A Study on the Effects of Servovalve Lap on the Performance of a Closed - Loop Electrohydraulic Position Control System

Transcription

1 Al-Balawi: A tuy on the Effects of eroale ap on the erformance of a Close.. A tuy on the Effects of eroale ap on the erformance of a Close - oop Electrohyraulic osition Control ystem Rafa A. H. A-Balawi Yahya A. Faraj Rawan E. J. Talabani Al-Mustansiriya niersity-college of Eng. Mechanical Eng. Department. niersity of irkuk \ College of Eng. etroleum Eng. Department. Abstract This paper eals with a close loop position control of a ouble acting an ouble ro actuator using an electrohyraulic seroale (EHV). This system is stuie by using symmetric critical center spool ale (zerolappe) an open center spool ale (unerlappe). The nonlinear ynamic behaior of each case is unertaken an simulate. The system is moele by using fie state ariables (piston position, piston elocity, actuator pressures, an seroale spool isplacement) an is teste uner ifferent step inputs. The EHV is moele with a first orer ifferential equation. The close-loop system stability is inestigate by introucing equilibrium state into Jacobian matri an etermining the eignalues. Viscous friction an compressibility of oil are inclue in the moeling of the system. Because the electrohyraulic position sero system is not ery sensitie to coulomb friction an piston leakage they are neglecte. The work showe that when the unerlappe seroale operates in the unerlap region, the hyraulic position control system has more stable operation an better transient responses. eywors: Zerolap, nerlap, EHV, teay-tate Characteristics, Dynamic Response, osition Control, Moeling, imulation. (ap).. (unerlap) (zerolap),) (tate ariable).. (.. Jacobian. (unerlap). Receie 4 August 8 Accepte 3 Dec. 8

2 Al-Rafiain Engineering Vol.7 No.5 October 9. Introuction: Flui power control, that is the transmission an control of energy by means of a pressurize flui, is an ol an well-recognize iscipline. The growth of flui power has accelerate with our esires to control eer increasing quantities of power an mass with higher spees an greater precision. More specifically, where precise motion control is esire an space an weight are limite, the conenience of small size-to-high power ratios an the ability to apply ery large forces an torques with fast response times, at the same time, achiee a high egree of both accuracy an performance to make the hyraulic seromechanism systems the ieal control elements. The eman to achiee more accurate an faster control at high power leels has prouce an ieal marriage of hyraulic seromechanisms with electronic signal processing. Information coul be transuce, generate, an processe more easily in the electronic meium than as pure mechanical or flui signals, while the eliery of power at high spees coul be accomplishe best by the hyraulic sero. This sophistication of electronic eices an hyraulic eices into electrohyraulic seromechanisms renere better hyraulic systems, more efficient, more reliable, an faster equipments than eer before. The key element in this family of mechanisms is the electrohyraulic seroale. With a great power gains, the seroale acts as a power amplifier that conerts a low-power electrical signal into high-power hyraulic signal. These characteristics of electrohyraulic sero system make it ery attractie for many applications, such as the control of inustrial robots, processing of plastics, aircraft, satellites, launch ehicles, flight simulations, turbine control, an numerous military applications[]. Because of the nonlinearity an uncertainty parameters in hyraulic systems (such as nonlinear flow/pressure characteristics, friction forces, flow forces an their effects on the spool position an unknown eternal isturbances) many researchers proi a nonlinear moel of the hyraulic sero system control. A nonlinear moel of electrohyraulic elocity sero system is introuce by Joanoic []. In this paper; the flow nonlinearities, internal friction, oil compressibility, an ale ynamic (as first orer transfer function) are presente. yapuno base esign is use to eelop a nonlinear controller. He shows a goo agreement between the analytical technique an eperimental results. The work of Yun an Cho [3] has consiere unknown loa isturbance as parameter uncertainty an esigne a yapunoe base controller to make the hyraulic system follows a gien secon orer linear moel. Rui [4] also eelops a nonlinear moel for single stage electrohyraulic seroale an prouce a nonlinear controller base on backstepping approach. His work inclue the effects of frictions (ry, Coulomb, an iscous), ale ynamic, an oil compressibility. eroale ynamics play an important part on system behaior oer a certain range of frequency response. Consierable efforts ha been gone into moeling the electrohyraulic seroale, which suggeste that, epening on the frequency range of interest, a seroale is best moele by a first or secon orer transfer function [5,6]. Mookherjee [7] has use a computer-aie esign an sensitiity analyses to stuy the effects of raial clearance, mismatch in the areas of the tractie air gaps, an port geometry on the ale performance in a single stage seroale. His work inclue a nonlinear fiel moeling of hyraulic flui in the spool ale an magnetic flu in the motor, showe qualitatie conformity with the results presente in Moog technical Bulletin. This work is concerne with stuying the performance of a hyraulic sero position control system using zerolappe an unerlappe spool seroale. The work aims at stuying the steay-state characteristics of the EHV (zerolappe an unerlappe) such as coefficients of ale flow gain, flow-pressure, an pressure sensitiity, etermining an simulating a mathematical moel for an electrohyraulic position sero control system, an fining out the influences of seroale lap on the performance of the hyraulic position seromechanism.

