Robust Extremum Seeking and Speed Ratio Control forHigh-Performance CVT Operation

Stan van der Meulen, Bram de Jager, Frans Veldpaus, and Maarten Steinbuch

Abstract— The variator in a pushbelt continuously variabletransmission (CVT) enables a stepless variation of the trans-mission ratio within a finite range. Nowadays, the variator iselectronically controlled and the variator control objectives aretwofold: 1) tracking a transmission ratio reference; 2) opti-mizing the variator efficiency. Recently, the extremum seekingcontrol (ESC) technique is exploited in view of optimizingthe variator efficiency, which only uses measurements fromsensors that are standard. However, the operating conditionsare fixed and tracking a transmission ratio reference is omitted,for simplicity. In this paper, extensions are proposed thatovercome these limitations. This is achieved via the constructionof a disturbance feedforward control design and a speedratio control design. Experiments illustrate the effectiveness ofthese extensions when the operating conditions are varied andtracking a transmission ratio reference is required.

I. INTRODUCTION

The pushbelt continuously variable transmission (CVT)incorporates several components, e.g., the variator and thehydraulic actuation system. The variator enables that thetransmission ratio is continuously varied in between twobounds, i.e., Low and High. The variator consists of ametal V-belt, i.e., a pushbelt, which is clamped betweentwo pairs of conical sheaves, i.e., two pulleys, see Fig. 1. Aprimary (input, subscript “p”) pulley and a secondary (output,subscript “s”) pulley are distinguished. Each pulley consistsof one axially moveable sheave and one axially fixed sheave.Each axially moveable sheave is connected to a hydrauliccylinder, which is pressurized by the hydraulic actuationsystem. Essentially, the hydraulic actuation system translatesa desired pressure pj,ref into a realized pressure pj , wherethe pressure pj in the hydraulic cylinder is directly related tothe clamping force Fj on the axially moveable sheave, wherej ∈ {p, s}. The level of the clamping forces determines thetorque capacity, whereas the ratio of the clamping forcesdetermines the transmission ratio. When the level of theclamping forces is too high, variator efficiency is compro-mized, since the friction loss is increased. When the level ofthe clamping forces is too low, variator damage is introduced,since the slip is increased. Hence, there exists a choice of theclamping forces that guarantees the functionality of the CVTand optimizes the efficiency of the CVT, which demands acontrol design in which both items are explicitly addressed.

The objective for the variator control system is twofold:1) tracking a speed ratio reference rs,ref , which is prescribed

S. van der Meulen, B. de Jager, F. Veldpaus, and M.Steinbuch are with the Department of Mechanical Engineering,Control Systems Technology Group, Eindhoven University ofTechnology, PO Box 513, 5600 MB Eindhoven, The Netherlands.S.H.v.d.Meulen@tue.nl, A.G.de.Jager@tue.nl,F.E.Veldpaus@tue.nl, M.Steinbuch@tue.nl

This research is partially funded by Bosch Transmission Technology,Tilburg, The Netherlands.

Fs Ts

Tp Fp

xp

xs

Rs

Rp

ωp

ωs

β

a

Fig. 1. Schematic illustration of pushbelt variator.

by the driveline control system; 2) optimizing the variatorefficiency η. The transmission ratio is represented by thespeed ratio rs, which is easily computed from the ratioof the measurements of the angular velocities. The variatorefficiency η is defined by the ratio of the powers, which arenot measured.

Traditionally, the majority of the approaches control thespeed ratio via the primary pulley with the primary hydrauliccircuit and the torque capacity via the secondary pulley withthe secondary hydraulic circuit, see, e.g., [1]. The primarypressure that is required in order to achieve the speed ratiois computed by means of a feedback controller (closedloop). Several feedback control designs are encountered, e.g.,PI(D) control [1], fuzzy control [2], robust control [3]. Thesecondary pressure that is required in order to transfer thetorque is computed by means of a variator model (open loop).Since the variator model is uncertain, a safety strategy isemployed, which utilizes a safety factor. Generally, the safetyfactor ranges from 1.2 [-] to 1.3 [-], which implies that thevariator efficiency is seriously compromized.

