When constructing the dynamic model of the MRF damper, it is imperative to conduct tests to assess its dynamic characteristics. The Instron E10000 electronic dynamic and static testing machine was used in this experiment. The MRF damper used in this study has a stroke of 35 mm, a tensile length of 230 mm, and a diameter of 40 mm. The sinusoidal signal with displacement amplitudes of 5 and 10 mm and frequencies of 1, 2, and 3 Hz were applied to the damper. The control currents of the experiment were 0, 1, 2, 3, 4 and 5 A. The experimental process is shown in Fig. 2. This paper selects the experimental results of 10 mm, 2 Hz working conditions to analyze, as shown in Fig. 3.

As evident from the experimental findings depicted in Fig. 3, the damping-force and displacement curves of the MRF damper are approximately elliptical and relatively full, indicating that the damper has good energy consumption and vibration attenuation performance during operation. The damping force of the MRF damper experiences a significant increase as the input current rises, indicating that the damper has good current control performance.

Common forward dynamic models of MRF dampers include the Bingham model (Stanway et al., 1987), the polynomial model (Choi et al., 2001), the Bouc–Wen model (Bouc, 1976), and the hyperbolic tangent model (Kwok et al., 2006). Among them, the Bouc–Wen model finds widespread use due to its ability to represent the nonlinear characteristics of MRF dampers more effectively. Therefore, this paper employs the Bouc–Wen model to represent the dynamics of MRF damper. The damping-force expression of the Bouc–Wen model is as follows (Wen, 1976): 1F=c0x˙+k0(x-x0)+αzz˙=-γx˙zzn-1-βx˙zn+Ax˙.

In this context, the variables are defined as follows: F represents the damping force of the MRF damper, c0 is the viscosity coefficient of the magnetorheological fluid after yielding, x signifies the relative displacement between the two ends of the damper, x˙ is the relative velocity of the two ends of the damper, x0 is the relative equilibrium position offset displacement, z is the hysteresis variable, n is the curve-rounding coefficient, γ is the adjustment coefficient of the hysteresis loop width, β is the adjustment coefficient of the hysteresis loop height, α is the adjustment coefficient of the proportion of the total damping force, and A is the scaling factor related to the maximum damping force.

In this research, the parameter identification of the Bouc–Wen model is conducted using the Simulink parameter estimation module in MATLAB. The initial step involves constructing the Bouc–Wen model in Simulink. After completing the modeling, the test data are imported.

After completing the modeling, we then import the test data. The input value was the displacement of the damper piston, and the output was damping force. Then we selected the parameters to be identified. Firstly, the damping-characteristic curves for the 10 mm, 2 Hz working condition were integrated into the Simulink model. Subsequently, the Bouc–Wen model's eight parameters were derived through the initial identification process. Various parameters exert distinct influences on the model. Further comparison shows that the fluctuations of the two parameters α and c0 are significant under different currents, and so the remaining parameters are held constant during the subsequent parameter identification process. The values of the two parameters α and c0 under different currents are identified separately, and, finally, the relationship curves between these two parameters and the current are fitted. The parameter identification results are shown in Table 1 (I is the current in Table 1), and a comparison of the model fitting results with the experimental results is shown in Fig. 4.

Figure 4 displays the force–displacement and force–velocity characteristic curves of the MRF damper. The solid and dashed lines indicate the experimental measurements and the calculated values of the Bouc–Wen simulation model under the currents of 0, 1, 2, 3, 4, and 5 A. The figures show that the damping force calculated by the MRF damper dynamic model established in this paper is consistent with the actual damping force obtained from the experimental tests, which can meet the requirements of semi-active control.

To further verify the universality of the identified model under different working conditions, the test data obtained under other working conditions should be input into the model for validation. The comparison results are shown in Fig. 5.

As can be seen from Fig. 5, the identified Bouc–Wen model can accurately reflect the displacement–damping-force characteristics and velocity–damping-force characteristics of the magnetorheological fluid damper and exhibits good universality under different working conditions

In the context of semi-active seat suspension control, the control force is determined through a control method but then needs to be transmitted via the magnetorheological fluid (MRF) damper. This entails adjusting the current of the MRF damper to achieve the desired control force, which requires the establishment of an inverse dynamics model for the MRF damper. The MRF damper's forward dynamics model is complex, presenting significant nonlinearity in the relationship between output damping force, input control current, relative piston displacement, and velocity. This complexity makes it challenging to derive the control current directly from the desired control force using a direct mathematical method (Gao et al., 2023). To address this, our approach employs a BP neural network to establish the inverse model for the MRF damper.

