The Horsepower Reserve Formulation of Driveability for a Vehicle Fitted With a Continuously Variable...

The Horsepower Reserve Formulation of Driveability

for a Vehicle Fitted with a Continuously Variable

Transmission

MICHAEL HENRY SMITH,2 ERIC J. BARTH,∗,1 NADER SADEGH3 ANDGEORGE J. VACHTSEVANOS4

SUMMARY

A systematic methodology is developed for choosing the optimum ratio trajectory of a continuouslyvariable transmission in a passenger vehicle. The optimum CVT ratio schedule is formulated as a con-strained optimization problem with maximum fuel economy as the objective function and driveability con-cerns and physical limitations included as the constraints. The key notion to achieving good driveabilityis the introduction and definition of a horsepower reserve function that creates a consistent and desirablevehicle response under different driving conditions. Simulation results compare the optimized schedule’sperformance with several other possible ratio schedules, including the minimum brake specific fuel con-sumption map. Results from the optimized schedule indicate only a mild tradeoff between driveability andfuel economy relative to the other ratio schedules. The ratio optimization problem formulation and solutionprovide a novel and unique approach for systematically addressing driveability and fuel economy consider-ations associated with a continuously variable transmission.

Keywords: cvt, continuously variable transmission, driveability, torque margin, ratio schedule,optimization.

1. NOMENCLATURE

TA Throttle angle or throttle position∆TA Change in throttle angleTA∗ Equilibrium throttle angleHp Engine output power∆Hp Change in engine power

1Address correspondance to: Vanderbilt University, Department of Mechanical Engineering, VU Station B351592, 2301 Vanderbilt Place, Nashville, TN 37235, [email protected], office: (615)322-1893,fax: (615)343-69252ETAS, inc., [email protected] Institute of Technology, George W. Woodruff School of Mechanical Engineering, Atlanta, GA30332–0405, [email protected]

4Georgia Institute of Technology, School of Electrical and Computer Engineering, Atlanta, GA 30332–0250, [email protected]

M.H. SMITH, E.J. BARTH, N. SADEGH AND G.J. VACHTSEVANOS 2

Hp∗ Equilibrium engine powerωe Engine speed, CVT input speedωdes

e Desired engine speedωmin

e , ωmaxe Minimum and maximum engine speed

Te Engine torqueωs CVT output speedi CVT ratio (i = ωe

ωs)

imin, imax Minimum and maximum CVT ratioides Desired CVT ratiogmin, gmax Minimum and maximum achievable CVT ratio ratesHpreserve Horsepower reserve functionhreserve Prescribed horsepower reserve valuevveh Vehicle speedBSFC Brake-specific fuel consumption

2. INTRODUCTION

Continuously variable transmissions (CVT) have recently received renewed interestas a viable alternative to discrete-ratio transmissions in passenger vehicles. Unfortu-nately, there does not currently exist a methodology to choose the appropriate trans-mission ratio to take full advantage of the CVT’s potential to offer both improvedfuel economy and driveability over discrete-ratio transmissions. Both production andresearch-oriented ratio schedules typically revolve around ad-hoc, hand-tuned, andtrial-and-error methods. Presented here is a well defined and systematic ratio opti-mization methodology.

There has been a substantial amount of research recently published related to CVTratio control. In almost every instance, the authors present very nicely developed con-trol algorithms to achieve a desired ratio, while the desired ratio is usually chosen to befunctions for fuel efficiency and/or performance [26, 8, 11]. The fuel efficient target ra-tio is fairly straightforward and well-defined, while the ‘performance’ mode is usuallysome arbitrary function commanding a relatively higher engine speed for all throttleinputs. However, in no publication does any author systematically address driveabil-ity concerns in the choice of a target ratio or engine speed; driveability is assessedafter the fact and the ratio schedule is iteratively modified until driveability is satisfac-tory. Most authors do acknowledge poor driveability from a maximum-fuel-efficiencyschedule—hence, the ‘performance’ or modified schedule. Most of the current workalso point out the potentially undesirable shift-shock or torque hole effects resultingfrom the negative torque contribution to vehicle acceleration from the non-minimumphase vehicle dynamics induced by the ratio rate of the CVT. To compensate, the com-

3 THE HORSEPOWER RESERVE FORMULATION . . .

manded ratio rate is often reshaped, as in [13], and again often in an ad-hoc manner, toenhance or address the driveability in terms of which mode the vehicle is in or how thethrottle changes; for example, a faster rate is allowed in performance mode. This ratiorate shaping can add significant complexity to the control problem. This is especiallytrue in [5], where the rate shaping is done with an extra fuzzy controller. This is aninteresting exercise, but the fuzzy controller adds a large number of parameters thatmust be set or tuned, and the claims of improved driveability are marginal at best. Asin most cases, the necessity for ratio rate shaping is the result of a poor choice of targetratio—steady-state, transient and/or a function of time—in the first place.

Despite the relative lack of research related to CVT ratio scheduling and driveabil-ity, one paper in particular must be cited for contributing important information. In[22], the authors Vahabzadeh and Linzell report on a study of drivetrain parametersfor an automatic transmission-equipped vehicle that correlated drivers’ throttle pedalinputs to engine power. They found that the driver commanded throttle position wasdirectly proportional to the driver’s desired engine power. Because the driver doeshave direct control of the throttle, and also will expect some level of familiarity froma new type of transmission (i.e. the CVT) for acceptance, this correlation result putsthe focus of driveability issues on engine power availability and delivery. Regardlessof this fact, many works incorrectly correlate the driver controlled throttle angle withthe demanded drive torque [21, 27]. Furthermore, scheduling the desired engine speedor ratio as a function of throttle only is almost guaranteed have bad ‘feel’ to the driver,since the engine’s power delivery additionally depends on the driving environment(hills, loads, etc.). However, this approach is taken in the majority of the efforts citedabove.

