Mitigating Mode Switching Oscillation in a One-Motor-One-Pump Motor-Controlled Hydraulic Cylinder via System Pressure Control: Simulation Study

Abstract

This study focuses on a hydraulic cylinder that is directly connected to a fixed-displacement hydraulic pump driven by an electric servo motor. This particular setup is referred to as a one-motor-one-pump motor-controlled hydraulic cylinder (MCC). This paper presents a new approach to address mode switching oscillation (MSO) in MCCs by incorporating system pressure control capabilities. It conducts a detailed investigation into the factors that contribute to MSO in standard MCCs and thoroughly evaluates the effectiveness of the proposed system in mitigating MSO. The simulation results demonstrate the successful suppression of MSO. In conclusion, the proposed MCC with system pressure control capabilities is validated and, furthermore, it shows great potential for practical applications involving small loads and rapid retraction.

1. Introduction

By directly connecting a hydraulic cylinder to a fixed-displacement hydraulic pump driven by an electric servo motor, the cylinder’s motion is regulated through the angular velocity of the servo motor. This configuration is referred to as a one-motor-one-pump motor-controlled hydraulic cylinder (MCC) in this study. The MCC is a promising technological alternative to a valve-controlled hydraulic cylinder (VCC).

Figure 1 depicts the typical layout of a MCC, which comprises an electric motor, a fixed-displacement hydraulic pump, an accumulator or reservoir, auxiliary valves, and a differential hydraulic cylinder. In this system, the hydraulic cylinder is directly coupled to the hydraulic pump, which is driven by an electric servo motor. The cylinder’s motion is therefore dictated by the rotational speed of the servo motor.

The auxiliary valves play a crucial role in balancing the differential flow rate and maintaining load stability. Unlike VCCs, MCCs prevent energy losses related to valve throttling, resulting in markedly improved energy efficiency. For example, a single-boom crane utilizing an MCC can achieve a 62% reduction in energy consumption compared to one employing a VCC over the same operational cycle [2]. Moreover, an experimental evaluation described in [3] demonstrated that integrating six MCCs into an excavator lowered energy usage by 47.8% relative to traditional valve-controlled systems under equivalent working conditions.

However, the occurrence of mode switching oscillation (MSO) may hinder the use of MCCs. The MSO was initially observed when a pump-controlled cylinder lowers a small load at a high speed [4]. This pump-controlled cylinder uses the same valve compensation mechanism for the differential flow rate as a standard MCC. The MSO is characterized by severe oscillations of the compensation valves between the on and off positions and the hydraulic pump/motor unit oscillating between pumping and motoring modes. A predictive observer was proposed for controlling the MSO. However, the observer lead time is a limitation for the performance [4]. In a subsequent study [5], further analysis was conducted to determine the root cause of the MSO. Nevertheless, the study’s conclusion only confirms the occurrence of MSO under specific conditions, without providing an explanation of the specific conditions or the underlying mechanism behind the phenomenon.

An approach using Lyapunov exponents theory revealed a significant influence of the cracking pressure of the rod-side pilot-operated check valve (POCV) on triggering the MSO [6]. Additionally, the MSO can be predicted based on the level of the inertial load [7]. Another way to predict the MSO is to identify the MSO region on the load pressure and velocity plane [8]. A specially designed 2/3 spool valve was installed on the cylinder bore-side line to eliminate the MSO when the system operates within the identified region [8]. However, this proposed spool valve is not commercially available, and its opening areas must be adjusted based on the loading conditions. In a subsequent study [9], two sequence valves instead of POCVs were used to compensate for the differential flow rate and suppress the MSO. To further enhance the performance of the two sequence valves, a new definition of four-quadrant representation and a separate hydraulic processing module based on the new representation were proposed in [10,11]. However, an open tank and a charge line are required in these studies, which is not applicable to MCCs.

A concept of using two electric on/off valves instead of POCVs to compensate for the differential flow rate in an MCC was introduced in [12]. This strategy, accompanied by an appropriate control algorithm, demonstrated the potential to eliminate MSO. Nonetheless, the requirement of a 30 bar accumulator pressure in this approach may pose practical challenges for MCCs because the pressure limitations of hydraulic pump housings are normally lower than 2 bar. A novel method of using controlled leakages generated by two flow control valves and dedicated control algorithms was proposed to eliminate the MSO in [13,14]. Furthermore, this study showed that the inverse shuttle valve outperforms POCVs in compensating for the cylinder’s differential flow rate. However, it should be noted that the commercially available inverse shuttle valves have limitations in providing high flow rates, which can limit their use in larger systems. This drawback also applies to the underlapped inverse shuttle valves proposed to mitigate the MSO [15,16]. Although different pressure sensing strategies and active hysteresis controls can be introduced to reduce the risk of MSO, a high-level reservoir pressure is required to ensure controllability [17].

From the above discussion, despite extensive research on MSO, further efforts are necessary to explore the root cause of the MSO. Furthermore, a suitable method for effectively mitigating the MSO in MCCs used in large-scale applications needs to be developed. To address these research gaps, this paper presents an in-depth analysis of MSO in MCCs. It proposes a new system aimed at mitigating this oscillation through the integration of system pressure control. Additionally, the paper presents simulations of the proposed system driving a single-boom crane to verify the functionality of the proposed system.

