Dynamic hybrid model of an electro-pneumatic clutch system

Mechatronics 23 (2013) 21–36

Contents lists available at SciVerse ScienceDirect

Mechatronics journal homepage: www.elsevier.com/locate/mechatronics

Dynamic hybrid model of an electro-pneumatic clutch system Barna Szimandl ⇑, Huba Németh Budapest University of Technology and Economics, Department of Automobiles, Stoczek Street 6. Floor 5, H-1111 Budapest, Hungary

a r t i c l e

i n f o

Article history: Received 25 April 2012 Accepted 13 October 2012 Available online 6 December 2012 Keywords: Electro-pneumatic clutch Solenoid magnet valve Dynamic hybrid model State space realisation Verification Validation

a b s t r a c t This paper deals with modelling an electro-pneumatic clutch system, which is used for medium- and heavy duty commercial vehicles. The mathematical model is built up for dynamic simulation, parameter estimation and control design/validation purposes, which is a phase of the design process of a new clutch system. These intended applications define the modelling goals and determine the modelling assumptions, which let one to reduce the model complexity. Since the model shows discrete–continuous behaviour, i.e. the model has hybrid properties, a nominal state domain or hybrid mode has been chosen for the sake of simplicity, where the model is continuous. In addition all the cases are given systematically, where the model has discrete transients. The model is constructed on the basis of the conservation principles such as mass, energy, momentum and magnetic linkage conservation and it is provided with constitutive equations to get a solvable set of equations. This final collection is then transformed into state space form for the given applications above. The verification of the developed model is carried out using extensive simulations against engineering perception and operation experience on the qualitative behaviour. Then for validation purposes the outputs of the model are compared to measurements on the real system to give a quantitative performance index about the model accuracy. Since for modelbased controller design the developed model is too complex it should be simplified. Hence possible model reduction methods are proposed, which omit all details that are weakly represented in the state variables/outputs and not coupled with the control aims. Ó 2012 Elsevier Ltd. All rights reserved.

1. Introduction For commercial vehicles one of the widely used actuation types are the pneumatics and electro-pneumatics, which are used for braking, levelling, gear shifting, clutching and so on. These actuators use the compressed energy of the gas as the source for the force transmission. Systems which are provided with pneumatic actuators have many attributes that make them attractive for use in automotive environments, such as the air is not subjected to the temperature limitations unlike the fluids in hydraulic actuators, the exhaust air from the pneumatic actuators do not need to be collected, so fluid return lines are unnecessary. Furthermore the long term pneumatic energy storage is not a problem because no chemical materials need be used as opposed to the electric energy. Moreover, the pneumatic actuator has a lower specific weight and a higher power density than an equivalent electro-mechanic actuator. Thus the research work and publications on modelling and control of electro-pneumatic systems for industrial application are widely used in practice [1–3]. In this paper the dynamic model of an electro-pneumatic clutch (EPC) system, used in medium- and heavy duty commercial ⇑ Corresponding author. Tel.: +36 1 3829 416; fax: +36 1 3829 810. E-mail address: [email protected] (B. Szimandl). 0957-4158/$ - see front matter Ó 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.mechatronics.2012.10.006

vehicles, is presented. This system has many merits, such as the easy maintenance and handling, relatively simple technology and low cost, clean, reliable and easy to install, as oppose to the conventional hydraulic and electro-hydraulic systems [4]. In case of any x-by-wire principle, the pedal characteristic can be set up arbitrarily or the pedal can be left out totally, unlike in a conventional hydraulic clutch system. Moreover EPC systems provide an opportunity to do additional clutch control functions such as start assistance and hill holding, which can reduce the clutch wear during vehicle start and can utilise the support of the brake system and traction control as well, e.g. disconnect the power train in an emergency situation and help on the synchronisation of the gearbox. The mentioned reasons give the main advantages of the EPC system, which makes it possible to increase the transmitted torque by increasing the clutch actuator force without changing the pedal characteristic. Since the clutch system is a safety critical part of the vehicle detailed knowledge of its input–output behaviour and effects of its parameter changes are essential for clutch control design and validation purposes. Moreover the failure modes and their effects, which can happen during the lifetime of the clutch, should be known as well. One possible way to collect this information is the mathematical modelling approach, where the solution of the model, i.e. the simulation results, can approximate the behaviour

22

B. Szimandl, H. Németh / Mechatronics 23 (2013) 21–36

of the real system. Besides, the simulations can help in the development phase of a new clutch system to reduce the number of the manufactured prototype versions, by this way influencing the cost and time of the design propitiously. It can be effectively used for calculating the values of certain parameters from given known inputs and desired outputs. This type of problem is normally solved using an optimisation technique, which finds the parameter values achieving the desired outputs [5]. In the literature, many models of electro-pneumatic clutch systems have been introduced. These models are developed for control design purposes only [6–8], and do not cover the overall dynamic of the system, e.g. the solenoid valve dynamics is not considered entirely. In the recent years the flow cross section of the on/off solenoid valves, applied in clutch actuators, are increased to achieve the desired performance. This allows fast dynamics, but causes difficulties to the control, since the increased throughput of the valves changes the opening and closing dynamics and consequently reduces the fine dosage of the compressed air. Hence, the earlier presented models in the literature are not enough accurate for the state of the art electro-pneumatic clutches. This phenomenon and the requirements above inspire to consider not only the piston and the gas dynamics, but the solenoid valve and the power stage dynamics as well. Thus a detailed EPC model has been developed and the following steps are considered for the systematic modelling procedure [9]. First of all the description of the system and its boundary (Section 2), then the definition of the modelling goals and approaches will be given (Section 3). In the next two steps the simplification assumptions will be considered (Section 4) and the nominal hybrid mode will be chosen (Section 5). In the main phase of the modelling the conservation equations will be derived (Section 6) and the constitutive equations will be constructed (Section 7). After that the model will be transformed into standard state space form (Section 9) and then the model verification and validation will be presented (Section 10). Finally concluding remarks will be collected (Section 11) regarding the resulted model type, complexity and possible applications.

2. System definition In a vehicle driveline, when gear change is demanded, the connection between the engine and the gearbox must be disengaged before any gear shifting procedure is started. This process along with the reconnection of the engine and the gearbox is done by the clutch. The connection is disengaged in the connection point of the engine shaft and the gearbox input shaft, where normally the clutch transmits the momentum through the clutch disc. The clutch friction disc, the pressure plate and the flywheel are rotating together due to the friction force between them. This force is caused by the normal force of a disc spring, which pushes the clutch pressure plate to the friction disc and the flywheel. When the clutching is demanded, Solenoid Magnet Valves (abbreviated as SMV throughout the paper) driven electro-pneumatic actuator pre-stress the disc spring, which lets the clutch pressure plate to moves apart from the friction disc, thus terminates the connection. The general layout of the EPC with its close surrounding to be modelled [10] can be seen in Fig. 1. The system is supplied by compressed air, thus for supply pressure an air reservoir (1) is applied. The actuator contains four SMVs, two of them (2, 3) can connect the supply pressure to the chamber (11), so they are called load valves and the remaining two (4, 5) can connect the chamber to the ambient pressure, so they are called exhaust valves. The geometry of the four SMVs are identical except for their cross section. The variables of the SMVs are denoted as follows: sl, bl, se and be for small- and big load and small- and big exhaust respectively. Each SMV has an own power stage (6–9), see

details later, which can transform the command signal to appropriate terminal voltage for the solenoid of the SMV. This structure ensures positive and negative direction displacement of the piston (10), which is the final element of the actuator that performs the clutching procedure. The variables of the piston are denoted by pst subscript. The actuator contains a holder spring (12), which pushes the piston to the clutch mechanism to reduce the clearance between them. The main load of the actuator comes from the disc spring (13) of the clutch mechanism and acts against the piston movement. The disc spring is slotted in the inner diameter and the release bearing of the piston (14) is connected to this area. The slots have the effect of reducing the spring load and increasing the deflection. In the outer diameter of the disc spring is connected to the pressure plate (15), which can push the clutch friction disc (16) to the flywheel(17). Moreover the clutch friction disc contains cushion springs (18). The non-linear stiffness of this set of springs has a paramount role in the controllability performance at small torques. The disc spring is compressed between the pressure plate and the housing (19). The disc spring is fixed to the housing with pins (20), which ensures a fulcrum ring (21), where the spring can bend. The pressure plate is also fixed to the housing with tangential leaf springs (22), these springs transform the torque to the housing, determine the radial position of the pressure plate and push it to the disc spring. Finally the flywheel is connected to the engine and the friction disc is connected to the splined gearbox input shaft (23). 3. Modelling-goals and approaches The modelling goals are specified by the intended use of the model moreover they have a major impact on the level of detail and the mathematical form of the model. One of the most important and widely used modelling goals in practice is the construction design, when the model is developed to represent the output change in time, with given inputs, model structure and parameters. An other widespread modelling goal is to develop the model for control system design and/or validation to produce an input for which the system responds in a prescribed way. Hence the modelling goal is a complex statement, where one assume it to be given in terms of a set of performance indices [v1, . . . , vn], where the performance index vi can be real and/or boolean quantity which defined for the model M as: vi : M ! R; B. In this instance the performance index represents a model characteristic that is captured as a real and/or boolean valued quantity, e.g. differential index, model accuracy and so on. Note that the boolean items can express the presence or absence of a characteristic. Furthermore each performance index can be stated with acceptance limits in the form of inequalities: vmin 6 vi 6 vmax ; i ¼ 1; . . . ; n. i i Through these the following properties of the EPC model are considered to achieve the modelling goals, which are dynamic simulation, clutch control design and validation. 3.1. Model properties

MP1. The model description should be based on the mechanisms of the EPC and the model variables and parameters should have physical meaning ðvMP1 2 BÞ. MP2. It should be a deterministic input–output model ðvMP2 2 BÞ. MP3: The model class should be restricted to index-1 model class. That is, the model should be a set of differential algebraic equations (DAEs), where the algebraic equations can be substituted into the differential ones. ðvMP3 2 BÞ.

