Position measurement and output feedback tracking method of hydraulic servo system based on pressure sensor

Abstract

Instead of using position sensor, a position measurement and output feedback tracking method is proposed based on pressure sensor for a class of hydraulic servo system. First, the desired accumulator pressure is calculated and obtained in an open-loop way according to the desired cylinder position. Then, the position tracking task can be indirectly realized by achieving the pressure control. As a result, only one pressure sensor is required instead of position sensor, which is different from almost all the traditional output feedback position controller of hydraulic system. For the closed-loop pressure tracking, a desired compensation method and an auxiliary variable-based state observer are integrated to form the nonlinear controller. The theoretical and numerical simulation results demonstrate that the presented controller can attain satisfactory pressure tracking performance.

Keywords

Output feedback hydraulic servo system hydraulic accumulator desired compensation auxiliary variable-based observer

Introduction

Electro-hydraulic systems offer distinct advantages over conventional electric or mechanical drives, including high-power density and substantial force/torque output.¹ Because of these features, electro-hydraulic servo systems have been utilized in hydraulic robots,² hydraulic suspensions,³ and various simulators.⁴ To achieve optimal control performance, numerous advanced control strategies have been developed, such as adaptive robust control,⁵ observer-based control,⁶ and intelligent control.⁷

Although many existing control methods can achieve expected tracking accuracy of hydraulic servo systems, they often require all the information of the system states.⁸ To measure signals of all the system states, large installation space and extra cost are inevitable. Moreover, the measurement noise caused by theses sensors may deteriorate the tracking performance. Owing to the above deficiency, many output feedback controllers are developed. The prevailing configuration commonly consists of the extended state observer (ESO) and the back-stepping method.⁹ In addition, many advanced control methods or functions, such as positive-increasing function,¹⁰ adaptive control,^11,12 valve dynamic compensation,^13,14 model predictive control,¹⁵ command filter,^16–18 and disturbance observer,^19–21 are incorporated with ESO to achieve satisfied performance. Alongside the aforementioned ESO-based state estimation approaches, there are other methods available for state estimation, such as fault-tolerant-based output feedback controller,²² high gain observer,^23,24 and fuzzy control.²⁵ However, despite the satisfactory performance of the aforementioned methods, most of them rely on position sensors instead of pressure sensors, even though position sensors are typically more expensive, bulkier, and harder to install than pressure sensors.

Therefore, if a technique could be developed to represent position using pressure signals and achieve output feedback, the benefits of output feedback control would be more pronounced. To achieve this, two issues need to be addressed: how to use pressure signals to represent cylinder position and how to achieve pressure control using only one pressure sensor based on the resulting system model.

For any system composed of hydraulic cylinders and accumulators, the direct relationship between accumulator pressure and position or velocity of hydraulic cylinder can always be found. In addition, some medium and large hydraulic systems are often equipped with a hydraulic accumulator to achieve many functions such as auxiliary driving and energy regeneration.^26,27 In this study, the mathematical correlation between the cylinder’s position and the accumulator’s pressure is deduced based on a specific hydraulic system with accumulator. Subsequently, the tracking problem is transformed from position control to accumulator pressure control.

Although the use of position sensor can be avoided based on the aforementioned method, the task of achieving output feedback pressure control based on the asymmetric-hydraulic-cylinder-based system model is still an ongoing challenge that needs to be addressed. Up to now, several related methods have been explored in the literature, such as the singular value perturbation approach,²⁸ the equivalent model transformation method,²⁹ state equation reconstruction,^30,31 and model linearization.³² As a promising control method, the combination of model predictive control and neural networks may enable precise control through online learning.^33,34 However, the aforementioned method cannot be applied to the proposed system. It is because the position control task is transformed into the pressure control, and these approaches are not suitable for the pressure signal-based state equation. To address this issue, an integrated approach incorporating a desired compensation method and a state observer is employed to solve the problem. Specially, the unknown third order state is estimated by an auxiliary variable-based observer. Furthermore, due to the unavailability of direct measurement or observation for the second-order state of the system, the corresponding desired value is used in the model compensation part both in the controller and the state observer. Through rigorous theoretical derivations and analysis, the system stability is confirmed. Besides, the resulting tracking accuracy of the proposed method, including pressure tracking and final position tracking, is validated by comparative results.

The primary contributions are listed as:

The position tracking objective is accomplished indirectly through the pressure control of the hydraulic accumulator.

An output feedback control method combining desired compensation and state observation method is proposed.

A position signal acquisition method based on pressure information is proposed for hydraulic systems with hydraulic accumulators.

Nonlinear model and control objective

Figure 1(a) shows the simplified configuration of the system, while Figure 1(b) depicts the conventional two-chamber hydraulic actuator. In the proposed actuator, there are three operational chambers: the storage chamber, the piston chamber and rod chamber. The last two chambers of the system are connected with a proportional directional valve (PDV), which serves as the control element.

Figure 1.

Simplified configuration of the proposed system (a) and the traditional two chamber hydraulic actuator (b).

Pressure model of the hydraulic accumulator

The first step is to deduce the pressure model for the accumulator. Assuming a scenario where the hydraulic cylinder undergoes rapid movement over a short duration, it is reasonable to consider the working process as an adiabatic process. According to Boyle’s theorem, one obtains

p_{a} V_{a}^{n} = p_{0} V_{0}^{n}

(1)

where $p_{a}$ and $V_{a}$ are the pressure and corresponding gas volume of the hydraulic accumulator, respectively, $p_{0}$ is the pre-charge pressure, $V_{0}$ is the total volume of the hydraulic accumulator, $n$ is the polytropic index, which is assumed as a constant value during the movement of the hydraulic cylinder.

Based on the assumption that a positive flow rate for the hydraulic accumulator indicates energy release, while a negative flow rate indicates energy storage, the pressure dynamic is deduced as follows by referring to equation (1).

\begin{matrix} {\overset{\cdot}{p}}_{a} = \partial (\frac{p_{0} V_{0}^{n}}{V_{a}^{n}}) / \partial t = - p_{0} V_{0}^{n} {nV}_{a}^{- n - 1} {\overset{\cdot}{V}}_{a} \\ = - \frac{n p_{0}}{V_{0}} Q_{a} {(\frac{V_{0}}{V_{a}})}^{n + 1} = - γ \frac{p_{0}}{V_{0}} Q_{a} = - γ \frac{p_{0}}{V_{0}} A_{a} \overset{\cdot}{L} \end{matrix}

(2)

where $γ = n {(\frac{V_{0}}{V_{a}})}^{n + 1} = n {(\frac{p_{a}}{p_{0}})}^{\frac{n + 1}{n}}$ , $Q_{a} = {\overset{\cdot}{V}}_{a} = A_{a} \overset{\cdot}{L}$ is the flow of hydraulic accumulator, $A_{a}$ is the area of the storage chamber, $L$ is the actual position of the cylinder.