2. Mode Switching Oscillation Analysis

In this section, the root cause of the MSO in MCCs is analyzed based on a standard MCC depicted in Figure 2. The analysis breaks down the standard MCC into two main assemblies: the MCC cylinder, highlighted with blue dashed lines, and the MCC drive unit, outlined with purple dashed lines. In the cylinder assembly, $p_{\mathrm{a}}$ represents the bore-side cylinder pressure, $p_{\mathrm{b}}$ represents the rod-side cylinder pressure, $A_{\mathrm{a}}$ represents the piston area, and $A_{\mathrm{b}}$ represents the cylinder annular area. Similar to the description provided in Section 1, the drive unit assembly comprises a four-quadrant hydraulic pump/motor unit (P) directly connected to the cylinder, an electric motor/generator unit (M) driving P, an accumulator (ACC) serving as the reservoir, and two pilot-operated check valves (POCVs) compensating the differential flow. The accumulator pressure is represented by $p_{\mathrm{acc}}$. In the absence of load-holding devices, $p_{\mathrm{b}}$ serves as the pilot pressure of POCVa, while $p_{\mathrm{a}}$ serves as the pilot pressure of POCV_b.

2.1. Oscillation Root Cause General Analysis

The fundamental cause of the MSO in a standard MCC is the inherent mismatch between the four-quadrant operation of the MCC cylinder and the four-quadrant operation of the MCC drive unit. This mismatch arises because $p_{\mathrm{a}}$ serves simultaneously as the cylinder bore-side pressure and the left-hand pressure of the hydraulic pump/motor unit, while $p_{\mathrm{b}}$ functions as both the cylinder rod-side pressure and the right-hand pressure of the hydraulic pump/motor unit. A more detailed analysis is presented in the remainder of this section.

Four-quadrant operations of the cylinder and the drive unit in a standard MCC are shown in Figure 3. The cylinder’s four-quadrant operation is defined based on the cylinder force ($F_{\mathrm{cyl}}$) and the piston velocity (${\dot{x}}_{\mathrm{p}}$). $F_{\mathrm{cyl}}$ is calculated via Equation (1).

$$F_{\mathrm{cyl}}=p_{\mathrm{a}}{A}_{\mathrm{a}}-p_{\mathrm{b}}{A}_{\mathrm{b}}=A_{\mathrm{a}}(p_{\mathrm{a}}-\alpha p_{\mathrm{b}})$$

(1)

The cylinder area ratio is represented by $\alpha =A_{\mathrm{b}}/A_{\mathrm{a}}$. Because $A_{\mathrm{a}}$ is a constant, $F_{\mathrm{cyl}}$ is proportional to $(p_{\mathrm{a}}-\alpha p_{\mathrm{b}})$. Therefore, the horizontal axis in cylinder four-quadrant representation is defined by $(p_{\mathrm{a}}-\alpha p_{\mathrm{b}})$ in Figure 3.

The four-quadrant operation of the drive unit is defined by the relationship between the pressure difference ($p_{\mathrm{a}}-p_{\mathrm{b}}$) across P and the angular velocity ($\omega$) of P. Because $\omega$ is proportional to ${\dot{x}}_{\mathrm{p}}$, it can be said that the four-quadrant representations of the cylinder and the drive unit have the same vertical axis. Apparently, $(p_{\mathrm{a}}-p_{\mathrm{b}})$ is not proportional to $(p_{\mathrm{a}}-\alpha p_{\mathrm{b}})$. Therefore, the horizontal axes of the two representations are different. When the drive unit changes operation modes from the right-half plane to the left-half plane in Figure 3, it crosses the vertical axis with a condition $p_{\mathrm{a}}=p_{\mathrm{b}}=p_{\mathrm{acc}}$, which is also the switching condition for the POCVs. However, the same condition on the cylinder four-quadrant representation is still on the right-half plane with a positive pressure of $(1-\alpha )p_{\mathrm{acc}}$ marked as a red dotted line. The cylinder force corresponding to this positive pressure is defined as the critical cylinder force ($F_{\mathrm{cyl},\mathrm{cr}}$), which is calculated via Equation (2). In Figure 3, the red dotted line and the Y-axis of the cylinder four-quadrant operation forms a hatched area. This area is also the difference between the MCC cylinder’s four-quadrant operation and the MCC drive unit’s four-quadrant operation. A comprehensive analysis of the effects of crossing the hatched area from each of the four quadrants is presented in the following section.

$$F_{\mathrm{cyl},\mathrm{cr}}=A_{\mathrm{a}}(p_{\mathrm{acc}}-\alpha p_{\mathrm{acc}})=A_{\mathrm{a}}{p}_{\mathrm{acc}}(1-\alpha )$$

(2)

2.2. Oscillation Analysis in Four Quadrants

In Q_I of the drive unit, P is in pumping mode and rotating anticlockwise. It should be noted that, in this paper, when P rotates anticlockwise, it pumps/motors oil from the right to the left side. Conversely, when rotating clockwise, it pumps/motors oil from the left to the right side. POCV_b is open, while POCV_a is closed. $p_{\mathrm{a}}$ is higher than $p_{\mathrm{b}}$. The cylinder is driven by P to extend. The force equilibrium of the piston is shown in Equation (3). m is the equivalent mass of the system. $F_{\mathrm{load}}$ is the load force applied on the piston, which typically depends on m. Therefore, it is given via Equation (5). Equation (4) shows the friction force ($F_{\mathrm{fr}}$) to be a function of the piston velocity ${\dot{x}}_{\mathrm{p}}$. The functions $F_{\mathrm{fr}}$ and $F_{\mathrm{load}}$ in Equations (4) and (5) are all positive defined. It should be noted that the acceleration term $m{\ddot{x}}_{\mathrm{p}}$ in Equation (3) is generally very small in comparison to other force terms for heavy hydraulic machinery. Therefore, $F_{\mathrm{cyl}}$ is mainly affected by $F_{\mathrm{fr}}$ and $F_{\mathrm{load}}$.