Vahabzadeh and Linzell, among others [17, 16, 9, 23], also state an explanation whythe most fuel-efficient ratio schedule has driveability problems. The reason is that atthe low engine speeds for good fuel economy, the engine has a low torque margin. Inthe work by Kasai and Morimoto [5], the authors formulate—without an explanationhow—a ratio schedule based on both vehicle speed and throttle input, claiming simplythat “this feels better to most drivers.” What Kasai and Morimoto are doing, perhapsunknowingly, is addressing both the torque margin problem and the consistent powerissue. Essentially, their ratio schedule prevents the torque margin from falling at highervehicle speed, thereby having consistent power delivery, or more accurately, changesin power delivery with respect to throttle inputs, which reflects positively on driverfeel. Others, too, utilize schedules with vehicle speed as an input [6, 20, 12].

Clearly, the ratio or engine speed needs to be scheduled based on more informationthan throttle angle. Having separate modes, such as performance and economy stilldoes not circumvent the inherent problems, since these modes are still only functionsof throttle angle and cannot deliver a consistent response. Indeed, even for traditionalautomatic transmissions, researchers (e.g. [14, 15, 25, 4]) are currently examiningways to base shift schedules on driving environment information, most notably by


intelligently identifying mountainous roads via measurable vehicle sensor feedback.Typically, acceleration forces are very closely related to how a vehicle ‘feels.’

Smooth power delivery equates to smooth acceleration forces (smooth in the senseof a constant acceleration derivative, or jerk), which is almost universally accepted asdesirable. For discrete ratio transmissions, an objective analysis of ‘good feel’ usuallyconsists of measuring the magnitude of the jerk or acceleration associated with a shift.For example, rapid clutch engagements cause large torque variations to be induced andtransmitted in the drivetrain, causing large variations or oscillations in accelerations ofthe engine and/or vehicle. A gear shifting analogy exists for a CVT, when the ratio ischanged. During a ratio change, it is important to manage the acceleration induced bynon-minimum phase dynamics [7, 2]. To counteract undesirable behavior, a constraintcould be included that limits the contribution of the undesirable effects, such as lim-iting the CVT’s ratio rate of change. However, variable limits could be required; forexample, limits based on the throttle rate, where a higher throttle rate may indicate aquicker response is acceptable in exchange for a larger initial deceleration. The exam-ples in this paragraph describe many of the factors associated with ratio-rate shaping;clearly, to address each operating condition can make the problem quite complex, andusually leads to an ad-hoc, trial-and-error design process. The same would be true forother measures that, although perhaps indicative of good driveability such as the ratioof a stepwise change to either engine acceleration of vehicle acceleration (∆TA/ω̇ e

or ∆TA/v̇veh), would need to be specified for too many different, i.e. all, operatingconditions. In addition, the values assigned to these targets must also be classified andidentified as ‘acceptable’ or ‘good,’ which would require exhaustive and expensivestudies, tests, and surveys.

Clearly from a design point-of-view, driveability measures are more useful for anal-ysis a posteriori rather than being included a priori [24]. This does not imply thatthe measures should be ignored, rather, what is much more practical is to address inthe design stage, those factors that influence the driveability measures and driver ex-pectations, and then create a flexible methodology so that those influences are easilymanipulated and analyzed by comparative tests and simulations.

In summary, it is impossible to fully quantify the subjective notion of driveability.The definition of driveability revolves around driver expectation, where one can cor-rectly assume that a smooth, quick, predictable, and repeatable response to the driver’sthrottle input is desirable. This type of response is the realistic goal of a CVT ratio con-troller, and it is the goal of the constrained optimization methodology presented here.Furthermore, engine power has been identified as a central influence to attaining thegoal. Based on this, a straight-forward, well-founded, and systematic methodology isformulated, and then claims of ‘good,’ or improved driveability will be made based onresponse comparisons with other techniques and ratio control schedules.


3. HORSEPOWER RESERVE DEFINITION

The horsepower reserve formulation is based on the fact that a driver is requesting acertain output power when pressing on the throttle pedal. Likewise, if the driver altersthe throttle, he/she is requesting a change in output power. To maintain driveability,it is essential that the same change in output power is delivered to the vehicle forthe same change in throttle input, no matter what the driving condition is. This is themeaning of ‘consistent power delivery’ in the context of this work. In order to achieveconsistent power delivery and therefore a desirable vehicle response across differentoperating conditions, it is necessary to characterize an engine’s response relative tothrottle inputs by examining its torque and horsepower characteristics. When design-ing shift strategies for conventional discrete-ratio automatic transmissions, engineerslook at the engine’s torque margin. The torque margin is fairly well represented forsmall changes in throttle position by the partial derivative of the torque productionwith respect to the throttle angle:

∂Te

∂TA= f(ωe, TA) (1)

The torque margin relates changes in throttle angle to changes in torque production,and helps a designer decide when to schedule a gear shift in a discrete-ratio transmis-sion. A gear shift would not be commanded if the new engine speed after the shiftresults in a low torque margin, yielding poor driveability and the possibility of lug-ging. A low torque margin means that when the driver depresses the throttle pedal, theengine’s torque cannot change by a sufficiently large value, so the vehicle’s responseis sluggish and delayed, and hence, results in poor driveability.

CVT-equipped vehicles can also suffer from the effects of low torque margins, asnoted by Vahabzadeh and Linzell [22]. As with conventional transmissions, the enginespeed must be raised to sufficiently increase torque by moving to a higher numericalratio, but this will delay the vehicle’s response. To overcome a low or undesirabletorque margin for the CVT optimization problem, the notion of torque margin is ex-tended by first examining the horsepower margin

∂Hp∂TA

= ωe∂Te

∂TA= ωef(ωe, TA). (2)

The horsepower margin is shown in Figure 1.


1000

2000

3000

4000

5000

6000

7000

020

4060

80

0

2

4

6

8

10

engine speed (rpm) throttle angle (degrees)

dHp/

dTA

(H

p/de

gree

)

Fig. 1. Horsepower margin for a typical gasoline engine.

