Design and Analysis of an Electro-Hydraulic Servo Loading System for a Pavement Mechanical Properties Test Device

Abstract

An electro-hydraulic servo loading system for a pavement mechanical properties test device was designed. The simulation analysis and test results showed that the PID control met the design requirements, but the output’s maximum error did not. Therefore, a fast terminal sliding mode control strategy with an extended state observer (ESO) was proposed. A tracking differentiator was constructed to obtain smooth differential signals from the input signals. The order of the system was reduced by considering the third and higher orders of the system as the total disturbance, and the states and the total disturbance of the system were estimated using the ESO. The fast terminal sliding mode control achieved fast convergence of the system within a limited time. The simulation results showed that the proposed control strategy improved the system accuracy and anti-disturbance ability, and system control performance was optimized.

1. Introduction

Pavement mechanical properties include the deflection and rebound modulus, which reflect the state characteristics of the intensity, stiffness, bearing capacity, and stability of the pavement structure [1,2]. At present, the commonly used measurement methods for the deflection include the Benkelman beam, auto deflectometer, and falling weight deflectometer (FWD) methods [3,4]. Commonly used measurement methods for the rebound modulus include the Benkelman beam, bearing plate, FWD, and portable falling weight deflectometer (PFWD) methods, etc. [5,6,7,8]. During the measurement, a simulated load is applied to the pavement, provided by a load-generating device, to simulate the actual force. The Benkelman beam method uses a truck with a standard axle load on the rear axle as the load-generating device, and the required load is achieved by counterweighting the truck. The auto deflectometer method uses an integrated carrier car with a rear-axle standard axle load of 10 tons as the load-generating device. The load of the FWD and PFWD methods is provided by a heavy hammer, and the impact load is changed by changing the hammer’s mass and height. The bearing plate method uses a hydraulic jack to provide the load, and the load is changed by adjusting the hydraulic jack. There are problems with the load-generating devices of these measurement methods: either only a static or dynamic load can be simulated, the simulated load type is single, the amplitude is limited, the loading accuracy is low, the loading speed is slow, and the loading steps are cumbersome.

An electro-hydraulic servo system has the merits of high control accuracy, a fast response speed, large output power, and flexible signal processing, and it is easy to determine the feedback of various parameters [9]. In a typical electro-hydraulic servo system, the regulated variable is the force or torque. At present, electro-hydraulic servo loading systems have been widely used in the reinforced concrete towers of large wind turbines, triaxial apparatus, helicopter-manipulating boosters, pumping units, and other fields [10,11,12,13].

An electro-hydraulic servo system has characteristics of nonlinearity, time variation, parameter uncertainty, and mechanical–hydraulic coupling [14], which increases the difficulty of system control. In order to achieve a better quality of the system control of an electro-hydraulic servo system, scholars have carried out extensive and in-depth research, introducing different control strategies, such as intelligent, PID, backstep, active disturbance rejection (ADRC), and sliding mode (SMC), which have achieved good control effects. Cao [15] optimized the coefficients of the PID controller using the genetic algorithm, which improved the dynamic response characteristics of the system. Feng et al. [16] proposed an improved particle swarm optimization algorithm to optimize the coefficients of the PID controller, and the system trajectory tracking accuracy was improved. Zaare et al. [17] proposed an optimal robust adaptive fuzzy backstepping control method, which suppressed the chattering problem and effectively reduced the disturbances, while ensuring the system performance was tracked. Yang et al. [18] proposed a backstepping control scheme based on the differentiator by using finite-time-convergent second-order differentiators, which reduced the position tracking error of the system.

ADRC regards the unmodeled dynamics, nonlinearity, disturbance, etc., of the system as the ‘total disturbance’ and estimates it through ESO, to carry out targeted dynamic compensation and optimize system performance [19]. Gao [20] proposed a linear ADRC form, which simplified the controller parameter setting and promoted the development of ADRC. In recent years, ADRC has been applied increasingly in the field of electro-hydraulic servo systems. Gao et al. [21] proposed a compound control strategy that combines velocity compensation with ADRC, which enhanced the anti-jamming ability and improved the control precision of the system. Wang et al. [22] improved the three parts of ADRC, and the gray wolf algorithm was used to set the parameters, which met the precision requirements and tracking performance of a continuous rotary motor electro-hydraulic servo system under unknown strong nonlinear and uncertain strong disturbance factors. Zhang et al. [23] integrated linear ESO and a neuro-adaptive controller for the precise position control of the system.