$$F_{\mathrm{cyl}}+F_{\mathrm{load}}+F_{\mathrm{fr}}=m{\ddot{x}}_{\mathrm{p}}$$

(3)

$$F_{\mathrm{fr}}=-f_{\mathrm{fr}}\left({\dot{x}}_{\mathrm{p}}\right)$$

(4)

$$F_{\mathrm{load}}=\left\{\begin{array}{cc}-f_{\mathrm{L}}\left(m\right)\hfill & \mathrm{in}{Q}_{\mathrm{I}}\mathrm{and}{Q}_{\mathrm{IV}}\hfill \\ f_{\mathrm{L}}\left(m\right)\hfill & \mathrm{in}{Q}_{\mathrm{II}}\mathrm{and}{Q}_{\mathrm{III}}\hfill \end{array}\right.$$

(5)

When the positive $F_{\mathrm{cyl}}$ needs to decrease below $F_{\mathrm{cyl},\mathrm{cr}}$ under specific operating conditions, it tends to cross the red dotted line into the upper part of the hatched area. According to Equation (3), the reason can be a reduction in m and/or ${\dot{x}}_{\mathrm{p}}$. This shift would require a transition from $p_{\mathrm{a}}>p_{\mathrm{b}}$ to $p_{\mathrm{a}}<p_{\mathrm{b}}$. In this case, the cylinder is still in Q_I, consuming energy but the drive unit has gone into Q_II, starting to regenerate energy. The accumulator becomes the only energy source. However, the accumulator in an MCC typically operates at a pressure below 3 bar and is primarily used for storage rather than as an active energy source. Furthermore, given the energy losses in both the cylinder and the drive unit during operations, the accumulator cannot provide sufficient energy to support the cylinder’s energy consumption and the drive unit’s energy regeneration simultaneously. As a result, the system remains on the red dotted line without crossing it, and both POCV_a and POCV_b are closed. However, the accumulator is used to supply the cylinder differential flow rate in Q_I and Q_II. Therefore, the two POCVs mainly work as check valves. As P continues pumping oil from the rod side to the bore side, POCV_b will reopen when $p_{\mathrm{b}}$ drops below $p_{\mathrm{acc}}$. There are no sudden changes in $p_{\mathrm{a}}$ and $p_{\mathrm{b}}$. Therefore, the MSO will not be triggered. The system remains stuck in this state until the operating conditions change. However, considering Equations (3)–(5), the $F_{\mathrm{cyl}}$ must be at least equal to the sum of $f_{\mathrm{fr}}\left({\dot{x}}_{\mathrm{p}}\right)$ and $f_{\mathrm{L}}\left(m\right)$, which is seldom lower than $F_{\mathrm{cyl},\mathrm{cr}}$. As a result, when the cylinder operates in quadrant Q_I, the system rarely approaches the red dotted line.

In Q_II of the drive unit, P is in motoring mode and rotating anticlockwise, and POCV_a is open. The energy is regenerated from the load by P. $p_{\mathrm{b}}$ is higher than $p_{\mathrm{a}}$. The piston is driven by $F_{\mathrm{load}}$ to extend. In Q_II, $F_{\mathrm{cyl}}$ is negative. When $F_{\mathrm{load}}$ decreases value or changes directions, and/or ${\dot{x}}_{\mathrm{p}}$ increases, the negative cylinder force $F_{\mathrm{cyl}}$ may need to cross the hatched area and the red dotted line. When $F_{\mathrm{cyl}}$ crosses the upper part of the hatched area, the drive unit will continue operating in its Q_II without any mode switch, and the POCV opening states will also remain unchanged. When $F_{\mathrm{cyl}}$ crosses the red dotted line, POCV_a and POCV_b close. But POCV_b will reopen as a check valve when necessary. As a result, there are no sudden changes in $p_{\mathrm{a}}$ and $p_{\mathrm{b}}$ and the MSO will not be triggered. Nevertheless, as previously analyzed, in most cases within Q_II, the system rarely transitions into the hatched area where the cylinder consumes energy while the drive unit regenerates energy.

In Q_III of the drive unit, P is in pumping mode and rotating clockwise, with POCV_a open and $p_{\mathrm{b}}$ higher than $p_{\mathrm{a}}$. The piston is driven by P to retract. When $F_{\mathrm{load}}$ decreases in value or changes direction, and/or ${\dot{x}}_{\mathrm{p}}$ decreases, the negative cylinder force $F_{\mathrm{cyl}}$ may need to cross the lower part of the hatched area. It is important to note that, in Q_III and Q_IV, the accumulator is used for storing the cylinder differential flow rate. Consequently, the two POCVs do not function as check valves, and sudden changes in $p_{\mathrm{a}}$ and $p_{\mathrm{b}}$ can easily occur. When $F_{\mathrm{cyl}}$ crosses the lower part of the hatched area, the drive unit remains in Q_III without any mode switch, and the POCV opening states stay unchanged. From an energy perspective, in the lower part of the hatched area, the accumulator acts as an energy storage device, capturing energy from both P and the cylinder, rather than supplying energy. As a result, it is easier for the system to enter the lower part of the hatched area than the upper part. When $F_{\mathrm{cyl}}$ continues to increase and crosses the red dotted line, $p_{\mathrm{b}}$ decreases to $p_{\mathrm{acc}}$, causing POCV_a to close. Consequently, the differential flow rate from the cylinder bore side has no available path for discharge. As P continues to pump oil from the bore side to the rod side, the undischarged cylinder differential flow rate causes a sudden increase in $p_{\mathrm{a}}$. This sudden increase in $p_{\mathrm{a}}$ causes a reduction in ${\dot{x}}_{\mathrm{p}}$ and the opening of POCV_b. The reduction in ${\dot{x}}_{\mathrm{p}}$ reduces $f_{\mathrm{fr}}$. According to Equation (3), a smaller $f_{\mathrm{fr}}$ leads to a greater $f_{\mathrm{cyl}}$. Therefore, the system can cross the red dotted line smoothly without inducing oscillations. However, in industrial hydraulic machinery, when a cylinder operates in Q_III, it is rare for $F_{\mathrm{cyl}}$ to cross zero, as it must overcome both $F_{\mathrm{fr}}$ and $F_{\mathrm{load}}$. Similarly, it is uncommon for the cylinder to shift from Q_III to Q_IV.

