Research on ride control strategy of electromagnetic suspension based on sliding mode algorithm

Abstract

This study proposes a main inner loop control technique to address the dynamic response of the electromagnetic suspension during vehicle driving. The genetic algorithm is employed to optimize the control parameters of the linear quadratic adjustment controller in the control main loop, with the aim of achieving the desired control force. In the control inner loop, a closed-loop sliding mode controller is designed based on the exponential approximation law to regulate the current. The Lyapunov function was utilized to assess the stability of the controller in terms of its theoretical feasibility. The simulation results of Matlab/Simulink show that under the given working conditions, compared with the passive suspension, sliding mode control and fuzzy sliding mode control for references, the root mean square value and peak value of the vertical acceleration of the body and the dynamic load of the tire with the improved sliding mode control are significantly reduced, the root mean square value of the dynamic deflection of the suspension was not improved. However, on the whole, the ride comfort of the vehicle has been improved.

Keywords

Electromagnetic suspension GA-LQR controller slide mode control ride comfort parameter optimization

Introduction

The vehicle’s driving circumstances are intricate, and the standard mechanical suspension system struggles to adequately respond to the unevenness of the road, failing to match consumers’ expectations for a comfortable ride. As a result, the car industry has focused on researching electromagnetic suspension systems (EAS) that have the ability to vary the damping force, making them a popular area of study.¹ The control method of EAS is a crucial determinant of the system’s benefits and drawbacks. A stable and dependable EAS control system can effectively enhance the ride comfort of the vehicle, as extensively investigated by researchers in the field.

Regarding the algorithm for generating the target damping force of EAS, to ensure the driving smoothness and reduce the road noise interference, Zhao² et al. developed a controller that combines LQR controller and proportional integral sliding mode (PI-SMC), in which The PI-SMC strategy is responsible for tracking the reference force, while the LQR control calculates the necessary control force in real-time. Considering that nonlinearity will reduce the performance of the suspension system and road disturbance, Khan³ et al. introduced an enhanced control algorithm for the suspension system in order to mitigate the negative effects of nonlinearity and road disturbance. The algorithm utilizes the LQR algorithm to calculate a robust gain matrix and damping force. Simulation results demonstrate that this control strategy enhances the dynamic performance of the suspension system and reduces the impact of road surface disturbance. In order to improve the performance of the vehicle, Bai⁴ et al. introduced a novel robust optimum control technique for the active suspension system with the aim of enhancing the vehicle’s performance. To address the challenges posed by uncertain parameters and external disturbances, a combination of LQR control and sliding mode control is employed. The effectiveness of this control strategy and its ability to enhance the overall performance of the vehicle are then validated by Carsim/Simulink co-simulation. Gysen BLJ⁵ et al. designed a prototype electromagnetic active suspension system that provides additional stability through linear active roll and pitch control during cornering and braking, thereby improving vehicle safety and ride comfort. The dynamic performance of the electromagnetic suspension system is proved by the experiment of 1/4 vehicle electromagnetic drive on the road and the comparison with the passive suspension system. Su⁶ et al. proposed a fuzzy controller design method for electromagnetic suspension system based on T-S model. By using the fuzzy state feedback controller, the target electromagnetic damping force was obtained through simulation experiments, which ensured the hybrid performance of the original electromagnetic suspension system and verified the effectiveness of the proposed method. MS Seong⁷ et al. examined the issue of damping force hysteresis and developed a control strategy that incorporates LQR feedforward compensation. They chose a damper that is appropriate for high-end passenger cars and conducted experiments to evaluate and identify the parameters of the damping force. The simulation results demonstrated an improvement in the problem of damping force hysteresis and enhanced the dynamic performance of the suspension. At present, the desired damping force of the EAS system is mostly achieved through the use of a LQR controller. However, determining the settings of the typical LQR controller is challenging, and obtaining the desired damping force is also problematic.

