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Huan LiSchool of Mechanical Engineering,
Beijing Institute of Technology,
Beijing 100081, China
e-mail: [email protected]
Ying HuangSchool of Mechanical Engineering,
Beijing Institute of Technology,
Beijing 100081, China
e-mail: [email protected]
Guoming Zhu1
Fellow ASME
Mechanical Engineering,
Michigan State University,
East Lansing, MI 48824
e-mail: [email protected]
Zheng LouJiangsu Gongda Power Technologies Ltd., Co.,
Changshu 215513, China
e-mail: [email protected]
Linear Parameter-Varying Modelof an Electro-Hydraulic VariableValve Actuator for InternalCombustion EnginesThis paper presents a novel linear parameter-varying (LPV) model of an electro-hydraulic variable valve actuator (EHVVA) for internal combustion engines that is capa-ble of continuously varying valve timing with dual-lift. The dual-lift is realized mechani-cally through a hydraulic lift control sleeve; valve opening (VO) terminal and closingseating velocities are regulated using a top or bottom snubber; and opening and closingtimings, as well as lift profile area, are controlled by the valve actuation timing andhydraulic supply pressure. First, nonlinear mathematical system model is developedbased on the Newton’s law, orifice flow equation, and fluid constitutive law, where thefluid dynamics of the actuation solenoid valve, actuation piston, passages, and orifices,that influence the engine valve profile, are considered in detail. Second, to have an LPVcontrol-oriented model, the order of nonlinear model is reduced and subsequently trans-formed into an LPV model with minimal deviation by carefully considering the systemnonlinearities, time delay, and time-varying parameters. Calibration and validationexperiments for both nonlinear and LPV models were performed on the test bench underdifferent operational conditions. The key time-varying parameters, the time constant ofthe actuation piston top pressure and the discharge coefficient, are highly nonlinear asfunctions of temperature-sensitive fluid viscosity and are determined using the test datathrough the least-squares optimization. With the identified and calibrated model parame-ters, simulation results of both nonlinear and LPV models are in good agreement with theexperimental ones under different operational conditions. [DOI: 10.1115/1.4037286]
1 Introduction
The increasingly stringent regulations for NOx, HC, and CO2
emissions lead to continuous improvement of vehicular technolo-gies. For more than 100 years, the internal combustion engineshave been the dominant power sources for automotive until theintroduction of hybrid electric vehicles (HEV) and electric vehicles(EV) in recent years. Since the battery power density, charging dura-tion, and infrastructure need to be improved to make EVs practical,most analysts agree that performance improvement of existing inter-nal combustion engines is necessary to reduce vehicle fuel consump-tion and emissions powered by conventional internal combustion(IC) engines for decades [1,2]. In HEV systems, innovation in enginetechnologies such as variable valve actuation (VVA) will also pro-vide significant benefits in fuel economy, leading to improved hybridpowertrain performance.
Among the three fundamental combustion control parameters(i.e., ignition timing, fuel and air), engine air management is theonly remaining one to be regulated in a cycle-by-cycle mannersince the other two are electronically controlled and updated everyengine cycle. The VVA is able to bring revolutionary air chargecontrol features to further improve IC engine performance. Theengine valve timing, lift, and duration can be freely adjusted bythe VVA system under different engine operational conditions foroptimized combustion efficiency, thus greatly improving the fueleconomy and output torque performance with satisfactory emis-sions [3–5]. These benefits include the pumping loss reductionunder throttleless operations, cylinder deactivation, trappedexhaust gas recirculation regulation, in-cylinder turbulence con-trol for stratified combustion, etc. VVA is also a key enabler forAtkinson (or Miller) cycle engines widely adopted by HEVs and
for new combustion modes such as homogenous charge combus-tion ignition (HCCI) [6] and turbulent-jet ignition.
The most commonly used VVA systems are the KSPG univalvesystem [7], the BMW valvetronic system [8], and the cam phasingvariable valve timing (VVT) systems (e.g., hydraulic vane typeVVT systems [9,10], and electrical VVT systems [11]). Using thethrottleless load control, Flierl et al. [7] demonstrated that the univalvesystem implemented on a 2.0 L turbocharged gasoline engine is ableto improve the fuel consumption by 14% at partial load and increasethe low speed peak torque by 10%. Negurescu et al. [5] showed thatSI engine efficiency can be increased by 29% due to VVT, comparedwith a conventional throttled engine. Ren and Zhu [11] and Zhanget al. [6,12] realized the mode transition between SI and HCCI com-bustion to achieve HCCI combustion at low and medium engine loadand SI combustion at high load using two electric motor-driven camphasing VVTs due to their fast transient responses. The cam-basedVVT systems are widely used due to their compatibility to traditionalvalve trains with known reliability and durability. However, the lim-ited ability of adjusting valve timing and lift for cam-based valve sys-tems prevents their application to nontraditional engines such ashydraulic free piston engines. Therefore, exploring noncam-basedVVA system and developing associated control strategies are crucialfor further improving engine performance to meet future fuel econ-omy and emission improvement demand.
Recently, intensive research has been focused on camless VVAsystems for internal combustion engines. The camless VVA sys-tem can be divided into three main groups: the electro-magnetic,electro-hydraulic, and electro-pneumatic actuators. The electro-magnetic VVA systems provide fully flexible valve timing andduration with relatively high efficiency, but its repeatability ishighly sensitive to valve opening (VO) load with fairly high elec-tric power demand [13]. In addition, it also has high valve seatingvelocity due to the nonlinear electromagnetic force and is limitedto a fixed valve lift [13–15]. Therefore, complicated real-timecontrol algorithms are required for soft seating control and liftcontrol, which increases the implementation complexity. The
1Corresponding author.Contributed by the Dynamic Systems Division of ASME for publication in the
JOURNAL OF DYNAMIC SYSTEMS, MEASUREMENT, AND CONTROL. Manuscript receivedDecember 21, 2016; final manuscript received June 21, 2017; published onlineAugust 29, 2017. Assoc. Editor: Zongxuan Sun.