Recently, the variator efficiency is explicitly addressed inthe control design that is proposed in [4]. The existence ofa certain optimum for the variator efficiency as a functionof the slip is shown by means of experiments. As a result, astraightforward approach is to control the slip in such a waythat a certain slip reference is tracked, which corresponds tothe optimum variator efficiency [4]. However, this approachinvolves two issues. First, the determination of the slip refer-ence [5, Section 7.2]. Since the optimum variator efficiencydepends on, e.g., the transmission ratio, the variator load,and the variator wear, the determination of the slip referenceis not straightforward and often time-consuming, which istypically caused by the unreliability of the available variator

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models. Second, the reconstruction of the slip in the variator.This typically requires a dedicated sensor, e.g., measurementof the pushbelt running radius [6] or measurement of theaxially moveable sheave position [4], which increases boththe complexity and the costs. In addition, the reconstructionof the slip in the variator on the basis of one of thesemeasurements is extremely sensitive to deformations in thevariator, which are unknown.

These drawbacks are avoided in the control design thatis proposed in [7], [8], which effectively improves thevariator efficiency and only uses the measurements of theangular velocities and the secondary pressure, which arestandard. The control design exploits the observation that themaximum of the (ps, rs) equilibrium map and the maximumof the (ps, η) equilibrium map are achieved for values ofps that nearly coincide. This motivates the consideration ofthe input-output map in which the secondary pressure ps isthe input and the speed ratio rs is the output, although thelocation of the maximum is unknown. For this reason, themaximum of the input-output map is found by means ofextremum seeking control (ESC) [9], which aims to adaptthe input in order to maximize the output.

In [8], however, two simplifications are adopted. First,the torques that are exerted on the variator, which areconsidered in terms of disturbances, are stationary of nature.Obviously, the torque disturbances that are encountered whenthe variator is installed in a vehicle are transient of nature.These torque disturbances possibly enforce a transition ofthe variator behavior from open loop stable to open loopunstable, since the torque capacity is nearly consumed. Thispossibly destabilizes the ESC feedback mechanism, sincethe ESC feedback mechanism is not robustified. Second,the control problem for optimizing the variator efficiencyis isolated from the control problem for tracking the speedratio reference. With this simplification, a single-input single-output (SISO) control problem is obtained (input: ps, output:rs). Without this simplification, a multi-input single-output(MISO) control problem is obtained (inputs: pp and ps,output: rs), which is not treated.

The main contribution of this paper is twofold. The firstcontribution concerns a solution for the robustness problemof the ESC feedback mechanism in view of a primary sidedisturbance that resembles a depression of the acceleratorpedal. This is achieved via the addition of a disturbance feed-forward control design. The second contribution concerns asolution for the MISO control problem that simultaneouslysatisfies both variator control objectives. This is achieved viathe integration of the ESC design with a speed ratio control(SRC) design. The remainder of this paper is organizedas follows. The preliminaries are addressed in Section II,which includes the definitions and the experimental setup. InSection III, the ESC design is described and the disturbancefeedforward control design is introduced. In Section IV, theESC design is optimized and the SRC design is introduced.Finally, the paper concludes with a discussion in Section V.

Notation: Consider the variator in Fig. 1. The torquesthat are exerted on the variator are denoted by Tp and Ts.Furthermore, the angular velocities are denoted by ωp andωs, the clamping forces by Fp and Fs, the axially moveablesheave positions by xp and xs, and the running radii by Rp

and Rs. Finally, a denotes the variator centre distance andβ denotes half the pulley wedge angle, i.e., β = 11 [deg].

II. PRELIMINARIES

A. DefinitionsThe geometric ratio rg and the speed ratio rs of the

variator are defined by:

rg =RpRs

(1)

rs =ωsωp. (2)

The relative slip ν is defined by:

ν =ωpRp − ωsRs

ωpRp= 1− rs

rg, (3)

see [4]. The variator efficiency η is defined by:

η =Pout

Pin=TsωsTpωp

, (4)

where Pin and Pout denote the input power and the outputpower, respectively.

B. Experimental SetupThe experimental setup is depicted in Fig. 2 and consists

of five main components. These are given by two identicalelectric motors, a pushbelt variator, a hydraulic actuationsystem, and a data acquisition system. The experimentalsetup incorporates additional sensors, which are primarilyused for analysis purposes.

