Control of an overlap-type proportional directional control valve using input shaping filter

Mechatronics 29 (2015) 87–95

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Control of an overlap-type proportional directional control valve using input shaping filter Ill-yeong Lee a,⇑, Dong-hun Oh b, Sang-won Ji c, So-nam Yun d a

Department of Mechanical and Automotive Engineering, Pukyong National University, Shinsunro 365, Nam-gu, Busan 608-739, Republic of Korea Test and Proto Operations Div., Volvo Construction Equipment Korea, Changwon, Republic of Korea c Department of Mechanical System Engineering, Pukyong National University, Busan, Republic of Korea d Extreme Mechanical Engineering Research Div., Korea Institute of Machinery and Materials, Daejeon, Republic of Korea b

a r t i c l e

i n f o

Article history: Received 5 May 2014 Accepted 5 October 2014 Available online 30 October 2014 Keywords: Proportional directional control valve Dead-zone compensation Input shaping filter

a b s t r a c t In a hydraulic control system (herein, termed a major control loop), a proportional directional control (PDC) valve with spool position feedback may work in a minor (or inner) control loop. In this study the authors propose control designs to improve the performance of the PDC valve control loop (a minor control loop). At first, the mathematical model for the PDC valve is developed through an experimental identification process. Then, a dead-zone compensator that facilitates jumping of the overlap zone in spool/sleeve combination is devised and applied. A reference input following controller (including PI-D and feed-forward controller) that enables robust control of the PDC valve under disturbances is devised using the pole placement method and the zero placement method. Subsequently, an input shaping filter is incorporated into the devised control system to improve the reference following characteristic in high frequency range. Finally, the effectiveness of the proposed control design is verified experimentally. Ó 2014 Elsevier Ltd. All rights reserved.

1. Introduction Proportional directional control (PDC) valves for hydraulic control systems are used to control the flow direction and flow rate. The PDC valve investigated in this study is a spool type direct-acting (single-stage) valve with a linear variable differential transformer (LVDT) for sensing the position of the spool [1]. The hydraulic actuator control system to which the PDC valve is applied has a major control loop and minor control loop(s). The major control loop is the entire control loop mainly utilizing feedback signal(s) from the actuator. An important minor (or inner) control loop in the major loop is the PDC valve control loop. Numerous previous researchers on PDC valves focused on improving the control performance in the major control loop (for example, [2]). The absence of satisfactory control performance in the PDC valve control loop that is a minor loop, will prevent excellent control performance in the major loop. As the researches on the PDC valve (a minor control loop), there were works on dead-zone compensation [3–5], flow force analysis [6–10], signal transforming in proportional amplifier [11–13], control scheme [14–16], and modeling & analysis [17,18]. ⇑ Corresponding author. Tel.: +82 51 629 6153; fax: +82 51 629 6150. E-mail addresses: [email protected] (I.-y. Lee), [email protected] (D.-h. Oh), [email protected] (S.-w. Ji), [email protected] (S.-n. Yun). http://dx.doi.org/10.1016/j.mechatronics.2014.10.003 0957-4158/Ó 2014 Elsevier Ltd. All rights reserved.

The PDC valve has an overlap zone (dead-zone) in the spool/ sleeve combination. Therefore, the dead-zone characteristic should be compensated for in order for the valve to be able to conduct continuous control action. In this study, the authors devised a dead-zone compensator that allows jumping movement of the spool in the overlap zone, and applied it to the valve. Prior to the controller design for the PDC valve, the mathematical model of the PDC valve was developed through a preliminary investigation of the frequency response characteristic of the valve using a simple closed loop valve control system. The governing equations of the PDC valve contain nonlinearities in the relations between electric current and solenoid force, and between solenoid force and spring displacement. These nonlinearities, together with flow force and viscous frictional force, act as disturbances to the valve control system. To achieve a robust reference following control while rejecting the effects of the disturbances, a reference following controller (including PI-D and feedforward controller) was devised using the pole placement method and the zero placement method [19]. Generally, the dynamic characteristics of hydraulic servovalves and proportional control valves are evaluated with bandwidth (the frequency at which the gain falls below 3 dB from DC gain) in Bode diagrams. Also, it is known that the input shaping filter (ISF) which is positioned outside the feedback control loop in control systems can modify the entire loop transfer function,