SMC is a widely used nonlinear control method that has received growing attention due to its advantages of a fast response, strong robustness to disturbances and unmodeled dynamics, and simple physical implementation [24,25,26,27]. The application of SMC in electro-hydraulic servo systems is also increasing. Cerman et al. [28] proposed an innovative approach to the adaptive fuzzy sliding mode control method, to adapt the sliding mode control parameters by introducing the fuzzy self-tuning mechanism; the chattering problem in classical sliding mode control was reduced, and the convergence speed of the system was improved. Du et al. [29] proposed integral sliding mode control and applied it to the electro-hydraulic servo loading system of a heavy vehicle steering test board; the pressure control performance of the system was improved. Chen et al. [30] proposed a control strategy combining feedback linearization theory and sliding mode variable structure theory, which effectively improved the pressure output precision and dynamic response characteristics of the direct-drive volume control electro-hydraulic servo system.

Based on the existing problems of the load-generating devices in pavement mechanical properties testing, a loading scheme using an electro-hydraulic servo loading system is proposed, which was applied to a pavement mechanical properties test device. Simulink simulations and tests were used to verify the effectiveness of the loading scheme. A fast terminal sliding mode control strategy with ESO is also proposed, which further improved the control performance of the electro-hydraulic servo loading system.

2. Pavement Mechanical Properties Test Device

Figure 1 shows the structure of the test device, which mainly includes moving mechanisms, a reaction bearing mechanism, a deflection loading device, a rebound modulus loading device, hydraulic pressure stands, a control system, and a data collection system. The moving mechanisms control the lateral and longitudinal movement of the test device along the road surface; the lifting hydraulic cylinders control the vertical road surface movement of the test device; the reaction force generated during loading is borne by the reaction bearing mechanism; the deflection and rebound modulus loading devices provide the loads required for the deflection and rebound modulus testing, respectively; the hydraulic pressure stands provide hydraulic power; the control system enables movement and loading; the data collection system enables data collection, storage, and analysis; the lateral movement mechanism is installed on the top of the lifting mechanism, which is movably connected to the loading device, and the lifting mechanism is fixed to the lifting hydraulic cylinders, so that the lifting mechanism and loading devices move up or down with the lifting hydraulic cylinders.

The testing process is as follows: First, the test device is moved above the detection area by the motor-driven longitudinal and lateral moving mechanism. Second, the lifting hydraulic cylinders extend downward via the hydraulic pressure stands to lower the lifting mechanism, the four outriggers on the lifting mechanism are in full contact with the road surface, and the test device is jacked up so that the longitudinal moving mechanism is suspended; then, the movement of the lifting hydraulic cylinders is stopped to prevent the longitudinal movement of the test device during the loading process. Third, the hydraulic pressure stands enable the deflection or rebound modulus loading device to press down to the pavement’s mechanical properties testing point. Fourth, two high-precision laser displacement sensors are placed at the testing point. Fifth, the control system controls the loading device to apply a load to the pavement, and the data collection system simultaneously collects sensor data and performs data processing. Sixth, after the loading stops, the control system enables the loading device to retract upward, to prevent it from contacting the road surface, and enables the lifting hydraulic cylinders to retract upward, so that the longitudinal moving mechanism lowers and contacts the road surface; then, the longitudinal moving mechanism travels to the next pavement mechanical properties detection area.

The test device can apply dynamic and static loads. Moreover, there are various ways to achieve the dynamic loading waveform, and the loading accuracy is high, which makes up for the shortcomings of other load-generating devices for testing pavement’s mechanical properties. The loading reaction force generated is borne by the reaction bearing mechanism and is not transmitted to other structures, which ensures smooth loading. High-precision laser displacement sensors are used to measure the displacement signal of the pavement, and the test device is equipped with an automatic collection device and analysis software of the dynamic and static loads and the corresponding deformation during the loading process, with functions such as signal collection and data storage and analysis. The test device enables automated and precise testing of the deflection and rebound moduli, greatly improving the test efficiency and the accuracy of the test results and eliminating the influence of human factors. The test device can move freely along the lateral, longitudinal, and vertical directions of the road surface; hence, it can realize the measurement of testing points for different mechanical properties.

3. Design of the Electro-Hydraulic Servo Loading System

The design requirements of the electro-hydraulic servo loading system are shown in

Table 1. System design requirements.

Loading waveform	Customizable static and dynamic loading waveforms
Loading amplitude	0 kN–50 kN
Loading direction	Vertical direction
Dynamic loading frequency range	0 Hz–15 Hz
Loading amplitude accuracy	0.5 kN

.

