Parameter Estimation of a Valve-Controlled Cylinder System Model Based on Bench Test and Operating Data Fusion

Abstract

The accurate estimation of parameters is the premise for establishing a high-fidelity simulation model of a valve-controlled cylinder system. Bench test data are easily obtained, but it is challenging to emulate actual loads in the research on parameter estimation of valve-controlled cylinder system. Despite the actual load information contained in the operating data of the control valve, its acquisition remains challenging. This paper proposes a method that fuses bench test and operating data for parameter estimation to address the aforementioned problems. The proposed method is based on Bayesian theory, and its core is a pool fusion of prior information from bench test and operating data. Firstly, a system model is established, and the parameters in the model are analysed. Secondly, the bench and operating data of the system are collected. Then, the model parameters and weight coefficients are estimated using the data fusion method. Finally, the estimated effects of the data fusion method, Bayesian method, and particle swarm optimisation (PSO) algorithm on system model parameters are compared. The research shows that the weight coefficient represents the contribution of different prior information to the parameter estimation result. The effect of parameter estimation based on the data fusion method is better than that of the Bayesian method and the PSO algorithm. Increasing load complexity leads to a decrease in model accuracy, highlighting the crucial role of the data fusion method in parameter estimation studies.

1 Introduction

A valve-controlled cylinder system is a typical hydraulic drive system consisting of a pilot handle, control valve and cylinder. It is widely used in construction, agriculture, aerospace and other fields, which require low energy consumption, stable control and a fast response. To obtain good valve-controlled cylinder system performance, it is often necessary to optimise the design and verification of system parameters and control algorithms. Due to the cost of a prototype and trial, a simulation model is usually used to replace a prototype to design and verify the optimisation scheme. The accuracy of a simulation model directly affects the effectiveness of the optimisation scheme, and the accurate estimation of parameters is the premise for establishing a high-fidelity valve-controlled cylinder system simulation model.

Valve-controlled cylinder system simulation model parameter estimation requires the support of system data. Parameter estimation research uses the load data as the model input condition to optimise the error between the simulation results and the system tested data. The smaller the error between the two, the more accurate the parameter estimation results. Parameter estimation is a simulation model parameter optimisation problem. Yan et al. [1] estimated the model parameters of a valve-controlled cylinder system using the recursive least squares method. Based on this model, the maximum error in predicting the general trajectory tracking speed was 13%. Wang et al. [2] estimated the parameters of a discrete model of a valve-controlled cylinder system using the Particle Swarm Optimization (PSO) algorithm. A comparison with the results obtained using the least squares method demonstrated the parameter estimation capability of the PSO algorithm. Aboelela et al. [3] estimated the parameters of a valve-controlled cylinder system using the Matlab/Simulink parameter identification toolbox, achieving satisfactory identification results. Similarly, Liu et al. [4] estimated the parameters of a hydraulic drive unit using the Matlab/Simulink parameter identification toolbox. Although the integrated error in this study was only 1.51%, the fitting between the simulation and experimental results was moderate. Among these parameter estimation studies, the Matlab/Simulink parameter identification toolbox was found to be the most convenient tool, and the PSO algorithm yielded the best results. Essa et al. [5] also discussed a parameter estimation method for a valve-controlled cylinder system based on a black box state space model. However, black box models lack any internal knowledge of the system. In other studies on parameter estimation of hydraulic systems, Nápoles-Báez et al. [6] estimated the parameters of a hydraulic actuator using the Matlab/Simulink parameter identification toolbox. Lu et al. [7] determined the optimal parameters of a Pump-Motor Servo System (PMSS) PID speed controller based on a hybrid Grey Wolf Optimization (GWO) and Particle Swarm Optimization (PSO) algorithm, achieving constant speed control of the PMSS.

