Flatness Based Control of a Throttle Plate

R. Rothfuß1, H.-M. Heinkel1, R. Sedlmeyer2, H.Schmidt3, S. Stoll4, J. Winkelhake41 Robert Bosch GmbH, Dept. FV/FLI,

P.O. Box 10 60 50, D-7 00 49 Stuttgart, Germany,2 Robert Bosch GmbH, Dept. K3/ESI,

Robert-Bosch-Str. 2, D-7 17 01 Schwieberdingen, Germany,3 S3-Process Control, Royal Inst. of Techn.

S-1 00 44 Stockholm, Sweden4 Inst. f. Elektrische Informationstechnik, Leibnizstr. 28

D-38678 Clausthal-Zellerfeld, GermanyRalf.Rothfuss@de.bosch.com

Hans-Martin.Heinkel@de.bosch.comRafael.Sedlymeyer@de.bosch.com

schmidt@e.kth.sesorn@iei.tu-clausthal.de

winkelhake@iei.tu-clausthal.de

Keywords: Injection systems, throttle plate, flat systems,tracking control

Abstract

In classical gasoline injection systems, the control of the throt-tle plate position is important for efficiency and emission ofthe engine. Depending on the current load and speed of theengine the angular position of the throttle plate has to followa desired trajectory. The tracking problem is treated by usingtheflatnessproperty of the plate together with different con-trol strategies. Experimental results from a hardware-in-the-loop test stand show that flatness-based control serves well forsolving the given control problem.

1 Introduction

Figure 1: Scheme of the throttle plate.

The considered system is the throttle plate shown in Fig. 1.The throttle plate consists of two parts: an electrical part (elec-tric motor) and a mechanical part (throttle flap, mechanicalparts of the motor, . . . ). It is assumed that the throttle flap isdirectly driven by the motor. Therefore, no gearbox model hasto be included. Furthermore, the joint between the motor andthe throttle flap is rigid.

The system is described by using a model with the statevector x = [I, φ, φ]T and the inputu: I is the motor cur-rent in Ampere,φ the position of the motor and throttle flapmeasured in radians andφ is the angular velocity of the motorand throttle flap measured in radians/second. The inputu is apulse-width-modulated (PWM) voltage. In the following, theeffects of the pulse-width-modulated input can be neglected,because it is assumed that the electrical subsystem is very slowcompared to the PWM frequency. The states are given by thefollowing differential equations:

x1 = −RL

x1 −Cm

Lx3 +

1L

u (1a)

x2 = x3 (1b)

x3 =Cm

Jx1 −

Cf (x2)J

x2 −DJ

x3 + M(x2, x3).(1c)

The coefficients have the following meaning:R is the resis-tance of the motor,Cm is the motor constant,L describes theinductance of the motor. The constantJ is the moment of in-ertia,Cf (x2) represents the mechanical spring constant, andD describes the mechanical damping. The torque in (1c) isdescribed as

M(x2, x3) =1J

(

Cf (x2)φLHP−Mpre(x2)

−Mfr(x2) sign(x3))

.(1d)

Because of the special safety demands in the automotive sec-tor, the electrical throttle plate has a defined position to whichthe flap returns when the power supply of the electric motorfails or an error of a higher level control occurs. This positionφLHP is calledlimp home position. A consequence of the spe-cial construction of the throttle plate is that some parametersof the state equation (1) of the model are different for angularblade positions over and under the limp home positionφLHP.Thus, one gets the following relations for the spring constantCf (x2), the pretension of the springsMpre(x2) atφLHP andfor the Coulomb frictionMfr(x2):

Cf (x2) ={

Cf1 : x2 > φLHPCf2 : x2 ≤ φLHP

(2a)

Mpre(x2) =

{

Mpre1 : x2 > φLHPMpre2 : x2 ≤ φLHP

(2b)

Mfr(x2) ={

Mfr1 : x2 > φLHPMfr2 : x2 ≤ φLHP

(2c)

Strictly speaking, the model (1), (2) is non linear. Due to theparticular structure of (2), it can be considered as being piece-wisely linear for different operation areas.