The horsepower margin is considered for analysis instead of the torque margin forseveral reasons. First, it has already been established that power, not torque, is thefocus from a driveability point of view. Second, while torque is multiplied througha transmission, power is conserved (minus some losses which can be ignored in thisanalysis), meaning that the power and horsepower margin from the engine is essen-tially the same as the drive power and power margin at the wheels. Finally, note fromEquation 2 that a constant horsepower margin for all speeds does not equate to a con-stant torque margin. As the CVT allows the engine to operate at any point (limited, ofcourse, by physical constraints), the advantage of maintaining a constant horsepowermargin can be realized. A constant horsepower margin is desirable because the ve-hicle will deliver a consistent power response for different operating conditions. Inthis way, the expectations of the driver will be met and thereby ‘good’ driveabilitywill be perceived. This notion is the main motivation behind the horsepower reservemethodology.

Instead of introducing an optimization constraint based on the horsepower margin(a local quantity), a more physically intuitive interpretation of the horsepower marginis defined; it is analogous to the more ‘global’ torque margin often defined for tran-sitions to wide-open-throttle. This function is called the horsepower reserve, and isdefined by the change in engine power for a given change in throttle angle.

Hpreserve(TA, ωe) � ∆Hp∣∣∆TA (3)

≈∫ TA+∆TA

x=TA

∂Hp∂x

dx (4)

≈ ∆TA∂Hp∂TA

(5)


Not only is the horsepower reserve more physically intuitive, but also more physi-cally descriptive and useful from a driveability point-of-view. Furthermore, as seenfrom Equation 4 this definition of horsepower reserve essentially averages the powermargin and describes the change in power for a finite change in the throttle angle (asopposed to an infinitesimal change for the horsepower margin function). Thus, thehorsepower reserve is formulated by choosing a meaningful change in throttle angle,∆TA, and then an associated change in engine power ∆Hp. The horsepower reservefunction is shown in Figure 2; note its shape is similar to the horsepower margin inFigure 1, as in Equation 5. Also note the important differences between the horse-power margin (Figure 1) and the horsepower reserve (Figure 2) functions at the edgesof their domains.

1000

2000

3000

4000

5000

6000

7000

020

4060

80

0

10

20

30

40

50

60

70

engine speed (rpm) throttle angle (degrees)

Hp

rese

rve

(Hp)

Fig. 2. Horsepower reserve surface for a typical gasoline engine.

4. HORSEPOWER RESERVE BASED SCHEDULING

With the definition of horsepower reserve, and some background on driveability anddriver expectation, this section presents a detailed analysis and support of the opti-mization problem formulation. The following assumption is made: the torque con-verter is considered to be locked, and is thus ignored in the ratio optimization proce-dure. The engine speed is therefore considered to be essentially equal to the CVT’sinput speed. The desired ratio is formulated here as a constrained optimization prob-lem, which is stated as follows:


Find a function, y(TA, Hp∗) = ides = desired CVT ratio, that minimizesfuel consumption and satisfies the constraints

imin ≤ ides ≤ imax (6)

ωmine ≤ ωdes

e ≤ ωmaxe (7)

∂ωdese

∂TA≥ 0 (8)

Hpreserve ≥ min{hreserve, highest Hpreserve available} ∀ Hp ≤ Hp∗ (9)

∂ωdese

∂TA= 0 ∀ TA ≤ TA∗ (10)

where TA∗ and Hp∗ are throttle angle and engine power required to main-tain a constant vehicle speed, Hpreserve is the horsepower reserve func-tion specific to the engine used (i.e. Figure 2), and h reserve is the pre-scribed minimum horsepower reserve available to the driver (in the sub-sequent design it is set as a constant). The fuel consumption and rangeof engine power are described by the functions fuelNN = f1(TA, ωe) andHpNN = f2(TA, ωe)[19].

To solve the constrained optimization problem, which is done off-line by an iter-ative, global search technique, knowledge of the engine’s horsepower and fuel con-sumption is explicitly required. A previous powertrain modeling effort [19], resultingin neural network models of engine torque, horsepower, and fuel consumption, will beused here for the functions required above. Simulation results will support the claimsof improved driveability. The constraints in the optimization problem fall into two cat-egories. The first category is physical constraints, and the second is driveability con-straints. This formulation of the optimization problem also allows further constraintsto be added, such as limits on emissions levels, although these are not considered here.It should be noted that the optimization problem presented here is not dependent onthe particular type of continuously variable transmission, the type of engine, or thetype of vehicle. Hence, the methodology is applicable and valid for traction drives andother CVTs.

4.1. Objective Function

In the optimization problem statement, the objective is chosen to be minimize fuelconsumption subject to the constraints. The fuel consumption objective can be equiva-lently restated as minimizing the brake-specific fuel consumption (BSFC). Other work[1, 8, 26] in the field often refer to the minimum BSFC solution as the ‘ideal’ operat-


ing line, maximum efficiency curve, or optimum-fuel schedule (see Figure 4(b)) andwould represent the solution to the unconstrained optimization problem.

Obviously, the objective could have been to maximize driveability, but this wouldbe a much more challenging problem considering the discussion and conclusionsof the previous sections. Conversely, fuel consumption is both a well-defined andwell-known function of throttle angle and engine speed, from a modeling perspec-tive at least [19]. In addition, it is much easier to include driveability considerationsas constraints, thereby allowing many driveability or driver expectation issues to beaddressed directly, and also permitting the inclusion of other, perhaps unforeseen con-straints in the future without completely redefining the problem.

Because the engine is actually the major fuel consuming and power producingelement—it should be clear by now that horsepower plays the essential role fordriveability—the focus of the optimization problem is shifted to the engine. Althoughthe CVT, like all transmissions, is a crucial factor in determining the engine’s speedand delivering power to drive the vehicle, its role can be delayed temporarily. By do-ing so, the necessity for including the vehicle speed (via the CVT’s output speed,recall i = ωe/ωs) in the optimization solution can also be ignored. Furthermore, theassumption that the engine speed is equal to the CVT’s input speed is also made. Theoptimization objective can thus be restated as: Find a function, y(TA, Hp ∗) = ωdes

e =desired engine speed, that minimizes BSFC, and satisfies the constraints of Equa-tions 6–10, or

minωe

|BSFC| ∀Hp (11)

where BSFC = f(TA, ωe), since engine torque and fuel consumption are only func-tions of throttle and speed. The solution will then yield a target engine speed, and thedesired ratio will be defined in Section 4.6.