In Q_IV of the drive unit, P is in motoring mode and rotating clockwise, and POCV_b is open. $p_{\mathrm{a}}$ is higher than $p_{\mathrm{b}}$. The piston is driven by $F_{\mathrm{load}}$ to retract. When $F_{\mathrm{load}}$ decreases in value indicating a reduction in the equivalent mass of the moving parts (m), and/or $F_{\mathrm{fr}}$ increases due to the increment in ${\dot{x}}_{\mathrm{p}}$, $F_{\mathrm{cyl}}$ can decrease to a level close to or below $F_{\mathrm{cyl},\mathrm{cr}}$, which lies on the red dotted line. When this occurs, particularly at high piston speeds, POCV_b closes abruptly, leading to a sudden and significant increase in $p_{\mathrm{b}}$. Notably, the volume on the rod side is much smaller than that on the bore side, causing the flow rate to the rod side to increase by a factor of $1/\alpha$. As a result, the increase in $p_{\mathrm{b}}$ in this case is significantly greater than the increase in $p_{\mathrm{a}}$ in Q_III. At a certain point, POCV_a is forced to suddenly open, causing $p_{\mathrm{a}}$ to drop immediately to $p_{\mathrm{acc}}$. The hydraulic cylinder then increases its retraction speed, surpassing the speed of P. Consequently, $p_{\mathrm{b}}$ drops to $p_{\mathrm{acc}}$, while $p_{\mathrm{a}}$ rises above $p_{\mathrm{acc}}$, leading to a change in the POCVs’ opening states. The drive unit shifts back to Q_IV, and $F_{\mathrm{cyl}}$ increases to a value beyond the red dotted line before gradually approaching it again. This sequence repeats itself rapidly, causing frequent mode switching in P and the POCVs, as well as significant pressure and position oscillations, commonly referred to as MSO. The MSO continues until $F_{\mathrm{cyl}}$ moves sufficiently far away from the red dotted line, as dictated by the new operating conditions, at which point the oscillations cease. The MSO triggered in Q_IV is commonly observed due to the typical behavior of differential cylinders used in industrial applications. These cylinders tend to extend slowly under heavy loads and retract rapidly once the load is unloaded [4].

2.3. Linear Model Analysis

To further investigate the MSO in Q_IV, a stability analysis on a simplified and linearized model of the standard MCC shown in Figure 2 is carried out. This simplified and linearized model is derived from the nonlinear model of the MCC presented in Section 4. During the simplification process, certain features are omitted from the model to streamline the analysis:

• dynamics of the electric servo motor,
• leakage in the hydraulic pump,
• variable bulk modulus based on the ratio of undissolved gas,
• cylinder friction,
• accumulator dynamics.

The model linearization is based on a steady state situation where the system has changed from Q_I (${\dot{x}}_{\mathrm{p}}>0$) to Q_IV (${\dot{x}}_{\mathrm{p}}<0$) and with an external load, $F_{\mathrm{load}}$, that ensures that $p_{\mathrm{a}}$ and $p_{\mathrm{b}}$ are almost equal. The following equations appear with tilde on the variables signifying linearized values:

$$m{s}^2{\tilde{x}}_{\mathrm{p}}={\tilde{p}}_{\mathrm{b}}\alpha A_{\mathrm{a}}-{\tilde{p}}_{\mathrm{a}}{A}_{\mathrm{a}}$$

(6)

$$s{\tilde{p}}_{\mathrm{b}}={\displaystyle \frac{1}{C_{\mathrm{b}}}}({\tilde{Q}}_{\mathrm{p}}-s{\tilde{x}}_{\mathrm{p}}\alpha A_{\mathrm{a}}-{\tilde{Q}}_{\mathrm{POCV},\mathrm{b}})$$

(7)

$$s{\tilde{p}}_{\mathrm{a}}={\displaystyle \frac{1}{C_{\mathrm{a}}}}(-{\tilde{Q}}_{\mathrm{p}}+s{\tilde{x}}_{\mathrm{p}}{A}_{\mathrm{a}})$$

(8)

$${\tilde{Q}}_{\mathrm{POCV},\mathrm{b}}=K_{\mathrm{qu}}{\tilde{u}}_{\mathrm{POCV},\mathrm{b}}+K_{\mathrm{qp}}{\tilde{p}}_{\mathrm{b}}$$

(9)

$${\tilde{u}}_{\mathrm{POCV},\mathrm{b}}={\displaystyle \frac{\psi {\tilde{p}}_{\mathrm{a}}-{\tilde{p}}_{\mathrm{b}}}{\Delta p_{\mathrm{open}}}}$$

(10)

In the above equations, the pilot operated check valve (POCV_b), connecting the rod side pressure, $p_{\mathrm{b}}$, with the accumulator pressure, $p_{\mathrm{acc}}$, is considered infinitely fast. Further, the usual flow and pressure gain coefficients have been introduced in Equation (9). In Equation (10), the dimensionless opening of the check valve is included, adding the pilot area ratio, $\psi$, and the required pressure difference to pilot the valve fully open, $\Delta p_{\mathrm{open}}$.

