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Mathematical Modeling and Analysis of a Hydrostatic Drive Train★
8th Vienna International Conference on Mathematical Modelling 8th Vienna International Conference on Mathematical Modelling 8th Vienna Vienna18International Conference on Mathematical Mathematical Modelling February - 20, 2015. Vienna University of Technology, Vienna, 8th Conference on Modelling February 18International - 20, 2015. Vienna University of Technology, Available online at Vienna, www.sciencedirect.com February Austria February 18 18 -- 20, 20, 2015. 2015. Vienna Vienna University University of of Technology, Technology, Vienna, Vienna, Austria Austria Austria
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IFAC-PapersOnLine 48-1 (2015) 508–513
Mathematical Modeling and Analysis Mathematical Modeling and Analysis Mathematical Modeling and Analysis ⋆⋆ Mathematical Modeling and Analysis Hydrostatic Drive Train ⋆ Hydrostatic Drive Train Hydrostatic Drive Train Hydrostatic Drive Train ⋆
of of of of
a a a a
∗ ∗ ∗ P. Zeman u ller ∗ , W. Kemmetm¨ ∗ , A. Kugi ∗ P. Zeman u ller ∗ , W. Kemmetm¨ ∗ , A. Kugi ∗ ∗ , W. Kemmetm¨ ∗ , A. Kugi ∗ P. Zeman u ller P. Zeman , W. Kemmetm¨ uller , A. Kugi ∗ ∗ Automation and Control Institute, Vienna University of Technology, ∗ Automation and Control Institute, Vienna University of Technology, ∗ Automation and Control Institute, Vienna Vienna, [email protected]) Automation andAustria Control(e-mail: Institute, Vienna University University of of Technology, Technology, Vienna, Austria (e-mail: [email protected]) Vienna, Austria (e-mail: [email protected]) Vienna, Austria (e-mail: [email protected])
Abstract: Hydrostatic Hydrostatic drives drives constitute constitute an an advantageous advantageous alternative alternative to to conventional conventional mechanimechaniAbstract: Abstract: Hydrostatic drives constitute an advantageous alternative to conventional mechanical gears with a fixed transmission ratio. The hydraulic part of the system decouples the input Abstract: Hydrostatic drives constitute an advantageous alternative to conventional mechanical gears with a transmission ratio. The hydraulic part of system decouples the input cal gears with a fixed fixed transmission ratio. The as hydraulic part of the thefor system decouples the input and output speeds of the drive and may serve an energy storage recuperation as well. For cal gears with a fixed transmission ratio. The hydraulic part of the system decouples the input and output speeds of the drive and may serve as an energy storage for recuperation as well. For and output speeds of the drive and may serve as an energy storage for recuperation as well. For this reason, hydrostatic drives are a well-established concept, in particular in the field of mobile and output speeds of the drive and may serve as an energy storage for recuperation as well. For this reason, hydrostatic drives are a well-established concept, in particular in the field of mobile this reason, hydrostatic a in in of working machines such drives as lift lift are trucks or excavators. excavators. concept, In this this work, work, a mathematical mathematical model for aa this reason, hydrostatic drives are a well-established well-established concept, in particular particular in the the field field of mobile mobile working machines such as trucks or In a model for working machines such as lift trucks or excavators. In this work, a mathematical model for hydrostatic drive train of a passenger vehicle is developed. The main focus of the contribution working machines such asalift trucks or excavators. In this The work, a mathematical model for aa hydrostatic drive vehicle is main the hydrostatic drive train train of of modeling a passenger passenger vehicle is developed. developed. The displacement main focus focus of ofaxial the contribution contribution lies on the mathematical of the self-supplied variable piston units hydrostatic drive train of a passenger vehicle is developed. The main focus of the contribution lies on the mathematical modeling of the self-supplied variable displacement axial piston units lies on the mathematical modeling of the self-supplied variable displacement axial piston units of the hydraulic system since they basically determine the performance of the overall torque lies on the mathematical modeling of the self-supplied variable displacement axial piston units of the hydraulic system since they basically determine the performance of the overall torque of the system since basically determine the of overall control. Starting from from a complex complex high-order dynamic model, systematic order order reduction of the the of the hydraulic hydraulic system since they they basicallydynamic determine the aaperformance performance of the thereduction overall torque torque control. Starting a high-order model, systematic of control. Starting from a complex high-order dynamic model, a systematic order reduction the nonlinear model is accomplished by means of singular perturbation theory. The accuracy of control. Starting from a complex by high-order dynamic model, a systematic reduction the nonlinear model is accomplished means of singular perturbation theory.order The accuracy of the nonlinear model accomplished by singular The derived models models isis evaluated by measurement measurement results onperturbation an industrial industrialtheory. test bench. bench. nonlinear modelis isevaluated accomplished by means means of of results singularon perturbation theory. The accuracy accuracy of of the the derived by an test derived models is evaluated by measurement results on an industrial test bench. derived models is evaluated by measurement results on an industrial test bench. © 2015, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. Keywords: serial serial hydraulic hydraulic hybrid hybrid vehicle, vehicle, infinitely infinitely variable variable transmission, transmission, variable variable Keywords: Keywords: serial hydraulic hybridelectro-hydraulic vehicle, infinitely infinitelysystems, variablesingular transmission, variablemethod displacement axial piston pump, perturbation Keywords: serial hydraulic hybrid vehicle, variable transmission, variable displacement axial piston pump, electro-hydraulic systems, singular perturbation method displacement axial piston pump, electro-hydraulic systems, singular perturbation method displacement axial piston pump, electro-hydraulic systems, singular perturbation method 1. INTRODUCTION INTRODUCTION underlying dynamics are not accounted for, neither in the 1. underlying dynamics are not accounted for, neither in the 1. INTRODUCTION INTRODUCTION underlying dynamics are not accounted for, neither in the model nor in the controller design. For the design of highly 1. underlying dynamics are not accounted for, neither in the model nor in the controller design. For the design of highly model nor in the controller design. For the design of highly dynamic torque control strategies, this simplification is not In a typical setup of a hydrostatic drive train, the energy model nor in the controller design. For the design of highly dynamic torque control strategies, this simplification is not In typical setup of hydrostatic drive train, the dynamic torque control strategies, this simplification is not In a typical setup of a hydrostatic drive train, the energy energy and yields aa limited control performance. For delivered by setup an internal internal combustion engine (ICE) on the the feasible dynamic torque control strategies, this simplification is this not In aa typical of aa hydrostatic drive train, the energy feasible and yields limited control performance. For this delivered by an combustion engine (ICE) on feasible and yields a limited control performance. For this delivered by an internal combustion engine (ICE) on the reason, the present contribution deals with the mathematprimary side of the system is transmitted across a hyfeasible and yields a limited control performance. For this delivered by an internal combustion engine (ICE) on the reason, the present contribution deals with the mathematprimary side system is across aa hythe contribution deals with the primarycircuit side of oftothe the system isontransmitted transmitted across hy- reason, ical modeling of an SHHV, where the essential nonlinear draulic the output the secondary side. reason, the present present contribution deals with the mathematmathematprimary side of the system is transmitted across a The hyical modeling of an SHHV, where the essential nonlinear draulic circuit to the output on the secondary side. The ical modeling of an SHHV, where the essential nonlinear draulic circuit circuit to the the output on on (IVT) the secondary secondary side. The dynamics of the axial piston units are taken into account. infinitely variable transmission allows to run the ical modeling of an SHHV, where the essential nonlinear draulic to output the side. The dynamics of the axial piston units are taken into account. infinitely variable transmission (IVT) allows to run the dynamics of the axial piston units are taken into account. infinitely variable transmission (IVT) allows to run the The mathematical modeling of variable axial piston pumps ICE in a set point which is optimal from an energetic point dynamics of the axial piston units are taken into account. infinitely variable transmission (IVT) allows to run the The mathematical modeling of variable axial piston pumps ICE in a set point which is optimal from an energetic point The mathematical modeling of variable axial piston pumps ICE in a set point which is optimal from an energetic point has been studied extensively in Ivantysyn and Ivantysynof view. Furthermore, recuperation of braking energy can The been mathematical modeling ofinvariable axialand piston pumps ICE in a Furthermore, set point which is optimal from an energetic point has studied extensively Ivantysyn Ivantysynof view. recuperation of braking energy can has been studied extensively in Ivantysyn and Ivantysynof view. Furthermore, recuperation of braking energy can ova (1993). Very detailed models are given there which be done in a very efficient way by loading a hydraulic achas been studied extensively in Ivantysyn and Ivantysynof view. Furthermore, recuperation of braking energy can ova (1993). Very detailed models are given there which be done in a very efficient way by loading a hydraulic ac(1993). Very detailed models are given there which be done done in in aasee, very efficient way by(2013). loadingCurrent a hydraulic hydraulic ac- ova describe, e.g., the pressure dynamics in the cylinder barrel, cumulator, e.g., Erkkil¨ a et al. research ova (1993). Very detailed models are given there which be very efficient way by loading a acdescribe, e.g., the pressure dynamics in the cylinder barrel, cumulator, see, e.g., Erkkil¨ a et al. (2013). Current research describe, e.g., the pressure dynamics