4.2. Constraints 1 and 2: Physical Limitations

The two physical constraints on engine speed and ratio are introduced first:

imin ≤ ides ≤ imax (12)

ωmine ≤ ωdes

e ≤ ωmaxe (13)

where imin, imax, ωmine , and ωmax

e are minimum and maximum ratios, and minimumand maximum engine speeds, respectively. The ratio limitations are constrained bythe CVT hardware. The maximum engine speed limit represents the safe operatingregion for the engine, while the minimum is set as idle speed.


4.3. Constraint 3: Driver Expectation of Engine Speed

The third constraint can be made without, hopefully, too much explanation or con-vincing for the reader:

∂ωdese

∂TA≥ 0. (14)

Physically, this constraint means that the engine’s speed will not decrease when thethrottle is depressed, which obviously could be disconcerting to the driver.

4.4. Constraint 4: Horsepower Reserve

The next constraint is crucial for good driveability in terms of a quick and consistentvehicle response under different driving conditions:

Hpreserve ≥ min{hreserve, highest Hpreserve available} ∀ Hp ≤ Hp∗ (15)

where Hp∗ is the engine power required to maintain a constant vehicle speed, and isreferred to as the equilibrium power. The horsepower reserve, Hp reserve = ∆Hp

∣∣∆TA,

is defined in Section 3 and its value, hreserve, must be chosen by the designer. From asimulation standpoint, it is difficult to identify what the best reserve value would be.However, based on existing ratio schedules for test vehicles, a value in the range of15 to 25 horsepower should be realistic for a heuristically determined ∆TA = 7 ◦ (seeSection 5.1). Furthermore, relatively high values—e.g. 25 horsepower—will providea quicker vehicle response to throttle inputs, and could be classified as more ‘sporty’than low reserve values.

The horsepower reserve reflects the sensitivity of the engine (and drive) power rel-ative to throttle inputs. It also is indicative of how much power is available ‘instanta-neously.’ Since by assumption, engine power is a function of only throttle and speed, itcan be increased by raising the throttle angle, the engine speed, or both. However, be-cause the transmission, engine inertia, and vehicle dynamics are much slower than thetorque production dynamics, engine power responds more quickly to throttle inputsthan to ratio changes. Thus, the horsepower reserve helps characterize the vehicle’s,and engine’s, responsiveness to throttle inputs.

4.5. Constraint 5: Engine Braking

The engine braking constraint is defined by

∂ωdese

∂TA= 0 ∀ TA ≤ TA∗ (16)

where TA∗ is defined in previous constraint as the equilibrium throttle angle. This con-straint is included for one obvious driveability reason, but will result in other benefits


that will be shown in the simulation studies. The driveability concern is to introducethe slight engine braking characteristic that is present in discrete-ratio transmission ve-hicles. Because the ratio is fixed in traditional transmissions, when the driver releasesthe throttle pedal, the engine speed does not drop until the vehicle speed decreases.Thus, for some duration, the throttle angle is small or zero while the engine speedis relatively high. In the regions of high engine speeds and small throttle, the enginetorque and power is negative, which actually aids driveability when the driver is mak-ing small speed adjustments and does not wish to use the brake pedal. However, for aCVT, where the engine speed can be and often is reduced, the engine braking charac-teristic is lost. Adding engine braking will also offer a familiar driving characteristicfor the driver of a CVT vehicle, which should increase the CVT’s acceptance. A com-mon complaint about CVT vehicles is that they ‘feel weird’, and a lack of enginebraking in one of the causes (excessive engine speed changes is another). No publi-cation has explicitly included engine braking in the engine speed schedule, althoughratio shaping can accomplish a similar goal (for example, a slow ratio change).

Enforcing a constant engine speed below the equilibrium throttle angle does notimply the engine speed will not decrease, however, simply because the vehicle speedwill likely decrease, thereby lowering the equilibrium power and throttle and thus theallowable engine speed. Note that this engine braking constraint is not the same as theengine braking necessary to maintain a constant speed while traveling downhill. Thatdesirable engine braking situation automatically falls out of the optimization solutionfrom the inclusion of the reserve horsepower constraint.

4.6. Solution Technique

The solution to the optimization problem presented in this section is not difficult tofind. Although many solution techniques are available to solve constrained optimiza-tion problems, for example gradient descent or Gauss-Newton methods, the combina-tion of a finite state space and the generally well-defined and known functions involvedmake an off-line, global-search, iterative solution the most practical. The iterative pro-cess consists of first choosing a load value, then evaluating the engine speed and throt-tle angle combinations that satisfy all the constraints, and finally choosing the pointscorresponding to minimum brake specific fuel consumption. The engine power andfuel consumption maps (functions) are required to evaluate and find the horsepowerreserve and equilibrium and optimum fuel points (throttle-speed pairs) for the solution.The ranges of load, throttle, and speed variables are divided into suitably small incre-ments to achieve an acceptable level of accuracy. Smaller increments were chosen forsmall throttle angles, small loads, and low engine speeds, since the combination ofthese ranges represents the significant majority of the vehicle’s operating region. Thedesired engine speeds can then be stored as look-up tables or approximated and storedas functions. The three-dimensional desired engine speed map is shown in Figure 3


for hreserve = 20 horsepower.

0

20

40

60

80

−50

0

50

100

150

1000

2000

3000

4000

5000

6000

throttle angle (o)load Hp

desi

red

engi

ne s

peed

(rp

m)

Fig. 3. Desired engine speed of a typical gasoline engine for 20 horsepower reserve.

The ratio schedule generated from the optimization solution essentially allows theratio rate to be controlled by the driver, via the throttle rate. Thus, the control systemdoes not have to ‘guess’ what the rate should be (e.g. how fast or slow, as is oftendone by other control methodologies [5, 22, 3]). If the throttle rate is high, for examplethe driver ‘floors’ the pedal, indicating a fast desired change in power, the schedulewill command a fast ratio rate. Although the non-minimum phase contribution mayworsen, the driver’s large and rapid throttle change signify a willingness to trade asmooth response for quicker power delivery. Thus, more direct control is given to thedriver in determining how the vehicle will react.