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without being influenced by the action of disturbance rejecting control loop [20,21]. ISF has been successfully implemented to reduce residual vibration, and thereby to shorten the settling time in the control of flexible structures [22]. Further, ISF is expected to be a very helpful measure to improve the dynamic characteristics of servovalves and proportional control valves in the high frequency range. Thus far, however, little research has applied ISF for improving the dynamic characteristics of hydraulic control valves. In this paper, therefore, the authors propose a control design using ISF for improving the dynamic characteristics of the PDC valve in the high frequency range. The effectiveness of the proposed control design is verified experimentally. 2. The structural features of the PDC valve, and experimental circuit for the PDC valve 2.1. The structural features of the PDC valve Fig. 1 shows the structure of the PDC valve investigated in this study. Fig. 2 shows photograph of the valve and the spool (enlarged), and the proportional amplifier. The valve has four ports (A, B, P & T ports) and consists of a spool, a sleeve, two proportional solenoids, two plungers, four centering springs and a position sensor (LVDT, not shown in Fig. 1). The valve is an overlap (closed-center) type one. Therefore, the dead-zone characteristic in the overlap-type valve should be compensated for in order that the valve can have continuous control action. As the valve uses two pairs of springs with different stiffness, there is a nonlinearity between the spool displacement and solenoid force. The solenoid force in the valve changes with the variation of not only electric current but also spool displacement, which also causes a nonlinearity. These nonlinearities, together with flow force and viscous frictional force, act as disturbances to the valve control system. 2.2. Experimental circuit for the PDC valve Fig. 3 shows the signal line’s connection in the experimental system for investigating the valve’s characteristics. Fig. 4 shows

Fig. 2. Photograph of the valve and the proportional amplifier [1]. (The spool is enlarged to show it clearly.)

the detailed configuration between the valve and a PC for control and signal measuring. The proportional amplifier shown in Fig. 4 is a PWM-type current driver that simply transforms the voltage input to current output. The signal from the LVDT is transformed through an independent signal conditioner. The control period for digital control using the PC is 1 ms all over the experiment in this study. Parameters’ values of the major components in the experimental system are shown in Table 1. 3. Compensation for the overlap characteristics of the PDC valve Fig. 5 is a conceptual diagram for designing a dead-zone compensator in this study. The left side diagram shows the relation between u (input voltage to the amplifier) and A (opening area of the PDC valve) before the compensation. The middle one depicts the characteristic of the compensator, and the right side one represents the relation between u and A after the compensation. In this figure, 2u0 is the dead-zone (expressed with voltage value) in the

Fig. 1. Structural view of the PDC valve (LVDT is not shown) [1].

Fig. 3. Signal lines’ connection in the experimental system.

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Fig. 4. Detailed configuration between the PDC valve and a PC for control and signal measuring.

Table 1 Parameters of the major components in the experimental system. Component

Parameter and value

Hydraulic pump The PDC valve

Displacement: 14 cm3/rev, speed: 1750 rpm, relief pressure: 10 bar Model: 4WRE 6E 16, load pressure: 0 bar, rated flow rate: 16 L/min, spool overlap: ±0.35 mm, rated displacement: ±2.33 mm Model: VT-VSPA2, gain: 0.25 A/V, freq. response (3 dB): over 2 kHz

PWM-type current driver for the PDC valve Oscillator/demodulator for LVDT PC & DAQ

Model: Trans-Tek 1000-0012, gain: 0.4115 V/mm, freq. response (3 dB): over 1 kHz [PC] Pentium 4 (Win XP) [DAQ] NI PCI-6024E (12 bit, 250 ks/s) [control program] Matlab/Simulink Real-Time Window Target

Fig. 5. Conceptual diagram in designing a dead-zone compensator.