3 Al-Balawi: A tuy on the Effects of eroale ap on the erformance of a Close... Dynamic Moels of Electrohyraulic ystem: Figure () shows a schematic of the actuator\seroale. As etaile by Merritt[8], the twostage electrohyraulic seroale is comprise of a coil wrappe armature connecte to a spool by a spring use for force feeback. The spool acts as a control ale, that regulates flow into the hyraulic actuator, which contains the two-actuator champers an a piston. The aboe control system can be moele as follows: I. eroale Flow Equations: The flow through the seroale, shown in figure (a), can be compare to the flow of a flui through a constricte point[9], as in figure (b). The equation, which escribes the flow through the ale, can be erie from Bernoulli s equation an the conseration of mass of the flui as it moes through the constricte orifice[9]. The olumetric flow rate through an orifice is gien by: = C A () Where C is the ischarge coefficient of the orifice, A is the area of the orifice, is the ensity of the flui, an an are the respectie pressure on either sie of the orifice. The area A aries linearly with the position of the spool, in case of critical center spool ale, the orifice area is only a function of spool position, A = A ( ) where is the ale spool isplacement while in open center ale, the orifice area is a function of spool ale isplacement an ale unerlap, A = A (, ), where is the ale unerlap. Electrohyraulic seroale Hyraulic Actuator s r s s supply pressure r return pressure Chamber Chamber poo Figure (): Detail of actuator/seroale[9] ist on -b- -a- Figure (): Flow through an office[9] 3

4 Al-Rafiain Engineering Vol.7 No.5 October 9 4 For the critical center spool ale, the flow equations are, as shown in figure (3a), (neglecting the leakage flow rate): C w,, C w, where w, which is calle the area graient of the ale, is the rate of change of orifice area with stroke (i.e. A = w ). For open center spool ale the flow equations, as shown in figure (3b), are (for unerlap region only):... ()

5 Al-Balawi: A tuy on the Effects of eroale ap on the erformance of a Close.. 5 An for :, w C, w C An for :, w C. w C If a matche an a symmetrical spool ale is use, = ( s + ) / an = (s - ) / can be use to write a general flow-pressure equations for both seroales in term of the supply pressure ( s ) an the loa pressure ( )[5] as follows: Zerolappe eroale: C w (4) nerlappe eroale: = = = C w ) ( ) ( (5) II. eroale Coefficients: A general epression for the loa flow is:, (6) If this function is epresse as a Taylor s series about a particular operating point = an when the higher orer infinitesimals is neglecte, it can be written as: (7) The partial eriaties require are obtaine by ifferentiation of the pressure-flow equation. These partials efine the two most important parameters of a ale.the flow gain is efine by: g (8) an the flow-pressure coefficient s is efine by: c (9) Another useful quantity is the pressure sensitiity c efine by:... (3)