Nonetheless, some ratio shaping is necessary from a controllability viewpoint; thatis, for the benefit of the low-level of the control hierarchy. The controllability con-siderations consist of limiting the ratio rate to a physically realizable value for thehydraulics actuating the CVT, whose dynamics are of course, finitely fast. The othermodification to the desired ratio is to add a small integral term to ensure that thesteady-state engine speed error is zero. Thus the desired ratio is defined as

ides =ωdes

e − α∫

ωerror

ωs(17)

gmin ≤∣∣∣∣dides

dt

∣∣∣∣ ≤ gmax (18)


where α is a suitably small positive constant, and ωerror = ωdese − ωe is the difference

between the desired engine speed and the actual engine speed. The rate limits g min

and gmax are determined based on experimental data and/or analysis of the low-levelhydraulic model. These limits can and should be functions of operating conditions,which will maximize the allowable ratio rate range while avoiding potentially damag-ing operation.

5. SIMULATION STUDIES AND ANALYSIS

To support the claims of an optimum ratio trajectory and show the validity of thesystematic optimization methodology, this section presents some comparisons for dif-ferent driving conditions typical of realistic driving scenarios. The comparisons arebased on several different target engine speed schedules which all have the same ratioshaping and limits of the previous section; that is, no schedule will violate the physicallimitations of the CVT, and all are subject to the same low-level hydraulic control.

The vehicle model used consists of lumped parameter models of the engine dynam-ics, drivetrain dynamics and gross vehicle dynamics. Neural network were utilized tomap engine torque production, emissions and fuel consumption based on data takenfrom an actual vehicle (1996 3.0 liter V-6 Ford Taurus). The driver was modeled asa PID controller generating throttle and braking commands to track a desired vehiclespeed. The CVT hydraulics and its low-level controller are simplified to a realisticrepresentation of the closed-loop hydraulic dynamics, modeled by a rate-limited first-order lag. Further details regarding the simulation model can be found in [2, 18]

5.1. Alternative Desired Engine Speeds for Comparison

Because claims of driveability are much more easily shown in relative terms, severalother desired engine speed schedules are introduced for comparisons. Figures 4(a) and4(b) show the curves plotted on the engine surface, with the dotted lines representingconstant engine powers. The arbitrary-torque schedule is linear in a speed-torque re-lationship, and the arbitrary-speed map is linear in a throttle-speed relationship. Thearbitrary-torque function is representative of what is often done by other researchers[22, 26]; that is, shifting the desired speed to the right of the optimum fuel line, asshown in Figure 4(a). The arbitrary-speed curve is a sensible and realistic possibilityfor a desired engine speed, since a linear function of desired speed versus throttle an-gle would give both a predictable engine speed, and an engine speed slightly abovethe optimum-fuel line that results in a better torque margin, as shown in Figure 4(b).The optimum-fuel curve is shown also, and will be used in the comparison tests.


1000 2000 3000 4000 5000 6000 70000

20

40

60

80

100

120

140

160

180

200

engine speed (rpm)

engi

ne to

rque

(lb−

ft)

arb−speed arb−torqueopt−fuel

(a) Speed-torque surface.

0 10 20 30 40 50 60 70 80

1000

2000

3000

4000

5000

6000

throttle angle (o)

engi

ne s

peed

(rp

m)

arb−speed arb−torqueopt−fuel

(b) Throttle-speed surface.

Fig. 4. Different desired engine speeds plotted as functions of throttle angle and torque for a typical gasolineengine.

Another desired speed schedule is shown in Figure 5. This schedule was hand-tunedand was tested and used in a prototype vehicle equipped with a CVT. The scheduleresults in both decent driveability and roughly a 10% increase in fuel economy over astandard discrete-ratio automatic transmission. Similar results regarding fuel economyare reported by Toyota [11] and ZF [10]. This schedule can and would be furthertuned (by hand) to possibly improve both fuel economy and driveability before beingconsidered production-ready. However, it provides a good basis for comparison sinceit has been ‘real-world’ tested with acceptable results. The desired engine speed is afunction of both throttle angle and vehicle speed.

020

4060

80

2040

6080

100

1000

2000

3000

4000

5000

6000

throttle anglevehicle speed

desi

red

engi

ne s

peed

Fig. 5. Hand-tuned desired engine speed schedule for a typical gasoline engine.


Some initial observations of the different desired speed schedules are made here tohelp support and clarify the simulation results. Figure 4(b) shows the arbitrary-torque

curve to have very steep slope, meaning that dωdese

dTA is high, and the desired engine speedis very sensitive to throttle inputs. In addition, the arbitrary-speed, optimum-fuel, andhand-tuned schedules also have a positive dωdes

e

dTA or ∂ωdese

∂TA for all throttle angles. Thesepositive slopes will result in a change of commanded engine speed for all throttlechanges. However, recall that the optimization-based schedule has a section of con-stant desired engine speed around the equilibrium throttle angle. This constant speedpart combined with the available horsepower margin (from the horsepower reserveconstraint), allow changes in engine power from small throttle changes to go directlyto vehicle acceleration instead of engine acceleration, resulting in a quicker vehicleresponse and reduced shift-shock.

6. CASE STUDIES

The three case studies presented in this section demonstrate the advantages gainedthrough the systematic ratio optimization problem formulation and its subsequent so-lution.

6.1. Case 1: Vehicle Acceleration

The first simulation example is a simple transient maneuver where the driver wishes toaccelerate from 35 to 45 mph in a fairly aggressive but reasonable fashion, implyingthat the pedal inputs should be less than full throttle, as is the case in Figure 6(a).Figures 6(b)–8(b) plot various vehicle outputs for this acceleration test. Figure 6(b)shows the desired vehicle speed and the simulated vehicle speed resulting from eachengine speed schedule. Because the driver is modeled only as a feedback controller,all speed responses show a slight delay.