The electro-hydraulic servo loading system is mainly composed of a controller, a servo amplifier, an electro-hydraulic servo valve, a hydraulic cylinder, and a force sensor, as shown in Figure 2. The control goal of the system is to ensure the output force of the hydraulic cylinder tracks the given signal quickly and accurately. The specific working process of the electro-hydraulic servo loading system is as follows: the force sensor detects the output force of the hydraulic cylinder and converts it into a voltage signal, which is compared with the voltage of the given signal to obtain the deviation; the controller calculates the control voltage value according to the deviation and then converts it into the corresponding current signal through the servo amplifier; the current signal controls the movement of the electro-hydraulic servo valve and then drives the hydraulic cylinder; the force sensor detects the output force of the hydraulic cylinder for feedback comparison, and the cycle is repeated to achieve precise control of the output force of the hydraulic cylinder. The PID control controls the system by combining the proportion, integral, and differential of the error signal; because of its simple structure, convenient implementation, good adaptability, and its wide use in engineering practice, the test device initially adopted the PID controller.

The schematic diagram of the hydraulic system of the electro-hydraulic servo loading system is shown in Figure 3. The system uses a fixed displacement vane pump as the power source and is driven by a three-phase asynchronous motor. The variation range in the oil supply pressure is controlled by the pressure relay and the unloading valve: when the system pressure rises to a certain value, the pressure relay sends a signal to prompt the unloading valve to unload the hydraulic pump, while the system pressure is maintained by the accumulator. When the system pressure drops to a certain value, the pressure relay sends a reverse signal to change the unloading valve to a loaded state, the hydraulic pump supplies oil to the system, and it fills the accumulator at the same time. The accumulator is used for energy storage and oil supply, reducing system noise, absorbing the energy generated when the system pressure changes suddenly, and improving the stability of the system. Check valves are used to prevent the reverse flow of oil. The electro-hydraulic servo valve is a key component in the system, which connects the electrical part of the system to the hydraulic part to enable the conversion and amplification of the electrical and hydraulic signals and control of the hydraulic cylinder. The system used MOOG’s dual-nozzle-baffle two-stage flow servo valve G761-3005B.

The test device has two sets of loading devices, one using an electro-hydraulic servo loading system and the other using an electro-hydraulic proportional loading system, whose principle is similar to that of the electro-hydraulic servo loading system, used to simulate static loads with higher load values. In this work, the electro-hydraulic servo loading system was studied. The forced structure includes the deflection or rebound modulus loading tooling, which is applied to the deflection and rebound modulus detection, respectively. Through tooling, the applied load is transferred to the subgrade or pavement. The connection mechanism is used to connect the tooling to the hydraulic cylinder; the structure and size of the two connection mechanisms are identical, which makes the tooling interchangeable. Hence, the electro-hydraulic servo loading system can be used in both the deflection and rebound modulus detection loading, as shown in Figure 4.

4. Model Establishment and Simulation Analysis of the Electro-Hydraulic Servo Loading System

4.1. Model Establishment of the Electro-Hydraulic Servo Loading System

The electro-hydraulic servo loading system adopted the form of a valve-controlled symmetrical cylinder. Assuming that the electro-hydraulic servo valve is a zero-opened and four-through slide valve, the four orifices are matched and symmetrical. The oil supply pressure $p_s$ is constant, the return oil pressure $p_0$ is zero, and the pressure loss in the pipeline and the pipeline dynamics are ignored. The pressure is assumed to be equal everywhere in each working cavity of the hydraulic cylinder. The oil temperature and bulk elastic modulus are considered constant, and both the internal and external leakage of the hydraulic cylinder are laminar flow. Under these assumptions, the flow equation of the control valve, the flow continuity equation of the hydraulic cylinder, and the force balance equation between the hydraulic cylinder and the load are established [31]:

$$\left\{\begin{array}{l}{Q}_L=K_q{x}_v-K_c{p}_L\\ Q_L=A_p{\displaystyle \frac{d{x}_p}{dt}}+C_t{p}_L+{\displaystyle \frac{V_t}{4{\beta }_e}}{\displaystyle \frac{d{p}_L}{dt}}\\ F_g=A_p{p}_L=m{\displaystyle \frac{d^2{x}_p}{d{t}^2}}+B_p{\displaystyle \frac{d{x}_p}{dt}}+K{x}_p\end{array}\right.$$

