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330 Beshahwired Ayalew, Bohdan T. Kulakowski / ISA Transactions 44 (2005) 329–343
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Beshahwired Ayalew, Bohdan T. Kulakowski / ISA Transactions 44 (2005) 329–343 331 Fig. 1. Schematic of test system ͑HPϭhigh pressure, LPϭlow pressure͒.supply and return lines are usually ﬂexible hoses output stage. The servovalve is close coupled withrated for the appropriate working pressure. a 10-kN, 102-mm-stroke symmetric actuator, Aside from the extensive presentation by Vier- which is mounted on a load frame. Pressure trans-sma ͓7͔, not much has been reported on the analy- ducers are used for sensing the pressures at thesis of a hydraulic servosystem including supply four ports of the servovalve. A linear variable dif-pressure variations at the servovalve. Viersma’s ferential transformer ͑LVDT͒ is mounted on theanalysis was done in the frequency domain and the actuator piston for position measurement.emphasis was to provide design rules for the loca- Control and signal processing is done with ation and sizing of the components of hydraulic ser- dSpace® 1104 single processor board, which in-vosystems. Yang and Tobler ͓10͔ developed a cludes onboard A/D and D/A converters and amodal approximation technique that enables slave digital signal processor ͑DSP͒. An ampliﬁeranalysis of ﬂuid transmission lines in the time do- circuit converts a 0–10-V control output from themain via state space formulations. Modal approxi- dSpace® D/A to a high-impedance Ϯ50-mA cur-mation results provide modular and simpler alter- rent input to the torque motor coils of the servov-natives to direct numerical time-domain solutions alve.of the ﬂow equations. The unit labeled hydraulic services manifold In this paper, the authors present and use modal ͑HSM͒ is connected to the servovalve using 3.048-approximation results within a model of an elec- m-long SAE-100R2 hoses. The hydraulic powertrohydraulic actuation system to investigate supply supply ͑HPS͒ unit, including its heat exchangerand return pressure variations at the servovalve and drive units, is housed separately and is con-due to transmission line dynamics. Open-loop and nected to the HSM via 3.048-m-long SAE-100R2closed-loop tests were conducted to validate the hoses. The HSM provides basic line pressure regu-model. The model was then used to make a quick lation via the accumulators. In addition, the HSMevaluation of alternative layouts of the supply and is equipped with a control manifold circuitry toreturn lines. permit selection of high- and low-pressure operat- ing modes, low-pressure level adjustment, slow2. Description of test system pressure turn-on and turn-off, and fast pressure un- loading. The drain line provides a path for oil that The hydraulic system shown schematically in seeps past the seals in the actuator and also forFig. 1 was designed for fatigue testing applica- draining oil from the HSM pressure gage.tions. The servovalve is a 5-gpm ͑19 lpm͒ two- During a normal fatigue testing operation, bothstage servovalve employing a torque motor driven the low-pressure and high-pressure solenoids aredouble nozzle-ﬂapper ﬁrst stage and a main spool energized, the main control valve is completely
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332 Beshahwired Ayalew, Bohdan T. Kulakowski / ISA Transactions 44 (2005) 329–343 Fig. 2. Simpliﬁed system.wide open, and the circuitry of the HSM allows pressed from the pump output pressure. Therefore,ﬂow at full system pressure ͓11͔. We therefore in the following analysis, it is assumed that themodel the HSM by considering the lumped non- output pressure just after the accumulator andlinear resistance arising from change of ﬂow di- pressure relief valve connection points in the HPSrections, ﬂow cross sections, as well as ﬂow in the unit can be set as a known pressure input to theﬁlter element. The total pressure drop between the rest of the system. In fact, this is not a very restric-pressure inlet and outlet ports of the HSM is given tive assumption, since the modeling approach pre-in manufacturer speciﬁcations. The available data sented here is modular and models of the upstreamsatisfy a nonlinear expression relating ﬂow rate to components of the whole HPS unit can be incor-pressure drop. While the HSM unit is rated for a porated if needed.wide range of ﬂow rate capacities, the rated ﬂow For the simpliﬁed system, we now have two sec-rate through the servovalve is within 10% of tions of transmission hoses to model. The ﬁrst sec-nominal ﬂow rate of the HSM. We therefore use a tion is for the supply and return hoses between thelocal linear approximation to account for losses in HPS and the HSM and the second is for the hosesthe HSM, between the HSM and the servovalve. A model applicable for each section is discussed next. ⌬p HSM ϭR HSM q. ͑1͒ We further assume that the check valve is an 3. Model of transmission linesideal one, so that its own dynamics are fast enoughto be neglected and its backﬂow restriction has a To model the hydraulic hoses for our system, welarge enough parallel resistance that the permitted refer to previous rigorous results on validated so-backﬂow is very small. We also consider the drain lutions of the mass and momentum conservationﬂow to be negligible. With these simpliﬁcations, equations governing ﬂow in one-dimensional ﬂuidthe system reduces to the one shown in Fig. 2. It transmission lines having a circular cross sectionshould be noted that the resistances were lumped ͓5,6,8͔. In