Two optimization-based schedules with different horsepower reserves (20 and 25horsepower) are shown for comparison. The speed and acceleration responses inFigures 6(b)–7(b) demonstrate a significant advantage of the optimization approach.Clearly, the optimization-based response is both more immediate and desirable sincethe vehicle acceleration is always positive. Each of the other schedules show an initialvehicle deceleration, which is not only opposite to what the driver desires, but also de-lays the onset of vehicle acceleration. Furthermore, even though the arbitrary-torqueschedule has a high horsepower reserve, its response displays a very large shift-shockand erratic vehicle acceleration resulting from the high throttle-engine speed sensitiv-ity of the desired speed curve; as with the simulated PID driver, a real driver wouldfind this vehicle difficult to control. However, the high reserves of the optimization ap-


proach display better driveability with their smooth responses and fast settling times.These optimization-based benefits result from the constant desired speed portion ofthe schedule which commands a smaller initial ratio change, as seen in Figure 8(a).Finally, Figure 8(b) shows that the two optimization-based engine speeds partly over-lap the optimum-fuel engine speed, as expected in the transient maneuver.

0 2 4 6 8 10 12 14 160

5

10

15

20

25

30

time (sec)

thro

ttle

angl

e (o )

tuned

opt−fuel

arb−torque

arb−speed

20 Hp reserve

25 Hp reserve

(a) Throttle angle.

0 2 4 6 8 10 12 14 1634

36

38

40

42

44

46

time (sec)

vehi

cle

spee

d (m

ph)

desired

tuned

opt−fuel

arb−torque

arb−speed

20 Hp reserve

25 Hp reserve

(b) Vehicle speed.

Fig. 6. Throttle angle and vehicle speed comparisons for 35–45 mph maneuver.

0 2 4 6 8 10 12 14 16

−0.02

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

time (sec)

vehi

cle

acce

lera

tion

(g)

tuned

opt−fuel

arb−torque

arb−speed

20 Hp reserve

25 Hp reserve

(a) Entire maneuver.

0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2−0.02

−0.01

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

time (sec)

vehi

cle

acce

lera

tion

(g)

tuned

opt−fuel

arb−torque

arb−speed

20 Hp reserve

25 Hp reserve

(b) Initiating maneuver.

Fig. 7. Vehicle acceleration comparison for 35–45 mph maneuver, and detail of initiating the maneuver.


0 2 4 6 8 10 12 14 160.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

2.2

2.4

time (sec)

ratio

tuned

opt−fuel

arb−torque

arb−speed

20 Hp reserve

25 Hp reserve

(a) CVT ratio.

0 2 4 6 8 10 12 14 16

1000

1500

2000

2500

3000

3500

4000

time (sec)

engi

ne s

peed

(rp

m)

tuned

opt−fuel

arb−torque

arb−speed

20 Hp reserve

25 Hp reserve

(b) Engine Speed.

Fig. 8. CVT ratio and engine speed comparisons for 35–45 mph maneuver.

6.2. Case 2: Small Vehicle Speed Changes

The second case study presents a subtle yet extremely important and very commondriving scenario. In real-world driving, the vehicle speed is rarely if ever actually con-stant. The driver is constantly making small throttle and speed adjustments to compen-sate for such things as road load variations or proximity to other vehicles. The overallresponse to these small changes plays a very significant role in determining how thedriver perceives the vehicle’s driveability in normal driving scenarios. Indeed, thesesituations are what define normal driving, not for example, wide-open-throttle maneu-vers that are more dependent on maximum engine power and are much less frequentlyencountered. As such, a small speed variation scenario is simulated and shown in Fig-ures 9(a)–11(b). The desired vehicle speed profile decreases then increases over a 2mph range.

In this small speed variation test, the shift-shock phenomenon is small but stillpresent for the non-optimized schedules. For these schedules, the initial accelerationis opposite to the desired acceleration, which also again contributes to the delayedresponse of vehicle acceleration and speed. Although the delay and shift-shock is ofsmaller magnitude than in the previous vehicle acceleration case study, it is actuallyof more relative significance. The importance of a relatively quicker and smootherresponse for small speed changes is because a driver is less willing to accept eithershift-shock behavior or delays for smaller throttle changes. To see this, consider awide-open-throttle maneuver, where the driver would expect, or even desire, both a‘harsher’ response and a vehicle acceleration delay while the engine speed is raised,because not only does this more rapidly increase engine power but it also sounds and


feels like the vehicle is performing. Furthermore, consider an analogy with a manualdiscrete-ratio transmission: for small speed variations, the driver does not shift gearsbut lets the throttle alone create speed changes. However, for more aggressive speedchanges, passing for example, the driver downshifts, creating a slight but acceptabledelay.

The discrete-ratio transmission analogy reveals another desirable benefit of theoptimized engine speed schedule. Consider the ratio and engine speed response ofFigures 11(b) and 11(a). These plots show that neither the ratio nor engine speedsvary much relative to the other schedules’ performance. The net result is that theoptimization-based schedule again allows engine power changes to be directed to ve-hicle acceleration instead of the engine acceleration. Furthermore, as this behavioris similar to standard transmissions, driver acceptance of the new CVT transmissionshould be increased while annoying and unnecessary engine speed and ratio variationsare eliminated.

This simulation also demonstrates the advantages of slight engine braking. Notonly does the engine braking allow for a quicker deceleration—more closely match-ing the desired speed—but also upon re-accelerating, the engine speed does not haveto be raised as far, further eliminating shift-shock and undesirable delays. This con-tributes to better driveability simply because the car is more closely doing what isdesired, which is also supported by the smaller required changes in throttle inputs ofFigure 9(a) and the quicker settling times and smaller vehicle speed tracking error inFigure 9(b).

0 2 4 6 8 10 12 14 16 180

2

4

6

8

10

12

time (sec)

thro

ttle

angl

e (o )

tuned

opt−fuel

arb−torque

arb−speed

20 Hp reserve

25 Hp reserve

(a) Throttle angle.

0 2 4 6 8 10 12 14 16 1833.5

34

34.5

35

35.5

36

36.5

time (sec)

vehi

cle

spee

d (m

ph)

desired

tuned

opt−fuel

arb−torque

arb−speed

20 Hp reserve

25 Hp reserve

(b) Vehicle speed.

Fig. 9. Throttle angle and vehicle speed comparisons for small speed variation.