valve, and u0 is the signal after the compensator (refer to Fjump(s) in upper block diagram of Fig. 6(a)). The dead-zone compensator devised with the concept shown in Fig. 5 allows the jumping movement of the spool in the overlap zone in the spool/sleeve combination. By using the compensator, the dead-zone can be reduced from 2u0 to 2au0. By determining the value of a closer to zero, though the dead-zone can approach to zero, severe chattering may occur when the spool stays in the neutral position. In this study, a was determined as 0.13 by considering both the compensation effect of the dead-zone and the chattering avoidance of the valve. Fig. 6(a) shows block diagrams of the proposed valve control system. In Fig. 6(a), xsr and xso denote the reference input and output of the spool displacement of the valve with the unit of voltage, respectively. FL is the disturbance force acting on the spool,

Kamp the gain of the current driver of the valve. Gsol(s) the transfer function of the solenoid element in the valve, Gs(s) the transfer function of the spool in the valve. Fjump(s) is the transfer function of the dead-zone compensator used to make up the deficiency due to the overlap structure in the spool/sleeve combination, and kxs the gain of the position sensor (LVDT). GPI(s), GD(s), GFF(s) and GISF(s) are transfer functions of the controller components, which are defined in Section 5 below. The difference between Gvalve(s) and G0v alv e ðsÞ arises due to the position of Fjump(s) in the block diagram, as shown in Fig. 6(a). Fig. 6(b) shows the experimental responses of spool displacement of the valve xso/kxs (mm) when xsr stays at 0 (V). In Fig. 6(b), two responses of xso in the cases of Gvalve(s) and G0v alv e ðsÞ were obtained from the two different valve control systems shown in Fig. 6(a). The controller components ‘GPI(s), GD, GFF and GISF(s)’ in Fig. 6(a) are same as those of the

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(a) Block diagrams of the valve control system

(b) Effects of the position of Fjump s in the valve control system

(c) Relation between x s r and xs o in the case of G 'valve s Fig. 6. Dead-zone compensation characteristic.

control system in Fig. 8. The procedure to design the controller is described in Section 5 below. In the case of Gvalve(s), a sustained vibration occurred in the signal of xso due to the signal passing through Fjump(s) from GD(s) (a differentiation controller), which was the differentiating noise signal from the LVDT. Fig. 6(c) shows the experimental result of the steady-state characteristics of the valve obtained in the control system G0v alv e ðsÞ shown in Fig. 6(a). Inspection of Fig. 6(b) and (c) reveals that the control system described by G0v alv e ðsÞ in Fig. 6(a) can accomplish dead-zone compensating function satisfactorily.

4. Mathematical modeling of the PDC valve To develop mathematical model of the PDC valve, the authors used a feedback control system with a proportional controller (control gain K 0p ¼ 10) described by the block diagram in Fig. 7. The frequency response characteristic was investigated experimentally to afford a Bode diagram. In the experiment, the amplitude of xsr was 25% of the rated value of xsr, the pressure drop (Dpv) across the valve was 10 bar, and the load pressure was set to zero based on ISO 10770-1 [23]. From the values on the Bode

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Fig. 7. A system for identifying the PDC valve.

diagram shown in Fig. 8, the closed loop transfer function Gv-closed (s) was approximated as follows.

Gv -closed ðsÞ ¼

s2

bx2n ; ½f ffi 0:8; xn ffi 176; b ffi 0:95 þ 2fxn s þ x2n

ð1Þ

The closed loop transfer function for the block diagram shown in Fig. 7, Gv-closed(s), is described as

Gv -closed ðsÞ ¼

K 0p Gv -open ðsÞ 1 þ K 0p Gv -open ðsÞ

ð2Þ

From Eqs. (1) and (2), Gv-open(s) was identified with the following equation.

Gv -open ðsÞ ¼

Gv -closed ðsÞ K 0p ð1  Gv -closed ðsÞÞ

bx2n =K 0p þ 2fxn s þ ð1  bÞx2n 2943 ¼ 2 s þ 281:6s þ 1548:8 ¼

(3) An input shaping filter ([20,21]) is arranged in front of the designed feedback control loop, to improve the input tracking ability in the higher frequency range. (4) Control outputs (frequency response and transient response) are checked experimentally, and the stability of the control system is verified. Fig. 9 shows a block diagram of the control system completed with the above-mentioned design procedure. In the figure, GPI(s), GD(s), GFF(s) and GISF(s) represent PI (proportional + integral) controller, differential controller, feed-forward controller and input shaping filter respectively. KFS is defined as 1/(Kamp  Gsol(s)), which can be approximated as a constant with KFS ffi 0.133 V/N at steady state. 5.1. Design of a PI-D controller considering disturbance rejecting ability

s2

ð3Þ

5. Control design for the PDC valve The authors design a feedback controller so as to obtain excellent control responses between the reference input xsr and the output xso. The design concepts and design procedure are summarized as follows.