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modeling and control of the electro-magnetic VVAs [16–18] havebeen intensively studied.
The electro-hydraulic VVAs are usually considered as the mostflexible VVA concept using the pressurized hydraulic fluid todrive the engine valves. Valve timing and duration can be continu-ously varied since the hydraulic flow rate can be tailored by sole-noid valves [19]. These valves also have very fast responses dueto high hydraulic power density and good soft landing perform-ance with low cost. The electro-hydraulic VVAs, designed by Louet al. [20,21], have a build-in soft-landing feature; thus, they donot require complicated control strategy for seating velocity con-trol. Gillella and Sun [14] achieved precise valve motion controlof an electro-hydraulic VVA using a low-cost feedback controlsystem. Sun and Kuo [15] developed a transient control strategyfor an electro-hydraulic VVA to ensure precise valve profile track-ing under both constant and varying engine speed. For electro-hydraulic VVAs, as well as the hydraulic VVT systems, one keychallenge is the nonlinear temperature-sensitive characteristics ofthe hydraulic fluid viscosity, which is critical during the engine’scold start [19]. Although extensive studies indicated that this canbe compensated by using low lifts at lower temperature, this prob-lem remains challenge for electric-hydraulic VVAs.
To overcome the temperature effect to the hydraulic oil viscos-ity, the electro-pneumatic VVAs are studied using the pressurizedair as actuation power source that is relatively insensitive to tem-perature variations. However, the high compressibility of the pneu-matic fluid makes it difficult to control valve motion and the seatingvelocity precisely. The pneumatic actuator is often incorporated witha latch mechanism using the magnet [22] or hydraulic fluid [23] todeal with the soft seating and energy consumption problems. Themain disadvantage is the requirement of a separate pneumatic fluidsupply system as part of the engine system with increased complex-ity and cost. Ma et al. [23–25] systematically studied the modelingand control of an electro-pneumatic valve actuator.
The linear parameter-varying (LPV) systems are a class offinite-dimensional linear parameter-varying plants whose systemcoefficients are functions of a set of measurable varying parame-ters. The LPV modeling and control techniques can be used todeal with nonlinear parameter-varying systems, have received agreat deal of attention in automotive applications, and wereapplied recently to both gasoline and diesel engines [26–29].White et al. [27] used a time-varying friction coefficient toaccount for the variation of the engine oil viscosity for a VVT sys-tem driven by an electric motor with a planetary gear train. Enginespeed and oil pressure are used as the varying parameters to forman LPV hydraulic VVT system [28]. The LPV control is also usedto compensate for the variable time delay presented in the air–fuelratio control loop of a lean burn spark ignition engine [29].
In this paper, a new electro-hydraulic variable valve actuator(EHVVA) [21] is presented. The EHVVA is capable of variablevalve timing, lift, and duration through control of actuation solenoidvalves and supply pressure with very low cost. Two frequently usedlifts (high and low lifts) are achieved mechanically using a hydrau-lic lift controlled sleeve. Simple top and bottom snubbers are usedto ensure soft seating for both opening and closing operations. Theenergy consumption of both high and low lift operations is provedto be comparable with the traditional cam-based valve trainsrequired by original equipment manufacturers [21]. This paperdevelops the nonlinear mathematical model for the EHVVA system.Due to the nature of high-order dynamics and system nonlinearitywith time delay and varying parameters, the high-order nonlinearsystem model is simplified to obtain a low-order control-orientednonlinear model with certain modeling error, and subsequentlytransformed into an LPV model without loss of modeling accuracy.
This paper is organized as follows: System dynamics isdescribed in Sec. 2, and a nonlinear mathematical model is devel-oped in Sec. 3. The nonlinear model reduction and LPV modelingare introduced in Sec. 4. In Sec. 5, both reduced-order nonlinearand LPV models are experimentally calibrated and validated.Finally, conclusions are drawn in Sec. 6.
2 System Dynamics
As shown in Fig. 1, the EHVVA system consists of an actuationpiston, an actuation cylinder, an actuation valve consisting of bothpilot and main valves, a lift valve, a lift control sleeve, a checkvalve, a relief valve with a pressure regulator, an orifice switchvalve with its control solenoid, and a returning spring (part ofengine valve system). The actuation cylinder has the top, bottom,and lift ports. The dash-dotted lines denote the hydraulic fluidpaths at hydraulic tank pressure that is close to the atmospherepressure. The bold-solid lines are connected to the hydraulic fluidsource, a high-pressure (Ph) accumulator, that is pressurized by amotor (or crank-shaft)-driven pump. The bold-dashed lines arelinked to the low-pressure fluid source with back pressure (Pl)that is pressurized by the returning fluid from the top port at valveclosing (VC) and maintained by the low-pressure accumulatorthrough a check valve. The top port is switched between the low-and high-pressure fluid sources via the actuation valve that con-sists of a three-way two-position hydraulic main valve controlledby a solenoid pilot valve. The serial design is able to achieveresponse fast enough for engine applications with much lowercost than that of the single-stage solenoid valve. The bottom portis connected to the low-pressure fluid source. The actuation pistonand engine valve are pushed downward when the top port is pres-surized by the high-pressure fluid once the actuation valve is acti-vated, and returned upward by the returning spring when the topport is switched to the default low-pressure fluid. The lift port isswitched between the tank and high-pressure fluid sources via thesolenoid controlled lift valve. The lift control sleeve is used tosupply two discrete lifts (S1 and S2) corresponding to its two posi-tions (high and low positions). The sleeve moves to its high posi-tion once the lift port is pressurized by high-pressure (Ph) fluidwhen the lift control valve is activated; and otherwise, the sleeveremains at the default low position by the low-pressure (Pl) fluidapplied on the top of the sleeve surface. Note that there is suffi-cient force to maintain the sleeve at its high or low positions,respectively, since the low-pressure (Pl) is designed to be faraway from both the high-pressure (Ph) and the tank pressure. Thetwo discrete lifts shall meet most of the engine operation demands
Fig. 1 EHVVA schematic
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with fast response due to high supply pressure and accurate lift dueto hard lift constrains. At the same time, continuous valve lifts can beachieved by accurately controlling the supply and back pressures.