(1)

where $Q_L$ is the load flow, $K_q$ is the servo valve flow gain, $x_v$ is the valve core displacement of the servo valve, $K_c$ is the servo valve flow-pressure coefficient, $p_L$ is the load pressure drop, $A_p$ is the effective area of the hydraulic cylinder piston, $x_p$ is the hydraulic cylinder piston displacement, $C_t$ is the total leakage coefficient of the hydraulic cylinder, $V_t$ is the total volume of the hydraulic cylinder oil chamber, ${\beta }_e$ is the effective bulk elastic modulus, $F_g$ is the output force of the hydraulic cylinder, $m$ is the load mass, $B_p$ is the viscous friction coefficient, and $K$ is the elastic load stiffness.(The complete parameter list and definition are shown in Appendix A: Symbol Table)

Usually, $B_p$ is very small and can be ignored. Laplace transform is performed on the above formula, the transfer function from the valve core displacement of the electro-hydraulic servo valve to the output force of the hydraulic cylinder can be obtained as follows:

$${\displaystyle \frac{F_g}{X_v}}={\displaystyle \frac{\frac{K_q}{K_{ce}}{A}_p\left(\frac{m}{K}{s}^2+1\right)}{\frac{V_{t}m}{4{\beta }_e{K}_{ce}K}{s}^3+\frac{m}{K}{s}^2+\left(\frac{V_t}{4{\beta }_e{K}_{ce}}+\frac{A_p^2}{K_{ce}K}\right)s+1}}$$

(2)

where $X_v$ is obtained by the Laplace transform of $x_v$, $K_{ce}=K_c+C_t$ and it is the total flow-pressure coefficient.

The error signal is

$$U=U_r-U_f$$

(3)

where $U_r$ is the voltage signal of the given signal, and $U_f$ is the voltage signal of the feedback signal.

The servo amplifier converts the deviation signal into a current signal and amplifies it, which can be equivalent to a proportional element [32]:

$${\displaystyle \frac{I}{U}}=K_a$$

(4)

where $K_a$ is the servo amplifier gain, and $I$ is the servo amplifier output current signal.

The electro-hydraulic servo valve can be approximately regarded as a second-order oscillation element:

$${\displaystyle \frac{X_v}{I}}={\displaystyle \frac{K_{sv}}{\frac{s^2}{{\omega }_{sv}^2}+\frac{2{\xi }_{sv}}{{\omega }_{sv}}+1}}$$

(5)

where $K_{sv}$ is the servo valve gain, ${\omega }_{sv}$ is the servo valve natural frequency, and ${\xi }_{sv}$ is the servo valve damping ratio.

The force sensor can be equivalent to a proportional element

$${\displaystyle \frac{U_f}{F_g}}=K_{fF}$$

(6)

where $K_{fF}$ is the force sensor gain.

Based on (1) to (6), the system block diagram is shown in Figure 5.

4.2. Simulation Analysis of the Electro-Hydraulic Servo Loading System

The relevant parameters of the components of the electro-hydraulic servo loading system and the actual project, are shown in

Table 2. System model parameters.

Parameter Name	Value
servo amplifier gain/(A·V−1)	7 × 10−3
servo valve gain/(m3·s−1·A−1)	2.5 × 10−2
servo valve natural frequency/(rad·s−1)	647
servo valve damping ratio	0.85
servo valve flow gain/(m2·s−1)	0.1
effective area of hydraulic cylinder piston/m2	4 × 10−3
load mass/kg	120
total flow-pressure coefficient/(m3·s−1·Pa−1)	2.5 × 10−8
elastic load stiffness/(N·m−1)	1.2 × 109
total volume of hydraulic cylinder oil chamber/m3	7.9 × 10−4
effective bulk elastic modulus/Pa	9 × 108
force sensor gain/(V·N−1)	9 × 10−4

[31].

The model of the electro-hydraulic servo loading system was established in MATLAB/Simulink software, and the PID controller parameters were as follows: $k_p=1.25$, $k_i=0.001$, $k_d=0$. Therefore, it was a PI controller.

Different loading waveforms were simulated. The simulation results are shown in Figure 6, among them, the step-by-step signal simulates the actual loading conditions, as shown in Figure 6b; the error analysis results are shown in

Table 3. Error analysis of simulations.

Load Signal	Absolute Value of Maximum Error/kN	Value of Mean Absolute Deviation/kN
Ramp	0.189	0.064
Step-by-step	0.443	0.187
$45+2\sin \left(2\pi \times 5t\right)$	0.240	0.130
$45+2\sin \left(2\pi \times 10t\right)$	0.444	0.257
Triangular wave 1	0.237	0.189
Triangular wave 2	0.434	0.363

.