ÀÁ

Â

Ã

Ä

Å

Æ

Ç

Fig. 2. Experimental setup with pushbelt variator (À: Pushbelt variator;Á: Primary torque sensor; Â: Secondary torque sensor; Ã: Primaryelectric motor; Ä: Secondary electric motor; Å: Hydraulic actuationsystem; Æ: Accumulator; Ç: Data acquisition system).

III. ROBUST EXTREMUM SEEKING CONTROL

The control configuration, which incorporates the ESCdesign, is introduced in Section III-A. The ESC design issubsequently described in Section III-B. The ESC designoptions are addressed in Section III-C. A detailed analysisof stability and performance of the ESC design is presentedin [8]. Then, the background of the torque disturbances is

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highlighted in Section III-D. Finally, the robustness of theESC design with respect to a primary side disturbance isanalyzed in Section III-E.

A. Control ConfigurationConsider the control configuration that is depicted in

Fig. 3. Here, GV denotes the relation between ps and rsand THp and THs denote the relations between pp,ref andpp and ps,ref and ps, respectively. Furthermore, ∆Tp and∆Ts denote the deviations that are possibly superposed tothe nominal values Tp and Ts.

pp,refTHp

pp

THs

ps,ref ps

rs[

.ESC

] [. GV

]

Tp

∆Tp

Ts

∆Ts

Tp

Ts

Fig. 3. Control configuration.

B. Extremum Seeking Control DesignThe feedback mechanism is depicted in Fig. 4. Obviously,

the feedback mechanism utilizes a sinusoidal perturbationαm sin(2πfmt), which is added to ps,ref , i.e., the estimateof the optimum input p∗s,ref . As a result, the input of thehydraulic actuation system ps,ref is defined by:

ps,ref(t) = ps,ref(t) + αm sin(2πfmt), (5)

where αm denotes the perturbation amplitude and fm de-notes the perturbation frequency. When ps,ref is on eitherside of p∗s,ref , the periodic perturbation enforces a periodicresponse of the output of the variator rs, which is either inphase or out of phase with the periodic perturbation. Withthis information, the feedback mechanism from rs to ps,ref

is designed, which consists of the following operations:

ξ1 = Hb(s)rs (6)ξ2 = Hb(s)ps (7)ξ3 = ξ1ξ2 (8)ξ4 = Hl(s)ξ3 (9)

ps,ref =1sIξ4. (10)

Here, Hb(s) denotes a band-pass filter, Hl(s) denotes a low-pass filter, and I denotes the integrator gain. The band-passfilter Hb(s) enforces the suppression of “DC components”and noise for rs and ps, which results in ξ1 and ξ2, respec-tively. As a result, ξ1 and ξ2 are approximately two sinusoids,which are out of phase for ps,ref > p∗s,ref and in phase forps,ref < p∗s,ref . In either case, the product of both sinusoidsξ3 has a “DC component”. The low-pass filter Hl(s) extractsthe “DC component” of ξ3, which results in ξ4. Finally, ps,ref

results from integration of ξ4, with integrator gain I . Theinitial condition for the integrator is equal to ps,ref , whichcorresponds to a stationary operating point. Observe that (9)contains the gradient information and (10) represents the

gradient update law, which enables the adaptation of ps,ref

towards the optimum input p∗s,ref .

THs GV

Hb Hb

ξ1ξ2αm sin(2πfmt)

ξ3

Hl

ξ41sI

ps,ref

ps,ref

ps,ref

rsps

Fig. 4. Feedback mechanism from rs to ps,ref for ESC.

Obviously, the feedback mechanism incorporates five de-sign options. These are the perturbation amplitude αm,the perturbation frequency fm, the band-pass filter Hb(s),the low-pass filter Hl(s), and the integrator gain I . Theselection of these design options is closely related to theproof of stability for the closed loop system, which isaddressed in [9]. The feedback mechanism in [9] is similarto the feedback mechanism in Fig. 4. However, a high-passfilter is employed instead of a band-pass filter. The mainreason for the application of a band-pass filter concernsthe suppression of noise. Three assumptions are required,see [9], which are satisfied for the operating conditions thatnormally occur, see [8]. Then, convergence of the solution(ps,ref(t), ξ4(t), rs(t)) towards a certain neighborhood of thepoint (p∗s,ref , 0, rs,max) is guaranteed by [9, Theorem 5.1] fora suitable choice of the design options. A suitable choice ofthe design options is made in Section III-C.

