A Novel Control Design for Realizing Passive Load-Holding Function on a Two-Motor-Two-Pump Motor-Controlled Hydraulic Cylinder

When a hydraulic cylinder connects two chambers directly to one or two hydraulic pumps driven by electric servo motors without any control valve in between, it can be called a motor-controlled hydraulic cylinder (MCC). Unlike valve-controlled cylinders, MCCs have no valve throttling, which significantly increases the energy efficiency. Among different MCC topologies, the two-motor-two-pump (2M2P) MCC has several advantages, such as cylinder pressure control and no mode switch oscillations. However, due to state coupling when controlling both piston position and minimum cylinder chamber pressure, the 2M2P MCC is a multi-input-multi-output (MIMO) system that usually requires advanced MIMO controller analysis and design. This paper presents a control algorithm for a 2M2P MCC with the minimum cylinder pressure control and passive load-holding function. This control algorithm is tested on a single-boom crane characterized by overrunning loads. It is designed based on the analysis of the system characteristics, requiring no MIMO controller analysis and design. A non-linear model of a single-boom crane driven by the proposed 2M2P MCC is created in MATLAB/Simulink and experimentally validated. Feedback controllers are designed and verified via simulations to realize position control, minimum cylinder pressure control, and load-holding under standstill command. For a given load and speed profile, the hydraulic system efficiency during pumping and motoring mode is 55-60 % and 20-25 %, respectively. The system’s overall efficiency can be enhanced with electrical regenerative drives, which feeds the generated power from potential energy to the grid or battery and reused in the next working cycle. The experimental results presented in this paper verifies the efficacy of the proposed control algorithm and demonstrates its superior performance in achieving the desired system response under various operating conditions.