in the cylinder barrel, cumulator, see, e.g., Erkkil¨ a et al. (2013). Current research the actual volume flow and the internal force distribution. is concerned with the exploitation of these advantages for describe, e.g., the pressure dynamics in the cylinder barrel, cumulator, see, e.g., Erkkil¨ a et al. (2013). Current research the volume flow and internal force is with exploitation of these advantages for the actual actual volume flowuseful and the the internal force distribution. distribution. is concerned concerned with the the exploitation ofdrive thesetrain advantages for the These models are not for the considered SHHV since passenger vehicles. The hydrostaticof represents actual volume flow and the internal force distribution. is concerned with the exploitation these advantages for These models are not useful for the considered SHHV since passenger vehicles. The hydrostatic drive train represents These models are not useful for the considered SHHV since passenger vehicles. The hydrostatic drive train represents these effects can be well described by mean-value models an alternative concept for automotive manufacturers comThese models are not useful for the considered SHHV since passenger vehicles. Thefor hydrostatic drive train represents these effects can be well described by mean-value models an alternative concept automotive manufacturers comthese effects can be well described by mean-value models an alternative concept for automotive manufacturers comwith an adequate accuracy, see, e.g., Manring (2005). In pared to conventional hybrid drives with electric engines. these effects can be well described by mean-value models an alternative concept for automotive manufacturers comwith an adequate accuracy, see, e.g., Manring (2005). In pared conventional hybrid drives with electric with an adequate accuracy, see, e.g., Manring (2005). In pared to to conventional hybrid drives with(SHHVs) electric engines. engines. Kugi et al. (2000), an online simulator for the swash plate These serial hydraulic hybrid vehicles require with an adequate accuracy, see, e.g., Manring (2005). In pared to conventional hybrid drives with electric engines. Kugi et al. (2000), an online simulator for the swash plate These serial hydraulic hybrid vehicles (SHHVs) require Kugi et et al. (2000), displacement an online online simulator simulator forpresented, the swash swashwhich plate These serial hydraulic hybrid vehicles (SHHVs) require angle of a variable pump is sophisticated control strategies in order to translate the Kugi al. (2000), an for the plate These serial hydraulic hybrid vehicles (SHHVs) require angle of a variable displacement pump is presented, which sophisticated control strategies in to the of aa variable displacement pump is which sophisticated control strategies in aorder order to translate translate the takes into account the essential dynamics of the swash torque requestcontrol of the the strategies driver into intoin suitable actuationthe of angle angle of variable displacement pump is presented, presented, which sophisticated to translate takes into account the essential dynamics of the swash torque request of driver aaorder suitable actuation of takes into account the essential dynamics of the swash torque request of the driver into suitable actuation of plate mechanism. The dynamic mathematical model of the system by simultaneously fulfilling performance and takes into account the essential dynamics of the swash torque request of the driver into a suitable actuation of plate mechanism. The dynamic mathematical model of the system by simultaneously fulfilling performance and plate mechanism. The dynamic mathematical model the system by simultaneously fulfilling performance and a variable axial piston pump, consisting of the equations energy efficiency demands. The The fulfilling basis for for performance the controller controllerand de- aplate mechanism. The pump, dynamic mathematical model of of the system by simultaneously variable axial piston consisting of the equations energy efficiency demands. basis the dea variable axial piston pump, consisting of the equations energy efficiency demands. The basis basis forwhich the controller controller de- aof variable motion of the pump, the swash plate mechanism and sign are suitable mathematical models are tailored axial piston pump, consisting of the equations energy efficiency demands. The for the deof motion of the pump, the swash plate mechanism and sign are suitable models motion of pump, the plate mechanism and sign areconsidered suitable mathematical mathematical models which which are are tailored tailored of the actuating valve, is presented Fuchshumer (2009). to the drive train. train. models of motion of the the pump, the swash swashin plate mechanism and sign are suitable mathematical which are tailored the actuating valve, is presented in Fuchshumer (2009). to the considered drive the actuating valve, is presented in Fuchshumer (2009). to the considered drive train. The models derived in Kugi et al. (2000) and Fuchshumer the actuating valve, is presented in Fuchshumer (2009). to the considered drive train. models derived in Kugi et al. (2000) and Fuchshumer Various configurations configurations of of SHHVs SHHVs exist exist in in the the literalitera- The The derived in et and Fuchshumer (2009) cannot be directly to the piston Various The models models derived in Kugi Kugi adapted et al. al. (2000) (2000) and axial Fuchshumer (2009) cannot be directly adapted to the axial piston Various configurations of SHHVs exist in the literature. Typically, like in Deppen and Alleyne (2012), (2009) cannot be directly adapted to the axial piston VariousTypically, configurations of Deppen SHHVs and existAlleyne in the (2012), litera- (2009) units of the SHHV because of the different hydraulic and ture. like in cannot be directly adapted to the axial piston units of the SHHV because of the different hydraulic and ture. Typically, like in Deppen and Alleyne (2012), swash plate controlled variable displacement axial piston units of the SHHV because of the different hydraulic and ture. Typically, like in Deppen and Alleyne (2012), mechanical setting of the displacement system, which reswash plate controlled variable displacement axial piston units of the SHHV because of the different hydraulic and mechanical setting of the displacement system, which reswash plate controlled variable displacement axial piston pumps/motors are used to control the hydraulic flow bemechanical setting of the displacement system, which reswash plate controlled variable displacement axial piston sults in a different structure of the mathematical model. pumps/motors are used to control the hydraulic flow bemechanical setting of the displacement system, which results in a different structure of the mathematical model. pumps/motors are used to control the hydraulic flow between the primary and secondary side of the drive. Tosults in a different structure of the mathematical model. pumps/motors are used to control the hydraulic flow beTherefore, also a dynamic model of the axial piston units tween the primary and secondary side of the drive. Tosults in a different structure of the mathematical model. Therefore, also a dynamic model of the axial piston units tween the the primary and secondary secondary side of the the drive. ToTogether withprimary high energy energy efficiency, fast dynamics are Therefore, also model of the axial piston tween and side of drive. with the displacement system has be derived. gether aaa high efficiency, aaa fast are also aa dynamic dynamic model ofto the axial piston units units with the displacement system has to be derived. gether with with high energy efficiency, fast dynamics dynamics are Therefore, obtained in this setup. In most contributions it is assumed with the displacement system has to be derived. gether with a high energy efficiency, a fast dynamics are obtained in this setup. In most contributions it is assumed with the displacement system has to be derived. obtained in this setup. setup. In axial most contributions contributions it is isperformed assumed In this contribution, two mathematical models of the conthat the in control of the the piston units units is is obtained this In most it assumed In this contribution, two mathematical models of the conthat the control of axial piston performed this two mathematical models of the that the control of the axial piston units is performed sidered drive train are both fulfilling by standard displacement controllers and therefore the In In this contribution, contribution, twodeveloped, mathematical models of different the conconthatstandard the control of the axial piston units is performed sidered drive train are developed, both fulfilling different by displacement controllers and therefore the drive train are developed, both fulfilling different by standard standard displacement displacement controllers controllers and and therefore therefore the the sidered requirements. A detailed model will be given in Section 2. sidered drive train are developed, both fulfilling different by requirements. A detailed model will be given in Section 2. requirements. A detailed model will be given in Section 2. ⋆ The authors from Vienna University of Technology highly appreThe purpose of this model is to validate, evaluate and comrequirements. A detailed model will be given in Section 2. ⋆ The authors from Vienna University of Technology highly appreThe purpose of this model is to validate, evaluate and com⋆ The purpose of this model is to validate, evaluate and comThe authors from Vienna University of Technology highly apprepare different control strategies. Moreover, the detailed ciate the technical and financial support provided by Robert Bosch ⋆ The purpose of this model is to validate, evaluate and comThethe authors fromand Vienna University Technology highly apprepare different control strategies. Moreover, the detailed ciate technical financial supportofprovided by Robert Bosch pare ciate the GmbH. pare different different control control strategies. strategies. Moreover, Moreover, the the detailed detailed ciate the technical technical and and financial financial support support provided provided by by Robert Robert Bosch Bosch GmbH. GmbH. GmbH.
Copyright 2015,IFAC IFAC (International Federation of Automatic Control) 508Hosting by Elsevier Ltd. All rights reserved. 2405-8963 ©© 2015, Copyright © 2015, IFAC 508 Copyright 2015, IFAC 508 Peer review© of International Federation of Automatic Copyright ©under 2015,responsibility IFAC 508Control. 10.1016/j.ifacol.2015.05.064
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ϕs ph
ICE
ϕs2
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ph cv
rp
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ωc2
ωc1
A2 , m2
rfb
Vh ϕs1
i0
cfs
509
ra
cfb
Fm (i) i
L
Ap , mp
u
R pa A1 , m1
pl see Figure 2
pl
accumulator Fig. 1. SHHV system topology for IVT operation mode.