(1) Expression of Xso(s)/FL(s) In Fig. 9, the transfer function Xso(s)/FL(s) is found as follows.

X so ðsÞ K FS Gv open ðsÞ ¼ F L ðsÞ 1 þ Gv open ðsÞðGPI ðsÞ þ GD ðsÞÞ

By defining GPI(s) = KP + KIs, GD(s) = KDs, and using Eq. (3) on Gv-open(s), Xso(s)/FL(s) is described as

ðK FS bx2n =K 0p Þs X so ðsÞ   h i ¼ bx2 bx2 bx2 F L ðsÞ s3 þ 2fxn þ 0 n K D s2 þ ð1  bÞx2 þ 0 n K P s þ 0 n K I Kp

(1) Incorporates a PI-D controller in order to compensate for external disturbances satisfactorily. D controller is positioned in the feedback loop to restrain any excessive overshoot which may arise in the response. The control gains are determined by the pole placement method. (2) A feed-forward controller is incorporated using the zero placement method to maximize the input tracking ability [19].

ð4Þ

n

Kp

Kp

ð5Þ (2) Determination of the reference model By reflecting the design specifications on system response, the reference model for a control system should be determined. Here, the reference model is constituted as the product of a standard second-order system, a first-order lag system and a differentiation element shown as Eq. (6). In Eq. (6), xnr, fr, k and Kr are constants. To obtain prompt response on disturbance rejection, and to reduce residual vibration under disturbance, the representative poles and the damping coefficient of the second-order system were set as xnr = 75 ± 56j and fr = 0.8, respectively. The first-order lag system was determined so that its pole become three times of the real part of xnr (i.e., k ¼ 3). Following the determination of these parameter values, [Xso(s)/FL(s)]ref is described as follows.

  X so ðsÞ x2nr kfr xnr ¼ 2 Krs F L ðsÞ ref s þ 2fr xnr s þ x2nr s þ kfr xnr

ð6Þ

  X so ðsÞ kfr x3nr K r s ¼ F L ðsÞ ref s3 þ ð2fr xnr þ kfr xnr Þs2 þ ðx2nr þ 2kf2r x2nr Þs þ kfr x3nr 1; 978; 000K r s ¼ 3 ð7Þ s þ 375s2 þ 42540s þ 1; 978; 000

Fig. 8. Frequency responses of the system shown in Fig. 7.

(3) Determination of PI-D controller’s gains By using the pole placement method, that is, by equating each coefficient of the denominators of Eqs. (5) and (7), we obtain the following design results.

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Fig. 9. The complete control system for the PDC valve.

GPI ðsÞ ¼ 13:9 þ 672=s; GD ðsÞ ¼ 0:0317s

ð8Þ

By equating the numerators of Eqs. (5) and (7), Kr was determined, and thus Xso(s)/FL(s) was obtained as

X so ðsÞ 391:5s ¼ 3 F L ðsÞ s þ 375s2 þ 42540s þ 1; 978; 000

ð9Þ

The response of xso from the designed system with Eq. (9) under stepwise disturbance of FL = 50 N was computed and the result is shown in Fig. 10, which shows comparatively the prompt disturbance rejection performance. 5.2. Design of a feed-forward controller using the zero placement method When we assume that GISF(s) = 1 and Fjump(s) is incorporated in Gv-open(s) in Fig. 9, the transfer function Xso(s)/Xsr(s) is described as follows.

X so ðsÞ ðGPI ðsÞ þ GFF ÞGv -open ðsÞ ¼ X sr ðsÞ 1 þ ðGPI ðsÞ þ GD ðsÞÞGv -open ðsÞ

ð10Þ

Let’s describe GFF(s) as

GFF ðsÞ ¼ c1 s þ c2

ð11Þ

From Eqs. (3), (8), (10) and (11), we obtain the following equation.