general, some assumptions are neces-in the HSM excluding the gas-charged accumula- sary for the basic results to hold. These assump-tors. tions include laminar ﬂow in the lines, negligible We make one more assumption before we pro- gravitational effects, negligible tangential velocity,ceed. Through extensive frequency domain analy- and negligible variations of pressure and densitysis, Viersma ͓7͔ has shown that, provided that the in the radial and tangential directions. We also as-accumulator and the pressure relief valve on the sume constant and uniform temperature and ignorehydraulic power supply unit are located sufﬁ- heat transfer effects in the ﬂuid line. We therebyciently close to the pump outlet ͑within 0.3 m͒, the limit the discussion to the linear friction model,pump ﬂow pulsation frequencies can be sup- which does not include distributed viscosity and
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Beshahwired Ayalew, Bohdan T. Kulakowski / ISA Transactions 44 (2005) 329–343 333 other input ͓12͔. Since we have already assumed the pressure just after the connection point of the pressure relief valve and ﬁrst accumulator to be taken as an input to the system, the desired causal Fig. 3. A ﬂuid transmission line. form of the four-pole equations for the supply line is the so-called pressure-input/pressure-output causality form ͓15͔. It can be derived by deﬁning the boundary conditions for the distributed param-heat transfer effects ͓6,10͔. Corrections are applied eter model as the upstream pressure and ﬂow rateto account for the frequency dependence of vis- ( p u ,q d ) and the downstream pressure and ﬂowcosity on the linear friction model, following the rate ( p d ,q u ) at the opposite ends of the line,work of Yang and Tobler ͓10͔. The ﬂow lines are assumed to have rigid wallsin some derivations ͓1,7͔. However, Blackburn etal. ͓12͔ and McCloy and Martin ͓13͔ arrive at the ͫ ͬ P d͑ s ͒ Q u͑ s ͒ ͫ ͬsame governing equations as the rigid wall case 1 Z c ͑ s ͒ sinh ⌫ ͑ s ͒͑for a frictionless ﬂow͒ by allowing for wall ﬂex- Ϫ cosh ⌫ ͑ s ͒ cosh ⌫ ͑ s ͒ibility and deﬁning an effective bulk modulus ϭcombining the ﬂexibility of the wall and that of sinh ⌫ ͑ s ͒ 1the oil. Their deﬁnition of effective bulk modulus Z c ͑ s ͒ cosh ⌫ ͑ s ͒ cosh ⌫ ͑ s ͒is the same as that derived by Merritt ͓4͔, wherethe effective bulk modulus is viewed as a seriesinterconnection of the ‘‘stiffness’’ of the oil, of a ͫ ͬ P u͑ s ͒ ϫ Q s͒ . d͑ ͑2͒container wall and even of entrapped air volume inthe oil. Following this approach, we consider ﬂex- The deﬁnitions of the propagation operator ⌫ ( s )ibility effects via the effective bulk modulus  e . and the line characteristic impedance Z c ( s ) de-The model parameters required for any section of pend on the friction model chosen ͓6,9͔. In thisthe transmission line reduce to the ones shown in paper, the authors use the approach of Yang andFig. 3. For the hydraulic hoses in this work, nomi- Tobler ͓10͔ that incorporated frequency-dependentnal values of the bulk modulus were taken from damping and natural frequency modiﬁcation fac-Ref. ͓13͔. tors into analytically derived modal representa- Using the above assumptions for the single tions of the four-pole equations for the linear fric-transmission line, the conservation laws can be in- tion model. ⌫ ( s ) and Z c ( s ) are deﬁned bytegrated in the Laplace domain to yield a well-known distributed parameter model commonly ex-pressed as a two-port matrix equation and ⌫ ͑ s ͒ ϭD n d 2s 4 ͱ ␣ 2ϩ 32␣ sd 2 , ͑3͒sometimes known as the four-pole equation ͓1,7͔.The four-pole equations can take four physicallyrealizable causal forms ͓6,9,12͔. Two of these fourare readily relevant to the problem at hand: one for Z c ͑ s ͒ ϭZ 0 ͱ 32␣ sd 2 ϩ ␣ 2. ͑4͒the supply line hoses and another one for the re-turn line hoses. The third one also ﬁnds use with The frequency-dependent correction factors ␣accumulator connection lines, as discussed by Ay- and  are obtained by comparing the modal un-alew and Kulakowski ͓14͔. damped natural frequencies and damping coefﬁ- Taking the supply line case ﬁrst, we notice that cients of the modal approximations of the dissipa-in most hydraulic servosystem applications, a con- tive ͑‘‘exact’’͒ model, which is described in detailtrol signal modulates the servovalve consumption in Ref. ͓16͔, against the modal representation ofﬂow rate downstream of the supply line, q d ( t ) , the linear friction model ͓10͔. Corrected kinematicfollowing the excursions of the ͑loaded͒ actuator viscosity ͑͒ values are used ͓8͔. The dimension-piston. Then q d ( t ) is a preferred input to the trans- less numbers D n and Z 0 are the dissipation num-mission line model, and a realizable causality form ber and the line impedance constant, respectively,requires that either p u ( t ) or q u ( t ) should be the and are given by