Introduction
The hydraulic linear actuation system is extensively used in the industry for heavy load-carrying applications.Marine and offshore are major sectors that use a large number of hydraulic actuators in largescale applications such as offshore cranes.The benefits of using hydraulic actuators in these applications are the high power-to-weight ratio, flexible power transmission, inherent damping quality, and high reliabil-ity of the hydraulic systems.However, conventional valve-controlled hydraulic cylinders have long been associated with low energy efficiency, primarily due to the throttling effect caused by control valves Zimmerman et al. (2007).Given the rising global energy demand, environmental concerns, and the trend towards electrification and sustainable industrial practices, the need for efficient high-power hydraulic systems has become increasingly important.Such systems must be designed to deliver equivalent performance while minimizing energy consumption and promoting sustainable operation.Many researchers working on linear hydraulic systems are incorporating pump-controlled technology to enhance system efficiency.Quan et al. (2014) has done a literature review presenting pumpcontrolled cylinders of various topologies.The authors show the replacement of valve-controlled cylinders with none-valve cylinders, where the flow is varied by either the electric motor speed (motor-controlled hydraulic cylinders) or the pump displacement (pumpcontrolled cylinders).This is to remove the throttling losses in the system.Various shown topologies are suitable for particular applications.It also shows that motor-controlled cylinders (MCC) can generate energy using regenerative electrical drives.This enhances the efficiency of the system significantly.Figure 1 illustrates the general structure of MCCs.An MCC comprises two parts: a cylinder and a drive unit.The MCC drive unit contains electric motors, hydraulic pumps, accumulators, and auxiliary valves.In an MCC, the hydraulic cylinder is connected to one or two fixed-displacement hydraulic pumps and an oil reservoir through auxiliary valves.The auxiliary valves are used for the differential flow rates compensation and the load-holding capability.Electric servo motors drive the fixed-displacement hydraulic pumps.The work done by Fresia et al. (2022) shows the implementation of a novel layout on a hybrid excavator.The working hydraulics and boom actuation of the system is modified while achieving throttle-less motion and energy recovery in supercapacitors.According to the research, it saves around 26-28 % of fuel consumption as compared to conventional loadsensing functions.However, the research is going on the electro-mechanical actuators in parallel because of their higher efficiency but are commercially limited to lower power applications.This is due to its excessive mechanical wear and difficulty in overload protection Hagen et al. (2020).Hence, for higher load applications, MCCs seem to be a promising solution.The class of energy-efficient MCCs is discussed by Schmidt et al. (2019).It is concluded that with different hydraulic topologies and standard hydraulic components, MCCs can be scaled up to different applications.The energy-saving potential of these kinds of systems is estimated by Ketelsen et al. (2018) using simulations for a knuckle boom crane.It is observed that when carrying a load of 5 tons for a particular cyclic operation, a valve-controlled system consumes 0.79 kWh of electrical energy, whereas the proposed drive consumes 0.06 kWh.Another work done by Agostini et al. (2020) evaluates the energy efficiency of an MCC for a crane at a laboratory scale with energy regeneration capability.It is shown that the proposed system results in an efficiency of 54 % and 38 % with and without electrical energy regeneration, respectively.Additionally, another study Qu et al. (2020) demonstrated even higher efficiency, reaching up to 80 % in a laboratory application utilizing an MCC.
Despite the greater efficiency improvements offered by MCCs compared to conventional valve-controlled cylinders, the enforcement of load-holding capability by legislation poses challenges for MCCs.The application of counterbalance valves (CBVs) within an MCC is shown in Jalayeri et al. (2015) and Imam et al. (2017).This gives the passive load-holding capability of the system.However, it reduces the system's efficiency due to throttle losses occurring in the CBV.In addition to counterbalance valves, two on/off electric valves can be employed in MCCs to provide an active load-holding function without introducing throttling losses Zhang et al. (2017); Jensen et al. (2021).While normal-closed on/off electric valves can offer load-holding capabilities during power blackouts, achieving load-holding functions during hose ruptures or sudden drops in line pressure can present challenges.Another approach to achieve load holding is to utilize two pilot-operated check valves Parker (Accessed July 07, 2023); Sweeney T. (2012).However, using only two pilot-operated check valves without additional devices does not allow for four-quadrant operations, which are often required for large-size offshore knuckle boom cranes.To address this challenge, two pilot-operated check valves and a solenoid valve as a loadholding device for cylinder locking are introduced by Hagen et al. (2018) andPadovani et al. (2019).The latter depresses the pilot line pressure during a power failure, preventing the load from falling.However, the operation of the system depends upon the condition of a solenoid valve, which only makes it partially passive.Also, the undesired oscillations are present during the opening/closing of the pilot-operated check valve.The solution to avoid these oscillations is developed by Hagen and Padovani (2020).The evaluation of a novel solution for a two-motor-two-pump (2M2P) MCC with fully passive load-holding capability is conducted in simulations by Ketelsen et al. (2020).Two 2/2 poppettype pilot-operated valves are proposed as load-holding devices, each at the cylinder inlet and outlet.These valves are opened by the minimum cylinder pressure.An inverse shuttle valve (ISV) is used for always connecting the cylinder's minimum pressure for opening the load-holding valve.This 2M2P MCC is considered a multi-input-multi-output (MIMO) system, offering the flexibility to control both the minimum cylinder pressure and the cylinder's position by a specially designed MIMO controller.Yet, it is worth noting that designing the MIMO controller for such a system can be complex, requiring state coupling analysis and decoupler design.Furthermore, the MIMO controller needs to be tailored and designed specifically for each application to ensure optimal performance and functionality.From the above discussion, it can be concluded that only the 2M2P MCC has the possibility to offer a fully passive load-holding capability.However, it is important to note that the existing research on the 2M2P MCC is primarily limited to simulation studies.Additionally, the design and implementation of the MIMO controller for the 2M2P MCC is a highly intricate and customized process specific to each application.Therefore, further efforts are necessary to design a more universally applicable controller for 2M2P MCCs.Moreover, experimental studies are needed to validate the functionality of the 2M2P MCC and its practical viability.
In this paper, a novel control algorithm is designed and implemented on a 2M2P MCC with a passive loadholding capability on a laboratory-scale single-boom crane, which can be scaled up to various lifting applications.The energy efficiency of the proposed system is also evaluated experimentally.The open-loop simulation model of the 2M2P system is validated for future analysis.The paper is organized in the following sections.In Section 2, the hydraulic architecture and control algorithm of the system under investigation are presented.In Section 3, the experimental setup is discussed.In Section 4, the mathematical model of the hydraulics and multi-body crane model in MAT-LAB/Simulink is made and validated for open-loop conditions.The experimental results obtained for the two-quadrant operation are described in Section 5 and discussed in Section 6.In Section 7, the outcome of the paper is concluded.

System under Investigation
In this paper, the novel solution named 2M2P MCC proposed by Ketelsen et al. (2020) is further explored with a novel control algorithm experimentally.The following subsections, System Architecture and Control Algorithm describe the system's topology and novel control algorithm with passive load-holding capability.The 2M2P MCC under investigation comprises the two fixed-displacement pumps, the main pump (P 1 ) and the assisting pump (P 2 ), driven by two servo motors, the main electric servo motor (M 1 ) and the assisting electric motor (M 2 ), as illustrated in Figure 2. It is a closed-circuit system that can operate in four quadrants, along with the passive load-holding capability.The low-pressure accumulator (ACC) is used as the pressurized reservoir (3 bar) for supplying the volumetric difference (rod side and bore side of the cylinder) of are used to relieve the system's overpressure.