Based on Equations (6)–(10), a 3rd order transfer function between the cylinder velocity and the pump flow, $F\left(s\right)={\displaystyle \frac{s{\tilde{x}}_{\mathrm{p}}\alpha A_{\mathrm{a}}}{{\tilde{Q}}_{\mathrm{p}}}}$, can be established. Applying the Routh-Hurwitz stability criterion to that yields the following inequalities:

$$K_{\mathrm{qp}}\Delta p_{\mathrm{open}}-K_{\mathrm{qu}}>0$$

(11)

$$C_{\mathrm{a}}(K_{\mathrm{qp}}\Delta p_{\mathrm{open}}-K_{\mathrm{qu}})-C_{\mathrm{b}}\psi K_{\mathrm{qu}}>0$$

(12)

Typical pilot operated check valves have an on/off behavior with very small opening pressures, $\Delta p_{\mathrm{open}}$. Therefore, it is not feasible, with practical values, to get either of the above expressions to be positive. The only solution is to change the characteristics of the pilot operated check valve and introduce an extremely high opening pressure. That will give stability; however, it will ruin the efficiency and the operational range of the system. Even with an unreasonably high resistance from the pilot operated check valve there would still be issues (because of Equation (12)) when the cylinder is near zero stroke, as $C_{\mathrm{b}}\gg C_{\mathrm{a}}$. Hence, the system is inherently oscillatory, and it is not feasible to mitigate that behavior from adjusting the regular system parameters.

2.4. Oscillation Analysis Summary

All the analyses and demonstrations of oscillations across the four quadrants, as well as the mismatch between the cylinder’s and the drive unit’s four-quadrant operations, are based on one fundamental fact: $p_{\mathrm{a}}$ and $p_{\mathrm{b}}$ act as shared links, serving both the cylinder and the drive unit, as illustrated in Figure 2. Decoupling these links would enable the pressures on the cylinder’s two sides to operate independently of those on the drive unit’s sides, resolving the inherent mismatch between their four-quadrant operations. Consequently, the switching conditions for the hydraulic pump/motor unit and the POCVs would no longer depend on cylinder pressures, effectively eliminating the MSO. Achieving this would require a novel system design to decouple these pressure links.

3. Proposed Oscillation-Free MCC

Based on the analysis in Section 2, decoupling the pressure links between the cylinder and the drive unit effectively eliminates the MSO. Following this principle, a new MCC design is proposed in this study. Notably, the proposed system also exhibits strong load-holding capabilities, as demonstrated in [1].

However, since the primary focus of this study is on mitigating MSO, the analysis and verification of the load-holding devices and their functions across four quadrants are not addressed here. Nonetheless, the load-holding devices are still described and modeled.

3.1. System Architecture

The proposed MCC, as shown in Figure 4, is designed based on the standard MCC shown in Figure 2. Hydraulically driven passive load-holding devices are added into to the cylinder component. PRV_a and PRV_b, as two pressure relief valves, are used to protect the cylinder from overpressure. LH_a and LH_b are two 2/2 normally closed load-holding valves, which are opened by the pilot pressure $p_{\mathrm{pi}}$ when it surpassed the cracking pressure $p_{\mathrm{lh}}=$ 10 bar. CV₁ and CV₂ are two check valves, which are installed beside LH_a and LH_b. This is to offer free flow for the cylinder. $p_{\mathrm{pi}}$ is deactivated by the minimum pressure between $p_{\mathrm{sva}}$ and $p_{\mathrm{svb}}$. The minimum pressure will be transferred by the inverse shuttle valve (ISV). Output pressures, $p_{\mathrm{sva}}$ and $p_{\mathrm{svb}}$, are from two shuttle valves SV_a and SV_b. $p_{\mathrm{sva}}$ is the maximum pressure between $p_{\mathrm{a}}$ and $p_{\mathrm{pa}}$. $p_{\mathrm{svb}}$ is the maximum pressure between $p_{\mathrm{b}}$ and $p_{\mathrm{pb}}$.

In the MCC drive unit component, four 2/2 proportional solenoid valves (PSV₁ to PSV₄), are deployed symmetrically around POCV_a and POCV_b to regulate the cylinder bore-side pressure $p_{\mathrm{a}}$, cylinder rod-side pressure $p_{\mathrm{b}}$, pump/motor unit pressure on the cylinder bore side $p_{\mathrm{pa}}$, and pump/motor unit pressure on the cylinder rod side $p_{\mathrm{pb}}$. This is done by controlling flows in one specific direction. Due to the new load-holding devices, the pilot pressures for POCV_a and POCV_b become line pressures $p_{\mathrm{la}}$ and $p_{\mathrm{lb}}$. ACC keeps the minimum pump/motor unit pressure. Check valves CV₇ and CV₈ are used to prevent cavitation. The check valve CV₉ is used for guiding the leakage of P to ACC. The ACC pressure $p_{\mathrm{acc}}$ is lower than 3 bar all the time. Check valves CV₃ to CV₆ are used to ensure free flow in the other direction.

3.2. System Working Analysis

In Q_I, the bore-side pressure, $p_{\mathrm{a}}$, represents the system’s highest pressure (indicated in red). M rotates counterclockwise, functioning as a servo motor, while P operates as a pump. Valves LH_a, LH_b, PSV₁, PSV₃, and PSV₄ remain fully open, ensuring that $p_{\mathrm{a}}$, $p_{\mathrm{la}}$, $p_{\mathrm{pa}}$, and $p_{\mathrm{sva}}$ are equal. The higher pressure at $p_{\mathrm{la}}$ compared to $p_{\mathrm{lb}}$ triggers the opening of POCV_b. The flow from the accumulator ($Q_{\mathrm{acc}}$) combines with the flow from the cylinder’s rod side and enters the inlet port of P to balance the differential volume. This results in $p_{\mathrm{pb}}=p_{\mathrm{lb}}=p_{\mathrm{acc}}=$ 3 bar (represented in green). PSV₂ acts as a dynamic control valve, throttling the flow to maintain $p_{\mathrm{b}}$ above the cracking pressure of the load-holding valve, $p_{\mathrm{lh}}=10$ bar. The selection mechanisms of ISV and SV_b ensure that $p_{\mathrm{pi}}$ equals $p_{\mathrm{b}}$, keeping LH_a and LH_b open [1].