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334 Beshahwired Ayalew, Bohdan T. Kulakowski / ISA Transactions 44 (2005) 329–343 ͫ ͬ n 4l D nϭ , ͑5͒ ͚ p di cd 2 ͫ ͬ pd qu ϭ iϭ1 n ϭ ͓ I 2 I 2 ¯I 2 ͔ 4c iϭ1 ͚ q ui Z 0ϭ , ͑6͒ d2 ϫ͓ p d1 q u1 p d2 q u2 ¯p dn q un ͔ . ͑10͒where c is the speed of sound in the oil, It should be noted that the truncation to ﬁnite number of modes will introduce steady-state er- cϭ ͱ . e ͑7͒ rors. Some methods have been suggested to re- cover the steady-state output based on the fact that at steady state the original four-pole equation, Eq. The three causal functions 1/cosh ⌫(s), ͑2͒, reduces toZ c ( s ) sinh ⌫(s)/cosh ⌫(s), and sinh ⌫(s)/Zc(s)cosh ⌫(s) can be represented as inﬁnite sumsof quadratic modal transfer functions. The goal is ͫ ͬ ͫ pd qu ss ϭ 1 0 Ϫ8D n p u 1 qd ͬͫ ͬ ss . ͑11͒to use a ﬁnite number of modes to approximate theotherwise inﬁnite sum of the modal contributions Hsue and Hullender ͓16͔ discussed rescaling thefor the outputs. Of particular interest for the time truncated sum of the modal approximation for thedomain models sought in this paper is the state dissipative model by its zero-frequency magnitudespace formulation ͓10,15͔, to bring about Eq. ͑11͒. Van Schothorst ͓9͔ and Hullender et al. ͓15͔ described an additive ap- ͫ ͬͫ ͬ 0 ͑ Ϫ1 ͒ iϩ1 Z 0 ci proach where the steady-state error is eliminatedͫ ͬ ˙ di p q ui ϭ Ϫ ˙ ͑ Ϫ1 ͒ iϩ1 ci 2 Ϫ 2 32  p di q ui by adding a corrective feed-through term on the output equation ͑10͒. However, the transfer func- tions so implemented will no longer be strictly Z 0␣ d ␣ ͫ ͬ proper. This may entail the need for off-line alge- 8Z0 braic manipulations when the transmission line is 0 Ϫ connected to static source and/or load linear resis- ϩ 8 d 2D n ͫ ͬ pu qd , tances or other transmission line models with their own direct feed-through gains. The eigenvalues of 0 the coupled system may then be altered by the d 2Z 0D n␣ 2 steady-state correction ͓15͔. Yang and Tobler ͓10͔ introduced methods that iϭ1,2,3...n. ͑8͒ modify the input-matrix or use a state similarity transformation matrix to affect the steady-stateThe ci are the modal undamped natural frequen- correction while preserving the modal eigenvaluescies of blocked line for the linear friction model of the truncated model. Since comparable resultsgiven by are obtained by the use of either method, we adopt the input-matrix modiﬁcation method here. Sup- ͩ ͪ pose matrices A i and B i represent, respectively, 1 4 iϪ the feedback and input matrices in the modal state 2 space Eq. ͑8͒. Introducing the input-matrix modi- ci ϭ , iϭ1,2,3,...n. ͑9͒ d 2D n ﬁer G,The modiﬁcation factors ␣ and  are given asfunctions of the dimensionless modal frequencies ͫ ͬ ͫ ͬ ˙ p di q p di pu ˙ ui ϭA i q ui ϩB i G q d . ͫ ͬ ͑12͒d 2 ci /4 ͓10͔. The output is the sum of the modal The steady-state value of the n-mode approxima-contributions and is expressed as tion is then
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Beshahwired Ayalew, Bohdan T. Kulakowski / ISA Transactions 44 (2005) 329–343 335 ͫ ͬ ͫ ͬ ͫ ͬ n n pd p di pu qu ϭ͚ q ui ϭϪ ͚ ͑ A Ϫ1 B i ͒ G q i . ss iϭ1 ss iϭ1 d ss ͑13͒Comparing with the desired steady-state valuegiven by Eq. ͑11͒ and solving for G, GϭϪ ͚ͩ n iϭ1 A Ϫ1 B i i ͪͫ Ϫ1 1 0 Ϫ8Z 0 D n 1 ͬ . ͑14͒The number of modes n to be chosen depends onthe frequency range of interest for the application. Similarly, for the return line from the servo-valve, we can deﬁne the ﬂow rate at the servov-alve end and the pressure at the downstream ͑to-ward the tank͒ end as inputs to the model of thereturn line based on the other four-pole equationof causality dual to Eq. ͑2͒ ͓6͔. Equally, we canuse the observation that switching the sign con-vention of the ﬂow direction for just the return lineand using the four-pole equation dual to Eq. ͑2͒yields the same set of four-pole equations as Eq. Fig. 4. Piston accumulator.͑2͒, provided the inputs to the model remain ﬂowrate toward the servovalve end and pressure at theother end. This fact can easily be shown math- plied after initial transients have died down. Inematically, but we omit it here for brevity and general ‘‘steady’’ simulations, like those involvingstate that for the return line model all of the deri- sinusoidal fatigue test wave forms, the modal ini-vations presented above for modeling the supply tial conditions of the interconnected system areline hold. The caveat is to exercise care in using less important.the proper signs for the input and output ﬂow rates As shown in Fig. 3, the model described in thisat both ends of the return line when forming inter- section requires few parameters to set up andconnections with other system components. simulate each transmission line section using lin- For step response simulations, it is desirable to ear state space models in the time domain. This ishave good estimates of the initial conditions of the particularly more convenient than ﬁnite differencemodal states, especially when the interconnected approaches, which may require rigorous descreti-system model contains nonlinearities. Usually, for zation methods.a single pipeline section, the derivative of themodal output can be assumed to be zero just be-fore the application of the step change in the input, 4. Modeling accumulatorsand modal initial conditions can be computedfrom For the nitrogen gas-charged accumulators, which are of piston type for our system ͑Fig. 4͒, ͫ p di ͑ 0 Ϫ ͒ q ui ͑ 0 Ϫ ͒ͬ i ͫ p u͑ 0 Ϫ ͒ ϭϪA Ϫ1 B i G q 0 Ϫ ͒ , d͑ ͬ ͑15͒ we assume that the piston mass and seal friction are negligible. Under this assumption, the gas pressure and the oil pressure are equal. We alsowhere the ͓ p u ( 0 Ϫ ) ,q d ( 0 Ϫ ) ͔ T are the inputs just neglect the compressibility of the oil volume inbefore the step change. For an interconnected the accumulator compared to the compressibilitypipeline system, the inputs to one pipeline section of the gas. Furthermore, we consider the gas tomay be outputs of another section, in which case undergo a polytropic expansion and compressionthe determination of proper initial values for the process with polytropic exponent m.modal states of each section can be done by trialand error. The step disturbances can also be ap- p g V m ϭconst. g ͑16͒