  K P þ KsI þ ðc1 s þ c2 Þ ðbx2n =K 0P Þs X so ¼ 3 X sr s þ 375s2 þ 42540s þ 1; 978; 000

ð12Þ

By applying the zero placement method [19] to Eq. (12), that is, by equating the numerator and the last three terms of the denominator of Eq. (12), GFF(s) can be determined to be 0

GFF ðsÞ ¼ 0:13s þ 0:52

ð11 Þ

Therefore, Eq. (12) is described as

X so ðsÞ 375s2 þ 42540s þ 1; 978; 000 ¼ 3 X sr ðsÞ s þ 375s2 þ 42540s þ 1; 978; 000

0

ð12 Þ

0

The response of the designed system with Eq. (12 ) on xso under stepwise reference input of xsr = 0.6 mm was computed and the result is shown in Fig. 11, from which it was confirmed that the controller design was desirable. 5.3. Design of input shaping filter (ISF) The dynamic response characteristics of PDC valves are generally evaluated with bandwidth in Bode diagrams. In this study, so as to improve the input tracking ability in a broader frequency range, an ISF ([20,21]) was applied to the front part of the feedback control system already designed in Sections 5.1 and 5.2. By using the ISF, the amplitude of the transfer function of a system could be modified in some frequency range. The transfer function of the ISF designed here is described as

GISF ðsÞ ¼

T2s þ 1 T1s þ 1

ð13Þ

GISF was designed so as to reform the Bode diagram of a FF-PI-D control system (shown in Fig. 13) into the desired form. To obtain ISFFF-PI-D control system shown in Fig. 9, GISF(s) was designed with T1 = 0.00318 and T2 = 0.00265. The procedure to determine the parameters in GISF(s) is described in Section 6 of this paper. In general, depending on the dynamic characteristics of the system to be reformed, a lead, lag, notch or other style filter can be applied. 5.4. Stability verification of the designed system As described in Section 5.1(2) of this paper, the closed loop poles of the FF-PI-D control system were placed on sufficiently stable positions on a complex plane. In the mathematical model, however, the modes in the higher frequency range that may exist in the real valve model are not reflected. Due to this uncertainty in the mathematical model, sufficient relative stability may not be secured in the experiment using the real PDC valve. Further, due

0.8

x s-o / K xs [mm]

x s-o / Kxs [mm]

2.5 2 1.5 1

0.6

0.4 0.2

0.5 0 0

0

0.05

0.1

0.15

0.2

Time [s]

0

0.05

0.1

0

Fig. 10. Computed response of Eq. (9) under stepwise disturbance of FL = 50 N.

0.15

0.2

Time [s] Fig. 11. Computed response of Eq. (12 ) under stepwise reference input of xsr = 0.6 mm.

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(a)

(b) Fig. 12. (a) Complete control system T(s) (Fig. 9) and (b) the system equivalent to T(s).

to the addition of the ISF to the FF-PI-D control system, the relative stability may deteriorate in some filter designs. In this study, as a measure to investigate the relative stability of the real PDC valve, the authors assumed an equivalent system with a unity feedback shown in Fig. 12(b) to the complete control system using the real PDC valve shown in Fig. 9 or Fig. 12(a). The open loop transfer function GO(s) in the equivalent system in Fig. 12(b) can be computed from the following equation. Fig. 13. Frequency responses of the system shown in Fig. 8.