B. Szimandl, H. Németh / Mechatronics 23 (2013) 21–36

23

Fig. 1. The layout of the electro-pneumatic clutch (EPC) system.

MP4: The model should be represented in state space form ðvMP4 2 BÞ. MP5: The model should be capable of describing the dynamic behaviour of the electro-pneumatic clutch system within 5% deviation in the whole operation domain, i.e. this accuracy should be valid for all the model outputs individually and for a collection of them ðvMP5ag 2 R; vmax MP5ag ¼ 0:05Þ. For accuracy validation criterion an L2 error has been used to measure the deviation of the model response, based on the entries of the model output vectors and the measurement results on the real system. In the literature several modelling approaches have been published (see a comprehensive collection with many examples in [11]). Considering the modelling goals, first of all, mechanistic modelling approaches should be used to satisfy vMP1 , but the most common form of models, which describe complex systems, are a combination of mechanistic and empirical parts. The advantage of the mechanistic modelling approaches is that the model parameters have physical meaning unlike the empirical one, but the empirical approaches are widely used, where the actual underlying phenomena are not known or understood well. In order to satisfy vMP2 deterministic modelling approach is used. The concentrated parameter models, called lumped models, are one of the most important and widespread class of dynamic models, moreover majority of dynamic model simulations and model based control techniques deal with lumped models [12]. Therefore the designed model is restricted to this case which satisfies vMP3 . The consideration of the valve and the power stage dynamics to achieve vMP5ag introduces discrete–continuous behaviour, e.g. the change of the flow cross section of the valves. Thus a hybrid, i.e. discrete–continuous approach is used [13–15]. Moreover non-linear relationships between the parameters and/or variables are considered. 4. Simplifying assumptions and input constraints When constructing the model of the EPC system assumptions have been made in order to reduce the complexity and to get a solvable set of equations. Publications on the representation of

modelling assumptions are available in the literature [16,17]. First assumptions are made to get a concentrated parameter model [18], then to reduce the model components which show discrete behaviour. In the operation domain some components can be considered with linear relation instead of nonlinear. Finally variable lumping and variable removal is used to decrease the number of the model ingredients [19]. The assumptions have been derived iteratively according to the model complexity and the achievement of the modelling goals using the seven step model building procedure [9]. Concluding the following assumptions are made: 4.1. Assumptions A1: The gas physical properties in the chamber of the actuator such as specific heats, gas constant and adiabatic exponent are assumed to be constant over the whole time, pressure and temperature domain. A2: The chamber pressure is higher or equal to the ambient pressure. A3: The gas in the chamber is perfectly mixed, no spatial variation is considered. A4: The heat radiation is neglected and the rate of the heat transfer is proportional to the temperature difference between the gas and its surroundings (Newton’s heat transfer law). A5: The kinetic and the potential energy of the gas can be neglected, since the gas density is low. A6: The air flow (r) of the SMVs are assumed to have non-negative values only (see the directions in Fig. 1). A7: The SMVs magnetic elements are modelled assuming linear magneto-dynamically homogeneous material and the physical properties assumed to be constant over the whole temperature domain.   A8: The maximal SMV body stroke xmax and the SMV port xx diameter (dxx) are assumed to satisfy the inequality for all dxx the four SMVs: xmax xx > 4 , where xx can be sl, bl, se and be. A9: The cross sections of the SMV ports are assumed to satisfy the following condition for all the four SMVs: Aout  Ain, where Ain and Aout are the in- and output cross sections of the SMVs respectively.

24

B. Szimandl, H. Németh / Mechatronics 23 (2013) 21–36

A10: The force generated by the static pressure, acting on the armature of the SMVs, is neglected since the magnetic force, which moves the armature, is significantly higher. A11: The aerodynamic resistance of the armature can be neglected due to the low density of the gas. A12: The armature friction is neglected since the forces acting on the armature have axial component only. A13: The armature mass of the SMVs assumed to be constant in time. A14: The high frequency of the pulse width modulated (PWM) control signals of the power stages ensure that the currents of the SMVs can be well approximated by the average value and the current ripples can be neglected. A15: The switching devices (MOSFET) in the power stages have a constant drain to source turned on resistance in the applied working range. A16: The clutch mechanism and the clutch actuator moving masses are lumped into the piston mass, which is assumed to be constant in time. A17: The clutch mechanism and the clutch actuator friction effects can be lumped together into one friction effect. A18: The clutch mechanism and the clutch actuator damping effects can be lumped together into one damping effect. A19: The nonlinear characteristic of the clutch disc spring, which has hysteresis loop, can be approximated with its empirical centre characteristic line. A20: The pretension of the disc spring does not change due to the wear of the friction disc, since the clutch mechanism contains wear compensation system.

completely closed. The gas speed in the flow process of the load SMVs are considered as subsonic due to the pressure ratio between the chamber and the reservoir. The piston stroke has an intermediate position and is moving in positive direction. The conditions, defining the chosen nominal hybrid mode, are summarised in Table 1, where ixx and xxx are the solenoid current and armature stroke of the SMVs, xlim;1 are the strokes, where the armature xx reaches the valve seat, pch is the chamber pressure, Pcrit is the critical pressure ratio, xpst and vpst are the piston position and velocity respectively and finally xlim;1 and xlim;2 are the piston pst pst limiting positions.

Assumptions A1–A13 have been validated earlier in [20]. The remaining assumptions are derived iteratively in order to achieve the prescribed modelling goals. Then the previously done and the new assumptions have been validated together as well (see in Section 10).

6.1. Conservation of gas mass

6. Conservation equations The dynamic equations describing the mathematical model of the clutch actuator are based on first engineering, i.e. conservation, principles. The region, in which the conserved quantity is contained, is a basic element of the model called balance volume, which is determined by the applied conservation principles. The model is considered as a lumped parameter dynamic model, since there are no spatial variations and the materials are homogeneous, thus the balances are obtained as ordinary differential equations. In order to derive the conservation equations ten balance volumes are defined; one for each SMV armature, one for each SMV solenoid winding, one for the clutch chamber and one for the piston. The balance equations are based on the conservation of mass, energy, momentum and magnetic linkage within the given balance volume.

The expression for mass balance [21], considering no generation and consumption terms, forms the following equation in case of lumped parameter systems with p input and q output:

X dm X ¼ rj  rk dt j¼1 k¼1 p

4.2. Input constraints IC1: Opening the load and the exhaust SMVs in the same time is not allowed. IC2: The disturbance variables are limited by the following constraints: 16 6 Usup 6 32 [V], 7  105 6 psup 6 12  105 [Pa], 233 6 Tsup 6 358 [K], 0.92  105 6 pamb 6 1.08  105 [Pa] and 233 6 Tamb 6 393 [K], where Usup is the supply voltage, psup is the compressed (supply) air pressure, Tsup is the compressed (supply) air temperature, pamb is the ambient air pressure and Tamb is the ambient air temperature respectively. IC3: The control input variables, i.e. the duty cycle of the PWM control signals (md,xx) are limited by the following constraints: 0 6 md,xx 6 1 [].

ð1Þ

where m is the mass and r is the mass flow rate. Since the clutch chamber has only one port (see Fig. 1), which serves as both in- and output port, its mass flow equals the sum of the four SMVs output ports mass flow:

dmch ¼ rsl þ rbl  rse  rbe dt

ð2Þ

where mch is the gas mass in the chamber.

Table 1 Conditions of the nominal hybrid mode (a) for power stages, (b) for SMVs and (c) for piston.

