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S adhan a Vol. 31, Part 5, October 2006, pp. 543556. Printed in IndiaEffect of bulk modulus on performance of a hydrostatictransmission control systemALI VOLKAN AKKAYAYildiz Technical University, Mechanical Engineering Department, 34349,Besiktas, Istanbul, Turkeye-mail: .trMS received 9 September 2005; revised 20 February 2006Abstract.In this paper, we examine the performance of PID (proportionalintegral derivative) and fuzzy controllers on the angular velocity of a hydrostatictransmission system by means of Matlab-Simulink. A very novel aspect is that itincludes the analysis of the effect of bulk modulus on system control. Simulationresultsdemonstratesthatbulkmodulusshouldbeconsideredasavariableparameterto obtain a more realistic model. Additionally, a PID controller is insufficient inpresence of variable bulk modulus, whereas a fuzzy controller provides robustangular velocity control.Keywords.Hydrostatic transmission; bulk modulus; PID (proportional integralderivative); fuzzy controller.1. IntroductionHydrostatic transmission (HST) systems are widely recognized as an excellent means ofpower transmission when variable output velocity is required in engineering applications,especially in field of manufacturing, automation and heavy duty vehicles. They offer fastresponse,maintainprecisevelocityundervaryingloadsandallowimprovedenergyefficiencyand power variability (Dasgupta 2000; Kugi et al 2000). A basic hydrostatic transmission isan entire hydraulic system. Generally, it contains a variable-displacement pump driven byan induction motor, a fixed or variable displacement motor, and all required controls in onesimple package. By regulating the displacement of the pump and/or motor, a continuouslyvariable velocity can be achieved (Wu et al 2004).Manufacturers and researchers continue to improve the performance and reduce the costof hydrostatic systems. Especially, modelling and control studies of hydrostatic transmissionsystemshaveattractedconsiderableattentioninrecentdecades.Somestudiesonthistopiccanbe found in the literature (Huhtala 1996; Manring & Luecke 1998; Dasgupta 2000; Kugi et al2000;Dasguptaetal2005).Variousrotationalvelocitycontrolalgorithmsforhydrostaticsys-temsaredevelopedandappliedbyLennevi&Palmberg(1995),Lee&Wu(1996),Piotrowska(2003). All these designs use the bulk modulus as a fixed value through a wide pressurerange. However, in practice, the bulk modulus is an essential part of dynamic behaviours of543544Ali Volkan Akkayathe hydraulic systems (McCloy & Martin 1980; Watton 1989). Due to temperature variationsand air entrapment, the bulk modulus may vary during the operation of the hydraulic sys-tems (Eryilmaz & Wilson 2001). A little entrapped air is enough to reduce the bulk modulussignificantly (Merrit 1967; Tan & Sepehri 2002). Moreover, system pressure plays an impor-tant role on the bulk modulus value (Wu et al 2004). Some effects of instabilities induced bybulk modulus nonlinearities such as pressure oscillations in the form of pressure waves canbe detrimental to operation of hydraulic systems and may result in reduced component life,loss of performance, disturbance in control systems, reduced efficiency and increased acous-tic noise. In spite of these adverse effects, there are few studies about bulk modulus withinhydrostatic transmission systems. Yu et al (1994) developed an on-line parameter identifica-tion method, determining the effective oil bulk modulus within an actual hydraulic system bymeasuring the propagation of a pressure wave through a long pipe. Marning (1997) devel-oped a linear relation between oil bulk modulus and pressure for a HST system. However, todate, nothing has appeared in the literature that addresses the effect of bulk modulus dynam-ics incorporated into a hydrostatic transmission model on control design process of the HSTsystem. In fact, models of hydrostatic transmission systems with variable bulk modulus havemore complex dynamic behaviour than normal. Moreover, having servo control of the sys-tem, dynamics of bulk modulus becomes more important because the closed-loop systemitself raises the issue of stability.Bulk modulus cannot be determined directly and hence needs to be estimated. Based onthis estimation, corrective actions may be taken in control applications for HST systems. Thecomplex dynamic interactions between variable bulk modulus and the