Fig. 2. Axial piston unit with displacement system.
model has to be able to capture parameter variations in order to verify the robustness of the developed control strategies. Additionally, a reduced-order model will be derived in Section 3 which is suitable for online computation and realtime applications. This model serves as a basis for controller and estimator design. Thus, the main focus is to minimize the computational costs by simultaneously preserving the accuracy of the detailed model as good as possible.
generalized coordinates as q = [ϕs , ϕc ]T , the equations of motion are obtained from d ∂T ∂T − = Qj , j ∈ {s, c} (1) dt ∂ ϕ˙ j ∂ϕj with the kinetic energy T = 1/2q˙ T Dq˙ and the generalized forces Qj . The positive definite mass matrix of the system is given by D = diag (Ds , Dc ) with 1 rp2 mp np + 2 (m1 + m2 ) ra2 (2a) Ds (ϕs ) = Js + 2 cos4 (ϕs ) 1 Dc (ϕs ) = Jc + rp2 mp np 2 + tan2 (ϕs ) , (2b) 2 where Js and Jc are the inertias of the swash plate and the cylinder barrel with respect to the corresponding axis of rotation. The np pistons, each one of mass mp , perform an axial movement in the cylinder barrel in a distance of rp from the shaft. Furthermore, m1 and m2 denote the masses of the pushing and pulling pistons which are both mounted in a distance of ra from the shaft. The equations of motion of the swash plate and the shaft can be written in the form =:Φs(ϕs ,ωs ,ωc ) ϕ˙ s = ωs (3a) 1 1 dDc 2 1 dDs 2 (3b) ω − ω +Qs ω˙ s = Ds (ϕs ) 2 dϕs c 2 dϕs s dDc 1 − (3c) ωs ωc + Qc , ω˙ c = Dc (ϕs ) dϕs with the angular velocities ωs and ωc and the centrifugal term Φs . The generalized forces can be obtained from Figure 2 1 (ra (A1 pa − A2 ph − Ffs ) + Ts ) Qs = cos2 (ϕs ) (4a) − kv ωs − kc sign(ωs ) Vp (ϕs ) ph − Td − Te , (4b) Qc = 2π with the areas A1 and A2 of the pushing and pulling piston. The force Ffs = cfs ra (tan(ϕs ) − tan(ϕˆs )) + F0 tanh(κ (ϕs − ϕˆs )) (5) results from the pre-stressed restoring springs with stiffness cfs , pre-tension F0 and rest position ϕˆs . The pretension F0 is approximated by the second term on the right-hand side of (5), with the tuning parameter κ. A mean-value model as, e.g., presented in Ivantysyn and Ivantysynova (1993) is employed to account for the resulting swivel torque Ts = cs ph which is the reaction torque
2. DETAILED MODEL A mathematical model of the SHHV drive train depicted in Figure 1 will be derived in this section. In IVT operation mode, the hydraulic accumulator on the high-pressure side is disconnected from the hydrostatic drive and may be neglected in the subsequent considerations. The first step is devoted to the modeling of a single axial piston unit (APU) as shown in Figure 2. The model is then combined with the remaining components to form the overall drive train model. 2.1 Axial Piston Unit The APU with the electro-hydraulic displacement system comprising two hydraulic cylinders and an electromagnetically actuated 3/3-way spool valve is depicted in Figure 2. The APU is self-supplied which means that the volume flow for the displacement is branched off the supply port of the pump with pressure ph . A displacement in the direction of the swash plate angle ϕs is realized by supplying hydraulic oil across the variable orifice of the valve to the pushing cylinder with pressure pa . Connecting the cylinder chamber with tank pressure pl moves the swash plate in the opposite direction, where the motion is supported by a hydraulic pulling cylinder connected to the supply port pressure ph . The angular position of the swash plate is fed back to the spool valve via a mechanical spring, which operates as a stabilizing mechanical controller for the pump displacement. As will be shown later, this electroproportional (EP) displacement system gives an angular position of the pump which is basically proportional to the magnetic force Fm . In a similar manner as described in Fuchshumer (2009), the APU is modeled as a rigid body system with two degrees of freedom which are determined by the swash plate angle ϕs and the shaft angle ϕc , see Figure 2. Defining the 509
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i=0 j=0
with the coefficients cij fitted to measurements of the torque by means of a least-squares identification. This approximation procedure is also of interest for an overall energy efficiency optimization. It can be seen from the upper part of Figure 3 that there is nearly a linear dependence between the speed and the torque loss, which is in accordance with theoretical considerations. A more detailed analysis of the internal hydraulic design of the APU would explain the nonlinear dependence on the pressure. Nevertheless, the measured data can be approximated very well in a wide operating range.
The mass balance of the pushing cylinder chamber reads as � � A1 ra ωs β p˙ a = − 2 + qa , (7) Va + A1 ra tan(ϕs ) cos (ϕs ) with the effective bulk modulus β of the hydraulic fluid, the volume Va of the cylinder chamber for ϕs = 0 and the volumetric flow rate qa into the cylinder chamber. In order to guarantee low leakages, the valve exhibits a closed-center design with overlap γ > 0, see the left hand side of Figure 4. Of course, no perfect sealing for −γ < sv < γ can be achieved due to the leakage along the small gaps between the valve spool and the body. An analytical model of these leakages is beyond the scope of this paper. Instead, a small opening of the valve is considered in the dead zone. Using the equations of a turbulent flow of an incompressible fluid from McCloy and 510
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Fig. 3. Measured and approximated torque and volumetric loss. The quantities are scaled with respect to the ideal torque T0 and volume flow q0 of the APU at nominal pressure p0 , angular speed ωc0 and full displacement ϕs = ϕ¯s . Martin (1980) and assuming pl ≤ pa ≤ ph , the following model can be obtained for the valve volume flow � αv 2 Av (sv ) √ph − pa , sv ≥ 0 �ρ qa = (8) −αv 2 Av (sv ) √pa − pl , sv < 0, ρ
with the orifice contraction coefficient αv and the density ρ of the hydraulic oil. The unique relationship between the valve spool position sv and the opening area Av (sv ) can be derived by a geometric consideration, see Fuchshumer (2009), and is depicted in Figure 4. It can be seen that the actual non-ideal characteristics near the dead band is obtained by a slight shaping of the ideal characteristics. pa
0.1
ph
pl
opening area Av /Av,max
A physical explanation of the torque loss Td , which takes into account the non-ideal behavior of the APU in terms of mechanical efficiency, is given, e.g., in Ivantysyn and Ivantysynova (1993). There, it is explained that the accumulated pressure drop across the internal resistance network of hydraulic pumps causes a difference between the ideal and actual engine torque. Besides, viscous friction occurring between the sliding parts results in a torque loss which depends on the different gap sizes and dynamic viscosity of the hydraulic oil. Because of the many different influencing factors it is beyond the scope of this contribution to derive a physically motivated model which captures all these effects separately. Instead, the reported significant dependence of the torque loss Td on the pressure ph and the angular speed ωc is approximated by the polynomial 2 2 � � cij pih ωcj , (6) Td (ph , ωc ) =
4 torque loss Td /T0 [%]
of the pistons acting on the swash plate with the constant cs . The viscous and Coulomb friction coefficients kv and kc account for the friction in the bearing of the swash plate. The ideal average engine torque Vp (ϕs ) /(2π)ph is determined by the adjustable geometric displaced volume Vp (ϕs ) = 2np rp Ap tan(ϕs ) of the pump with the effective area Ap per piston, cf. Manring (2005). Finally, the right hand side of (4b) comprises the torque loss Td and the external load torque Te on the pump shaft. Remark 1. The swash plate dynamics (3a) and (3b) are only valid between the limit stops ±ϕ¯s of the swash plate, |ϕs | ≤ ϕ¯s . If this condition is violated, the model equations (3a) and (3b) are switched to ϕ˙ s = ω˙ s = 0 and ϕs is set to the corresponding limit stop. In this state, the force acting on the swash plate is evaluated, which decides when to switch to the original system dynamics (3a) and (3b).