System Architecture
The four-quadrant operation of the system is shown in Figure 3. P 1 functions as a pump in quadrants I and III, while P 2 serves as a pump in quadrant I and a motor in quadrant III.This is due to the excess oil that exits the bore side compared to the rod side for a given actuator speed.P 1 acts as a motor in quadrants II and IV, and P 2 acts as a pump in quadrant II and as a motor in quadrant IV.As in quadrant II, the extra amount of oil is to be pumped into the bore side from ACC to control the minimum cylinder pressure.This is also explained mathematically with the following equations.
Referring to Equation 1, ω 1 and ω 2 are positive, this implies that both P 1 and P 2 operates as a pumps against positive pressure difference p a > p b and p a > p acc .D 1 is the displacement of P 1 .D 2 is the displacement of P 2 .ω 1 is the shaft speed of M 1 .ω 2 is the shaft speed of M 2 .A a and A b are the bore-side and rod-side areas.ẋp is the piston velocity.F load is the external load force acting on the cylinder.When F load is in the same direction as ẋp , it is an overrunning load.Otherwise, it is a resistant load.
Referring to Equation 2, ω1 and ω2 are positive, this implies that the rotation of P1 and P2 are similar as quadrant I. P1 operates under negative pressure difference pa < p b , thus acts as a motor, and P2 operates under positive pressure difference pa > p b , thus acts as a pump.
Referring to Equation 3, ω1 and ω2 are negative, this implies that the rotation of P1 and P2 are opposite as quadrant I. Also, the rod-side pressure is higher than the boreside pressure.The P2 works as a motor as it operates in negative pressure difference pa < pacc.However, P1 works against the positive pressure difference p b > pa and operates as a pump.
Referring to Equation 4, ω1 and ω2 are negative and pa > p b .Therefore, P1 and P2 act as motors as they work against the negative pressure difference.

Control Algorithm
The 2M2P MCC under investigation is a MIMO system, where inputs ω1,input and ω2,input of M1 and M2 are the system inputs, and piston position xp and the minimum cylinder pressure ppa or p pb are the required system outputs.A control algorithm is developed based upon the steady-state analysis of the system in four quadrants as explained in Section 2.1.The control algorithm designed for the proposed 2M2P MCC is demonstrated in Figure 4.The switching logic of the mode and inputs selection module is shown in Figure 5.This control algorithm consists of four control loops for a smooth transition between motion and load-holding modes.These four loops are the position control loop, cylinder pressure control loop, load-holding control loop, and pressure level control loop.Each control loop generates two reference speeds ω 1,ref and ω 2,ref for M1 and M2.These speed references are fed into the mode and inputs selection algorithm to generate the final speed inputs to M1 and M2 according to four-quadrant operations.There are two working modes: motion mode and load-holding mode.When the cylinder speed reference ẋref is zero, the system is in load-holding mode.Otherwise, the system is in motion mode.Four control loops are described in the following subsections.

Position Control Loop
The green area in Figure 4 shows the position control loop.The position control loop is activated in motion mode.A feedforward controller estimates the required speed of M1 (u f f ) via Equation 5.
The position controller, a proportional-integral controller, corrects the prediction of the feedforward controller.The proportional gain (kP ) is 70 rev/min/m and the integral gain (kI ) is 15 rev/min/m/s.A load-pressure feedback signal (pL) calculated via the bore-side pressure (pa), the rod-side pressure (p b ), and the area ratio (A b /Aa) is filtered by a high-pass filter (GHP ) shown in Equation 6.The filter cut-off frequency (ωHP ) is 10 rad/s.The filter gain (kHP ) is 20 rev/min/bar.The negative filtered load-pressure feedback signal is added to increase the system damping by canceling out pressure oscillations.
In the proposed 2M2P MCC, the main hydraulic pump (P1) moves hydraulic oil in the annular volume from one side to the other side of the cylinder.When the piston moves, the assist pump (P2) compensates for the cylinder differential volume, which is also the rod volume.When two pumps' flow rates match the cylinder area ratio, the system works smoothly without cavitation or overpressure.Therefore, the total position control loop output (upos) is multiplied by a speed gain to make the two pumps' flow rates match the cylinder area ratio.For the system under investigation, the pump displacement ratio (D1/D2) is 2.0 and the area ratio of rod area over annulus area (Ar/A b ) is 1.04, hence the speed gain (k speed ) is 2.08.

Cylinder Pressure Control Loop
The purple area in Figure 4 shows the cylinder pressure control loop.The cylinder pressure control loop is activated in motion mode.As described in Section 2.1, the minimum cylinder pressure needs to be over 10 bar to activate the load-holding valves in motion mode and to be less than 10 bar to deactivate the load-holding valves in load-holding mode.The minimum pressure between ppa and p pb is chosen as the feedback signal through the minimum cylinder pressure selection algorithm.The cylinder pressure reference signal in the loop is set to 25 bar.