6 Al-Rafiain Engineering Vol.7 No.5 October 9 t () which is relate to the other quantity by the well-known relation from calculus: or g t () c The coefficients g, c an t are calle ale coefficients, that are etremely important in etermining stability, frequency response, an other ynamic characteristics. The flow gain irectly affects the loop gain constant in a system, therefore, it has a irect influence on system stability. The flow-pressure coefficient irectly affects the amping ratio of ale- motor combinations, while the pressure sensitiity coefficient of ales is quite large, which accounts for the ability of ale-motor combinations to break away large friction loas with little error [8]. The alues of ale coefficients ary with operating point. The most important operating point is the origin of the pressure-flow cures (i.e. = = = ), as in figure 4, because system operation usually occurs near this region an the ale coefficients ealuate at this operating point are calle the null ale coefficients [8]. Critical center spool ale coefficients The ale coefficients of an ieal zerolappe spool ale (leakage flow is zero) can be obtaine by ifferentiation of equation. (4). The flow gain is g C w () a- For zerolappe seroale 6

7 Al-Balawi: A tuy on the Effects of eroale ap on the erformance of a Close.. b- For unerlappe seroale Figure (4): eroale pressure-flow characteristics for a supply pressure of pci The flow pressure coefficient is : c Cw an the pressure sensitiity is t The null ale coefficients are: (3) (4) C w (5) g c (6) (7) t Open center spool ale coefficients By ifferentiating esquation (5), the ale coefficients can be obtaine for unerlappe ale in the unerlap region, an by ealuating the eriaties at = = =, the null ale coefficients are: 7

8 Al-Rafiain Engineering Vol.7 No.5 October 9 C w (8) g C w c (9) t () III. Flow Into an Out of the Actuator: The epression of the net flow into an out of the actuator consists of three terms: - Flow ue to piston moement A /t, - Flow ue to laminar leakage across the piston, C 3- Flow ue to compressibility of the working flui. Thus the flow rate into the riing sie of the actuator is: ( A ) A C ( ` )...() t t An the flow rate out of the actuator is: ( A ) A C ( )...() t t where an are the initial olume of the riing an other sies of the linear actuator, A = A = A p, A p is surface area of piton hea. is the bulk moulus of the flui, an C is leakage coefficient across the actuator hea. IV. oa Dynamic Equation: Many researchers hae written the loa equation in ifferent forms. The most general form of the equation is[, 9]:...(3) t t t A M Ffc F where is iscous friction constant, F fc is moele coulomb friction force, is loa spring constant, M is the actuator an loa mass, an F is the eternal isturbance. V. Electrohyraulic eroale Equation: It is often conenient in sero analysis or in system synthesis work to represent an electrohyraulic seroale by a simplifie, equialent transfer function[]. If the effects of hysteresis an flow forces on the seroale are neglecte, then the ynamic behaior of the seroale can be escribe by a first-orer approimation[4], as follows: u...(4) where is ale spool isplacement, is time constant of the seroale, is seroale gain, an u is input current. 8

9 Al-Balawi: A tuy on the Effects of eroale ap on the erformance of a Close.. VI. Error ignal Equation: It can be seen from the block iagram of the control system, figure (5), that the summing amplifier ifferentially compares the input, an feeback oltage, which is obtaine from the input an feeback potentiometers respectiely; the resulting signal being fe to the amplifier. The output from the amplifier is use to moe the spool of the electrohyraulic seroale an subsequently moe the loa to its require position. If a potentiometer (input an feeback) gain ( p ) an a sero amplifier gain (G a ) are use, then the error current signal equation can be written as: u = p G a ( esire )...(5) In orer to close the system moel, it becomes necessary to replace the system input u with equation (5); therefore, equation (4) is rewritten as: p. Ga esire...(6) esire + - Amplifier G a u eroale Cyliner an the loa Figure (5): EHV osition control ystem. Deriation of the tate Equations: A number of assumptions are mae in orer to simplify the ynamic moel of the position control system. They are as follows: -The effect of Coulomb friction is neglecte an it is assume that there are no eternal isturbance an spring force. -The seroale has a symmetrical spool, an the control is ominate to be within unerlap region, i.e. < < for unerlappe spool. 3- The piston is locate in the center of the actuator so that = = t / = an the piston leakage is neglecte. 4- The hyraulic pump eliers a constant supply pressure. Fie state ariables are use to escribe the system operation. The first four state ariables escribe the operation of the hyraulic actuator: - the position of the piston (), - the elocity of the piston (/t), an 3 & 4 - the pressures in the actuator chambers. The final state ariable is from the seroale: 5 - the position of the control ale spool ( ). On the basis of the preceing assumptions an the state ariables efinitions, the equations of the close-loop control system can be written as: 9