Regarding the management of dynamic response in EAS, Wang⁸ et al. developed a Variable Damping Control Strategy (EFVD) that utilizes electromagnetic suspension energy recovery to optimize suspension performance and minimize energy loss. This strategy enhances suspension dynamics and improves energy efficiency, as demonstrated through simulation. Xiong⁹ et al. connected the linear motor in parallel with the MR damper, and simulated the system using backstep control and model predictive control. The results demonstrated that the suspension system’s energy efficiency was higher when using backstep control compared to PID control. This improvement effectively enhanced the vehicle’s dynamic performance in low power situation. Ding¹⁰ et al. designed a hybrid electromagnetic suspension system and conducted experiments to recover energy and compare vibration reduction of the 1/4 suspension system. The findings showed that the hybrid electromagnetic suspension, when combined with an enhanced ceiling control strategy, effectively ensured vehicle comfort and facilitated energy recovery. Zhang¹¹ et al. devised a range of electromagnetic components using rotating machine structures. They employed group control to regulate the quantity of shunt resistors in the suspension system, so enabling the system to generate a substantial electromagnetic damping force and achieve the desired damping coefficient. The test findings indicate that the highest efficiency and average efficiency are 54.98% and 44.24%, respectively. In order to enhance the ride comfort and energy recuperation, Montazeri-GH¹² adopts a switch-off hybrid control strategy. Simulation results indicate that this strategy improves suspension performance compared to the ceiling control strategy. However, the energy recovery value of the canopy control strategy surpasses that of the switch-switch hybrid control. The aforementioned academics have primarily concentrated on assessing the energy recovery efficiency of EAS. However, further investigation is required to validate its impact on the dynamic performance of EAS. Due to the presence of several unpredictable disturbances in the suspension system, the typical linear control algorithm has challenges in efficiently suppressing them. Sliding mode control (SMC) is a commonly utilized control algorithm due to its exceptional ability to resist interference and maintain internal stability. For example, in order to solve the problems of low accuracy and serious jitter of traditional sliding mode control, ZOU X et al.¹³ adopted a sensor less position control scheme based on fuzzy sliding mode motor to improve the stability of the system in order to solve the problems of low accuracy and serious jitter of traditional SMC. The jitter phenomena occurring inside the SMC is influenced by its structural properties, so affecting the output. Consequently, additional optimization of the design is essential.

Overall, the parameters of the LQR controller are intricate, which are frequently employed in the control strategy of the EAS main loop and the precision of the resulting target damping force is not optimal. The inner loop control approach poses challenges for the control algorithm to effectively consider the dynamic response characteristics and robustness. Considering the aforementioned factors, the uniqueness of this study can be summarized as follows:

(1) By utilizing the global search capability of the genetic algorithm (GA), the optimal weight coefficient matrix is automatically sought in order to achieve automated adjustment of the parameters of the LQR controller and enhance the precision of the desired electromagnetic damping force.

(2) Selecting the appropriate approach law during the design process of the SMC to reduce the chatter.

The dynamic modeling of EAS

In this study, the EAS is simplified into a 1/4 vehicle two-degree-of-freedom model, the basic structure and working principle are shown in Figure 1.

Figure 1.

Schematic diagram of the EAS dynamic model.

Figure 1 illustrates the primary components of the EAS, which consists of shock absorbers, ECUs, linear motors, sensors and other elements. The sensor collects and transmits the vehicle vibration information to the ECU when the tire is exposed to road excitation. The ECU utilizes the received status information to generate the desired electromagnetic damping force. Simultaneously, it regulates the linear motor to provide the appropriate actual electromagnetic damping force to enhance the ride comfort of the vehicle.

According to Newton’s second law, the differential equation corresponding to the dynamical model in Figure 1 is as follows:

{\begin{cases} m_{s} {\ddot{x}}_{s = k_{s} (x_{u} - x_{s}) + c_{s} ({\dot{x}}_{u} - {\dot{x}}_{s}) + F} \\ m_{u} {\ddot{x}}_{u = k_{t} (x_{t} - x_{u}) + k_{s} (x_{u} - x_{s}) - c_{s} ({\dot{x}}_{u} - {\dot{x}}_{s}) - F} \end{cases}

(1)

Where m_s denotes the sprung mass, m_u denotes the unsprung mass, k_s is the suspension stiffness coefficient, k_t is the tire stiffness coefficient, x_t is the pavement excitation, c_s is the damping coefficient of the shock absorber, F denotes the electromagnetic damping force output for a linear motor.