From Figure 6 and Table 3, it can be seen that the system can better track different loading waveforms, and the values of the maximum error and mean absolute deviation are less than 0.5 kN, which meet the requirements of the loading amplitude, loading amplitude accuracy, and dynamic loading frequency range.

5. Test of the Electro-Hydraulic Servo Loading System

Figure 7 shows the components of the electro-hydraulic servo loading system, which mainly include hydraulic pressure stands, a force application mechanism, an electro-hydraulic servo valve, and a computer control system. The hydraulic pressure stands provide hydraulic power and are connected to the electro-hydraulic servo valve through tubing. The force application mechanism is composed of a hydraulic cylinder and tooling (the rebound modulus loading tooling was used in this test). The computer control system is composed of an industrial control computer, a PCI-1716 data acquisition board, and a programmable logic controller. The data acquisition board completes the collection of the output force signal of the hydraulic cylinder, and the range of the force sensor is 0–50 kN. The programmable logic controller was an S7-200 SMART for PI control, which outputs the control quantity to the servo amplifier. Then, the servo amplifier converts the voltage signal into a current signal and power amplification, which drives the valve core of the electro-hydraulic servo valve and then the piston of the hydraulic cylinder to move.

The electro-hydraulic servo loading system usually has two loading waveforms in practical application: one is step-by-step loading, the other is sine wave loading. Both step-by-step and sine wave loading signals were used in this test.

5.1. Static Loading Test

Step-by-step loading can test the stability and rapidity of the system and the ability to track mutation signals. Therefore, step-by-step loading was adopted as the static loading test, and two step-by-step loading signals with small and large loading spans were carried out, respectively; the results are shown in Figure 8.

From Figure 8, we observe that the system tracks the step-by-step loading signals well and has good performance both in stability and rapidity without overshoot. The maximum error and mean absolute deviation between the reference and the real signal were calculated, and the results are shown in

Table 4. Error analysis of tests.

Load Signal	Absolute Value of Maximum Error/kN	Value of Mean Absolute Deviation/kN
Step-by-step 1	0.225	0.143
Step-by-step 2	0.475	0.294
$15+2\sin \left(2\pi \times 1t\right)$	0.231	0.098
$45+2\sin \left(2\pi \times 1t\right)$	0.490	0.309
$15+2\sin \left(2\pi \times 5t\right)$	0.485	0.155
$45+2\sin \left(2\pi \times 5t\right)$	0.491	0.225
$15+2\sin \left(2\pi \times 15t\right)$	0.472	0.185
$45+2\sin \left(2\pi \times 15t\right)$	0.482	0.212

. The maximum error was 0.225 kN and 0.475 kN, respectively, and the mean absolute deviation was 0.143 kN and 0.294 kN, respectively.

5.2. Dynamic Loading Test

The sine wave loading was adopted for the dynamic loading test. The amplitude of the sine wave signals was 2 kN, and the tests were carried out under three different frequencies (1 Hz, 5 Hz, 15 Hz) and two different upward offsets (15 kN, 45 kN). The process of the signal rising from the initial to the upward offset was ignored, and the real signal was collected when the sine wave signal began to appear; the test results are shown in Figure 9.

From Figure 9, we observe that the system tracks the sine wave loading signals well. Since the first few sine waves of the real signals did not reach the loading requirements, they were discarded from the analysis. Therefore, the data of 4–10 s were selected for the signal with a frequency of 1 Hz, the data of 0.8–3 s were selected for the signal with a frequency of 5 Hz, and the data of 0.267–1 s were selected for the signal with a frequency of 15 Hz. The error analysis results are shown in Table 4. Under the test frequencies of 1, 5, and 15 Hz, the maximum errors and the mean absolute deviations of the loading system were all less than 0.5 kN.

When the electro-hydraulic servo loading system adopted PI control, although the system loading amplitude accuracy of 0.5 kN was satisfied, the maximum output error was still relatively large; so, it is necessary to improve the system control strategy.

6. Fast Terminal Sliding Mode Control with ESO

In order to improve the control accuracy of the system, the proposed fast terminal sliding mode control strategy with ESO was used, as shown in Figure 10.