Training data samples are generated to ensure comprehensive coverage of the MRF damper's operational range, with consideration of the structural parameters used in this study. These samples involve random excitation signals with displacements of less than 10 mm as input training data and their derivatives as velocity input training data. For control input current, random white-noise signals in the range of 0–5 A serve as training samples. These training samples are then fed into the Bouc–Wen dynamic model of the MRF damper, producing the corresponding output damping force as training data.

Once the BP neural network inverse model for the MRF damper has been successfully trained, a selected dataset is utilized to test the neural network's ability to predict the control current. Then the error between predicted current and actual current is calculated. The prediction results and error analysis are presented in Fig. 6.

It can be seen from Fig. 6 that the current values predicted by the BP neural network model established in this paper are in good agreement with the actual current values. A further analysis of the current prediction error shows that the maximum current prediction error is 0.71 A. The root mean square (rms) value of the prediction error for the BP neural network is remarkably low, measuring at just 0.168 A. This level of accuracy fulfills the control requirements of the MRF damper. Following the establishment of both the dynamic and inverse dynamic model of the MRF damper, the calculation of the MRF damper's control current can be carried out based on the expected control force derived from the semi-active control algorithm.

When establishing the human-seat suspension model, this paper adopts the 5-degree-of-freedom human-seat model (Wang et al., 2018) for semi-active control research, as shown in Fig. 7. In the figure, M1–M5, respectively represent the mass of the seat itself, the mass of the human pelvis, the mass of the human viscera, the mass of the upper torso, and the mass of the head. k1–k5, respectively, represent the stiffness of the seat suspension, the stiffness of the human pelvis, the stiffness of the human viscera, the stiffness of the upper torso, and the stiffness of the head; fMRF is the damping force of the MRF damper when no current is added to it; c2–c5 are the damping coefficients of the human pelvis, the damping of the upper torso, the damping of the human viscera, and the damping of the head, respectively; k2c and c2c are the stiffness and damping coefficient of the seat cushion, respectively, x0–x5 represent the displacement of the vehicle floor, seat cushion, pelvis, upper torso, viscera, and head. In addition, the MRF damper can directly generate control force after current is applied, and the generated control force is represented by f(t). The system dynamic equation is shown in Eq. (2). 2M1x¨1+k1(x1-x0)+fMRF+f(t)+c2⋅c2cc2+c2c(x˙1-x˙2)+k2⋅k2ck2+k2c(x1-x2)=0M2x¨2+c2⋅c2cc2+c2c(x˙2-x˙1)+k2⋅k2ck2+k2c(x2-x1)+k4(x2-x4)+c4(x˙2-x˙4)=0M3x¨3+k3(x3-x4)+c3(x˙3-x˙4)=0M4x¨4+k4(x4-x2)+c4(x˙4-x˙2)+k3(x4-x3)+c3(x˙4-x˙3)+k5(x4-x5)+c5(x˙4-x˙5)=0M5x¨5+k5(x5-x4)+c5(x˙5-x˙4)=0

To precisely control the multi-degree-of-freedom human-seat suspension system, it is necessary to determine each part's stiffness and damping coefficients. Due to the nonlinear structure of the human body, its stiffness and damping coefficients will constantly change with different postures during practical operation, making it difficult to measure accurately. The design of semi-active control algorithms based on multiple-degree-of-freedom systems is complex and difficult to apply in practical engineering. By simplifying the human-seat suspension system model, we can simplify the controller design and make it easier to apply in practical engineering. However, during model simplification and actual seat operation, a series of disturbances may occur, and so it is necessary to estimate these interferences to improve the accuracy of control. In this paper, the human body model was simplified for the control strategy, where the mass parameters from m1 to m5 were considered as a whole. Herein, m1 refers to the mass of the seat cushion, and the sum of m2 to m5 represents the mass of the human body supported by the seat in the sitting posture. In this paper, the values of m1 to m5 are taken from the model parameters in the reference (Wang et al., 2018), where m1 = 22 kg, m2 = 27 kg, m3 = 20 kg, m4 = 9 kg, m5 = 5.5 kg, and the sum of m2 to m5 is 61.5 kg. The seat stiffness k1 is set to 15 000 Nm-1. We simplify the 5-degree-of-freedom human-seat suspension system model established in this paper as shown below. 3Mx¨1=k1(x0-x1)+fMRF-fds-f(t)4fds=f(s)+Δf(t)5M=m1+m2+m3+m4+m56f(s)=M2(x¨2-x¨1)+M3(x¨3-x¨1)+M4(x¨4-x¨1)+M5(x¨5-x¨1)

In the above equations, f(s) represents the disturbance generated during the simplification process of the model, and Δf(t) represents the disturbance generated externally by the seat during actual operation. We select X=[x1-x0,x˙1]T, Y1=x1-x0,Y2=x¨1, and then 7X˙=AX+B(u+fds-fMRF),8Y1=C1X,9Y2=C2X+D(u+fds-fMRF).

