
==== Front
Sci Rep
Sci Rep
Scientific Reports
2045-2322
Nature Publishing Group UK London

39294255
72583
10.1038/s41598-024-72583-w
Article
Nonlinear flow modeling of electro hydrostatic pump unit based on Gauss Newton iterative method for high performance control
Zhang Tiangui 3
Yan Guishan yangsh235@mail.sysu.edu.cn

12
Liu Xianhang 3
Yao Chong 3
Ai Chao 3
1 https://ror.org/0064kty71 grid.12981.33 0000 0001 2360 039X School of Intelligent Systems Engineering, Sun Yat-Sen University, Shenzhen, 510275 China
2 https://ror.org/01h0g5x34 The State Key Laboratory of Fluid Power and Mechatronic Systems, Hangzhou, 310000 China
3 https://ror.org/02txfnf15 grid.413012.5 0000 0000 8954 0417 School of Mechanical Engineering, Yanshan University, Qinhuangdao, 066004 China
18 9 2024
18 9 2024
2024
14 217505 6 2024
9 9 2024
© The Author(s) 2024
2024
https://creativecommons.org/licenses/by-nc-nd/4.0/ Open Access This article is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License, which permits any non-commercial use, sharing, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if you modified the licensed material. You do not have permission under this licence to share adapted material derived from this article or parts of it. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by-nc-nd/4.0/.
For the electric hydrostatic pump unit (EPU), which is a crucial component in the electro hydraulic servo pump control system (EHSPCS), nonlinear flow modelling is a key challenge to achieve high performance operation, especially in position control. For the nonlinear flow of EPUs, simple linearization combined with compensation has become a popular control method recently. However, the control performance is constrained by the reliance on deviation correction. This paper proposes utilizing the Gauss Newton iterative method to investigate the nonlinear flow modeling in the EPU, aiming to obtain the mapping relationship between the load flow and operating parameters in the EPU, thereby directly improving the control performance. A nonlinear flow model is built by utilizing the Gauss Newton iterative method based on test data obtained from various operating conditions, and the mapping relationship is presented accordingly. Experimental studies show that compared to well-established methods, this model exhibits high accuracy and practicality in improving the control performance, particularly in position control of the EPU.

Keywords

Electro hydrostatic pump unit (EPU)
Electro hydraulic servo pump control system (EHSPCS)
Nonlinear flow
Gauss Newton iterative method
Subject terms

Mechanical engineering
Applied mathematics
http://dx.doi.org/10.13039/501100001809 National Natural Science Foundation of China 52305082 Yan Guishan Funded by Open Foundation of the State Key Laboratory of Fluid Power and Mechatronic SystemsGZKF-202332 Yan Guishan Science and the Technology Research Program of Higher Education Institutions in Hebei ProvinceCXY2024034 Ai Chao issue-copyright-statement© Springer Nature Limited 2024
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pmcIntroduction

As a crucial core component of the electro hydraulic servo pump control system (EHSPCS), the electric hydrostatic pump unit (EPU) can realize high performance control by directly adjusting the speed and torque of the servo motor1–3, and the control effect is affected by the nonlinear flow4. For the EPU, due to the dual influence of the motor speed and load pressure, the output flow exhibits strong nonlinearity5–7, and the time-varying soft parameters, especially the internal leakage and the bulk elastic modulus of oil, make the nonlinear mechanism of flow more complex8,9. Experts have researched these nonlinear flow issues and recognized that nonlinear flow presents difficulties in achieving precise control of the EHSPCS10–13. Thus, nonlinear flow stands in the way of enhancing the control performance of the EHSPCS14,15.

An important direction of research on nonlinear flow is to investigate the influence of internal and external disturbances on the convective nonlinearity, such as the impact of the oil compression, oil temperature, external load interference, and other factors on the flow nonlinearity. Researchers16 from Chang’an University in China, led by Gu, delved into the influence of the oil characteristics on the volumetric efficiency and flow stability of a quantitative pump. Their investigations uncovered an intricate and nonlinear association between the compressibility and elastic modulus of oil with the pump flow output. Yao17 from Zhejiang University in China investigated the nonlinear disturbances of the output flow caused by pressure pulsations at low speeds of the hydraulic pump and conducted relevant experimental verification. Mccullough18 from McMaster University in Canada examined the effect of the oil temperature on the nonlinear flow of the EHSPCS. Through detailed experimental methods, he summarized the impact of the oil temperature on the crucial operating parameters, such as the leakage coefficient and damping coefficient of the system.