control action is inves-tigated using modelling and simulation analysis. Simulation tests are particularly beneficialwhen preparing a model of a real system is complicated and time-consuming. A servo hydro-statictransmissioncontrolsystemisagoodexampleforthisissue.Thedeterminationofstaticand dynamic behaviours using simulation tests is possible without expensive prototypes. Thesimulation also makes a shorter product-designing cycle possible.This study focuses on control performance of a typical HST system. A nonlinear modelof the system is studied by means of Matlab-Simulink software. The system model is acombination of each individual component model consisting of pump, valve, hydraulic hoseand hydraulic motor. In addition, the variable bulk modulus is presented to describe theeffects of this phenomenon on system dynamics and control algorithm. For this purpose, twodifferent hydraulic hose Simulink models are incorporated separately into the system model.In addition, the models are utilized in the control design process. The control of the angularvelocity of the hydraulic motor coupled with load is achieved by PID (proportional integralderivative) and fuzzy types of controller. In the first model, bulk modulus is assumed to havea fixed value and angular velocity control of the HST system is carried out with the classicalPID control algorithm. In the second model, bulk modulus is defined as a variable parameterdependingonentrappedairandsystempressure.Thisnewmodelisappliedonvelocitycontrolof the HST system under the same PID control parameters. In the following, fuzzy controllerisimplementedinthisnewmodelinordertojudgeitscapabilityagainstvariablebulkmodulusnonlinearity. The simulation results of two control approaches are then compared to analysethe differences in the performance of the HST system in terms of bulk modulus dynamics.2. Mathematical modelThe physical model of the HST system considered for this study is shown in figure 1. Thevariable displacement pump driven by an induction motor supplies hydraulic power to a fixedEffect of bulk modulus on performance of a transmission control system545Figure 1.Hydrostatic transmission system.displacementhydraulicmotorfordrivingload.Toprotectthesystemfromexcessivepressure,a pressure relief valve is used.From a research objective point of view, the descriptions of a system mathematical modelshould be as simple as possible. At the same time, it must include important characteristics ofthe real event. One way to understand the system is to separate the system into componentsfor the purpose of modelling. Using a fundamental knowledge of physics, for instance themomentequilibriumandcontinuityequation,amodelthatrepresentsthedynamicsbehaviourofeachcomponentcanbederivedatthecomponentlevels.Havingunderstoodeachindividualcomponent,wecanunderstandtheoverallsystembyinterconnectingthecomponentstogetherto obtain an overall system model (Prasetiawan 2001). In this paper, the model of eachcomponent used for the HST system is developed using earlier methods (Jedrzykiewicz et al1997, 1998).2.1 Variable-displacement pumpIt is assumed that the angular velocity of the prime mover (induction motor) is constant.Therefore, angular velocity of the pump shaft is constant. Pump flow rate can be adjustedwith variable displacement via the swashplate displacement angle and can be given asQp= kpvp,(1)where, Qpis pump flow rate (m3/s), is displacement angle of swashplate (), kpis pumpcoefficient (m3/s), vpis pump volumetric efficiency () which is assumed not to depend onpump rotation angle.2.2 Pressure relief valveTo simplify, pressure relief valve dynamics is not taken into consideration. Therefore, twoequation as below are given for passing flow rate through pressure relief valve (m3/s) in thestate of opening and closing.Qv= kv(P Pv), if P Pv,(2)Qv= 0, if P Pv,(3)546Ali Volkan Akkayawhere, kvis slope coefficient of valve static characteristic (m5/Ns), P is system pressure (Pa)and Pvis valve opening pressure (Pa).2.3 Hydraulic hoseAs in traditional modelling, the pressurized hose that connects the pump to the motors ismodelled as volume with a fixed bulk modulus in this section. Variable bulk modulus arediscussed in the following subsection.The fluid compressibility relation can be given as in (4). Equation (5) provides the pressurevaluefromagivenflowrate.Itisassumedthatpressuredropinthehydraulichoseisnegligible.Qc= (V/)(dP/dt),(4)(dP/dt) = (/V)Qc,(5)where, Qcis