5. Nominal hybrid mode (a)

As mentioned in the introduction the mechanism of the system shows not only continuous but discrete behaviour. In consequence, the equations describing the overall dynamic behaviour of the system vary according to certain circumstances, which correspond to the conservation- and constitutive equations as well. To keep the model description as simple as possible the system equations are shown in one dedicated domain only. Hence certain circumstances are fixed to obtain a model with well defined unique structure. This dedicated domain or hybrid mode corresponds to a phase of the disengagement process of the clutch. In this case the load valves are excited and the exhaust valves are not, thus the load SMVs are partially opened and the exhaust SMVs are

q

(b)

(c)

No.

Condition

HM1,1 HM2,1 HM3,3 HM4,3

md,sl > 0 md,bl > 0 md,se = 0, ise = 0 md,be = 0, ibe = 0

HM5,1

1 P ppch > Pcrit

HM6,1

1 P ppamb > Pcrit

HM7,2

xlim;1 6 xsl < d4sl sl

HM8,2

xlim;1 6 xbl < d4bl bl

HM9,1

xse < xlim;1 se

HM10,1

xbe < xlim;1 be

HM11,2

lim;2 xlim;1 pst 6 xpst < xpst

HM12,3

vpst > 0

sup

ch

B. Szimandl, H. Németh / Mechatronics 23 (2013) 21–36

6.2. Conservation of gas energy The general form of total energy (E) for a given balance volume [21] with p input and q output flows is written as:

X dE X ¼ rj ðh þ ek þ ep Þ  rk ðh þ ek þ ep Þ þ Q þ W dt j¼1 k¼1 p

q

ð3Þ

where h, ek and ep denotes the mass specific-enthalpy, kinetic energy and potential energy terms respectively. Q is the heat transfer and W is the work term. According to Assumption A5 the potential and kinetic energy terms are neglected. In conclusion the simplified energy balance equation is written as:

X dU X rj hj  rk hk þ Q þ W ¼ dt j¼1 k¼1 p

q

ð4Þ

where the extensive conserved quantity is the internal energy ðUÞ on the left hand side that dominates the total energy content of the gas. The above introduced extensive form of the conservation balance equation should be transformed into its intensive form, in order to have a measurable intensive variable as its differential variable. For this purpose the chamber pressure has been selected. The chamber pressure change can be expressed using the definition of the internal energy and the ideal gas equation (pV = mRT) as follows: p V dU ch dðcv mch T ch Þ dðcv chR ch Þ cv dp cv pch dV ch ¼ ¼ R ch þ ¼ dt dt dt V ch dt R dt V ch dpch pch dV ch ¼ þ j  1 dt j  1 dt

where U ch is the gas internal energy in the chamber, cv is the specific heat at constant volume, cp is the specific heat at constant pressure, Tch is the gas temperature in the chamber, R is the specific gas constant, Vch is the instantaneous volume of the chamber, j is the adiabatic exponent and

cv ¼

R R j; thus ; c ¼ j1 p j1

j¼

cp cv

ð6Þ

The mass specific enthalpy term h is defined as the product of the coefficient of specific heat at constant pressure and the source side temperature (T) as:

h ¼ cp T

ð7Þ

The source side is determined by the air flow direction, but according to assumption A2, in which the SMVs air flows are assumed to have non-negative values only, the source side do not change. Using Eqs. (4)–(7) the pressure change in the chamber is written as follows:

dpch j R p dV ch ðrsl T sup þ rbl T sup  rse T ch  rbe T ch Þ  ch ¼ V ch dt V ch dt j1 j1 Q ch  W ch  V ch V ch

ð8Þ

According to Newton’s law the momentum (M) is the product of mass and velocity, thus the general form of momentum balance volume with p forces acting on the system is written:

ð9Þ

ð10Þ

where mpst is the lumped mass of the moving parts ðA16Þ, F pch is the pressure-, Fhsp is the holder spring-, Ffr is the friction-, Fdmp is the damping- and Flim is the limiting force acting on the piston. The load of the piston Fload(xpst) comes from the clutch mechanism. In accordance to A16, in which assuming a lumped constant mass, the velocity and the stroke change of the piston are obtained as follows:

dv pst F pch þ F hsp  F fr  F dmp  F lim  F load ðxpst Þ ¼ mpst dt dxpst ¼ v pst dt

ð11Þ ð12Þ

6.4. Conservation of SMV armature momentum Similarly to the momentum balance of the piston, the SMV armature balance is derived considering Assumption A10–A12 in which the pressure force, the aerodynamic resistance and the friction force are neglected. Thus the momentum balance of the SMV armature is written as:

ð13Þ

where Fmg is the magnetic force generated by the magnetic field of the solenoid, Frsp is the force coming from the return spring and Flim is the stroke limiting force of the SMV armature. Since the SMV armature mass is constant in time according to A13, the equations for armature velocity and position change are obtained as:

dv F mg  F rsp  F lim ¼ dt m dx ¼v dt

ð14Þ ð15Þ

6.5. Conservation of magnetic linkage The balance of the magnetic linkage is determined by Maxwell’s second equation (Faraday’s law), which describes how a time varying stand still magnetic field induces an electric field:

I

E dl ¼ 

C

d dt

Z

B n da

ð16Þ

S

where E is the electric field intensity, B is the magnetic flux density moreover the surface S is enclosed by the contour C and the positive direction of the normal vector n is defined by the usual right-hand rule. In regions, where the magnetic field is either static or negligible the electric field intensity can be derived as the gradient of a scalar potential / as follows:

ð17Þ

The difference in potential between two points, say a and b, is a measure of the line integral of E, for

Z a

p

where Fk denotes the forces acting on the system.

dðmpst v pst Þ ¼ F pch þ F hsp  F fr  F dmp  F lim  F load ðxpst Þ dt

E ¼ r/

6.3. Conservation of piston momentum

dM X ¼ Fk; dt k¼1

Considering the forces acting on the system generated by the pressure, spring, limitations, etc. the momentum balance of the piston is obtained as follows:

dðmv Þ ¼ F mg  F rsp  F lim dt ð5Þ

25

b

E dl ¼

Z

b

r / dl ¼ /a  /b

ð18Þ

a

The potential difference /a  /b is referred to as the voltage of point a with respect to b. Thus Faraday’s law yields the induced voltage (uind) of the solenoid as follows:

26

B. Szimandl, H. Németh / Mechatronics 23 (2013) 21–36

uind ¼

d dt

Z

B n da ¼

S

dk dk di dk dx ¼ þ dt di dt dx dt

ð19Þ

where k is the flux linkage of the circuit. Assuming a magnetically linear system according to A7 whose flux linkage can be expressed in terms of an inductance L as k = L i. Through these the induced voltage becomes:

uind ¼ L

di dL dx þi dt dx dt

ð20Þ

The terminal voltage of the SMVs (uterm) is dropped on the ohmic resistance (R) and the inductive parts as follows according to Kirchoff’s second law:

uterm ¼ ures þ uind

ð21Þ

Using Ohm’s law and substitution Eq. (20) into Eq. (21) the current change can be obtained as follows:

di uterm R 1 dL dx ¼ i  i dt L L dx dt L

ð22Þ

mined by the contraction coefficient (a) of the stream, the flow cross section (Ain), the source side-pressure (pin) and temperature (Tin) and the pressure ratio (P = pout/pin) between the in- and output ports [22].

vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi " u 2  jþ1 # u 2j 1 pout j pout j t T r ¼ aAin pin  pin j  1 R in pin

ð28Þ

The flow cross section of the SMV is determined by its orifice between the valve seat and the armature. If the armature stroke is less to the value, where the armature reaches the valve  or equal  seat xlim;1 see in Fig. 2, then there is no flow. If the stroke is above xx this value the smallest orifice is determined by a cylindrical surface. If there is a big stroke then the orifice is limited by the circular surface of the SMV seat. This implies hybrid behaviour depending on the SMV armature position. According to the nominal hybrid mode conditions (see Table 1), in which partially opened load valves are considered, the flow cross section of the load SMVs are written as follows:

Asl ¼ xsl dsl

p p

ð29Þ

7. Constitutive equations

Abl ¼ xbl dbl

To complete the above equations some additional algebraic constraints are needed to be defined such as transfer rates, property relations, equipment constraints and defining equations for other characterising variables.

