Observer based feedforward/feedback control of electro-pneumatic clutch systems

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Observer based feedforward/feedback control of electro-pneumatic clutch systems

Ph.D. Thesis

Barna Szimandl

Supervisor: Huba Németh

Kálmán Kandó Doctoral School of Transportation Engineering Transportation and vehicle engineering sciences

Budapest University of Technology and Economics Faculty of Transportation Engineering

Department of Automobiles and Vehicle Manufacturing Budapest, Hungary

2015

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Abstract

This dissertation deals the with control of electro-pneumatic clutch systems applied in medium and heavy duty commercial vehicles. The main goal of the thesis is to build up a dynamic model of the system, to prepare and apply a model simpliﬁcation approach, to analyze the dynamic properties of the simpliﬁed models and to develop a clutch controller using the nonlinear model of the electro-pneumatic clutch.

It is shown that the electro-pneumatic clutch system can be described as a mixed thermodynamic, mechanic and electro-magnetic system and its model can be build and verified by using a systematic modeling methodology. The developed model structure is valid for clutch systems applied with concentric and forked lever type electro-pneumatic clutch actuators as well, only the model parameters differ from each other. The model exhibits hybrid, i.e. discrete-continuous behavior caused by different elements with inherently discrete behavior. The model has been verified and then validated against laboratory measurements. It has been shown that it is able to describe the dynamic behavior of the modeled system within the predefined tolerance limit.

A systematic model simplification process has been applied to the detailed model of the electro-pneumatic clutch. Two simplified models have been constructed: one for fast simulation and an other one for control design purpose. The size of the state vector has been reduced and the structure of the algebraic equations has been simplified considerably in both cases. The discrete components of the models have been eliminated completely in case of the control oriented model.

It has been shown that all retained system variable entries of the simpliﬁed models preserved their physical meaning and the control oriented model can be rewritten into standard input aﬃne form.

When performing model analysis for the control oriented model, it has been proved that this model is jointly reachable and detectable, thus it is minimal. The stability analysis has shown that the global asymptotic stability of the open loop model depends on model parameters. The asymptotic stability of the zero dynamics has been proved, thus the system is locally asymptot- ically stabilizable and asymptotic output tracking is achievable with appropriate feedback.

Based on the control aims and the input signal constraints an observer based feedforward/feedback controller structure has been developed. It includes three blocks in form of a state observer supplying the unmeasurable states, a feedforward controller unit producing the mass flow rate control and the I/O linearization of the solenoid magnet valves, and a model based feedback controller unit provides a piston position control. For the state observer the high-gain observer method has been used. For the feedforward and the feedback controllers several approaches have been considered, such as the static- and dynamic mass flow rate de- composition approach, the linear quadratic, robustH∞and sliding mode control approach. The obtained closed loops are investigated by extensive simulation, bench- and vehicle tests to verify the properties of the different controls.

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Tartalmi kivonat

A disszertáció közepes és nehéz haszongépjárművekben alkalmazott elektro-pneumatikus tengelykapcsolók irányításával foglalkozik. A vizsgálat tárgya az elektro-pneumatikus tengelykapcsolók dinamikus modellezése, a modell meghatározott célra való egyszerűsítése, a modell dinamikus analízise valamint egy modell alapú tengelykapcsoló szabályozó tervezése és a tervezett szabályozó tulajdonságainak ellenőrzése.

A vizsgálat megmutatta, hogy az elektro-pneumatikus tengelykapcsoló működtető egy vegyes termodinamikai, mechanikai és elektro-dinamikai rendszer, amelynek modellje szisztematikus modellezési eljárással felépíthető és veriﬁkálható. A különböző elektro-pneumatikus tengelykapcsoló kialakítások modellstruktúrája azonos, csak a modell paraméterekben különböznek egymástól. Az így felépített nemlineáris dinamikus hibrid modell speciális struktúrájú, diszkrét és folytonos elemeket is tartalmaz. A modell érvényességének ellenőrzése laboratóriumi mérések segítségével történt. A vizsgálat bebizonyította, hogy a modell alkalmas a valós rendszer dinamikus viselkedésének megadott tolerancia szinten belüli leírására.

A fizikai törvényszerűségek felhasználásával megalkotott modellt a szerző tovább egyszerűsítette. E célból egy szisztematikus modellegyszerűsítési eljárást alkalmazott és két egyszerűsített modellt alkotott meg: egyet gyors szimulációs célokra és egy másikat szabályozó-tervezés céljára. A modell egyszerűsítése révén csökkent az állapot vektor dimenziója és jelentősen egyszerűsödött az egyenletek algebrai alakja mind a két esetben. A vizsgálat kimutatta, hogy a rendszer változói megtartják fizikai jelentésüket. A diszkrét-folytonos elemeket a szabályozó-tervezés céljára egyszerűsített modell esetén a szerző teljesen kiküszöbölte, valamint a modellt standard input affin alakra hozta.

A modell analízise során a szerző kimutatta, hogy a szabályozó-tervezés céljára egyszerűsített modell együttesen elérhető és detektálható, azaz a modell minimális reprezentációjú. A nyitott rendszer stabilitási vizsgálata megmutatta, hogy a modell globális asszimptotikus stabilitása függ a modell paramétereitől. Ezenkívül a szerző azt is igazolta, hogy az egyszerűsített modell maximális relatív fokszámmal rendelkezik, így a zérus dinamika aszimptotikusan stabil. Ezzel a rendszer lokálisan aszimptotikusan stabilizálható és aszimptotikus jelkövetés valósítható meg egy megfelelő visszacsatolással.