Load-Holding Control Loop
The orange area in Figure 4 shows the load-holding control loop.This control loop is activated in load-holding mode.As shown in Figure 3, two scenarios exist when the 2M2P MCC turns from motion mode to load-holding mode.In one scenario, the 2M2P MCC turns from quadrants I and IV to piston-side load-holding mode shown in Figure 3(e).
In this case, pressure ppa carries the load, and pressure p pb controls the load-holding valves.Therefore, the loadholding pressure controller needs to decrease the pressure p pb to deactivate the load-holding valves without affecting the pressure ppa, which is holding the piston position.This is done by adding the negative controller output signal on the main and assist servo motors, where the signal on the assist servo motor needs to be doubled because the pump displacement ratio (D1/D2) is 2.0.In this way, the oil in the chamber forming pressure p pb is pumped by P1 and P2 to the accumulator, while the oil in the chamber forming   pressure ppa stays the same.Therefore, the cylinder position is maintained during the transient period.

Is the speed equal to zero?
System in loadholding mode

System in operation mode
Reference Speed (x ref ) . In the other scenario, the 2M2P MCC turns from quadrants II and III to rod-side load-holding mode shown in Figure 3(f).In this case, pressure p pb carries the load, and pressure ppa controls the load-holding valves.The loadholding pressure controller only needs to be applied on M2 while M1 stands still.Therefore, pressure ppa is reduced by the controller to deactivate the load-holding valves and pressure p pb stays the same to hold the cylinder position during the transient period.The load-holding pressure controller is a proportional-integral controller.The proportional gain (kP ) is 25 rev/min/bar, and the integral gain (kI ) is 3 rev/min/bar/s.

Pressure Level Control Loop
The blue area in Figure 4 shows the pressure level control loop.This control loop is activated during the transition from load-holding mode to motion mode.There are two reasons for designing this control loop.One reason is that pressures on two sides of the load-carrying side load-holding valve, pa and ppa in quadrants I and IV or p b and p pb in quadrants II and III, should be the same when the loadholding valves are activated.In this case, the piston position is maintained during the transient period from loadholding to motion mode.
The other reason is that when the system is in transition from load-holding mode to motion mode, the SV oscillates severely if the magnitude of ppa is similar to the magnitude of p pb .Therefore, the controller needs to quickly increase the pressure before the load-carrying side load-holding valve (ppa in quadrants I and IV or p pb in quadrants II and III) to the load pressure (pa in quadrants I and IV or p b in quadrants II and III) to prevent oscillations of SV.
The pressure level controller is a proportional-integral controller.The proportional gain (kP ) is 92 rev/min/bar, and the integral gain (kI ) is 4 rev/min/bar/s.The controller output is applied to M2 during the system's transition from the piston-side load-holding mode (shown in Figure 3(e)) to the motion modes in quadrants I and IV (shown in Figures 3(a) and 3(d), respectively).In contrast, the controller output is applied to M1 during the system's transition from the rod-side load-holding mode (shown in Figure 3(f)) to the motion modes in quadrants II and III (shown in Figures 3(b) and 3(c), respectively).

Experimental Setup
The experiments are performed on a single-boom crane driven by the proposed 2M2P MCC located at Aalborg University, Denmark.The 2M2P MCC drive unit, the singleboom crane, and the control cabinet are presented in Figure 6.The single-boom crane has the capability to operate in quadrants I and IV, i.e., it provides resisting and assisting loads during cylinder extending and retracting motions, respectively.This experimental testbench also gives the possibility to validate the passive load-holding functionality of the system.The testing of the 2M2P MCC on the singleboom crane is challenging and interesting to validate due to its inherited enhanced oscillations because of its design.
The 2M2P test bench comprises the two electric servo motors, M1 and M2, which drive the two fixed-displacement axial piston pumps, P1 and P2.The flow rate from the pumps is adjusted by varying rotational speeds of M1 and M2.The rotational speeds of M1 and M2 are changed by the dedicated electrical drives, which get speed commands from the programmable logic controller (PLC) provided by Bosch Rexroth.The 3/2 hydraulic pilot-operated SV is deployed in the circuit to provide the pilot pressure (ppi) to normally closed load-holding valves (LHa and LH b ).The load-holding valves are the spring return (set at 10 bar) logic elements that isolate the fluid supply to the cylinder during the pressure loss from the pump.In real crane applications, the load-holding valves have to be installed as an integral part of the cylinder inlet and outlet to prevent the crane from falling during hose rupture conditions.
The two check valves (cracking pressure 0.2 bar), CV1 and CV2, are installed parallel to the LH valves to improve the system response.The hydraulic accumulator, sized according to the cylinder area ratio and length, is installed to provide the differential flow to the cylinder and avoid cavitation.The cross port pressure relief valves (Maximum set pressure 240 bar) are also installed in the circuit to avoid over-pressurization.To make the system compact, all the valves deployed are of cavity design which is kept inside the manifold.
The sensors used are draw-wire sensors for monitoring the piston position, and pressure sensors of different ranges are used for monitoring the pressure in the lines.The speed and position of M1 and M2 are monitored through inbuilt standard encoders.For executing the control algorithms, the PLC (XM22) with digital and analog I/O's extensions by Bosch Rexroth with SERCOS communication is used.The data acquisition from the sensors and programming of PLC are made on Indraworks Engineering software.The technical specifications of the major components used in the system are described in Table 1 in Appendix B.

