Nonlinear Control of an Electromechanical Valve Actuator

Copyright © IFAC Advances in Automotive Control Salemo, Italy, 2004

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NONLINEAR CONTROL OF AN ELECTROMECHANICAL VALVE ACTUATOR MarceUo Montanari *, Fabio Roncbi * , Carlo Rossi *

* CASY - Center for Research on Complex Automated Systems

DEIS - University of Bologna Viale Risorgimento 2, 40136 Bologna, ITALY Tel. +39051 2093020, Fax. +39051 2093073 E-mail: {mmontanari.fronchi.crossi}@deis.unibo.it

Abstract. Camless internal combustion engines offer major improvements over traditional engines in terms of efficiency, pollutant emissions, maximum torque and power. Electromechanical valve actuators are very promising in this context, but still present significant control problems. Low valve seating velocity, small transition time for valve opening and closing are conflicting objectives that need to be jointly considered when designing the valve control system. The paper presents a hybrid control system architecture capable to deal with all these issues. The design of the valve position controller is addressed in detail, in the framework of backstepping approach. Simulation results show the effectiveness of the proposed solution. Copyright © 2004IFAC Keywords: Automotive control, Control oriented models, Internal combustion engines, Nonlinear control, Piston valves, State feedback, Statecharts

1. INTRODUCTION

ers (see (Gray, 1988) (Schechter and Levin, 1996) and the references therein). Camless engines allow for the implementation of several control strategies. The achievable major advantages are: fuel savings, nearly flat torque characteristic, pollutant emission reduction, increased burn rate, variable compression ratio, improved combustion stability at low speed and reduced energy consumption.

Internal combustion engines traditionally use mechanically driven camshaft to actuate intake and exhaust valves. Their lift's profiles are direct function of the engine crank angle and cannot be adjusted to optimize engine performances in different operating conditions. Hence, a trade-off between achievable engine efficiency, pollutant emission, maximum torque and power must be considered. Growing needs to improve fuel economy and reduce exhaust emissions lead to the development of alternative valve operating methods, which aim to alleviate or completely avoid the limitations imposed by a fixed valve timing (Ahmad and Theobald, 1989), (Gray, 1988), (Schechter and Levin, 1996).

Electro-hydraulic (Kim et aI., 1997) or electromechanical (Butzmann et aI., 2000), (Wang et al., 2000b) valve actuators have been proposed for camless engines. In this paper the focus is on an electromechanical actuator based on a lever-type structure. With respect to linear actuators considered in the literature (Tai et al., 2(01), (Wang et al., 2000a), the considered lever-type actuator gives improvements in terms of efficiency.

Variable Valve Timing (VVT) solutions use a wide spectrum of different technologies. Camshaft-based variable-valve mechanisms offer significant improvements on the engine performances (see (Jankovic and Magner, 2002) and the references therein). To fully exploit the possibilities offered by a complete VVT system, camless engine valve-trains, in which the valve motion is completely independent of the piston motion, are currently the topic of an intensive research activity, both from academia and several manufactur-

Different tasks must be taken into account in the design of the valve actuator controller. The biggest difficulty comes from the valve seating velocity, i.e. the valve velocity when it comes against the closing position. It should be very low to avoid acoustical noise and wear and tear of mechanical components, with typical values between 0.05 rnIs and 0.1 mls at idle speed. Solving this issue is complicated by the fact

179

that the system itself is unstable near valve's tenninal positions, hence closed loop control of this highly nonlinear system is mandatory (Tai et al., 2(01). Dynamical response of the system, especially the travel time required for a transition between the two tenninal positions of the valve, is another key point. At high engine speed, the time available for the intake and exhaust process becomes low, hence opening and closing of the valve should be very fast to keep engine performances. Typical values for transition time range between 3 ms and 5 ms, with an average valve velocity between 1.5 mJs and 2.5 mJs. The above factors, i.e. limited seating velocity and short transition time, make the closed loop control system design a very complex task. In this paper the architecture of a hybrid control system and the design of the position controller for the electromechanical valve actuator are considered. As stated before, closed loop position controller is mandatory at narrow air-gaps. On the other hand, since the two magnets are not able to deliver significant forces at large air-gaps, in this situation the position feedback control is useless and therefore a flux feedback control is implemented. Considering the different objectives and main limitations of the plant, the valve transition is divided into some steps and the hybrid controller architecture is defined. Assuming that all the state variables are available, the position tracking controller is designed in the framework of the backstepping approach. The availability of position and speed measurements is justified by the compatibility of the proposed control architecture with a position reconstructor (Ronchi et al., 2(02) based on flux and current measurements.