A szabályozási célok és a bemeneti jelre előírt korlátozás alapján egy megfigyelő alapú előrecsatolt/visszacsatolt szabályozó struktúra került kifejlesztésre. A szabályozási struktúra három blokkot tartalmaz: egy állapot megfigyelőt mely a nem mérhető állapotot állítja elő, egy előrecsatoló egységet mely az elektro-pneumatikus tengelykapcsoló mőködtetőben alkalmazott szelepek légtömegáram vezérlését és I/O linearizálását valósítja meg és egy visszacsatoló egységet mely a mőködtető dugattyú pozíciójának szabályozását biztosítja. Az állapot megfigyelő esetén egy nagy-erősítésű megfigyelő került kidolgozásra. Az előrecsatolt és visszacsatolt irányítások esetén különböző megoldásokat vizsgált meg a szerző, úgymint statikus és dinamikus légtömegáram felbontás, lineáris kvadratikus, robusztus H∞ és csúszó mód irányítás. A kidolgozott szabályozási rendszereket kiterjedt szimulációs-, tesztpadi és járműves tesztekkel ellenőrizte a szerző.

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Foreword

This thesis summarizes the contributions of my research work for obtaining Ph.D. degree in Kálmán Kandó Doctoral School of Transportation Engineering at the Faculty of Transportation Engineering of the Budapest University of Technology and Economics. The scientiﬁc part of the studies has been undertaken at the Knorr-Bremse Research and Development Centre Budapest.

This work would have never been written without the help, continuous support and encour- agement of several people. First of all, I want to express my sincere gratitude to my supervisor, head of the Advanced Engineering Group at Knorr-Bremse R&D Center Budapest and asso- ciate professor of Budapest University of Technology and Economics Faculty of Transportation Engineering Department of Automobiles and Vehicle Manufacturing, Dr. Huba Németh, for his patient guidance throughout my studies and support for realizing the opportunities for the experiments.

I would like to express my gratitude to Professor József Bokor, the head of Systems and Control Laboratory and his colleagues for providing me with the essential ideas and literature on dynamic systems and control. I am also grateful to my colleagues, Zoltán Geiszt and Balázs Trencséni for the joint work.

Finally, I am grateful to my wife Ági, my parents and my friends for supporting my studies in many ways for such a long time.

The undersigned, Barna Szimandl declares that this Ph.D. thesis has been prepared by himself as well as that the indicated sources have been used only. All parts that have been taken over literally or by content are cited unambiguously.

Alulírott Szimandl Barna kijelentem, hogy ezt a doktori értekezést magam készítettem és abban csak a megadott forrásokat használtam fel. Minden olyan részt, amelyet szó szerint, vagy azonos tartalomban, de átfogalmazva más forrásból átvettem, egyértelműen, a forrás megadásával megjelöltem.

Budapest, 2015.06.30. . . . Szimandl Barna

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List of Figures

2.1 The layout of the electro-pneumatic clutch (EPC) system . . . 8

2.2 Free body diagram of the piston. . . 14

2.3 Free body diagram of the solenoid magnet valve (SMV) armature . . . 15

2.4 Magnetic linkage in the solenoid. . . 15

2.5 Two way two port on/oﬀ SMV layout. . . 17

2.6 Characteristic of the clutch mechanismF_l(x_pst) . . . 19

2.7 Transient of the valve signals during disengagement . . . 27

2.8 Transient of the chamber/piston states during disengagement . . . 28

2.9 Transient of the valve signals during engagement . . . 29

2.10 Transient of the chamber/piston states during engagement . . . 30

2.11 Transient of the terminal voltages and currents during disengagement. . . 31

2.12 Transient of the terminal voltages and currents during engagement . . . 32

2.13 Transient of the pressure and position during disengagement . . . 32

2.14 Transient of the pressure and position during engagement . . . 33

2.15 Real clutching procedure in case of gear shifting . . . 34

3.1 Structure graph of the diﬀerential variables of the EPC model . . . 37

3.2 Hierarchical structure of the detailed nonlinear dynamic hybrid model . . . 38

3.3 Model simpliﬁcation procedure . . . 39

3.4 EPC measurement and simulation results in case of model M0,M1 and M2 . . 43

3.5 Hierarchical structure of the simpliﬁed model M₁ . . . 44

3.6 Structure graph of the diﬀerential variables of the simpliﬁed model M₁ . . . 45

3.7 Model performance and size indices in case of model M0,M1 andM2 . . . 46

3.8 Hierarchical structure of the simpliﬁed model M₂ . . . 48

3.9 Structure graph of the diﬀerential variables of the simpliﬁed model M₂ . . . 48 4.1 Equilibrium points over the pressure-position plane with increasing dead volume 57

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4.2 Sensitivity of the states regarding the model parameters . . . 60