System Modelling and Validation
Multiple dynamic models of the 2M2P MCC driving the single-boom crane were created and simulated in MATLAB-Simulink ® .

3D Crane Model
A multibody 3D model of the single-boom crane with rigid body components is constructed in the Simulink-Simscape environment by importing the crane CAD model.In this model, a cylindrical joint works as a hydraulic cylinder.The actuation force to the cylindrical joint is from the 2M2P MCC model.The position and velocity measurements of the cylindrical joint are fed back to the 2M2P MCC model.This rigid 3D multibody crane model and the connection to the 2M2P MCC model are shown and further discussed in Appendix A.

Electric Servo Motor
Two electric servo motors (M1 and M2) are modeled by second-order transfer functions shown in Equation 7 and 8.These transfer functions are from M1 and M2 input signals (ω 1,ref and ω 2,ref ) to hydraulic pump shaft speeds (ω1 and ω2).The natural frequencies and the damping ratios are determined using the embedded parameter estimator toolbox in Simulink against experimental data.The natural frequencies are ωem1 = 23.8Hz and ωem2 = 12.6 Hz.The damping ratios are ζem1 = 0.73 and ζem2 = 0.79.

Fixed-Displacement Hydraulic Pumps
Two fixed-displacement hydraulic pumps (P1 and P2) are the hydraulic power source in the circuit.Because they are rigidly connected to the electric motor, they have the same angular speeds (ω1 and ω2) as the electric servo motors.The pump's dynamics is combined with the electric motor's dynamics and represented via Equation 7 and 8.The pump flow rates (Qpa1 and Qpa2) are modeled via Equation 9 and 10.D1 = 6 cc/rev and D2 = 3 cc/rev are hydraulic pump displacements.
The leakages (Q le,high and Q le,low ) in P1 are modeled via Equations 11 and 12. ∆p h is the pressure difference between the pump high-pressure side and the accumulator.
∆p l is the pressure difference between the pump low-pressure side and the accumulator.12.
Because the P2 low-pressure side is connected to the accumulator, the leakage (Q le,assist ) in P2 is modeled via Equation 14. ∆passist is the pressure difference between two sides of the assist hydraulic pump.k le2 = 5.6 • 10 −3 L/min/bar is the assist hydraulic pump leakage coefficient.

Effective Bulk Modulus
The hydraulics is modeled utilizing a consolidated approach.The effective fluid's bulk modulus (β chb,i ) in the i-th hydraulic chamber is calculated via Equations 15, 16, and 17. β oil = 7000 bar is the hydraulic oil's bulk modulus.p chb,i is the pressure in i-th hydraulic chamber.patm = 1 bar is the atmospheric pressure.kair = 1.4 is the air adiabatic constant.β chb,air,i is the i-th chamber entrapped air bulk modulus.V atm,%air = 0.01 is the entrapped air relative volume when the oil is in the open air.V chb,%air,i is the entrapped air relative volume in i-th chamber oil.
The pressure-force conversion (F cyl ) is modeled via Equation 20.
The cylinder stroke limit contact forces (F lower and Fupper) are modeled via Equations 21 and 22. kc = 6 • 10 10 N/m is the contact spring constant.cc = 9 • 10 10 Ns/m is the contact damping coefficient.
The friction between cylinder and piston is modeled via the Stribeck curve Equation 23. fv = 4000 Ns/m is the viscous friction coefficient.FC = 75 N is the coulomb friction force.γ = 250 s/m is the hyperbolic tangent coefficient.
FS = 500 N is the static friction force.τ = 0.02 s/m is the static friction time constant.