C. ESC Design OptionsThe perturbation amplitude αm, the perturbation frequency

fm, the band-pass filter Hb(s), the low-pass filter Hl(s), andthe integrator gain I are given by:

αm = 0.7 (11)fm = 3 (12)

Hb(s) =2

2πfm0.01s

1(2πfm)2 s

2 + 22πfm

0.01s+ 1(13)

Hl(s) =1

12π1s+ 1

(14)

I = 250. (15)

D. Background of Torque DisturbancesThe torque disturbances are induced by the internal com-

bustion engine (ICE) (primary side) or the road (secondaryside). The primary torque disturbance ∆Tp is typically im-posed by the driver who depresses or releases the acceleratorpedal. The secondary torque disturbance ∆Ts is typicallyinduced by obstacles or unevennesses. Regarding the primaryside, the engine control unit (ECU) of the ICE measuresseveral variables, e.g., the accelerator pedal position or thethrottle valve angle and the crankshaft angular velocity.These variables in combination with the performance map ofthe ICE enable the estimation of the torque. Subsequently,this estimate of the torque is transferred to the transmission

112

control unit (TCU). Hence, a priori information with respectto the primary torque disturbance ∆Tp is typically available.Regarding the secondary side, a priori information withrespect to the secondary torque disturbance ∆Ts is typicallyunavailable.

E. Primary DisturbanceA specific experiment is performed in order to analyze

the robustness of the ESC feedback mechanism with respectto a primary torque disturbance ∆Tp,ref . The experiment isstarted from a certain stationary operating point. The ESCfeedback mechanism converges towards a small neighbor-hood of the extremum rs = rs,max, which is approximatelyreached for t ≈ 100 [s]. Finally, the deviation ∆Tp,ref =15 [Nm] is superposed to the nominal value Tp,ref = 67[Nm] for t ≥ 400.0 [s]. The chosen deviation ∆Tp,ref

is representative for a depression of the accelerator pedalwith moderate impact. The experiment is performed twice.First, without feedforward of the primary torque disturbance∆Tp,ref . Second, with feedforward of the primary torquedisturbance ∆Tp,ref . A contribution ∆ps,ref is added to theestimate of the optimum input ps,ref , which is defined by:

∆ps,ref = 0 (16a)

∆ps,ref =1As

∆Tp,ref cos(β)2µRp

, (16b)

for the case without feedforward and the case with feedfor-ward, respectively. Here, As denotes the secondary hydrauliccylinder pressure surface and µ denotes the traction coeffi-cient, i.e., µ = 0.09 [-]. The presentation of the experimentalresults shows the case without feedforward on the left andthe case with feedforward on the right.

399 399.5 400 400.5 40160

65

70

75

80

85

90

Tp

[Nm

]

399 399.5 400 400.5 40120

25

30

35

40

t [s]

Ts

[Nm

]

399 400 401 402 40360

65

70

75

80

85

90

399 400 401 402 40320

25

30

35

40

t [s]

Fig. 5. Experimental results for High and primary disturbance (Left:Without feedforward; Right: With feedforward) (black: Measurement;grey: Reference).

The torques are depicted in Fig. 5. For the case withoutfeedforward, the variator is unable to transfer the additionaltorque, which is given by the deviation ∆Tp (Fig. 5 (topleft)). For the case with feedforward, the variator is able totransfer the additional torque, which is given by the deviation∆Tp (Fig. 5 (top right)).

399 399.5 400 400.5 40175

80

85

90

95

100

η[%

]

399 399.5 400 400.5 4010

5

10

15

t [s]

ν[%

]

399 400 401 402 40390

92

94

96

98

100

399 400 401 402 4030

0.2

0.4

0.6

0.8

1

t [s]

Fig. 6. Experimental results for High and primary disturbance (Left:Without feedforward; Right: With feedforward).

399 399.5 400 400.5 401−0.1

0

0.1

∆p

s,ref

[bar

]

399 399.5 400 400.5 4014.22

4.23

4.24

ps,ref[b

ar]

399 399.5 400 400.5 4013.5

4

4.5

5

ps,ref

[bar

]

399 399.5 400 400.5 4013.5

4

4.5

5

t [s]

ps

[bar

]399 400 401 402 4030

0.5

1

399 400 401 402 4034.18

4.19

4.2

399 400 401 402 4033

4

5

6

399 400 401 402 403

4

5

6

t [s]

Fig. 7. Experimental results for High and primary disturbance (Left:Without feedforward; Right: With feedforward).