5.1 Structure of the observer based feedforward/feedback control . . . 62

5.2 LQ servo block structure . . . 64

5.3 Closed loop interconnection structure . . . 65

5.4 The ∆-P-K structure . . . 66

5.5 Singular values with full order controller regarding RS, NP and RP . . . 67

5.6 Singular values with full and reduced order controllers . . . 67

5.7 High-gain anti-windup structure. . . 68

5.8 Available mass ﬂow rates plotted against diﬀerent chamber pressure. . . 71

5.9 Mass ﬂow rates of ideal, proportional and real valves . . . 72

5.10 Characteristics of small SMVs . . . 73

5.11 Characteristics of big SMVs . . . 74

5.12 Magnitude and phase of the frequency response from y to x,ˆ x˙ˆ and tox¨ˆ . . . 77

5.13 Graphical interpretation of the performance indices . . . 78

5.14 Step responses with 100% stroke in case of simpliﬁed model . . . 79

5.15 Clutch engagement test functions in case of simpliﬁed model . . . 79

5.16 Step responses with 100% stroke in case of detailed model . . . 81

5.17 Clutch engagement test functions in case of detailed model . . . 81

5.18 Electro-pneumatic clutch system layout . . . 82

5.19 Step responses with 100% stroke in case of test bench . . . 83

5.20 Clutch engagement test functions in case of test bench . . . 83

5.21 The sliding surface and the state trajectory in the phase plane. . . 84

5.22 Smooth launch of the vehicle . . . 85

5.23 Dynamic launch and gearshifting . . . 86

A.1 Friction disc, clutch mechanism and concentric clutch actuator from ZF SACHS . 100 A.2 Engine, gearbox and clutch system . . . 101

A.3 Clutch systems applied with forked lever- and concentric type EPC actuators . . 101

B.1 Block diagram of the exact linearization via state feedback . . . 106

List of Tables

2.1 Balance volumes and conserved quantities . . . 12

2.2 Hybrid modes of the power stage voltage drop . . . 21

2.3 Hybrid modes of the air ﬂow. . . 22

2.4 Hybrid modes of the armature stroke dependent terms . . . 22

2.5 Hybrid modes of the clutch actuator piston limiting forces . . . 22

2.6 Hybrid modes of the Friction forces . . . 23

2.7 Accuracy and range of measured signals . . . 30

2.8 The modeling errors in case of nonlinear dynamic hybrid model . . . 35

5.1 Performance indices of step response and clutch slipping tests . . . 80

A.1 List of parameters . . . 102

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Chapter 1

Introduction

”The science of today is the technology of tomorrow.”

Ede Teller

1.1 Problem statement and motivation

The automotive industry is one of the leading industry branches all around the world. The main reason of this fact is that this is the primary field of ”civil” application of the newest scientific results reached in the space, aviation and military research, as well as a good trial opportunity for the new innovations in other scientific areas. No doubt, the passenger car development, application of new ideas and technology is the leading area compared to the other road vehicle systems. The explanation for it is obvious: the price of passenger cars, usually bought for pleasure rather than making profit, can incorporate the extra costs of the advanced systems. This is the ground for the wide application of controlled vehicle systems in passenger cars: anti-lock braking system, traction control system, electronic engine control, semi-active and/or adaptive suspension controls are all standard in even medium size passenger cars. The application of advanced, electronically controlled systems in commercial vehicles somehow has not been as fast as in the passenger cars in the past. The explanation of this situation shows the constraints for the development and marketing of these systems:

1. The primary reason why a commercial vehicle is purchased in business like: making proﬁt, which means low price of the vehicle, low maintenance cost, reliability throughout the life cycle of the vehicle. This fact is contradictory to the application of any advanced system, since normally they make the vehicle more expensive, although their impact on the vehicle safety and on the costs of operation is obviously advantageous.

2. The commercial vehicle market is more conservative, does not like to accept new systems unless it is convinced about the deﬁnitive advantages. Typical example is the reluctance of the market concerning the electro-pneumatic brake systems for heavy commercial vehicles, whereas the advantages are obvious, but people ”would not see” the brake actuation (i.e.

there are no pneumatic lines, tubes, valves to control the wheel brake) since it is done electronically. This was the reason (besides the legislation) that redundant pneumatic circuits had to be installed in parallel to the otherwise very safe electronic brake system.

However, with growing number of the vehicles all around the world, the demand of the society on the traffic safety is also increasing. Since the transportation infrastructure cannot keep up with rising number of vehicles there is a severe task for the transportation as well as control and mechanical engineers to control the traffic flow in the way of enhancing traffic safety and, at the

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same time, increasing the efficiency of the transportation, i.e. increasing the traffic density. As seen, there is an obvious contradiction between the mentioned two facts, since increasing the traffic density will result in growing probability of traffic accidents. This contradiction cannot be relieved, but it can be optimized by a certain way, giving intelligence both to the vehicle itself, and also to the infrastructure, making the information flow between the road and the car possible.

These and similar requirements explain the need of the society for safer, less polluting, less dangerous and last but not least less expensive heavy vehicles, which have no significantly different performance as the passenger cars. These fact make the development of commercial vehicle advanced systems more interesting and more challenging for development engineers and scien- tists, since to fulfill all the technical conditions, at relatively lower price, resulting in a less complex system is not an easy task [1].

An important part of this innovation of commercial vehicles is to improve the control of electron-pneumatic clutch systems to achieve high dynamics and accurate signal tracking.

1.2 Electro-pneumatic clutch systems and their control

The requirements of the clutch control function are determined by the controllability of the torque, transmitted by the clutch. The transmitted torque is a function of the piston position of the clutch actuator realized by a disc spring, therefore the position control can provide clutch torque control function.

In the last decade, several papers have been published on the topic of the position control of electro-pneumatic clutch actuators. These actuators are driven by proportional- or on/oﬀ valves, which yield that the control signals can be continuous and have to be quantized respectively.

One of the proposed control methods are the PID type controls, extended with self-tuning ability using parallel feedforward compensator, neural network or fuzzy proportional summation and derivative (PSD) methods [2, 3, 4]. These controls are designed for clutch actuators with proportional valves. Other published methods are the switching controls, in which a control Lyapunov function is defined to achieve exponential stability in the operation domain using quantized control input, where the quantization came from applying on/off valves instead of proportional ones [5,6, 7,8]. Explicit model predictive control techniques are also used, where the minimization of a cost function in a finite horizon is performed off-line [9,10,11].