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336 Beshahwired Ayalew, Bohdan T. Kulakowski / ISA Transactions 44 (2005) 329–343 Fig. 5. Model interconnections for the supply line ͑Version I͒.The exponent m approaches 1 for a slow ͑nearly can be estimated as the HSM pressure minus theisothermal͒ process and the speciﬁc heat ratio of pressure drop in the connection lines.the gas for a rapid ͑adiabatic͒ process on an idealgas. However, it should be noted that if heat trans-fer effects were to be included, then real gas equa- 5. Model interconnections for the supply linetions of state should be used together with appro-priate energy conservation laws. For the work The components on the supply line of the sim-presented here, the present model was found to be pliﬁed model shown in Fig. 2 can be intercon-sufﬁciently accurate. nected as shown in Figs. 5 or 6. The arrows indi- Given initial gas pressure p g0 and gas volume, cate the input-output causality assigned for eachV g0 , the gas pressure is computed from Eq. ͑16͒, subsystem. Each of the blocks section I and sec-which is equivalent to tion II implement Eqs. ͑8͒–͑14͒ for the corre- sponding sections of the supply line. mpg Integration causality is the desired form for the ˙ p gϭ qa , p g ͑ 0 ͒ ϭp g0 , ͑17͒ model of the accumulator, which is given by Eq. V g0 Ϫ ͐ t0 q a dt ͑17͒. It was pointed out by Viersma ͓7͔ that thewhere q a is the ﬂow rate of the hydraulic oil to the ﬂow dynamics in the short branch-away connec-accumulator. For simulations involving distur- tion lines to the accumulators are signiﬁcant inbances applied at the servovalve, it is reasonable most cases. Under the linear resistance assumptionto assume that the accumulator already develops given by Eq. ͑1͒, the subsystem ‘‘manifold andan initial gas pressure through a slow isothermal check valve loss’’ can be conﬁgured either as aprocess1 ( mϭ1 ) . The initial gas volume V g0 can pressure-input/ﬂow rate-output subsystem ͑in ver-be estimated by applying Eq. ͑16͒ between the sion I, Fig. 5͒ or as a pressure-input/pressure-pre-charge state ͑the gas pre-charge pressure at ac- output subsystem ͑in version II, Fig. 6͒. As a con-cumulator capacity͒ and the initial state at the on- sequence, the model of the short accumulatorset of the disturbance. The initial gas pressure p g0 connection line changes between version I and version II. The model of the short connection line 1 in version I has the same structure as the one de- The present system has an additional slow turn-on/turn-off accumulator that enables the HSM to come to full sys- scribed above for the sections of the main supplytem pressure from an off state in a slow and controlled line. The model for the short connection line inmanner. The servovalve disturbances considered in this version II is derived using the modal approxima-study are applied after the whole system has reached nor- tion for the relevant four-pole equations withmal operating conditions. ( p u ,p d ) as input and ( q u ,q d ) as output, as de-
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Beshahwired Ayalew, Bohdan T. Kulakowski / ISA Transactions 44 (2005) 329–343 337 Fig. 6. Model interconnections for the supply line ͑Version II͒.scribed by Ayalew and Kulakowski ͓14͔. It was 6. Model for servovalve and actuatorshown there that the dynamics of the short connec-tion line can be approximated by a ﬁrst-order term Physical models of electrohydraulic servoactua-that reduces to a series interconnection of hydrau- tors are quite widely available in the literaturelic resistance and inertance. This result goes along ͓1– 4,7,9,17͔. The model presented here is adaptedwith the convenient assumption that the oil side to apply to a four-way servovalve close coupledcompressibility in the accumulator is negligible with a double-ended piston actuator.compared to that of the gas side. This result also Fig. 7 shows a double-ended translational pistonmakes version II preferable to version I, since it actuator with hydraulic ﬂow rates q t from the topreduces the order of the overall system and veriﬁes chamber and q b to the bottom chamber of the cyl-a physically supported model order reduction. inder. Leakage ﬂow between the two chambers is The model for the return line is developed in a either internal ( q i ) between the two chambers orsimilar way noting the reverse direction of the external from the top chamber ( q e,t ) and from theﬂow, as mentioned earlier. It should be noted that bottom chamber ( q e,b ) . A t and A b represent thethe modularity of the subsystem model intercon- effective piston areas of the top and bottom face,nections allows changes to be made to the system respectively. V t and V b are the volumes of oil inmodel with ease. the top and bottom chamber of the cylinder, re- spectively, corresponding to the center position ( x p ϭ0 ) of the piston. These volumes are also considered to include the respective volumes of oil in the pipelines between the close-coupled servo- valve and actuator as well as the small volumes in the servovalve itself. We assume that the pressure dynamics in the lines between the servovalve and the actuator are negligible due to the close coupling.2 Furthermore, even for a long-stroke actuator used in a ﬂight simulator application, Van Schothorst ͓9͔ has 2 This is to say that any resonances introduced by the short-length lines are well above the frequency range of interest for our system. In fact, this can be veriﬁed using the model development presented earlier or referring to theFig. 7. Schematic of a rectilinear actuator and servovalve. results of Van Schothorst ͓9͔.