More importantly, data are the benchmark for parameter estimation of a valve-controlled cylinder system model. Only accurate data corresponding to the actual operating conditions can ensure the accuracy of parameter estimation results. The aforementioned research on parameter estimation employed various optimization algorithms, all conducted based on bench tests, making the bench test data the reference for parameter estimation. While bench test data offers the advantage of convenient collection, it falls short in simulating actual operating conditions of the system, leading to a discrepancy between the test data and real operating conditions [8]. Consequently, there is an inherent limitation in benchmarking parameter estimation research solely based on bench test data. Operating data, on the other hand, originates from the actual system operation processes of the system and includes real load conditions and personalized environmental information [9, 10]. Nevertheless, acquiring direct measurements of operating data from control valves poses a complex and formidable challenge. This task entails the installation of various sensors, including flow rate sensors, and the intricate design and arrangement of external piping systems. External piping introduces substantial pressure losses, leading to consequential measurement inaccuracies. Compounding the challenge are limitations in operational space, intricate equipment structures, and adverse working conditions, further complicating the acquisition of precise control valve operating data. Consequently, attaining accurate control valve operating data results necessitates substantial time and financial resources. Existing research on construction machinery operating data does not involve collecting control valve operating data [11, 12]. Feng et al. [13] estimated the model parameters of a valve-controlled cylinder system through cylinder operating data. Since cylinder displacement data are not the direct control valve data, this method causes deviations in control valve parameter estimation. Whether bench test or operating data are used in parameter estimation research, the estimation results accuracy is affected to varying degrees. Suppose a reasonable method can be selected to fuse the two data so the parameter estimation includes both the control valve output data and the actual load information. In that case, the system parameter estimation results will be more accurate.

In studying the model parameter estimation of valve-controlled cylinder systems, model parameter estimates obtained based on bench test or operating data can be regarded as prior information for the final parameter estimates. Research on model parameter estimation using these two sets of prior information meets the requirements of bench test and operating data fusion. If there are multiple priors regarding the parameters to be estimated, a common approach for fusing prior information is to average all the multiple priors [14]. However, the deviations of the different opinions are not well quantified in the averaging approach. Pooling methods such as the linear and geometric pooling methods allow unequal weights for each prior and emphasize the diversity of multiple priors [15,16,17]. Based on the geometric pooling method, Poole et al. [18] pioneered the incorporation of Bayesian theory to combine distinct prior information, enabling the estimation of system performance parameters. Specifically, Bayesian theory is highly regarded in parameter estimation research because it can update the probability distribution of unknown variables by integrating information from multiple data sources [19, 20]. Yang et al. [21] proposed an adaptive Bayesian method based on geometric pooling for evaluating the performance of products with hierarchical structural characteristics. Yang et al. [22] fused inconsistent prior information using Bayesian theory and geometric pooling methods. The proposed approach shows significant advantages in parameter estimation and reliability assessment. Jia et al. [23] proposed a Bayesian-based multi-level system analysis approach for fusing multi-source data at the lower level, resulting in the posterior distribution of model parameters. As mentioned earlier, combining pooling methods and Bayesian theory has proven to be a reliable approach for system data fusion. Currently, this method is widely used in system reliability estimation. Furthermore, it has also been applied in animal population studies [18, 24], urban simulation [25], climate forecasting [26], and other fields. However, its application in parameter estimation research on valve-controlled cylinder systems has not been reported thus far. Addressing the limitations in the research of model parameter estimation for valve-controlled cylinder systems, as well as leveraging the advantages of data fusion method, we propose a data fusion method based on pooling fusion and Bayesian theory for estimating the parameters of valve-controlled cylinder systems.

In summary, this study aims to fuse bench test and operating data using the pooling method and Bayesian theory to utilize the information from both data fully. This data fusion method will be used to estimate and analyse model parameters for valve-controlled cylinder systems. The remainder of this paper is arranged as follows: Section 2 establishes a simulation model of the valve-controlled cylinder system and analyzes the model parameters; Section 3 introduces the test bench and operating data fusion method; Section 4 data acquisition; Section 5 results and discussion. Section 6 summarises this paper.

2 Valve-Controlled Cylinder System Modelling and Parameter Analysis

The research object of this study is the valve-controlled cylinder system of an excavator boom. It is a typical complex nonlinear system composed of a pilot handle, control valve and cylinder, as shown in Figure 1. During boom lifting, the pilot handle controls the pilot pressure signal P_XAb to drive the control valve spool to move and open the valve. The oil flows into the cylinder through the P-A circuit, drives the cylinder to move, and returns to the tank through the B-T circuit. Similarly, during the boom lowering process, the pilot pressure signal is P_XBb, and the control valve and cylinder move opposite to the lifting process. This research studies the mathematical model of the control valve and cylinder in the system without considering the pilot handle dynamic model.

Figure 1

Valve-controlled cylinder system

The dynamic equation of the control valve is as follows:

$$P_{pilot} A_{v} - F_{0} = m_{v} \frac{{\text{d}^{2} x_{v} }}{{\text{d}t^{2} }} + B_{v} \frac{{\text{d}x_{v} }}{\text{d}t} + K_{v} x_{v},$$

(1)

where P_pilot is the pilot pressure, A_v is the valve spool cross-sectional area, F₀ is the external force applied to the control valve. The valve spool is primarily subjected to static pressures, such as spring preload, and the impact of transient fluid dynamics and frictional forces on the control valve was ignored [27]. And m_v is the mass of the valve spool, B_v is the damping coefficient, K_v is the spring stiffness, and x_v is the displacement of the valve spool.