The exhaust SMVs are closed, i.e. the armature of the SMVs lies on the valve seat, thus the exhaust flow cross sections (Ase and Abe) are equal to zero. Since the streaming conditions are defined as subsonic, the mass flow through the load SMVs are written as:

7.1. Chamber and gas properties The volume   of the chamber is obtained from a constant dead volume V dch and an additive volume set by the moving piston of the system, where the dead volume of the chamber is defined as the minimum volume that the chamber may have, independent of the current application. With these the chamber current volume- and the volume change of the clutch actuator can be written as follows:

V ch ¼ V dch þ xpst Apst

ð23Þ

dV ch ¼ v pst Apst dt

ð24Þ

where Apst is the cross section area of the piston. The temperature of the gas in the chamber (Tch) is obtained using the ideal gas equation, thus the chamber gas temperature is written:

T ch

  d pch V ch pch V ch þ xpst Apst ¼ ¼ mch R mch R

ð25Þ

The heat transfer in the gas chamber is calculated according to Newton’s heat transfer law (see in Assumption A4) that gives the following equation for the chamber:

Q ch ¼ kht Aht ðT ch  T amb Þ

ð30Þ

vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 3 u !j2 !jþ1 u j u 2j 1 p p ch ch 5 T 4 rsl ¼ asl xsl dsl p psup t  psup j  1 R sup psup vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 3 u !j2 !jþ1 u j u 2j 1 pch pch 4 5 T rbl ¼ abl xbl dbl p psup t  psup j  1 R sup psup

dV ch ¼ pch v pst Apst dt

7.3. Forces acting on the piston The force generated by the chamber pressure, acting on the piston surface and used to pre-stress the disc spring, can be written as:

F pch ¼ ðpch  pamb ÞApst

ð33Þ

The holder spring pushes the piston towards the disc spring, thus the holder spring force acting on the piston can be written as follows:

F hsp ¼ chsp ðxhsp0  xpst Þ

ð26Þ

ð27Þ

7.2. SMV airflow properties In general the local gas speed in the SMVs at vena contracta, the point in a flow, where the diameter of the flow is the least, is deter-

ð32Þ

where asl and abl are the contraction coefficients of the flow in the small- and big load SMVs respectively. The mass flow of the exhaust SMVs rse and rbe are equal to zero due to the zero flow cross sections of the exhaust valves.

where kht is the heat transfer coefficient and Aht is the surface area of the chamber. The work term can be calculated using the general gas work equation in case of changing volume:

W ch ¼ pch

ð31Þ

Fig. 2. Two way two port SMV layout.

ð34Þ

27

B. Szimandl, H. Németh / Mechatronics 23 (2013) 21–36

where chsp is the stiffness of the holder spring and xhsp0 is the spring pretension stroke. According to Assumption A17 in which the clutch mechanism and the clutch actuator friction effects are lumped together into one friction effect, the magnitude of the friction force depends on a lumped friction coefficient lpst and the piston pressure force. The actuator friction comes from the friction of the pressure amplified piston sealing. The friction of the clutch mechanism comes from the contact of the release bearing and the disc spring moreover the contact of the disc spring and the pressure plate (see Fig. 1). The clamping forces of these contacts are also proportional with the pressure force. The friction force acts against the piston movement and introduces three hybrid items depending on the piston velocity, which is positive in the nominal hybrid mode, so the friction force is written as:

F fr ¼ lpst ðpch  pamb ÞApst

ð35Þ

RR ¼ Rpl þ Rfr þ Rc1 þ Rc2 þ Rarm :

ð39Þ

The constant part of RR is denoted by R as the constant part of the magnetic loop so the total magnetic resistance can be given as:

x

RR ¼ R þ Rc2 ¼ R þ

ð40Þ

l0 Aarm 2

where l0 is the permeability of vacuum and Aarm ¼ darm p=4 is the cross section of the SMV armature body. Since there is only one stroke dependent component, the derivative function with respect to x is written as:

dRR dRc2 1 ¼ ¼ dx dx l0 Aarm

ð41Þ

With this the magnetic force can be expressed as follows:

ðN iÞ2 F mg ¼  2 2 R þ l Axarm

1

l0 Aarm

ð42Þ

Considering A18, in which assuming that the clutch mechanism and the clutch actuator damping effects are lumped together into one damping effect kpst, the damping force acting against the piston movement is obtained as follows:

The force coming from the armature return spring is obtained as follows:

F dmp ¼ v pst kpst

F rsp ¼ srsp ðx þ xrsp0 Þ þ krsp v

ð36Þ

The stroke limiting force of the piston is modelled as a stiff spring if the stroke exceeds the limits. This introduces three hybrid modes, two limiting positions (xlim;1 and xlim;2 pst pst ) at the stroke ends (see Fig. 1) and the third one corresponding to the intermediate position. According to the selected nominal hybrid mode the piston is in intermediate position, so the stroke limiting force is zero:

F lim ¼ 0

ð37Þ

The Fload(xpst), acting against the piston movement, comes from the clutch mechanism. This force is a highly non-linear function of the displacement, generated by the disc spring, the cushion springs of the friction disc and the leaf springs. The explicit formula of the load characteristic (see Fig. 3), which can provide the prescribed accuracy, is too complex, therefore in the model a realisation through the empirical centre characteristic line will be used. The large hysteresis of the force characteristics, generated by the friction, is a result of the friction force defined above. In accordance with Assumption A20 the pretension of the disc spring does not change in spite of the fact that the friction disc wears, hence the characteristic cannot change during the lifetime of the clutch mechanism. 7.4. Forces acting on the SMV armature The magnetic force (Fmg) can be calculated as the partial derivative of the energy of the magnetic field (E) with respect to the armature stroke as:

F mg ¼ 

@E H2 dRR ðN iÞ2 dRR ¼ ¼ ; @x 2R2R dx 2R2R dx

ð38Þ

where H is the excitation (magnetic voltage), RR is the magnetic resistance and N is the number of solenoid turns. The connected magnetic resistances are related to the frame ðRfr Þ, the plug ðRpl Þ, the SMV armature ðRarm Þ, the air clearance between the overlapping coaxial cylindrical surfaces of the SMV armature and the frame ðRc1 Þ and resistance in the air clearance ðRc2 Þ between the plug and the armature (see Fig. 2). The only component that depends on the stroke is Rc2 , which is proportional to x. The change of Rarm is negligible small so it is considered constant as well. The magnetic resistance can be calculated as a function of armature stroke from the magnetic circuit as:

0

ð43Þ

where srsp is the stiffness, xrsp0 is the pretension and krsp is the damping of the return spring. Similarly to the piston stroke limitation the SMV stroke limiting force is modelled as a stiff spring if the stroke exceeds the limits (xlim;1 and xlim;2 xx xx ). This introduces three more hybrid modes for each SMV the same way as it already been discussed. Since the load SMVs have a small, but intermediate stroke, in the nominal hybrid mode therefore the stroke limiting forces for the load SMVs are zero Flim,sl = Flim,bl = 0. The exhaust SMVs are closed thus the limiting forces can be obtained as follows for small   exhaust SMV F lim;se ¼ cse xse  xlim;1 and for big exhaust SMV se   lim;1 , where cse and cbe are the end stop or stroke F lim;be ¼ cbe xbe  xbe limiting stiffness for the small- and big exhaust SMV respectively. 7.5. Electro-magnetic relations The inductance of the SMV is written as the following equality of the number of solenoid turns and the magnetic resistance:

L¼

N2 RR

ð44Þ

Its derivative with respect to the armature position can be written as:

dL dL dRR ¼ dx dRR dx

ð45Þ

and derivative with respect to the magnetic resistance is given as follows:

dL N2 ¼ 2: dRR RR

ð46Þ

7.6. Power stage relations The electric circuits of the SMVs (see Fig. 1) are low side driven, this means that one of the terminals of the SMVs is connected to the supply voltage (Usup) and the other is connected to the switching element of the power stage. The power stages are driven by the clutch control unit with high frequency PWM signals (uin). In accordance with Assumption A14 the PWM frequency is high enough to approximate the currents with its average values, thus

28

B. Szimandl, H. Németh / Mechatronics 23 (2013) 21–36

10000 9000 8000 7000

5000

F

load

[N]

6000

4000 3000 2000 1000 0

Measurement data Centre characteristic line 0

2

4

6

8

10

x

12

pst

14

16

18

20

22

[mm]

Fig. 3. Characteristic of the clutch mechanism.

the terminal voltage of the SMVs depends only on the supply voltage and on the voltage drop of the power stage (upws) as follows:

uterm ¼ U sup  upws

ð47Þ

The voltage drop of the power stages introduce three hybrid states depending on whether the duty cycle (md) of the PWM signal and the SMV current are equal or greater than zero. This is the consequence of the unclamped switching [23]. As the SMVs are switched off a high induced voltage appears between the terminals of the SMVs, due to the self inductance, which are limited by the power stage. Hence the voltage drop of the power stages can be written as follows:

upws

8 > < U sup ð1  md Þ þ r DSðonÞ  i; if md > 0 ¼ U BR ; if md ¼ 0 and i > 0 > : U sup ; otherwise

ð48Þ

where UBR is the breakdown voltage of the MOSFET and rDS(on) is the drain to source turned on resistance according to Assumption A15. According to the selected nominal hybrid mode the md,sl = md,bl > 0 and md,se = md,be = 0, thus the power stage voltages can be considered as follows:

upws;sl ¼ U sup ð1  md;sl Þ þ r DSðonÞ;sl  isl upws;bl ¼ U sup ð1  md;bl Þ þ r DSðonÞ;bl  ibl upws;se ¼ U sup upws;be ¼ U sup