In the above equations, 10A=01-k1M0,B=0-1M,C1=10,C2=-k1M0,D=-1M.

Conventional sliding-mode control is difficult to apply in practice due to the high cost of the required state variables; to overcome the above problems, most of the current sliding-mode controllers are designed by using the model reference method. The principle is that an ideal model is selected as the reference model, and a suitable control force is output to control the actual controlled system with a suitable control force so that it can achieve the same control effect as the ideal control system. Since this paper adopts an MRF damper to realize the semi-active control of seat suspension, it needs the output of semi-active control force through variable damping force, and so, combining with the working characteristics of MRF damper, this paper selects the ideal skyhook control as the reference model and added the dynamics model of MRF damper (established in the second part of the paper) into the reference model, as shown in Fig. 8. Since the controller designed in this paper is implemented based on an MRF damper, the dynamic model of the MRF damper is incorporated into the control system to ensure that the control force calculated by the controller can be rapidly output by the MRF damper. Specifically, the dynamic model of the MRF damper is included in both the actual model and the reference model for sliding-mode control. The MRF damper's output damping force consists of two components: the fixed damping force and the variable damping force. The fixed damping force arises when the input current is zero and depends on the damper piston's velocity. This remains unalterable. The variable damping force, on the other hand, is modifiable through adjustments to the input current. In the reference model, the fixed damping force is denoted as fMRFi, and, in the actual model, the fix damping force is denoted as fMRF.

The motion state error between the actual system and the reference model is chosen as the input into the sliding-mode controller. This controller acts to reduce the dynamic error between the actual model and the reference model and to keep it in the sliding mode. Consequently, the motion state of the actual model can effectively track the reference model, which particularly enhances the frequency response of the seat suspension in the low-frequency range.

The kinetic equation of the reference model is as follows: 11Mx¨1i+k1(x1i-x0)+fMRFi+f(t)=0, where fMRFi and fMRF are the fixed damping force of the reference model and actual model, respectively; x0 is the input vibration displacement of the cab floor; x1i and x1 are the seat platform displacements of the reference and actual models, respectively; k is the seat suspension stiffness of the reference and actual models; Csky is the skyhook damping coefficient; and f(t) is the skyhook damping force of the reference model, expressed as follows: 12f(t)=-csky⋅x˙1i.

The fundamental concept behind the sliding-mode controller developed in this paper is to enable the motion state of the actual model to track the reference model's motion state as closely as possible. Therefore, the error vector is defined as the combination of the velocity error, the displacement error, and the integral of the displacement error between the actual model and the reference model, as shown in Eq. (13) (Liu et al., 2008). 13e=e1=∫x1-x1ie2=x1-x1ie3=x˙1-x˙1i

Differentiating Eq. (13), the kinetic equations of the system are obtained as follows: 14e˙=A(t)e+B(t)w+E(t)fds+G(t)f(t).

In this equation, 15A(t)=0100010-k1M0,B(t)=00cskyM,E(t)=00-1M,G(t)=00-1M,w=[x˙1i].

Then, sliding-mode control is applied to the above error dynamic system, and the sliding-mode surface is designed as follows: 16s=ce=[c1c21][e1e2e3]T=c1e1+c2e2+e3, where c1 and c2 are parameters to be determined and should satisfy the polynomial p2+c2p+c1 as Hurwitz polynomials; to satisfy the above condition, take (p+λ)2 = 0 = 0 and λ>0, where c1=λ2, and c2 = 2λ.

A convergence law is used to design the sliding-mode surface: 17s˙=ce˙=c(A(t)e+B(t)w+E(t)fds+G(t)f(t)).

The index convergence law is designed as follows: 18s˙=-εsgn(s)-ks.

In the equation, s˙=-ks is the exponential convergence term, which can ensure that, when s is large, the system state can approach the switching surface s at a relatively high speed, with k > 0; s˙=-εsgn(s). The constant velocity approach term ensures that the moving point can reach the switching surface within a finite time; ε > 0 is a symbolic function that can be represented as follows: 19sgn(s)1,s>00,s=0-1,s<0.

From Eqs. (17) and (18), the following can be obtained: 20f(t)=-(cG(t))-1[c(A(t)e+B(t)w+E(t)fds)+εsgn(s)+ks].

Bringing the matrices G(t), A(t), B(t), and E(t) and the vector c into Eq. (20), the result is shown as follows: 21f(t)=Mc1e2+Mc2e3+k1(x0-x1)+fMRF-fds-Mx¨1i+Mεsgn(s)+Mks.

Define the Lyapunov function as V1=12s2, taking the derivative of V1, the result is as follows: 22V˙1=ss˙=sc(A(t)e+B(t)w+E(t)fΔ+G(t)u)=-Mεssgn(s)-Mks2=-Mεs-Mks2<0.