Accumulator
The bladder accumulator is modeled using the method suggested in Hansen (2023).As shown in Equation 24, the gas pressure (pg) is equal to the fluid pressure (p f ) when the fluid pressure is higher than the pre-charge gas pressure (pg0 = 0.96 bar).
When the gas pressure changes, the gas volume (Vg) is modeled via Eq.25. n = 1.4 is the ratio of the gas heat.
The fluid volume (V f ) is calculated as the difference in total accumulator volume (Vg0 = 2.8 L) and gas volume (Vg): The fluid pressure build-up ( ṗf ) is modeled via Eq.27.
The accumulator pressure (pacc) is equal to the fluid pressure (p f ).

Pressure Relief Valves (PRV)
The bore and rod side PRVs are modeled via Equation 28to 31.Qprv1 and Qprv2 are the flow rates passing through two PRVs.kprv = 500 L/min/bar is the PRV flow rate constant.pprv1 = pprv2 = 250 bar are the PRV cracking pressures.∆pa = pa − pacc (28) PRV3 is on the pilot pressure line (ppi).The flow rate passing through PRV3 is modeled via Equation 33.The cracking pressure (pprv3) is set to 100 bar.

Check Valves (CV)
As shown in Figure 2, there are seven CVs in the 2M2P MCC.They are all modeled via Equation 34.Qcv,i is the flow rate passing through the i-th CV. ∆pcv,i is the pressure drop across the i-th CV.The cracking pressure pcv = 0.2 bar is the same for all CVs.The CV flow rate constant kcv = 500 L/min/bar is also the same for all CVs.

Inverse Shuttle Valve (ISV) and Load-Holding Valves (LH)
The ISV is modeled as a single selection logic shown in Equation 35.ppa is the pressure between LHa and the assist hydraulic pump (P2).p pb is the pressure between LH b and the main hydraulic pump (P1).
The LHa and LH b are modeled as orifices.Flow rates (Q lha and Q lhb ) passing through LHa and LH b are calculated via Equations 36 and 37. C d = 0.7 is the orifice discharge coefficient.u lha and u lhb are valve opening commands.
A lha = A lhb = 200 mm 2 are valve maximum opening areas.ρ = 870 kg/m 3 is the hydraulic oil density.
u lha and u lhb are provided by the logic: p lh = 10 bar is the load-holding cracking pressure.

Open-Loop Model Validation
The nonlinear dynamic model of the 2M2P MCC was validated against experimental measurements.The open-loop speed input commands for the main and assist electric servo motors, as shown in Figure 7, drive the cylinder to extend and retract.Figures 8 to 13 show good agreements between measured and simulated variables, confirming the high-fidelity nature of the dynamic model.Therefore, it is a good simulation tool for further control analyses.

Experimental Results
A representative working trajectory consisting of extending and retracting the hydraulic cylinder across three loadholding operating sections is used as the piston position command.This command corresponds to a piston velocity of 20 mm/s, which is the typical working speed for this type of crane.By analyzing system performance when controlled by the algorithm shown in Figure 4, the chosen trajectory allows for an assessment of the algorithm's effectiveness.The experimental results are presented and analyzed in subsequent subsections.Note that LH in the following plots represents the load-holding mode, while pumping and motoring refer to pumping and motoring modes, respectively.

Position Tracking Performance
The commanded and measured piston position and the position tracking error are presented in Figure 14a and 14b.
The measured position follows the commanded position well.The tracking error falls within -3.2 mm to 2.3 mm.
The error peaks at points A, B, and C, appear when the system turns from load-holding to motion mode.The average error of the whole period is 0.39 mm.The position tracking performance is satisfactory since the maximum position error of hydraulic cranes is not uncommon to be bigger than 15 mm Kjelland and Hansen (2015).

Minimum Cylinder Pressure Control Performance
The proposed 2M2P MCC offers the advantage of cylinder pressure control, a capability not present in the 1M1P MCC Padovani et al. (2019).The minimum pressure between ppa and p pb is controlled to activate or deactivate LH  are shown in Figure 16.In motion mode, ppa and p pb are almost equal to pa and pa.In load-holding mode, because of the activated load-holding valves, pa stays at a certain pressure level to hold the load while ppa drops to approximately 5 bar due to the internal leakages across the two hydraulic pumps.Because p pb is the minimum pressure activating and deactivating load-holding valves, it is controlled in both motion and load-holding modes.In motion mode, p pb is settled at approximately 26 bar after a short period, where the control reference is 25 bar.In load-holding mode, after a short period, p pb is settled at approximately 4 bar, the same as the control reference.System pressures show stable behavior and smooth transitions between load-holding and motion modes, except for two pressure peaks at points A and B in the figures.Furthermore, it is noteworthy to highlight the stability and smoothness of system pressure behavior exhibited in MCC.This stands in contrast to hydraulic cranes equipped with standard load-holding valves, which may encounter oscillatory behaviors Sørensen et al. (2016).