Expressing equation (1) as an equation of state yields:

{\begin{cases} \dot{X} = A X + {B U}_{1} \\ Y = C X + {D U}_{1} \\ X = [x_{s} - x_{u}, {\dot{x}}_{s}, x_{u} - x_{t}, {\dot{x}}_{u}] \\ \dot{Y} = [{\ddot{x}}_{s}, x_{s} - x_{u}, k_{t} (x_{u} - x_{t}), {\dot{x}}_{u}] \end{cases}

(2)

Where X denotes the system state variable, Y is the output variable, x_s-x_u represents the suspension dynamic displacement, k_t(x_u-x_t) represents the tire dynamic load, x_t is the road input excitation, A is the system state matrix, B is the control input matrix, C is the output matrix, and D is the direct transfer matrix.

The matrices A , B , C , D , and U ₁ are obtained, respectively:

A = [\begin{array}{l} 0 & 1 & 0 & - 1 \\ - \frac{k_{s}}{m_{s}} & - \frac{c_{s}}{m_{s}} & 0 & \frac{c_{s}}{m_{s}} \\ 0 & 0 & 0 & 1 \\ \frac{k_{s}}{m_{u}} & \frac{c_{s}}{m_{u}} & - \frac{k_{t}}{m_{u}} & - \frac{c_{s}}{m_{u}} \end{array}], B = [\begin{array}{l} 0 & 0 \\ 0 & \frac{1}{m_{s}} \\ - 1 & 0 \\ 0 & - \frac{1}{m_{u}} \end{array}],

(3)

C = [\begin{array}{l} - \frac{k_{s}}{m_{s}} & - \frac{c_{s}}{m_{s}} & 0 & \frac{c_{s}}{m_{s}} \\ 1 & 0 & 0 & 0 \\ 0 & 0 & k_{t} & 0 \\ 0 & 0 & 0 & 1 \end{array}], D = [\begin{array}{l} 0 & \frac{1}{m_{s}} \\ 0 & 0 \\ 0 & 0 \\ 0 & 0 \end{array}], U_{1} = [\begin{array}{l} {\dot{x}}_{1} \\ F \end{array}]

(4)

The actuator in EAS is a linear motor that can be equivalent to a series structure consisting of an inductor, resistor, and controlled source when analyzing its circuit. According to Kirchhoff’s law, the expression between the voltage and the electromagnetic damping force in a linear motor is:

{\begin{cases} U = L \frac{d i}{d t} + R i + ϕ \frac{d z}{d t} \\ F = ϕ i \end{cases}

(5)

Where U is the voltage, L is the motor inductance, z is the vertical displacement of the ball screw, i is the motor current, ϕ is the motor constant, R is the motor resistance, F is the electromagnetic damping force.

EAS control strategy

The EAS control strategy designed in this study includes a target signal generation module and a motor tracking control module, and the main inner loop control process is shown in Figure 2.

Figure 2.

Electromagnetic suspension main inner loop control block diagram.

The GA-LQR controller in the ECU computes the desired damping force by analyzing the suspension state variable in response to road surface stimulation. The intended damping force is subsequently passed to the SMC, which is tasked with regulating the damping force of the linear motor. In the end, the linear motor produces the necessary damping force to limit the impact of road vibration on the suspension system. The following text provides a detailed explanation of the design process for each individual component of the controller.

GA-LQR controller

As a full-state feedback controller, the LQR controller reduces the control error by selecting the appropriate gain matrix and objective function.¹⁴ For EAS system, the objective function of designing the LQR controller is as follows:

J = \int_{0}^{\infty} [r_{1} {\ddot{x}}_{s}^{2} + r_{2} {(x_{s} - x_{u})}^{2} + r_{3} {(x_{u} - x_{t})}^{2}] d t

(6)

Where r_1, r₂, r₃ represents the weight coefficient of the body acceleration, the weight coefficient of sprung dynamic deflection and the weight coefficient of tire’s dynamic load, respectively.

Rewriting the above indicators into a standard quadratic type is:

J = \int_{0}^{\infty} (x^{T} Q x + u^{T} R u + 2 x^{T} N u) d t

(7)

Where matrix Q represents the gain of the state variable, matrix N represents the gain of the correlation between the two variables, and matrix R represents the gain of the control variable.