By combining equation (2), the time derivative of equation (2) is

{\overset{\cdot\cdot}{p}}_{a} = - γ \frac{p_{0}}{V_{0}} A_{a} \overset{\cdot\cdot}{L} - \overset{\cdot}{γ} \frac{p_{0}}{V_{0}} A_{a} \overset{\cdot}{L} = κ γ \overset{\cdot\cdot}{L} + \frac{n + 1}{n} {(\frac{p_{a}}{p_{0}})}^{- 1} \frac{{\overset{\cdot}{p}}_{a}^{2}}{p_{0}}

(3)

where $κ = - \frac{p_{0}}{V_{0}} A_{a}$ .

According to the expression of $γ$ , Figures 2 and 3 can be drawn. (The relationship between $γ$ , $p_{a} / p_{0}$ , and $n$ ). It can be inferred from the two figures that the relationship curves between $γ$ and $p_{a} / p_{0}$ approach straight line under a certain polytropic index. Therefore, the following equation can be obtained.

γ = θ_{1} + θ_{2} \frac{p_{a}}{p_{0}}

(4)

where $θ_{1}$ and $θ_{2}$ are constants, and the values can be calculated based on Figure 2 when $n$ is determined.

Figure 2.

The relationship between $γ$ and $p_{a} / p_{0}$ under different $n$ .

Figure 3.

The relationship between $γ$ , $p_{a} / p_{0}$ , and $n$ .

Based on equation (4) and consider the model uncertainties, equation (3) can be simplified as follows

{\overset{\cdot\cdot}{p}}_{a} = κ (θ_{1} + θ_{2} \frac{p_{a}}{p_{0}}) \overset{\cdot\cdot}{L} + θ_{3} {(\frac{p_{a}}{p_{0}})}^{- 1} \frac{{\overset{\cdot}{p}}_{a}^{2}}{p_{0}} + d_{1}

(5)

where $θ_{3} = \frac{n + 1}{n}$ , $d_{1}$ represents the model uncertainties.

State equation of the system

The subsequent step is to obtain the complete state equation of the system by combining the pressure dynamic with the load balance equation and valve model. Considering that the friction force and damping force are relatively negligible compared with the load, the dynamic of the hydraulic piston is

m \overset{\cdot\cdot}{L} = p_{1} A_{1} - p_{2} A_{2} + p_{a} A_{a} - G (t) + d_{2}

(6)

where $m$ is the piston mass, $p_{1}$ , $p_{2}$ are the pressures of piston rod chamber respectively, $A_{1}$ , $A_{2}$ are the corresponding effective areas, $G$ is the gravity of load, $d_{2}$ represents the lumped uncertainties, including parametric uncertainties, unmodelled dynamics (such as friction force and damping force) and external disturbance.

The flow equation of the cylinder is expressed as

{\begin{matrix} Q_{1} = k_{q} k_{x} u [\underset{g_{1}}{\underset{︸}{s_{g} (u) \sqrt{p_{s} - p_{1}} + s_{g} (- u) \sqrt{p_{1} - p_{t}}}}] \\ Q_{2} = k_{q} k_{x} u [\underset{g_{2}}{\underset{︸}{s_{g} (u) \sqrt{p_{2} - p_{t}} + s_{g} (- u) \sqrt{p_{s} - p_{2}}}}] \end{matrix}

(7)

s_{g} (•) = {\begin{matrix} 1 if • \geq 0 \\ 0 if • < 0 \end{matrix}

(8)

where $k_{q}$ and $k_{x}$ are the parameters of the valve, $p_{s}$ is the pump pressure, $p_{t}$ is the tank pressure, $Q_{1}$ is the flow rate of the forward chamber and $Q_{2}$ is the flow rate of backward chamber, $u$ is the control signal generated by the PDV.

The pressure dynamics of the piston and rod chamber are given as follows by neglecting the leakage.

{\begin{matrix} {\overset{\cdot}{p}}_{1} = \frac{β_{e}}{V_{1} (L)} (Q_{1} - A_{1} \overset{\cdot}{L}) + d_{3} \\ {\overset{\cdot}{p}}_{2} = \frac{β_{e}}{V_{2} (L)} (A_{2} \overset{\cdot}{L} - Q_{2}) + d_{4} \end{matrix}

(9)

where $V_{1} (L) = A_{1} L$ , $V_{2} (L) = A_{2} (L_{max} - L)$ are the volumes of piston chamber and rod chamber respectively, $L_{max}$ is the stroke of the hydraulic cylinder, $β_{e}$ is the effective bulk modulus of the system, $d_{3}$ and $d_{4}$ denote the lumped uncertainties such as the neglected items related to leakage.

The state variables vector is

[x_{1}, x_{2}, x_{3}]^{T} = [p_{a}, {\overset{\cdot}{p}}_{a}, A_{1} p_{1} - A_{2} p_{2}]^{T}

(10)

Based on equations (5)–(10), the pressure signal-based state equation is given as follows

{\begin{matrix} {\overset{\cdot}{x}}_{1} = x_{2} \\ {\overset{\cdot}{x}}_{2} = φ_{1} (x_{1}) S_{t} (x_{1}, x_{3}) + φ_{2} S_{f} (x_{1}, x_{2}) + D_{1} \\ {\overset{\cdot}{x}}_{3} = S_{g} (p_{1}, p_{2}, L) u - S_{c} (L, \overset{\cdot}{L}) + D_{2} \end{matrix}

(11)

where

$c_{1} = \frac{β_{e}}{V_{1} (L)}$ , $c_{2} = \frac{β_{e}}{V_{2} (L)}$ , $S_{t} (x_{1}, x_{3}) = x_{3} + A_{a} x_{1} - G$ , $S_{f} (x_{1}, x_{2}) = {(\frac{x_{1}}{p_{0}})}^{- 1} \frac{x_{2}^{2}}{p_{0}}$ ,