The flow rate equation at the outlet of the control valve is as follows:

$$q_{1} = \left\{ \begin{gathered} C_{q} W\sqrt {2(p_{p} - p_{A} )/\rho }, \, \,x_{v} > 0, \hfill \\ 0, \,\, x_{v} = 0, \hfill \\ C_{q} W\sqrt {2(p_{A} - p_{T} )/\rho }, \, \,x_{v} < 0. \hfill \\ \end{gathered} \right.$$

(2)

The flow rate equation at the inlet of the control valve is as follows:

$$q_{2} = \left\{ \begin{gathered} C_{q} W\sqrt {2(p_{B} - p_{T} )/\rho }, \,\, x_{v} > 0, \,\hfill \\ 0, \, x_{v} = 0, \hfill \\ C_{q} W\sqrt {2(p_{p} - p_{B} )/\rho }, \, \,x_{v} < 0, \hfill \\ \end{gathered} \right.$$

(3)

where C_q is the flow rate coefficient, W is the overflow area, p_p is the main pump outlet pressure, p_T is the tank pressure, p_A is the cylinder large cavity pressure, and p_B is the cylinder small cavity pressure.

The overflow area W is calculated using the area equivalent formula [28], as shown in Figure 2.

Figure 2

Overflow area of the control valve

Considering the cylinder leakage characteristics and the compressibility of hydraulic oil, the flow rate equation at the cylinder inlet is:

$$\left\{ {\begin{array}{*{20}l} {q_{1} = A_{A} \frac{\text{d}y}{{\text{d}t}} + C_{i} \left( {p_{A} - p_{B} } \right) + C_{e} p_{A} + \frac{{V_{{\text{A}}} }}{{\beta_{e} }}\frac{{\text{d}p_{A} }}{\text{d}t}}, \hfill \\ {V_{A} = V_{A0} + A_{A} y}. \hfill \\ \end{array} } \right.$$

(4)

The cylinder outlet flow rate equation is

$$\left\{ {\begin{array}{*{20}l} {q_{2} = A_{B} \frac{\text{d}y}{{\text{d}t}}{ - }C_{i} \left( {p_{A} - p_{B} } \right) - C_{e} p_{B} - \frac{{V_{B} }}{{\beta_{e} }}\frac{{\text{d}p_{B} }}{\text{d}t}}, \hfill \\ {V_{B} = V_{B0} - A_{B} y}, \hfill \\ \end{array} } \right.$$

(5)

where A_A and A_B are the piston areas of cylinder's large and small cavities, y is the displacement of the cylinder, C_i and C_e are the cylinder's internal and external leakage coefficients, respectively, β_e is the equivalent bulk elastic modulus of hydraulic oil, V₀₁ and V₀₂ are the initial volumes of large and small cavities, and P_A and P_B are the pressures in cylinder's large and small cavities.

The expressions of V_A0 and V_B0 are as follows:

$$\left\{ {\begin{array}{*{20}l} {V_{A0} = V_{Ad} + A_{A} L_{0} }, \hfill \\ {V_{B0} = V_{Bd} + A_{B} \left( {L - L_{0} } \right)}, \hfill \\ \end{array} } \right.$$

(6)

where V_Ad and V_Bd are the dead zone volumes of the large and small cavities of the cylinder, respectively, L is the maximum stroke of the cylinder, and L₀ is the initial position of the cylinder.

Based on the load characteristics of the cylinder, the force balance equation of the cylinder is described as follows:

$$A_{A} p_{A} - A_{B} p_{B} = M\frac{{\text{d}y^{2} }}{\text{d}t} + B_{p} \frac{\text{d}y}{{\text{d}t}} + Ky + F_{f} + F_{l},$$

(7)

where M is the total mass of the cylinder and the load, B_p is the damping coefficient of the cylinder, K is the load stiffness, F_f is the coulomb friction, F_l is the load force. In this paper, the minor friction force F_f relative to the load force F_l is ignored in a valve-controlled cylinder system.

According to Eqs. (1) to (7), a valve-control cylinder system block diagram [29,30,31] is established, as shown in Figure 3, from which the valve-controlled cylinder system simulation model is established.