In Q_II, the external load provides assistance and may become overrunning. The rod-side pressure, $p_{\mathrm{b}}$, is the highest in the system (highlighted in red). M rotates counterclockwise, functioning as a generator, while P operates as a hydraulic motor. Valves LH_a, LH_b, PSV₁, PSV₂, and PSV₄ remain fully open, ensuring that $p_{\mathrm{b}}$, $p_{\mathrm{lb}}$, $p_{\mathrm{pb}}$, and $p_{\mathrm{svb}}$ are equal. The higher pressure at $p_{\mathrm{lb}}$ compared to $p_{\mathrm{la}}$ triggers the opening of POCV_a. The flow from the accumulator ($Q_{\mathrm{acc}}$) combines with the return flow from P and is directed to the cylinder’s piston side to balance the differential volume. This results in $p_{\mathrm{a}}=p_{\mathrm{la}}=p_{\mathrm{acc}}$ (represented in green). PSV₃ acts as a dynamic control valve, throttling the flow to maintain $p_{\mathrm{pa}}$ above the cracking pressure of the load-holding valve, $p_{\mathrm{lh}}$. The selection mechanisms of ISV and SV_a, ensure that $p_{\mathrm{pi}}$ matches $p_{\mathrm{pa}}$, keeping LH_a and LH_b open [1].

In Q_III, the rod-side pressure, $p_{\mathrm{b}}$, is the system’s highest pressure (highlighted in red). M rotates clockwise, functioning as a servo motor, while P operates as a pump. Valves LH_a, LH_b, PSV₂, PSV₃, and PSV₄ are fully open, ensuring that $p_{\mathrm{b}}$, $p_{\mathrm{lb}}$, $p_{\mathrm{pb}}$, and $p_{\mathrm{svb}}$ are equal. The pressure difference between $p_{\mathrm{lb}}$ and $p_{\mathrm{la}}$ causes POCV_a to open. Due to the high flow rate from the bore side, the pump inlet cannot accommodate it entirely, resulting in excess flow being redirected to the accumulator as $Q_{\mathrm{acc}}$. This flow adjustment equalizes $p_{\mathrm{la}}=p_{\mathrm{pa}}=p_{\mathrm{acc}}$ (represented in green). PSV₁ acts as an active control valve, managing the flow to ensure that $p_{\mathrm{a}}$ remains above the cracking pressure of the load-holding valve, $p_{\mathrm{lh}}$. The selection mechanisms of ISV and SV_a align $p_{\mathrm{pi}}$ with $p_{\mathrm{a}}$, maintaining LH_a and LH_b in the open position [1].

In Q_IV, the external load provides assistance and may become overrunning. The bore-side pressure, $p_{\mathrm{a}}$, reaches the system’s maximum pressure level (indicated in red). M rotates clockwise, functioning as a generator, while P operates in motor mode.Valves LH_a, LH_b, PSV₁, PSV₂, and PSV₃ are fully open, resulting in equal pressures for $p_{\mathrm{a}}$, $p_{\mathrm{la}}$, $p_{\mathrm{pa}}$, and $p_{\mathrm{sva}}$. The pressure difference between $p_{\mathrm{la}}$ and $p_{\mathrm{lb}}$ causing POCV_b to open. Due to the high flow rate from the bore side passing through P, there is an excess flow relative to the rod side. This differential flow is directed to the accumulator as ($Q_{\mathrm{acc}}$), leading to $p_{\mathrm{b}}=p_{\mathrm{la}}=p_{\mathrm{acc}}$ (indicated in green). PSV₄ acts as an active control valve, throttling the flow to ensure $p_{\mathrm{pb}}$ remains higher than the cracking pressure of the load-holding valve, $p_{\mathrm{lh}}$. The selection functions of ISV and SV_b, ensure that $p_{\mathrm{pi}}$ matches $p_{\mathrm{pb}}$, which keeps LH_a and LH_b open [1].

3.3. Mitigation of Mode Switching Oscillation

In the pumping mode, P pumps the oil from the low-pressure side to the high-pressure side, and the energy in the system transfers from P to the piston. In the motoring mode, P motors the oil from the high-pressure side to the low-pressure side, and the energy in the system transfers from the piston to P.

In contrast to the standard MCC depicted in Figure 3, the proposed system deviates from the equalization of cylinder pressures ($p_{\mathrm{a}}$ and $p_{\mathrm{b}}$) with the pump/motor unit pressures ($p_{\mathrm{pa}}$ and $p_{\mathrm{pb}}$) due to the incorporation of four PSVs, as shown in Figure 5. Furthermore, due to the PSVs, $p_{\mathrm{a}}$, $p_{\mathrm{b}}$, $p_{\mathrm{pa}}$, and $p_{\mathrm{pb}}$ can be controlled separately. As a result, a required $F_{\mathrm{cyl}}$ smaller than $F_{\mathrm{cyl},\mathrm{cr}}$ can be generated without interfering with the working state of P and POCV_a and POCV_b.

In Q_IV presented in Figure 5, $p_{\mathrm{pb}}$ is controlled over 10 bar by PSV₄. P is in motoring mode. $p_{\mathrm{a}}$ is higher than $p_{\mathrm{pb}}$. Furthermore, since POCV_b is open, $p_{\mathrm{b}}$ and $p_{\mathrm{lb}}$ are equal to 3 bar. Therefore, the minimum cylinder force ($F_{\mathrm{cyl},\mathrm{mini}}$) generated in this scenario is calculated via Equation (13).

$$F_{\mathrm{cyl},\mathrm{mini}}=(p_{\mathrm{pb}}-\alpha p_{\mathrm{acc}})A_{\mathrm{a}}$$

(13)