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338 Beshahwired Ayalew, Bohdan T. Kulakowski / ISA Transactions 44 (2005) 329–343shown that the pressure dynamics in the actuatorchambers need not be modeled using distributedparameter models. It is therefore assumed that thepressure is uniform in each cylinder chamber andis the same as the pressure at the respective port ofthe servovalve. Starting with the continuity equation and intro-ducing the state equation with the effective bulkmodulus for the cylinder chambers, it can be Fig. 8. Actuator with a simple load model.shown that the pressure dynamics are given by͑see, for example Ref. ͓2͔͒ dpb c ϭ ˙ ͑ q ϪA b x p ϩq i Ϫq e,b ͒ , K v ,i ϭc d,i w i ͱ2/ , iϭ1,2,3,4. ͑20͒ dt V b ϩA b x p b ͑17a͒ These coefﬁcients could be computed from data for the discharge coefﬁcient c d,i , port widths w i , dpt c and oil density . If we assume that all oriﬁces are ϭ ˙ ͑ Ϫq t ϩA t x p Ϫq i Ϫq e,t ͒ . dt V t ϪA t x p identical with the same coefﬁcient K v , then the ͑17b͒ value of K v can also be estimated from manufac- turer data for the rated valve pressure drop ( ⌬p N ) ,These equations show that the hydraulic capaci- rated ﬂow ( Q N ) , and maximum valve stroketance, and hence the pressure evolution in the two ( x v max) using the following equation ͓1,4͔:chambers, depends on the piston position. Theleakage ﬂows q i , q e,b , and q e,t are considered QNnegligible. K v ,i ϭK v ϭ , iϭ1,2,3,4. The predominantly turbulent ﬂows through the x v maxͱ1/2⌬p Nsharp-edged control oriﬁces of a spool valve, to ͑21͒and from the two sides of the cylinder chambers, As an approximation of the servovalve spool dy-are modeled by nonlinear expressions ͓1,2,4͔. As- namics, a second-order transfer function orsuming positive ﬂow directions as shown in Fig. equivalently a second-order state space was ex-7, we have tracted from manufacturer speciﬁcations,q b ϭK v ,1sg ͑ x v ϩu 1 ͒ sgn͑ p S Ϫp b ͒ ͱ͉ p S Ϫp b ͉ X v͑ s ͒ G v 2 v n, ϭ . ͑22͒ ϪK v ,2sg ͑ Ϫx v ϩu 2 ͒ sgn͑ p b Ϫp R ͒ ͱ͉ p b Ϫp R ͉ , I v ͑ s ͒ s 2 ϩ2 v n, v sϩ 2 n, v ͑18a͒ The state equations governing piston motion are derived considering the loading model for the ac- q t ϭK v ,3sg ͑ x v ϩu 3 ͒ sgn͑ p t Ϫp R ͒ ͱ͉ p t Ϫp R ͉ tuator. For the test system, the actuator cylinder is rigidly mounted on a load frame, which we use as ϪK v ,4sg ͑ Ϫx v ϩu 4 ͒ sgn͑ p S Ϫp t ͒ ͱ͉ p S Ϫp t ͉ , an inertial frame as shown in Fig. 8. ͑18b͒ The upward force on the actuator piston due to the oil pressure in the two cylinder chambers iswhere the sg ( x ) function is deﬁned by given by sg ͑ x ͒ ϭ ͭ x, xу0 0, xϽ0 . ͑19͒ F p ϭA b p b ϪA t p t . The friction force on the piston in the cylinder is ͑23͒ denoted by F f and the external loading includingThe parameters u 1 , u 2 , u 3 , u 4 are included to specimen damping and stiffness forces are lumpedaccount for valve spool lap conditions as shown in together in F ext . The equations of motion are eas-Fig. 7. Negative values represent overlap while ily derived by applying Newton’s second law:positive values represent underlap. The valve co-efﬁcients K v ,i are given by ˙ x pϭ v p , ͑24͒
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Beshahwired Ayalew, Bohdan T. Kulakowski / ISA Transactions 44 (2005) 329–343 339 1 ˙ v pϭ ͓ A p ϪA t p t ϪF ext ϪF f Ϫm p g ͔ . mp b b ͑25͒Eqs. ͑17a͒, ͑17b͒, ͑24͒, and ͑25͒ with q b and q tgiven by Eqs. ͑18a͒ and ͑18b͒ constitute the statespace model for the servovalve and loaded actua-tor subsystem under consideration. These equa-tions also contain the major nonlinearities in thesystem: the variable capacitance and the squareroot ﬂow rate versus pressure drop relations. Non-linearity is also introduced in Eq. ͑25͒ by the non-linear friction force, which includes Coulomb,static, and viscous components ͓1,18͔ as discussednext. Fig. 9. Piston friction force.7. Modeling and identiﬁcation of friction behavior of friction. For simplicity, we use the Friction affects the dynamics of the electrohy- common memory-less analytical model of frictiondraulic servovalve as well as the dynamics of the ͑without hysteresis͒ as a function of velocity,actuator piston. Friction in the servovalve is gen- given by Eq. ͑26͒ and included in Fig. 9,erally considered to be of Coulomb type acting on ˙the spool of the valve and can in practice be suf- F f ϭF Ϯ x p ϩsgn͑ x p ͒͑ F Ϯ ϩ ͑ F s ϪF Ϯ ͒ e ͑ x p /C s ͒ ͒ . v ˙ ˙ c Ϯ cﬁciently eliminated by using dither signals ͓9͔. ͑26͒The particular friction effect of interest in this sec- The asymmetry of the friction force with respecttion is the model of the frictional forces that ap- to the sign of the velocity observed in the mea-pear in the equations of motion of the