Another direction of research on nonlinear flow involves exploring the characteristics and compensation methods of flow nonlinearity, including compensation for flow dead zones and leakage nonlinearity. Zad19 from the University College Cork in Ireland, summarized the parameter characteristics of an EHSPCS, such as the parameter nonlinearity, flow dead zone, sensor noise, and system time-varying characteristics, and analyzed the nonlinear flow of the system. Ren20 from the University of Manitoba in Canada designed a leakage compensation controller to address the nonlinear disturbances caused by oil leakages in electro hydraulic pump control systems and performed relevant experimental verification. Sakuma21,22 from Saitama University in Japan proposed a control compensation approach based on a feedback regulator to address the nonlinear flow problem present in EHSPCSs. Yuan, Na and Kim Y B23. from Chonnam National University in South Korea took into account nonlinear disturbances, including external load disturbances and oil compression. They then formulated a nonlinear mathematical model and employed a gray box system approach to identify the crucial parameters of the model.

Currently, the prevalent approach to tackling the nonlinearity in the EPU flow involves simplifying the system model through appropriate linearization techniques, coupled with the compensation mechanisms to compensate for the shortcomings in simple linearization24. However, when the EPU runs far from equilibrium points or experiences large disturbances, pure linearization frequently falls short of adequately capturing all of its dynamic characteristics, which has a substantial impact on the system control performance. Moreover, existing methods addressing flow nonlinearity rely heavily on the accuracy and timeliness of deviation correction. In cases where deviation correction is insufficiently precise or prompt, the system may experience significant steady-state errors or dynamic fluctuations, directly compromising the overall control effectiveness. While the combination of simple linearization and compensation offers a practical solution for controlling nonlinear flow in the EPU, it fundamentally lacks the capability to explain and resolve the EHSPCS nonlinear phenomena comprehensively. Consequently, there remains a pressing need for further research into more advanced methodologies for addressing the nonlinearity in the EPU flow, aiming to achieve a deeper understanding and more robust control strategies.

Based on the limits of previous research, this article proposes utilizing the Gauss Newton iterative method to establish a nonlinear model of the fluid of the EPU and obtain the mapping relationship between the load flow and the operating parameters of the EPU, thereby directly improving the control performance of the EPU. Specifically, the idea is to employ the Gauss Newton iterative method to fit experimental data, including the no-load flow, no-load build pressure, and flow output of the EPU, and finally obtain the nonlinear flow model in the EPU. With the proposed nonlinear flow model, the mapping relationship between the EPU load flow, operating speed, and load pressure can be accurately reflected, laying the groundwork for enhanced control capabilities of the system. The research content of this paper provided a new solution to the nonlinear flow problem, especially in the EHSPCS. More importantly, the proposed model was utilized to enhance the performance of the EPU positional control, and the feasibility and effectiveness of the model were verified.

Introduction to the EHSPCS and the EPU

Introduction to the EHSPCS

The EHSPCS consists of the servo motor, quantitative pump, hydraulic cylinder and functional valve group, etc. It is a kind of energy-saving and efficient hydraulic control mode, which controls the action of the hydraulic cylinder through the constant change of the output flow of quantitative pump.

Introduction to the EPU

In the EHSPCS, the EPU integrates the servo motor and quantitative pump, which are connected coaxially to achieve precise speed control of the pump by the motor. The composition of the EPU is depicted in Fig. 2.Fig.1 Working principle diagram of the EHSPCS. The configuration and working principle of the EHSPCS in this study is depicted in Fig. 1. 1-servo motor, 2-quantitative pump, 3-accumulator, 4.1-one-way valve of chamber A, 4.2-one-way valve of chamber B, 5.1/5.2-relief valves, 6.1-A cavity pressure sensor, 6.2-B cavity pressure sensor. 7-displacement sensor, and 8-hydraulic cylinder.

Fig.2 Composition of the EPU.

The servo motor controller regulates the rotational speed of the servo motor, thereby governing the output flow of the hydraulic pump. On one hand, as the system energy source, the EPU generates the hydraulic force required to power the entire system. On the other hand, as the control element of the system, the EPU achieves precise control over the system output, serving as the pivotal component for the EHSPCS high performance operation.