The matrices Q , N and R are as follows:

Q = [\begin{array}{c} 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & r_{2} + \frac{{K_{s}}^{2}}{{m_{b}}^{2}} & - r_{2} - \frac{{K_{s}}^{2}}{{m_{b}}^{2}} & 0 \\ 0 & 0 & - r_{2} - \frac{{K_{s}}^{2}}{{m_{b}}^{2}} & r_{1} + r_{2} + \frac{{K_{s}}^{2}}{{m_{b}}^{2}} & - r_{1} \\ 0 & 0 & 0 & - r_{1} & r_{1} \end{array}], N = \frac{1}{{m_{b}}^{2}} [\begin{array}{c} 0 \\ 0 \\ - K_{s} \\ K_{s} \\ 0 \end{array}], R = \frac{1}{{m_{b}}^{2}}

(8)

Drawing upon optimal control theory, the optimal control force exerted by the actuator is given as follows:

F_{1} = - K x (t)

(9)

Where

x (t)

is the feedback state variable at any time point, and K is the optimal feedback gain matrix.

The performance index function can be obtained by designing the linear quadratic optimal controller:

[K, S] = L Q R (A, B, Q, R, N)

(10)

When J is taken as a minimum, the corresponding optimal control force F_excellent is:

F_{e x c e l l e n t} = - R^{- 1} B^{T} P x

(11)

The P matrix satisfies the following Riccati differential equation,¹⁵ namely:

\dot{P} = - P A - A^{T} P + P B R^{- 1} B^{T} P - Q

(12)

The solution P of the Riccati differential equation will approach a constant matrix such that $\dot{P} = 0$ .

The formula for solving the matrix for P is as follows:

P A + A^{T} P - P B R^{- 1} B^{T} P + Q = 0

(13)

The design of the LQR controller is influenced by the choice of weight coefficients r₁, r₂ and r₃, as shown in equations (6)–(11). Typically, the control parameters are obtained through a trial method. However, the parameters obtained through this method are highly subjective, and there is potential for improving the control effectiveness. The evolutionary algorithm is employed to optimize the weighting coefficient in the design of the LQR controller, aiming to minimize the objective function J and determine the ideal value.

The evolutionary algorithm that has been constructed, which is used to optimize the LQR control process, as depicted in Figure 3.

Figure 3.

Genetic algorithm optimization LQR control flow chart.

At the beginning of the iteration, the genetic algorithm is employed to create the initial population. Each individual in the population is then assigned to r₁, r₂ and r₃ in the LQR controller sequentially. This ensures that each individual can contribute to the optimization process by representing a specific set of control parameters. The fitness function value of each individual is calculated to determine if it meets the termination condition of the genetic algorithm. If the termination conditions are not satisfied, the process of selection, preservation, crossing, and mutation is performed to create a new population, and this cyclical process is repeated. Upon satisfying the specified conditions, the execution of the genetic algorithm is terminated and the most optimal solution is generated as output.

Considering that the units and orders of magnitude of r₁, r₂ and r₃ are different, the fitness function of the genetic algorithm can be obtained by dividing the three by the corresponding passive suspension performance index value:

\min i m i z e L = \frac{H (X)}{H_{p a s}} + \frac{I (X)}{I_{p a s}} + \frac{N (X)}{N_{p a s}}

(14)

The optimization variables are:

X = (r_{1,} r_{2}, r_{3}), 0.1 < X_{i} < 10^{6}, i = 1, 2, 3

(15)

The constraints are:

s . t . {\begin{cases} H < H_{p a s} \\ I < I_{p a s} \\ N_{0} < {N_{0}}_{p a s} \end{cases}

(16)

Where H, I, and N₀ represents the RMS values of the body acceleration, the dynamic deflection of the suspension sprung and the tire’s dynamic load, respectively.

During the simulation in Matlab/Simulink, the initial parameter for the gain matrix K ₁ is set at [3596.97, −1028.94, −292.00, −4863.43,6638.76]. By inputting this specific set of parameters into the MDL simulation file, one can receive the simulation results and values of the fitness function. Subsequently, the obtained value is sent back to the M file responsible for parameter optimization. The genetic optimization process is then executed on the population to be adjusted based on the fitness function value. The resulting new population is subsequently fed into the simulation model once again. Continue doing the aforementioned actions until the ideal outcome is achieved or the termination condition is met. The alteration in the fitness function during the optimization process may be observed in Figure 4.