flow rate deal with fluid compressibility (m3/s), V is the fluid volume (m3)subjected to pressure effect, is fixed bulk modulus (Pa).2.3a VariablebulkmodulusFluidisanimportantelementofhydrostaticsystemsandenablespower transmission, hence it can influence the dynamic behaviours of the system and thecontrol system. The bulk modulus of non-aerated hydraulic oil depends on temperature andpressure, for mineral oils with additives its value ranges from 1200 to 2000MPa. Moreover,system pressure and entrapped air affect the bulk modulus value. If a hydraulic hose is usedratherthanasteelpipe,thebulkmodulusofthissectionmaybeconsiderablyreduced.Owingto these reasons, the parameters influencing bulk modulus value must be included in the HSTmodel for more accurate system dynamics.The equation which gives the variable bulk modulus of fluid-air mixture in a flexiblecontainer is as follows (McCloy & Martin 1980):1v=1f+1h+VaVt1a,(6)where, the subcripts a, f and h refer to air, fluid, and hose respectively. It is assumed that theinitial total volume Vt= Vf+Va, and that f? a. Thus bulk modulus will be less than anyf, h, or Vt/Vaa. The bulk modulus of the fluid fis obtained from the manufacturersdata. The adiabatic bulk modulus used for air is (Cp/Cv)P = 14P. With these assumptions,(6) can be rewritten as in,1v=1f+1h+s14 P,(7)where, s is entrapped air percent in the total volume (s = Va/Vt).2.4 Hydraulic motor and loadFlow rate used in the hydraulic motor (m3/s) can be written as inQm= km/vm,(8)where, kmis hydraulic motor coefficient (m3), is angular velocity of hydraulic motor (1/s)andvmisvolumetricefficiencyofthemotor().Itisassumedthathydraulicmotorefficiencydoes not depend on its shaft rotation angle. Hydraulic motor torque (Nm) can be written as,Mm= kmt?Pmm,(9)Effect of bulk modulus on performance of a transmission control system547where, kmtis motor torque coefficient (m3), ?P is pressure drop in hydraulic motor (Pa)and mmis mechanical efficiency of hydraulic the motor (). The torque produced in thehydraulic motor (Nm) is equal to the sum of the moments from the motor loads and can begiven as,Mm= MI+ MB+ Mo,(10)where,MI,MBandMoarethemomentsresultingfromloadinertia,frictionforceandmachineoperation respectively. These moments can be denoted asMm= Im(d/dt) + B + Mo,(11)where, Imis the inertia of the hydraulic motor shaft (Nms2), B is viscous friction coefficientof motor and its shaft (Ns/m), and is angular velocity of motor shaft (1/s). Equation (11)can be used to determine the angular velocity of the hydraulic motor shaft. This equation isrearranged for angular velocity asd/dt = (Mm B Mo)/Im.(12)2.5 Hydrostatic transmission systemThe fundamental mathematical models of the system components and phenomena occurringin hydrostatic systems are conveniently combined to obtain the overall HST system model.Accordingly, a hydrostatic transmission is modelled as a lumped system. In the developmentof the dynamic model of the system, it is assumed that static and dynamic features of thetransmissiondonotdependuponthedirectionofhydraulicmotorrotationandthetransmissionis a state of thermal balance. Leakage flows in pump and motor are not taken into accountduring the modelling.The mathematical model of the HST system consists of two equations as below:equality of flow rate:Qp= Qm+ Qc+ Qv,(13)moment:Mm= MI+ MB+ Mo.(14)Using (5) and (12), the following are then obtained,dP/dt = (/V)(Qp Qm Qv),(15)d/dt = (Mm B Mo)/Im.(16)A commonly available general purpose simulation package Matlab/Simulink is used tosolve the nonlinear equations. The Simulink model based on the component mathematicalmodels of HST system is given in figure 2. The component models can be easily modifiedin accordance width specific constructions. Accordingly, when bulk modulus is rebuilt in thehydraulic hose component with regard to (7), the second model can be generated.548Ali Volkan AkkayaFigure 2.Simulink model of hydrostatic transmission system.3. Control applicationsMost publications related to the HST control are related to the speed control of the hydraulicmotorconnectedtotheload.Inordertoachievethisgoal,differentclosed-loopcontroldesignstrategiescanbeused.However,Lee&Wu(1996)showedthatusingonlypumpdisplacementto regulate load speed is the most effective of all the methods they tested. In addition, Re et al(1996) concluded that the sole use of pump displacement actuation to control one load