EPU flow nonlinear model

The EPU flow nonlinearity must be addressed for the system to achieve high performance control. To understand the essence of the EPU flow nonlinearity, a detailed analysis of the EPU nonlinear flow model is presented in this section. After the design and selection of the servo motor and the quantitative pump are finalized, the primary factors influencing the EPU load flow are the motor rotational speed and the applied load pressure. The EPU load flow can be represented as1 Qp=f(ωp,pL)

The mathematical model of the quantitative pump shows that the load flow of the EPU can be simplified as2 Qp=Dpωp-CppL

where Dp is the displacement of the quantitative pump (mL/r), ωp is the speed of the quantitative pump (rad/s), Cp is the quantitative pump total leakage coefficient (m3/(s∙Pa)), and pL is the quantitative pump load port pressure (Pa). Eq. (2) can macroscopically represent the variations of the EPU load flow with the motor speed and load pressure. However, this model is a local linearization model and cannot be used to express the overall nonlinear characteristics of the system.

In this section, based on the nonlinear characteristics test data, a multivariate nonlinear fitting method is proposed to obtain a nonlinear flow model of the EPU through the Gauss Newton iterative method, which accurately reflects the mapping relationship between the EPU load flow and its operating speed. The load pressure lays the foundation for the precise servo control of the EHSPCS. The framework is depicted in Fig. 3.Fig.3 Framework of the research idea of the nonlinear flow model.

Nonlinear flow test

To obtain the nonlinear data of the EPU flow, we first built a nonlinear test platform of the EPU flow of the EHSPCS. The detailed experimental principle is depicted in Fig. 4. Two high-precision flow sensors (3.1 and 3.2) were set in the system to test the EPU load flow and leakage flow. The throttle valve (11) was set as a load simulation to adjust the pressure load of the system. The test bench included the EPU, functional valve group, high-precision flow sensor, and electronic control system. The hydraulic test platform and electrical control cabinet are depicted in Fig. 5. Using this test platform, no-load flow, load pressure buildup, and flow nonlinear tests were performed on the EPU.Fig.4 Schematic diagram of the EPU flow nonlinearity test. 1-servo motor, 2.1/2.3-high-pressure sensors, 2.2-low-pressure sensor, 3.1/3.2-flowmeters. 4-shuttle valve, 5-one-way valve, 6-relief valve, 7-solenoid reversing valve, 8-accumulator. 9-temperature sensor, 10-damping screw, and 11-throttle valve.

Fig.5 Photographs of the EPU nonlinear flow test bench.

No-load flow test

To perform this test, the throttle valve (11) was opened to its fullest extent, it was ensured that the EHSPCS operated under a no-load state, and the rotational speed of the servo motor was adjusted. The load flow and leakage flow of the EPU were measured, and subsequently, the no-load characteristics of the system were derived, as depicted in Fig. 6. The flow rate of the EPU increased as the speed increased, and there was a specific dead zone at low speeds. Although there was a slight increase in the leakage flow as the speed increased, the change was not significant and tended to remain relatively stable.Fig.6 No-load flow characteristic curve of the EPU.

Load pressure buildup test

To perform this test, the load simulation throttle valve (11) was turned off completely, the speed was gradually increased, and the maximum pressure of the load chamber of the quantitative pump was recorded. When the system pressure remained stable, the hydraulic pump generated flow to accommodate both internal pump leakage and oil compression. The experimental results are depicted in Fig. 7.Fig.7 Load pressure characteristic curve of the EPU.

Figure 7 shows that the output flow of the EPU was small within a small speed range, which was insufficient to maintain the leakage and precompression of the quantitative pump. Hence, the pressure could not be established. As the speed increased, the output flow could overcome the leakage of the quantitative pump, producing oil volume compression, and the system pressure increased. With the increase in the system pressure, the leakage of the quantitative pump further increased. When the quantitative pump load pressure reached the equilibrium point, the output flow was used to provide hydraulic pressure.

Nonlinear flow test

To perform this test, the load simulation throttle valve (11) was adjusted to increase the system load pressure step-by-step, the EPU speed input was adjusted through the servo motor, and the load flow at each operating point was recorded. The quantitative pump overcame its own internal leakage and oil compression to achieve load flow output. The nonlinear flow test model obtained is depicted in Fig. 8.Fig.8 Nonlinear flow test diagram of the EPU.

As shown in Fig. 8, the EPU load flow exhibited strong nonlinear characteristics owing to the effects of the speed and load pressure. Overall, the load flow increases with speed, but when subjected to load pressure, it exhibits sagging, resulting in a decrease in load flow output as load pressure increases.