Figure 4.

Change in fitness function.

The population size is set at 100, the elite individuals is 20, the crossover rate is 0.5 and the mutation rate is 0.1. At the beginning of the iteration, the fitness function value was the maximum value of 21.6322, and after 10 iterations, the fitness function value stabilized at 19.1436. The current optimal gain matrix, K ₂ = [4028.3, −392.4, 8492.3, 13,727, −19,847]. By equation (5), the optimal target electromagnetic damping force F can be determined and the target current i₀ can be derived. Figure 5 shows the comparison of target control force before and after optimizing LQR controller with GA.

Figure 5.

Target control before and after genetic algorithm optimization.

As can be seen from Figure 5, the implementation of LQR control allows for more precise adjustment of the suspension system compared to the previously optimized target control force, resulting in an enhanced level of system accuracy.

Inner loop control strategy

To provide precise tracking of the target control force, SMC is implemented in the inner loop to track the target current, thereby controlling the linear motor to generate stable electromagnetic damping force. The design of the SMC generally includes two aspects:

(1) Designing the sliding mode surface function to make the state trajectory of the system with appropriate dynamic characteristics;

(2) The stability analysis of the system in the sliding mode state is convenient during designing.

The tracking error of the SMC can be expressed as:

e = i - i_{0}

(17)

Where i is the output current of the linear motor; i₀ is the target current of the linear motor.

The switching function of the SMC is designed as follows:

s = e

(18)

The derivatives of the sliding surface is:

\dot{s} = \dot{e}

(19)

From equation (17), it can be obtained:

\dot{e} = \dot{i} - {\dot{i}}_{0}

(20)

It can be obtained from the linear motor equation (5):

\dot{s} = \dot{i} - {\dot{i}}_{0} = \frac{U - R_{i} - ϕ \frac{d z}{d t}}{L} - {\dot{i}}_{0}

(21)

The structure of the sliding form controller is as follows:

U = U_{s} + U_{e q}

(22)

Where U_s is the sliding mode controller switching term and U_eq is the equivalent control term.

U_eq expression is:

U_{e q} = L i + R i + ϕ \frac{d z}{d t}

(23)

The improved SMC equivalence condition is a necessary condition for equation (23) to be established.

The convergence law determines the mass of the system entering the sliding surface from outside during normal motion. The laws governing SMC primarily consist of the general approximation law, power approximation law, constant velocity approximation law and exponential approximation law. The exponential component serves to expedite the convergence time, causing the approach speed to gradually decline from high to low until it tends to 0. This reduction in speed allows for a decrease in the rate at which the switching surface is reached, thus lowering chatter. Therefore, choose the law of exponential convergence:

\dot{s} = - η sgn (s) - λ s

(24)

Where

η

is the switching gain,

λ

is the exponential coefficient,

η > 0, λ > 0

The switching term expression of the controller is:

U_{s} = L (η sgn (s) + λ s)

(25)

Where s is the sliding surface,

\dot{s}

is the error differential and sgn(s) is the symbolic function.

Combined with the Lyapunov equation to verify the stability of the control system¹⁶:

V = \frac{1}{2} s^{2}

(26)

Caculate the derivative of the Lyapunov function:

\dot{V} = s \dot{s} = s (- η sgn (s) - λ s) = (- η | s | - λ s^{2})

(27)

Where

s \cdot sgn (s) = | s |

and

η > 0, λ > 0

So:

\dot{V} \leq 0

(28)

Therefore, the system satisfies Lyapunov stability.

By adjusting the parameters $η, λ$ , the control system not only improves its ability to reach a stable state, but also reduces the time it takes to make adjustments, thus achieving the desired control outcome. The parameter adjustment strategy aims to simultaneously reduce $η$ and increase $λ$ , which can not only ensure the approach speed, but also reduce the system jitter problem as much as possible. Combining equations (5), (22) and (23), the control law U of the sliding mode controller is as follows:

U = {L \dot{i}}_{0} + R i + ϕ \frac{d z}{d t} + L η sgn (s) + L λ S

(29)

To reduce the chattering problem of the system, a saturation function is defined:

s a t (s) = {\begin{cases} 1, s > Δ \\ k s, | s | \leq Δ \\ - 1, s < Δ \end{cases}

(30)

Where

k = \frac{1}{Δ}, Δ > 0

Δ

is the boundary of the boundary layer.