Measured Servo Motor Speeds
Figure 17 presents the measured speeds of M1 and M2.In motion mode, the speed of M2 is around 2.08 (k speed ) times the speed of M1, and they have the same rotational direction.In load-holding mode, these two speeds settle to zero when p pb reaches 4 bar.The system turns from load-holding to motion mode at transition points A and C. The pressure level control loop rapidly increases (positive direction) the speed (ω2) of M2 to increase the pressure ppa to pa.The system pressure control loop decreases (negative direction) the speed (ω1) of M1 to increase the minimum pressure (p pb ) over 10 bar to activate the load-holding valves.In this case, the pressure ppa must be increased much faster than p pb to prevent SV oscillating.Therefore, ω2 has a much higher peak value than ω1 at points A and C. The system turns from motion to load-holding mode at transition points B and D. The load-holding control loop commands two servo motors to decrease the pressure (p pb ) to deactivate the loadholding valves.ω2 is two times and opposite direction of ω1.

2M2P MCC Energy Efficiency
The 2M2P MCC system power transfer and energy efficiency are addressed in this section.When the piston extends, the system works in pumping mode (quadrant I).
When the piston retracts, the system works in motoring mode (quadrant IV).The total servo motor shaft power (PM) and the cylinder output power (PC) are presented in Figure 18.PM is calculated via Equations 39, 40, and 41.
T1 and T2 are the measured shaft torques of M1 and M2.PM1 and PM2 are the calculated M1 and M2 shaft powers.PC is calculated by the measured bore-side pressure (pa) and rod-side pressure (p b ) via Equation 20. Figure 18 illustrates that the power differences (|PC − PM|) in both pumping and motoring modes are of comparable magnitudes.This observation suggests that the power losses in both modes are similar.The energy efficiency is calculated via η = PC/PM in pumping mode, and via η = PM/PC in motoring mode.It should be noted that the cylinder friction is not included in the PC calculation because friction is not directly measurable in the experimental test.Furthermore, servo motor efficiencies are not included in the PM calculation.Therefore, η represents the hydraulic power transfer efficiency in the 2M2P MCC.A moving average filter is used on η to remove noise.The filtered η is presented in Figure 19.η is approximately 56 % in pumping mode, much higher than the conventional valve-controlled hydraulic circuits used in a similar crane system Hagen et al. (2019).
η is approximately 24 % in motoring mode, representing the hydraulic power transfer efficiency in energy regeneration.As mentioned earlier, the power losses in both pumping and motoring modes are comparable.Moreover, the powers exerted by the cylinder in both modes are equal.Consequently, the regenerative efficiency is significantly lower than the efficiency observed in pumping mode.In theory, the regenerated energy can be fed to the grid or a battery.However, the regenerated energy is dissipated by an electric resistor because of lacking required facilities.It is important to note that, as shown in Figure 20, the M2 outputs and regenerates more power than M1 in pumping and motoring modes.Therefore, M2 and P2 should have higher rated powers than M1 and P1 to maximize the 2M2P MCC's overall output power limit.However, this factor was not revealed when designing the test facility.The M2 rated power (1.7 kW) is much lower than the M1 rated power (2.7 kW).Therefore, the overall output power of the 2M2P MCC testbench is limited.

Discussion
Based on the data presented in Figure 20, it shows that the power output of the M2 shaft is significantly greater than that of the M1 shaft in pumping mode.This can be attributed to the higher flow rate and the greater pressure increment across P2 compared to P1.The higher flow rate across P2 is due to the area ratio of the rod area to the annular area (Ar/A b ), which is 1.04.Additionally, the greater pressure increment across P2 results from the system pressure control loop controlling the rod-side pressure, the low-pressure side of P1, to the reference pressure.In contrast, the low-pressure side of P2 is connected to the accumulator (3 bar).Therefore, as the control reference pressure increases, the power ratio (P2/P1) of the M2 shaft power over the M1 shaft power also increases in pumping mode.This finding is further supported by the validated simulation model, as depicted in Fig. 21.When the control reference pressure increases from 10 to 40 bar, the power ratio increases from 1.44 to 3.20.Because the load-holding valve activating pressure is 10 bar, the M2 rated power needs to be at least 1.44 times higher than the M1 rated power in the proposed 2M2P MCC.However, M2 and P2 can only contribute to the cylinder output power in quadrant I or cylinder power regenera-tion in quadrant IV.In quadrant II, the cylinder regenerated power comes solely from M1 and P1, while M2 and P2 compensate for the cylinder differential flow and control the minimum cylinder pressure (ppa).In quadrant III, the cylinder output power comes solely from M1 and P1.Although M2 and P2 can regenerate a small amount of cylinder power when compensating for the cylinder differential flow and controlling the minimum cylinder pressure (ppa), they cannot contribute to the cylinder output power in quadrants II and III.Therefore, the proposed 2M2P MCC is more suitable for applications that operate only in quadrants I and IV, such as the crane used in this paper, than for applications that operate in all four quadrants, such as the knuckle boom cylinder circuit on offshore cranes Zhao et al. (2022).Application of the proposed 2M2P MCC in the knuckle boom cylinder circuit on offshore cranes results in greater energy losses than the 1M1P counterpart when the drive unit is mounted on the knuckle boom cylinder.This is due to the larger mass of the 2M2P MCC drive unit, even at the same output power level as the 1M1P MCC, provided that M2 and P2 offer higher output power than M1 and P1 Zhao and Bhola (2023).