Nonlinear flow model

Based on the highly nonlinear characteristic test data of the EPU load flow, this study adopted a multivariate nonlinear fitting method to construct a theoretical model for nonlinear flow regression. This model profoundly unveils the intricate relationship between load flow and its critical operating parameters: rotational speed and load pressure, while incorporating an estimable parameter vector and inevitable random errors. To precisely solve the parameters of this nonlinear regression model, the efficient Gauss Newton iterative algorithm was employed. Grounded on the optimization principle of least squares, this algorithm iteratively minimizes the sum of squared residuals, gradually approaching the true parameter values. Through a series of meticulous calculations and iterations, a nonlinear model for the EPU flow was successfully established. This model not only accurately reflects the complex mapping relationship between the EPU load flow and its operational rotational speed, load pressure, but also demonstrates a high level of predictive capability and robustness.

To comprehensively validate the effectiveness and feasibility of the constructed model, an in-depth comparative analysis was conducted between the predictive results of the EPU flow nonlinear model and actual experimental data. The results reveal an extremely high degree of agreement, fully substantiating the outstanding performance and wide applicability of the adopted multivariate nonlinear fitting method in practical applications. This achievement provides solid theoretical support for optimizing the performance of the EPU.

Construction of theoretical model

According to the analysis of the working mechanism of the EPU, the nonlinear regression model can be expressed as3 Y=fX1,X2,β+ε

where the independent variables are the rotational speed vector X1 (X1 = (ωp1, ωp2,…, ωpn)) and the pressure vector X2 (X2 = (pL1, pL2,…, pLn)), while the dependent variable is the load flow Y (Y = (QL1, QL2,…, QLn)). β is the vector of parameters that need to be estimated, and ε is the random error.

The first part of Eq. (3) reflects the deterministic functional relationship between the dependent variable load flow Y and the independent variable speed vector X1 and pressure vector X2. The second part of Eq. (3) reflects uncertainties such as the data observation error and model error. The above theoretical model was solved as a multivariate nonlinear regression problem.

Parameter solution of regression model

Due to the high nonlinearity of the EPU load flow, it is difficult to convert it into a linear model by function transformation to perform linear regression analysis. This section introduces the Gauss Newton iterative method to obtain the parameters of the regression model through data iteration. First, the residual sum of squares is defined as4 L(β)=∑i=1nεi2=∑i=1nyi-f(x1i,x2i,β)2

The residual sum of squares is minimized based on the least-squares regression principle. If the regression function f(x1i, x2i, β) is continuously differentiable for the parameter β, then the derivative of L(β) can be obtained. If the first derivative is 0, then we can obtain5 dLβdβ=2∑i=1nyi-fx1i,x2i,βdf(x1i,x2i,βdβ=0

The Gauss Newton iterative method is utilized to deal with the aforementioned nonlinear equation. The starting point of the specific method is to linearly approximate the nonlinear model and perform the iterative calculation.

g(0) = (g(0) 0, g(0) 1, g(0) 2,…, g(0) p-1) are set as the initial values of the regression coefficients β = (β0, β1,…, βp-1) to be estimated. The regression model f(x1i, x2i, β) is Taylor expanded near the point g(0) to obtain6 fx1i,x2i,β≈fx1i,x2i,g(0)+∑k=0p-1∂fx1i,x2i,β∂βkβ=g(0)βk-gk(0)

By substituting (6) into (3), we obtain7 yi≈f(x1i,x2i,g(0))+∑k=0p-1∂f(x1i,x2i,β)∂βkβ=g(0)(βk-gk(0))+εi

In practical application, the variables X1i and X2i should be replaced with the actual measured speed and pressure data, and Yi should be replaced with the corresponding load flow data. The parameter vector βk contains all the estimated coefficients in the model, which are updated during iteration. εi refers to residuals, the difference between the predicted value and the actual value.