The control problem for (1), (2) is to follow a desired tra-jectory φd(t) for the angular positionx2(t). This trackingproblem has an elegant solution if the system is differentiallyflat in the sense of [2, 3, 4, 7, 1].

Roughly speaking, this means that there exists a (ficticious)outputy (dim y = dim u), called aflat (or linearizing) output,with the following properties:

(i) The components ofy can be calculated from the statevariablesx, the inputsu, and a finite number of theirtime derivatives

y = Φ(

x, u1, . . . ,(α1)u1 , . . . , um, . . . ,

(αm)um

)

= Φ(

x, u, u, . . . ,(α)u

)

.(3)

(ii) Any system variablez, i. e., the state variablesx, the in-put variablesu, and their time derivatives and functionsof them, can be calculated fromy and a finite number ofits time derivatives

x = ψ1

(

y1, . . . ,(β1)y1 , . . . , ym, . . . ,

(βm)ym

)

= ψ1

(

y, y, . . . ,(β)y

)

(4a)

u = ψ2

(

y1, . . . ,(β1+1)

y1 , . . . , ym, . . . ,(βm+1)

ym

)

= ψ2

(

y, y, . . . ,(β+1)

y)

. (4b)

Since (1), (2) is considered to be a linear system, the flat-ness property is equivalent to the controllability of the sys-tem [2, 4]. The controllability of (1) can be shown by using

well-established criteria [6, 5]. Moreover, the flatness basedapproach that is presented here offers a straight-forward solu-tion to the given tracking problem exploiting the controllabil-ity property.

In section 2, the flatness of the system is shown. The flat-ness property forms the basis for all subsequent open andclosed loop schemes. In section 3, suitable trajectories aredesigned and an open loop control is discussed satisfying in-put constraints. In section 4, two control schemes for theclosed loop are presented: a linear output feedback includingthe open loop control strategy, and a flatness-based controlscheme based on state feedback. Experimental results for theopen and closed loop are presented in section 5.

2 Flat output

For the model (1), (2), it can be shown that the angle

y = x2 (5)

is a flat output of the throttle plate. In order to show the flat-ness of the system, all state variables as well as the input haveto be expressed as a function ofy and a finite number of itstime derivatives. By differentiating (5), one gets

x3 = y. (6)

Using the second time derivative ofy

y =Cm

Jx1 −

Cf (y)J

y − DJ

y + M(y, y), (7)

the currentx1 can be calculated:

x1 =1

Cm

[

Cf (y)y + Dy + J(

y −M(y, y))]

. (8)

Finally, with the third time derivative

(3)y = ϕ(y, y, y, u) (9)

one gets the input

u =−DL (JM(y, y) + Cmx1 − Cf (y)y) + D2Ly−

CmJ

J(

CmRx1 + C2my + CfLy + JL

(3)y

)

CmJ.

(10)The last relation describes theinverse systemfrom which theopen and closed loop control are derived in the following.

3 Trajectory Planning and Open loop control

In order to determine the open loop control law, a suitabledesired trajectoryyd(t) = φd(t) has to be defined. Accord-ing to (10), this trajectory must have smooth derivatives upto the order three. To this extent, a polynomial approach ora third-order time-delay element can be used. Both options

-φd(t)

Inverse system (10) -ud(t)

Throttle plate (1,2) -y(t) != yd(t)

Figure 2: Structure of the open loop control for the throttle-plate.

have been tested and give similar results. A third-order time-delay element is implemented more easily on a test bench andis used in the following experimental examinations (Section5). Nevertheless, it might be advantageous to use a polyno-mial approach, because of the reduced computational effort inthe real-time environment. Therefore, the procedure to definea sufficiently smooth desired trajectory

yd(t) =3

∑

i=0

kiti, 0 ≤ t ≤ T (11)

with a suitably chosen transient timeT is described here.The coefficientski, i = 0, . . . , 3 of (11) are calculated in

such a way that (11) satisfies the conditions

yd(0) = φ0 yd(T ) = φT

yd(0) = 0 yd(T ) = 0 (12)

depending on the transient timeT . With (10)–(12), the openloop control can be calculated

ud = ϕ(

yd, yd, yd,(3)yd

)