Conclusions
A 2M2P MCC can achieve precise control over the piston position and the minimum cylinder pressure.However, its status as a MIMO system presents significant challenges in MIMO control algorithm analysis and design.Moreover, implementing a passive load-holding function further complicates the matter.These challenges make it difficult to apply a 2M2P MCC to real-world applications.This paper proposes a novel motor-controlled cylinder and a control algorithm to overcoming these challenges through the following key aspects: • A novel two-motor-two-pump motor-controlled cylinder with passive load-holding function was implemented on a laboratory crane system.
• A dynamic model of the system was developed and experimentally validated, providing a valuable simulation tool for future analysis and control design.
• A control algorithm that consists of four control loops to achieve precise control over the piston position and minimum cylinder chamber pressure, as well as a smooth transition between motion and load-holding modes was designed.
• The position tracking error falls within -3.2 mm to 2.3 mm, and the average value is 0.39 mm.The minimum cylinder pressure (p b ) is well controlled above 25 bar.
• The overall system energy efficiency is about 56 % in pumping mode and 24 % in energy regeneration mode.
• Upon analysis, it is found that the output powers of the main electric servo motor and hydraulic pump exceed those of the main electric servo motor and hydraulic pump.Therefore, 2M2P MCCs are more suit-able for applications that operate in quadrants I and IV.
In conclusion, the experimental results presented in this paper verify the efficacy of the proposed control algorithm and the functionality of the 2M2P MCC.This technique has the potential to be extended to a wide range of industrial applications, particularly those that require a smooth transition between motion and load-holding modes, such as industrial cranes and pitch angle control systems on wind turbines.Overall, the 2M2P MCC represents a promising approach for improving control and energy efficiency in linear hydraulic systems.

Future Work
The control algorithm presented in this paper is designed for four-quadrant operations.However, due to limitations of the test facility, only two-quadrant operation was experimentally verified.Furthermore, the load on the system is small; therefore, the cylinder working pressure is low.Hence, the efficiency results can be more convincing if pumps operate at higher pressures.Future work will involve implementing the proposed system and control algorithm in an application that operates in four quadrants, such as the knuckle boom cylinder circuit of a knuckle boom crane.This will enable a more comprehensive evaluation of the system's performance and energy efficiency and further demonstrate its potential for industrial applications.Furthermore, the potential for even greater system energy efficiency is present, as the regenerated energy in the experiments was not able to be reused due to the limitations of the test facility.If this energy could be effectively stored and reused in subsequent operation cycles, the overall energy efficiency of the system could be further improved.Thus, future work will include a simulation study that investigates the impact of stored and reused regenerated energy on the overall energy efficiency of the system.
Finally, as shown in Figure 17, there are motor speed jerks during the transition from load-holding to motion mode due to the pressure level control loop.These jerks harm the rotating components when the drive is at high power levels.Therefore, future work should cover a control algorithm improvement to remove these jerks.

Figure 1 :
Figure 1: Structure of Motor-controlled cylinders

Figure 2 :
Figure 2: System architecture of the 2M2P MCC.

Figure 4 :
Figure 4: Realization of the control algorithm.

Figure 5 :
Figure 5: Switching logic of the mode and inputs selection module.

k
le1 = 2.5 • 10 −3 L/min/bar is the main hydraulic pump leakage coefficient.The total leakage of the main hydraulic pump is modeled via Equation 17)4.2.4.Hydraulic cylinderA hydraulic cylinder model comprises four parts: pressure build-up, pressure-force conversion, stroke limits, and cylinder friction.Pressure build-ups on the bore side ( ṗa) and rod side ( ṗb ) of the cylinder are modeled via Equations 18 and 19.Qa and Q b are the flow rates going in or out of the cylinder bore and rod sides.xp and ẋp are the piston position and velocity, respectively.Aa and A b are the bore and rod side areas.x end = 400 mm is the cylinder stroke end.V0 is the cylinder dead volume plus line volume.

Figure 21 :
Figure 21: Ratio of servo motor shaft powers.