The opening terminal and closing seating velocities are keysources of the valve-train noise and vibrations, and high terminaland seating velocities could reduce the engine valve reliabilityand durability. The valve bouncing due to high seating velocitycan affect the engine performance. The top and bottom snubbersare designed for EHVVA to reduce both valve opening terminaland closing seating velocities. The top snubber is used for softseating and consists of a notch on the top edge of the actuationpiston, an undercut ðLs1Þ inside the top of actuation cylinder, acheck valve, and a relief valve with two parallel arranged top ori-fices. Note that one of the orifices can be opened and closed by aswitch valve. When the actuation piston moves up and past thetop undercut into the snubbing zone, the fluid above the piston isforced to escape only through the orifices and the notch. The notcharea gets small as the piston goes up and is designed to provide asmooth transition at the beginning of the snubbing process whilethe orifices are designed to provide more precise control during theseating process. The soft seating is basically achieved by the topundercut, the notch, and the orifices. The switch valve is designedto adjust the flow area of the top orifices to achieve different levelsof soft seating under different operational conditions. The reliefvalve is pressure-sensitive and provides a fast pressure relief whenthe snubbing process is overpressurized to prevent the valve frombouncing back. The check valve is designed to open at the valveopening for a fast opening by allowing more fluid flow into the cyl-inder and to be closed to reduce the flow area to ensure soft seating.
The bottom snubber is designed for a soft landing at the termi-nal stage of the valve opening. It consists of a notch on the bottomedge of the actuation piston and an undercut with a distance ðLs2Þ tothe top of the lift control sleeve. The bottom snubber is simpler thanthe top snubber mainly due to the additional counter force providedby the returning spring that reduces the opening terminal velocity.
Figure 2 illustrates the engine valve dynamic parameters includingthe valve lift, valve velocity, the high-pressure (Ph), low-pressure(Pl), pilot valve outlet pressure, actuation piston top pressure, andthe associated control signal.
For the high-lift operation with 7.2 MPa supply pressure, thevalve opening and closing durations, defined as the time durationbetween 1% and 99% of the stroke, are 3.7 ms and 2.8 ms, respec-tively. Due to the top and bottom snubber design, there is a signifi-cant deceleration during the landing (opening terminal) andseating processes. The sharp rise of the piston top pressure at time“d” indicates that the top snubber provides an effective snubbingforce during the seating process. The dash-dotted and solid liftcurves represent for the seating process with (S-On) and without
(S-Off) the switch valve activated. The corresponding velocitycurves are also shown in dash-dotted and solid lines. The dash-dotted line has a much lower seating velocity at 0.25 m/s (absolutevalue) compared with 0.69 m/s when the switch valve is turned off.However, it comes with a penalty of a 1 ms long tail when theswitch valve is turned on, that is, it takes additional 1 ms forthe valve to be fully closed. Note that the additional 1 ms tail makesthe EHVVA seating profile compatible with the conventional camprofile, leading to a compatible air charge performance in this case.
The time duration td is the valve actuation delay includingthe pilot valve delay t0 and the main valve delay t1. Note that t0is a combination of the solenoid electromagnetic dynamics, themechanical and hydraulic fluid dynamics, and t1 is the requiredtime for the hydraulic fluid at the pilot outlet to push the mainvalve open and the time-associated filling and draining processes.That is why the pilot pressure increases and decreases during theopening and closing processes, respectively. After the main valveis fully open, the piston top pressure and the valve lift rise rapidlyalmost at the same time. The variation of the supply pressure Ph
and back pressure Pl is mainly affected by charging and discharg-ing the accumulators due to the valve motion.
As a summary, the EHVVA dynamics includes the time delays,the soft landing and seating processes, and system hydraulic pres-sure variations that affect the valve actuation profile (e.g., theopening and closing time, lift area, etc.). Note that the valve pro-file affects the engine combustion process significantly and ismodeled in detail in Sec. 3.
3 Nonlinear Mathematical Modeling
In this section, a detailed dynamic model of the EHVVA isdeveloped component by component based on the governingequations of the fluid dynamics in the actuation cylinder, accumula-tors, passages and orifices, as well as the valve dynamics. TheEHVVA components can be divided into four groups including theactuation piston, the actuation valve, the top and bottom snubbers,and the accumulators for the high- and low-pressure fluid sources.
According to Fig. 1, the actuation piston is balanced by the topand bottom hydraulic pressures, the system returning spring forceand the associated friction force, assuming the returning springforce is large enough and will keep the valve stem following theactuation piston movement. The actuation piston dynamics can beobtained by applying Newton’s second law; see below
me€LþCf
_LþksðLþdsÞ¼AtPt�AbPb;
€L¼ _L¼ 0;
0<L< S
L¼ 0;S
((1)
where me is the effective mass of total moving parts; L is theengine valve displacement; and Cf is the friction coefficient; ks
and ds are the returning spring stiffness and preload. Note that At
and Ab are the cross areas of the actuation piston top and bottom;Pt and Pb are the actuation piston top and bottom pressures to bederived in the top and bottom snubber model later; and S is theengine valve lift switched between the high lift S1 (7.5 mm) andlow lift S2 (4 mm) through the lift control valve. When Lapproaches the lower or upper limits (i.e., L ¼ 0 or S), the bottomor top seat in the actuation cylinder provides additional counter-force that balances the hydraulic and spring force difference andstops the valve immediately. The engine valve remains in the limitposition (L ¼ 0 or S) until the piston top pressure Pt is changeddue to activating or deactivating the actuation valve.