Figure 3

Valve-controlled cylinder system block diagram

In the system model, the control valve parameters include A_v, F₀, m_v, B_v, K_v, C_q,$\rho$, while q₁ (q₂) represent the control valve output variables, corresponding to the valve spool displacement and flow rate at the valve port, respectively. Parameters in the cylinder include A_A, A_B, C_i, C_e,${\beta }_{e}$, V_Ad, V_Bd, L₀, M, B_p and K. S is the output variable of the cylinder, representing the displacement of the cylinder. The description, experience or measured value of each parameter is listed in Table 1.

Table 1 Parameters of the valve-controlled cylinder system

The empirical values of F₀, B_v, K_v, C_q, C_i, C_e,${ \beta }_{e}$ and B_p in the valve-controlled cylinder system simulation model are inaccurate, so accurate values need to be obtained through parameter estimation research. A_v, A_A, A_B, V_Ad and V_Bd are measurable design parameters of the component, and L₀ is the initial value of the cylinder, which shall be measured and recorded before each test.

The pump-valve bench test can collect data x_v, q₁ (q₂), p_p and p_A through which the control valve model parameters can be estimated. However, p_p and p_A are simulated loads; the difference between them and the operation conditions causes deviations in the control valve parameter estimation. In addition, the pump-valve bench test cannot collect cylinder data, and the cylinder parameters cannot be estimated through the bench test data. On the other hand, under the operation conditions, x_v and q₁ (q₂) are challenging to collect, and only the cylinder operating data can be used to estimate the parameters in the valve-controlled cylinder system model. Since the cylinder operating data are indirect control valve data, this leads to a deviation in the control valve parameter estimation.

Whether bench test or operating data are used, a deviation in parameter estimation results will occur. Consider the fusion of the two parameter estimation results to make the parameter estimation include both actual load data and control valve data. A comparison between the three methods is shown in Figure 4.

Figure 4

Comparison of different parameter estimation methods

3 Parameter Estimation Method Based on Bench Test and Operating Data Fusion

The proposed data fusion method in this study is based on the pooling method and Bayesian theory. For the continuous parameter vector θ, the posterior supported by the system tested data D can be expressed by the Bayesian formula as follows [32,33,34]:

$$p({\varvec{\theta}}\left| D \right.) = \frac{{p(D\left| {\varvec{\theta}} \right.)p({\varvec{\theta}})}}{p(D)}$$

(8)

where p(θ) is the prior probability distribution of the parameter vector ${\varvec{\theta}}$, p $\left(D|{\varvec{\theta}}\right)$ is the joint probability density of the parameter vector ${\varvec{\theta}}$ and the system tested data D, and $p\left(D\right)=\int p\left({\text{D}}|\theta \right)p\left(\theta \right){\text{d}}\theta$ is the normalisation constant. When a simulation model is a group of parameter vectors θ*, the relationship between the system tested data and the simulation results is:

$$T_{i} = S_{i} + \varphi_{i}, \, i = 1, \, 2,..., \, N,$$

(9)

where T_i and S_i are the system tested data and simulation results, ${\varphi }_{i}$ is a Gaussian random number with a mean value of zero and variance of ${\sigma }^{2}$, i is the time sample node number, and N is the total number of time sample nodes.

The maximum likelihood function of parameter estimates is as follows:

$$p\left( {D|\theta ,\sigma ^{2} } \right) = \frac{1}{{\sqrt {2\pi \sigma ^{2} } }}\exp \left( {\frac{{ - 1}}{{2\sigma ^{2} }}\sum\limits_{{i = 1}}^{N} {\frac{{(T_{i} - S_{i} )^{2} }}{N}} } \right).$$

(10)

The mean square error (MSE) function between the simulation output data and system tested data is defined as:

$$Q = \frac{1}{N}\sum\limits_{i = 1}^{N} {(T_{i} - S_{i} )^{2} }.$$

(11)

Based on the likelihood function, the posterior estimation result of the parameters obtained by continuously updating the prior data is:

$$\begin{gathered} p\left( {\theta _{j} |D} \right) = \smallint p\left( {\user2{\theta },\sigma ^{2} |D} \right){\text{d}}\user2{\theta }_{{ - j}} {\text{d}}\sigma ^{2} \propto \hfill \\ \smallint p\left( {D|\user2{\theta },\sigma ^{2} } \right)p(\user2{\theta }){\text{d}}\sigma ^{2} {\text{d}}\user2{\theta }_{{ - j}}, \hfill \\ \end{gathered}$$

(12)

where j represents the parameter index, take as 1, 2, 3,…, p, p is the total number of parameters.