. (13)

The structure of the open loop control is shown in Fig. 2.Since (13) is an explicit expression for the control input,

input constraints

umin ≤ ud(t) ≤ umax (14)

may be considered off-line. This can be done by a simpleevaluation of the functional relation (13) — without requiringthe integration of differential equations. If the constraints areviolated due to the choice of the transient timeT , this param-eter has to be increased. Accordingly, it may be decreased toget faster motion in the opposite case. An iterative procedureeasily leads to a trajectory and open loop control respectingthe constraints with approriate margins (e. g.,10% of umax).

Due to parameter variations in automotive systems and dis-turbances during operation of a throttle plate, the open loopcontrol by itself is not sufficient to control the throttle plate.Nevertheless, different step responses of the flatness-basedopen loop control are shown in section 5.1 to document thisstatement.

4 Closed loop control

To ensure a steady state accurate step response and reducethe influence of parameter variations, the plate has to be oper-ated in closed loop. Using the flatness-based open loop strat-egy (13), an additional feedback can be determined in order

to achieve desired dynamic behaviour and to compensate theexternal disturbances.

In the following, two possibilities for a closed loop schemeare presented:

1st strategy:

For control of the throttle plate, the flatness-based open loop(13) with an additionalPI-controller can be used. Theopen loop controlud(t) is to impose the desired trajectory tothe system, whereas thePI-controller deals with deviationscaused by external disturbances and parameter changes. Thedesign of thePI-Controller (15) is based on the root-locusmethod and is primarily optimized to enhance the disturbancereaction of the closed loop. Using an integral part in the struc-ture of the controller

e(t) = φd(t)− φ(t)

upi(t) = Kpe(t) + KI

t∫

0

e(τ)dτ, (15)

a steady state accurate step response of the control loop is en-sured. The step response of the closed loop control at differentsteps for the desired value is shown in section 5.1.

Note that the proposed control scheme is a pure output feed-back, i. e., no state estimation is necessary.

2nd strategy:

The second control scheme comprises a flatness-based closedloop that can be obtained as follows: For (9), a new input isdefined

(3)y != v. (16)

By setting

v =(3)yd +

2∑

k=0

pk

(

(k)yd −

(k)y

)

(17)

the closed loop (1), (10) is stabilized asymptotically by a suit-able choice of the coefficientspk, k = 0, 1, 2. However, thistype of feedback is a state feedback and requires knowledgeof the state variables either by measurement or by a state ob-server.

The reconstruction of the state variables is done on the basisof the measured quantities

η1 = x1, η2 = x2. (18)

In order to reduce the computational effort, no state observerfor the estimation of the non-measured state variablesy = φ

-r?

+

-e6

φd(t) r-

Inverse system (10) - -ud(t)

PI-Controller

6e

+Throttle plate (1,2) -

y(t) != yd(t)

Figure 3: Structure of the first closed loop control for the throttle-plate.

PI-Controller(15)

6

φ(t)

φd(t)

State Feedback(17),(21)-

-r

-

φd, φd, φd,(3)φd

-v InverseSystem

(10)

e+-

u Throttle Plate(1),(2)

x

�Measurementη1,2 = h1,2(x)

�

�

r

-

η1 = Imot

η2 = φApproximation

(19),(20)

y, yap, yap

Figure 4: Scheme of the flatness-based control loop for the throttle plate with estimation of the state variables.

and y = φ is used designing the closed loop control. Here-with, the implementation in an electronic control unit is eased.The experimental results in the next section show the validityof this approach.