0 2 4 6 8 10 12 14 16 18−0.03

−0.02

−0.01

0

0.01

0.02

0.03

0.04

0.05

time (sec)

vehi

cle

acce

lera

tion

(g)

tuned

opt−fuel

arb−torque

arb−speed

20 Hp reserve

25 Hp reserve

(a) Entire maneuver.

0.5 1 1.5 2 2.5 3 3.5 4 4.5 5−0.02

−0.015

−0.01

−0.005

0

0.005

0.01

time (sec)

vehi

cle

acce

lera

tion

(g)

tuned

opt−fuel

arb−torque

arb−speed

20 Hp reserve

25 Hp reserve

(b) Initiating maneuver.

Fig. 10. Vehicle acceleration comparison for small speed variation, and detail of initiating the maneuver.

0 2 4 6 8 10 12 14 16 180.4

0.6

0.8

1

1.2

1.4

1.6

time (sec)

ratio

tuned

opt−fuel

arb−torque

arb−speed

20 Hp reserve

25 Hp reserve

(a) CVT ratio.

0 2 4 6 8 10 12 14 16 18

1000

1500

2000

2500

3000

time (sec)

engi

ne s

peed

(rp

m)

tuned

opt−fuel

arb−torque

arb−speed

20 Hp reserve

25 Hp reserve

(b) Engine Speed.

Fig. 11. CVT ratio and engine speed comparisons for small speed variation.

6.3. Case 3: EPA/FTP75 Cycle Simulation

Although the preceding sections have put significant emphasis on driveability, fueleconomy is an equally important topic, and indeed, was the original driving forcebehind CVT development. Fuel consumption is also a more straight-forward, measur-able, and indisputable quantity by which to evaluate and compare the success of theoptimization-based schedule. Hence, this case study analyzes the engine-speed sched-


ules on part of the EPA/FTP75 test cycle, which is used to calculate published fueleconomy and emission figures. Instead of the complete EPA cycle, a modified EPAcycle is simulated, where desired vehicle speeds below 10 mph are eliminated be-cause of the assumption that the CVT mode is only above that minimum speed. Themodified cycle is shown in Figure 12.

0 200 400 600 800 1000 1200 1400 160010

15

20

25

30

35

40

45

50

55

60

time (sec)

desi

red

spee

d (m

ph)

Fig. 12. Modified EPA cycle (no speeds below 10 mph).

The test cycle is simulated with the four alternative engine speed schedules as inSection 5.1, but two different optimization-based schedules are used. The first useshigh-level information regarding the driving environment (city or highway classifi-cation) to modulate between a 22 horsepower reserve for the normal mode and a 15horsepower reserve for cruise mode. The other optimization-based schedule does notuse high-level information and remains at a constant 20 horsepower reserve. Thesetwo different schedules are intended to show both the benefits and flexibility of theoptimization-based approach, while also allowing independent analysis of the ratiooptimization approach. Specifically, the variable horsepower reserve schedule demon-strates the advantage of having one meaningful design parameter, as opposed to re-tuning the entire schedule for different driving environments.

The EPA cycle consists mainly of the type of driving scenarios simulated in cases1 and 2 from Section 6.1 and 6.2. Therefore, instead of showing the various instan-taneous vehicle responses over the cycle, cumulative results are presented. Figure 13presents the total, normalized fuel consumption over the modified EPA cycle. Not sur-prisingly, the optimum-fuel schedule has the lowest total fuel consumption, and thehand-tuned schedule shows decent fuel economy also. The optimization-based sched-ule without high-level information shows increased fuel consumption for the highwayportion (second half) of the drive cycle, resulting in slightly worse overall consump-tion. However, using the high-level information to switch modes in cruise conditions


allows good overall fuel consumption to be regained. In addition, the normal mode re-serve of 22 horsepower is higher than the 20 horsepower reserve of the no high-levelinformation case.

0 200 400 600 800 1000 1200 1400 16000

0.1

0.2

0.3

0.4

0.5

0.6

time (sec)

fuel

con

sum

ptio

n

tuned

opt−fuel

arb−torque

arb−speed

reserve w/o high

reserve with high

Fig. 13. Fuel consumption comparison for modified EPA cycle.

7. CONCLUDING REMARKS

The ratio optimization problem formulation presented here systematically addressesboth fuel economy and driveability concerns for a CVT-equipped vehicle. The resultreveals a counter-intuitive solution where maximum fuel efficiency is found in tran-sient maneuvers, while driveability is gained from properly choosing a steady-stateoperating point. This conclusion can be drawn from observing that the equilibriumpower point Hp∗, representative of a constant vehicle speed for a given road incline,results in an engine speed higher than the most fuel efficient point. For engine speedsabove equilibrium which necessarily represent a transient state (vehicle acceleration),the engine is operating on a more fuel efficient curve. This is most evident in Fig-ure 8(b). However, this is both necessary and desirable because, although driveabilityis associated with transient conditions, the equilibrium operating conditions determinehow quickly and smoothly a move to a transient state can be made. That is, the currentoperating condition determines how well a move is made to a new operating condition,which is where opinions about a vehicle’s driveability are formed.

Although driveability and fuel economy are usually considered to be a trade-off, the


optimization-based solution has shown that neither needs to be significantly sacrificedto improve the other. This result is a consequence of formulating the problem as a wellposed constrained optimization problem that incorporates known and estimated engineand CVT information, equilibrium power (load), and a set of constraints that reflectthe driver’s expectations. The methodology enforces a constant horsepower reservefor different operating conditions and thus delivers consistent performance and drive-ability. The solution also results in a constant desired engine speed for small changesaround the equilibrium throttle position and helps to diminish shift-shock effects, in-creases vehicle response, and reduces unnecessary ratio variations. The horsepowerreserve formulation is a generalized approach which is not specific to the type ofCVT, vehicle, or engine, and eliminates the need for ad-hoc ratio or ratio-rate shapingby providing the designer with a meaningful and intuitive design parameter (horse-power reserve). Finally, the simulation model presented is based on and correlateswell to an actual test vehicle and therefore real-world performance should match thepredicted results on a relative/comparative basis.

REFERENCES

1. C. CHAN, T. VOLZ, D. BREITWEISER, A. FRANK, AND F. JAMZADAH, System design and controlconsiderations of automotive continuously variable transmissions, Tech. Report 840048, SAE, 1984.