10 3 4 Al-Rafiain Engineering Vol.7 No.5 October 9 ( ( 3 A A 4 p p. A p ) ) Ga. p. esire. Where equations. ( & 3) are use for &.. ystem tability: A p m. A p. It is epecte that the stability criteria of the linear systems coul be applie to nonlinear systems, if the eiations from the equilibrium state are sufficiently small an the signals inole are small, therefore, the nonlinearity has only a minor effect. Thus a linearization principle can be use to etermine the stability of nonlinear systems [,]. The equilibrium state e of the nonlinear unforce, u =, autonomous system: f ( e )...(8) is asymptotically stable if the eigenalues of the matri: f A...(9) e has negatie real parts. It is unstable if at least one eigenalue of A has a positie real part. It is completely unstable if all eigenalues of A hae positie real parts[]. [f()] enotes the Jacobian matri an equation (9) is calle the linearize system of equation (8) about the equilibrium state e. The equilibrium state of the current hyraulic position sero control system escribe by the equation () (for both zerolappe & unerlappe), for zero input, is: ( e = = = 5 = ) &( 3 = 4 = / ) 3. imulation Results: 3. ystem Response: The mass ensity of oil (the transmission meium use in hyraulic components) is a function of both pressure an temperature = f (,T). When one assumes a constant temperature an neglects the change in mass ensity () ue to change in pressure, a constant alue of (.78 Ib-sec /in 4 ) can be consiere for oil ensity. The ata that ha been use in the simulation are shown in tables () an (). The isplacement step responses obtaine from the simulation for both critical an open center spool seroales are shown in figure (4) with the step input signals of.in. &.3in. amplitues. The pressure responses for both sies of the riing cyliner are shown in figure (5) for zerolappe together with unerlappe seroale an the elocity responses for the actuator are shown in figures (6)....(7)

11 Al-Balawi: A tuy on the Effects of eroale ap on the erformance of a Close.. Table (): HM58-OBE chneier two-stage seroale parameters ymbol Notation Value nit Rate flow at 8. 3pci rate. iter/min. Operating pressure pci Null leakage flow at 8. 3pci nl.74 iter/min. Rate input current ---- ; ma. eroale gain.955 in./a. Area graient w.53 in. /in. Typical spool trael rate.9 in. Vale time constant. s Table (): ystem parameters ymbol Notation Value nit iston hea insie iameter Di.5 in. iston ro outsie iameter Dr.375 in. urface area of piston A p in. iston stroke t 6 in. Current amplifier gain Ga 4 ma./v Mass of piston an loa M.775 lb.s /in. Viscous friction coefficient lb.s/in. upply pressure s pci Mass ensity of hyraulic flui.78 lb.s /in 4 Effectie bulk moulus, lb/in. Input an feeback (VDT) gain p 5.5 V/in. a- For. in. step Input. Figure (6): osition Transient Response a- For. in. step input. b- For.3 in. step input. Figure (6): osition Transient Response