Comparing these methods above, it can say that, nonlinear methods can achieve higher disturbance rejection performance and wider stability margin versus linear ones. In Lyapunov function based switching controls a stabilizing controller is defined first and a switching surface is found from this. It is more straightforward to provide a predefined dynamic behavior on a switching surface, e.g. in case of sliding mode control, to achieve the desired performance of the system easily. Model predictive controls may not provide acceptable code length, memory claim and computing costs for embedded applications, since these could become high if the number of the partitions of the explicit piecewise constant approximate solution increases e.g. considering not only the state variables but the disturbance inputs as well.

Moreover the requirements of the clutch control are changed. In the recent years, the flow cross section of the on/off solenoid valves, applied in clutch actuators, are increased to achieve the increased performance. Obviously, this performance requirement is dictated by the additional clutch control functions to further improve the fuel efficiency, the maintenance period and the safety of the trucks. Although, increasing the cross section of the valves 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 potential of the fine application

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of the compressed air and through this the ﬁne application of the torque transmitted by the clutch. Hence, there is an obvious development opportunity to improve the performance of the electro-pneumatic clutch control. This was the initial motivation for the research studies of the author.

In the literature, several models of electro-pneumatic clutch systems have been introduced.

These models are developed for control design purposes only [2, 8, 11], and do not cover the overall dynamics of the system e.g. the solenoid valve dynamics and the hybrid behavior of the system are not entirely considered. Hence, the earlier presented models in the literature are not accurate enough for the state of the art electro-pneumatic clutch systems due to the increased flow cross section of the applied on/off solenoid valves. This phenomenon and the requirements 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 nonlinear dynamic hybrid electro-pneumatic clutch model should be developed firstly based on a systematic modeling procedure. Then, based on the developed model an appropriate control method should be elaborated.

1.3 The aim of the work

Considering the initial motivations and the results of the literature review the target of the study described in this thesis is to design, tune and compare appropriate controllers for electro- pneumatic clutch systems to achieve high dynamics and accurate signal tracking. For this research aim first a nonlinear lumped parameter dynamic model had to be derived with appropriate dimension and complexity level for control design purposes. The model had to be verified and validated for the above application aim. Therefore one had to investigate the dynamic properties of the model by means of dynamic model analysis and finally after defining the control aims appropriate controllers had to be designed tuned and compared with each other.

1.4 Layout of the thesis

The thesis consists of 6 chapters (including this Introduction) and an Appendix of 2 parts. Each chapter begins with a motivation part that describes the main problem statement and aim of the corresponding part. The chapters are ﬁnished with a summary where the conclusions are drawn. The layout of the thesis and the main scientiﬁc contributions are described below.

Chapter 2. The nonlinear hybrid model of the electro-pneumatic clutch is derived in this part utilizing thermodynamic, mechanic and electro-magnetic first engineering principles. This is described as conservation and constitutive equations inSection 2.4-2.5. These equations form a set of differential-algebraic equations. The model parts that exhibit switching behavior are discussed in Section 2.6. Then the model is given in state space form in Section 2.7. Finally the model verification and validation is presented in Section 2.8.

Chapter 3. The model based on first engineering principles fromChapter 2has been considered too complex for the intended uses, which are on one hand fast dynamic simulation with reduced computational effort and on the other hand control design. This chapter deals with a model simplification procedure. First the structure of the detailed model is examined in Section 3.1. Then a systematic model simplification approach is given inSection 3.2. The criteria and the simplification steps are shown in Subsection 3.3.1 in case of simplified nonlinear dynamic model for simulation purposes, while in Subsection 3.3.2 in case of simplified nonlinear dynamic model for control design purposes.

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Chapter 4. The chapter contains the dynamic analysis of the control oriented model. The investigations are divided into ﬁve main parts. The reachability and observability are discussed in Section 4.1 and in Section 4.2. The asymptotic stability of the model is assessed inSection 4.3. The zero dynamics is examined inSection 4.4. Finally the system sensitivity to parameter uncertainty and disturbances is investigated in Section 4.5.

Chapter 5. This part shows a control design method of a clutch controller for the electro- pneumatic clutch actuator. The requirements of the clutch control are gathered in Section 5.1. The designed observer based feedforward/feedback controller structure is discussed inSection 5.2. The feedback module is synthesized inSection 5.3. The feedforward module of the controller is given in Section 5.4. The controller utilizes a state observer that is designed in Section 5.5. The experimental results are presented and discussed in Section 5.6.

Chapter 6. This chapter contains the ﬁnal conclusions and the related publications of the thesis, moreover it describes the possible directions for future research.

Appendix A This part of the Appendix contains ﬁgures and tables that could not be ﬁt to the main text due to space limitations.

Appendix B This part includes the model transformations of the control oriented model such as linearization and coordinate transformation.

1.5 Nomenclature

The notation list contains all the commonly used symbols and abbreviations throughout the thesis. The units of the physical variables are given in brackets that refer to the SI standard.