actuator sured data is taken into account by taking differentpiston. The literature offers various empirical coefﬁcients in Eq. ͑26͒ for the up and down mo-models applied to speciﬁc hydraulic actuators tions. The strong scatter in the estimation data is͓1,3,18,19͔. In the most general case, friction in typical of friction phenomena.the actuator cylinder is considered to be a functionof the position and velocity of the piston, thechamber pressures ͑the differential pressures when 8. Experimental validation of the modelthe piston is sticking near zero velocity͒, the localoil temperature and also running time. The model described in the previous sections In this study, open-loop and closed-loop tests was implemented in MATLAB/SIMULINK, and base-were performed to identify the friction force on line open-loop and closed-loop experiments werethe actuator piston by assuming it to be a function conducted to validate it. In these experiments, aof velocity. Open-loop tests involved changing the simple load mass is rigidly attached to the pistonset current input to the servovalve while measur- rod. For the models of each of the sections of theing the steady-state piston velocity and cylinder supply and return line hoses, only six modes arechamber pressure responses. The system behaves retained in the modal approximation. This was de-as a velocity source in the open loop. The closed- cided considering the actuator hydraulic naturalloop test involved tracking a 2-Hz 35-mm sine frequency of 172 Hz computed using formulaswave piston position command under P control from linear models ͑see Merritt ͓4͔͒ and selectingwhile measuring acceleration, velocity, and cham- the natural frequency of the highest mode of theber pressures. In both the open-loop and closed- approximation for each section to be close to twiceloop tests, the friction is then estimated using this value. From manufacturer frequency responseNewton’s second law. Fig. 9 shows the result of data for the servovalve, the natural frequency formultiple open-loop tests and a closed-loop test. the servovalve was estimated to be 140 Hz with aThe closed-loop test clearly shows the hysteretic damping ratio of 1.
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340 Beshahwired Ayalew, Bohdan T. Kulakowski / ISA Transactions 44 (2005) 329–343Fig. 10. Supply and return pressure following a step change Fig. 11. Open-loop cylinder chamber pressure responses.in servovalve current. namics introduced by the long sections of hoses are reﬂected in the individual cylinder chamber To measure the supply and return pressure ﬂuc- pressures.tuation, pressure transducers were mounted on the In the open-loop piston velocity response shownsupply and return ports at the servovalve and a in Fig. 12, the velocity signal was obtained by50-mA rated step current was supplied to the ser- low-pass ﬁltering and differentiating the LVDTvovalve in the open-loop. Fig. 10 shows a com- position signal. The ﬁgure shows that the modelparison of the supply and return pressure from does a fairly good job of predicting the piston ve-measurement and simulation. Two observations locity response as well. The differences are againcan be made from this ﬁgure. First, the supply and attributed to uncertainties in the servovalve modelreturn pressure ﬂuctuations contain the fundamen- and also errors in friction estimation, which has atal periods of 25 and 32 ms, respectively. These considerable scatter, as shown in Fig. 9.correspond to fundamental frequencies of about 40and 31 Hz, respectively. The implication of these 9. Predicting system performanceﬂuctuations is that the bandwidth of the actuator islimited by the dynamics of the supply and return It should be recalled that, for the present system,hoses, since the other dynamic elements including the fundamental frequencies of oscillation of thethe servovalve and the actuator have higher cornerfrequencies. Second, the model follows the mea-surement well. In particular, the frequency con-tents match nicely. The discrepancies can be at-tributed to errors in the estimation of effectivebulk moduli for the different hose sections, andthe estimation of manifold pressure drops as wellas unknown but estimated parameters in theadopted simpliﬁed model of the servovalve. The open-loop response of the system can beinvestigated further by looking at the cylinderchamber pressures shown in Fig. 11 and the pistonvelocity response shown in Fig. 12 following thesame step-rated current input to the servovalve. Itcan be seen from Fig. 11 that the simulation pre-dictions of chamber pressures follow the measure-ments and that the supply and return pressure dy- Fig. 12. Open-loop piston velocity response.