$D_{1} = d_{1} + (θ_{1} + θ_{2} \frac{x_{1}}{p_{0}}) \frac{κ}{m} d_{2}$ , $S_{g} (p_{1}, p_{2}, L) = k_{q} k_{x} (A_{1} c_{1} g_{1} + A_{2} c_{2} g_{2})$ ,

$S_{c} (L, \overset{\cdot}{L}) = A_{1}^{2} c_{1} \overset{\cdot}{L} + A_{2}^{2} c_{2} \overset{\cdot}{L}$ , $D_{2} = A_{1} d_{3} - A_{2} d_{4}$ , $φ_{1} (x_{1}) = (θ_{1} + θ_{2} \frac{x_{1}}{p_{0}}) \frac{κ}{m}$ , $φ_{2} = θ_{3}$

Transformation of the system model

From equation (11), one can find that the cylinder position is in the third state equation, which is unknown and is not represented by state variables. Therefore, the model should be transformed, thereby facilitating the subsequent derivation.

Based on equation (1), one can obtain

p_{a} = \frac{p_{0} V_{0}^{n}}{V_{a}^{n}} = \frac{p_{0} V_{0}^{n}}{{(V_{min} + A_{a} L)}^{n}}

(12)

where $V_{min} = {(\frac{p_{0} V_{0}^{n}}{p_{max}})}^{\frac{1}{n}}$ is the minimum gas volume of the hydraulic accumulator, $p_{max}$ is the maximum working pressure of the hydraulic accumulator.

Based on equation (12), the relationship between the position of the hydraulic cylinder and the pressure of the hydraulic accumulator can be given as.

p_{a} = \frac{p_{0} V_{0}^{n}}{V_{a}^{n}} = \frac{p_{0} V_{0}^{n}}{{({(\frac{p_{0} V_{0}^{n}}{p_{max}})}^{\frac{1}{n}} + A_{a} L)}^{n}}

(13)

From equation (13) and Figure 4, the position of the hydraulic cylinder is directly related to the pressure of the hydraulic accumulator once the parameters are determined.

Figure 4.

The relationship between $γ$ , $p_{a} / p_{0}$ , and $n$ .

Based on equation (2), equation (13) and noting the definition of the state variables, it can be inferred that $L$ and $\overset{\cdot}{L}$ can be represented by $x_{1}$ and $x_{2}$ . Therefore, equation (11) is reconstructed as follows

{\begin{matrix} {\overset{\cdot}{x}}_{1} = x_{2} \\ {\overset{\cdot}{x}}_{2} = φ_{1} (x_{1}) S_{t} (x_{1}, x_{3}) + φ_{2} S_{f} (x_{1}, x_{2}) + D_{1} \\ {\overset{\cdot}{x}}_{3} = S_{g} (p_{1}, p_{2}, x_{1}) u - S_{c} (x_{1}, x_{2}) + D_{2} \end{matrix}

(14)

It can be seen that unmeasured two chamber pressures are still in the third state equation of equation (14). Then, the relationship between two chamber pressures and state variables should be deduced.

By neglecting oil compression, leakage, lumped uncertainties and assuming the tank pressure is zero, the following equations can be obtained from equations (7) and (9).

\frac{A_{1}}{A_{2}} = {\begin{matrix} \frac{\sqrt{p_{s} - p_{1}}}{\sqrt{p_{2}}} extension \\ \frac{\sqrt{p_{1}}}{\sqrt{p_{s} - p_{2}}} retraction \end{matrix}

(15)

Because $x_{3} = A_{1} p_{1} - A_{2} p_{2}$ , $p_{1}$ and $p_{2}$ can be represented based on equation (15).

p_{1} = (A_{2} p_{2} + x_{3}) / A_{1}

(16)

p_{2} = {\begin{matrix} (A_{1} p_{s} - x_{3}) / (A_{1}^{3} / A_{2}^{2} + A_{2}) extension \\ (A_{1}^{3} p_{s} / A_{2}^{2} - x_{3}) / (A_{1}^{3} / A_{2}^{2} + A_{2}) retraction \end{matrix}

(17)

Therefore, equation (14) is reconstructed as

{\begin{matrix} {\overset{\cdot}{x}}_{1} = x_{2} \\ {\overset{\cdot}{x}}_{2} = φ_{1} (x_{1}) S_{t} (x_{1}, x_{3}) + φ_{2} S_{f} (x_{1}, x_{2}) + D_{1} \\ {\overset{\cdot}{x}}_{3} = S_{g} (x_{1}, x_{3}) u - S_{c} (x_{1}, x_{2}) + D_{2} \end{matrix}

(18)

The modeling error caused by approximate estimation of pressure can be regarded as a part of $D_{2}$ .

Assumption 1: Unmatched and matched disturbances are bounded, that is,

| D_{1} | \leq D_{1 max}, | D_{2} | \leq D_{2 max}

(19)

Where $D_{1 max}$ and $D_{2 max}$ are positive constants.

Control objective and scheme

The equations (11) and (18) are state equations based on accumulator pressure. To achieve control of hydraulic cylinder position, it is necessary to establish a numerical relationship between the accumulator pressure and the cylinder position. Fortunately, it has been done by equation (13). Therefore, the desired hydraulic cylinder position can be brought into equation (13) to obtain the corresponding expected accumulator pressure. The polytropic index in equation (13) remains uncertain, playing a pivotal role in directly setting the target pressure and, in turn, indirectly influencing the attainable precision. Importantly, the determination of this index is not incorporated within the closed-loop pressure tracking system. As such, the choice of the polytropic index constitutes an open-loop aspect, resulting in a combined control approach that integrates open-loop parameter selection with closed-loop pressure control, as illustrated in Figure 5.

Figure 5.

Schematic diagram of the proposed method.

The control task is concluded as: the expected pressure of the hydraulic accumulator $x_{1 d}$ is calculated according to the expected cylinder position $L_{d}$ . Then, the control law is derived based on equation (18) to achieve precise pressure tracking while the actual position of the hydraulic cylinder $L$ will approach $L_{d}$ automatically and simultaneously.

Remark 1: To simplify the analysis process, the following notations are utilized in the paper: $\hat{•}$ is used to represent the estimate of $•$ , $\tilde{•} = • - \hat{•}$ represents the estimation error.