6.1. Design of ESO

The ESO estimates system states and the total disturbance. The electro-hydraulic servo loading system is a fifth-order system, and the third-order and above are regarded as part of the total disturbance. The system state equation is

$$\left\{\begin{array}{l}{\dot{x}}_1=x_2\\ {\dot{x}}_2=f+bu\\ y=x_1\end{array}\right.$$

(7)

where $x_2$ is the system output differential value, $f$ is the system total disturbance, including external disturbances (road surface reaction forces), unmodeled dynamics (valve nonlinearity), parameter uncertainties (changes in the elastic modulus of oil), and system coupling effects. It is the sum of all deviations between the actual system and the ideal mathematical model. The total disturbance $f$ disrupts the ideal mathematical model of the system, causing deviation terms in the state equations. $b$ is the compensation factor, $u$ is the system input control quantity, $x_1$ is the system output.

Assuming that the derivative of $f$ exists and is bounded, it is regarded as the third state variable by the ESO:

$$\left\{\begin{array}{l}{\dot{x}}_1=x_2\\ {\dot{x}}_2=x_3+bu\\ {\dot{x}}_3=\dot{f}\\ \mathrm{y}=x_1\end{array}\right.$$

(8)

In this paper, a nonlinear ESO is used, which is constructed as follows [33]:

$$\left\{\begin{array}{l}\epsilon =z_1-y\\ {\dot{z}}_1=z_2-{\beta }_1\epsilon \\ {\dot{z}}_2=z_3-{\beta }_{2}fal\left(\epsilon ,{\alpha }_1,\delta \right)+bu\\ {\dot{z}}_3=-{\beta }_{3}fal\left(\epsilon ,{\alpha }_2,\delta \right)\end{array}\right.$$

(9)

where $z_1$ is the observed value of the system output $x_1$, $z_2$ is the observed value of the system output differential value $x_2$, $z_3$ is the observed value of the system total disturbance; ${\beta }_1,{\beta }_2,{\beta }_3,\delta ,{\alpha }_1,{\alpha }_2$ are adjustable parameters for ESO, where generally, ${\alpha }_1$ is taken as 0.5, and ${\alpha }_2$ is taken as 0.25. $fal(·)$ is defined as

$$fal\left(\epsilon ,\alpha ,\delta \right)=\left\{\begin{array}{l}{\left|\epsilon \right|}^{\alpha }\mathrm{sgn}\left(\epsilon \right), \left|\epsilon \right|>\delta \\ {\displaystyle \frac{\epsilon }{{\delta }^{1-\alpha }}}, \left|\epsilon \right|\le \delta \end{array}\right.$$

(10)

The ESO is discretized to obtain

$$\left\{\begin{array}{l}\epsilon =z_1\left(k\right)-y\left(k\right)\\ z_1\left(k+1\right)=z_1\left(k\right)+h\left[z_2\left(k\right)-{\beta }_1\epsilon \right]\\ z_2\left(k+1\right)=z_2\left(k\right)+h\left[z_3\left(k\right)-{\beta }_{2}fal\left(\epsilon ,{\alpha }_1,\delta \right)+bu\left(k\right)\right]\\ z_3\left(k+1\right)=z_3\left(k\right)-h{\beta }_{3}fal\left(\epsilon ,{\alpha }_2,\delta \right)\end{array}\right.$$

(11)

where $h$ is the sampling period.

6.2. Design of the Tracking Differentiator

The nonlinear tracking differentiator in discrete form is designed as

$$\left\{\begin{array}{l}{v}_1\left(k+1\right)=v_1\left(k\right)+h{v}_2\left(k\right)\\ v_2\left(k+1\right)=v_2\left(k\right)+hfst\left(v_1\left(k\right)-r\left(k\right),v_2\left(k\right),\phi ,h\right)\\ v_3\left(k+1\right)=fst\left(v_1\left(k\right)-r\left(k\right),v_2\left(k\right),\phi ,h\right)\end{array}\right.$$

(12)

where $r\left(k\right)$ is the input signal at time $k$, and $\phi$ is the parameter that determines the tracking speed. $fst\left(\cdot \right)$ is a fast optimal control function, and its definition is as follows:

$$fst\left(x_1,x_2,\phi ,\delta \right)=\left\{\begin{array}{l}-\phi \mathrm{sgn}\left(a\right), \left|a\right|>d\\ -\phi {\displaystyle \frac{a}{d}}, \left|a\right|\le d\end{array}\right.$$

(13)

$$a=\left\{\begin{array}{l}{x}_2+{\displaystyle \frac{a_0-d}{2}}\mathrm{sgn}\left(x_1+h{x}_2\right), \left|x_1+h{x}_2\right|>d_0\\ x_2+{\displaystyle \frac{x_1+h{x}_2}{h}}, \left|x_1+h{x}_2\right|\le d_0\end{array}\right.$$

(14)

where $d=\phi h$, $d_0=hd$, and $a_0=\sqrt{d^2+8\phi \left|x_1+h{x}_2\right|}$.