By setting yi(0)=yi-f(x1i,x2i,g(0)),Dik(0)=∂f(x1i,x2i,β)∂βkβ=g(0),bk(0)=βk-gk(0)

we obtain8 yi(0)≈∑k=0p-1Dik(0)bk(0)+εi

The matrix expression of Eq. (8) is9 Y(0)≈D(0)b(0)+ε

Each matrix can be further expressed asYn×1(0)=y1-f(x11,x21,g(0)).....yn-f(x1n,x2n,g(0))Dn×p(0)=D10(0)..D1p-1(0)::Dn0(0)..Dnp-1(0)bp×1(0)=b0(0).....bp-1(0)

Furthermore, using the least-squares method to revise Eq. (9), we have10 b(0)=(D(0)TD(0))-1D(0)TY(0)

By letting g(0) be the first iteration value, we obtain11 g(1)=g(0)+b(0)

When the iteration is repeated s times, we can obtain12 b(s)=(D(s)TD(s))-1D(s)TY(s)g(s+1)=g(s)+b(s)

Test of accuracy

Let the residual sum of squares be13 SSR(s)=∑i=1nyi-f(x1i,x2i,g(s))2

where s is the number of repeated iterations. For a given allowable error rate k, the judgment condition of the iteration is14 SSR(s)-SSR(s-1)SSR(s)≤k

When Eq. (14) is satisfied, the iteration is stopped; otherwise, it is repeated until the condition is satisfied. The Gauss Newton iterative solution is established based on the MATLAB® control system toolbox. The least-squares estimation value β can be obtained, thereby obtaining the EPU nonlinear flow mapping model.

Using the MATLAB toolbox to model and solve the above multivariate nonlinear regression method, the EPU nonlinear flow equation was obtained. Based on this equation, the EPU nonlinear flow theoretical envelope diagram was generated, and the experimental envelope diagram in Fig. 8 was constructed. The comparison of the results is shown in Fig. 9.Fig.9 Comparison of the theoretical and experimental models for the EPU nonlinear flow.

Figure 9 shows that an accurate nonlinear theoretical model of the EPU flow can be obtained using the multivariate nonlinear regression method proposed in this study. The theoretical and experimental models showed good agreement at each pressure and speed operating point. The overall error rate k was less than 2%.

Nonlinear analysis of flow

Research on nonlinear characteristics of flow rate under rotational speed excitation

According to the analysis of the quantitative pump mechanism model, the gain sensitivity coefficient of the load flow to the rotational speed is defined as15 Kω=∂QL∂ωp

The EPU load flow output variations with the rotational speed were studied by a partition processing, with three distinct zones: the flow dead zone, flow load zone, and flow saturation zone. The characteristics of the flow output are depicted in Fig. 10. The variations of the EPU load flow with rotational speed can be divided into three regions: (I) low-speed characteristic region, (II) flow load region, and (III) flow saturation region.Fig.10 Zoning diagram of the EPU load flow and speed.

Low-flow area

When the EHSPCS performed position/force control, the hydraulic cylinder entered a steady state and reached the desired value. The quantitative pump needed to operate at a lower speed to maintain system leakage and oil compression. Therefore, the low-speed characteristics of the EPU played a crucial role in ensuring high performance control of the EHSPCS. The low-speed characteristic zone of the EPU was manifested in two areas: the flow dead zone and the low-speed unstable zone.

Within the flow dead zone, the operational flow of the quantitative pump was utilized to sustain both the leakage flow and the oil compression flow of the EPU while it rotated at a low speeds. The EPU had a no-load flow output. The EPU speed corresponding to the flow dead zone was defined as the dead zone speed, which can be expressed as16 ωp0=CppLDp+V0pDpβedpLdt

It can be seen from the above formula that the dead zone speed of the EPU was dependent on the leakage coefficient and load pressure of the quantitative pump. Within a certain range, the larger the leakage coefficient and load pressure were, the higher the dead zone speed was. The flow dead zone was an important indicator for measuring the low-speed characteristics of the EPU. To a certain extent, a smaller value indicates a superior low-speed performance of the system.

The low-speed instability zone means that when the EPU crossed the flow dead zone and began to produce flow because of the internal structure of the quantitative pump: the quantitative pump rotated at a low frequency, and the pressure flow pulsation generated by the internal plunger was more prominent, resulting in an unstable EPU flow output. In addition, the structure and control process of the servo motor determined that the torque pulsation generated by the motor at low speeds led to significant speed pulsations, which caused the motor to shake and affected the low-speed stability.

Traffic load area

As the speed of the servo motor increased, the EPU entered the flow load zone from the low-speed flow zone. Under the flow load zone condition, the EPU nonlinear flow was primarily affected by the load pressure and operating temperature of the oil. As the load pressure continued to increase, the leakage flow of the quantitative pump and the oil compression flow increased. Hence, there was a certain attenuation of the load flow output of the quantitative pump. The increase in the operating temperature of the oil changed the leakage coefficient of the EPU and the bulk elastic modulus of the oil, leading to a certain attenuation of the load flow output of the quantitative pump.