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13 Al-Balawi: A tuy on the Effects of eroale ap on the erformance of a Close.. 3. ystem tability : Accoring to the theorem presente in section., the eigenalues of the mathematical moel, (equation.7) aroun the equilibrium state e are shown in figure (9) an below. For zerolappe seroale system.47e 8.45e.66e.76e 6.5ei.76e 6.5ei For unerlappe seroale system 5.84e e 9.3e 4.5e 6.ei 4.5e 6.ei For zerolappe system For unerlappe system Figure (9): Eigenalues of close-loop electrohyraulic position control system. 4. Discussion: As the uner lappe sero ale works in the uner lap region (i.e. ), the position control system has ynamic results better than the same system with zero lappe sero ale. The sero ale flow gain has a irect influence on the system stability[8] an because the uner lappe sero ale has a flow gain that is twice that of zero lappe sero ale, (eqs. 5 & 8 ), in the uner lap region, therefore, the system with uner lappe sero ale is relatiely more stable as it can be note in figure (9).The sero ale flow-pressure coefficient irectly affects the amping ratio of the hyraulic system[8]; the uner lappe sero ale has a greater flowpressure coefficient than the zero lappe sero ale, (eqs. 6 & 9 ), therefore, the control system with uner lappe sero ale has a amping ratio less than the control system with zero lappe sero ale (if the system is consiere a linear system), so the position response of the control system, with uner lappe sero ale, reaches the steay state faster than the system with zero lappe sero ale as shown in the figure (6) which also shows that the position control system (of both sero ales) has a secon orer behaior for position response to a step input with amping ratio >. It can be seen in figures (7) that both sies steay-state cyliner s pressure are the same an equal to.5 of for both spool ales. It can also be seen that the 3

14 Al-Rafiain Engineering Vol.7 No.5 October 9 pressure responses of the system with uner lappe sero ale are faster than the system with zero lappe sero ale (which is more oscillatory). 5. Conclusion: The effects of seroale lap on the performance of a close - loop electrohyraulic position control system are stuie. The relate equations of the control system are erie an simulate using zerolappe an unerlappe seroale. eroale null coefficients are also etermine. The stuy showe that when the unerlappe seroale operates in the unerlap region, the position control system has more stable operation an better transient responses than the position control system with zerolappe seroale an the control system has a secon orer behaior for isplacement transient response to step input with >. 5. References: []. Vilenius, M. J. The Application of ensitiity Analysis to Electrohyraulic osition Control eros AME Journal of Dynamic ystems, Measurements, an control, Vol. 5, No., pp. 77-8, June 983. []. Joanoic M. Nonlinear Control of an Electrohyraulic Velocity erosystem, niersity of California,. [3]. Yun, J.., an Cho H.., Application of an Aaptie Moel Following Control Technique to a Hyraulic ero ystem ubjecte to nknown Disturbances AME Journal of Dynamic ystems, Measurement, an Control, Vol. 3, No. 3, pp , eptember 99. [4]. Rui. Nonlinear Control of Electro-hyraulic erosystems: Theory an Eperiment, Master egree thesis, niersity of Illinois, 998. [5].Al-Balawi R.A.H. Optimum erformance of Electrohyraulic osition Control ystems sing an nerlappe eroale, h.d. Thesis, niersity of Wales College of Cariff. October 99. [6]. Martin, D.J. an Burrows, C.R. The Dynamic Characteristics of an Electro-hyraulic eroale Trans of AME, Journal of Dynamic ystem, Measurement an Control December 976, pp 395. [7]. Mookherjee. Design an ensitiity Analysis of a ingle-stage Electro-hyraulic eroale roc. of First FNI-hD ymposium Hamburg, pp [8]. Merritt, H. E. Hyraulic control systems John wiley an ons, Inc., 967. [9]. Manhartsgruber B. Application of ingular erturbation Theory to Hyraulic ero Dries-ystem Analysis an Control Design, Department of mechanics & Machine Design, Johannes epler niersity.998. [].Thayer W. J. Transfer function for Moog seroales Technical Bulletin 3, Moog serocontrols Inc. New York, December 958, Re. January 965. []. Ogata. Moern Control Engineering 4th Eition rentice-hall, Inc.,. []. Willems J.. tability Theory of Dynamical ystems Thomas Nelson an ons i. 97. The work was carrie out at the college of Engg. niersity of Mosul 4