speedof a system with variable-displacement pump and motor is the most efficient, and should bealways preferred whenever possible. For this reason, in the HST systems being considered inthis study, the output angular velocity is controlled by the flow rate supplied to the hydraulicmotor, and this flowrate is adjusted by the swashplate angle of the variable-displacementpump. Swashplate dynamics are not taken into consideration in the control application inthis study for the sake of simplicity. In addition, the swashplate control system usually hasfaster dynamics than the rest of the system, and therefore neglecting its dynamics is justified(Watton 1989).Topreciselycontroltheangularvelocityofthehydraulicmotorinhydrostatictransmissioncontrol systems, an appropriate controller must be designed in advance. In industrial appli-cations, classical control methods such as PI, PID are being used for velocity control of HSTsystems.ItiscrucialtodeterminecontrollerparametersaccuratelybecausePIDcontrolmeth-ods have linear characteristics. They are sometimes insufficient to overcome nonlinearitieswhich exist in the nature of the HST systems for high precision applications (Tikkanen et al1995; Prasetiawan 2001). In particular, the bulk modulus ought to be regarded as a source ofsignificant nonlinearity for this type of controller. Thus, the controller has to be very robustto account for such wide variation. Use of knowledge-based systems in process control isincreasing, especially in the fields of fuzzy control (Tanaka 1996). Unlike classical controlmethods, the fuzzy controller is designed with linguistic terms to cope with the nonlineari-ties. Therefore, this control method is also applied to judge its capacity to reduce the adverseeffect of variable bulk modulus.3.1 PID controlThe structure of the PID control algorithm used for the angular velocity control of HSTsystem is given in (17) and (18) below. Ziegler-Nichols method is implemented for tuningcontrol parameters, such as proportional gain (Kp), derivative time constant (d) and integraltime constant (i) (Ogata 1990). After fine adjustments, the optimal control parameters areEffect of bulk modulus on performance of a transmission control system549Figure 3.Simulink model of HST system for PID control.determined for the reference angular velocity. Figure 3 shows the Simulink model of thePID-controlled HST system.uv(t) = Kpe(t) + Kpdde(t)dt+Kpi?e(t) dt,(17)e(t) = r .(18)3.2 Fuzzy controlFuzzy logic has come a long way since it was first presented to technical society, whenZadeh (1965) first published his seminal work. Since then, the subject has been the focusof many independent research investigations. The attention currently being paid to fuzzylogic is most likely the result of present popular consumer products employing fuzzy logic.The advantages of this method are its applicability to nonlinear systems, simplicity, goodperformance and robust character. These days, this method is being applied to engineer-ing control systems such as robot control, flight control, motor control and power systemssuccessfully.In fuzzy control, linguistic descriptions of human expertise in controlling a process arerepresented as fuzzy rules or relations. This knowledge base is used by an inference mecha-nism, in conjunction with some knowledge of the states of the process in order to determinecontrol actions. Unlike the conventional controller, there are three procedures involved in theimplementation of a fuzzy controller: fuzzification of inputs, and fuzzy inference based onthe knowledge and the defuzzification of the rule-based control signal. The structure of thefuzzy controller is seen in figure 4.An applied fuzzy controller needs two input signals. These signals are error (e) and deriva-tive of error (de) respectively. The usual overlapped triangular fuzzy membership functionsare used for two input signals (e,de/dt) and the output signal (u). Figure 5 shows the struc-ture of the membership functions of input and output signals. Input signals are transformedat intervals of 1,1 by scaling factors which are Ge and Gde.In the fuzzification process, all input signals are expressed as linguistic values which are:NB negative big, NM negative medium, NS-negative small, ZE-zero, PS-positive small,PM-positive medium, PB-positive big. After input signals are converted to fuzzy linguisticvariables, these variables are sent to the inference