Figure 1. Valve sketch via simulation experiments, shown in section 6. In section 7 conclusions are reported.

2. SYSTEM DESCRIPTION AND MATHEMATICAL MODEL The considered actuator is depicted in fig. 1. It is composed by a mobile mechanical part (valve, levers), a spring B counteracted by a torsion bar A, two mechanical stops and two electromagnets that develop the force needed to move the valve along the vertical axis x. Electromagnet EMl moves the valve toward the upper mechanical stop, then closing the valve, whereas electromagnet EM2 opens the valve. Torsion bar A delivers a force in the x positive direction, and spring B gives force in the opposite direction. The spring/torsion bar system is preloaded to an amount such that when no other forces are present, the valve is in the center of its stroke (x = 0). The maximum voltage and current feeding the electromagnets are limited by vehicle battery to 40 V and to 50 A, respectively.

Correct definition of the hybrid controller architecture and trajectory design allows to deal with actuator characteristics (i.e. magnetic saturation and control input constraints) and control requirements (i.e. "soft touch" and transition time), as deeply discussed in (Montanari et al., 2(03).

Neglecting magnetic hysteresis and Foucault currents and supposing that the magnetic circuits of the two electromagnets are decoupled, the simplified model of the actuator is drawn. The equations of the j-th electromagnet are

From the modelling point of view, main features of the proposed solution are the following. Differently from many works (Wang et al., 2000a), (Tai and Tsao, 2(03), the model of the actuator is expressed choosing as state variables position, velocity and fluxes (instead of currents). This state space representation allows to better deal with magnetic saturation, which must be taken into account for this kind of actuator. Moreover, a more general expression of the reluctance curve (with respect to the classical linear one) can be treated in a simple way.

N

:Pj

= Uj

-

rj

ij

,

j

= 1, 2

(1)

where Uj, i j are voltage and current, N is the number of turn of the coil, N!pj is the linked flux, rj is the electrical resistance. Voltage Uj is generated by a switching converter, whose DC-link voltage VDC is supplied by the vehicle battery and whose maximum current is [m=: .

In section 2, the mathematical model of the system is introduced. In section 3 the control system requirements are presented. In section 4 the control system architecture is described. The position closed loop control is derived in section 5. The performances achievable with the proposed control architecture are verified

Current and flux in each winding can be related (see (Filicori et al., 1993) for a detailed discussion) by splitting the magneto motive force (MMF) into two components, as follows Nij=:Fj(cpj)+~j(x)!pj,

180

j=I,2.

(2)

soo--~------~--~--------------~

1000r-------~------~--_.--~

:s 500 ~"

/

OL--=====~=-~----~ o 0.5 1.5 'I' (mWb)

Figure 2. Functions Fl (x) (solid) and F 2 (x) (dashed) -~~~-3~~-2~~-1--~O--~--~--~--~· • (mm)

Figure 4. Forces Fl (x) (solid) and F 2 (x) (dashed) delivered by the electromagnets for different values of fluxes, compared with spring force (dotted) Table I. Table of model parameters Symbol

/

o~----~----~----~----~

-4

-2

o

2

L M

4

x (mm)

k

b

Figure 3. Functions ~1(X) (solid) and ~2(X) (dashed)

VDC lmaz

The first term F j (cpj) accounts for the MMF drop in the iron portion of the magnetic flux path, which is assumed to be independent of the air-gap and therefore of the position x . The nonlinear dependance on the flux is shown in fig. 2 for both the electromagnets. The second term takes into account the air-gap portion of the magnetic flux path. The MMF drop in the air-gap is linear in the flux CPj, since no saturation is possible in the air. The reluctance function ~j (x) nonlinearly depends on the position x as shown in fig. 3 for both the electromagnets. Note that the common assumption of linear reluctance does not match with the experimentally identified curves.