The actuation valve has a serial design consisting of a solenoidpilot valve and a hydraulic operated main valve. Once the pilotvalve is activated, the high-pressure fluid activates the main valveby pushing the spool open to connect the top port to the high-pressure fluid source. The pilot valve can be modeled as threeparts: a spool model, a solenoid model, and an orifice model. Thespool model shown in Eq. (2) is used to describe the spool motionand calculate the associated displacement xp based on Newton’ssecond law
Fig. 2 Engine valve dynamic at high lift (initial Ph 5 7.2 MPa,22 �C)
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mep€xp þ Cfp _xp þ ks1ðxp þ ds1Þ ¼ Fmeg;
€xp ¼ _xp ¼ 0;
0 < xp < xp0
xp ¼ 0; xp0
((2)
where mep is the effective mass of moving parts for pilot valve;Cfp is the friction coefficient; ks1 and ds1 are the stiffness and thepreload of the spring, respectively; and xp0 is the maximumallowed displacement of the pilot valve spool. When the spoolapproaches its limit (i.e., xp ¼ 0 or xp0), it stops immediately dueto the additional counterforce provided by the pilot valve cylinderand remains at that position until the solenoid is activated ordeactivated.
The solenoid model [23] is used to determine the electro-magnetic force Fmeg by applying Kirchhoff law as shown in thebelow equation
U � IR� Li_I ¼ 0; Fmeg ¼ bI2ð1þ xp=aÞ�1
(3)
where U, I, R, and Li are the solenoid input voltage, current,resistance, and inductance, respectively; and a and b are constantsused to curve fit the empirical data provided by the manufacture.
The orifice flow model with a varying orifice area ðAorip ¼ wpxp)determined by the pilot spool displacement xp and calibrated con-stant coefficient wp is used to describe the fluid dynamics throughthe pilot valve orifice and calculate the delayed outlet pressure ~Ppo
using the nonlinear orifice flow equation given below
QVp ¼ CdAorip
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2sgn _Lð Þ u Ph � Patmð Þ þ Patm � ~Ppo tþ t0ð Þ
� �qoil
s
¼ Aspoolm _xm (4)
where QVp and Cd are the volumetric flow rate and discharge coef-ficient of the fluid across the pilot valve orifice; qoil is the densityof the fluid; Patm is the tank pressure; ~Ppo is the delayed Ppo bytime t0 (i.e., ~PpoðtÞ ¼ Ppoðt� t0Þ; see Fig. 2 for description); u isthe pilot solenoid on-and-off control input (on: 1 and off: 0); termuðPh � PatmÞ þ Patm represents the pilot valve inlet pressure andPh will be calculated by the accumulator model later; Aspoolm andxm are the stressed area and displacement of the main valve spool,respectively; and the sign function represents the flow direction.
The delayed pilot outlet pressure ~Ppo will then push the mainvalve open and determine the main valve displacement xm basedon the main valve model, which, similarly, can be modeled as aspool valve and an orifice model applying the same principlesused in Eqs. (2) and (4) for the pilot valve model to determine thedelayed main valve outlet pressure ~PmoðtÞ ¼ Pmoðt� t1Þ.
The top and bottom snubbers are designed for engine valvesoft-landing and seating, respectively, by controlling the top and
bottom orifice flow areas. The snubbers can be described as equiv-alent orifice models using the same orifice flow equation as thepilot valve to obtain the actuation piston top pressure Pt and bot-tom pressure Pb as follows:
Pt ¼ ~Pmo � sgn _Lð Þ At_L
CdAorit
!2
qoil
2(5)
Pb ¼ Pl þ sgn _Lð Þ Ab_L
CdAorib
!2
qoil
2(6)
where At and Ab are the actuation piston top and bottom surfaceareas, respectively; Aorit is the equivalent top orifice area for thetop port orifice, top notch orifice, switch valve orifice, check valveorifice, and relief valve orifice. Therefore, Aorit is a function of thevalve displacement and the associated control valve state. It canbe seen from Eq. (5) that Aorit is a key parameter for the soft-seating since very small orifice leads to vary large top pressurechange, which significantly affects the seating process. Similar tothe equivalent top orifice, Aorib is the equivalent bottom orificearea for the bottom port and bottom notch orifices and it signifi-cantly affects the opening terminal (landing) velocity.
Two hydraulic accumulators are used for temporarily storingpressurized fluids, maintaining pressures, and absorbing pressurepulsations. Note that the accumulator pressures fluctuate cycli-cally due to the nature of periodic motion of engine valves. Byoptimizing the spring-accumulator parameters, the pulsation canbe limited to the acceptable level. The linear fluid accumulatormodel, that describes the accumulator pressure variation due tothe net flow rate change, is used to describe the flow dynamics forboth supply and back pressure sources
Cacl_Pl ¼
1� sgn _Lð Þ2
At þ Ab
� �_L � Qrel (7)
Cach_Ph ¼ �
1þ sgn _Lð Þ2
At_L þ Qpump � Qreh (8)
where Cacl and Cach are the capacitances of the high- and low-pressure accumulators, respectively; Qpump is the pump volumetricflow rate; and Qrel and Qreh are leakage flow rates of the reliefvalves for high and back pressure accumulators, respectively.
Rearranging Eqs. (1)–(8), along with the associated two counter-part Eqs. (2) and (4), for the main valve model, leads to a ninth-order detailed nonlinear model for the EHVVA system as below
me€L þ Cf
_L þ ks Lþ dsð Þ ¼ At Pmo t� tdð Þ �qoilsgn _Lð Þ
2
At_L
CdAorit
!224
35� Ab Pl þ
qoilsgn _Lð Þ2
Ab_L
CdAorib
!224
35
mep€xp þ Cfp _xp þ ks1 xp þ ds1ð Þ ¼ bI2 1þ xp=a� ��1
; �Li_I � RI þ U ¼ 0
Ppo ¼ u Ph � Patmð Þ þ Patm �qoilsgn _Lð Þ
2
Aspoolm _xm
Cdwpxp
2
mem€xm þ Cfm _xm þ ks2 xm þ ds2ð Þ ¼ Ppo t� t0ð ÞAspoolm
Pmo ¼ u Ph � Plð Þ þ Pl �qoilsgn _Lð Þ
2
At_L
Cdwmxm
!2
Cacl_Pl ¼
1� sgn _Lð Þ2
At þ Ab
� �_L � Qrel
Cach_Ph ¼ �
1þ sgn _Lð Þ2
At_L þ Qpump � Qreh
8>>>>>>>>>>>>>>>>>>>>>>>>>>><>>>>>>>>>>>>>>>>>>>>>>>>>>>:
(9)
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where mem, Cf m, ks2, ds2, and wm are the equivalent moving mass,friction coefficient, spring stiffness, spring preload, and the con-stant orifice area coefficient for the main valve, respectively.