volumetric loss ql /q0 [%]
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Fig. 4. Considered spool valve with overlap γ and normalized opening area Av /Av,max . Av,max denotes the opening area for the full deflection sv = ±¯ sv of the spool valve. The overall volumetric flow of the APU can be formulated as Vp (ϕs ) A2 ra ωs qp = ωc − ql (ph , ωc ) + − qd , (9) 2π cos2 (ϕs ) with the ideal average volume flow Vp (ϕs ) /(2π)ωc , the volumetric loss ql (ph , ωc ) according to Figure 3 and the volume flow
qd =
αv
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√ 2 ρ Av (sv ) ph
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of the displacement system. The volumetric loss ql (ph , ωc ) accounts for the hydraulic efficiency of the APU due to leakage flows and is discussed, e.g., in Ivantysyn and Ivantysynova (1993). For the same reasons as outlined in the context of the torque loss, the volumetric loss can be approximated by fitting a polynomial similar to (6) to measured data of the volume flow. The lower part of Figure 3 shows the result of this approximation. The valve is actuated by two proportional magnets with input voltages u1 and u2 and currents i1 and i2 . By introducing the effective current i = i1 − i2 , a relationship Fm (i) for the resulting bidirectional magnetic force is obtained, cf. Figure 2. The current dynamics are assumed to be linear d 1 i = (−Ri + u) , (11) dt L with the electric resistance R, the inductance L and the effective voltage u = u1 − u2 , which is regarded as the control input to the system. The equations of motion of the valve are given in the form s˙ v = wv (12a) 1 w˙ v = (−c0 sv − Fd − Ffb − Fjet + Fm (i)) , (12b) mv where mv denotes the mass of the valve spool and c0 = cv + cfb is the combined spring stiffness of the centering and the feedback spring. A static friction model of the form Fd = kv wv + kvc + kvh − kvc exp −kvb wv2 sign(wv ) is used with the viscous damping coefficient kv and the friction parameters kvh , kvc and kvb , see Figure 5. Flow forces acting on the valve spool are modeled according to McCloy and Martin (1980) by 2αv cos(ϕjet ) Av (sv ) (ph − pa ) , sv ≥ 0 Fjet = (13) −2αv cos(ϕjet ) Av (sv ) (pa − pl ) , sv < 0 with the jet angle ϕjet . The force (14) Ffb = cfb rfb sin(ϕs ) + F0 on the right hand side of (12b) results from the feedback spring attached to the valve spool at radius rfb from the shaft of the pump. The introduction of F0 is motivated as follows: The equilibrium of (12) and (7) for sv = 0 reads as (15) cfb rfb sin(ϕs ) + F0 = Fm (i) . This implies a stationary nonlinear relationship between the current i and the swash plate angle ϕs , also denoted as EP (electro-proportional) map. For an ideal setup, Fm = 0 would hold which implies ϕs = 0. The real system, however, suffers from unknown asymmetries of the feedback system which are summarized in the parameter F0 . Finally, the limit stops ±¯ sv of the valve spool are modeled by switching the dynamics (12) to s˙ v = w˙ v = 0 in the same manner as described in Remark 1. 2.2 Overall Drive Train Model The mathematical model for the SHHV drive train from Figure 1 consists of the equations (3a) and (3b), (7), (11) and (12) of the swash plate, the cylinder chamber, the 511
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velocity wv /wv0 [%]
Fig. 5. Friction force on the valve spool. The velocity is scaled to the maximum value wv0 occurring at a typical setpoint change of the APU. current and the valve for the primary and secondary side with the corresponding swash plate angles ϕs1 and ϕs2 . The secondary APU is directly coupled to the vehicle via a differential gear. Thus, the angular speed ωc2 is proportional to the vehicle speed. In this work, no model for the vehicle is included. Instead it is assumed that ωc2 is a measured external input to the system. The primary APU is coupled to the ICE, which gives dDc1 1 Vp1 (ϕs1 ) ω˙ c1 = ph − Td1 + i0 Ti , − ωs1 ωc1 + Dc1 dϕs1 2π (16) see (3c), with the external torque Ti of the ICE transformed by the gear transmission ratio i0 . Subsequently, an index 1 or 2 is always used to refer to the primary and secondary APU, respectively. The ICE is assumed to be torque-controlled with the desired torque Tˆi as control input and the closed-loop dynamics T˙i = ai Tˆi − Ti , (17)
with the time constant 1/ai . Finally, the coupling between the two hydrostatic engines is modeled by a constant hydraulic volume Vh with the pressure dynamics given by β (qp1 + qp2 ) , (18) p˙ h = Vh with the primary and secondary volumetric flow qp1 and qp2 according to (9). The pressure pl at the low-pressure side of the hydrostatic drive is assumed to be constant. 3. REDUCED-ORDER MODEL The nonlinear model of the drive train is of order fifteen and thus entails a very high computational burden, in particular if it serves as a basis for nonlinear model predictive control. Therefore, in this section a reduced-order model is derived by successively reducing the complexity of the APU model presented in Section 2.1. Thereby, the singular perturbation theory is exploited, see, e.g., Kokotovic et al. (1999) for a detailed discussion on singularly perturbed systems. In a first step, the valve spool (12) is considered. In real applications, a sinusoidal dither signal is superimposed to the actuating voltage u to overcome the static friction of the spool valve. Therefore, the frictional components related to kvh and kvc can be neglected and (12) can be transformed to the standard form of singular perturbation theory
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s˙ v = wv
(19a) 1 ǫw˙ v = −wv + (−c0 sv − Ffb − Fjet + Fm (i)) , (19b) kv with the perturbation parameter ǫ = mv /kv . For ǫ → 0, which means that the damping is high compared to the mass of the valve spool, the solution of (19b) by wv used in (19a) results in the reduced slow dynamics of the valve spool 1 s˙ v = (−c0 sv − Ffb − Fjet + Fm (i)) . (20) kv In a second step, the equations of motion (3a) and (3b) of the swash plate together with the pressure dynamics (7) are rewritten in the form ϕ˙ s = ωs (21a) 1 (Φs + Qs ) (21b) Ds (ϕs ) A1 ra ωs β − 2 + qa (pa ) . (21c) p˙a = Va + A1 ra tan(ϕs ) cos (ϕs ) As has been shown in Fuchshumer (2009), the states ωs and pa correspond to the fast dynamics and ϕs is the slow part of the system. Using the quasi-stationary assumption ω˙ s ≈ 0 (22a) (22b) p˙ a ≈ 0 in (21b) and (21c) gives the quasi-stationary states ω ¯ s and p¯a . Using these results in (21a) yields the slow dynamics cos2 (ϕs ) ϕ˙ s = qa (¯ pa ) (23) A1 ra for the swash plate angle. Bearing in mind the nonlinear behavior of qa (¯ pa ), the nonlinear equation 1 (ra (A1 p¯a − A2 ph − Ffs ) + Ts ) 0 = Φs + 2 cos (ϕs ) (24) − kv ω ¯ s − kc sign(¯ ωs ) has to be solved for the stationary pressure p¯a . An approximate solution can be found by neglecting the centrifugal term Φs and the friction terms in the second line of (24) leading to the explicit solution 1 Ts A2 ph + Ffs − . (25) p¯a = A1 ra The simplifications (19) to (20) and (21) to (23) yield a model order reduction by three states. Combining this reduced model with the current dynamics (11) and the shaft dynamics (3c), the reduced-order model for the APU is of order four. The fast dynamics of the original nonlinear 7th order model is considered in a quasi-stationary manner, thus leading to a reduced-order model suitable for the controller design. ω˙ s =
4. EXPERIMENTAL RESULTS The objective of this section is the validation of the detailed and reduced-order model derived in Sections 2.1 and 3. As the main components of the system are the APUs, various measurements are carried out on an experimental test bench for an APU. The following assumptions hold throughout the whole analysis: • The APU is coupled to a speed-controlled load engine and therefore the dynamics of the engine shaft are not considered in the analysis. 512
• To overcome the friction of the spool valve, a sinusoidal dither signal is always superimposed to the actuating voltage u. • The simulation results are obtained by feeding the models with the actuating voltage u of the test bench system. • The voltage, current and pressure are normalized to typical set point values u0 , i0 and p0 and the swash plate displacement and valve position are normalized to the corresponding limit stops ϕ¯s and s¯v . In a first step, the quasi-stationary behavior of the model is examined. For this purpose, triangular reference signals are fed to the system which can be considered slow compared to the whole system dynamics. The stationary behavior of the model is adapted to the measurements by fitting the unknown relationship Fm (i) from the EP map (15). A comparison between the two models and the measurement signals is shown in Figure 6. In this figure, the dither signal of the voltage u, the current i and the valve position sv is filtered out of the signals because only the mean values are of interest. It can be seen that both models give a very good approximation of the normalized displacement α = tan(ϕs ) / tan(ϕ¯s ), which is of primary interest for a higher level swash plate control. A small deviation in the simulated current results in a slight deviation for positive displacements which is mainly due to a different amplitude of the dither signal. The dither signal is visible in the pressure pa , which is also predicted well by the detailed model. Although only the mean value can be approximated by the reduced-order model, it is still a good basis for controller design since the frequency of the dither signal is considerably higher than the desired closedloop dynamics. It follows from the inspection of the valve position sv that its mean value is in the range of the dead band of the valve because of the rather slow displacement velocities. Despite the lack of a detailed leakage model the mean value of the valve position can be predicted very well by both models. Figure 7 shows a measurement scenario with a displacement over a wide range in a short time compared to the system dynamics. Both displacement directions are considered and the signals are not filtered for the dither frequency. Both models are able to predict the dynamic behavior of the swash plate in a very good way. Even in this scenario exploiting the whole stroke of the valve, a good model prediction outside the dead band is accomplished. The pressure in the pushing cylinder is again very well approximated by its quasi-stationary approximation. 5. CONCLUSION In this paper, a mathematical model of a serial hydraulic hybrid vehicle was derived. By means of a systematic model order reduction scheme, the nonlinear model was simplified and a reduced-order model for the underlying self-supplied variable displacement axial piston units was obtained. Both models were validated by measurements on an industrial test bench and it was shown that the reduced-order model almost preserves the accuracy of the detailed model. In contrast to Fuchshumer (2009), the valve dynamics are incorporated into the controller design model since they
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Fig. 6. Simulated and measured data for a slow displacement of the APU. have a significant influence on the overall system dynamics. Besides, the system is subject to constraints such as the limit stop of the valve or the constraints of the maximum voltage and current. Neglecting these nonlinearities in the controller design degrades the performance of classical nonlinear controllers. MPC (Model Predictive Control ) strategies offer the possibility to systematically take into account such constraints in the controller design by online predicting the system trajectories and finding the optimal control with respect to a cost functional, see, e.g., Gr¨ une and Pannek (2011). Because of the high computational burden of MPC, it is impossible to apply the detailed model in a real-time MPC controller. On the contrary, the reduced-order model can serve as a perfect basis for such a controller. Current research is directed to the application of nonlinear MPC to the considered system. REFERENCES Deppen, T.O. and Alleyne, A.G. (2012). Optimal energy use in a light weight hydraulic hybrid passenger vehicle. Journal of Dynamic Systems, Measurement and Control, 134(4). Erkkil¨ a, M., Bauer, F., and Feld, D. (2013). Universal Energy Storage and Recovery System - A Novel Approach for Hydraulic Hybrid. In Proceedings of the 13th 513
Fig. 7. Simulated and measured data for a fast displacement of the APU. Scandinavian International Conference on Fluid Power, SICFP2013, 92, 45–52. Link¨oping, Sweden. 3 – 5 June 2013. Fuchshumer, F. (2009). Modellierung, Analyse und nichtlineare modellbasierte Regelung von verstellbaren Axialkolbenpumpen. Shaker, Aachen, Germany. Gr¨ une, L. and Pannek, J. (2011). Nonlinear Model Predictive Control: Theory and Algorithms. Communications and Control Engineering. Springer, London. Ivantysyn, J. and Ivantysynova, M. (1993). Hydrostatische Pumpen und Motoren: Konstruktion und Berechnung. Vogel, W¨ urzburg, Germany. Kokotovic, P., Khali, H., and O’Reilly, J. (1999). Singular Perturbation Methods in Control: Analysis and Design. Society for Industrial and Applied Mathematics, Philadelphia, USA. Kugi, A., Schlacher, K., Aitzetm¨ uller, H., and Hirmann, G. (2000). Modeling and simulation of a hydrostatic transmission with variable-displacement pump. Mathematics and Computers in Simulation, 53, 409–414. Manring, N. (2005). Hydraulic Control Systems. John Wiley & Sons, New Jersey, USA. McCloy, D. and Martin, H. (1980). Control of Fluid Power: Analysis and Design (2nd edition). John Wiley & Sons, New York, USA.