Closed-loop pressure control

From equation (18), it is evident that the conventional state observer, such as ESO, cannot be applied to the system. Therefore, an auxiliary variable-based observer is only used to obtain the estimate of the third state variable, and the time derivative of the accumulator pressure is regarded as the value of the second state variable. However, this will inevitably bring about measurement noise and affect achievable control performance. To mitigate the impact of measurement noise, the observer and controller design process incorporate the desired compensation method.³⁵ Specially, the desired value of $x_{2}$ , namely ${\overset{\cdot}{x}}_{1 d}$ , is used in the model compensation part both in the observer and the nonlinear controller so as to attenuate the measurement noise. Hence, the closed-loop pressure control can be seen as a combination of state observation and desired compensation method (as is shown in Figure 5).

Design of the state observer

An auxiliary variable is defined as³⁶

ξ = x_{3} - k_{o} {\overset{\cdot}{x}}_{1 d}

(20)

where $k_{o} < 0$ is the observer gain.

In this equation, the desired value of $x_{2}$ , namely ${\overset{\cdot}{x}}_{1 d}$ , is used for desired compensation. By doing so, the noise contaminated signal is replaced by the smooth signal.

Based on equations (18) and (20), the dynamic of the auxiliary variables is

\overset{\cdot}{ξ} = S_{g} (x_{1}, x_{3}) u - S_{c} (x_{1}, x_{2}) + D_{2} - k_{o} {\overset{\cdot\cdot}{x}}_{1 d}

(21)

It can be seen from the above equation that the unmeasured signal $x_{2}$ still exist. To address this issue, desired compensation method is also utilized. By using the desired value of $x_{2}$ , equation (21) can be rewritten as

\begin{matrix} \overset{\cdot}{ξ} = S_{g} (x_{1}, {\hat{x}}_{3}) u - S_{c} (x_{1}, {\overset{\cdot}{x}}_{1 d}) + D_{2} + ς_{1} - k_{o} {\overset{\cdot\cdot}{x}}_{1 d} \\ = S_{g} (x_{1}, {\hat{x}}_{3}) u - S_{c} (x_{1}, {\overset{\cdot}{x}}_{1 d}) + D_{2} + ς_{1} + ς_{2} - k_{o} {\overset{\cdot}{x}}_{2} \\ = S_{g} (x_{1}, {\hat{x}}_{3}) u - S_{c} (x_{1}, {\overset{\cdot}{x}}_{1 d}) + D_{2} + ς_{1} + ς_{2} \\ - k_{o} [φ_{1} (x_{1}) (ξ + k_{o} {\overset{\cdot}{x}}_{1 d} + A_{a} x_{1} - G) + φ_{2} S_{f} (x_{1}, x_{2}) + D_{1}] \\ = S_{g} (x_{1}, {\hat{x}}_{3}) u - S_{c} (x_{1}, {\overset{\cdot}{x}}_{1 d}) + D_{2} + ς_{1} + ς_{2} \\ - k_{o} [φ_{1} (x_{1}) (ξ + k_{o} {\overset{\cdot}{x}}_{1 d} + A_{a} x_{1} - G) + φ_{2} S_{f} (x_{1}, {\overset{\cdot}{x}}_{1 d}) + ς_{3} + D_{1}] \\ = - k_{o} φ_{1} ξ + o + D_{3} \end{matrix}

(22)

where

$ς_{1} = ς_{11} + ς_{12}$ , $ς_{11} = S_{g} (x_{1}, x_{3}) u - S_{g} (x_{1}, {\hat{x}}_{3}) u$ , $ς_{12} = - S_{c} (x_{1}, x_{2}) + S_{c} (x_{1}, {\overset{\cdot}{x}}_{1 d})$ ,

$ς_{2} = k_{o} {\overset{\cdot}{x}}_{2} - k_{o} {\overset{\cdot\cdot}{x}}_{1 d}$ , $ς_{3} = φ_{2} S_{f} (x_{1}, x_{2}) - φ_{2} S_{f} (x_{1}, {\overset{\cdot}{x}}_{1 d})$ , $ς_{4} = ς_{3} + D_{1} - {\overset{\cdot}{x}}_{2} + {\overset{\cdot\cdot}{x}}_{1 d}$ ,

\begin{matrix} o = S_{g} (x_{1}, {\hat{x}}_{3}) u - S_{c} (x_{1}, {\overset{\cdot}{x}}_{1 d}) - k_{o} φ_{1} (x_{1}) (k_{o} {\overset{\cdot}{x}}_{1 d} \\ + A_{a} x_{1} - G) - k_{o} φ_{2} S_{f} (x_{1}, {\overset{\cdot}{x}}_{1 d}), \end{matrix}

D_{3} = D_{2} + ς_{1} + ς_{2} - k_{o} (ς_{3} + D_{1}) = D_{2} + ς_{1} - k_{o} ς_{4}

The dynamic of $\hat{ξ}$ is defined as

\overset{\cdot}{\hat{ξ}} = - k_{o} φ_{1} \hat{ξ} + o

(23)

The estimate of $x_{3}$ can be given as follows based on equation (20).

{\hat{x}}_{3} = \hat{ξ} + k_{o} {\overset{\cdot}{x}}_{1 d}

(24)

Based on equations (20) and (24), the estimation error of $x_{3}$ is equal to that of $ξ$ (namely, ${\tilde{x}}_{3} = \tilde{ξ}$ ). Therefore, the time derivative of the estimation error is deduced according to equations (22) and (23).

\overset{\cdot}{{\tilde{x}}_{3}} = \overset{\cdot}{\tilde{ξ}} = - k_{o} φ_{1} \tilde{ξ} + D_{3}

(25)

Remark 2: It can be inferred that $ς_{12}$ and $ς_{4}$ are bounded based on Assumption 1 and mean value theorem, namely $| ς_{12} | \leq {ς_{12}}_{max}$ , $| ς_{4} | \leq {ς_{4}}_{max}$ . Besides, according to the mean value theorem, we have $ς_{11} \leq λ_{9} | {\tilde{x}}_{3} | = λ_{9} | \tilde{ξ} |$ , and the parameter $λ_{9}$ is a positive constant which will be invoked later.