HM5,x and HM6,x for load and exhaust SMVs respectively. The third is the flow cross-section and armature stroke limiting force of the SMVs, marked with HM710,x for the four SMVs. The fourth term is the piston stroke limiting force HM11,x and the last is the friction force of the piston HM12,x. 8.1. Power stage voltage drop The power stage of the SMVs can be considered as a switching element which inherently causes hybrid behaviour. Moreover the unclamped inductive switching, where the power stage itself limits the induced voltage in the avalanche state, introduces three hybrid states according to the duty cycle of the control PWM signal and the solenoid current weather they are equal or grater than zero. In this way the three states are as follows: switched on state, after switch of state (avalanche) and released state. The corresponding hybrid modes are shown in Table 2. 8.2. SMV airflow The air flow on a port between two chambers (see Eq. 28) is governed by the pressure ratio (P). In connection with the flow, four cases can be distinguished that can be subsonic and sonic in both directions (assuming that no Laval geometry is met). The sonic flow conditions are determined by the critical pressure ratio as:

 ð49Þ

8. Hybrid items The above defined equations describe the system in a special hybrid mode only. To generalise the model all of the cases that describe the changes in the conservation and constitutive equations of the model have to be collected. The model includes five subsystem types that exhibit discrete– continuous behaviour. The first is the voltage drop of the power stages marked with HM14,x for the four SMVs. The next is the air flow term of the SMVs, the related hybrid modes marked with

Pcrit ¼

jj1 2 jþ1

ð50Þ

Assumption A6 states that the SMVs air flow have non-negative values only. This implies only two hybrid modes in the same flow direction. So there is only one part in Eq. (28) that depends on the

Table 2 Hybrid modes of the power stage voltage drop. No.

Condition

upws

HM14,1 HM14,2 HM14,3

md,xx > 0 md,xx = 0 and ixx > 0 md,xx = 0 and ixx = 0

Usup(1  md) + rDS(on)  i UBR Usup

29

B. Szimandl, H. Németh / Mechatronics 23 (2013) 21–36

hybrid mode, namely the pressure ratio under the exponents. The corresponding hybrid modes are shown in Table 3.

Table 4 Hybrid modes of the SMV cross sections and limiting forces. No.

8.3. SMV flow cross-section and stroke limiting forces The flow cross section expressions and stroke limiting forces are depending on the armature stroke of the SMVs, in this way they are dependent hybrid modes regarding to the same SMV. Assumption A8 considers that the armature stroke can be bigger than dxx/4, which implies three hybrid modes regarding to the flow cross section such as zero-, cylindrical- and circular cross section. The stroke limitation is modelled by stiff springs if the stroke exceeds the limits. In intermediate position this limiting force is absent. In conclusion the SMVs have three hybrid modes that are stroke dependent due to the stroke limiting force equations. The hybrid modes regarding to the flow cross sections and stroke limiting forces of the four SMVs are shown in Table 4, where cxx is the stroke limitation stiffness of the SMVs and xx can be sl, bl, se, and be.

Condition

Axx

HM710,1

xxx <

HM710,2

xlim;1 6 xxx < d4xx xx

HM710,3

dxx 4

HM710,4

xlim;2 6 xxx xx

Flim,xx   cxx xxx  xlim;1 xx

0

xlim;1 xx

xxxdxxp 2

6 xxx < xlim;2 xx

dxx 4 2

dxx 4

0 0

p

  cxx xxx  xlim;2 xx

p

Table 5 Hybrid modes of the clutch actuator piston limiting forces. No.

Condition

Flim   cpst xpst  xlim;1 pst

HM11,1

xpst <

HM11,2

xlim;1 6 xpst < xlim;2 pst pst

HM11,3

xlim;2 6 xpst pst

xlim;1 pst

0   cpst xpst  xlim;2 pst

Table 6 Hybrid modes of the friction forces.

8.4. Piston stroke limiting forces

No.

Condition

Ffr

HM12,1 HM12,2 HM12,3

vpst < 0 vpst = 0 vpst > 0

lpst(pch  pamb)Apst 0 lpst(pch  pamb)Apst

The stroke limitation of the piston is modelled similarly to the SMV armature stroke, if the piston stroke exceeds its limits. In intermediate position this limiting force is zero as well. Thus the piston of the clutch actuator has three hybrid modes that are stroke dependent due to the stroke limiting force equations. The hybrid modes of the piston limiting force are shown in Table 5, where cpst is the stroke limitation stiffness of the piston.

The uncontrollable inputs form the disturbance vector including the supply voltage, compressed (supply) air- pressure and temperature, ambient- pressure and temperature respectively:

8.5. Piston friction force

d ¼ U sup

Since the friction force always acts against the movement the direction of the movement introduces three new hybrid modes for the piston friction force (see Eq. 35), which are summarised in Table 6.

One of the most widespread model representation for analysis and control design purposes is the state space realisation based on a set of coupled first-order differential algebraic equations including a set of system variables in vector format [24]. Since this realisation is not unique, many equivalent representations with the same dimension can be found, giving rise to the same input– output description of a given system. Hence for the EPC model the following system variables are composed to retain the physical meaning of the variables. From the conservation equations the state vector is composed of their differential variables as follows:

v sl

xsl

ibl

v bl

xbl

ise

v se

xse

ibe

v be

xbe

mch

pch

v pst

xpst

T

ð51Þ

Table 3 Hybrid modes of the air flow (a) for load SMVs and (b) for exhaust SMVs.

(a)

(b)

No.

Condition

Pl

HM5,1

1 P ppch > Pcrit

pch psup

HM5,2

pch psup

Pcrit

No.

Condition

Pe

HM6,1

1 P ppamb > Pcrit

pamb pch

HM6,2

pamb pch

Pcrit

sup

6 Pcrit

ch

6 Pcrit

T sup

pamb

T amb

T

ð52Þ

The control input vector includes the duty cycle of the PWM control signals:

u ¼ ½ md;sl

md;bl

md;se

md;be 

T

ð53Þ

The output vector includes the current of the SMVs, the chamber pressure and the piston position:

9. Model equations in state space form

x ¼ isl

psup

y ¼ isl

ibl

ise

ibe

pch

xpst

T

ð54Þ

9.1. State equation For sake of simplicity the state space equation is given for the same nominal hybrid mode as used at the equation definition. Substituting the constitutive equations into the differential conservation balances the following state space input-affine model is obtained: m X dx gi ðx; d; rÞui ¼ fðx; d; rÞ þ dt i¼1

ð55Þ

where m is the number of the control inputs and r : Rn ! N is a piece-wise constant switching function mapping from the state space to N. The integer set N is finite, i.e. N ¼ 1; 2; . . . ; n, where n ¼ P8i¼1 ni is the total number of the hybrid modes and ni is the number of the individual hybrid modes of the subsystem (n = 2  3  3  2  2  2  4  4  3  3 = 20736). The values of r are composed by the conditions defined in Tables 3–6. Let the value of the switching function mapping r be 1 for the nominal hybrid mode, whose hybrid items are shown in Table 1. Through these the nonlinear state functions with all constitutive relations substituted for the nominal hybrid mode are obtained as (the entries that depend on the hybrid operation modes are boxed):

30

B. Szimandl, H. Németh / Mechatronics 23 (2013) 21–36

9.2. Output equation Since the output is linear with respect to the state vector, the measured output is written as the following equation:

y ¼ Cx:

ð78Þ

where the C matrix is determined by the intended application of the model, e.g. on purpose to do simulations for verification C can be equal to the identity matrix to reach all the state variables. As opposed to the simulation the instrumental conditions do not make it possible to reach all the state variables thus for validation purposes the model output matrix has been selected as the following way C = hci,ji, where c1,1 = c2,4 = c3,7 = c4,10 = c5,14 = c6,16 = 1 and the remained elements are equal to zero. The performance output is generated from the measured output by the following simple equation:

z ¼ ½0 0 0 0 0 1y:

ð79Þ

10. Model verification and validation The verification of the developed model is performed by extensive simulations using MATLAB/SIMULINK model. The simulation results are compared to the real system behaviour such as trends-, operation domain- and relationship of the variables. These attributes above are gathered from operation experiences on the real system. To obtain the model solutions the stiff ODE15s solver, with variable step size has been used, where the relative tolerance of the solver has been set up to 108. In the simulation calculations two typical operating cases have been considered. One of them have been executed with load SMVs excitations only in order to see the activation effect of these valves on the chamber pressure. In this operating case, called disengagement, the pressure force distorts the disc spring, hence the pressure plate can separate from the friction disc. In the other the exhaust SMVs excitations have been investigated in order to see the activation effect of the exhaust SMVs on the chamber pressure. This operating case, when the pressure is decreased in the chamber, is used for the engagement of the clutch. The parameters, considered in simulation calculations, can be seen in Table 7. 10.1. Disengagement process verification This process has been simulated with constant disturbances, starting from the engaged state of the clutch mechanism (currentless valves and exhausted chamber). The initial state vector is as follows:

x ¼ ½0 0 0 0 0 0 0 0 0 0 0 0 6:82  104 105 0 0T

ð80Þ

The disturbance vector has been considered with its nominal values as follows:

d ¼ 24 9:5  105 The elements of the gi(x, d, 1) = < gj,i(x, d, 1)> functions are equal to zero except the jth as:

coordinate

293 105

293

T

ð81Þ

Using the given boundary conditions above, from Eq. (22) which describes the current change of the SMVs, the time function of the currents can be derived, but the switching moments are shown only in the time graphs to focus attention on the details of the transient behaviour (see Fig. 4 left side). It can be seen that the current goes towards to its steady state in exponential way after the solenoid terminal voltage becomes higher than zero. When the electro magnetic force, generated by the solenoid current (see Eq. (42)), exceeds the return spring force (see Eq. (43)) the armature of the SMVs starts to move. The armature movement causes induced voltage due to the mutual induc-

B. Szimandl, H. Németh / Mechatronics 23 (2013) 21–36 Table 7 List of parameters. Parameter name

Symbol

Value

Unit

Adiabatic exponent Permeability of vacuum Specific gas constant Drain to source on resistance Effective breakdown voltage Inlet diameter of sx SMV Inlet diameter of bx SMV Armature diameter of sx SMV Armature diameter of bx SMV Stiffness of sx SMV spring Stiffness of bx SMV spring Pretension of sx SMV spring Pretension of bx SMV spring Mass of sx SMV armature Mass of bx SMV armature Solenoid turns of sx SMVs Solenoid turns of bx SMVs Electric resistance of sx SMVs Electric resistance of bx SMVs Contraction coefficient of SMVs Magnetic loop resistance of SMVs Stroke limiting stiffness of SMVs Damping coefficient of SMVs Stiffness of helper spring Pretension stroke of helper spring Area of piston Dead volume of chamber

j l0

1.4 4p  107 287.14 0.071 71 0.0015 0.0035 0.010 0.012 489 567 0.0102 0.0063 0.012 0.016 600 910 11.3 9.1 0.6 12,000,000 107 30 1  104 0.06 0.0227 5.5982  104

– Vs/Am J/kgK X V m m m m N/m N/m m m kg kg – – X X – A/Vs N/m Ns/m N/m m m2 m3

9.3922 0.01892 0.0689 0.1391 2251.3825 108

kg W/m2 K m2 – Ns/m N/m

Lumped mass Heat transfer coefficient of chamber Heat transfer area of chamber Friction coefficient of piston Damping coefficient of piston Stroke limiting stiffness of piston

R rDS(on),xx UBR,xx dsx dbx darm,sx darm,bx ssx sbx xsx,0 xbx,0 msx mbx Nsx Nbx Rsx Rbx

axx Rxx cxx kxx chsp xhsp0 Apst V dch mpst kht Aht

lpst kpst cpst

tance, which decreases the current of the solenoid according to the 1 dL dx i term of Eq. (22). When the armature reaches its limitation L dx dt and its velocity becomes zero, the current starts to increase towards the steady state in an exponentially way again. This means that the SMV is opened completely. The second exponential current change is faster due to the change of the inductance of the solenoid (see Eqs. (44) and (41)). The current level in the steady state, from Eq. (22), where di/dt = 0, is uterm/R as expected. When the solenoid terminal voltage is switched off the current starts to decrease immediately and this current change induces voltage due to the self inductance of the solenoid. This voltage is limited by the power stage of the SMV to protect its switching circuit. The SMV armature is returned by the return spring when the electro magnetic force decreases below a certain level and the SMV becomes closed again. The responses of the load SMVs are similar, since only some parameters are different from each other. The deviations originated from the different solenoid type, which causes different current dynamics and steady state. The chamber pressure starts increasing when the load SMVs become open (see Fig. 4 right side). The load SMVs are activated for constant 200 ms intervals. First the solenoid of the small load SMV is excited only (red lines), then the big load SMV (green lines) and in the end both of them are activated (black lines) in order to see the dynamic responses of the disengagement with different flow cross sections. In all the three cases 100% PWM duty cycle are applied. As expected the bigger flow cross section causes higher mass flow rate, more dynamic pressure build up and higher piston velocity, which causes shorter disengagement process. After the SMVs have been opened completely the sum of the valve mass flow rates (r), which equal to the change of the gas mass in the chamber (see Eq. (2)), have constant values due to the constant supply pressure, temperature and the overcritical pressure ratio, i.e. sonic flow (see Eq. (28) and Table 3 for details). Moreover it can be seen that

31

the pressure curves are not increased monotonically. On one hand this is the consequence of the piston dynamic depending on its inertia, on the other hand arising from the nonlinearity of the static characteristic of the clutch mechanism (see Fig. 3). 10.2. Engagement process verification This process has been simulated with constant disturbances also, starting from the disengaged state of the clutch mechanism (currentless valves but filled up chamber) as opposed to the previous case. The initial state vector is as:

x ¼ ½0 0 0 0 0 0 0 0 0 0 0 0 43:93  104 4  105 0 0:01543T

ð82Þ

The disturbance vector is considered as the same as before. The responses of the exhaust and load SMVs are similar to each other (see Fig. 5 left side). As expected, the chamber pressure starts to decrease when the exhaust SMVs become open. The exhaust SMVs are activated for constant 800 ms intervals and for all the cases 100% PWM duty cycle has been applied. First the solenoid of the small exhaust SMV is excited (blue lines), then the big exhaust SMV (magenta lines) and in the end both of them are activated (black lines) in order to see the dynamic responses of the engagement with different flow cross sections similarly as before. Although the flow cross sections of the load and exhaust valves are equal, the disengagement processes show slower dynamic piston movement compareded to the engagement, caused by the available mass flow rate. This is influenced by the pressure ratio differences (see Eq. (28)) between the chamber-reservoir and ambient-chamber pressures. The pressure curves do not change in a monotonic way, similarly to the disengagement process, due to compression effects. 10.3. Validation In order to validate the model, the model output vector has been compared to measurements on the real EPC system. The measurement set-up includes a measurement PC with data acquisition unit that is connected to the measurement sensors. The data acquisition unit is able to measure analogue-to-digital converted (ADC) channels and has digital I/O channels that can be used as control outputs to switching the power stages of the SMVs properly. The resolution of the ADC channels is 14 bits and the sampling frequency can be set up to 20 kHz. The target is to control the real system by its input vector to execute predefined test sequences and to measure all the characterising signals which are available. Hence, all the members of the input and the output vector are measured on the test bench, which introduces ten signals to the measurement system that are as follows: uterm,sl, uterm,bl, uterm,se, uterm,be, isl, ibl, ise, ibe, pch and xpst. Note that, instead of the real inputs (md,xx) the terminal voltage of the SMVs are measured, since they are represent not only the presence or absence of the supply voltage between the terminals of the SMVs but also show the brake down voltage (UBR) in the avalanche state. To predict the behaviour of the real system the disturbance variables should be known as well. From the five signals of the disturbance vector three can be considered as quasi-constant signals (Tsup, pamb and Tabm) that changes slowly, so these values are assumed to be constants (293 K, 105 Pa and 293 K). The remained (usup and psup) can change dynamically, so they should be acquired as well. The accuracy and the measurement range, according to the specification of the sensor suppliers, are seen in Table 8. For validation purpose the simulated test cases above are acquired on a real EPC bench. Then an additional test case has been

32

B. Szimandl, H. Németh / Mechatronics 23 (2013) 21–36

SMV

SMV

sl

SMVsl

bl

SMVbl

SMVsl+bl

−3

50

x 10 6

u

−50

0.05

0.06

ch

−50 0.245

0.07

4

m

term

0

u

term

0

[kg]

[V]

[V]

50

0.25

0.255

2 0

0.26

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.1

0.2

0.3

0.4

0.5

0.6

0.7

2

2

0.05

0.06

0 0.245

0.07

0.255

pst

0

0.05

0.06

−0.5 0.245

0.07

−3

x 10

0.255

0.26

x 10

x 10

0.5 0.05

0 −3

1.5

0.06

20

1

x [m]

x [m]

0

−3

1

0

0.25

0.3 0.2 0.1 0 −0.1

xpst [m]

1.5

2 0

0.26

v

0

−0.5

4

0.5

v [m/s]

v [m/s]

0.5

0.25

[m/s]

0

1

p

1

x 10 6

[Pa]