Eq. (12) is a general expression of the posterior probability distribution of θ_j, and it is not easy to obtain the analytical solution of independent samples through it [35, 36]. The relevant sample sequence representing a posterior distribution can be obtained by the Markov Chain Monte Carlo (MCMC) sampling method. The process is as follows [32, 37]:

• (1) Set the total number of iterations N_n, 'burn in' iteration number N_b and parameter vector length p. Initialize the Markov chain k=0, extract the parameter vector from the parameter prior distribution N(θ): θ^*(0)=(θ₁^*,θ₂^*,…,θ_p^*), define the random walk interval length L_j of each parameter, calculate Q(θ^*(0)) and the sample variance subject to the inverse gamma distribution [38]: ${\sigma }^{2}(0)$=IG(N/2+1, Q(θ^*(0))/2);

• (2) k=k+1, update each parameter in the parameter vector according to θ_j^*=θ_j (k−1) +2L_j $\times$ U (−1 1). Calculate the acceptance function α=p(${{\varvec{\theta}}}_{{\varvec{j}}}^{\boldsymbol{*}}$)/p(${{\varvec{\theta}}}_{{\varvec{j}}-1}$)exp((−0.5/σ²(k−1))×(Q(θ^*)−Q(${{\varvec{\theta}}}_{{\varvec{j}}-1}$))), where θ^*=(θ₁(k),…, θ_j^*, θ_j+1(k−1),…, θ_p(k−1)), ${{\varvec{\theta}}}_{{\varvec{j}}-1}$=(θ₁(k),…, θ_j−1(k), θ_j(k−1), θ_j+1(k–1),…, θ_p(k−1));

• (3) Generate a random number α_n in uniform distribution U (0 1). When α_n<α, θ_j(k)=θ^*, L_j=1.01L_j; otherwise, θ_j(k)=θ_j(k−1), L_j=L_j/ 1.007;

• (4) After each iteration completes the parameter update, calculate Q(θ(k)), and extract the random number from the inverse gamma distribution IG(N/2 +1, Q(θ(k))/2) as the variance σ²(k) of the kth iteration, where θ(k)=(θ₁(k), θ₂(k), θ₃(k),…, θ_p(k));

• (5) Repeat steps (2) to (4). Stop L_j adjustment when k>N_b; when k>N_n, stop the update of θ_j^*. Remove the results of the previous N_b iteration and record the remaining parameter vector θ as the posterior distribution p (θ) of the parameter.

Based on Bayesian theory and the MCMC sampling method, the prior distribution (prior₁, prior₂ and prior₃) for each parameter is obtained separately using bench test and operating data as benchmarks. The pooling method is then applied to fuse the different prior information, followed by another round of Bayesian inference to estimate the model parameters. This process yields the final estimates of the model parameters (θ₁, θ₂) and determines the contribution of each type data through weighting coefficients k₁. This constitutes the methodology for estimating model parameters by combining bench test and operating data. The specific implementation steps are as follows:

Step 1: Assume that the initial prior distribution of the control valve model parameters is a uniform distribution U1 (${\theta }_{{1j}_{1}}^{L}{ \theta }_{{1j}_{1}}^{U}$), according to the empirical values in Table 1. Here, ${\theta }_{{1j}_{1}}^{L}$ represents the minimum values for the control valve j₁th parameter, ${\theta }_{{1j}_{1}}^{U}$ represents the maxmum values for the control valve j₁th parameter, and j₁ takes values of 1, 2,…, 4. Based on the bench test data of the control valve, the distribution of the control valve model parameters (prior₁ (${{\varvec{\theta}}}_{1})$) is obtained through Bayesian theory, which is an estimated value lacking actual load information.

Step 2: Similarly, the control valve and hydraulic cylinder parameters are set as uniform distributions U1 (${\theta }_{{1j}_{1}}^{L}{ \theta }_{{1j}_{1}}^{U}$) and U2 (${\theta }_{{1j}_{2}}^{L}{ \theta }_{{1j}_{2}}^{U}$), respectively, based on the empirical values provided in Table 1. Here, ${\theta }_{{1j}_{2}}^{L}$ represents the minmum value for the cylinder j₂th parameter, ${\theta }_{{1j}_{2}}^{U}$ represents the maxmum value for the cylinder j₂th parameter, and j₂ takes values of 1, 2,…, 4. Based on the operating data of the cylinder, the distribution of the control valve and cylinder parameters (prior₂ (${{\varvec{\theta}}}_{1}$) and prior₃ (${{\varvec{\theta}}}_{2})$) are obtained through Bayesian theory. These distributions incorporate actual load information.