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IFAC-PapersOnLine 48-1 (2015) 508–513
Mathematical Modeling and Analysis Mathematical Modeling and Analysis Mathematical Modeling and Analysis ⋆⋆ Mathematical Modeling and Analysis Hydrostatic Drive Train ⋆ Hydrostatic Drive Train Hydrostatic Drive Train Hydrostatic Drive Train ⋆
of of of of
a a a a
∗ ∗ ∗ P. Zeman u ller ∗ , W. Kemmetm¨ ∗ , A. Kugi ∗ P. Zeman u ller ∗ , W. Kemmetm¨ ∗ , A. Kugi ∗ ∗ , W. Kemmetm¨ ∗ , A. Kugi ∗ P. Zeman u ller P. Zeman , W. Kemmetm¨ uller , A. Kugi ∗ ∗ Automation and Control Institute, Vienna University of Technology, ∗ Automation and Control Institute, Vienna University of Technology, ∗ Automation and Control Institute, Vienna Vienna, [email protected]) Automation andAustria Control(e-mail: Institute, Vienna University University of of Technology, Technology, Vienna, Austria (e-mail: [email protected]) Vienna, Austria (e-mail: [email protected]) Vienna, Austria (e-mail: [email protected])
Abstract: Hydrostatic Hydrostatic drives drives constitute constitute an an advantageous advantageous alternative alternative to to conventional conventional mechanimechaniAbstract: Abstract: Hydrostatic drives constitute an advantageous alternative to conventional mechanical gears with a fixed transmission ratio. The hydraulic part of the system decouples the input Abstract: Hydrostatic drives constitute an advantageous alternative to conventional mechanical gears with a transmission ratio. The hydraulic part of system decouples the input cal gears with a fixed fixed transmission ratio. The as hydraulic part of the thefor system decouples the input and output speeds of the drive and may serve an energy storage recuperation as well. For cal gears with a fixed transmission ratio. The hydraulic part of the system decouples the input and output speeds of the drive and may serve as an energy storage for recuperation as well. For and output speeds of the drive and may serve as an energy storage for recuperation as well. For this reason, hydrostatic drives are a well-established concept, in particular in the field of mobile and output speeds of the drive and may serve as an energy storage for recuperation as well. For this reason, hydrostatic drives are a well-established concept, in particular in the field of mobile this reason, hydrostatic a in in of working machines such drives as lift lift are trucks or excavators. excavators. concept, In this this work, work, a mathematical mathematical model for aa this reason, hydrostatic drives are a well-established well-established concept, in particular particular in the the field field of mobile mobile working machines such as trucks or In a model for working machines such as lift trucks or excavators. In this work, a mathematical model for hydrostatic drive train of a passenger vehicle is developed. The main focus of the contribution working machines such asalift trucks or excavators. In this The work, a mathematical model for aa hydrostatic drive vehicle is main the hydrostatic drive train train of of modeling a passenger passenger vehicle is developed. developed. The displacement main focus focus of ofaxial the contribution contribution lies on the mathematical of the self-supplied variable piston units hydrostatic drive train of a passenger vehicle is developed. The main focus of the contribution lies on the mathematical modeling of the self-supplied variable displacement axial piston units lies on the mathematical modeling of the self-supplied variable displacement axial piston units of the hydraulic system since they basically determine the performance of the overall torque lies on the mathematical modeling of the self-supplied variable displacement axial piston units of the hydraulic system since they basically determine the performance of the overall torque of the system since basically determine the of overall control. Starting from from a complex complex high-order dynamic model, systematic order order reduction of the the of the hydraulic hydraulic system since they they basicallydynamic determine the aaperformance performance of the thereduction overall torque torque control. Starting a high-order model, systematic of control. Starting from a complex high-order dynamic model, a systematic order reduction the nonlinear model is accomplished by means of singular perturbation theory. The accuracy of control. Starting from a complex by high-order dynamic model, a systematic reduction the nonlinear model is accomplished means of singular perturbation theory.order The accuracy of the nonlinear model accomplished by singular The derived models models isis evaluated by measurement measurement results onperturbation an industrial industrialtheory. test bench. bench. nonlinear modelis isevaluated accomplished by means means of of results singularon perturbation theory. The accuracy accuracy of of the the derived by an test derived models is evaluated by measurement results on an industrial test bench. derived models is evaluated by measurement results on an industrial test bench. © 2015, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. Keywords: serial serial hydraulic hydraulic hybrid hybrid vehicle, vehicle, infinitely infinitely variable variable transmission, transmission, variable variable Keywords: Keywords: serial hydraulic hybridelectro-hydraulic vehicle, infinitely infinitelysystems, variablesingular transmission, variablemethod displacement axial piston pump, perturbation Keywords: serial hydraulic hybrid vehicle, variable transmission, variable displacement axial piston pump, electro-hydraulic systems, singular perturbation method displacement axial piston pump, electro-hydraulic systems, singular perturbation method displacement axial piston pump, electro-hydraulic systems, singular perturbation method 1. INTRODUCTION INTRODUCTION underlying dynamics are not accounted for, neither in the 1. underlying dynamics are not accounted for, neither in the 1. INTRODUCTION INTRODUCTION underlying dynamics are not accounted for, neither in the model nor in the controller design. For the design of highly 1. underlying dynamics are not accounted for, neither in the model nor in the controller design. For the design of highly model nor in the controller design. For the design of highly dynamic torque control strategies, this simplification is not In a typical setup of a hydrostatic drive train, the energy model nor in the controller design. For the design of highly dynamic torque control strategies, this simplification is not In typical setup of hydrostatic drive train, the dynamic torque control strategies, this simplification is not In a typical setup of a hydrostatic drive train, the energy energy and yields aa limited control performance. For delivered by setup an internal internal combustion engine (ICE) on the the feasible dynamic torque control strategies, this simplification is this not In aa typical of aa hydrostatic drive train, the energy feasible and yields limited control performance. For this delivered by an combustion engine (ICE) on feasible and yields a limited control performance. For this delivered by an internal combustion engine (ICE) on the reason, the present contribution deals with the mathematprimary side of the system is transmitted across a hyfeasible and yields a limited control performance. For this delivered by an internal combustion engine (ICE) on the reason, the present contribution deals with the mathematprimary side system is across aa hythe contribution deals with the primarycircuit side of oftothe the system isontransmitted transmitted across hy- reason, ical modeling of an SHHV, where the essential nonlinear draulic the output the secondary side. reason, the present present contribution deals with the mathematmathematprimary side of the system is transmitted across a The hyical modeling of an SHHV, where the essential nonlinear draulic circuit to the output on the secondary side. The ical modeling of an SHHV, where the essential nonlinear draulic circuit circuit to the the output on on (IVT) the secondary secondary side. The dynamics of the axial piston units are taken into account. infinitely variable transmission allows to run the ical modeling of an SHHV, where the essential nonlinear draulic to output the side. The dynamics of the axial piston units are taken into account. infinitely variable transmission (IVT) allows to run the dynamics of the axial piston units are taken into account. infinitely variable transmission (IVT) allows to run the The mathematical modeling of variable axial piston pumps ICE in a set point which is optimal from an energetic point dynamics of the axial piston units are taken into account. infinitely variable transmission (IVT) allows to run the The mathematical modeling of variable axial piston pumps ICE in a set point which is optimal from an energetic point The mathematical modeling of variable axial piston pumps ICE in a set point which is optimal from an energetic point has been studied extensively in Ivantysyn and Ivantysynof view. Furthermore, recuperation of braking energy can The been mathematical modeling ofinvariable axialand piston pumps ICE in a Furthermore, set point which is optimal from an energetic point has studied extensively Ivantysyn Ivantysynof view. recuperation of braking energy can has been studied extensively in Ivantysyn and Ivantysynof view. Furthermore, recuperation of braking energy can ova (1993). Very detailed models are given there which be done in a very efficient way by loading a hydraulic achas been studied extensively in Ivantysyn and Ivantysynof view. Furthermore, recuperation of braking energy can ova (1993). Very detailed models are given there which be done in a very efficient way by loading a hydraulic ac(1993). Very detailed models are given there which be done done in in aasee, very efficient way by(2013). loadingCurrent a hydraulic hydraulic ac- ova describe, e.g., the pressure dynamics in the cylinder barrel, cumulator, e.g., Erkkil¨ a et al. research ova (1993). Very detailed models are given there which be very efficient way by loading a acdescribe, e.g., the pressure dynamics in the cylinder barrel, cumulator, see, e.g., Erkkil¨ a et al. (2013). Current research describe, e.g., the pressure dynamics in the cylinder barrel, cumulator, see, e.g., Erkkil¨ a et al. (2013). Current research the actual volume flow and the internal force distribution. is concerned with the exploitation of these advantages for describe, e.g., the pressure dynamics in the