TðsÞ GO ðsÞ ¼ 1  TðsÞ

ð14Þ

If GO(s) is a minimum-phase system, then the stability of the system can be verified easily by analyzing the characteristics of GO(s). In Section 6, the relative stability of the PDC valve is checked using GO(jx), which is obtained by substituting the experimental values of T(jx) into Eq. (14). 6. Control performance of the PDC valve Fig. 13 shows experimental results of frequency response with the control systems of PI-D, FF-PI-D and ISF-FF-PI-D designed in Section 5 using the experimental system shown in Figs. 2 and 3. In the experiment, the amplitude of xsr was 25% of the rated value of xsr, the pressure drop (Dpv) across the valve was 10 bar, and the load pressure was zero based on ISO 10770-1 [23]. In the experimental results shown in Fig. 13, the bandwidth in the case of the PI-D control system is about 30 Hz. In the case of the FF-PI-D control system, bandwidth reaches about 90 Hz, however a +2.7 dB gain appears near 60 Hz, which confines the actual operating frequency range of the FF-PI-D control system to below 50 Hz. In addition, excessive response may occur in the high frequency range in the FF-PI-D control system. On the other hand, it is known that the ISF-FF-PI-D control system can operate safely up to 90 Hz. The design procedure of the ISF is depicted in Fig. 14. A gain diagram of Xso(jx)/Xsr(jx) is shown in Fig. 14. In the enlarged diagram, No. (1) and No. (2) are the asymptotic lines of the denominator and the numerator of Eq. (13), respectively. No. (3) is the asymptotic line of Eq. (13) formed by combining lines No. (1) and No. (2). On the Bode gain diagram, time constants T1 and T2 in Eq. (13) can be selected according to the control designer’s intuition so as to determine an ISF that induces appropriate gain modification effect on the FF-PI-D control system. If the system 0 to be reformed is a minimum-phase system such as Eq. (12 ), then a unique transfer function of ISF can be found by inspection of the Bode gain diagram. Let’s denote each frequency transfer functions of the closed loop systems controlled by the PI-D, FF-PI-D and ISF-FF-PI-D controller shown in Fig. 12 with the common symbol T(jx). By substituting the T(jx) values of each system in Fig. 12 to Eq. (14), we obtain GO(jx) for each T(jx) as shown in Fig. 14. If GO(jx) is a minimum-phase system, then the relative stability can be verified

Fig. 14. Gain diagram of Fig. 12 for explaining the design procedure of the ISF.

on the Bode diagram as in Fig. 14. From Fig. 9 together with Eqs. 0 (3), (8), (11 ) and (13), it is easily deduced that each GO(jx) can be categorized to a minimum-phase system. In Fig. 14, phase margin of each GO(jx) of the control systems with the PI-D, FF-PI-D and ISF-FF-PI-D controller appeared to be 68.6°, 50.5° and 57.3°, respectively. These results on stability verification revealed that the closed loop control system for the PDC valve using the ISFFF-PI-D controller is stable with appropriate margin. (See Fig. 15). Fig. 16 shows the experimental results of the step responses of the control systems for the PDC valve with the PI-D, FF-PI-D and

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ISF-FF-PI-D shows the influence of a gain decrease in the high frequency range. In Fig. 16(b), the PI-D and FF-PI-D controlled systems presented an overshoot of about 17%. The overshoot was reduced to about 8% in the ISF-FF-PI-D controlled system. No steady state error was observed in any of the three cases. In conclusion, the dynamic characteristics of the PDC valve were greatly improved by using the proposed ISF-FF-PI-D controller. 7. Conclusions

Fig. 15. Frequency responses of GO(jx) in Eq. (14).

This study has presented several control designs to improve the control response characteristics between the reference input and the output of the spool position in a PDC valve. The valve has features such as dead-zone characteristics in the neutral position of the spool, severe nonlinearities due to nonlinear spring force, nonlinear solenoid force and disturbances arising from flow force working on the spool. To overcome the intrinsic handicaps of the valve in terms of controllability, the authors proposed a dead-zone compensation design, a disturbance rejecting control design, and a control design for improving the reference tracking ability. In particular, an ISF was applied to reform the control characteristic in the higher frequency range. The study results demonstrated that the application of the proposed control design using ISF can satisfactorily compensate for the dead-zone, and greatly improve the dynamic response characteristics of the PDC valve. Finally, the stability verification results of the PDC valve’s control system supported the applicability of the proposed control design. Acknowledgement This work was supported by a Research Grant of Pukyong National University (2014 year). References

Fig. 16. Step responses of the system in Fig. 8.

ISF-FF-PI-D controller under step inputs with 25% of the rated valve of xso. The control gains in the case of Fig. 16 are the same as in the case of Fig. 13. In Fig. 16(a), the signal x0sr in the case of the

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