Step 3: After obtaining the distributions of the control valve parameters (prior₁ (${{\varvec{\theta}}}_{1}$) and prior₂ (${{\varvec{\theta}}}_{1})$), a geometric pooling method [17] is employed to fuse these distributions, ensuring that the parameter estimates encompass both the control valve output data and the actual load information. The pooled result is denoted as prior*:

$$\begin{gathered} prior*({{\varvec{\theta}}}_{{\mathbf{1}}} ) = \prod\limits_{i = 1}^{2} {prior_{i} ({{\varvec{\theta}}}_{{\mathbf{1}}} )^{{k_{i} }} } \hfill \\ = prior_{1} ({{\varvec{\theta}}}_{{\mathbf{1}}} )^{{k_{1} }} prior_{2} ({{\varvec{\theta}}}_{{\mathbf{1}}} )^{{1 - k_{1} }}, \hfill \\ \end{gathered}$$

(13)

where k_i is the weight coefficient, representing the contribution of prior_i to prior*, $\sum\limits_{i = 1}^{2} {k_{i} } = 1$, and its prior distribution can be set as a uniform distribution U (0, 1) [21].

Step 4: Take the distributions prior*(${{\varvec{\theta}}}_{1}$) and prior₃ (${{\varvec{\theta}}}_{2}$) as new prior information and take the cylinder operating data as the benchmark to estimate the system model parameters through Bayesian theory. Obtain the final estimated values of the system model parameters vector θ₁, θ₂ and weight coefficient vector k₁. The process is shown in Figure 5.

Figure 5

Valve-controlled cylinder system parameter estimation based on the data fusion method

In Figure 5, prior₁ (${{\varvec{\theta}}}_{1}$) and prior₂ (${{\varvec{\theta}}}_{1}$) and prior₃ (${{\varvec{\theta}}}_{2})$ are obtained using the MCMC sampling method based on the Bayesian posterior estimation in Eq. (12). The Bayesian posterior estimation expression in Step 4 is updated to Eq. (14). The estimation results of the system model parameters vector θ₁, θ₂ and weight coefficient vector k₁ are also obtained through the MCMC sampling method.

$$\begin{gathered} p\left( {\theta _{j} |D} \right) = \smallint p\left( {\theta ,\sigma ^{2} |D} \right)\text{d}\theta _{{ - j}} \text{d}\sigma ^{2} \hfill \\ \propto \smallint p\left( {D|\theta ,\sigma ^{2} } \right)prior*(\theta _{{j1}} ) \hfill \\ prior_{3} (\theta _{{j2}} )\text{d}\sigma ^{2} \text{d}\theta _{{ - j1}} \text{d}\theta _{{ - j2}}. \hfill \\ \end{gathered}$$

(14)

4 Data Acquisition

4.1 Operating Data Acquisition

The operating data acquisition test was conducted at an excavator manufacturing company's standard test site, which complies with testing standards. A prototype of a medium-sized positive flow control system excavator [39, 40] was employed for the experiment, primarily targeting data acquisition such as main pump pressure, boom cylinder pressure and displacement. Detailed information regarding the measurement points is provided in Table 2. Data was recorded and stored via a Dewe-43 data acquisition device with a frequency of 500 Hz.

Table 2 Measurement point details

The test conditions comprised three types: single action of the boom using fast operation, single action of the boom using slow operation, and excavation operation. They are known simply as fast boom operation, slow boom operation, and excavation. During fast boom operation conditions, the pilot signal resemble a step, while slow boom operation conditions resemble a ramp signal. The excavation condition utilises ordinary soil as the working material. Each test consisted of three cycles, and a total of 20 tests were performed for each operation condition. The experimental arrangement is depicted in Table 3.

Table 3 Experimental conditions description

Among the three operation conditions, the excavation operation load is the worst, followed by the fast boom operation, while the slow boom operation is the most stable. The experimental site and sensor layout are illustrated in Figure 6.