2. D. C. FOLEY, N. SADEGH, E. J. BARTH, AND G. J. VACHTSEVANOS,Model identification and back-stepping control of a continuously variable transmission system, in Proceedings of the American Con-trol Conference, June 2001, pp. 4591–4596.

3. K. FUNATSU, H. KOYAMA, AND T. AOKI, Electronic control system of honda for CVT, in Proceed-ing of the International Conference on Continuously Variable Power Transmissions, September 1996,pp. 43–52.

4. K. HAYASHI, Y. SHIMIZU, S. NAKAMURA, Y. DOTE, A. TAKAYAM, AND A. HIRAKO, Neuro fuzzyoptimal transmission control for automobile with variable loads, in Proceedings of the Industrial Elec-tronics Conference (IECON), vol. 1, 1993, pp. 430–434.

5. Y. KASAI AND Y. MORIMOTO, Electronically controlled continuously variable transmission (ECVT-II), in Proceedings of the International Congress on Transportation Electronics, October 1988, pp. 33–42.

6. H. KIM, H. SONG, T. KIM, AND J. KIM, Metal belt CVT and engine optimal operation by PWMelectro-hydraulic control, in Proceeding of the International Conference on Continuously VariablePower Transmissions, September 1996, pp. 157–160.

7. S. LIU AND B. PADEN, A survey of today’s CVT controls, in Proceedings of the 36th IEEE Conferenceon Decision & Control, December 1997, pp. 4738–4743.

8. N. LUO, N. BEACHLEY, AND F. FRONCZAK, Fuel economy optimization of a hydrostatic CVT pas-senger automobile, in Proceedings of the National Conference on Fluid Power, 1986, pp. 313–320.

9. MAUGHAM, ET AL, Preliminary investigation of dilution strategies to control engine torgue duringtransient events, in ASME Fall Technical Conference Proceedings, Argonne Illinois, 2001, Paper No.2001-ICE-428, 2001, pp. 35–43.

10. H. MOZER, A. PIEPENBRINK, AND S. SOMMER, The technology of the ZF CVT – CFT 23, Tech.Report 2001-01-0873, SAE, 2001.


11. M. NISHIGAYA, T. TAMURA, H. YASUE, S. KASUGA, AND M. SUGAYA, Development of Toyota’snew “Super CVT”, Tech. Report 2001-01-0872, SAE, 2001.

12. K. OSAMURA, H. ITOYAMA, AND H. IWANO, Improvement of drive torque response by applying anintegrated control algorithm for a diesel engine and CVT, Tech. Report 20014344, JSAE, 2001.

13. , Study of an integrated diesel engine-CVT control algorithm for improving driveability andexhaust emission performance, Tech. Report 2001-01-3452, SAE, 2001.

14. S. SAKAGUCHI, I. SAKAI, AND T. HAGA, Application of fuzzy logic to shift scheduling method forautomatic transmission, in Proceedings of the IEEE Second International Fuzzy Systems Conference,vol. 1, 1993, pp. 52–58.

15. I. SAKAI, Y. ARAI, Y. HASEGAWA, S. SAKAGUCHI, AND Y. IWAKI, Shift scheduling method ofautomatic transmission vehicles with application of fuzzy logic, Tech. Report 905049, SAE, 1990.

16. SCHMIDT AND BITZENBERGER, Example of powertrain management: A vehicle with CVT, in InstMech Engrs Conference: Advanced Vehicle Transmissions and Powertrain Management, London,1997, Paper 1.

17. SERRARENS AND VELDPAUS, Dynamic modeling for dimensioning and control of a flywheel assisteddriveline, in EAEC Conference IV: Energy Saving and Emissions Reduction, Barcelona, 1999, pp. 261–268.

18. M. H. SMITH, Vehicle Powertrain Modeling and Ratio Optimization for a Continuously Variable Trans-mission, PhD thesis, Georgia Institute of Technology, 1998.

19. M. H. SMITH, J. ZHANG, W. TOBLER, AND P. KOLLER, Powertrain modeling for control of a continu-ously variable transmission using neural networks and analytical techniques, Tech. Report SR-97-165,Ford Motor Company, 1997.

20. H. SONG, T. KIM, J. KIM, AND H. KIM, Shifting characteristics of PWM electro-hydraulic controlledmetal belt CVT for the engine optimal operation, in Proceeding of the Power Transmission and GearingConference, vol. 88, September 1996, pp. 81–88.

21. S. TAKAHASHI, Fundamental study of low fuel consumption control scheme based on combination ofdirect fuel injection engine and continuously variable transmission, in Proceedings of the 37th IEEEConference on Decision & Control, December 1998, pp. 1522–1529.

22. H. VAHABZADEH AND S. M. LINZELL, Modeling, simulation, and control implementation for a split-torque, geared neutral, infinitely variable transmission, Tech. Report 910409, SAE, 1991.

23. VAN DER LAAN AND LUH, Model based variator control applied to a belt type CVT, in CVT ’99Eindhoven, 1999, pp. 105–116.

24. V. WICKE, C. J. BRACE, AND N. D. VAUGHAN, The potential for simulation of driveability of CVTvehicles, Tech. Report 2000-01-0830, SAE, 2000.

25. H. YAMAGUCHI, Y. NARITA, H. TAKAHASHI, AND Y. KATOU, Automatic transmission shift schedulecontrol using fuzzy logic, Tech. Report 930674, SAE, 1993.

26. D. YANG, Z. GUO, AND A. FRANK, Control and response of continuously variable transmission (CVT)vehicles, in Proceedings of the 1985 American Control Conference, June 1985, pp. 1438–1444.

27. M. YASUOKA, M. UCHIDA, S. KATAKURA, AND T. YOSHINO, An integrated control algorithm foran SI engine and a CVT, Tech. Report 1999-01-0752, SAE, 1999.

The Horsepower Reserve Formulation of Driveability for a Vehicle Fitted With a Continuously Variable...

Documents

Transcript of The Horsepower Reserve Formulation of Driveability for a Vehicle Fitted With a Continuously Variable...