Given that $p_{\mathrm{pb}}>10$ bar and $p_{\mathrm{acc}}=3$ bar, the value of $F_{\mathrm{cyl},\mathrm{mini}}$ corresponds to a relatively large magnitude. Considering Equation (13), when the load $F_{\mathrm{load}}$ is smaller than $F_{\mathrm{cyl},\mathrm{mini}}$, P and all valves in the proposed system changes to the working states in Q_III when the piston starts to retract. In this case, the relative magnitudes of the pressures are shown in Equation (14). As a result, the working state of P and POCVs stays unchanged. The pump mode oscillation is, therefore, prevented.

$$p_{\mathrm{a}}>p_{\mathrm{b}}=p_{\mathrm{lb}}=p_{\mathrm{pb}}>p_{\mathrm{la}}=p_{\mathrm{pa}}=p_{\mathrm{acc}}$$

(14)

When the load $F_{\mathrm{load}}$ is greater than $F_{\mathrm{cyl},\mathrm{mini}}$, the required $F_{\mathrm{cyl}}$ can be decreased to a level below $F_{\mathrm{cyl},\mathrm{mini}}$ as $F_{\mathrm{fr}}$ is increased to a certain level by raising ${\dot{x}}_{\mathrm{p}}$. As a result, P and all valves in the proposed system transition to the working state in Q_III. However, the retracting speed ${\dot{x}}_{\mathrm{p}}$ is high at this moment. Consequently, the flow rate to the cylinder rod side experiences a sudden increase by $1/\alpha$ when changing mode. This abrupt change results in a rapid rise of $p_{\mathrm{b}}$, leading to the occurrence of MSO as discussed in Section 2. Nevertheless, due to the additional damping introduced by PSV₁, the MSO triggered in the proposed system is less severe than that observed in conventional MCC configurations.

4. Systems Modeling

The dynamic system models of both the standard and proposed MCCs are derived from the experimentally validated system model of the 2M2P MCC utilized in [18]. The models for the main components, including the electric servo motor (M), hydraulic pump (P), accumulator (ACC), hydraulic cylinder, variable bulk modulus, pressure relief valves (PRV), and check valves (CV), from the 2M2P MCC are reused in the standard and proposed MCCs, ensuring consistency and accuracy in representing system dynamics.

The selected off-the-shelf components of the proposed MCC are listed in

Table 1. Off-the-shelf components of the proposed MCC in Figure 4.

System	Components	Manufacturer	Product Number
1 & 2	M	Bosch Rexroth	MS2N07-D
1 & 2	P	Bosch Rexroth	A10FZG
1 & 2	ACC	Bosch Rexroth	HAD3,5-250-2X
1 & 2	Cylinder	LJM	NH41-0-SD
1 & 2	POCV	Sun Hydraulics	CKEBXCN
1 & 2	PRV	Bosch Rexroth	RE 25402
2	ISV	Bucher	HOSV-10
2	LH	Sun Hydraulics	DKHSXHN
2	CV	Bosch Rexroth	RE20380
2	PSV	Bosch Rexroth	KKDSR1PB
2	SV	Bosch Rexroth	MHSU2KA1X/420

. The modeling parameters are from the datasheet of these components. The diameters of the cylinder bore and rod are 63 mm and 45 mm, respectively. The cylinder stroke is 400 mm. The maximum flow rate in the simulations can reach 19 L/min. More details on the modeling parameters can be found in [18]. The standard MCC is marked as system 1, while the proposed MCC is marked as system 2. Notably, six components are common to both systems. As a result, the dynamic models of these systems can be converted into one another by simply adding or removing components unique to the proposed MCC, allowing for a seamless transition between the two configurations.

The modeling approach for each component listed in Table 1 has already been extensively covered in [1] and, therefore, will not be elaborated upon in this paper. For a more comprehensive explanation of the time-domain simulation model, readers are directed to [1]. This decision avoids redundancy while ensuring that all necessary details are available for those seeking a deeper understanding of the modeling process.

This study utilizes the single-boom crane previously examined in [18] to demonstrate the effectiveness of the proposed MCC in mitigating MSO. The crane is modeled within the Simulink-Simscape environment, where it interacts dynamically with the hydraulic system model. The hydraulic system provides the cylinder force to the crane model, while the crane returns the corresponding piston position and velocity.

Figure 6 depicts the crane, accompanied by key specifications. The combined mass of the crane boom and payload is 300.8 kg, with the center of mass positioned 1.77 m from the hinge point.

5. Controls

This section outlines the control algorithms for both the proposed and standard MCC configurations. The control framework, depicted in Figure 7, comprises three primary control loops: the position control loop, the line pressure control loop, and the PSV control loop. These loops work in tandem to ensure the MCC transitions seamlessly between operation mode and load-holding mode, adapting to varying working conditions. Since the same algorithms have been previously detailed in [1], the individual control loop descriptions will not be repeated here.

As described in Section 4, the dynamic model of the new proposed MCCs can be obtained by adding pressure control and load holding components to the model of the standard MCCs. Consequently, the control algorithm for the standard MCC is developed using only the position control loop from Figure 7, omitting the additional loops required for pressure control and load holding.

6. Simulation Results

As analyzed in Section 2, the MSO is often triggered when moving a small load in Q_IV with increasing friction. Therefore, adjustments were made in simulations by increasing the friction force of the cylinder model and decreasing the mass of the crane beam and payload to replicate the MSO. As demonstrated in [18], the friction force of the cylinder is modeled by Stribeck friction. The increase in cylinder friction force is realized by increasing the viscous friction coefficient. These adjustments induce the MSO in the standard MCC, as depicted in Figure 3.