mechanism to create output signals.550Ali Volkan AkkayaFigure 4.Structure of a fuzzy controller.Theinferenceprocessconsistsoffortyninerulesdrivenbythelinguisticvaluesoftheinputsignals. These fuzzy rules written as a rule base are shown given in table 1. The rule base isdeveloped by heuristics with error in motor angular velocity and derivation of error in thisvelocity. For instance, one of the possible rules is: IF e = PS and de = NB THAN u = NM.Thisrulecanbeexplainedasinthefollowing:Iftheerrorissmall,angularvelocityofhydraulicmotor is around the reference velocity. Significantly big negative value of derivation of errorshows that the motor velocity is rapidly approaching the reference position. Consequently,controller output should be negative middle to prevent overshoot and to create a brake effect.Asarule-inferencemethod,theMamdaniMethodisselectedbecauseofitsgeneralacceptance(Tanaka 1996).Defuzzification transforms the control linguistic variables into the exact control output. Indefuzzification, the method of centre of gravity is implemented (Tanaka 1996), asu =n?i=1WiBi/n?i=1Wi(19)Figure 5. Triangular fuzzy member-shipfunctions,(a)einputsignal,(b)deinput signal, (c) u output signals.Effect of bulk modulus on performance of a transmission control system551Table 1. Rule base for fuzzy control.deeNBNMNSZEPSPMPBNBNBNBNBNMNMNSZENMNBNBNMNSNSZEPSNSNBNMNSNSZEPSPMZENMNSNSZEPSPSPMPSNMNSZEPSPSPMPBPMNSZEPSPSPMPBPBPBZEPSPMPMPBPBPBwhere, u is the output signal of the fuzzy controller, Wiis the degree of the firing of the ithrule,Biisthecentroidoftheconsequentfuzzysubsetofithrule.Realvaluesofcontroloutputsignal (uv) are determined by the scaling factor of Guv. As a result, the fuzzy controllerbuilt-in Fuzzy Logic Toolbox of the Matlab program has been added to the Simulink modelof hydrostatic transmission system for simulation analysis (figure 6).4. Simulation results and discussionThe validity of the influence of bulk modulus dynamics on HST control system has beentestedincomputersimulations.Inordertocarryoutsimulation,somephysicalandsimulationparameters corresponding to HST system are taken from work of McCloy & Martin (1980)and Jedrzykiewicz et al (1997, 1998), and other control parameters are as given in table 2.OpenlooppressureandangularvelocityresponsesoftheHSTsystemaregiveninfigures 7aand b respectively, under fixed bulk modulus and variable bulk modulus. Comparing the sim-ulation results shows that the model including variable bulk modulus shows flexible dynam-ics and decreasing system stiffness (figure 7a). Moreover, a degree of aeration less than 1%brings about considerable changes of velocity and pressure responses because the aeration ofthe working fluid results in decrease of fluid bulk modulus and changes its characteristics asa function of pressure.The dynamic behaviour patterns in figure 8 are obtained from PID control. It is observedfrom figure 8a that the system model including the fixed bulk modulus is in good agreementwith reference velocity. In contrast to this, the simulation result of the model with variablebulk modulus show oscillations in the transient regime. The reason is that the bulk modulusFigure 6.Simulink model of HST system for fuzzy controller.552Ali Volkan AkkayaTable 2. Physical and simulation parameters.Moments resulting from machine operationMo(Nm)150Viscous friction coefficientB (Ns/m)15Displacement angle of swashplatea ()16 for open-loopInertia of hydraulic motor shaftIm(Nms2)004Opening pressure value of valvePv(Pa)12 106Fluid volume subjected to pressure effectV (m3)14145 104Pump coefficientkp(m3/s)2688 105Hydraulic motor coefficientkm(m3)3979 105Motor torque coefficientkmt(m3)3979 105Slope coefficient of relief valvekv(m5/Ns)02 109Pump volumetric efficiencyvp(%)97Mechanical efficiency of hydraulic motormm(%)95Bulk modulus of fluidf(Pa)149 109Bulk modulus of hoseh(Pa)47 107Degree of entrapped airs (%)05Reference angular velocityr(1/s)7Proportional gain constantKp()70Derivative time constantd()00002875Integral time constanti()000115Scaling factor of errorGe ()01Scaling factor of derivative of errorGde ()00001Scaling factor of control signalGu ()1500Figure7. OpenloopresponsesofHSTsystemunderfixedandvariable(a)systempressure,(b)angularvelocity.Effect of bulk modulus on performance of a transmission control system553Figure 8.PID control responses of HST system for fixed and variable bulk modulus (a) angularvelocity, (b) system