For the case without feedforward, the relative slip νsharply rises (Fig. 6 (bottom left)). The experiment is stoppedwhen the relative slip exceeds the predefined value ν = 15[%]. Obviously, the variator enters in the macro-slip region,i.e., the variator behavior is open loop unstable. Observe thatthe variator efficiency η sharply drops (Fig. 6 (top left)).The contribution ∆ps,ref and the estimate of the optimuminput ps,ref are depicted in Fig. 7 (left), together withthe secondary pressure reference ps,ref and the secondarypressure ps. Apart from the sinusoidal perturbation, ps,ref isonly determined by ps,ref , since the case without feedforwardis considered, i.e., ∆ps,ref is equal to (16a). It is observed thatthe estimate of the optimum input ps,ref slightly decreases,which is undesired, since the deviation ∆Tp is obviouslypositive. This behavior is also observed from the signals inthe feedback loop (Fig. 8 (left)), especially from ξ4, whichcontains the gradient information. Hence, the ESC feedbackmechanism without feedforward is unable to adapt to the

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399 399.5 400 400.5 401−2

0

2x 10

−3

ξ 1[-]

399 399.5 400 400.5 401−0.5

0

0.5

ξ 2[-]

399 399.5 400 400.5 401−2

0

2x 10

−4

ξ 3[-]

399 399.5 400 400.5 401−2

0

2x 10

−4

t [s]

ξ 4[-]

399 400 401 402 403−2

0

2x 10

−3

399 400 401 402 403−0.5

0

0.5

399 400 401 402 403−2

0

2x 10

−4

399 400 401 402 403−2

0

2x 10

−4

t [s]

Fig. 8. Experimental results for High and primary disturbance (Left:Without feedforward; Right: With feedforward).

change of the operating condition.For the case with feedforward, the relative slip ν slightly

increases (Fig. 6 (bottom right)). Obviously, the variatoroperates in the micro-slip region, i.e., the variator behavioris open loop stable. Observe that the variator efficiencyη slightly increases (Fig. 6 (top right)). The contribution∆ps,ref and the estimate of the optimum input ps,ref aredepicted in Fig. 7 (right), together with the secondarypressure reference ps,ref and the secondary pressure ps.Apart from the sinusoidal perturbation, ps,ref is mainlydetermined by ∆ps,ref , since the case with feedforward isconsidered, i.e., ∆ps,ref is equal to (16b). It is observed thatthe estimate of the optimum input ps,ref slightly changes,since the contribution ∆ps,ref is apparently inaccurate. Thisbehavior is also observed from the signals in the feedbackloop (Fig. 8 (right)), especially from ξ4, which contains thegradient information. Hence, the ESC feedback mechanismwith feedforward is able to adapt to the change of theoperating condition.

IV. INTEGRATION WITH SPEED RATIO CONTROL

The control configuration, which incorporates the ESCdesign, is introduced in Section IV-A. The SRC design issubsequently described in Section IV-B. A detailed analysisof stability and performance of the SRC design is presentedin [10]. The ESC design options are addressed in Section IV-C. In Section IV-D, a closed loop experiment is performed,which shows that the variator control objectives are simulta-neously satisfied.

A. Control Configuration

Consider the control configuration that is depicted inFig. 9. A cascade control design is employed. The inner loopincludes the hydraulic actuation system, which is closed loopcontrolled [11]. The desired pressures pp,ref and ps,ref are theinputs and the realized pressures pp and ps are the outputs.The outer loop includes the variator, which is controlled bythe combination of ESC and SRC.

[SRCESC

]rs,ref pp,ref

ps,ref

[THpp

THps

THspTHss

]

Ts

ωp

rs[GVp

GVs

]pp

ps

Fig. 9. Control configuration.