By using a tracking differentiator, $v_1\left(k\right),v_2\left(k\right),v_3\left(k\right)$ can track, respectively. The tracking differentiator enables the transition arrangement of the input signal and the extraction of the differential signal. When the input signal is mutated, it can provide a smooth output signal to the controller as an input, so that the output varies continuously, and overshoot will not be generated due to the mutations. At the same time, when the input signal exists outside disturbances, the tracking differentiator can also act as a filter [34].

6.3. Design and Analysis of Fast Terminal Sliding Mode Controller

The outputs of the tracking differentiator and ESO are input to the controller to replace the corresponding parameters required by the controller design:

$$\left\{\begin{array}{l}e=r-y=r-x_1=v_1-z_1\\ \dot{e}=\dot{r}-{\dot{x}}_1=\dot{r}-x_2=v_2-z_2\\ \ddot{e}=\ddot{r}-{\dot{x}}_2=\ddot{r}-f-bu=v_3-z_3-bu\end{array}\right.$$

(15)

In order to make the system stable in finite time have rapid convergence, and improve the robustness and stability, a fast terminal sliding surface is constructed by combining the characteristics of the traditional linear sliding mode control, which converges quickly near the equilibrium point, and the terminal sliding mode control, which converges in finite time. The constructed sliding surface is defined as [35]

$$s=c_1\dot{e}+c_{2}e+c_3{\left|e\right|}^{\sigma }\mathrm{sgn}\left(e\right)$$

(16)

where $c_1,c_2,c_3$ are greater than 0, $0<\sigma ={\displaystyle \frac{m}{n}}<1$, and $m,n$ are positive odd integers.

We differentiate the sliding surface as

$$\dot{s}=c_1\ddot{e}+c_2\dot{e}+c_3\sigma {\left|e\right|}^{\sigma -1}\dot{e}\mathrm{sgn}\left(e\right)$$

(17)

The exponential reaching law is adopted. In order to reduce chattering, a nonlinear function is used to replace the sign function:

$$\dot{s}=-k_{1}s-k_2{\displaystyle \frac{s}{\left|s\right|+\Delta }}$$

(18)

where $k_1,k_2$ are greater than 0, and $\Delta$ is a tiny quantity.

Based on (15) to (18), the control quantity is calculated as

$$u={\displaystyle \frac{1}{c_{1}b}}\left[c_2\dot{e}+c_3\sigma {\left|e\right|}^{\sigma -1}\dot{e}\mathrm{sgn}\left(e\right)+c_1{v}_3-c_1{z}_3+k_{1}s+k_2{\displaystyle \frac{s}{\left|s\right|+\Delta }}\right]$$

(19)

The Lyapunov function is constructed to verify the stability of the system, defined as follows:

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

(20)

Differentiating the Lyapunov function, we obtain

$$\dot{V}=s\dot{s}=s\left(-k_{1}s-k_2{\displaystyle \frac{s}{\left|s\right|+\Delta }}\right)=-k_1{s}^2-k_2{\displaystyle \frac{s^2}{\left|s\right|+\Delta }}<0$$

(21)

As $\dot{V}<0$, the proposed fast terminal sliding mode controller meets the Lyapunov stability condition, and the system under the action of the controller has asymptotic stability.

The convergence time of the fast terminal sliding mode control is calculated, where $s=0$, we have

$${\displaystyle \frac{de}{dt}}={\displaystyle \frac{-c_{2}e-c_3{\left|e\right|}^{\sigma }\mathrm{sgn}\left(e\right)}{c_1}}$$

(22)

The definite integral is calculated for Formula (22):

$${\displaystyle \frac{-1}{c_2\left(1-\sigma \right)}}\ln \left\{c_2{e}^{1-\sigma }\left(0\right)+c_3\mathrm{sgn}\left[e\left(0\right)\right]\right\}=-{\displaystyle \frac{1}{c_1}}{t}_s$$

(23)

where $e\left(0\right)$ is the initial error value.