A Lyapunov function is selected as $V_{o} = {\tilde{ξ}}^{2} / 2$ . Therefore, based on the Young’s inequality, the time derivative of $V_{o}$ is given as follows.

\begin{matrix} {\overset{\cdot}{V}}_{o} = - k_{o} φ_{1} {\tilde{ξ}}^{2} + \tilde{ξ} D_{3} = - k_{o} φ_{1} {\tilde{ξ}}^{2} + \tilde{ξ} (D_{2} + ς_{1} - k_{o} ς_{4}) \\ \leq - k_{o} φ_{1} {\tilde{ξ}}^{2} + λ_{9} {\tilde{ξ}}^{2} + k_{o} {\tilde{ξ}}^{2} / 2 + (D_{2 max} + ς_{12 max} - k_{o} ς_{4 max})^{2} / (2 k_{o}) \\ = - [k_{o} (φ_{1} - 1 / 2) - λ_{9}] {\tilde{ξ}}^{2} + (D_{2 max} + ς_{12 max} - k_{o} ς_{4 max})^{2} / (2 k_{o}) \\ = - γ V_{o} + κ \end{matrix}

(26)

where $γ = 2 k_{o} (φ_{1} - 1 / 2) - 2 λ_{9}$ , $κ = (D_{2 max} + ς_{12 max} - k_{o} ς_{4 max})^{2} / (2 k_{o})$ .

Based on equation (26), it can be claimed that $V_{o} (t) \leq V_{o} (0) e^{- γ t} + κ / γ$ so that $\tilde{ξ}$ will converge to a small compact set. Specially, $\tilde{ξ} \to 0$ as $t \to \infty$ holds for $k_{o} \to - \infty$ . It should be noted that the observer gain is designed as a negative value because $φ_{1} < 0$ according to its expression.

Desired compensation-based nonlinear controller

Based on the above analysis, the estimate of the third state variable, namely ${\hat{x}}_{3}$ , is obtained. In this part, the desired compensation method and the state observer is combined to construct the nonlinear controller, that is, the state variables $x_{2}$ and $x_{3}$ are replaced by ${\overset{\cdot}{x}}_{1 d}$ and ${\hat{x}}_{3}$ .

Step 1: A set of error variables is defined as

\begin{matrix} z_{1} = x_{1} - x_{1 d}, z_{2} = {\overset{\cdot}{z}}_{1} + k_{1} z_{1} \\ = x_{2} - α_{1}, α_{1} \overset{Δ}{=} {\overset{\cdot}{x}}_{1 d} - k_{1} z_{1} \end{matrix}

(27)

where $z_{1}$ is tracking error, $α_{1}$ is virtual control input, $k_{1}$ is a positive control gain.

By substituting the unmeasured signal $x_{2}$ with desired signal ${\overset{\cdot}{x}}_{1 d}$ , the time derivative of $z_{2}$ can be calculated as follows based on equations (18) and (27)

\begin{matrix} {\overset{\cdot}{z}}_{2} = φ_{1} (x_{1}) (x_{3} + A_{a} x_{1} - G) + φ_{2} S_{f} (x_{1}, x_{2}) + D_{1} - {\overset{\cdot}{α}}_{1} \\ = φ_{1} (x_{1}) (x_{3} + A_{a} x_{1} - G) + φ_{2} S_{f} (x_{1}, {\overset{\cdot}{x}}_{1 d}) \\ + D_{1} - {\overset{\cdot\cdot}{x}}_{1 d} + k_{1} (z_{2} - k_{1} z_{1}) + ς_{3} \end{matrix}

(28)

The virtual control input $α_{2}$ for $x_{3}$ is constructed as

\begin{matrix} α_{2} = α_{2 a} + α_{2 s}, α_{2 s} = - k_{2} {φ_{1}}^{- 1} z_{2} \\ α_{2 a} = - (A_{a} x_{1} - G) - {φ_{1}}^{- 1} [φ_{2} S_{f} (x_{1}, {\overset{\cdot}{x}}_{1 d}) - {\overset{\cdot\cdot}{x}}_{1 d}] \end{matrix}

(29)

Where $k_{2}$ is a positive control gain, $α_{2 a}$ is a model compensation term, $α_{2 s}$ act as a robust control law to stabilize the system.

Substituting equation (29) into equation (28), one obtains

{\overset{\cdot}{z}}_{2} = φ_{1} z_{3} + ς_{3} + D_{1} + k_{1} (z_{2} - k_{1} z_{1}) - k_{2} z_{2}

(30)

where $z_{3} = x_{3} - α_{2}$ is the discrepancy between $x_{3}$ and $α_{2}$ .

Step 2: Differentiating $z_{3}$ and noting equation (18), we have

\begin{matrix} {\overset{\cdot}{z}}_{3} = S_{g} (x_{1}, x_{3}) u - S_{c} (x_{1}, x_{2}) + D_{2} - {\overset{\cdot}{α}}_{2 c} - {\overset{\cdot}{α}}_{2 u} \\ = S_{g} (x_{1}, {\hat{x}}_{3}) u - S_{c} (x_{1}, {\overset{\cdot}{x}}_{1 d}) + D_{2} + ς_{1} - {\overset{\cdot}{α}}_{2 c} - {\overset{\cdot}{α}}_{2 u} \end{matrix}

(31)

In this equation, the third state variable is replaced by its corresponding estimate according to the result of equation (24), thereby canceling the measurement of the third state variable.

To avoid using pressure differentiation, ${\overset{\cdot}{α}}_{2}$ will be split into two parts.

where ${\overset{\cdot}{α}}_{2 u} = \frac{\partial α_{2}}{\partial x_{1}} (z_{2} - k_{1} z_{1}) + \frac{\partial α_{2}}{\partial x_{2}} [{\overset{\cdot}{z}}_{2} - k_{1} (z_{2} - k_{1} z_{1})]$ denotes the incalculable part of ${\overset{\cdot}{α}}_{2}$ , ${\overset{\cdot}{α}}_{2 c} = \frac{\partial α_{2}}{\partial t} + \frac{\partial α_{2}}{\partial x_{1}} {\overset{\cdot}{x}}_{1 d} + \frac{\partial α_{2}}{\partial x_{2}} {\overset{\cdot\cdot}{x}}_{1 d}$ represents the calculable part of ${\overset{\cdot}{α}}_{2}$ .

The resulting control law can be defined as

u = [S_{c} (x_{1}, {\overset{\cdot}{x}}_{1 d}) + {\overset{\cdot}{α}}_{2 c} - k_{3} ({\hat{x}}_{3} - α_{2})] / S_{g} (x_{1}, {\hat{x}}_{3})

(32)

where $k_{3}$ is a positive control gain.