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Variables Indices A area, surface [m²]

a acceleration [m/s²] α contraction coeﬃcient [−]

B magnetic induction [V s/m²] c_p speciﬁc heat of

constant pressure [J/kgK] c_v speciﬁc heat of

constant volume [J/kgK]

c stiﬀness [N/m]

d diameter [m]

E energy [J]

E electric ﬁeld strength [V /m]

i electric current [A]

F force [N]

h speciﬁc enthalpy [J/K]

k heat transfer coeﬃcient [W/m²K]

k damping coeﬃcient [N s/m]

κ adiabatic exponent [−]

L inductance [V s/A]

m mass [kg]

m_d duty [−]

µ permeability [V s/Am]

µ₀ permeability of free space [H/m]

N solenoid turns [−]

Q heat ﬂux [J/s]

p absolute pressure [P a]

Φ magnetic ﬂux [V s]

Ψ magnetic linkage [V s]

T temperature [K]

R electric resistance [Ω]

R magnetic resistance [A/V] R speciﬁc gas constant [J/kgK]

s spring coeﬃcient [N/m]

σ air ﬂow [kg/s]

t time [s]

T absolute temperature [K]

Θ excitation (magnetic voltage) [A]

U voltage [V]

U internal energy of gas [J] v speed [m/s]

V volume [m³] x stroke [m]

0 refers to initial state 1,2 refers to ends amb refers to ambient arm refers to armature BR refers to breakdown c1 refers to air clearance 1 c2 refers to air clearance 2 ch refers to clutch chamber crit refers to critical

dmp refers to damping

DS(on) refers to Drain to source on dsp refers to disc spring

exh refers to exhaust pl refers to plug f r refers to friction f rm refers to frame

gap refers to clearance or gap HM refers to hybrid mode hsp refers to helper spring ht refers to heat transfer in refers to inlet

l refers to load lim refers to limitation out refers to output p refers to pressure Π refers to pressure ratio pst refers to piston

pws refers to power stage Σ refers to magnetic resultant sup refers to supply

term refers to terminal

sl refers to small load valve bl refers to big load valve se refers to small exhaust valve be refers to big exhaust valve

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Notation for state space models

d disturbance vector (d:A∈R→D∈R^v) u input vector (u:A∈R→U ∈R^p)

k hybrid mode mapping (k:X∈Rⁿ→K ∈N) x state vector (x:A∈R→X ∈Rⁿ)

y=h(x) measured output vector (y:A∈R→Y ∈R^m) z performance output vector (z:A∈R→Z ∈R^r) f(x),g(x),h(x) coordinate functions of the nonlinear model

˙

x=dx/dt time derivative of the state vector x

df(x) =∂f/∂x Jacobi-matrix of the functionf :Rⁿ→Rⁿ,x→f(x) L^k_fh(x) repeated derivative of h(x) along vector fieldf L_gL^k_fh(x) repeated derivative of h(x) first along vector fieldf

and then along vector ﬁeld g A,B,C,D matrices of the linear model

Acronyms

ADC analogue-to-digital converted AMT automated mechanical transmission CAU clutch actuator unit

CBW clutch-by-wire CCU clutch control unit CM clutch mechanism

DAE diﬀerential algebraic equation ECU electronic control unit

EPC electro-pneumatic clutch HIL hardware in the loop I/O input/output

L₂ Euclidean norm LTI linear time-invariant LQ linear quadratic

MIMO multiple-input multiple-output PID proportional, integral and derivative PSD proportional, summation, derivative PWM pulse width modulated

SIL software in the loop SISO single-input single-output SMC sliding mode control SMV solenoid magnet valve

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Chapter 2

Nonlinear dynamic hybrid models of electro-pneumatic clutch systems

The aim of this chapter is to construct a systematically developed model of the electro-pneumatic clutch (EPC) systems.

A model is a simpliﬁed description of a real world object for a given application aim. The real processes of the modeled object are ﬁrst translated into mathematical forms which is then solved. The solution helps the user to understand the real world system better or design an appropriate control or diagnostic method to the corresponding object of the modeling.

The model is prepared using ﬁrst engineering principles such as thermodynamic, mechanical and electro-magnetic laws. It is then equipped with constitutive equations to obtain a solvable set of equations. This ﬁnal set is then transformed into the form required or convenient to the given application. The following steps are considered in this chapter for a systematic modeling procedure [12]:

• Description of the system and its boundary. This gives the components that are needed to be included, all the inputs/outputs that occur on the system boundary and all the processes within this boundary

• Deﬁnition of the modeling goals that prescribe the aim of the model and the required accuracy.

• Supplying of simpliﬁcation assumptions that enable to eliminate unimportant phenomena and thus to obtain simpler mathematical forms.

• Derivation of conservation equations that are the core equations of the model and are based on ﬁrst engineering principles.

• Construction of constitutive equations.

• Transformation of the model into state space form for control design applications.

2.1 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 at the link of the engine crank shaft and the gearbox input shaft, where normally

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the clutch transmits the torque through the clutch disc. The clutch friction disc, pressure plate and ﬂywheel are rotating together due to the friction force between them. This force is caused by normal force of a disc spring, which pushes the clutch pressure plate to the friction disc and the ﬂywheel. When the clutch actuation is demanded, solenoid magnet valves (SMV) driven electro-pneumatic actuator pre-stress the disc spring, which lets the clutch pressure plate to moves apart from the friction disc, thus disengaging the connection. The general layout of the EPC system with its close surrounding to be modeled [13,14] can be seen inFig. 2.1.

Figure 2.1: The layout of the electro-pneumatic clutch (EPC) system

The system is supplied by compressed air, thus for supplying pressure an air reservoir (1) is applied. The actuator contains four SMVs, two of them (2, 3) can connect the chamber (11) to the supply pressure, therefore called load valves and the remaining two (4, 5) can connect the chamber to the ambient pressure, called exhaust valves. The indices of the SMVs are denoted as follows: sl,bl, se andbe for small- and big load and small- and big exhaust, respectively. Each SMV has an own power stage (6-9), which can transform the command signal to an appropriate terminal voltage.

