600kts−11βm23.3×10−4ApS1.30τN/m54.51×1013αPa10342500PsN/m5/2kg1/21.54×109γTo simulate, the terrains of the ground as the uncertainty are applied to the wheels as(38)zr=−0.05sint,0.3≤t≤1.8,0.05sint,4.8≤t≤6.3,0,otherwise.The simulation was performed with the MATLAB software with sample time of 0.01 for a period of 7 seconds in the active and inactive doses, and the results are shown in Figures6 to 15. In the simulations, two types of roughness have been applied on the road. The first case is shown in Figure 6.Figure 6
Terrain of the road 1.Figure 7
Body displacement.Figure 8
Body velocity.Figure 9
Body acceleration.Figure 10
Control signal.Figure 11
Terrain of the road 2.Figure 12
Body displacement (step).Figure 13
Body velocity (step).Figure 14
Body acceleration (step).Figure 15
Control signal (step).Figure7 shows the results of the car body displacement control signal for both passive and active controller (sliding mode control with type-1 fuzzy and type-2 fuzzy systems).Figure8 shows the results of the vehicle body speed with all three control systems.The acceleration of the vehicle body is very important in the comfort of the occupants and also in maintaining stability. The performance results of the control systems for the acceleration of the body are shown in Figure9.The control signals for both active controllers (sliding model with type-1 and type-2 fuzzy systems) are shown in Figure10.To further explore and challenge the control system, a step roughness is also applied, as shown in Figure11.The performance of all three control systems for roughness is shown in Figures11 and Figure 12.Figures13 and 14 show the speed and acceleration of the body for the roughness of Figure 11, respectively.Finally, Figure15 shows the control signals from both active systems for stepped unevenness.Observing the simulation results, we find that an active control system is essential for vehicles. The active controller has been able to prevent the vehicle body from oscillating well and provide the comfort for the passengers. A 10 cm high step rug, which is one of the most challenging cases in the field of suspension systems, was also applied to our system, and the results show the high capability of the proposed control method.


## 7. Conclusion

The vehicle suspension system has a fundamental and direct effect on the comfort and convenience of the occupants. Therefore, better and more accurate control of this system is necessary and important. In the proposed approach, dynamic sliding mode control is used to improve the performance of the active suspension so that both mechanical and electrical equations of the suspension are considered, this is why the equations used are so complex. Type-1 and type-2 fuzzy systems were used to estimate the sliding mode control parameters. An electric valve is used, and the backlashing valve outlet activates the hydraulic actuator. That is, despite the uncertainties in the system (ground terrains), the only available signal is the inlet current of the valve. Therefore, the smooth flow without chattering of the solenoid valve is calculated in two steps such that the suspension has the desired behavior. The proposed method is without chattering and is also resistant to road uncertainties. New methods such as type-3 fuzzy sliding mode technique can be used to improve this work (as future research). This method has a higher degree of freedom and it has more control parameters. Other uncertainties such as the looseness of the control valve and the nonadjustment of the tire pressure can be considered.



---

*Source: 2685573-2022-05-24.xml*