Figure 6

Experiment for operating data acquisition

4.2 Bench Test Data Acquisition

Bench test data collection for the control valve was carried out in accordance with the parameter estimation requirements of the model for the valve-controlled cylinder system. A pump-valve test bench was utilized to perform load simulation on the pump and control valve, enabling concurrent measurement of the control valve spool displacement (x_v) and flow rates (q₁ (q₂)) through the valve ports. The pump-valve test bench principle is depicted in Figure 7(a). In the test, the motor drives the pump, while the boom valve and pump are controlled by the pilot pressure (P_pilot). The control valve load is set using Po1, measured via P2, while the pump's outlet pressure is measured by pressure sensor P1. Valve flow rate and spool displacement are collected by flow rate sensor Q1 and displacement sensor Xs, respectively. Pilot pressure (Ppilot) is collected by pressure sensor P3. All test data are recorded using Dewe-43 data acquisition equipment. Figure 7(b) illustrates the arrangement of the test bench.

Figure 7

Test bench description: (a) Test bench principle diagram, (b) Test bench picture

Similarly, the test bench primarily collected experimental data on the boom cylinder control valve under fast boom operation conditions, slow boom operation conditions, and excavation conditions. The pilot signals for each condition are the same as those used during operating data acquisition experiments. The maximum load of the control valve is set to 300 bar. Each test consisted of 3 cycles, and 20 tests were performed for each operation condition. Detailed information about the measurement points of the bench test is presented in Table 4.

Table 4 Bench test measurement point details

5 Results and Discussion

5.1 Parameter Estimation Results for Slow Operating Conditions

Using the slow operating condition as a case study, this paper conducts an in-depth analysis and discussion on the estimation of model parameters in the valve-controlled cylinder system. The obtained parameter estimation results are fitted to a normal distribution. When the bench test and operating data are taken as references respectively, the estimation results of the control valve parameters are shown in Figure 8 as prior₁ and prior₂. The parameter estimation result of the control valve, obtained using the data fusion method, is shown as fusion_v in Figure 8. It can be seen from Figure 8 that prior₁, prior₂ and fusion_v of each parameter have different distributions. Prior₂ exhibits the highest variance, displaying a dispersed distribution, while prior₁ demonstrates a comparatively smaller variance and a more centralized distribution. Fusion_v exhibits a mean value in proximity to that of prior₂, while its variance closely resembles that of prior₁. High consistency exists between the frequency distribution histogram and the probability density curve of fusion_v, indicating the stability and accuracy of parameter estimation results derived from the data fusion method. Adopting the mean value of fusion_v as the final estimated parameters for control valve.

Figure 8

Estimation results for the control valve parameters

The 95% confidence interval of the control valve parameter estimation results before and after the fusion method is shown in Figure 9. It can be seen from Figure 9 that the confidence intervals of prior₁, prior₂ and fusion_v are different, and the relationship between their confidence intervals is prior_{1_}interval<fusion_{v_}interval <prior_{2_}interval. Although the B_v confidence interval result differs from the above conclusion, the B_v confidence interval range is small, and the gap between them can be ignored.

Figure 9

95% confidence intervals of control valve parameter estimation results

To obtain the mean values and 95% confidence intervals for each weight coefficient, as shown in Figure 10. Taking the mean value as the estimation result of the weight coefficient, k_1Cq is 0.382, indicating that the bench test data has a small contribution to the estimation result of C_q. In contrast, the operating data have a significant contribution to them. This result is consistent with the characteristics that C_q is greatly affected by the load. However, k_1Bv and k_1Kv are all close to 0.5, indicating that the contributions of the bench test and operating data to the estimation results of B_v and K_v are equal. This seems to be related to the fact that they are the inherent attribute parameters of the control valve. The value of k_1F0 is 0.456, and the contribution of the operating data to the estimated result is slightly higher than that of the bench test data.

Figure 10

95% confidence interval of k₁

Before and after applying the data fusion method, the estimation of cylinder parameters is based on cylinder operating data, while the prior information remains the same. The estimation result of the cylinder parameters prior₃ and fusion_c exhibit little change, as shown in Figure 11. The slight change in cylinder parameters is caused by the change in parameters C_q before and after the data fusion method. They are connected through the flow rate continuity equation.

Figure 11

Estimation result for cylinder parameters

Figure 12 presents the 95% confidence intervals of the cylinder estimated parameters. Similarly, the confidence intervals exhibit small deviation, the variation in C_q likewise influences the differences between them.

Figure 12

Cylinder parameter estimation results’ 95% confidence intervals

5.2 Parameter Estimation Results of Other Conditions

Using the same method, the parameters of the valve-controlled cylinder system model were estimated and analyzed under fast boom operation and excavation conditions. The estimation results under fast boom operation conditions are shown in Tables 5, 6 and 7. While the parameter estimates and weight coefficients slightly differ from those obtained under slow boom operation conditions, it can be observed from the tables that they exhibit similar trends.