Therefore, U_s can be rewritten as:

U_{s} = L (η s a t (s) + λ s)

(31)

Combined with equations (17)–(31), the mathematical expression of the sliding mode variable structure controller is obtained:

U = {L \dot{i}}_{0} + R i + ϕ \frac{d z}{d t} + L η s a t (s) + L λ S

(32)

Simulation results and analysis

In order to further illustrate the effectiveness of the designed controller, the suspension performance of SMC, the fuzzy SMC used in the literature reference¹³ and the controller with improved SMC approximation law were simulated and compared, so as to verify the effectiveness of the main inner loop control strategy. Table 1 shows the simulation parameters for setting up the model in this study.

Table 1.

Simulation parameters.

The name of the parameter	Description	Numeric value
m_s (kg)	Sprung mass	320
m_u (kg)	Unsprung mass	40
K_s (Ns·m⁻¹)	Suspension stiffness	20,000
K_t (N·m⁻¹)	Tire stiffness	200,000
v (m/s)	Velocity	20
c_s (N·m⁻¹)	Suspension damping factor	1000
L (H)	Inductance constant	0.3
ϕ (N·A⁻¹)	Constant of the machine	90
R ( $Ω$ )	The resistance value of the energy -feeding circuit	10

In this study, the pavement road input excitation selects the filtered white noise, and the time-domain expression¹⁷ is.

{\dot{x}}_{t} (t) = - 2 π f_{0} x_{t} (t) + 2 π \sqrt{G_{0}} v w (t)

(33)

Where G₀ is the unevenness coefficient of the road surface, v is the speed of the vehicle, w is the Gaussian random road inputs and f₀ is the lower cut-off frequency and the value is 0.1 Hz.

At the speed of 20 m/s and with C-Class input excitation, the simulation was conducted for a duration of 10 seconds. The results of the road input excitation can be seen in Figure 6.

Figure 6.

C -Class input excitation.

The inner loop control employs SMC to track the target current in real time, thereby tracking the target control force. The comparison of the current magnitudes for SMC, fuzzy SMC, and improved SMC with the target current is shown in Figure 7; the current error between these three controls and the passive suspension is shown in Figure 8; and the comparison of the actual control force and the target control force obtained by these three different controls is shown in Figure 9.

Figure 7.

The magnitude of the tracking current for different controls.

Figure 8.

Current error.

Figure 9.

The magnitude of the control force of the different control tracks.

From Figure 7, it can be seen that compared with SMC and fuzzy SMC control, the output current of improved SMC can better track the target current and can more accurately track the suspension performance. Figure 8 shows the current error between SMC control, fuzzy SMC, improved SMC and passive suspension, with root mean square value (RMS) of 0.2385, 0.2148 and 0.1759, respectively. This indicates that the improved SMC performs better, validates the accuracy of the model and meets the requirements of real-time current adjustment. From Figure 9, it can be seen that compared with SMC and fuzzy SMC control, the improved SMC can better regulate the electromagnetic damping force to track the target electromagnetic damping force. In this study, the main loop control is used to obtain the target control force for the suspension through control parameters, while the inner loop control adjusts the linear motor output current to track the target control force. The ride comfort of vehicle is evaluated using the metrics of the body acceleration, the dynamic deflection of suspension sprung and the tire dynamic load. Simulations were performed using a 1/4 suspension model built in Matlab/Simulink. The simulation results for evaluating the suspension system performance under the condition of a C-Class pavement input and at the speed of 20 m/s are shown in Figures 10, 11 and 12.

Figure 10.

Vertical acceleration of the body (Figure 11).

Figure 11.

Dynamic deflection of suspension sprung.

Figure 12.

Tire dynamic load (Figure 12).

As can be seen from Figures 10, 11 and 12, compared with the passive suspension, SMC and fuzzy SMC, the vertical acceleration of the body and the fluctuation range of the tire dynamic load with the improved SMC was smaller, the improvement effect of the dynamic deflection of the suspension was not significant and has little impact, but the ride comfort of the electromagnetic suspension has improved as a whole.