pressure, (c) variable bulk modulus.value becomes lower than that of the fixed one (figure 8b) and changes with system pressure(figure 8c). The same characteristic is seen for increasing load moment at 003s. The aboveresults indicate that with change in the bulk modulus, peak pressure as well as fluctuation ofthe fluid pressure increase in closed loop control applications. This increases the settling timeof the systems responses. This may cause to failure of the stability of the system. Therefore,it is necessary to revise the control parameters or apply a more robust controller in terms ofvariable bulk modulus.Fuzzy and PID control responses of HST system under the variable bulk modulus aredepicted in figure 9. Simulation results of fuzzy controller show good performance comparedwith PID controller in tracking referenced velocity (figure 9a). In addition, fuzzy controllerrejects the effect of loading on pressure dynamics (figure 9b). Figure 9c indicates that fuzzycontroller minimizes the fluctuations of bulk modulus value. This is the reason why thefuzzy response is more robust. Figure 9d shows that PID controller can work safely up to200Hz,whereasafuzzycontrolleriscapableofenduring400Hz.Therefore,afuzzycontroller554Ali Volkan AkkayaFigure 9.Fuzzy and PID control responses of HST system under variable bulk modulus. (a) Angularvelocity, (b) system pressure, (c) variable bulk modulus, (d) velocity bode diagram.increases the stability conditions of a hydraulic transmission system in presence of variablebulk nonlinearity.5. ConclusionThe effects of bulk modulus nonlinearity on the performance of a hydrostatic transmissioncontrol system have been analysed through system modelling and simulation. This study hasEffect of bulk modulus on performance of a transmission control system555demonstrated that omitting the bulk modulus dynamics in hydrostatic transmission controlsystems may lead to major errors in system response and have implications on the safety ofoperation. Therefore, bulk modulus should be considered as a variable parameter to obtaina more realistic overall model and to determine more accurate control parameters in PIDcontroller. Analysis including bulk modulus dynamics in an HST-control system model withthis control design feature has not been described in the literature to date. Therefore, it maybe useful for early design of an HST system used for PID control application. In addition, itis clearly seen that a fuzzy controller has the capability of eliminating the adverse effects ofvariablebulkmodulus.Thiswillalsobenefitthecontroldesignprocessintermsofdevelopinga robust controller. For future research, model development will be expanded to includeswashplate dynamics, valve dynamics and more complex flow and torque models of thepump and the motor. Furthermore, an adaptive control method will be applied for changeablevelocity reference and load moment.List of symbolsBviscous friction coefficient of motor and its shaft Nms;Bicentroid of the consequent fuzzy subset of ith rule;HSThydrostatic transmission;Iminertia of hydraulic motor shaft Nms2;kppump coefficient m3/s;kmhydraulic motor coefficient m3;kmtmotor torque coefficient m3;kvslope coefficient of static characteristic of relief valve m5/Ns;MBmoments resulted from friction force Nm;MImoments resulted from load inertia Nm;Mmhydraulic motor torque Nm;Momoments resulted from machine operation Nm;Psystem pressure Pa;Pvopening pressure value of valve Pa;Qcflow rate deal with compressibility m3/s;Qmflow rate used in hydraulic motor m3/s;Qpflow rate of pump m3/s;Qvpassing flow rate through relief valve m3/s;uvoutput signal of fuzzy controller;Vfluid volume subjected to pressure effect m3;Widegree of firing of ith fuzzy rule;displacement angle of swashplate ;bulk modulus Pa;mmmechanical efficiency of hydraulic motor ;vppump volumetric efficiency ;vmvolumetric efficiency of motor ;angular velocity of motor 1/s;?Pmpressure drop in hydraulic motor Pa.556Ali Volkan AkkayaReferencesDasgupta K 2000 Analysis of a hydrostatic transmission system using low speed high torque motor.Mech. Mach. Theory 35: 14811499Dasgupta K, Chattapadhyay A, Mondal S K 2005 Selection of fire-resistant hydraulic fluids throughsystem modelling and simulation. Simul. Model. Pract. Theory 13: 120Eryilmaz B, Wilson B H 2001 Improved tracking control of hydraulic systems. Trans. ASME: J.
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