4 Reduced-Order Nonlinear and Linear Parameter-
Varying Models
As mentioned in the previous EHVVA publication [21], the sys-tem is highly nonlinear with respect to the system operationalparameters, such as fluid pressure and temperature, and the designedvalve profile, such as the lift-time area, and opening and closing tim-ings. Note that valve profile is very important to the engine intakeand exhaust processes and engine performance. The nonlinearitiesare mainly from the orifice equations, the temperature-sensitive vis-cosity of the hydraulic fluid, the time-varying actuation valve delay,etc. It is difficult to design a nonlinear controller based on the devel-oped nonlinear model with guaranteed performance due to lack ofassociated nonlinear control theory with guaranteed performance.Therefore, model reduction was conducted to simplify the originalnonlinear model with acceptable modeling error and an LPV modelis also obtained based on the reduced-order nonlinear model to beused in future for the LPV controller design.
4.1 Reduced-Order Nonlinear Model. According to Fig. 2,both piston top pressure and valve displacement start rising almostat the same time, as well as the second rise of the pilot pressure.This implies that a very short duration is needed for the mainvalve to open and its outlet pressure to reach the threshold that
overcomes the preload force of the valve spring. Therefore, theactuation valve (both pilot and main valves) model can be simpli-fied into the first-order dynamics with time constant sp and anequivalent time delay td is used to account for the sum of the pilotvalve delay t0 and main valve delay t1; see below
Pmo sð Þ¼Pmi sð Þspsþ1
¼u Ph�Plð ÞþPl
spsþ1(10)
~PmoðtÞ ¼ Pmoðt� tdÞ (11)
where Pmi ¼ uðPh � PlÞ þ Pl represents the main valve inlet pressurethat is switched between Ph and Pl by the pilot solenoid control signal;and ~Pmo is now the equivalent delayed main valve outlet pressure.Time delay td is explicitly modeled in the system instead of approxi-mated by linear transfer functions such as the Fourier–Laguerre series,Pad�e approximation, and shift-based approximations [30,31] widelyused in time delay systems. Note that these approximations couldintroduce significant modeling error during the start and end of thevalve motion due to the pressure-sensitive offshore and landing proc-esses of the actuation piston. Therefore, the time delay td will be usedexplicitly for the system state and subsequently used as a varyingparameter directly, resulting in an LPV time-delayed system.
Replacing the actuation valve model (pilot and main valvemodels) by Eqs. (10) and (11), and rearranging Eqs. (1) and(5)–(8), the original full-order nonlinear model (Eq. (9)) reducesto a fifth-order nonlinear model as shown below
me€L þ Cf
_L þ ks Lþ dsð Þ ¼ At Pmo t� tdð Þ �qoilsgn _Lð Þ
2
At_L
CdAorit
!224
35�Ab Pl þ
qoilsgn _Lð Þ2
Ab_L
CdAorib
!224
35
sp_Pmo ¼ �Pmo þ u Ph � Plð Þ þ Pl
Cacl_Pl ¼
1� sgn _Lð Þ2
At þ Ab
� �_L � Qrel
Cach_Ph ¼ �
1þ sgn _Lð Þ2
At_L þ Qpump � Qreh
8>>>>>>>>>>>>><>>>>>>>>>>>>>:
(12)
4.2 Linear Parameter-Varying Model
4.2.1 Time-Varying Parameters. By analyzing both full- andreduced-order models and comparing the model simulation resultswith the experimental ones, it is clear that the system time delaytdðPh;TÞ and time constant spðPh;TÞ for the piston top pressurevary with the supply fluid pressure and temperature; and the dis-charge coefficient, as a function of fluid viscosity, is mainly influ-enced by the fluid temperature. Generally, the local dischargecoefficient is also affected by the top and bottom equivalent ori-fice areas, AoritðLÞ and AoribðLÞ, and they are functions of the valvedisplacement. However, the accurate relationship between the ori-fice flow area and the discharge coefficient is usually difficult to
obtain, and as a result, it often relies on the empirical equation fora given orifice type [32]. Therefore, an area-irrelevant mean effec-tive discharge coefficient CdðTÞ is used to account for the viscos-ity as a function of temperature in the EHVVA model. The high-and low-pressure sources Ph and Pl are assumed to be measurablevarying parameters. As a summary, for the LPV model, the lasttwo accumulator equations in Eq. (12) can be eliminated, resultingin a third-order time delay system. Therefore, in the nonlinearmodel Eq. (12), tdðPh;TÞ, spðPh; TÞ, CdðTÞ, AoritðLÞ, AoribðLÞ, Ph,and Pl are varying parameters. Define the state vector x ¼½ x1 x2 x3 �T ¼ ½Pmo L _L �T and the system output equationy ¼ x2 ¼ L. The resulting third-order nonlinear model is
_x1 tð Þ ¼ � 1
sp Ph; Tð Þ x1 tð Þ þ 1
sp Ph;Tð Þ u tð Þ Ph � Plð Þ þ Pl
� �_x2 tð Þ ¼ x3 tð Þ
_x3 tð Þ ¼ At
mex1 t� tdð Þ �
ks
mex2 tð Þ �
qoilsgn x3ð Þ2meCd Tð Þ2
A3t
A2orit x2ð Þ
þ A3b
A2orib x2ð Þ
!x3 tð Þ2
� Cf
mex3 tð Þ � AbPl þ ksds
me
y tð Þ ¼ x2 tð Þ
8>>>>>>>>>>>><>>>>>>>>>>>>:
(13)
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4.2.2 Linear Parameter-Varying System Model With Input andParameters Delay. Although the time delay tdðPh;TÞ is treated asa varying parameter, it is feasible to design LPV controllers for theEHVVA system with time delay in control input and parameters[33] for the convenience of online implementation. Therefore, anequivalent transformation is applied to the nonlinear model (13).Since x1 is only a function of the input and varying parameters, thefirst equation in model (13) can be replaced by a dynamic equationwith the delayed state ~x1ðtÞ ¼ x1ðt� tdðPh;TÞÞ by substituting theinput and varying parameters with their delayed counterparts
_~x1 tð Þ ¼ � 1
sp ~Ph;Tð Þ
~x1 tð Þ þ 1
sp ~Ph;Tð Þ
~u tð Þ ~Ph � ~Pl
� �þ ~Pl
h i(14)
where ~uðtÞ ¼ uðt� tdð ~Ph;TÞÞ is the delayed input, and ~PhðtÞ ¼Phðt� tdð ~Ph;TÞÞ and ~PlðtÞ ¼ Plðt� tdð ~Ph; TÞÞ are the delayedvarying parameters. Note that the fluid temperature TðtÞ changesmuch slower than any other parameters so that it does not need tobe delayed.