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Beshahwired Ayalew, Bohdan T. Kulakowski / ISA Transactions 44 (2005) 329–343 341 Table 1 Summary of step response simulation results. Fundamental fre- quency of oscillation Peak amplitude of in oscillation step response ͑Hz͒ ͑Mpa͒ Lengths of sections II Supply Return Supply Return ͑m͒ pressure pressure pressure pressure 3.048 40 31 1 0.94 2.286 49 39 0.97 0.91 1.524 76 57 0.94 0.88 0.7512 147 115 0.87 0.71 0 N/A N/A 0.0 0.0Fig. 13. Supply and return pressures with 1.524-m-longlines. return line. The settling times are reduced by about 20 ms or about 22% compared to the origi- nal setup with 3.048 m lengths of sections II of thesupply and return line pressures are lower than the supply and return line hoses. The peak amplitudenatural frequencies of both the actuator and the of the oscillation is also slightly reduced, as shownservovalve. This implies that the bandwidth of the in Table 1. The bandwidth of the hydraulic actua-test system is limited by supply and return line tor can therefore be expected to be improved ac-dynamics. The model described in this paper was cordingly. However, the accumulators were notused to predict pressure ﬂuctuations in the supply shown to be effective in ﬁltering the ﬂuctuationsand return lines by running simulations for con- completely.templated modiﬁcations to the layout of the test As a second case, the sections II of the hoses aresystem. In particular, it was desired to see if re- considered to be removed from the system and byducing the lengths of sections II of both the supply so doing the HSM is close-coupled with the actua-and return hoses in the layout ͑Fig. 2͒ produces tor. This consideration would need signiﬁcantsigniﬁcant changes in the supply and return line changes to the physical design of the load frame topressure dynamics at the servovalve. The lengths allow the HSM to be mounted on it together withof sections I were kept unchanged since the physi- the actuator. In the model, the corresponding sub-cal constraints of housing the HPS required about3.048 m of hose lengths for sections I of both thesupply and return lines. The goal of the study wasto see if actually there will be signiﬁcant gainsfrom these measures in terms of increasing the ef-fective bandwidth for the actuator. As a ﬁrst case, the hoses of sections II werereduced to half the original length of 3.048 m. Alength of 1.524 m was considered sufﬁcient to stillplace the HSM unit on the ground while the ac-tuator was mounted on the load frame. Fig. 13shows the supply and return line pressures at theservovalve following the open-loop test describedpreviously with the lengths now at 1.524 m. It can be seen from Fig. 13 that the fundamentalperiod of pressure ﬂuctuation shifts to 13.3 and18.5 ms, respectively, for the supply and returnlines. These correspond to fundamental frequen- Fig. 14. Supply and return pressures with close-coupledcies of 75 Hz on the supply line and 54 Hz on the HSM and shorter lines.
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342 Beshahwired Ayalew, Bohdan T. Kulakowski / ISA Transactions 44 (2005) 329–343system of section II was removed from the sche- without ignoring supply and return line pressurematic in Fig. 6. Fig. 14 shows simulation results variations introduced by transmission line dynam-for this case as well as a hypothetical case for ics.which the hoses are shortened further to 0.7512 m. 3. The model of the overall hydraulic systemIt can be seen that the effect of the accumulators in was modular. Subsystem models can easily beﬁltering out the dynamics of the sections I from changed for desired emphasis observing only thethe supply and return lines is particularly evident input-output causality. For example, a detailedin the close-coupled HSM case. In addition, the model of the servovalve can be used in place ofsteady-state pressure drops in the supply and re- the simpliﬁed model used in this paper, therebyturn line are smaller in this case compared to the improving the predictive power of the overalloriginal setup ͑Fig. 10͒ and to the case with model.1.524-m lengths for sections II ͑Fig. 13͒. The im- Simulation results for some open-loop step re-provements predicted by the case of lengths 1.524 sponses were compared against experimental re-and 0.7512 m seem rather small compared to the sults for the test system. It is felt that the modelcase of close coupling the HSM with the actuator. performs well, given the minimal amount of infor- Table 1 summarizes these results, including one mation it uses to enable a time domain simulation.more additional length for sections II. It can be The use of constant bulk modulus values for theseen that the peak amplitudes of the supply and hose material, which in reality depends on pres-return pressure oscillations are progressively re- sure and frequency, may explain some of the dif-duced as the length of the hoses becomes shorter ference between the simulation and measurement.while the fundamental frequency of the oscilla- A more detailed servovalve model with measuredtions for the step responses becomes larger. In the operating characteristics, instead of the ratedlimit case of 0-m length for sections II, which speciﬁcations, could also improve the accuracy ofmeans the HSM accumulators are close coupled the overall electrohydraulic system model pre-with the servovalve, the oscillations are elimi- sented here.nated. In this case, the accumulators effectively A simulation analysis of the effects of thesuppress the oscillations arising out of the sections lengths of sections of the supply and return linesI of the supply and return line hoses. between the HSM accumulators and the servo- valve was done. It was concluded that reducing10. Conclusions the lengths of these lines progressively reduces the supply and return pressure ﬂuctuation at the servo- In this paper, a model of an electrohydraulic ac- valve during dynamic excursions by the actuator.tuator system focusing on supply and return line The test system bandwidth was correspondinglydynamics was presented. A model of hydraulic improved. The best scenario was shown to be onetransmission lines based on modal approximation where the accumulators were close coupled withtechniques was adopted yielding state space de- the servovalve, thereby employing the accumula-scriptions suitable for time domain simulations. tors effectively in ﬁltering out the effects of theThe interconnection model of the electrohydraulic sections of the supply and return lines between thesystem developed here is an attractive simulation pump ͑HPS͒ and the accumulators.tool for the following reasons. In conclusion, it is recommended that hydraulic 1. The distributed parameter transmission line control system design and analysis include themodel requires only aggregate line parameters modeling approach presented here, to account forsuch as line length, diameter, and effective bulk the effects of supply and return line dynamicsmodulus in state space dynamic models. It there- rather than assuming constant values for the sup-fore enables a quick investigation of line effects in ply and return pressure as has been usually done ina time domain analysis. It was used here to model the literature. It was also demonstrated that thehydraulic hoses with experimental validation. approach could be used to make a quick assess- 2. The time domain simulation, in turn, makes ment of alternative layouts for supply and returnit possible to include nonlinear actuator models. lines in terms of minimizing transmission line dy-This is particularly useful for the study of actuator namics. The method can easily be adopted to othermodels with linear and nonlinear control systems applications of hydraulic machinery and test sys-
15.