For this, the first derivative is approximated using the dif-ference

yap(t) ≈η2(t)− η2(t−∆t)

∆t. (19)

With this difference, the measurements (18) and (1c), an ap-proximation for the second derivative ofy is found

yap(t) ≈Cm

Jη1 −

Cf (η2)J

η2 −DJ

yap+ M(η2, yap). (20)

The scheme of the flatness-based control structure is shownin Fig. 4. In order to achieve steady state accuracy, aPI-controller according to (15) is used. Furthermore, the coeffi-cientspk, k = 0, 1, 2 of the state feedback controller have tobe determined in order to create an acceptable input responseof the closed loop. This task can be solved by assigning the

eigenvalues of the error dynamics. The eigenvalues of the er-ror dynamics are chosen to be a conjugate complex pair ofeigenvalues and an additional one on the real-axis with thefollowing parameters:

P1,2 = −80± j30[

1s

]

, P3 = −100[

1s

]

. (21a)

Based on these eigenvalues, the coefficientspk, k = 0, 1, 2 ofthe state feedback controller are given by the following values:

p0 = 730000[

1s

]

, p1 = 23300[

1s2

]

, p2 = 260[

1s3

]

.

(21b)The experimental results of the closed loop control using astate feedback controller are shown in section 5.3.

5 Experimental results

Traditionally, in the automotive industry step inputs are usedto verify the dynamic behaviour of a component like thethrottle-plate. That is why different step inputs of the desiredvalueφd(t) have been carried out to present and to comparethe responses of the open and closed loop controls. For eachcontrol method, the plots of three different step responses ofthe desired value are shown:

• φ(t1) = 20◦ to φ(t2) = 80◦

• φ(t1) = 45◦ to φ(t2) = 55◦

• φ(t1) = 5◦ to φ(t1) = 65◦

This means that the first two steps are above the limp-home-position, which is located at12◦. The third step of the de-sired value shows the input response of the loop when crossingthe dominant non-linearity of the system — the limp-home-position. To ensure good comparability of the systems reac-tions, the step-height is always the same.

For the flatness-based open and closed loop controls the de-sired trajectoryφd(t) is calculated on the basis ofφ(t) de-scribed above. This can be interpreted as the movement of theaccelerator pedal in conjunction with the trajectory element(third-order time-delay element or polynomial approach – seeSection 3).

For a comparisons, a step response of a common outputfeedback with a linearPID-Controller is shown in Fig. 5.These step responses can be considered as state of the art tech-

1.4 1.6 1.8 2 2.2 2.40

10

20

30

40

50

60

70

80

90

Time [s]

Des

ired−

/Act

ual V

alue

[°D

K]

φd(t)

y(t)

1.4 1.6 1.8 2 2.2 2.40

10

20

30

40

50

60

70

80

90

Time [s]

Des

ired−

/Act

ual V

alue

[°D

K]

φd(t)

y(t)

Figure 5: Input response of a closed loop control above theLHP using a conventionalPID-Controller.

nology and thereby serve as reference for comparison withthe step responses of the flatness-based control schemes ofthe throttle plate. The parametersKP , KI and KD of thePID-Controller (22) are designed and optimized using theroot-locus method. It can be assumed, that it is not possibleto improve the quality of the control significantly when usingthis control structure

upid(t) = Kpe(t) + Ki

t∫

0

e(τ)dτ + Kde(t). (22)

In Fig. 5, it can be seen that — using thePID-controller —the actual value of the angleφ(t) approaches the desired valueasymptotically. There is no overshooting, although the actualvalue is oszillating slightly while moving towards the desiredvalue. When increasing the proportional gainKp of thePID-controller, the oscillation rapidly grows and the control almostreaches the limit of stability. The input response of the closedloop finally achieves steady state accuracy because of the in-tegral part of the controller.

The input response of thePID-controller crossing the LHPis shown in Fig. 6. Note that the linear control across the LHP,the main nonlinear part of the throttle plate, obviously leads toa less desirable response of the closed loop. The actual angleof the throttle plate remains at the LHP for 300ms until theintegral part of the controller is getting strong enough to movethe plate across the LHP (note the stretched time axis). Then,the desired value is approached asymptotically.

1 1.5 2 2.5 30

10

20

30

40

50

60

70

80

90

Time [s]

Des

ired−

/Act

ual V

alue

[°D

K]

φd(t)

y(t)

Figure 6: Input response of a closed loop control across theLHP using aPID-Controller.