The convergence time $t_s$ is

$$t_s={\displaystyle \frac{c_1}{c_2\left(1-\sigma \right)}}\ln \left\{c_2{e}^{1-\sigma }\left(0\right)+c_3\mathrm{sgn}\left[e\left(0\right)\right]\right\}$$

(24)

Therefore, the output tracking error will converge to zero along the sliding surface in finite time, and the electro-hydraulic servo loading system has the characteristics of convergence in finite time under the proposed control strategy.

7. Simulation Verification

The simulation was carried out using MATLAB(2022b)/Simulink software. The performance of the proposed control strategy was verified by the comparative analysis of the PI control, ADRC, and proposed FTSMC-ESO control strategy.

The PID parameters were obtained through trial and error. Adjusted the parameters of the three components of ADRC separately. First, the parameters of the differential tracker were adjusted until it could track the input signal and its differential signal well. Then, the parameters of the extended state observer were adjusted in sequence until the observer performed well in terms of observing the respective states. Finally, adjusted the parameters of the nonlinear combined controller in sequence until the system output error was minimized. The parameters of the three components of FTSMC-ESO were also adjusted separately. The adjustment steps and values for the parameters of the differential tracker and the extended state observer were consistent with those of ADRC. Finally, the parameters of the fast terminal sliding mode controller were adjusted sequentially based on experience until the system control performance reached an ideal state. Ultimately, the parameter values for each controller were presented in

Table 5. Controller parameters.

Controller	Parameters
PI	$k_p=1.25,k_i=0.001$
ADRC	Tracking differentiator: $h=1\times {10}^{-5},\phi =2.95\times {10}^5$ ESO: $b=175785,{\beta }_1=5\times {10}^4,{\beta }_2=9\times {10}^6,{\beta }_3=4\times {10}^8,\delta =0.0001$ Nonlinear feedback controller: $k_p=155,k_d=1.5,{\alpha }_3=0.98,{\alpha }_4=1.1$
FTSMC-ESO	Tracking differentiator: $h=1\times {10}^{-5},\phi =2.95\times {10}^5$ ESO: $b=175785,{\beta }_1=5\times {10}^4,{\beta }_2=9\times {10}^6,{\beta }_3=4\times {10}^8,\delta =0.0001$ Fast terminal sliding mode controller:$c_1=8,c_2=2730,c_3=900,\sigma ={\displaystyle \frac{3}{5}},k_1=5020,k_2=0.8,\Delta =0.05$

.

The sine wave signal was selected for the loading simulation to verify the dynamic performance of the controllers, and the simulation results are shown in Figure 11. The system tracks the signal well under the three control strategies, and the error under FTSMC-ESO is smaller than that under PI control and ADRC.

A half-sine wave loading simulation was carried out, and the simulation results are shown in Figure 12. There was almost no difference in the effect of the three control strategies in the output curve (a), and the signal was well tracked. In the error curve (b), it can be clearly seen that the errors under the three control strategies increased sequentially from the PI control to the ADRC, and to the FTSMC-ESO.

The anti-disturbance ability of the proposed strategy was verified by applying external disturbance to the electro-hydraulic servo loading system; step disturbance $15step(t-4)KN$ was applied to the system at the fourth second. The simulation results are shown in Figure 13.

From Figure 13, it can be seen that the three control strategies can better track the ramp signal. After the external disturbance was applied to the system, the PI control did not have the ability to resist the disturbance; ADRC and FTSMC-ESO effectively suppressed the disturbance and returned to the equilibrium state. Under the proposed FTSMC-ESO, the fluctuation of the system after external disturbance was smaller than that in ADRC, and the time required to return to the equilibrium state was shorter than that in ADRC; hence, the control system has good robustness to disturbance.

8. Conclusions

In this work, an electro-hydraulic servo loading system was proposed to apply to a pavement mechanical properties test device. PI control was initially used, and static and dynamic loading simulation analysis and testing of the loading system were carried out. The results analysis showed that the maximum output errors and the mean absolute deviations were less than 0.5 kN, and other design requirements of the system were met. Compared with existing load-generating devices, the electro-hydraulic servo loading system can provide simulated loads of various types with high loading accuracy and fast loading speed, which provides more comprehensive support and a basis for the detection of pavement mechanical properties.

Although the designed electro-hydraulic servo loading system met the requirement for the loading amplitude accuracy of 0.5 kN, the maximum output error was close to 0.5 kN in many cases. In order to obtain better control performance for the electro-hydraulic servo loading system, the fast terminal sliding mode control strategy with ESO was proposed. The simulation results showed that the proposed control strategy was superior to PI control and ADRC both in control accuracy and in anti-disturbance ability.