cylinder barrel, cumulator, see, e.g., Erkkil¨ a et al. (2013). Current research the volume flow and internal force is with exploitation of these advantages for the actual actual volume flowuseful and the the internal force distribution. distribution. is concerned concerned with the the exploitation ofdrive thesetrain advantages for the These models are not for the considered SHHV since passenger vehicles. The hydrostaticof represents actual volume flow and the internal force distribution. is concerned with the exploitation these advantages for These models are not useful for the considered SHHV since passenger vehicles. The hydrostatic drive train represents These models are not useful for the considered SHHV since passenger vehicles. The hydrostatic drive train represents these effects can be well described by mean-value models an alternative concept for automotive manufacturers comThese models are not useful for the considered SHHV since passenger vehicles. Thefor hydrostatic drive train represents these effects can be well described by mean-value models an alternative concept automotive manufacturers comthese effects can be well described by mean-value models an alternative concept for automotive manufacturers comwith an adequate accuracy, see, e.g., Manring (2005). In pared to conventional hybrid drives with electric engines. these effects can be well described by mean-value models an alternative concept for automotive manufacturers comwith an adequate accuracy, see, e.g., Manring (2005). In pared conventional hybrid drives with electric with an adequate accuracy, see, e.g., Manring (2005). In pared to to conventional hybrid drives with(SHHVs) electric engines. engines. Kugi et al. (2000), an online simulator for the swash plate These serial hydraulic hybrid vehicles require with an adequate accuracy, see, e.g., Manring (2005). In pared to conventional hybrid drives with electric engines. Kugi et al. (2000), an online simulator for the swash plate These serial hydraulic hybrid vehicles (SHHVs) require Kugi et et al. (2000), displacement an online online simulator simulator forpresented, the swash swashwhich plate These serial hydraulic hybrid vehicles (SHHVs) require angle of a variable pump is sophisticated control strategies in order to translate the Kugi al. (2000), an for the plate These serial hydraulic hybrid vehicles (SHHVs) require angle of a variable displacement pump is presented, which sophisticated control strategies in to the of aa variable displacement pump is which sophisticated control strategies in aorder order to translate translate the takes into account the essential dynamics of the swash torque requestcontrol of the the strategies driver into intoin suitable actuationthe of angle angle of variable displacement pump is presented, presented, which sophisticated to translate takes into account the essential dynamics of the swash torque request of driver aaorder suitable actuation of takes into account the essential dynamics of the swash torque request of the driver into suitable actuation of plate mechanism. The dynamic mathematical model of the system by simultaneously fulfilling performance and takes into account the essential dynamics of the swash torque request of the driver into a suitable actuation of plate mechanism. The dynamic mathematical model of the system by simultaneously fulfilling performance and plate mechanism. The dynamic mathematical model the system by simultaneously fulfilling performance and a variable axial piston pump, consisting of the equations energy efficiency demands. The The fulfilling basis for for performance the controller controllerand de- aplate mechanism. The pump, dynamic mathematical model of of the system by simultaneously variable axial piston consisting of the equations energy efficiency demands. basis the dea variable axial piston pump, consisting of the equations energy efficiency demands. The basis basis forwhich the controller controller de- aof variable motion of the pump, the swash plate mechanism and sign are suitable mathematical models are tailored axial piston pump, consisting of the equations energy efficiency demands. The for the deof motion of the pump, the swash plate mechanism and sign are suitable models motion of pump, the plate mechanism and sign areconsidered suitable mathematical mathematical models which which are are tailored tailored of the actuating valve, is presented Fuchshumer (2009). to the drive train. train. models of motion of the the pump, the swash swashin plate mechanism and sign are suitable mathematical which are tailored the actuating valve, is presented in Fuchshumer (2009). to the considered drive the actuating valve, is presented in Fuchshumer (2009). to the considered drive train. The models derived in Kugi et al. (2000) and Fuchshumer the actuating valve, is presented in Fuchshumer (2009). to the considered drive train. models derived in Kugi et al. (2000) and Fuchshumer Various configurations configurations of of SHHVs SHHVs exist exist in in the the literalitera- The The derived in et and Fuchshumer (2009) cannot be directly to the piston Various The models models derived in Kugi Kugi adapted et al. al. (2000) (2000) and axial Fuchshumer (2009) cannot be directly adapted to the axial piston Various configurations of SHHVs exist in the literature. Typically, like in Deppen and Alleyne (2012), (2009) cannot be directly adapted to the axial piston VariousTypically, configurations of Deppen SHHVs and existAlleyne in the (2012), litera- (2009) units of the SHHV because of the different hydraulic and ture. like in cannot be directly adapted to the axial piston units of the SHHV because of the different hydraulic and ture. Typically, like in Deppen and Alleyne (2012), swash plate controlled variable displacement axial piston units of the SHHV because of the different hydraulic and ture. Typically, like in Deppen and Alleyne (2012), mechanical setting of the displacement system, which reswash plate controlled variable displacement axial piston units of the SHHV because of the different hydraulic and mechanical setting of the displacement system, which reswash plate controlled variable displacement axial piston pumps/motors are used to control the hydraulic flow bemechanical setting of the displacement system, which reswash plate controlled variable displacement axial piston sults in a different structure of the mathematical model. pumps/motors are used to control the hydraulic flow bemechanical setting of the displacement system, which results in a different structure of the mathematical model. pumps/motors are used to control the hydraulic flow between the primary and secondary side of the drive. Tosults in a different structure of the mathematical model. pumps/motors are used to control the hydraulic flow beTherefore, also a dynamic model of the axial piston units tween the primary and secondary side of the drive. Tosults in a different structure of the mathematical model. Therefore, also a dynamic model of the axial piston units tween the the primary and secondary secondary side of the the drive. ToTogether withprimary high energy energy efficiency, fast dynamics are Therefore, also model of the axial piston tween and side of drive. with the displacement system has be derived. gether aaa high efficiency, aaa fast are also aa dynamic dynamic model ofto the axial piston units units with the displacement system has to be derived. gether with with high energy efficiency, fast dynamics dynamics are Therefore, obtained in this setup. In most contributions it is assumed with the displacement system has to be derived. gether with a high energy efficiency, a fast dynamics are obtained in this setup. In most contributions it is assumed with the displacement system has to be derived. obtained in this setup. setup. In axial most contributions contributions it is isperformed assumed In this contribution, two mathematical models of the conthat the in control of the the piston units units is is obtained this In most it assumed In this contribution, two mathematical models of the conthat the control of axial piston performed this two mathematical models of the that the control of the axial piston units is performed sidered drive train are both fulfilling by standard displacement controllers and therefore the In In this contribution, contribution, twodeveloped, mathematical models of different the conconthatstandard the control of the axial piston units is performed sidered drive train are developed, both fulfilling different by displacement controllers and therefore the drive train are developed, both fulfilling different by standard standard displacement displacement controllers controllers and and therefore therefore the the sidered requirements. A detailed model will be given in Section 2. sidered drive train are developed, both fulfilling different by requirements. A detailed model will be given in Section 2. requirements. A detailed model will be given in Section 2. ⋆ The authors from Vienna University of Technology highly appreThe purpose of this model is to validate, evaluate and comrequirements. A detailed model will be given in Section 2. ⋆ The authors from Vienna University of Technology highly appreThe purpose of this model is to validate, evaluate and com⋆ The purpose of this model is to validate, evaluate and comThe authors from Vienna University of Technology highly apprepare different control strategies. Moreover, the detailed ciate the technical and financial support provided by Robert Bosch ⋆ The purpose of this model is to validate, evaluate and comThethe authors fromand Vienna University Technology highly apprepare different control strategies. Moreover, the detailed ciate technical financial supportofprovided by Robert Bosch pare ciate the GmbH. pare different different control control strategies. strategies. Moreover, Moreover, the the detailed detailed ciate the technical technical and and financial financial support support provided provided by by Robert Robert Bosch Bosch GmbH. GmbH. GmbH.
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ϕs ph
ICE
ϕs2
diff. gear
ph cv
rp
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ωc1
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rfb
Vh ϕs1
i0
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509
ra
cfb
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L
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u
R pa A1 , m1
pl see Figure 2
pl
accumulator Fig. 1. SHHV system topology for IVT operation mode.