The RMS value and peak value of the body acceleration, the dynamic deflection of the suspension sprung and the tire dynamic load of the suspension are:

{\begin{cases} x_{RMS} = \sqrt{\frac{1}{n} \sum_{i = 1}^{n} {x_{i}}^{2}} \\ x_{MAX} = \max_{i = 1, \dots, n} (| x_{i} |) \end{cases} i = 1, \dots, n

(34)

Where i is the number, x_i is the value of the body acceleration, the dynamic deflection of the suspension sprung and the tire’s dynamic displacement corresponding to the ith one, and x_RMS is the corresponding RMS value, x_MAX is the corresponding peak value.¹⁸

The RMS value and peak value of the body acceleration, dynamic deflection of the suspension sprung and the tire’s dynamic load are shown in Tables 2 and 3.

Table 2.

RMS value for suspension system response.

Performance	Vertical acceleration of the body/(m·s⁻²)	Dynamic deflection of the suspension sprung/(m)	Tire dynamic load/(N)
Passive suspension	1.4344	0.0256	621.0607
SMC control	0.6738	0.0445	488.8867
Improvement rate/%	53.02%	−61.70%	21.28%
Fuzzy SMC control	0.6473	0.0275	486.9633
Improvement rate/%	54.87%	−7.42%	21.59%
Improved SMC control	0.5911	0.0267	481.8097
Improvement rate/%	58.79%	−4.30%	22.42%

Table 3.

Peak value for suspension system response.

Performance	Vertical acceleration of the body/(m·s⁻²)	Dynamic deflection of the suspension sprung/(m)	Tire dynamic load/(N)
Passive suspension	4.1018	0.0597	1857.09
SMC control	2.1344	0.1546	1813.13
Improvement rate/%	47.96%	−75.20%	2.36%
Fuzzy SMC control	1.9119	0.0746	1837.13
Improvement rate/%	53.38%	−24.95%	1.07%
Improved SMC control	1.8180	0.0665	1823.89
Improvement rate/%	55.68%	−11.39%	0.17%

Based on the data shown in Table 2, compared with the passive suspension, the RMS improvement rates of vertical acceleration of the body of SMC, fuzzy SMC and the improved SMC were 53.02%,54.97% and 58.79%, respectively. The improvement rates of the RMS value of the dynamic deflection of the suspension sprung were −61.70%, −7.42% and −4.30%, respectively, the improvement effect was not obvious. The improvement rates of RMS value of tire dynamic load were 21.28%, 21.59% and 22.42%, respectively. It can be concluded that adopting improved SMC can better improve the performance of the suspension.

According to the data in Table 3, compared with the passive suspension, the peak value improvement rates of vertical acceleration of the body of SMC control, fuzzy SMC control and the improved SMC were 47.96%,53.38% and 55.68%, respectively. The peak value improvement rates of the dynamic deflection of the suspension spring were −75.2%, −24.95% and −11.39%, respectively, the effect was slightly worse. The peak value improvement rates of tire dynamic load were 2.36%,1.07% and 0.17%, respectively. From the data in Table 2 and Tables 3 and it can be obtained that in the electromagnetic suspension, the GA-LQR control can obtain the optimal damping force according to the driving state and road conditions of the vehicle; the inner loop control compare SMC, fuzzy SMC and improved SMC with passive suspension, it was concluded that the improved SMC strategy can improve the ride comfort of the vehicle as a whole, which verified the effectiveness of this method.

Conclusion

(1) Due to the challenges associated with altering the conventional LQR control parameters, the primary loop employed a genetic algorithm to optimize the LQR controller. This optimization aimed to establish the balance between several performance indicators and enhance the overall system performance as soon as possible. The inner loop utilized SMC to optimize the adjustment of current in order to accurately follow the target control force. This approach not only boosted the system’s resistance to external interference, but also guaranteed precise control and rapid reaction. Furthermore, the stability study of the constructed controller certificate was conducted to validate the efficacy of the controller, in accordance with Lyapunov’s stability theorem.