5.1 Experimental results of the flatness based open loopcontrol

The input response of the open loop control above the limp-home-position (LHP)is shown in Fig. 7. It can be seen that

0 1 2 3 40

10

20

30

40

50

60

70

80

90

100

Time [s]

Des

ired−

/Act

ual V

alue

[°]

φ(t) φ

d(t)

y(t)

0 1 2 3 40

10

20

30

40

50

60

70

80

90

100

Time [s]

Des

ired−

/Act

ual V

alue

[°]

φ(t) φ

d(t)

y(t)

Figure 7: Input response of the open loop control above theLHP.

the desired value can not been reached for all times. Thisis due to model errors and parameter uncertainty. The openloop control is not steady state accurate and the response ofthe60◦ step input shows a permanent angular deviation of8◦

(lower limit of φd(t)) in respect to12◦ (upper limit ofφd(t))from the desired value. The targetted dynamics of the desiredtrajectory, though, is well followed by the actual angle of thethrottle plate.

This result is reached using a desired trajectory, which iscalculated by a third-order time-delay element as indicated be-fore. It has the time constant T=40ms in combination with arate limiter of500◦/s. This means that a relatively slow de-sired trajectory is used with the open loop control to reducesaturation effects of the inductance. Thus, the setting voltageu does not exceed6V in operation.

The input response of the open loop control is getting evenworse if the step-height is getting smaller (right plot of thefigure). This is a result of nonlinear friction of the systemwhich is comparatively small and considered a disturbance.Therefore, it is not possible to compensate the stiction of thethrottle plate with an open loop control. Futhermore, both fig-ures demonstrate, that the open loop control is very sensitivetowards even slight changes of system parameters.

0 1 2 3 40

20

40

60

80

100

Time [s]

Des

ired−

/Act

ual V

alue

[°]

φ(t) φ

d(t)

y(t)

Figure 8: Input response of the open loop control across theLHP.

The step of the desired value through the LHP and the reac-tion of the flatness-based open loop control is shown in Fig. 8.It is shown, that the influence of the LHP can be compensatedto a great extent using the flatness-based open loop control.Compared to the input response of the linearPID-controller(6) the actual value only stays 130ms at the LHP before reach-ing the desired value below the LHP asymptotically.

5.2 Experimental results of the flatness-based outputfeedback control

To improve the steady state accuracy and disturbance reactionof the open loop control, a linear feedback controller with in-tegral part is used. It is shown that the input response of thisclosed loop is improved considerably (Fig. 9) using the samedesired trajectory. Thus, the maximum level of the settingvoltage is increased slightly compared to the open loop con-trol, but does not reach the input constraints. This graph shows

0 1 2 3 40

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Time [s]

Des

ired−

/Act

ual V

alue

[°]

φ(t) φ

d(t)

y(t)

0 1 2 3 40

10

20

30

40

50

60

70

80

90

100

Time [s]

Des

ired−

/Act

ual V

alue

[°]

φ(t) φ

d(t)

y(t)

Figure 9: Input response of the closed loopPI-Control abovethe LHP.

that the actual value of the angleφ(t) overshoots the desiredvalue. This happens independently of the step height and leadsto an asymptotical approach of the actual valuey(t) towardsthe desired valueφd(t). Furthermore, the actual value followsthe desired value more accurately with this control conceptthan with the open loop control.

As the parameters of the linear controller (15) are not op-timized thoroughly, the reactions to both step-inputs may beimproved slightly. A problematic circumstance in this case isthe tendency of the integral part of the controller to destabi-lize the closed loop if not designed carefully. For this reason,the absolute value of the controllers integral part could not beincreased significantly. Furthermore, the proportional gain ofthe linear controller has to stay small in relation to the inte-gral part for stability reasons, as well. These circumstancesrestrict the performance of an output feedback with a linearPI-control.

The step of the desired value crossing the LHP and the re-sponse of the flatness-based closed loop control is shown inFig. 10.

0 1 2 3 4

0

20

40

60

80

100

Time [s]

Des

ired−

/Act

ual V

alue

[°]

φ(t) φ

d(t)

y(t)

Figure 10: Input response of the closed loopPI-Controlacross the LHP.