Fig. 2. Axial piston unit with displacement system.
model has to be able to capture parameter variations in order to verify the robustness of the developed control strategies. Additionally, a reduced-order model will be derived in Section 3 which is suitable for online computation and realtime applications. This model serves as a basis for controller and estimator design. Thus, the main focus is to minimize the computational costs by simultaneously preserving the accuracy of the detailed model as good as possible.
generalized coordinates as q = [ϕs , ϕc ]T , the equations of motion are obtained from d ∂T ∂T − = Qj , j ∈ {s, c} (1) dt ∂ ϕ˙ j ∂ϕj with the kinetic energy T = 1/2q˙ T Dq˙ and the generalized forces Qj . The positive definite mass matrix of the system is given by D = diag (Ds , Dc ) with 1 rp2 mp np + 2 (m1 + m2 ) ra2 (2a) Ds (ϕs ) = Js + 2 cos4 (ϕs ) 1 Dc (ϕs ) = Jc + rp2 mp np 2 + tan2 (ϕs ) , (2b) 2 where Js and Jc are the inertias of the swash plate and the cylinder barrel with respect to the corresponding axis of rotation. The np pistons, each one of mass mp , perform an axial movement in the cylinder barrel in a distance of rp from the shaft. Furthermore, m1 and m2 denote the masses of the pushing and pulling pistons which are both mounted in a distance of ra from the shaft. The equations of motion of the swash plate and the shaft can be written in the form =:Φs(ϕs ,ωs ,ωc ) ϕ˙ s = ωs (3a) 1 1 dDc 2 1 dDs 2 (3b) ω − ω +Qs ω˙ s = Ds (ϕs ) 2 dϕs c 2 dϕs s dDc 1 − (3c) ωs ωc + Qc , ω˙ c = Dc (ϕs ) dϕs with the angular velocities ωs and ωc and the centrifugal term Φs . The generalized forces can be obtained from Figure 2 1 (ra (A1 pa − A2 ph − Ffs ) + Ts ) Qs = cos2 (ϕs ) (4a) − kv ωs − kc sign(ωs ) Vp (ϕs ) ph − Td − Te , (4b) Qc = 2π with the areas A1 and A2 of the pushing and pulling piston. The force Ffs = cfs ra (tan(ϕs ) − tan(ϕˆs )) + F0 tanh(κ (ϕs − ϕˆs )) (5) results from the pre-stressed restoring springs with stiffness cfs , pre-tension F0 and rest position ϕˆs . The pretension F0 is approximated by the second term on the right-hand side of (5), with the tuning parameter κ. A mean-value model as, e.g., presented in Ivantysyn and Ivantysynova (1993) is employed to account for the resulting swivel torque Ts = cs ph which is the reaction torque
2. DETAILED MODEL A mathematical model of the SHHV drive train depicted in Figure 1 will be derived in this section. In IVT operation mode, the hydraulic accumulator on the high-pressure side is disconnected from the hydrostatic drive and may be neglected in the subsequent considerations. The first step is devoted to the modeling of a single axial piston unit (APU) as shown in Figure 2. The model is then combined with the remaining components to form the overall drive train model. 2.1 Axial Piston Unit The APU with the electro-hydraulic displacement system comprising two hydraulic cylinders and an electromagnetically actuated 3/3-way spool valve is depicted in Figure 2. The APU is self-supplied which means that the volume flow for the displacement is branched off the supply port of the pump with pressure ph . A displacement in the direction of the swash plate angle ϕs is realized by supplying hydraulic oil across the variable orifice of the valve to the pushing cylinder with pressure pa . Connecting the cylinder chamber with tank pressure pl moves the swash plate in the opposite direction, where the motion is supported by a hydraulic pulling cylinder connected to the supply port pressure ph . The angular position of the swash plate is fed back to the spool valve via a mechanical spring, which operates as a stabilizing mechanical controller for the pump displacement. As will be shown later, this electroproportional (EP) displacement system gives an angular position of the pump which is basically proportional to the magnetic force Fm . In a similar manner as described in Fuchshumer (2009), the APU is modeled as a rigid body system with two degrees of freedom which are determined by the swash plate angle ϕs and the shaft angle ϕc , see Figure 2. Defining the 509
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with the coefficients cij fitted to measurements of the torque by means of a least-squares identification. This approximation procedure is also of interest for an overall energy efficiency optimization. It can be seen from the upper part of Figure 3 that there is nearly a linear dependence between the speed and the torque loss, which is in accordance with theoretical considerations. A more detailed analysis of the internal hydraulic design of the APU would explain the nonlinear dependence on the pressure. Nevertheless, the measured data can be approximated very well in a wide operating range.
The mass balance of the pushing cylinder chamber reads as � � A1 ra ωs β p˙ a = − 2 + qa , (7) Va + A1 ra tan(ϕs ) cos (ϕs ) with the effective bulk modulus β of the hydraulic fluid, the volume Va of the cylinder chamber for ϕs = 0 and the volumetric flow rate qa into the cylinder chamber. In order to guarantee low leakages, the valve exhibits a closed-center design with overlap γ > 0, see the left hand side of Figure 4. Of course, no perfect sealing for −γ < sv < γ can be achieved due to the leakage along the small gaps between the valve spool and the body. An analytical model of these leakages is beyond the scope of this paper. Instead, a small opening of the valve is considered in the dead zone. Using the equations of a turbulent flow of an incompressible fluid from McCloy and 510
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Fig. 3. Measured and approximated torque and volumetric loss. The quantities are scaled with respect to the ideal torque T0 and volume flow q0 of the APU at nominal pressure p0 , angular speed ωc0 and full displacement ϕs = ϕ¯s . Martin (1980) and assuming pl ≤ pa ≤ ph , the following model can be obtained for the valve volume flow � αv 2 Av (sv ) √ph − pa , sv ≥ 0 �ρ qa = (8) −αv 2 Av (sv ) √pa − pl , sv < 0, ρ
with the orifice contraction coefficient αv and the density ρ of the hydraulic oil. The unique relationship between the valve spool position sv and the opening area Av (sv ) can be derived by a geometric consideration, see Fuchshumer (2009), and is depicted in Figure 4. It can be seen that the actual non-ideal characteristics near the dead band is obtained by a slight shaping of the ideal characteristics. pa
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A physical explanation of the torque loss Td , which takes into account the non-ideal behavior of the APU in terms of mechanical efficiency, is given, e.g., in Ivantysyn and Ivantysynova (1993). There, it is explained that the accumulated pressure drop across the internal resistance network of hydraulic pumps causes a difference between the ideal and actual engine torque. Besides, viscous friction occurring between the sliding parts results in a torque loss which depends on the different gap sizes and dynamic viscosity of the hydraulic oil. Because of the many different influencing factors it is beyond the scope of this contribution to derive a physically motivated model which captures all these effects separately. Instead, the reported significant dependence of the torque loss Td on the pressure ph and the angular speed ωc is approximated by the polynomial 2 2 � � cij pih ωcj , (6) Td (ph , ωc ) =
4 torque loss Td /T0 [%]
of the pistons acting on the swash plate with the constant cs . The viscous and Coulomb friction coefficients kv and kc account for the friction in the bearing of the swash plate. The ideal average engine torque Vp (ϕs ) /(2π)ph is determined by the adjustable geometric displaced volume Vp (ϕs ) = 2np rp Ap tan(ϕs ) of the pump with the effective area Ap per piston, cf. Manring (2005). Finally, the right hand side of (4b) comprises the torque loss Td and the external load torque Te on the pump shaft. Remark 1. The swash plate dynamics (3a) and (3b) are only valid between the limit stops ±ϕ¯s of the swash plate, |ϕs | ≤ ϕ¯s . If this condition is violated, the model equations (3a) and (3b) are switched to ϕ˙ s = ω˙ s = 0 and ϕs is set to the corresponding limit stop. In this state, the force acting on the swash plate is evaluated, which decides when to switch to the original system dynamics (3a) and (3b).