Substituting equation (32) into equation (31), one obtains

\begin{matrix} {\overset{\cdot}{z}}_{3} = S_{g} (x_{1}, {\hat{x}}_{3}) u - S_{c} (x_{1}, {\overset{\cdot}{x}}_{1 d}) + D_{2} + ς_{1} - {\overset{\cdot}{α}}_{2 c} - {\overset{\cdot}{α}}_{2 u} \\ = - k_{3} z_{3} + k_{3} {\tilde{x}}_{3} + D_{2} + ς_{1} - \frac{\partial α_{2}}{\partial x_{1}} (z_{2} - k_{1} z_{1}) \\ - \frac{\partial α_{2}}{\partial x_{2}} [{\overset{\cdot}{z}}_{2} - k_{1} (z_{2} - k_{1} z_{1})] \end{matrix}

(33)

Main results

According to the mean value theorem and equations (19), (27), (30), and Remark 2, the following inequality can be obtained.

| ς_{3} | \leq λ_{1} | z_{1} | + λ_{2} | z_{2} |

(34)

| ς_{2} | \leq λ_{3} | z_{1} | + λ_{4} | z_{2} | + λ_{5} | z_{3} | + λ_{6} D_{1 max}

(35)

| ς_{1} | \leq | ς_{11} | + | ς_{12} | \leq λ_{7} | z_{1} | + λ_{8} | z_{2} | + λ_{9} | \tilde{ξ} |

(36)

where $λ_{i}, i = 1 ~ 9$ are positive constants.

Remark 3: From equations (34) to (36), it can be seen that the system exhibits uncertain nonlinearity. In this study, it is suppressed through robust terms. However, when the system suffers significant uncertain nonlinearity, it is necessary to combine online adaptive compensation method to obtain the estimates of the above items, thereby compensating them in a forward way. Similar advanced methods can refer to Yang et al.^37,38

The Lyapunov function is defined as

V = \frac{1}{2} z_{1}^{2} + \frac{1}{2} z_{2}^{2} + \frac{1}{2} z_{3}^{2} + \frac{1}{2} {\tilde{ξ}}^{2}

(37)

In addition, it can be inferred that ${\tilde{x}}_{3} = \tilde{ξ}$ based on equation (24). Considering the fact that $φ_{1} < 0$ and $k_{o} < 0$ , the time derivative of equation (37) is given as follows based on equations (19), (27), (30), (33)–(36).

\begin{matrix} \overset{\cdot}{V} = z_{1} (z_{2} - k_{1} z_{1}) + z_{2} [φ_{1} z_{3} + ς_{3} + D_{1} + k_{1} (z_{2} - k_{1} z_{1}) - k_{2} z_{2}] \\ + z_{3} [- k_{3} z_{3} + k_{3} \tilde{ξ} + D_{2} + ς_{1} - \frac{\partial α_{2}}{\partial x_{1}} (z_{2} - k_{1} z_{1}) \\ - \frac{\partial α_{2}}{\partial x_{2}} (φ_{1} z_{3} + ς_{3} + D_{1} - k_{2} z_{2})] \\ + \tilde{ξ} (- k_{o} φ_{1} \tilde{ξ} + D_{2} + ς_{1} + ς_{2} - k_{o} ς_{3} - k_{o} D_{1}) \\ \leq - k_{1} z_{1}^{2} - (k_{2} - k_{1} - λ_{2}) z_{2}^{2} - (k_{3} - | \frac{\partial α_{2}}{\partial x_{2}} φ_{1} |) z_{3}^{2} \\ - (k_{o} φ_{1} - λ_{9}) {\tilde{ξ}}^{2} + (1 + k_{1}^{2} + λ_{1}) | z_{1} | | z_{2} | \\ + (| φ_{1} | + | \frac{\partial α_{2}}{\partial x_{1}} | + k_{2} | \frac{\partial α_{2}}{\partial x_{2}} | + λ_{8} + λ_{2} | \frac{\partial α_{2}}{\partial x_{2}} |) | z_{2} | | z_{3} | \\ + (k_{1} | \frac{\partial α_{2}}{\partial x_{1}} | + λ_{7} + λ_{1} | \frac{\partial α_{2}}{\partial x_{2}} |) | z_{1} | | z_{3} | \\ + | z_{2} | | D_{1 max} | + (k_{3} + λ_{9} + λ_{5}) | z_{3} | | \tilde{ξ} | + | z_{3} | | D_{2 max} | \\ + | \frac{\partial α_{2}}{\partial x_{2}} | | z_{3} | | D_{1 max} | + | \tilde{ξ} | | D_{2 max} | + (| k_{o} | + λ_{6}) | \tilde{ξ} | | D_{1 max} | \\ + (λ_{3} + λ_{7} + | k_{o} | λ_{1}) | z_{1} | | \tilde{ξ} | + (λ_{4} + λ_{8} + | k_{o} | λ_{2}) | z_{2} | | \tilde{ξ} | \\ = - k_{1} z_{1}^{2} - ϕ_{1} z_{2}^{2} - ϕ_{2} z_{3}^{2} - ϕ_{3} {\tilde{ξ}}^{2} + ϕ_{4} | z_{1} | | z_{2} | + ϕ_{5} | z_{2} | | z_{3} | \\ + ϕ_{6} | z_{1} | | z_{3} | + ϕ_{7} | z_{1} | | \tilde{ξ} | + ϕ_{8} | z_{2} | | \tilde{ξ} | + ϕ_{9} | z_{3} | | \tilde{ξ} | + ϕ_{10} \\ = - η^{T} Λ η + ϕ_{10} \end{matrix}

(38)

where

$ϕ_{1} = k_{2} - k_{1} - λ_{2} - \frac{1}{2}$ , $ϕ_{2} = k_{3} - | \frac{\partial α_{2}}{\partial x_{2}} φ_{1} | - \frac{1}{2} - \frac{1}{2} | \frac{\partial α_{2}}{\partial x_{2}} |$ , $ϕ_{3} = k_{o} φ_{1} - λ_{9} - \frac{| k_{o} | + λ_{6} + 1}{2}$

$ϕ_{4} = 1 + k_{1}^{2} + λ_{1}$ , $ϕ_{5} = | φ_{1} | + | \frac{\partial α_{2}}{\partial x_{1}} | + k_{2} | \frac{\partial α_{2}}{\partial x_{2}} | + λ_{8} + λ_{2} | \frac{\partial α_{2}}{\partial x_{2}} |$ ,