Substituting Eq. (14) into the dynamic equation for x1 in Eq.(13) and replacing x1ðt� tdðPh; TÞÞ with ~x1ðtÞ, the time-delayedsystem can be converted into an input and parameter-delayed sys-tem that can be easily handled by directly using the delayed inputand parameters in the control system in future. Meanwhile, to con-vert the nonlinear model into the LPV format, additional transfor-mation and parameter merging are required to deal with thenonlinearities in the model.
First, define the new input, states, and outputs as:u ¼ ½ u1 u2 �T ¼ ½ ~u 1� ~u �T, x¼½x1 x2 x3 �T¼½x1 x2þds x3 �T,y ¼ x2. Second, in the dynamic equation of x3 in Eq. (13), mergingone x3 in the quadratic term and the sign function into the coeffi-cient, a lumped varying parameter K can be obtained. That is afunction of the top and bottom equivalent orifice areas, the dis-charge coefficient, and the engine valve velocity. The nonlinearmodel is then converted into an LPV time delay system below
_xðtÞ ¼ AðtÞxðtÞ þ BðtÞuðtÞ; yðtÞ ¼ CðtÞxðtÞ (15)
where
A ¼
� 1
sp ~Ph;Tð Þ
0 0
0 0 1
At
me� ks
meK
26666666664
37777777775; B ¼
~Ph
sp ~Ph;Tð Þ
~Pl
sp ~Ph;Tð Þ
0 0
AbPl
me
AbPl
me
26666666664
37777777775;
C ¼ 0 1 0� �
;
K ¼ � Cf
me� qoilsgn x3ð Þx3
2meCd Tð Þ2A3
t
A2orit x2ð Þ
þ A3b
A2orib x2ð Þ
!
Define the scheduling parameters as follows:
h ¼ h1 h2 h3 h4 h5
� �T¼
1
sp ~Ph;Tð Þ
K~Ph
sp ~Ph ;Tð Þ
~Pl
sp ~Ph;Tð Þ
Pl
24
35
T
The system and input matrices of the LPV system can be writteninto the following affine parameter-dependent form suitable forfuture LPV control design [34]:
A ¼
�1 0 0
0 0 0
0 0 0
26664
37775h1 þ
0 0 0
0 0 0
0 0 1
26664
37775h2 þ
0 0 0
0 0 1
At
me� ks
me0
266664
377775
B ¼
1 0
0 0
0 0
26664
37775h3 þ
0 1
0 0
0 0
26664
37775h4 þ
0 0
0 0
Ab
me
Ab
me
266664
377775h5
Among the scheduling parameters, h1; h3; h4; and h5 can beobtained directly or indirectly based on the measurable parametersunder different operational conditions. h2 can be constructed usingthe measurable state x2 and parameter T.
5 Model Calibration and Validation
5.1 Experiment Setup. A prototype EHVVA, GD-VVA-2[21], designed for the 2.0 L Ford Duratec engine was used for cali-brating and validating developed models; see Fig. 3. A constant-displacement pump driven by a direct current motor was used toregulate the high system pressure Ph. The high-pressure is contin-uously adjustable via the direct current motor speed controlled byan NI myRIO controller used for valve control and data acquisi-tion. A micro-epsilon point range laser sensor was used to mea-sure the valve displacement. The pressure sensors are locatedclose to the actuation cylinder to avoid additional pressure lossalong the oil path. A peak and hold drive circuit, similar to thedirect fuel injector drive circuit, was used for driving the pilotvalve solenoid. A low side drive circuit was used to drive the slowsnubber orifice switch and lift valve solenoids.
The experiments were conducted under a combination of vari-ous control parameters including supply pressure Ph, back pres-sure Pl, fluid temperature T, the switch valve, and lift valve states(on/off) for the purpose of calibrating the model parameters andvalidating the developed models. Table 1 shows 15 typical experi-ments conducted for the model validation process. The supply andback pressure ranges of 5–8 MPa and 1–4 MPa were selected basedon the required valve performance (max lift, opening/closingresponse, etc.), and a common oil temperature range of �20 to100 �C was selected. Two typical engine valve opening durationswere selected in the experiments: 8 ms (short dwell) for low loadoperations and 20 ms (long dwell) for high load. The obtainedengine valve profiles were compared with the simulation results ofnonlinear reduced-order and LPV models.
Fig. 3 Test bench
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5.2 Model Calibration. The varying parameters, need to becalibrated, include the time constant sp for the main valve outletpressure, the time delay td , and the discharge coefficient Cd .Recalling the system dynamics described in Sec. 2 and Fig. 2,td ¼ t1 þ t2 can be calibrated by comparing the control signal andthe pressure response at pilot valve outlet. Figure 4 shows the cali-brated time delay td under different supply pressures and tempera-tures for VC and VO, respectively.