This structure ensures positive and negative direction displacement of the piston (10), which is the final element of the actuator that performs the clutch activation procedure. The variables and the parameters of the piston are denoted by pstsubscript. The actuator contains a holder spring (12), which pushes the piston to the clutch mechanism to eliminate the clearance. 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 nonlinear stiﬀness of this

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set of springs has a paramount role in the controllability performance at low 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 by tangential leaf springs (22), these springs transfer the torque to the housing and determine the radial position of the pressure plate. Finally the flywheel is connected to the engine and the friction disc is connected to a splined gearbox input shaft (23).

In Appendix A in Fig. A.1 a picture about the friction disc, the clutch mechanism and the concentric clutch actuator from ZF SACHS is shown. InFig. A.2the engine, the gearbox and the complete clutch system with position sensor, valve block and transmission ECU are presented.

2.2 Modeling: goals and approaches

The modeling 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. A widely used modeling 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 modeling 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 modeling goal is a complex statement where one assume it to be given in terms of a set of performance indices [χ₁. . . χ_n], where the performance index χ_i can be real and/or Boolean quantity which deﬁned for the model M as: χi : 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: χ^min_i ≤χ_i≤χ^max_i , i= 1, . . . , n.

Through these the following properties of the EPC model are considered to achieve the modeling goals, which are dynamic simulation, clutch control design and validation.

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 (χMP1 ∈B).

MP2. It should be transformed to a deterministic input-output model (χMP2 ∈B).

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. (χMP3∈B).

MP4. The model should be represented in state space form (χMP4 ∈B).

MP5. The model should be capable of describing the dynamic behavior of the EPC 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 (χMP5a-g ∈ R, χ^maxMP5a-g = 0.05).

For accuracy validation criterion anL₂ error is 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 modeling approaches have been published (see a comprehensive collection with many examples in [15]). Considering the modeling goals, ﬁrst of all, mechanistic

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modeling approaches should be used to satisfy χMP1, but the most common form of models, which describe complex systems, are a combination of mechanistic and empirical parts. The advantage of the mechanistic modeling 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 χMP2, a deterministic modeling 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 [16].

Therefore the designed model is restricted to this case which satisﬁesχMP3. The consideration of the valve and the power stage dynamics to achieveχMP5a-gintroduces discrete-continuous behavior, e.g. the change of the ﬂow cross section of the valves.Thus a hybrid i.e. discrete-continuous approach is used [17, 18,19]. Moreover nonlinear relationships between the parameters and/or variables are considered.

2.3 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 modeling assumptions are available in the literature [20,21]. First assumptions are made to get a concentrated parameter model [22], then to reduce the model components which show discrete behavior. 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 [23]. The assumptions have been derived iteratively according to the model complexity and the achievement of the modeling goals using the seven step model building procedure [15]. As a conclusion the following assumptions are made:

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 than 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 (σ) of the SMVs are assumed to have non-negative values only (see the directions in Fig. 2.1).

A7. The SMVs magnetic elements are modeled 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 (x^max_xx =x^lim,2_xx −x^lim,1_xx ) and the SMV output port diameter (dxx) are assumed to satisfy the inequality for all the four SMVs: x^max_xx > ^d^xx₄ , where xx

can be_sl,_bl,_se and _be (see the layout of the SMV in Fig. 2.5).

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A9. The cross sections of the SMV ports are assumed to satisfy the following condition for all the four SMVs: A_in>> A_out, whereA_in andA_out are the in- and output cross sections of the SMVs respectively.

A10. The aerodynamic resistance of the armature can be neglected due to the low density of the gas.

A11. The armature friction is neglected since the forces acting on the armature have axial component only.

A12. The armature mass of the SMVs assumed to be constant in time.

A13. 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.

A14. The switching devices (MOSFET) in the power stages have a constant drain to source turned on resistance in the applied working range.

A15. The clutch mechanism and the clutch actuator moving masses are lumped into the piston mass, which is assumed to be constant in time.

A16. The clutch mechanism and the clutch actuator friction eﬀects can be lumped together into one friction eﬀect.

A17. The clutch mechanism and the clutch actuator damping eﬀects can be lumped together into one damping eﬀect.

A18. The nonlinear characteristic of the clutch mechanism, which has hysteresis loop, can be approximated with its empirical center characteristic line (seeFig. 2.6).

A19. 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.

AssumptionsA1-A12have been validated earlier in [24]. The remaining assumptions are derived iteratively in order to achieve the prescribed modeling goals. Then the previously constructed and the new assumptions have been validated together as well (see inSection 2.8).

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 ≤ Usup ≤ 32 [V], 7·10⁵≤p_sup≤12·10⁵[P a],233≤T_sup≤358 [K],0.92·10⁵≤p_amb ≤1.08·10⁵[P a]and 233≤T_amb≤393[K], whereU_supis the supply voltage,p_supis the compressed (supply) air pressure,Tsupis the compressed (supply) air temperature,p_ambis the ambient air pressure andT_amb is the ambient air temperature, respectively.

IC3. The control input variables i.e. the duty cycle of the PWM control signals (m_d,xx) are limited by the following constraints: 0≤m_d,xx ≤1 [−].

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2.4 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 diﬀer- ential equations. In order to derive the conservation equations ten balance volumes are deﬁned;

one for each SMV armature, one for each SMV magnetic circuit, 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 balance volumes and the corresponding conserved quantities are shown inTab. 2.1.