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of the displacement system. The volumetric loss ql (ph , ωc ) accounts for the hydraulic efficiency of the APU due to leakage flows and is discussed, e.g., in Ivantysyn and Ivantysynova (1993). For the same reasons as outlined in the context of the torque loss, the volumetric loss can be approximated by fitting a polynomial similar to (6) to measured data of the volume flow. The lower part of Figure 3 shows the result of this approximation. The valve is actuated by two proportional magnets with input voltages u1 and u2 and currents i1 and i2 . By introducing the effective current i = i1 − i2 , a relationship Fm (i) for the resulting bidirectional magnetic force is obtained, cf. Figure 2. The current dynamics are assumed to be linear d 1 i = (−Ri + u) , (11) dt L with the electric resistance R, the inductance L and the effective voltage u = u1 − u2 , which is regarded as the control input to the system. The equations of motion of the valve are given in the form s˙ v = wv (12a) 1 w˙ v = (−c0 sv − Fd − Ffb − Fjet + Fm (i)) , (12b) mv where mv denotes the mass of the valve spool and c0 = cv + cfb is the combined spring stiffness of the centering and the feedback spring. A static friction model of the form Fd = kv wv + kvc + kvh − kvc exp −kvb wv2 sign(wv ) is used with the viscous damping coefficient kv and the friction parameters kvh , kvc and kvb , see Figure 5. Flow forces acting on the valve spool are modeled according to McCloy and Martin (1980) by 2αv cos(ϕjet ) Av (sv ) (ph − pa ) , sv ≥ 0 Fjet = (13) −2αv cos(ϕjet ) Av (sv ) (pa − pl ) , sv < 0 with the jet angle ϕjet . The force (14) Ffb = cfb rfb sin(ϕs ) + F0 on the right hand side of (12b) results from the feedback spring attached to the valve spool at radius rfb from the shaft of the pump. The introduction of F0 is motivated as follows: The equilibrium of (12) and (7) for sv = 0 reads as (15) cfb rfb sin(ϕs ) + F0 = Fm (i) . This implies a stationary nonlinear relationship between the current i and the swash plate angle ϕs , also denoted as EP (electro-proportional) map. For an ideal setup, Fm = 0 would hold which implies ϕs = 0. The real system, however, suffers from unknown asymmetries of the feedback system which are summarized in the parameter F0 . Finally, the limit stops ±¯ sv of the valve spool are modeled by switching the dynamics (12) to s˙ v = w˙ v = 0 in the same manner as described in Remark 1. 2.2 Overall Drive Train Model The mathematical model for the SHHV drive train from Figure 1 consists of the equations (3a) and (3b), (7), (11) and (12) of the swash plate, the cylinder chamber, the 511
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Fig. 5. Friction force on the valve spool. The velocity is scaled to the maximum value wv0 occurring at a typical setpoint change of the APU. current and the valve for the primary and secondary side with the corresponding swash plate angles ϕs1 and ϕs2 . The secondary APU is directly coupled to the vehicle via a differential gear. Thus, the angular speed ωc2 is proportional to the vehicle speed. In this work, no model for the vehicle is included. Instead it is assumed that ωc2 is a measured external input to the system. The primary APU is coupled to the ICE, which gives dDc1 1 Vp1 (ϕs1 ) ω˙ c1 = ph − Td1 + i0 Ti , − ωs1 ωc1 + Dc1 dϕs1 2π (16) see (3c), with the external torque Ti of the ICE transformed by the gear transmission ratio i0 . Subsequently, an index 1 or 2 is always used to refer to the primary and secondary APU, respectively. The ICE is assumed to be torque-controlled with the desired torque Tˆi as control input and the closed-loop dynamics T˙i = ai Tˆi − Ti , (17)
with the time constant 1/ai . Finally, the coupling between the two hydrostatic engines is modeled by a constant hydraulic volume Vh with the pressure dynamics given by β (qp1 + qp2 ) , (18) p˙ h = Vh with the primary and secondary volumetric flow qp1 and qp2 according to (9). The pressure pl at the low-pressure side of the hydrostatic drive is assumed to be constant. 3. REDUCED-ORDER MODEL The nonlinear model of the drive train is of order fifteen and thus entails a very high computational burden, in particular if it serves as a basis for nonlinear model predictive control. Therefore, in this section a reduced-order model is derived by successively reducing the complexity of the APU model presented in Section 2.1. Thereby, the singular perturbation theory is exploited, see, e.g., Kokotovic et al. (1999) for a detailed discussion on singularly perturbed systems. In a first step, the valve spool (12) is considered. In real applications, a sinusoidal dither signal is superimposed to the actuating voltage u to overcome the static friction of the spool valve. Therefore, the frictional components related to kvh and kvc can be neglected and (12) can be transformed to the standard form of singular perturbation theory
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(19a) 1 ǫw˙ v = −wv + (−c0 sv − Ffb − Fjet + Fm (i)) , (19b) kv with the perturbation parameter ǫ = mv /kv . For ǫ → 0, which means that the damping is high compared to the mass of the valve spool, the solution of (19b) by wv used in (19a) results in the reduced slow dynamics of the valve spool 1 s˙ v = (−c0 sv − Ffb − Fjet + Fm (i)) . (20) kv In a second step, the equations of motion (3a) and (3b) of the swash plate together with the pressure dynamics (7) are rewritten in the form ϕ˙ s = ωs (21a) 1 (Φs + Qs ) (21b) Ds (ϕs ) A1 ra ωs β − 2 + qa (pa ) . (21c) p˙a = Va + A1 ra tan(ϕs ) cos (ϕs ) As has been shown in Fuchshumer (2009), the states ωs and pa correspond to the fast dynamics and ϕs is the slow part of the system. Using the quasi-stationary assumption ω˙ s ≈ 0 (22a) (22b) p˙ a ≈ 0 in (21b) and (21c) gives the quasi-stationary states ω ¯ s and p¯a . Using these results in (21a) yields the slow dynamics cos2 (ϕs ) ϕ˙ s = qa (¯ pa ) (23) A1 ra for the swash plate angle. Bearing in mind the nonlinear behavior of qa (¯ pa ), the nonlinear equation 1 (ra (A1 p¯a − A2 ph − Ffs ) + Ts ) 0 = Φs + 2 cos (ϕs ) (24) − kv ω ¯ s − kc sign(¯ ωs ) has to be solved for the stationary pressure p¯a . An approximate solution can be found by neglecting the centrifugal term Φs and the friction terms in the second line of (24) leading to the explicit solution 1 Ts A2 ph + Ffs − . (25) p¯a = A1 ra The simplifications (19) to (20) and (21) to (23) yield a model order reduction by three states. Combining this reduced model with the current dynamics (11) and the shaft dynamics (3c), the reduced-order model for the APU is of order four. The fast dynamics of the original nonlinear 7th order model is considered in a quasi-stationary manner, thus leading to a reduced-order model suitable for the controller design. ω˙ s =
4. EXPERIMENTAL RESULTS The objective of this section is the validation of the detailed and reduced-order model derived in Sections 2.1 and 3. As the main components of the system are the APUs, various measurements are carried out on an experimental test bench for an APU. The following assumptions hold throughout the whole analysis: • The APU is coupled to a speed-controlled load engine and therefore the dynamics of the engine shaft are not considered in the analysis. 512
• To overcome the friction of the spool valve, a sinusoidal dither signal is always superimposed to the actuating voltage u. • The simulation results are obtained by feeding the models with the actuating voltage u of the test bench system. • The voltage, current and pressure are normalized to typical set point values u0 , i0 and p0 and the swash plate displacement and valve position are normalized to the corresponding limit stops ϕ¯s and s¯v . In a first step, the quasi-stationary behavior of the model is examined. For this purpose, triangular reference signals are fed to the system which can be considered slow compared to the whole system dynamics. The stationary behavior of the model is adapted to the measurements by fitting the unknown relationship Fm (i) from the EP map (15). A comparison between the two models and the measurement signals is shown in Figure 6. In this figure, the dither signal of the voltage u, the current i and the valve position sv is filtered out of the signals because only the mean values are of interest. It can be seen that both models give a very good approximation of the normalized displacement α = tan(ϕs ) / tan(ϕ¯s ), which is of primary interest for a higher level swash plate control. A small deviation in the simulated current results in a slight deviation for positive displacements which is mainly due to a different amplitude of the dither signal. The dither signal is visible in the pressure pa , which is also predicted well by the detailed model. Although only the mean value can be approximated by the reduced-order model, it is still a good basis for controller design since the frequency of the dither signal is considerably higher than the desired closedloop dynamics. It follows from the inspection of the valve position sv that its mean value is in the range of the dead band of the valve because of the rather slow displacement velocities. Despite the lack of a detailed leakage model the mean value of the valve position can be predicted very well by both models. Figure 7 shows a measurement scenario with a displacement over a wide range in a short time compared to the system dynamics. Both displacement directions are considered and the signals are not filtered for the dither frequency. Both models are able to predict the dynamic behavior of the swash plate in a very good way. Even in this scenario exploiting the whole stroke of the valve, a good model prediction outside the dead band is accomplished. The pressure in the pushing cylinder is again very well approximated by its quasi-stationary approximation. 5. CONCLUSION In this paper, a mathematical model of a serial hydraulic hybrid vehicle was derived. By means of a systematic model order reduction scheme, the nonlinear model was simplified and a reduced-order model for the underlying self-supplied variable displacement axial piston units was obtained. Both models were validated by measurements on an industrial test bench and it was shown that the reduced-order model almost preserves the accuracy of the detailed model. In contrast to Fuchshumer (2009), the valve dynamics are incorporated into the controller design model since they
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Fig. 6. Simulated and measured data for a slow displacement of the APU. have a significant influence on the overall system dynamics. Besides, the system is subject to constraints such as the limit stop of the valve or the constraints of the maximum voltage and current. Neglecting these nonlinearities in the controller design degrades the performance of classical nonlinear controllers. MPC (Model Predictive Control ) strategies offer the possibility to systematically take into account such constraints in the controller design by online predicting the system trajectories and finding the optimal control with respect to a cost functional, see, e.g., Gr¨ une and Pannek (2011). Because of the high computational burden of MPC, it is impossible to apply the detailed model in a real-time MPC controller. On the contrary, the reduced-order model can serve as a perfect basis for such a controller. Current research is directed to the application of nonlinear MPC to the considered system. REFERENCES Deppen, T.O. and Alleyne, A.G. (2012). Optimal energy use in a light weight hydraulic hybrid passenger vehicle. Journal of Dynamic Systems, Measurement and Control, 134(4). Erkkil¨ a, M., Bauer, F., and Feld, D. (2013). Universal Energy Storage and Recovery System - A Novel Approach for Hydraulic Hybrid. In Proceedings of the 13th 513
Fig. 7. Simulated and measured data for a fast displacement of the APU. Scandinavian International Conference on Fluid Power, SICFP2013, 92, 45–52. Link¨oping, Sweden. 3 – 5 June 2013. Fuchshumer, F. (2009). Modellierung, Analyse und nichtlineare modellbasierte Regelung von verstellbaren Axialkolbenpumpen. Shaker, Aachen, Germany. Gr¨ une, L. and Pannek, J. (2011). Nonlinear Model Predictive Control: Theory and Algorithms. Communications and Control Engineering. Springer, London. Ivantysyn, J. and Ivantysynova, M. (1993). Hydrostatische Pumpen und Motoren: Konstruktion und Berechnung. Vogel, W¨ urzburg, Germany. Kokotovic, P., Khali, H., and O’Reilly, J. (1999). Singular Perturbation Methods in Control: Analysis and Design. Society for Industrial and Applied Mathematics, Philadelphia, USA. Kugi, A., Schlacher, K., Aitzetm¨ uller, H., and Hirmann, G. (2000). Modeling and simulation of a hydrostatic transmission with variable-displacement pump. Mathematics and Computers in Simulation, 53, 409–414. Manring, N. (2005). Hydraulic Control Systems. John Wiley & Sons, New Jersey, USA. McCloy, D. and Martin, H. (1980). Control of Fluid Power: Analysis and Design (2nd edition). John Wiley & Sons, New York, USA.