Flow saturation area

The maximum speed of the quantitative pump set a limit on the EPU performance. Even if the motor speed control command continued to increase, the quantitative pump remained at the highest allowable operating speed, and the load flow had a bottleneck saturation value.

Nonlinear characteristics of flow under pressure excitation

According to the quantitative pump mechanism model analysis, the gain sensitivity coefficient of the load flow to the load pressure is defined as17 Kp=∂QL∂pL

The EPU load flow output characteristics with pressure variations were studied by partition processing. The flow output characteristics are depicted in Fig. 11. The load flow variations of the EPU with pressure were segmented into two distinct areas: (I) light-load smooth area and (II) heavy-load attenuation area.Fig.11 Zoning diagram of the EPU load flow and pressure.

Light-load smooth area

Under the light-load condition of the EHSPCS, the leakage flow and oil compression flow corresponding to the EPU were small, and the total operating flow of the quantitative pump was mainly output in the form of a load flow.

Heavy-load attenuation area

When the EHSPCS was under heavy load conditions, the leakage and oil compression characteristics corresponding to the EPU were prominent, and the load flow of the quantitative pump decreased. The attenuation characteristics were dependent on the leakage coefficient of the EPU and the oil compression characteristics. The size of the pressure attenuation sensitivity Kp determined the speed of the quantitative pump when the EHSPCS entered the steady-state control process. In addition, the pressure attenuation sensitivity Kp also showed the efficiency of EPU in converting mechanical energy into hydraulic energy to a certain extent.

Application of the EPU nonlinear flow model

The EPU nonlinear flow model was applied to the high performance position control of the EHSPCS. The specific control framework is depicted in Fig. 12.Fig.12 Position control block diagram of the EHSPCS.

The hydraulic cylinder position command xpd was compared with the actual feedback xpf. The resulting deviation was obtained through the action of a proportional–integral–derivative (PID) controller to determine the load flow command QL. The nonlinear flow mapping model was used to obtain the speed loop control command uc of the servo motor, and the high-precision control of the hydraulic cylinder displacement was finally realized. The control methods mentioned above were tested in subsequent experiments to verify the validity and feasibility of the obtained nonlinear flow model.

Introduction to test bench

The EPU was installed on the function valve block and connected to the hydraulic cylinder of the actuator through the hydraulic pipeline. The test bench of the EHSPCS is depicted in Fig. 13.Fig.13 Physical map of the test bench.

During the test, the established nonlinear flow model was compiled with Simulink and downloaded to the axis controller, based on which the precise control of the system position was performed. Figure 14 illustrates the electrical control system of the EHSPCS test bench. Table 1 outlines the operating parameters of the EHSPCS test platform. Fig.14 Physical diagram of the electrical control part of the test bench.

Table 1 Operating parameters of the test platform.

Serial number	Name	Parameter	Unit	
1	Servo motor rated torque	22	Nm	
2	Servo motor rated speed	3000	r/min	
3	Hydraulic pump displacement	8	mL/r	
4	Hydraulic pump maximum speed	4500	r/min	
5	System operating pressure	35	MPa	
6	Hydraulic cylinder specifications	Ø50/Ø30	mm	
7	Hydraulic cylinder piston stroke	300	mm	

Analysis of test data

Utilizing the aforementioned test platform as a foundation, the high performance position control of the EHSPCS was carried out. During the test, given the step command of the hydraulic cylinder position, the test data were obtained, as depicted in Fig. 15. The EPU nonlinear flow model was used to control the system position and generate load flow commands. The servo motor exhibited prompt responses in both the speed and torque, ultimately actuating the hydraulic system.Fig.15 Experimental curves of the position step control. (a) Hydraulic cylinder displacement curves. (b) Load flow command curve. (c) Servo motor speed curve. (d) Servo motor torque curve. (e) High-pressure chamber pressure curve. (h) Low-pressure chamber pressure curve.

As shown in Fig. 15, the maximum pressure reached in the high-pressure chamber was approximately 47 bar. The low-pressure chamber was stable, and the hydraulic cylinder position reached a steady state within 1.2 s. The hydraulic cylinder positional accuracy during steady-state conditions was ± 0.01 mm.