$ϕ_{6} = k_{1} | \frac{\partial α_{2}}{\partial x_{1}} | + λ_{7} + λ_{1} | \frac{\partial α_{2}}{\partial x_{2}} |$ , $ϕ_{7} = λ_{3} + λ_{7} + | k_{o} | λ_{1}$ , $ϕ_{8} = λ_{4} + λ_{8} + | k_{o} | λ_{2}$

$ϕ_{9} = k_{3} + λ_{9} + λ_{5}$ , $ϕ_{10} = (1 + | k_{o} | + λ_{6} + | \frac{\partial α_{2}}{\partial x_{2}} |) {| D_{1 max} |}^{2} / 2 + {| D_{2 max} |}^{2}$

η = [| z_{1} |, | z_{2} |, | z_{3} |, | ξ |]^{T}

Λ = [\begin{matrix} k_{1} & - ϕ_{4} / 2 & - ϕ_{6} / 2 & - ϕ_{7} / 2 \\ - ϕ_{4} / 2 & ϕ_{1} & - ϕ_{5} / 2 & - ϕ_{8} / 2 \\ - ϕ_{6} / 2 & - ϕ_{5} / 2 & ϕ_{2} & - ϕ_{9} / 2 \\ - ϕ_{7} / 2 & - ϕ_{8} / 2 & - ϕ_{9} / 2 & ϕ_{3} \end{matrix}]

When the matrix $Λ$ is positive definite, it can be concluded as

A. When $D_{i} \neq 0, i = 1, 2$ , according to equation (38) and the comparison law, it can be concluded that $V$ is bounded by the following expression, and the tracking error and state estimation error are also bounded.

V (t) \leq V (0) \exp (- τ t) + \frac{ϕ_{10}}{τ} [1 - \exp (- τ t)]

(39)

where $τ = 2 λ_{min} (Λ)$ , $λ_{min} (Λ)$ is the minimal eigenvalue of $Λ$ .

B. When $D_{i} = 0, i = 1, 2$ , there is

\overset{\cdot}{V} (t) \leq - η^{T} Λ η

(40)

It can be inferred from equation (40) and Barbalat’s lemma that asymptotic tracking can be achieved, that is, $z_{1} \to 0$ as $t \to \infty$ .

Comparative results

For verifying the effectiveness of the proposed system, comparative simulations have been conducted. The main parameters of the proposed system and controller are given in Table 1.

Table 1.

Main parameters of the proposed system and controller.

Item	Parameter	Symbol	Value	Unit
Pump	Supply pressure	$p_{s}$	$60$	Bar
Accumulator	Pre-charge pressure	$p_{0}$	$2.7 \times 10^{6}$	Bar
Accumulator	Total volume	$V_{0}$	$0.63$	L
Cylinder	Area of piston chamber	$A_{1}$	$1.57 \times 10^{- 3}$	m²
	Area of rod chamber	$A_{2}$	$0.864 \times 10^{- 3}$	m²
	Area of storage chamber	$A_{a}$	$1.256 \times 10^{- 3}$	m²
Controller	Observer gain	$k_{o}$	$- 1 \times 10^{- 5}$	—
	Control gain 1	$k_{1}$	$200$	—
	Control gain 2	$k_{2}$	$200$	—
	Control gain 3	$k_{3}$	$200$	—

The following three controllers are used for comparison.

NOFC: The combination of the nonlinear output feedback controller [equation (32)] and the state observer [equation (23)] proposed in this paper is abbreviated as NOFC. The controller gains are presented in Table 1. Basically, the control gains can be chosen by using two approaches. One is to choose a set of control gains according the stability analysis to ensure the matrix $Λ$ is positive definite. However, this method needs a large amount of calculation, resulting in the increase of design complexity. Alternatively, a pragmatic way to select the control gains is trial and error. To be specific, the control gains are increased gradually from a small value until the tracking performance is satisfied.³⁹

PI: The proportional-integral controller is widely employed in various applications due to its advantages such as concise and easy-to-implement. In addition, because only the output signal is required in this kind of controller, it can also be regarded as an output feedback controller for comparison. The control gains are $k_{p} = - 1 \times 10^{- 10}$ , $k_{I} = - 3 \times 10^{- 5}$ , which are also determined by trial-and-error method to achieve the optimal tracking performance.

RC: This is the robust controller which is like NOFC but has no desired compensation part. Namely, the time derivative of the accumulator pressure is used in the model compensation part and model stabilization part of the observer and controller. The control gains of RC are consistent with the gains of NOFC.

Remark 4: As observed from equation (18), the conventional state observers such as ESO and Levant’s differentiator cannot be directly applied to the proposed system due to the specific form of the state equation. Consequently, it becomes challenging to compare the proposed method with other state observer-based combinations. To assess the performance of the proposed method in terms of output feedback, a PI controller is employed. Furthermore, an RC is constructed to validate the impact of desired compensation. Artificial measurement noise is added to the simulation model to verify the effectiveness of the proposed method compared with RC.

Three performance indexes are used to evaluate the performance of each control algorithm, that is, the maximum absolute value of tracking errors $M_{e}$ , average tracking errors $μ$ , and standard deviation of the tracking errors $σ$ . Their definitions can be found in Yao et al.⁹

The tracking performance of the proposed controller is first tested for a smooth sinusoidal-like motion trajectory $L_{d} = 0.02 \sin (2 π t / 2) + 0.03$ under large inertia load. Based on this desired position, the desired pressure signal $x_{1 d}$ can be obtained according to equation (13), which is also a smooth trajectory. Due to the complexity of the variation law of the polytropic index, this paper assumes that it randomly changes over time between 1 and 1.4 in the simulation. In the first test, the polytropic index is set as 1.35. It should be noted that the nominal value of the polytropic index in equation (13) should be carefully selected. This is because the position tracking in this paper is indirectly achieved through precise pressure tracking, and the pressure tracking process is closed-loop control, while position tracking is essentially open-loop. Although the tracking accuracy of the pressure can be guaranteed by the proposed robust controller regardless of the variation of the polytropic index, the position accuracy of the cylinder is directly related to the precision of this value.