Table 2.1: Balance volumes and conserved quantities

Symbol Balance volume Conserved quantity

V1 Clutch actuator chamber Gas mass

Gas energy

V₂ Clutch actuator piston Momentum

V₃ Armature of small load SMV Momentum

V₄ Armature of big load SMV Momentum

V₅ Armature of small exhaust SMV Momentum V₆ Armature of big exhaust SMV Momentum V₇ Magnetic circuit of small load SMV Magnetic linkage V₈ Magnetic circuit of big load SMV Magnetic linkage V₉ Magnetic circuit of small exhaust SMV Magnetic linkage V₁₀ Magnetic circuit of big exhaust SMV Magnetic linkage

2.4.1 Conservation of gas mass in the clutch actuator chamber V1

The expression for mass balance [25], considering no generation and consumption terms, forms the following equation in case of lumped parameter systems with pinput andq output:

dm dt =

Xp

j=1

σ_j− Xq

k=1

σ_k, (2.1)

wherem is the mass andσ is the mass ﬂow rate.

Since the clutch chamber has only one port (seeFig. 2.1), which serves as both in- and output port, its mass ﬂow equals the sum of the four SMVs output ports mass ﬂow:

dm_ch

dt =σ_sl+σ_bl−σ_se−σ_be, (2.2)

wherem_ch is the gas mass in the chamber.

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2.4.2 Conservation of gas energy in the clutch actuator chamber V₁

The general form of total energy (E) for a given balance volume [25] with pinput and q output ﬂows is written as:

dE dt =

Xp

j=1

σj(h+e_k+ep)− Xq

k=1

σ_k(h+e_k+ep) +Q+W, (2.3) whereh,e_kande_p denotes the mass speciﬁc-enthalpy, kinetic energy and potential energy terms respectively. Qis the heat transfer andW 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:

dU dt =

Xp

j=1

σ_jh_j− Xq

k=1

σ_kh_k+Q+W, (2.4)

where the extensive conserved quantity is the internal energy (U) 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:

dU_ch

dt = d(c_vm_chT_ch)

dt = d(c_v^p^ch^V^ch

R )

dt = c_vV_ch R

dp_ch

dt +c_vp_ch R

dV_ch

dt = V_ch κ−1

dp_ch dt + p_ch

κ−1 dV_ch

dt , (2.5) whereU_ch is the gas internal energy in the chamber, c_v is the specific heat at constant volume, c_p is the specific heat at constant pressure, T_ch is the gas temperature in the chamber, Ris the specific gas constant,V_chis the instantaneous volume of the chamber,κis the adiabatic exponent and

c_v= R

κ−1, c_p= R

κ−1κ, thusκ= c_p

c_v. (2.6)

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

h=c_pT. (2.7)

The source side is determined by the air ﬂow direction, but according to assumptionA2, in which the SMVs air ﬂows are assumed to have non-negative values only, the source side do not change.

Using Eq. (2.4)-(2.7)the pressure change in the chamber is written as follows:

dp_ch dt = κR

V_ch(σ_slT_sup+σ_blT_sup−σ_seT_ch−σ_beT_ch)−

−p_ch V_ch

dV_ch

dt − κ−1

V_ch Q_ch−κ−1 V_ch W_ch.

(2.8)

2.4.3 Conservation of clutch actuator piston momentum V2

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

dM dt =

Xp

F_k, (2.9)

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whereF_k denotes the forces acting on the system.

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

d(mpstvpst)

dt =F_p_ch+F_hsp−F_{f r}−F_dmp−F_pst,lim−F_l(x_pst), (2.10) where m_pst is the lumped mass of the moving parts (A15), F_p_ch is the pressure-, F_hsp is the holder spring-,F_{f r} is the friction-, F_dmp is the damping- and F_pst,lim is the limiting force acting on the piston. The load of the piston F_l(x_pst) comes from the clutch mechanism. The forces acting on the piston can be seen in Fig. 2.2.

Fp_ch Fhsp

Fl(xpst) Ff r

Fdmp

Fpst,lim

xpst, vpst

Figure 2.2: Free body diagram of the piston

In accordance to A15, in which assuming a lumped constant mass, the velocity of the piston is obtained as follows:

dv_pst

dt = Fpch+F_hsp−F_{f r}−F_dmp−F_pst,lim−F_l(xpst)

m_pst . (2.11)

The stroke change of the piston can be written as follows:

dx_pst

dt =v_pst. (2.12)

2.4.4 Conservation of SMV armature momentum V3−6

The design of the four SMVs are identical except some parameters, therefore the relations below are valid for all of them.

Similarly to the momentum balance of the piston, the SMV armature balance is derived considering assumption A10-A11in which the the aerodynamic resistance and the friction force are neglected. Thus the momentum balance of the SMV armature is written as:

d(mv)

dt =Fmg −Frsp−F∆p−F_arm,lim, (2.13)

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whereFmg is the magnetic force generated by the magnetic ﬁeld of the solenoid,Frsp is the force coming from the return spring,F_∆p is generated by the pressure diﬀerence on the cross section of the closed valve seat andF_arm,lim is the stroke limiting force of the SMV armature. The forces acting on the SMV armature can be seen in Fig. 2.3.