N rl rll

!RI (x) Fl ('1'1) N r2

rl2 !R2 (X) F2 ('1'2)

I

Value

8 0.10 117 * 103 6 40 35 EMl 50 0.20 0.58

I

Units mm

kg N/m N s/m

V A

turns

n adimensional

H- l

Fig.3 Fig.2

A

EM2 50 0.24 0.45

adimensional

Fig.3 Fig.2

A

turns

n H- l

Based on preliminary experimental tests, it results that collisions between the two moving masses (valve and lever) and modelization of the impact between valve and valve seat can be neglected in the control model describing the valve during transitions. Hence, supposing all the moving masses concentrated in the equivalent constant mass .111 moving along the valve vertical axis, the mathematical model of the complete model for the system is:

Assuming that positive forces are delivered in the direction of increasing positions, the forces F1 , F2 can be calculated by means of the D' Alembert principle (Woods on and Melcher, 1968):

(3) As typical in variable reluctance actuators, the sign of the force developed by a single actuator does not depend on the flux sign. It is worth noting that the decomposition (2) makes the force expression independent of the magnetic saturation. In fig . 4, forces Fl , F2 and spring force are compared. Note that forces F1 , F2 are significant only for narrow air-gaps. Forces needed during transitions require fluxes in the range of weak saturation. Hence, the magnetic saturation phenomenon can not be neglected. Since reluctances of the two actuators shown in fig. 3 are strictly increasing (~2) and decreasing (~1) functions of the position, the upper magnet can generate positive only forces, while the lower magnet can generate negative only forces.

x=v v = ..!... (-kX _ lru _ ! O~l cp2 _ ! oR 2922) M

2

ox

1

2

ox

,~ {U1 - ~ [~I(X)CPl + F 1 (cpdl}

01

=

02

= ~ {u2 - ~ [~2(X)cp2 + F 2(92)J}.

(4)

Note that the effect of gravity force can be neglected. Disturbance forces acting on the valve due to cylinder pressure are not considered at this stage of the controller design. The numerical parameters of the considered valve and electromagnets are reported in Table 1.

181

3. CONTROL SYSTEM REQUIREMENTS The main goal of the control system is to move the valve from the fully-closed to the fully-opened position (and vice-versa) avoiding noisy and wearing hits against the hard mechanical stops. Since the system is provided with a Hydraulic Lash Adjustment (HLA) device, which makes the position of the upper mechanical stop uncertain and variable during life-time of the actuator, the "soft-landing" behavior can only be achieved by controlling the speed of the valve when it is approaching the mechanical stops. This calls for a position tracking control along a reference trajectory, which has to be suitably designed (Montanari et

Figure 5. Valve transition diagram, opening

Traj.I--''''''=----------,

sen·

al.,2003). Figure 6. Control scheme

Trajectories characterized by low velocity when the valve is near the mechanical stops are unstable. This directly follows from the instability of the equilibrium points at narrow air-gaps (Tai et al., 2001). Closed loop position control is therefore mandatory when approaching or leaving the seating position. On the other side, closed loop position control is useless when the valve is in the central part of the transition, characterized by wide air-gap with respect to both magnets. In fact, forces supplied by the magnets in these positions can not be significant due to small values of 8Rj / 8x, as it can be seen in fig. 4. Hence, very high fluxes should be imposed to obtain significant forces, leading to magnetic saturation and high currents. Moreover, singularity occurring at zero flux must be avoided not to impair control performances. The motivation of this operation can be easily understood differentiating (3) with respect to time:

'-Pj

=

F·j

1 82~

x

When the central region of the transition is approached, control enters FLY state. This is recognized when the valve position is lower than a fixed threshold x o!! ' In the FLY state, open loop position control is applied (switches at position a in fig. 6) to switch off the releasing magnet EMl and to switch on the catching magnet EM2 • Closed loop flux control on the catching magnet is used to guarantee that the system enters the LANDING state with a proper flux level. This is crucial to ensure that a proper force can be delivered to the valve. When the valve position approaches the final part of the trajectory, recognized by a valve position lower than the threshold x on , the LANDING state is entered. Here closed loop position control is activated on the magnet EM2 until the mechanical stop is reached (switches at position c in fig. 6), tracking the position reference X r .

2

+ 28x'E
active actuator (switches at position b in fig. 6) and the reference generator feeds the controller with position reference Xr and its time derivatives Xr , Xr , r'

j = 1, 2.