3

ch

3

i [A]

i [A]

5

0.5 0 0.245

0.07

time [s]

0.25

0.255

15 10 5 0

0.26

0

time [s]

time [s] Fig. 4. Simulation result of disengagement process. SMVse

SMVbe

SMVse

SMVbe

SMVse+be

50

50

0

0

−50

0.05

0.06

6

mch [kg]

uterm [V]

−3

−50 0.845

0.07

0.85

0.855

4 2 0

0.86

x 10

0

0.2

0.4

0.6

0.8

1

1.2

1.4

0

0.2

0.4

0.6

0.8

1

1.2

1.4

0

0.2

0.4

0.6

0.8

1

1.2

1.4

0.2

0.4

0.6

0.8

1

1.2

1.4

3

5

2

2

4

1 0.05

0.06

0 0.845

0.07

0.06

−0.5 0.845

0.07

0.86

1 0.5 0.06

time [s]

0.07

−3

x 10

x 10

20

[m]

1.5

x [m]

x [m]

0.855

−3

x 10

0.05

0.85

2

0.1 0 −0.1 −0.2 −0.3 −0.4

1

pst

0.05

0.5 0 0.845

x 10

3

1

0.86

0

−3

0

0.855

vpst [m/s]

0

1.5

0.85

0.5

v [m/s]

v [m/s]

0.5

−0.5

1

x

0

pch [Pa]

3

i [A]

i [A]

5

0.85

0.855

0.86

15 10 5 0

0

time [s] Fig. 5. Simulation result of engagement process.

time [s]

33

B. Szimandl, H. Németh / Mechatronics 23 (2013) 21–36

force characteristics is approximated with its empirical centre curve. The remained assumptions for the clutch mechanism (Assumption A16–A18 and A20) have no significant effect on the actuator dynamics. The last test case is a complete clutching procedure applied for gear shifting (see Fig. 8). The deviations in this test case are accumulated during the operation, which cause higher differences in the piston position. For comparison purposes individual errors are calculated for each test case based on the entries of the output vectors and measurements results as follows:

Table 8 Accuracy and range of measured signals. Signal

u (V)

i (A)

p (Pa)

x (m)

Minima Maxima Accuracy (%)

60 40 ±1

0 4 ±1

0 16  105 ±1.6

0 0.04 ±2

acquired that includes a real clutching procedure. Through these the first six test cases investigate short term dynamic behaviour with one excitation of different SMV variation. The seventh one checks the long term dynamics with SMV excitation for a real clutching procedure. The disengagement and engagement with various SMV variations are presented below. The measured and the simulated signals are represented with continuous- and dashed lines respectively. In the next two graphs the transient states of the SMVs in the switching moment are shown (see Fig. 6), then the pressure and the position signals are presented (see Fig. 7). The deviations in the SMV currents between the measurement and simulation caused by the simplifying assumptions, in which the nonlinearity and the dynamics of the SMVs are neglected and reduced respectively (see Assumptions A7 and A10–A12). In spite of the deviations between the simulated and the measured currents in opening direction the valve opening times are close to each other. This can be seen from the change of the current curve, as it has been described in the verification above, where the armature of the valve reaches its stroke limit, i.e. opened fully. In closing direction the currents have no major deviations, thus the closing times are close to each other as well. The remained assumptions related to the model of the SMVs (Assumptions A6, A13) have no significant influence on the SMV model accuracy. Since the resulted opening and closing time differences between the measurement and simulation have small impact on the remained part of the system, the simplification assumptions for the SMVs can be accepted. The deviations between the measured and simulated pressure and piston position signals are caused mainly by the simplifying assumptions (Assumption A19) in which, the clutch nonlinear

uterm [V]

SMVsl on

SMVbl on

k;j

Partial;k ¼

Total ¼

y

1;1

SMVbl off

−20

−20

−40

−40 0.41

0.6

0.605

3

3

2.5

2.5

2

2

1.5

1.5

1

1

0.5

0.5

0

0.045 0.05 0.055

time [s]

0.25

0.255

time [s] mes

SMVsl

0.4

0.41

time [s] sim

SMVsl

0.6

0

0.605

time [s] mes

SMVbl

ð84Þ

rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 Xn n



2 k¼1 Partial;k

ð85Þ

 y3;1  y4;3  y6;3 < y5;4  y6;4 < y2;2  y3;2 SMVse on

0

0.4



2 j¼1 yk;j

Analysing the individual errors (see Table 9), in the test case 1– 6, it can be seen that the small SMVs have smaller current deviations than the big ones, moreover the big load SMV has larger current deviations than the big exhaust SMV, i.e.

0

0.255

rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Xm

where m is the number of the output signals. This error shows the complete error of the corresponding test case. Finally the total error is calculated on the individual errors of the n piece of test cases using the squared mean as follows:

20

0.25

ð83Þ

where the suffix j refers to the jth output of the model, k refers to the kth test case, mes and sim refer to the corresponding output vector of the measurement and simulation respectively. The over line refers to the integral mean of the particular signal and T is the duration of the test case. Through this each individual error is an Euclidean signal norm of the error in the particular output compared to the measurement. Moreover a partial error is calculated based on these individual error terms for each test case as:

20

0.045 0.05 0.055

i [A]

SMVsl off

y

vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ! ffi u Z u1 T ymes ðtÞ  ysim ðtÞ 2 k;j k;j ¼t dt mes T 0 y k;j

sim

SMVbe on

SMVbe off

1.1

1.11

1.9

1.905

2.1 2.105 2.11

2.9

2.905

1.1

1.11

1.9

1.905

2.1 2.105 2.11

2.9

2.905

time [s] SMVbl

SMVse off

ð86Þ

mes

SMVse

time [s] sim

SMVse

time [s] mes

SMVbe

Fig. 6. Transient of the terminal voltages and currents during disengagement (left) and engagement (right).

time [s] sim

SMVbe

34

B. Szimandl, H. Németh / Mechatronics 23 (2013) 21–36 5

x 10 5.5 5 4.5 4 3.5 3 2.5 2 1.5 1 0.5 0 0 0.1

pch [Pa]

p

ch

[Pa]

5

x 10 5.5 5 4.5 4 3.5 3 2.5 2 1.5 1 0.5 0 0

0.05

0.1

0.15

0.2

0.25

0.3

0.4

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

0.5

0.6

0.7

0.8

0.9

x 10 20

15

15

xpst [m]

20

[m] pst

0.3

−3

−3

x 10

x

0.2

10

10

5

5

0 0

0.05

0.1

0.15

0.2

0.25

0

0.3

0

0.1

time [s] mes sl

SMV

sim sl

mes bl

SMV

time [s] sim bl

SMV

mes sl+bl

SMV

SMV

SMV

SMVmes

sim sl+bl

SMVsim

se

SMVmes

se

SMVsim

be

SMVmes

be

se+be

SMVsim

se+be

Fig. 7. Disengagement (left) and engagement (right) processes with various SMV combination.

mes

SMVsl

sim

mes

SMVsl

SMVbl

sim

SMVbl

mes

SMVse

sim

SMVse

mes

SMVbe

sim

SMVbe

[V]

50

u

term

0 −50

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

i [A]

3 2 1 0

0 5

mes ch

4

p

sim ch

p

2

p

ch

[Pa]

x 10

0

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

−3

x

mes pst

sim pst

x

5

x

pst

[m]

x 10 10

0

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

time [s] Fig. 8. Clutching procedure for gear shifting.

It is the consequence of the unmodelled nonlinear magnetic behaviour of the SMV material, which nonlinearity depends on the magnitude of the applied current [25]. Another reason of the

deviations is the neglected pressure force acting on the SMVs armature. Hence the higher current deviations are caused by the applied large SMV current magnitudes and the high pressure ratios