Fmg

Frsp

F∆p

Farm,lim

x, v

Figure 2.3: Free body diagram of the solenoid magnet valve (SMV) armature

Since the SMV armature mass is constant in time according toA12, the equations for armature velocity is obtained as:

dv

dt = F_mg−F_rsp−F_∆p−F_arm,lim

m . (2.14)

The stroke change of the armature can be written as follows:

dx

dt =v. (2.15)

2.4.5 Conservation of magnetic linkage in the SMVs V₉−10

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

C

Edl=−d dt

Z

S

B nda, (2.16)

whereE 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 (see Fig. 2.4).

n B

Figure 2.4: Magnetic linkage in the solenoid

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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 φas follows:

E=−∇φ. (2.17)

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

Z b a

Edl= Z b

a

−∇φ dl=φ_a−φ_b. (2.18)

The potential diﬀerence φa−φ_b is referred to as the voltage of point a with respect to b.

Thus Faraday’s law yields the induced voltage (u_ind) of the solenoid as follows:

u_ind= d dt

Z

S

B nda= dλ dt = dλ

di di dt +dλ

dx dx

dt, (2.19)

whereλis the ﬂux linkage of the circuit.

Assuming a magnetically linear system according toA7 whose ﬂux linkage can be expressed in terms of an inductance Lasλ=L i. Through these the induced voltage becomes:

u_ind=Ldi

dt +idL dx

dx

dt. (2.20)

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

u_term=u_res+u_ind. (2.21)

Using Ohm’s law and substitution ofEq. (2.20)intoEq. (2.21)the current change can be obtained as follows:

di

dt = u_term L −R

Li− 1 L

dL dx

dx

dt i. (2.22)

2.5 Constitutive equations

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 characterizing variables.

2.5.1 Chamber and gas properties

The volume of the chamber is obtained from a constant dead volume (V_ch^d) 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 by of the current application.

With these the chamber current volume of the clutch actuator can be written as follows:

Vch=V_ch^d +xpstApst, (2.23)

whereA_pst is the cross section area of the piston.

The volume change of the chamber is obtained as:

dV_ch

dt =v_pstA_pst. (2.24)

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The temperature of the gas in the chamber (T_ch) is obtained using the ideal gas equation, thus the chamber gas temperature is written:

T_ch= p_chV_ch

m_chR = p_ch V_ch^d +x_pstA_pst

m_chR . (2.25)

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=k_htA_ht(T_ch−T_amb), (2.26) wherek_ht is the heat transfer coeﬃcient andA_ht 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=p_chdV_ch

dt =p_chv_pstA_pst. (2.27)

2.5.2 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 determined by the contraction coefficient (α) of the stream, the flow cross section (A_in), the source side- pressure (p_in) and temperature (T_in) and the pressure ratio (Π =p_out/p_in) between the in- and output ports [26] as follows:

σ=α Ain pin

vu ut 2κ

κ−1 1 RT_in

"

pout

p_in _κ²

− pout

p_in

^κ+1_κ #

. (2.28)

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 or equal to the value, where the armature reaches the valve seat (x^lim,1) see inFig. 2.5, then there is no flow. If the stroke is above this value the smallest orifice is determined by a cylindrical surface. If there is a big stroke then the orifice is limited by the area of the outlet hole. This implies hybrid behavior depending on the SMV armature position.

Outlet port

Inlet port Frame

Plug

Solenoid

Armature Return spring

m

x v x x^lim,2

lim,1

Seat

Figure 2.5: Two way two port on/oﬀ SMV layout

2.5.3 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_p = (p_ch−p_amb)A_pst. (2.29)

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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=c_hsp(x_hsp₀ −x_pst), (2.30) wherec_hsp is the stiﬀness of the holder spring andx_hsp₀ is the spring pretension stroke.

According to assumptionA16in which the clutch mechanism and the clutch actuator friction effects are lumped together into one friction effect, the magnitude of the friction force (F_{f r}) depends on a lumped friction coefficientµ_pst 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. 2.1). 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.

Considering A17, in which assuming that the clutch mechanism and the clutch actuator damping eﬀects are lumped together into one damping eﬀect k_pst, the damping force acting against the piston movement is obtained as follows:

Fdmp=vpstkpst. (2.31)

The stroke limiting force (F_pst,lim) of the piston is modeled as a stiﬀ spring if the stroke exceeds the limits. This introduces three hybrid modes, two limiting positions (x^lim,1_pst andx^lim,2_pst ) at the stroke ends (see Fig. 2.1) and the third one corresponding to the intermediate position.

The F_l(x_pst), acting against the piston movement, comes from the clutch mechanism. This force is a highly nonlinear 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 (seeFig. 2.6), which can provide the prescribed accuracy, is too complex, therefore in the model a realization through the empirical center 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 assumptionA19the 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.

2.5.4 Forces acting on the SMV armature

The magnetic force (F_mg) can be calculated as the partial derivative of the energy of the magnetic ﬁeld (E) with respect to the armature stroke as:

F_mg =−∂E

∂x =− Θ² 2R²_Σ

dR_Σ

dx =−(N i)² 2R²_Σ

dR_Σ

dx , (2.32)

whereΘis the excitation (magnetic voltage),RΣis the magnetic resistance andN is the number of solenoid turns.

The connected magnetic resistances are related to the frame (R^{f rm}), the plug (R^pl), the SMV armature (R^arm), the air clearance between the overlapping coaxial cylindrical surfaces of the SMV armature and the frame (R^c1) and ﬁnally resistance in the air clearance (R^c2(x)) between the plug and the armature (see Fig. 2.5).

The only component that depends on the stroke is R^c2(x) and it is considered as follows:

R^c2(x) = x^gap−x

µ A^arm, (2.33)

Observer based feedforward/feedback control of electro-pneumatic clutch systems