- 8x
When flux
5. CLOSED LOOP POSmON CONTROLLER In this section the closed loop position controller is designed using the backstepping approach (Khalil, 1996). The voltage control law to be applied to magnet EMl (in RELEASE state when opening and in LANDING state when closing) is calculated. Since only the electromagnet EMl is energized during these phases, the subscript 1 is omitted. Defining the position error

4. HYBRID CONTROL SYSTEM ARCHITECTURE

x=

x -

Xr

where Xr is the position reference, the velocity reference is chosen as follows

Based on considerations of previous section, the valve transition can be divided into several steps. As an example consider the transition from closed to opened valve position. Referring to figs . 5 and 6, the controller first enters RELEASE state. When in this state, closed loop position control is mandatory to prevent unpredictable valve movement due to instability. EMl is the

V·

= xr -

kxx

where kx > 0 is a control parameter. Defining the velocity error

182

and assuming the force p. as control output, model (4) can be rewritten as follows

Choosing the following control law

u =ri + N cp. - k
i = -kxx + v .

=

v

,

1

+ M k;

- (k

- bk x )

x-

= kX r + bXr + MX r -

bkxx - k,/v

with kv > (M kx - b) > 0, it follows that the resulting system is globally asymptotically stable.

sat+ (

1~;

-2"Fx(x

))

(5)

where sat+(z) z if z > 0 and sat+(z) = 0 if z ::; O. In the following, it is assumed that p. is always greater than zero and therefore

18R -2 Bx (x)'P· 2 =kxr

6. SIMULATION RESULTS

+ bXr + MXr - bkxx - kv v ,

Simulations are performed in Matlab/Simulink environment, taking into account several effects which characterize the experimental setup.

(6)

Vxr , xr , xr , X, v

This assumption is justified by proper definition of position reference and it is always largely satisfied in the performed simulations.

• The controllers are implemented in a timediscrete environment having sampling time T. = SOf-LS and an computational time delay of two sampling periods. • The control voltages U1, U2 are applied to the actuator by means of a symmetrical PWM technique with carrier of frequency f. = 20kHz. • The vehicle battery specifications limit the magnitude of control voltages U1 , U2 to VM = 40 V and the magnitude of currents i 1 , i2 to IM =

Defining the flux error

rp = 'P - 'P. and considering the actual control output u, model (4) can be rewritten as follows

i

= - kxx

~ =~

+v

+ Mk;)x -

[-(k

(b + kv - Mkx)v+

SOA.

_~ BR (x)(2 'P. + rp)rp]

Considering position, velocity and fluxes expressed in S.l. system in the controller equations, the chosen control parameters are: kx = 970, kv = 200, k", l = k
2 Bx

.

1

rp=N(u-ri-Ncp· ) where

,1 ,2

(7)

The control law (8) requires the computation of cp •. This term is a complex function of measurements and model parameters. In order to obtain an easy formulation of the controller, thus reducing the code execution time, cp. is calculated differentiating (in discrete time) the flux reference 'P. with respect to time.

~ [R(x) 'P + F ('P )]

i =

and the flux reference time derivative 1 cp. = -2 f(xr , xr , x r , x·r, x , v , rp)

'P.

can be calculated from known model parameters, measurements and reference time derivatives.

Simulation results for an opening valve transition at 3000rp

Differentiating the Lyapunov candidate function

W _ ~ -2 M -2 N _2 - 2 x + 2 (k + kx ) v + 2, 'P where , > 0 is a control parameter, along the trajectories of system (7), it follows that

= - k xx-2 -

W· -

b + kv - Mkx -2 rp [ . v + - u - rt+ k + Mk~ ,

TV . • , BR ( ) (2· 'P - 2(k + Mk~ ) Bx x 'P

1

-) -]

+ 'P

(8)

Differentiating (5) with respect to time it follows that if the flux reference 'P. is zero then it must be p. = 0 to obtain a finite cp. and ther~fore a not saturated control voltage u. This means that system is not able to track a force trajectory having p. = 0 and p. > 0, confirming the necessity to energize the catching magnet early in the FLY state, as enlightened in section 3.

The flux reference 'P. is calculated from the force reference p. in the following way

'P.