B. Speed Ratio Control DesignThe SRC design K(s) computes the input pp,ref on the

basis of the tracking error rs,ref − rs. The SRC design K(s)is given by the product of the following parts:

Kgain = 3 · 105 (17a)

Kint(s) =1s

(17b)

Klead(s) =

(1.12π1s+ 1

)2

1(17c)

Knotch(s) =1

(2πfm)2 s2 + 2

2πfm0.002s+ 1

1(2πfm)2 s

2 + 22πfm

0.05s+ 1(17d)

Kroll-off(s) =1

(s+ 2π6)2 . (17e)

The notch filter (17d) is implemented in order to reduce thesuppression of the periodic response that is enforced by thesinusoidal perturbation of the ESC design.

C. ESC Design OptionsThe perturbation amplitude αm, the perturbation frequency

fm, the band-pass filter Hb(s), the low-pass filter Hl(s), andthe integrator gain I are given by:

αm = 0.7 (18)fm = 10 (19)

Hb(s) =2

2πfm0.003s

1(2πfm)2 s

2 + 22πfm

0.003s+ 1(20)

Hl(s) =1

12π1s+ 1

(21)

I = 1250. (22)

The design options are changed in comparison with thedesign options in Section III-C. Both the perturbation fre-quency fm and the integrator gain I are increased in orderto improve the convergence speed, see [12]. The increase ofthe perturbation frequency fm enforces a change of the band-pass filter Hb(s). Finally, the damping ratio of the band-passfilter Hb(s) is modified.

D. Closed Loop ExperimentThe operation of the combination of the ESC design

and the SRC design is evaluated by means of a closedloop experiment, see Fig. 9. The operating conditions aregiven by ωp = 1000 [rpm] and Ts = 20 [Nm], whereasthe speed ratio reference is equal to rs,ref = 1.2 [-].The closed loop experiment is started from a stationary

114

operating point, which is defined by the initial condition forthe integrator of the ESC feedback mechanism. The initialcondition for the integrator is equal to ps,ref = 7.2 [bar].Actually, this stationary operating point corresponds to thesecondary pressure reference ps,ref that is achieved by theabsolute safety strategy, where the absolute safety factor isequal to 1.3 [-]. This absolute safety strategy is commonlyused by the automotive industry [13]. For this reason, thisabsolute safety strategy is adopted for comparison purposes.The experimental results for the proposed strategy and theabsolute safety strategy are depicted in Figs. 10, 11, and 12.From Fig. 11, it follows that the speed ratio reference rs,ref

is accurately tracked. The variator efficiency η for boththe proposed strategy and the absolute safety strategy isdepicted in Fig. 12. Obviously, when the pressure referencesfor the proposed strategy decrease in comparison with theabsolute safety strategy, see Fig. 10, the variator efficiencyincreases. The gain with respect to the variator efficiency isapproximately equal to 2.5 [%].

100 150 200 250 300 3500

2

4

6

8

pp

[bar

]

100 150 200 250 300 3500

2

4

6

8

100 150 200 250 300 3500

2

4

6

8

10

t [s]

ps

[bar

]

100 150 200 250 300 3500

2

4

6

8

10

t [s]

Fig. 10. Experimental results for pressures (Left: Proposed strategy; Right:Absolute safety strategy) (black: Measurement; grey: Reference).

100 150 200 250 300 3501.19

1.195

1.2

1.205

1.21

t [s]

r s[-]

100 150 200 250 300 3501.19

1.195

1.2

1.205

1.21

t [s]

Fig. 11. Experimental results for speed ratio (Left: Proposed strategy;Right: Absolute safety strategy) (black: Measurement; grey: Reference).

V. DISCUSSION

In this paper, a variator control system for a pushbeltCVT is proposed on the basis of the extremum seekingcontrol (ESC) technique, which effectively optimizes thevariator efficiency. A disturbance feedforward control designis proposed in order to deal with a primary torque disturbance

100 150 200 250 300 35090

92

94

96

98

t [s]

η[%

]

Fig. 12. Experimental results for variator efficiency (black: Proposedstrategy; grey: Absolute safety strategy).

that is typically imposed by the driver. A speed ratio controldesign is proposed in order to deal with a speed ratioreference that is typically imposed by the driveline controlsystem. The effectiveness of both extensions is successfullydemonstrated by means of experiments.

Two opportunities for future research are highlighted.First, the acceleration of the convergence of the ESC design,via optimization or extension of the ESC design. Second, theintegration of the separate control designs and the evaluationfor a driving cycle.

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