According to the Lyapunov stability criterion, the system is thus stable.

To obtain the traditional sliding-mode control law, it is first necessary to determine the perturbation fds generated by the model during the simplification process and external disturbances. In practical systems, this disturbance fds tends to be undetermined and cannot be given an accurate value, but it cannot be neglected either. In order to avoid affecting the control input f(t) of the system, it is therefore necessary to treat fds adaptively. The online estimation of fds is performed by designing a reasonable adaptive law to reduce its impact on the system performance. The adaptive control method is used to estimate the fds, and then the control law is given by 23f(t)=Mc1e2+Mc2e3+k1(x0-x1)+fMRF-f^ds-Mx¨1i+Mεsgn(s)+Mks.

Substituting Eq. (23) into Eq. (17) yields 24s˙=-f̃dsM-εsgn(s)-ks.

We define the Lyapunov function as follows: V2=V1+12γf̃ds2, and then 25V˙2=V˙1+1γf̃dsf̃˙ds=-f̃dsMs-εs-ks2+1γf̃dsf^˙ds=f̃ds1γf^˙ds-sM-εsgn(s)-ks2, where, assuming that 1γf̃˙ds-sM=0 and f^˙ds=rsM, it can be obtained that V˙2=-εssgn(s)-ks2,ε>0,k>0,and then 26V˙2=-εssgn(s)-ks2≤-ks2≤0.

The adaptive law is chosen to be 27f^˙ds=γsM,

where f̃ds=fds-f^ds; f^ds is the estimated value of fds; and γ is the parameter to be determined, with γ>0.

The sign function sgn(s) in Eq. (26) will aggravate the chattering phenomenon of the sliding-mode control; this discontinuity arises from the switching control. To mitigate this phenomenon, the hyperbolic tangent function tanh(s) is employed as a replacement for the sign function sgn(s) in the switching term, and it is represented as follows (Gao et al., 2004): 28tanh(x)=ex-e-xex+e-x, where σ is a positive constant, and its value dictates the rate at which the inflection points of the hyperbolic tangent function change. Replacing the discontinuous sign function with the smooth and continuous hyperbolic tangent function renders the switching process of sliding-mode control continuous, thereby enhancing the stability of the control system. The characteristics distinguishing the hyperbolic tangent function from the sign function are depicted in Fig. 9.

Owing to the advantages of strong robustness and high resistance to external interference, sliding-mode control finds diverse applications in the field of control. However, due to the “variable-structure” nature of sliding-mode control and its inherent control discontinuity, along with other characteristics, the whole control process produces chatter and instability. The fuzzy control method incorporates the expert control experience into the fuzzy controller through fuzzy logic reasoning and the fuzzy rule design. This control strategy does not require the construction of an accurate model of the controlled object, and so it is a commonly used method in solving the control of uncertainty systems. Therefore, in this paper, the fuzzy algorithm is employed to design a convergence law for sliding-mode control, ameliorating the chattering issue of the control system.

According to Eq. (18), when the moving point is close to the sliding-mode surface s=0, there is s-=±ε, which means that the convergence speed of the moving point when it reaches the sliding-mode surface is ε. When ε is chosen to be larger, the overshoot across the sliding-mode surface will be more significant, which will make the system vibrate violently, and the system will be unstable. Although a small value of ε can reduce the vibration, it will also slow down the convergence speed of the system. Therefore, to reduce the system's vibration and ensure the system's convergence speed, the value of k should be increased while decreasing ε. However, it can be seen from Eq. (20) that, when the motion point is faraway from the sliding-mode surface, too large a value of k will lead to a larger damping-force output from the semi-active damper. Due to the limited range of the semi-active damper's damping-force output, the value of k should not be selected to be too large. Considering the above factors, this paper uses fuzzy control to optimize the coefficients ε and k of the speed-reaching law in sliding-mode control and constructs a time-varying reaching law, which ensures the system's response speed while improving its stability. The fuzzy adaptive approach to design the convergence law is as follows: 29s˙=-u1εsgn(s)-ku2s, where u1 and u2 are the fuzzy coefficients of ε and k in the speed reaching law. This can keep the speed-reaching law of sliding-mode control within a reasonable range at all times. We select a triangular membership function with overlap as the membership function for the input variable s and the output variables α1 and α2, as illustrated in Fig. 10.

In the figure, PB (positive large), PM (positive middle), PS (positive small), and ZO (zero) are fuzzy subsets. The absolute value of the sliding-mode function, denoted as s, serves as the input, leading to fuzzy output variables u1 and u2. The following four fuzzy rules are proposed: R1 – if s is PB then u1 is PB and u2 is PSR2 – if s is PM then u1 is PM and u2 is PMR3 – if s is PS then u1 is PS and u2 is PBR4: if s is ZO then u1 is ZO and u2 is PM.