-) 'P + 'P v

the function Hr is negative definite and therefore the origin of the error model (7) is asymptotically stable. This is a local stability result because assumption (6) must hold.

(b - M k x ) v + F*] .

Defining the force control law

p.

BR ( ) (2.

+ 2(k+Mki) Bx x

M [-kxr - bX r - Mx r +

v

183

less than 0.1 m / s, ensuring the "soft-touch" behavior. Oscillations in the flux are due to discrete time implementation of the controller with saturated output and time delay.

4

2

E

.sx

0

-2 7. CONCLUSIONS 0.016 0.018 0.02 0.022 0.024 0.026

A control system architecture for position control of an electromechanical valve actuator has been proposed. Owing to system characteristics, closed loop position control is performed only when valve is approaching the mechanical stops. The position feedback control has been derived via Lyapunov techniques in the backstepping framework. Simulation results in discrete time environment enlighten that the proposed solution is suitable to obtain the desired performances and in particular the "soft-touch" behavior.

o

-3 _4L-~--------~--~--~--~~

0.016 0.018 0.02 0.022 0.024 0.026

REFERENCES

:c ~

T. Ahmad. M.A. Theobald (1989). A survey of variable valve actuation technology. SAE paper no. 891674. S. Butzmann. J . Melbert. A. Koch (2000). Sensoriess conlrol of eleclromagnetic actuators for variable valve train. SAE paper no. 2000-01-1225. F. Filicori. C. Guarino 1.0 Bianco. A. Tonielli (1993). Modeling and conlrol slrategies for a variable reluctance direct-drive motor. IEEE Trans. Ind. Electron.. voL 40. no. I. pp. 105-115. Gray. C. (1988). A review of variable engine vavle timing. SAE paper no. 880386. M. Jankovic. S. W. Magner (2002). Variable cam timing: Consequences to automotive conlrOl design. Proc. 15th IFAC World Congress. Barcelona, Spain . H.K. Khalil (1996). NoniinearSystems. Prentice Hall. Upper Saddle River. NJ. D. Kim. M . Anderson. T. Tsao. M. Levin (1997). A dynamic model of a springless eleclrohydraulic camless valvetrain systems. SAE paper no. 970248. M. Montanari. F. Ronchi. C. Rossi (2003). Trajectory generation for camless internal combustion engine valve conlroL 2003 IEEE International Symposium on Industrial Electronics, Rio de Janeiro, Brasil. F. Ronchi. C. Rossi. A. TlIli (2002). Sensing device for camJess engine eleclrornagnetic actuators . Proc. IECON·02. Seville. Spain. M. Schechter. M.B . Levin (1996). CamIess engine. SAE paper no. 860581 . C. Tai. A. Stubbs. T. Tsao (2001). Modeling and conlroller design of an eleclrOmagnetic engine valve. Proc. American Control Conference. Arlington. VA . C. Tai. T. Tsao (2003). Conlrol of an eleclromechanical actuator for camJess engines. Proc. American Control Conference. Denver. CO. Y. Wang. A. Stefanopoulou. K. Peterson. T. Megli . M. Haghgooie (20000). Modeling and conlrol of eleclromechanical valve actuator. SAE paper no. 2002-01 -1106. Y. Wang. A. Stefanopoulou. M. Haghgooie. I. Kolmanovsky. M. Hammoud (2000b). Modeling of an eleclromechanical valve actuator for a camless engine. Proc. AVEC 2000.

\

E :;:-{).5

\

..

. ,11. .

0.016 0.018 0.02 0.022 0.024 0.026 t (5)

Figure 7. Opening at 3000 rpm, position x (solid), velocity v (solid), fluxes !Pl (dashed), !P2 (solid) and relative references (dash-dotted). Position and velocity references are set to zero when not in closed loop position control, whereas open loop flux reference is set to zero when in closed loop position control. 15r-~--~----~--~--~--~-

10 51

0.016 0.018 0.02 0.022 0.024 0.026

l-

V

H. H. Woodson. 1. R. MeIcher (1968). Electromechanical Dynamics

0.016 0.018 0.02 0.022 0.024 0.026 t (5)

vo/. 1. WlIey. Kew York.

Figure 8. Opening at 3000 rpm, Currents i l (dashed), i2 (solid) and control voltages Ul (dashed), U2 (solid)

\ 84