35

B. Szimandl, H. Németh / Mechatronics 23 (2013) 21–36 Table 9 Model errors (%). k

Test case

yk;1

yk;2

yk;3

yk;4

yk;5

yk;6

Partial

Total

1 2 3 4 5 6 7

Disengage 1 Disengage 2 Disengage 3 Engage 1 Engage 2 Engage 3 Clutching

1.88 0.06 1.88 0.06 0.06 0.06 1.02

0.03 3.84 3.78 0.03 0.03 0.03 1.72

0.06 0.06 0.06 1.58 0.09 1.83 3.31

0.04 0.04 0.04 0.04 2.81 2.45 3.80

1.80 2.46 2.98 0.78 1.07 0.83 1.47

0.76 3.00 2.71 1.16 1.15 1.09 1.38

2.71 5.45 5.84 2.11 3.21 3.35 5.78

4.32

< vMP5a

< vMP5b

< vMP5c

< vMP5d

< vMP5e

< vMP5f

between the in- and outlet ports of the SMVs. These are mainly relevant to the big load- and much less to the small exhaust SMV. The magnitude distribution of the current deviations, in the seventh test case, is different. In this test case the deviations distribution has a relation with the switching number of the SMVs, since the steady state current errors are small. Moreover the deviations between the measured and simulated pressure and position signals, on one hand, are caused by the SMV model deviations, on the other hand, are caused by the reduced dynamics of the gas energy and the reduced dynamics of the piston momentum. Nevertheless, the total error in the validation is below the specified tolerance limit, thus the model accuracy can be accepted, vMP5:ag are satisfied. 11. Conclusion The multi-physical nature of the EPC is considered as a mixed thermodynamic, electro-magnetic and mechanic system. The model of this system has been built and verified using a systematic modelling methodology. Moreover the effects of the simplifying assumptions on the model– accuracy and on the behaviour have been verified and validated. The model can be classified as a mechanistic, i.e. ‘‘white box’’. Although it contains empirical part, such as the disc spring characteristic, the mechanisms are evident in the model description. Therefore the most of the model variables and parameters retain its physical meaning ðvMP1 : trueÞ. The model includes several parts that exhibit discrete–continuous behaviour. There are five independent groups of these parts as follows: power stage voltage drop, air flow terms, flow cross sections and stroke limiting force terms for the SMVs, stroke limiting force and friction force terms for the piston. Nevertheless the model based on cause-effect analysis instead of considering probabilistic elements ðvMP2 : trueÞ. The characteristics of the system volumes are considered with concentrated parameters, since there are no spatial variations and the materials are homogeneous. This generally lead to simpler equation systems which are easier to solve as opposed to the distributed parameter case. The conservation equations for gas mass and energy, momentum and magnetic linkage are derived. These equations are composed for ten predefined balance volumes, where the characterising extensive and intensive quantities are perfectly mixed. The conservation balance equations form a set of nonlinear ordinary DAEs and the algebraic part of the model contains all relationships to make the model complete. Hence, it has been shown that the model is given by a set of differential– algebraic equations, namely state equations, where all the algebraic equations can be substituted into the differential ones ðvMP3 : trueÞ. This differential–algebraic equation system can be represented in state space description by Eq. (55) and Eq. (78) with 16 state-, 4 input-, 5 disturbance- and 6 output variables ðvMP4 : trueÞ. Moreover the nonlinear state equations of the model Eq. (55) are in standard input affine form since the input coordinate function is

< vMP5g

affine with respect to the state vector. It has also been shown that the system output equations Eq. (78) are linear. In conclusion the developed model is able to predict the dynamic behaviour of the real system within the accuracy required for dynamic simulation, and control validation purposes   vmax MP5ag < 0:05 . In order to increase the accuracy a detailed model parameter calibration/identification should be executed using special test cases [26,27]. Besides, the parameter and disturbance input sensitivity- and the failure mode and effect analysis could be done as well. The model can be used for further development to achieve advantageous properties such as fast dynamics, lower production cost, smaller geometry, longer lifetime and so on. For model-based controller design the developed model is too complex, e.g. the number of the state variables is 16 and the total number of the hybrid modes is 20736. Therefore it should be simplified to retain all major dynamics of the real plant but omit all details that are weakly represented in the state variables/outputs and not coupled with the control aims. For lumped dynamic models several simplification/reduction methods can be found in the literature. E.g. using the hierarchical structure of model elements and applying model simplification methods by performing sensitivity analysis on the model with respect to any of its model elements to find out if the omission of the model element leaves the performance index within the acceptable range and reduces the model complexity [28,29]. A top-down way of traversing the model elements using their hierarchy tree offers a systematic way of doing model reduction [30], which can be made more efficiently instead of using heuristics. References [1] Mehmood A, Laghrouche S, Bagdouri ME. Modeling identification and simulation of pneumatic actuator for VGT system. Sens Actuat A: Phys 2011;165:367–78. [2] Khayati K, Bigras P, Dessaint L-A. LuGre model-based friction compensation and positioning control for a pneumatic actuator using multi-objective outputfeedback control via LMI optimization. Mechatronics 2009;19:535–47 [Robotics and Factory of the Future, New Trends and Challenges in Mechatronics INCOM 2006]. [3] Xiang F, Wikander J. Block-oriented approximate feedback linearization for control of pneumatic actuator system. Control Eng Practice 2004;12:387–99 [UKACC Conference Control 2002]. [4] Jungmann T. Die Clutch Servo Assistance von Luk. ATZ – Automobiltechnische Zeitschrift Ausgabe 2005;2. [5] Harris S, Elliott L, Ingham D, Pourkashanian M, Wilson C. The optimisation of reaction rate parameters for chemical kinetic modelling of combustion using genetic algorithms. Comput Methods Appl Mech Eng 2000;190:1065–90. [6] Kaasa G-O, Takahashi M. Adaptive tracking control of an electro-pneumatic clutch actuator. Model, Identif Control 2003;24:217–29. [7] Langjord H, Johansen T. Dual-mode switched control of an electropneumatic clutch actuator. IEEE/ASME Trans Mechatronics 2010;15:969–81. [8] Grancharova A, Johansen T. Design and comparison of explicit model predictive controllers for an electropneumatic clutch actuator using on/off valves. IEEE/ASME Trans Mechatronics 2011;16:665–73. [9] Perkins J, Stephanopoulos G. 2 A systematic approach to model building. In: Hangos K, Cameron I, editors. Process modelling and model analysis. Process systems engineering, vol. 4. Academic Press; 2001. p. 19–40. [10] Förster B, Lindner J, Steinel K, Stürmer W. Kupplungssysteme für schwere Nutzfahrzeuge. ATZ – Automobiltechnische Zeitschrift Ausgabe 2004;10.

36

B. Szimandl, H. Németh / Mechatronics 23 (2013) 21–36

[11] Perkins J, Stephanopoulos G. 1 The role of models in process systems engineering. In: Hangos K, Cameron I, editors. Process modelling and model analysis. Process systems engineering, vol. 4. Academic Press; 2001. p. 3–18. [12] Lakner R, Hangos K, Cameron I. On minimal models of process systems. Chem Eng Sci 2005;60:1127–42. [13] Mosterman PJ, Biswas G. A comprehensive methodology for building hybrid models of physical systems. Artif Intell 2000;121:171–209. [14] Wen C. Modeling problems of hybrid event dynamic systems. Simul Practice Theory 1998;6:413–22. [15] Dogruel M, Adli MA. Hybrid state approach for modelling electrical and mechanical systems. Math Comput Modell 2005;41:759–71. [16] Hangos K, Cameron I. The formal representation of process system modelling assumptions and their implications. Comput Chem Eng 1997;21(Supplement): S823–8. [17] Hangos K, Cameron I. A formal representation of assumptions in process modelling. Comput Chem Eng 2001;25:237–55. [18] Hangos K, Szederkényi G, Tuza Z. The effect of model simplification assumptions on the differential index of lumped process models. Comput Chem Eng 2004;28:129–37. [19] Hangos K, Tuza Z. Process model structure simplification. Comput Chem Eng 1999;23(Supplement):S343–6. [20] Németh H, Ailer P, Hangos KM. Nonlinear modelling and model verification of a single protection valve. Periodica Polytech Ser Trans Eng 2002;30:69–96. [21] Zucker RD. Fundamentals of gas dynamics. 2th ed. New Jerse: John Wiley & Sons Inc.; 2002.

[22] Young DF, Okiishi TH. Fundamentals of fluid mechanics. 5th ed. New Jerse: John Wiley & Sons Inc.; 2006. [23] Fischer K, Shenai K. Dynamics of power MOSFET switching under unclamped inductive loading conditions. IEEE Trans Electron Dev 1996;43:1007–15. [24] Zadeh LA, Desoer CA. Linear system theory; the state space approach. New York: McGraw-Hill; 1963. [25] Woodson HH, Melcher JR. Electromechanical dynamics: Part I. John Wiley & Sons Inc.; 1968. [26] H. Németh, L. Palkovics, K.M. Hangos, System identification of an electropneumatic protection valve, Technical report SCL-001/2003, Budapest (Hungary): Computer and Automation Research Institute; 2003. [27] Nyarko EK, Scitovski R. Solving the parameter identification problem of mathematical models using genetic algorithms. Appl Math Comput 2004;153:651–8. [28] Leitold A, Hangos KM, Tuza Z. Structure simplification of dynamic process models. J Process Control 2002;12:69–83. [29] Lakner R, Hangos K, Cameron I. Construction of minimal models for control purposes. In: Kraslawski A, Turunen I, editors. European symposium on computer aided process engineering. Computer aided chemical engineering, vol. 14. Elsevier; 2003. p. 755–60. [30] Németh H, Palkovics L, Hangos KM. Unified model simplification procedure applied to a single protection valve. Control Eng Practice 2005;13:315–26 (Aerospace IFAC 2002).