As shown in Fig. 4, the time delay decreases continuously asthe supply pressure increases for both VO and VC due to theincreased hydraulic driving force. However, the time delay forVC is slightly lower than that for VO because the oil dischargingresponse during valve closing is faster than that during valveopening. Note that the temperature affects the time delay throughthe discharge coefficient Cd . The time delay during VO increasesconsistently as the temperature decreases and the rate increasessignificantly when the temperature gets lower than 0 �C. The timedelay during VC behaves in a similar way to that in the low tem-perature region; however, it is almost not sensitive to the tempera-ture change in the high temperature region. As discussed in Sec.4.2.1, Cd is affected by both hydraulic fluid temperature and itsflow area (orifice). In the low temperature region, Cd is dominatedby the temperature due to the fluid viscosity property [32], whilein high temperature region, it is dominated by the flow area, whichis fairly large at the beginning of VC but very small for VO.
The time constant sp cannot be calibrated directly since themain valve outlet pressure is not measurable. At the same time,the temperature-sensitive discharge coefficient Cd is a case-dependent parameter and considered as a mean effective value forthe orifices in EHVVA model. To calibrate sp and Cd , the nonlin-ear least-squares optimization method was used to identify the
multiple parameters under different operational conditions byminimizing the error over the entire valve operational range.
The nonlinear least-squares optimization problem is describedbelow
minx1 ;x2
J ¼ minx1 ;x2
Xn
i¼1
ðyiðx1; x2Þ � yiÞ2
s:t: x1 > 0; 0 < x2 < 1
(16)
where the objective function J is the summation of the squares oferrors between the experimental valve lift yi and model output yi;x ¼ ½x1; x2�T ¼ ½sp;Cd�T is the parameter vector to be optimized;and n is the total number of data points used. This nonlinear least-squares minimization problem can be solved using numericalmethods such as the Gauss–Newton, the trust region, and theLevenberg–Marquardt methods [35]. For this optimization, notethat sp and Cd (included in K; see Eq. (15)) are two independentparameters. When minimizing the cost function (16) using sp andCd , it may lead to a local minimum due to nonlinear nature of theminimization problem. Due to their independence, there may notexist multiple pairs of sp and Cd that lead to the same local mini-mal. To make the solution within the feasible range, the con-straints x1 > 0 and 0 < x2 < 1 are used.
The trust region method is used in this study due to its simplic-ity and effectiveness. In this method, a region D is defined foreach iteration within which a quadratic model function M is usedas an adequate representation of the objective function below
J xþ hð Þ ffi M hð Þ ¼ J xð Þ þ hTgþ 1
2hTHh (17)
where h is the iteration step; g and H are the gradient vector andHessian matrix of the objective function, respectively; and thestep h is chosen by solving the following optimization problem[35]:
minh
M hð Þ ¼ J xð Þ þ hTgþ 1
2hTHh s:t: khk � D (18)
The trust region D is adjusted in each iteration according to theratio r between the actual and predicted decrements of the objec-tive function value; see below
r ¼ J xð Þ � J xþ hð ÞM 0ð Þ �M hð Þ
(19)
Dkþ1 ¼ cDk; c ¼ 2 r > 0:75
0:25 r < 0:25
�(20)
Table 1 Test matrix
Test no. Ph (MPa) Pl (MPa) T (�C) Switch valve Valve duration (ms) Lift
1 6 1 22 OFF 8 Low2 7 1 22 OFF 8 Low3 8 1 22 OFF 8 Low4 6 1 22 ON 8 High5 7 1 22 ON 8 High6 8 1 22 ON 8 High7 8 1 22 OFF 8 High8 5 1 �20 OFF 8 Low9 5 1 22 OFF 8 Low10 5 1 60 OFF 8 Low11 5 1 100 OFF 8 Low12 8 1 22 ON 20 High13 8 2 22 ON 20 High14 8 3 22 ON 20 High15 8 4 22 ON 20 High
Fig. 4 Equivalent time delay under different supply pressuresand temperatures
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where k is the number of iterations, and c is the adjusting coeffi-cient of the trust region based on r. Equation (20) indicates thatwhen a good agreement between the model M and the objectivefunction J is achieved using the step solution of Eq. (18), a largercoefficient is chosen to enlarge the trust region D for the next iter-ation, and otherwise, the region will be reduced. The solution h inEq. (18) is valid if the objective function value decreases (i.e.,r > 0); otherwise, the h is rejected and Eq. (18) will be solvedagain in the current iteration using the reduced region.
Using the nonlinear least-squares optimization approach, sp andCd shown in Fig. 5 are determined under different operationalconditions. The top graph in Fig. 5 shows that the time constant(sp) for the main valve outlet pressure decreases linearly as a func-tion of the supply pressure and remains almost unchanged as thetemperature varies. It also can be noticed that the mean of sp isclose to 0.1 ms, indicating that the main valve is very fast com-pared to the total valve response time. The discharge coefficientCd shown on the bottom graph of Fig. 5 decreases as the tempera-ture does and the rate increases significantly when the temperatureis low. This is consistent with the nonlinear relationship ofhydraulic fluid viscosity and its temperature [32] and indicatesthat the developed model captures this characteristic under differ-ent temperatures and pressures.
5.3 Simulation Results and Validation. With all the modelparameters calibrated, both reduced-order nonlinear and LPVmodels can be implemented in SIMULINK. The experimental andsimulated responses are presented in this subsection.