Figure 6 shows that the NOFC controller consistently maintains lower tracking errors compared to the PI and RC controllers throughout the sinusoidal trajectory. This indicates the robustness of the NOFC controller in handling dynamic variations and its superior ability to maintain stability under fluctuating conditions. The PI controller, while simple and easy to implement, lacks the sophistication needed for precise model-based compensation, which becomes evident in scenarios with significant inertia. The main reasons lies in several aspects. The first is the response speed. Although the proportion term (P term) in the PI controller can quickly adjust the control output based on the size of the error signal, the response speed of the system may be limited when dealing with large inertial loads. This means that even if the PI controller attempts to accelerate the system response by increasing the control output, the inertia of the load may hinder this rapid response. Additionally, even if the effect of the proportional term can make the system respond faster, it may also cause overshoot of the system. For high inertia loads, this overshoot and oscillation phenomenon may be more pronounced. Moreover, the performance of the PI controller largely depends on the setting of its parameters. However, for large inertial loads, finding a suitable set of parameters may be challenging. Inappropriate parameter settings may lead to system instability or performance degradation. Due to the existence of model compensation terms, NOFC and RC can offset the adverse effects by introducing a compensation term related to inertia, reduce the dynamic error of the system. The RC controller, despite being robust, fails to match the NOFC’s performance due to the absence of desired compensation. This is mainly due to the measurement noise contained in the pressure dynamic of the hydraulic accumulator, which affects the final tracking accuracy.

Figure 6.

Tracking errors of the three controllers in test 1.

Figure 7 further validates the superior performance of the NOFC controller by comparing key performance indices such as maximum absolute tracking error, average tracking error, and standard deviation. The NOFC controller not only shows lower values across all indices but also demonstrates more consistent and reliable performance. This consistency is crucial for applications requiring high precision and reliability. The PI controller, with its higher variability, highlights its limitations in maintaining stable performance under varying conditions. The RC controller, while better than PI in some aspects, still lags behind NOFC due to its lack of desired compensation. Moreover, as the trajectory period becomes shorter, the tracking error of both controllers increases, but the tracking errors of PI and RC are always greater than that of NOFC.

Figure 7.

Comparison of performance indices in test 1.

The estimation result is demonstrated in Figure 8. It can be seen that the proposed observer is able to obtain accurate estimation result of the state variable. By using the observer, output feedback can be achieved. The accurate state estimation is critical for the success of the NOFC controller. The observer’s capability to provide precise state feedback ensures that the NOFC controller can perform model-based compensation effectively. This capability is essential for maintaining control accuracy and stability, especially in systems with inherent uncertainties and dynamic changes.

Figure 8.

Estimation result in test 1.

The position tracking result is shown in Figure 9. As discussed above, the position tracking accuracy can be adjusted by choosing the nominal value of the polytropic index carefully. The ability to indirectly track the position through precise pressure control highlights the controller’s sophistication in handling multi-variable control tasks. This indirect approach, while complex, allows for more flexible and robust control strategies, particularly useful in systems where direct position tracking might be challenging or impractical.

Figure 9.

Position tracking result in test 1.

To further validate the applicability of the proposed method, a sawtooth-like motion trajectory is used as the desired position trajectory, which can be seen in Figure 13 (In this case, the polytropic index is set as 1.31 and the load condition is the same as test 1). Figures 10 and 11 demonstrate the tracking errors and performance indices respectively. These two figures provide a clear comparison of controller performances under more challenging conditions due to a faster motion trajectory. It is evident that the accuracy of the proposed NOFC is superior than that of the PI and RC controllers, as indicated by the provided performance indices. The maximum tracking error consistently occurs at the turning point of the trajectory. The sawtooth trajectory introduces sharp changes that test the responsiveness and robustness of each controller. The NOFC controller’s superior performance in handling these changes, along with its resilience to disturbances, underscores its practical applicability in such severe working scenarios.

Figure 10.

Tracking errors of the three controllers in test 2.

Figure 11.

Comparison of performance indices in test 2.

The estimation results of the state variables are depicted in Figure 12, showcasing the acceptable estimation capability of the proposed observer even under significant disturbances. The position tracking results are illustrated in Figure 13, demonstrating that the proposed method can indirectly obtain the position of the cylinder by utilizing the pressure signal of the hydraulic accumulator. Notably, it should be mentioned that the stroke of the hydraulic cylinder is 7 cm. Therefore, the straight section of the curve seems to exhibit better tracking performance. The consistent performance of the NOFC controller, even under large disturbances, as shown in Figures 12 and 13, highlights its robustness. The observer’s accurate state estimation ensures that the controller can adapt to disturbances effectively, maintaining control accuracy. The effective tracking of the hydraulic cylinder position, despite the short stroke, demonstrates the practical feasibility of using pressure signals for position control, offering a versatile solution for various hydraulic systems.

Figure 12.

Estimation result in test 2.

Figure 13.

Position tracking result in test 2.

Conclusions

Hydraulic servo control often adopts output feedback control to save costs and overcome the influence of sensor noise as much as possible. This method usually only requires one displacement sensor to achieve the control goal in position control. However, compared with pressure sensors, displacement sensors have higher costs and installation difficulties. Therefore, this paper proposes a position output feedback control method based on pressure sensor to further highlight the advantages of output feedback control. Therefore, the method proposed in this paper can provide a new solution for the industrial application of hydraulic servo systems. Additionally, a nonlinear output feedback pressure controller is developed, incorporating a state observer and desired compensation method. The system stability is guaranteed using the Lyapunov method, demonstrating that the proposed controller achieves the desired tracking performance in the presence of time-varying uncertainties and asymptotic tracking performance in their absence.

However, the proposed method also has certain limitations in practical applications. On the one hand, it is necessary to analyze or establish the relationship between the displacement of hydraulic cylinder and the accumulator pressure in advance, and the numerical relationship varies for different hydraulic systems. On the other hand, the polytropic index is not a constant in practical applications, and the accuracy of this parameter will directly affect the accuracy of displacement estimation. Therefore, how to establish a universal relationship between displacement and pressure and obtain accurate values of the polytropic index through online parameter estimation is the future research direction and challenge of this method.

Footnotes

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 is supported by the basic science research projects of higher education institutions in Jiangsu Province (natural science) [Grant No. 23KJB460013]; [Grant No. 23KJB460014]; Basic research plan of Xuzhou city [Grant No. KC23005]; Innovation and entrepreneurship training program for college students in Jiangsu Province [Grant No. 202310320142Y].

ORCID iDs

Wang Weiping

Zhang Jialu

Data availability statement

Data sharing not applicable to this article as no datasets were generated or analyzed during the current study.

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