Table 5 Parameter estimation results of control valve under fast boom operation conditions

Table 6 Parameter estimation results of weight coefficients under fast boom operation conditions

Table 7 Parameter estimation results of cylinder under fast boom operation conditions

The estimation results under excavation operation conditions are shown in Tables 8, 9 and 10. Although the estimated values differ between fast and slow boom operation conditions, they exhibit similar trends.

Table 8 Parameter estimation results of control valve under excavation operation conditions

Table 9 Parameter estimation results of weight coefficients under excavation operation conditions

Table 10 Parameter estimation results of cylinder under excavation operation conditions

5.3 Comparison of Different Parameter Estimation Methods

Based on the data obtained in Section 4, the parameter estimation of the system model is studied through the data fusion method, the Bayesian method [33, 38] and the PSO algorithm [2]. Notably, the latter two methods utilize cylinder operating data as their benchmark. Subsequently, leveraging the outcomes of the parameter estimation, a simulation model is developed to compute the cylinder displacement across diverse working conditions, and compare the model calculation results with the test results, as shown in Figure 13. It can be seen from Figure 13 that for all working conditions, the results of the parameter estimation method using the PSO algorithm have the worst coincidence with the test results. The above research has demonstrated that bench test and operating data contribute significantly to estimating control valve parameters. Due to the difficulty in collecting operating data of the control valve, the parameter estimation process based on operating data and the Bayesian method lacks information regarding the control valve. Additionally, the displacement data of the hydraulic cylinder is not a direct measurement of the control valve, leading to biases in the estimated control valve parameters and affecting the accuracy of the valve-controlled cylinder system model. The data fusion method integrates the actual load data and the control valve output data, making the parameter estimation results more accurate. Consequently, the simulation results of the valve-controlled cylinder system model-based data fusion method exhibit good agreement with the test results.

Figure 13

Comparison of cylinder displacement simulation results

Each parameter estimation method's model accuracy value Q is calculated using Eq. (11), and the results are listed in Table 11.

Table 11 Model accuracy value results comparison

In all working conditions, the relationship between the model accuracy obtained using the three methods is Q_fusion<Q_bayesain<Q_PSO, indicating that the model accuracy based on the data fusion method is the highest. Moreover, $\Delta {Q}_{maxfusion}$<$\Delta {Q}_{maxBayesain}$<$\Delta {Q}_{maxPSO}$, indicating that the parameter estimation result based on the data fusion method is the most stable. In addition, the relationship between the model accuracy values of different operation conditions in all methods is Q_Slow< Q_Fast< Q_Excavation, indicating that the worse the working conditions, the worse the model accuracy. This proves the critical role of the data fusion method.

6 Conclusions

To effectively use the advantages of bench test and operating data in research on the estimation of simulation model parameters, this paper proposed a model parameter estimation method for bench test and operating data fusion based on Bayesian theory and pool method. By using the data fusion method, valve-controlled cylinder system model parameters estimation were studied, and the following research conclusions were drawn:

• 1 Taking slow boom operation conditions as an example, variance in estimation results based on operating data (prior₂) is the largest, while for bench test data (prior₁), it is small. Prior₁ and prior₂ have varying degrees of influence on fusion_v. The relationship between the confidence intervals of prior₁, prior₂ and fusion_v is prior₁_interval< fusion_v_interval< prior₂_interval.

• 2 The weighting coefficient vector k₁ was obtained for slow boom operation conditions. From the weight coefficient estimation results, the bench test data contribute less to the estimation results of C_q, while the operating data contribute more. The contribution of bench test and operating data to B_v and K_v estimation results is similar, and the contribution of operating data to F₀ estimation results is slightly higher than that of the bench test data.

• 3 Using operating data as a benchmark, it is possible to obtain relatively accurate estimates of the cylinder parameters, and there is a high level of consistency between prior₃ and fusion_c. After the data fusion method, the few changes in the parameter results are caused by changes in the C_q estimation results.

• 4 Parameters of the valve-controlled cylinder system model were estimated under fast boom operation and excavation conditions simultaneously. Estimated values of parameters slightly differ among the three conditions, yet exhibit consistent patterns and trends.

• 5 Compared with the Bayesian method and the PSO algorithm, the simulation model results based on the data fusion method coincide best with the test results, and the model accuracy is the highest. In addition, the worse the operation conditions, the more inaccurate the parameter estimation results. This proves the importance of data fusion in parameter estimation research.