(2) The 1/4 electromagnetic suspension system was simulated and assessed by Matlab/Simulink under the conditions of C-class pavement and the speed of 20 m/s. Comparing with three control strategies, it can be found that the RMS value of body acceleration, suspension sprung dynamic deflection and tire dynamic load of the electromagnetic suspension with the improved SMC were improved by 58.79%, −4.30% and 22.42%, respectively. The peak value were improved by 55.68%, −11.39% and 0.17% respectively, while the effect of the dynamic deflection of the suspension sprung was not significantly improved, but on the whole, the ride comfort of the vehicle was improved.

Footnotes

Author contributions

Methodology, XX; software, JC; formal analysis, JC; resources, XX; data curation, JC, JW; writing—original draft preparation, JC; writing—review and editing, JC, JW; project administration, JC; funding acquisition, FX. All authors have read and agreed to the published version of the manuscript.

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was supported by the National Natural Science Foundation of China (NSFC) (Ref. 52102443) to conduct the research work presented in this article.

ORCID iDs

Xin Xiong

JiaLing Chen

References

Duan

Sun

, et al. Equalization of lithium-ion battery pack based on fuzzy logic control in electric vehicle[J]. IEEE Trans Ind Electron 2018; 65(8): 6762–6771.

Zhao

Wang

Zhao

, et al. LQR force command planning-based sliding mode control for active suspension system[J]. Proc IME J Syst Control Eng 2024; 238(2): 373–385.

Khan

Abid

Ahmed

, et al. Nonlinear control design of a half-car model using feedback linearization and an LQR controller[J]. Applied Sciences-Basel 2020; 10(9): 17.

Bai

Wang

. Robust optimal control for the vehicle suspension system with uncertainties[J]. IEEE Trans Cybern 2022; 52(9): 9263–9273.

Gysen

BLJ

Paulides

JJH

Janssen

JLG

, et al. Active electromagnetic suspension system for improved vehicle dynamics[J]. IEEE Trans Veh Technol 2010; 59(3): 1156–1163.

Yang

Shi

, et al. Fuzzy control of nonlinear electromagnetic suspension systems[J]. Mechatronics 2014; 24(4): 328–335.

Seong

M-S

Sung

K-G

Han

Y-M

, et al.

Vibration control of MR suspension system considering damping force hysteresis

2008, 315–322.

Wang

Kou

Zhang

, et al. A new energy-feeding variable damping control strategy for electromagnetic hybrid suspension systems[J]. Proc Inst Mech Eng - Part D J Automob Eng 2023; 1: 1.

Xiong

Abi

, et al. Step-control and vibration characteristics of a hybrid vehicle suspension system considering energy consumption[J]. Veh Syst Dyn 2022; 60(5): 1531–1554.

10.

Ding

Wang

Meng

, et al. Study on coordinated control of the energy regeneration and the vibration isolation in a hybrid electromagnetic suspension[J]. Proc Inst Mech Eng - Part D J Automob Eng 2017; 231(11): 1530–1539.

11.

Zhang

Chen

, et al. A high-efficiency energy regenerative shock absorber using supercapacitors for renewable energy applications in range extended electric vehicle. Appl Energy 2019: 178; 254.

12.

Montazeri-Gh

Kavianipour

. Investigation of the semi-active electromagnetic damper[J]. Smart Struct Syst 2014; 13(3): 419–434.

13.

Zou

Ding

JJJOEE

, et al.

Sensorless control strategy of permanent magnet synchronous motor based on fuzzy sliding mode controller and fuzzy sliding mode observer

2023; 18(3): 2355–2369.

14.

Meng

Chen

. Research on optimal control for the vehicle suspension based on the simulated annealing algorithm[J]. J Appl Math 2014: 5: 2.

15.

Liu

Shen

S-Y

Wang

Z-H

. The rational solutions to a generalized riccati equation and their application[J]. Int J Mod Phys B 2013; 27(5): 2.

16.

Chen

W-H

. Sliding mode control for a class of uncertain nonlinear system based on disturbance observer[J]. Int J Adapt Control Signal Process 2010; 24(1): 51–64.

17.

Shi

Cai

Wang

, et al. A new aircraft taxiing model based on filtering white NoiseMethod[J]. IEEE Access 2020; 8: 10070–10087.

18.

El Majdoub

Giri

Chaoui

F-Z

. Adaptive backstepping control design for semi-active suspension of half-vehicle with magnetorheological damper[J]. Ieee-Caa Journal of Automatica Sinica 2021; 8(3): 582–596.