In Fig. 10, it is shown that it is not possible to compensatethe discontinuity of the spring characteristics at the LHP andimprove the input response with a linearPI-controller. Al-

though the time that the throttle plate stayed at the LHP is re-duced to 50ms, the use of this linear controller leads to a fairlylarge under- and overshooting of the actual value. Therefore,it has to be noticed, that the input response of this closed loopcrossing the LHP is not improved compared to the flatness-based open loop control (Fig. 8).

5.3 Experimental results of the flatness-based state feed-back control

In order to solve the evident problems of the linear positioncontrol and the flatness-based output feedback control, a statefeedback controller is used in addition to the output feedback[8]. In Fig. 11, the step inputs of the desired value above theLHP are shown. Because of the improved dynamics due tothe state-feedback control, the time constant (note the timeaxes) of the desired trajectory is reduced from 40ms to 12ms.Additionally, the rate limit is increased to600◦/s. Using thisdesired trajectory, the setting voltage reaches the input limit at12V.

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40

50

60

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80

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100

Time [s]

Des

ired−

/Act

ual V

alue

[°]

φ(t) φ

d(t)

y(t)

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10

20

30

40

50

60

70

80

90

100

Time [s]

Des

ired−

/Act

ual V

alue

[°]

φ(t) φ

d(t)

y(t)

Figure 11: Input response of the flatness-based closed loopcontrol above the LHP.

According to Fig. 11, it is obvious that the input response ofthe closed loop using a state feedback controller is enhanced.The actual value of the angle follows the desired trajectoryvery accurately. The actual value is accurate at steady-stateconditions and although the desired trajectory is faster, thereis no overshooting. As well, the input response of the closedloop does not depend on the step-height. Since this flatness-based state control is able to follow the desired trajectory, thedynamic qualities of the closed loop are defined by the choiceof the transient time. The dynamics are primarily limited bythe physical properties of the throttle plate (motor-torque, mo-ment of inertia, etc.). With this desired trajectory, the throttleplate needs 175ms for a step-height of60◦ without overshoot-ing. This value can certainly be decreased in further examina-tions, if necessary.

The input response of the closed loop using a desired tra-jectory across the LHP is shown in Fig. 12. In this graph it hasto be noticed, that the step response is improved compared tothe linear feedback control, but it doesn´t follow the trajectoryas accurately as the step responses above the LHP. This re-sult, caused by the physical properties, might be improved by

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100

Time [s]

Des

ired−

/Act

ual V

alue

[°]

φ(t) φ

d(t)

y(t)

Figure 12: Input response of the flatness-based closed loopcontrol across the LHP.

different enhancements of the control concepts and is topic ofactual research efforts.

6 Conclusions

For the control of a throttle plate, an open and closed loopstrategy have been developed on the basis of a nonlinearmodel. It has been shown, that a flatness-based open loopcontrol on its own is not suitable to control the position of thenonlinear throttle plate. This is demonstrated by different stepresponses which showed a destinct deviation of steady stateaccuracy and a strong sensitivity towards variations of systemparameters.

For the closed loop control, two different control strate-gies have been developed. First, a flatness-based output feed-back control using a linear controller with integral part (PI-controller) is examined. This closed loop structure shows animproved input response compared to the open loop strategy.Because of thePI-controller, the input response of the closedloop reaches steady state accuracy. This structure shows slightovershooting of the actual value independently of the step-height followed by an asymptotical approach towards the de-sired value. It was not possible to improve the dynamics ofthe input response of the system and enhance the plate´s re-sponse when crossing the so-called limp-home-position of thethrottle flap.

Therefore, a flatness-based state feedback control structurein combination with a linear output feedback has been de-signed. This strategy shows the best experimental results us-ing the same step inputs of the desired value. Because ofthe increased dynamic ability by using this control structure,the desired trajectory could be designed a lot faster. In thehardware-in-the-loop environment, this flatness-based controlof the throttle plate follows the desired trajectory very accu-rately and fulfills the stability and dynamic requirements.

References

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