The dynamic characteristics of the proposed control algorithm were further verified with the hydraulic cylinder position command amplitude of 5 mm and frequency of 2 Hz. The findings of the test are depicted in Fig. 16.Fig.16 Dynamic tracking curve of the system position. (a) Position tracking curves. (b) Control command curve.

As shown in Fig. 16, the position command was promptly followed by the displacement of the hydraulic cylinder. The amplitude deviation was controlled within ± 0.01 mm, while the phase deviation was controlled within ± 0.1°, which indicated good dynamic following characteristics.

To comprehensively and systematically evaluate the enhancement of system control performance through the adoption of the nonlinear flow model, this study had devised and executed a comparative experimental analysis. In this experiment, when the hydraulic cylinder was in the initial position (+ 20.8 mm), the displacement command of the hydraulic cylinder -5.8 mm was given at 1.5s, and then the displacement command of the hydraulic cylinder -5 mm was given for three consecutive times, and the hydraulic cylinder returned to the zero position. Two distinct control methodologies were employed: a high performance control strategy that is based on the nonlinear flow model, and the traditionally prevalent PID control method. The two control methods were separately applied to the same control task to ensure consistency and comparability of experimental conditions. In addition, the high performance control strategy based on nonlinear flow model was also simulated. The experimental and simulation results are shown in Fig. 17.Fig.17 Position control curves under different control strategies. (a) 20.8–15 mm position control response curves. (b) 15–10 mm position control response curves. (c) 10–5 mm position control response curves. (d) 5 mm–0 position control response curves.

As shown in Fig. 17, the high performance control strategy designed upon the nonlinear flow model exhibits marked performance advantages over the conventional PID control strategy. Specifically, this high performance control approach demonstrates a significantly faster capability to stabilize the system (stabilize within 2 s after applying position control command), showcasing superior dynamic response characteristics that outperform PID control. Crucially, during the regulation process, the proposed strategy elicits a smaller overshoot, effectively mitigating system overshoot phenomena, which is paramount to ensuring smooth system operation and enhancing control precision. This underscores its heightened sensitivity prowess in adapting to the system dynamic fluctuations. Meanwhile, the steady-state accuracy of the system has been significantly improved through the application of high performance control strategy, with its maximum steady-state accuracy strictly controlled within 0.01 mm. This level of precision is notably superior to that achieved by traditional PID control strategy, which may have a maximum steady-state error of up to 0.2 mm. To further verify the reliability of above conclusion, this study simulated the displacement response using the high performance control strategy. The simulation results were highly consistent with subsequent experimental data, jointly demonstrating the effectiveness and superiority of the high performance control strategy based on the nonlinear flow model in improving system control performance.

Conclusions

In this study, a nonlinear flow model of the EPU was proposed to investigate the nonlinear flow variations and enhance the high performance control. The proposed model was developed using the Gauss Newton iteration method to fit the test data of no-load flow, load building, and flow output of the EPU. This model accurately reflected the mapping relationship between the load flow and operating parameters of the EPU. Moreover, experimental results demonstrated that applying this model enhanced the accuracy of the EPU position control. Compared with popular methods, the proposed technique exhibited a superior performance in terms of nonlinear flow modeling and compensation for the EPU. This study focused on validating the effectiveness of this nonlinear flow model for improving the high performance control of the EPU position. In the EPU, there may exist complex coupling relationships among various parameters. Although the model proposed in this study shows its superiority in position control, it may not fully consider the influence of these coupling effects on flow characteristics, and its effectiveness in force control needs to be further tested. Future research can introduce more sophisticated system identification methods to more accurately describe the nonlinear flow characteristics of the EPU.

Acknowledgements

This paper is supported by the National Natural Science Foundation of China (NSFC) (52305082), the Funded by Open Foundation of the State Key Laboratory of Fluid Power and Mechatronic Systems (GZKF-202332) and Science and the Technology Research Program of Higher Education Institutions in Hebei Province (CXY2024034).

Author contributions

Tiangui Zhang: Conceiving ideas, writing original draft, review and editing. Guishan Yan: Funding acquisition, review and editing. Xianhang Liu: Writing original draft, review and editing. Chong Yao: Supervision, review and editing. Chao Ai: Funding acquisition, project administration. All authors had read and approved the final manuscript.

Data availability

The data that supports the findings of this study are available from the corresponding author on request.

Competing interests

The authors declare no competing interests.

Publisher's note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
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