Figure 6 shows the results of both high- and low-lift operationsunder the supply pressure ranging between 6 and 8 MPa. Thereduced-order nonlinear model responses in gray dotted lines arevery close to the experimental ones shown in black-solid lines. Itis very important for the model to be able to predict in-cycle tra-jectories for the valve opening and closing slops since they areclosely related to the charging air quantity. Both slops can be con-trolled by regulating the supply pressure Ph and back pressure Pl
in Eq. (13), along with the Boolean solenoid control input (u)used for the valve timing control. As a result, the trajectory pre-diction performance is evaluated by the absolute error and meanrelative error (MRE) using model and experimental data. Theabsolute error between experimental and model (nonlinear andLPV) responses of all tests are shown in black-dashed lines with0.68 mm maximum value in test 4, leading a 1.7% relative errorof the lift area (an integration of valve lift over time), and theMRE of each test is shown in Table 2. The obtained good
agreement is mainly due to the utilization of nonlinear least-squares optimization approach to obtain the set of optimized vary-ing parameters (sP, Cd) over the entire operational range.
Also, the simulated LPV model responses are plotted with “þ”and they are the same as these from the reduced-order nonlinearmodel due to the direct transformation from the reduced-ordernonlinear model to the LPV one without introducing any addi-tional model error. Again, time delay (td) approximation error,introduced by linear transfer functions, is eliminated by express-ing the time delay explicitly and used as a transport delay in themodel. In this paper, the open-loop control is used for experimen-tal validation with calibrated valve-opening and closing delay td
(that is dependent on system parameter such as supply pressure).The modeled and experimental valve timings are fairly close; seeFig. 7.
The opening time for tests 1–6 are compared in Table 3. It canbe seen that the supply pressure is the key factor affecting thevalve opening time, implying that closed-loop supply pressurecontrol shall be used for tracking the desired opening time.
The soft seating responses, with and without the switch valveactivated (test 6 and test 7), are validated and shown in Fig. 8. Alow seating velocity of 0.22 m/s was achieved when the switchvalve was activated to close the second top orifice; and a longclosing tail indicates a later closing as expected. This demon-strates that the snubber model is satisfactory.
The temperature effect to the EHVVA was studied in tests 8–11under the same supply pressure (5 MPa) with a temperaturerange between �20 �C and 100 �C. The experimental and modelresponses are shown in Fig. 9 and Table 4. Both opening and clos-ing responses get slower as the temperature decreases, and theydeteriorate more significantly when the system is operated at verylow temperature (e.g., �20 �C). That coincides with the response
Fig. 5 Least-squares optimization results
Fig. 6 Experimental and model valve lift profiles for tests 1–6
Table 2 Mean relative errors for tests 1–6
Test no. 1 2 3 4 5 6
MRE (%) 4.2 2.2 2.0 5.1 2.6 2.3
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time delay results shown in Fig. 4 and the associated dischargecoefficient in Fig. 5. Note that the temperature effect to the valveclosing time is much smaller than the opening time, especially inthe high temperature region. This is because the opening process
is dominated by the hydraulic force, and the closing process, how-ever, is dominated by the returning spring force. The low-viscosity (34.5 mm2/s at 40 �C) automatic transmission fluid wasused in these experiments, and it is expected to have much higheffect to the response time at low temperature when high viscosityhydraulic fluid is used [21,32].
The experimental and simulated valve responses under differentback pressures are illustrated in Fig. 10 for tests 12–15. Duringthese tests, it is shown that valve closing tail increases as the backpressure goes up, resulting in later valve closing. The full valveclosing time is increased by about 7 ms at 4 MPa back pressure,compared to the case with back pressure at 1 MPa, and however,the opening time is reduced by 2 ms. That is because the valveopening force is dominated by the high supply pressure; and theclosing force is dependent on both back pressure and returningspring stiffness.
As expected, both reduced-order nonlinear and LPV models arefairly accurate under different supply pressure, back pressure,switch valve, and temperature in terms of opening and closingresponses under different seating velocities.
6 Conclusions
This paper presents both reduced-order nonlinear and LPVmodels for an electro-hydraulic variable valve actuator. Bothmodels are capable of capturing the valve dynamics accuratelysince they model the fluid dynamics of each subsystem componentand are calibrated using experimental data under different opera-tional conditions. Satisfactory model accuracy is achieved for thereduced-order nonlinear model due to utilizing the nonlinearleast-squares optimization/calibration, and minimal deviationfrom the reduced-order model is guaranteed for the LPV model
Fig. 7 Experimental and model valve opening/closing timingfor all tests
Table 3 Opening time comparison
Ph (MPa) 6 7 8
Low-lift operation (ms) 4.0 3.1 2.7High-lift operation (ms) 5.2 3.8 3.6
Fig. 8 Soft seating with switch valve on (test 6) and off (test 7)
Fig. 9 Experimental and model valve responses under differ-ent temperatures
Table 4 Temperature sensitivity
T(�C) Cd
VO delay(ms)
VC delay(ms)
VO time(ms)
VC time(ms)
�20 0.21 4.2 4.0 8.2 4.022 0.40 3.7 3.0 6.2 3.560 0.65 3.5 3.0 4.8 3.0100 0.70 3.2 3.0 4.4 2.8
Fig. 10 Valve opening and closing responses under differentback pressure for tests 12–15
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due to the usage of explicit time delay and direct transformation.The maximum mean relative error between experimental andmodel (nonlinear reduced-order and LPV) is 5.1% among all testresults. The model validation results also show that the varyingparameters of the LPV model have significant influences on thesystem dynamics. The varying fluid discharge coefficient thataccounts for the fluid viscosity affects the valve response timeunder low temperature operations; and it can be compensated byregulating the supply pressure due to the positive correlationbetween the supply pressure and valve response time. The timedelay due to the variations of supply pressure and fluid tempera-ture is the key for the accurate valve opening and closing timingcontrol. The back pressure and the switch valve state can be opti-mized for the desired seating velocity. It can be concluded that thedeveloped models, reduced-order nonlinear and LPV ones, pro-vide the dynamic relationship between control inputs and systemresponses that can be used for developing model-based controlstrategies for the electric-hydraulic variable valve actuator.
Acknowledgment
The authors would like to thank the China Scholarship Council(CSC) for supporting Mr. Huan Li’s study at Michigan State Uni-versity as a joint Ph.D. student.
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