6.1. Triggered Mode Switching Oscillation in the Standard MCC

The piston velocity input reference used to trigger the MSO in the standard MCC is shown in Figure 8a. In Q_I, the piston velocity ${\dot{x}}_{\mathrm{p}}$ undergoes a sequence of changes. It starts from zero and gradually increases to 100 mm/s, then decreases to zero by the 6-s mark. This behavior is characterized by an initial acceleration, followed by deceleration and ultimately coming to a complete stop. In Q_IV, ${\dot{x}}_{\mathrm{p}}$ starts from zero and gradually decreases to −100 mm/s, after which it begins to increase, eventually returning to zero by the 12 s mark. MSO represents the period of the triggered mode switching oscillation in result plots. The position tracking performance of the standard MCC under the velocity input reference is shown in Figure 8b,c. The position feedback signal (FB in plots) tracks the position reference (Ref in plots) well before 7.4 s when the MSO is triggered. The tracking error remains within ±1.5 mm before 7.4 s. When the MSO is triggered, the tracking error range increases to approximately −9 mm to 12 mm.

In the standard MCC, the bore-side cylinder pressure $p_{\mathrm{a}}$ and rod-side cylinder pressure $p_{\mathrm{b}}$ are pilot pressures for POCVs. They are illustrated in Figure 8d. Before the MSO is triggered, $p_{\mathrm{b}}$ equals $p_{\mathrm{acc}}$. $p_{\mathrm{a}}$ carries the load and follows the trend of ${\dot{x}}_{\mathrm{p}}$. In Q_IV, $p_{\mathrm{a}}$ decreases gradually due to the speed input reference and the friction force. When $p_{\mathrm{a}}=p_{\mathrm{b}}=p_{\mathrm{acc}}$ the MSO is triggered. Consequently, $p_{\mathrm{a}}$ and $p_{\mathrm{b}}$ oscillate severely. The POCV opening signals are shown in Figure 8e,f. When the MSO is triggered, the POCV opening signals oscillate. The standard MCC pump modes in the four-quadrant plane are illustrated in Figure 8g. It is worth highlighting that when the MSO is triggered, the pump mode oscillates between Q_I, Q_III, and Q_IV.

6.2. Mitigated MSO in the Proposed MCC

As discussed in Section 3.3, the proposed MCC has the capability to mitigate or prevent MSO when subjected to the same piston speed reference as the standard MCC. The simulation results presented in Figure 9a,b show the position tracking performance of the proposed MCC. The position feedback signal tracks the position reference well. The tracking error falls well within ±2 mm. It should be noted that the system operation moves from Q_I to Q_III after 6 s without moving into Q_IV, due to the function of PSVs. Therefore, the MSO is mitigated.

$p_{\mathrm{a}}$, $p_{\mathrm{b}}$, $p_{\mathrm{la}}$, and $p_{\mathrm{lb}}$ are shown in Figure 9c,d. The pressure control references for PSVs are 15 bar to keep the load-holding valves open. Because $p_{\mathrm{a}}$ and $p_{\mathrm{b}}$ are controlled and not the pilot pressures of POCVs, $p_{\mathrm{a}}$ can become lower than $p_{\mathrm{b}}$, which happens at about 5.2 s. The opening states of POCV_a and POCV_b are switched at the transition at about 6 s without triggering oscillations, as shown in Figure 9e,f. In contrast, the opening states of POCV_a and POCV_b experience severe oscillations in Figure 8e,f. The proposed MCC hydraulic pump/motor unit working modes in the four-quadrant plane are illustrated in Figure 9g. Notably, in the proposed MCC, the hydraulic pump/motor unit operates only in Q_I and Q_III as a hydraulic pump, consuming energy without frequent transitions to the hydraulic motor mode, thereby avoiding MSO.

7. Discussion

The proposed system builds upon the testing prototype presented in [18] by incorporating four PSVs and two logical shuttle valves, enhancing its practical applicability.

Although the proposed 1M1P MCC can effectively mitigate the MSO, the presence of PSVs introduces additional energy losses to the system. Nevertheless, as detailed in [1], these losses incurred by the PSVs remain unaffected by changes in output power and system pressures. Notably, with substantial increases in output power and system pressures, the relative proportion of PSV losses diminishes.

As shown in Figure 8g, when the MSO is triggered in Q_IV the system transitions among Q_I, Q_III, and Q_IV. The hydraulic pump/motor unit frequently alternates between motoring and pumping modes, resulting in severe oscillations. Conversely, Figure 9g demonstrates that the introduction of four PSVs effectively mitigates MSO in the proposed system by preventing the hydraulic pump/motor unit from entering motoring mode under MSO-triggering conditions. Notably, the hydraulic pump/motor unit operates as a hydraulic pump, consuming energy in Q_I and Q_III, and as a hydraulic motor, regenerating energy in Q_II and Q_IV.

However, the system’s operation is restricted to Q_I and Q_III, without transitioning into Q_IV. Consequently, the proposed system lacks the ability to regenerate energy when rapidly lowering a small load.

8. Conclusions

A standard one-motor-one-pump motor-controlled hydraulic cylinder experiences mode switching oscillation under certain operating conditions. This paper proposes a new one-motor-one-pump motor-controlled hydraulic cylinder to overcome this challenge through the following key aspects:

• The root cause of mode switching oscillation in a standard one-motor-one-pump motor-controlled hydraulic cylinder is identified as the inherent difference between the four-quadrant operation of the cylinder component and that of the drive unit component.
• A stability analysis on a simplified and linearized model of a standard one-motor-one-pump motor-controlled hydraulic cylinder is carried out, confirming that the system in Q_IV is inherently oscillatory, and it is not feasible to mitigate that behavior from adjusting the regular system parameters.
• A new one-motor-one-pump motor-controlled hydraulic cylinder with system pressure control in four quadrants is proposed. Its capability to mitigate mode switching oscillation is described and analyzed.
• The effectiveness of the proposed one-motor-one-pump motor-controlled hydraulic cylinder and the control algorithm in mitigating the mode switching oscillation under certain operating conditions is demonstrated through the simulation results.

In conclusion, the proposed system offers a new solution for mitigating the mode switching oscillation in one-motor-one-pump motor-controlled hydraulic cylinders. Future work will cover the experimental test to verify the simulation results.