Beshahwired Ayalew, Bohdan T. Kulakowski / ISA Transactions 44 (2005) 329–343 343tems where hydraulic oil supply and return lines of domain simulation of ﬂuid transmission lines usingsigniﬁcant length are involved. minimum order state variable models. Fluid Transmis- sion Line Dynamics. ASME special publication, 1983, pp. 78 –97.References ͓16͔ Hsue, C. Y.-Y. and Hullender, D. A., Modal approxi- mations for the ﬂuid dynamics of hydraulic and pneu- ͓1͔ Jelali, M. and Kroll, A., Hydraulic Servo-systems: Modelling, Identiﬁcation and Control ͑Advances in In- matic transmission lines, Fluid Transmission Line Dy- dustrial Control͒. Springer-Verlag, London, 2003. namics, edited by M. E. Franke and T. M. Drzewiecki. ͓2͔ Kugi, A., Non-linear Control Based on Physical Mod- ASME special publication, 1983, pp. 51–77. els, Lecture Notes in Control and Information Sci- ͓17͔ Dransﬁeld, P., Hydraulic Control Systems: Design and ences No. 260. Springer-Verlag, London, 2001. Analysis of their Dynamics, Springer-Verlag, Berlin, ͓3͔ Sohl, G. A. and Bobrow, J. E., Experiments and Simu- 1981. lations on the Nonlinear Control of a Hydraulic Ser- ͓18͔ Alleyne, A. and Liu, R., A simpliﬁed approach to force vosystem. IEEE Trans. Control Syst. Technol. 7͑2͒, control for electro-hydraulic systems. Control Eng. 238 –247 ͑1999͒. Pract. 8, 1347–1356 ͑2000͒. ͓4͔ Merritt, Herbert E., Hydraulic Control Systems. Wiley, ͓19͔ Manhartsgruber, B., Singular perturbation analysis of New York, 1967. an electrohydraulic servo-drive with discontinuous re- ͓5͔ D’Souza, A. F. and Oldenburger, R., Dynamic re- duced dynamics. Proceedings of the 1999 ASME De- sponse of ﬂuid lines. J. Basic Eng. 86͑3͒, 589–598 sign Engineering Technical Conference, Las Vegas, ͑1964͒. Nevada, 1999, pp. 2001–2011. ͓6͔ Goodson, R. E. and Leonard, R. G., A survey of mod- eling techniques for ﬂuid line transients. J. Basic Eng. 94͑2͒, 474 – 482 ͑1972͒. ͓7͔ Viersma, T. J., Analysis, Synthesis and Design of Hy- Beshahwired Ayalew was draulic Servosystems and Pipelines, Studies in Me- born in Goba, Ethiopia in chanical Engineering I. Elsevier Publishing Co., Am- 1975. He earned his BS degree in mechanical engineering sterdam, the Netherlands, 1980. from Addis Ababa University, ͓8͔ Woods, R. L., Hsu, C. H., and Chung, C. H., Compari- Ethiopia and his MS degree, son of theoretical and experimental ﬂuid line re- also in mechanical engineer- sponses with source and load impedance. Fluid Trans- ing, from the Pennsylvania mission Line Dynamics, edited by M. E. Franke and T. State University, USA. He is M. Drzewiecki. ASME Special Publications, New currently a Ph.D. candidate in mechanical engineering at the York, 1983, pp. 1–36. Pennsylvania State University. ͓9͔ Van Schothorst, G., Modeling of Long-Stroke Hydrau- His research interests include lic Servo-Systems for Flight Simulator Motion Control dynamic systems and control, and System Design. Ph.D. thesis, Delft University of energy systems, mechanical design, and vehicle dynamics. Technology, Delft, The Netherlands 1997.͓10͔ Yang, W. C. and Tobler, W. E., Dissipative modal ap- proximation of ﬂuid transmission lines using linear Bohdan T. Kulakowski ob- friction model. J. Dyn. Syst., Meas., Control 113, tained his Ph.D. from the Pol- 152–162 ͑March 1991͒. ish Academy of Sciences in͓11͔ MTS Corporation, Hardware Product Manual, 290.1X, 1972. He has been with Penn 290.2X, and 290.3X Hydraulic Service Manifold, State University since 1979, where he is currently a profes- 1985. sor of mechanical engineering.͓12͔ Blackburn, J. F., Reethof, G., and Shearer, J. L., Fluid Dr. Kulakowski teaches under- Power Control. MIT Press, Cambridge, MA, 1960. graduate and graduate courses͓13͔ McCloy, D. and Martin, H. R., Control of Fluid on engineering measurements, Power: Analysis and Design, second edition. Halsted system dynamics, and controls. Press, Chichester, UK, 1980. He is a co-author of the book ‘‘Dynamic Modeling and Con-͓14͔ Ayalew, B. and Kulakowski, B. T., Transactions of the trol of Engineering Systems.’’ ASME, Journal of Dynamic Systems, Measurement Dr. Kulakowski is a Fellow of ASME. His research interests are in and Control ͑in press͒. vehicle testing, vehicle dynamics, and vehicle interaction with envi-͓15͔ Hullender, D. A., Woods, R. L., and Hsu, C.-H., Time ronment.
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