Analysis and Control of a Flywheel Hybrid Vehicular Powertrain Shuiwen Shen and Frans E

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IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY, VOL. 12, NO. 5, SEPTEMBER 2004 645 Analysis and Control of a Flywheel Hybrid Vehicular Powertrain Shuiwen Shen and Frans E. Veldpaus

Abstract—Vehicular powertrains with an internal combustion Zero inertia ratio. engine, an electronic throttle valve, and a continuously variable External disturbance torque. transmission (CVT) offer much freedom in controlling the engine Vehicle road load torque. speed and torque. This can be used to improve fuel economy by operating the engine in fuel-optimal operating points. The main Desired engine torque. drawbacks of this approach are the low driveability and, possibly, Induced engine torque. an inverse response of the vehicle acceleration after a kick-down Roll resistance. of the drive pedal. This paper analyzes a concept for a novel pow- Torque in drive shafts. ertrain with an additional flywheel. The flywheel plays a part only Driver pedal angle. in transient situations by (partly) compensating the engine inertia, making it possible to optimize fuel economy in stationary situations Powertrain efficiency. without loosing driveability in transients. Two control strategies Engine speed. are discussed. The first one focuses on the engine and combines Desired engine speed. feedback linearization with proportional control of the CVT ratio. Flywheel speed. The CVT controller has to be combined with an engine torque con- Wheel speed. troller. Three possibilities for this controller are discussed. In the second strategy, focusing on control of the vehicle speed, a bifur- Desired wheel speed. cation occurs whenever a downshift of the CVT to the minimum Throttle opening. ratio is demanded. Some methods to overcome this problem are introduced. All controllers are designed, using a simple model of I. INTRODUCTION the powertrain. They have been evaluated by simulations with an advanced model. ECENT developments in design and control of vehicular Index Terms—Bifurcation, continuously variable transmission R powertrains, combined with ever tightening regulations on (CVT), flywheel hybrid powertrains, fuel economy, I/O lineariza- exhaust emissions, have prompted a renewed interest in the fuel tion, inverse behavior, nonlinear control, powertrain control. consumption of internal combustion engines (see, for instance, [9], [17], [20], [21], [24], [26], [29], [32], [33], [39], and [46]). The fuel mass flow per unit engine power in stationary situa- NOMENCLATURE tions strongly depends on the operating point, i.e., on the en- Air drag coefficient. gine speed and the torque or, alternatively, on and the Moment of inertia of engine, converter, and primary throttle opening . The fuel efficiency in stationary situations pulley. can be improved by operating the engine along the E-line, being Equivalent moment of inertia at engine side. the set of operating points in which a required engine power Moment of inertia of flywheel. is delivered with minimal fuel consumption ([11], Total moment of inertia. [18], [36], [41], [45]). Some papers [30], [33] not only take into Equivalent moment of inertia of wheel and vehicle. account the efficiency of the engine but also of other power- Equivalent moment of inertia at wheel side. train components (torque converter, transmission, etc.). In this Engine power. integrated powertrain control [4], [24], [33], [46], [49], the sta- Desired engine power. tionary operating points lie on the optimal operating line (OOL), Power at the wheels. being the set of operating points in which a required power at Desired power at the wheels. the wheels is delivered with minimal fuel consumption.1 The Overall transmission ratio. OOL will not completely coincide with the E-line. This is trivial Geared neutral ratio. for powertrains with a stepped transmission [17], [40], but is true Maximum transmission ratio. also for powertrains with a continuously variable transmission Minimum transmission ratio. (CVT) because the ratio coverage of current CVTs is fairly limited [24], [35]. Manuscript received October 9, 2001; revised December 18, 2002. Manu- The CVT and throttle controllers [1], [7], [11], [21], [35], script received in final form June 23, 2003. Recommended by Associate Editor [45], [49], aim to operate the engine in stationary situations in M. Jankovic. This work was supported by the Dutch Governmental Program points on or close to the OOL. In general, the engine speed in Economy, Ecology, and Technology (E.E.T.) S. Shen is with University of Leeds, Leeds LS2 9JT, U.K. these points is low (large CVT ratio) and the engine torque is F. E. Veldpaus is with the Dynamics and Control Technology Group, De- high (large throttle opening), meaning that the power reserve partment of Mechanical Engineering, Technische Universiteit Eindhoven, Eind- hoven 5600 MB, The Netherlands (e-mail: [email protected]). Digital Object Identifier 10.1109/TCST.2004.824792 1Sometimes [21], [27] the E-line is also called the optimal operating line.

(the difference between the power in the chosen operating point and the power at the same engine speed with a wide open throttle) is small. This can result in an unacceptable driveability, where driveability is seen as a measure for the promptness of the vehicle reaction on drive pedal motions. Suppose that, in a stationary situation, the drive pedal is suddenly kicked down completely, meaning that the driver wants the engine to deliver the maximum power as soon as possible. By opening the throttle as fast as possible, a prompt increase of the engine power of magnitude is obtained. However, a further, fast increase is possible only if the engine is speeded up quickly by a fast downshift of the CVT [41]. If is too small to realize the enforced large engine acceleration, power will be withdrawn from the vehicle to accelerate the engine [11], [24], [26], [27], [35], [41] and the vehicle will decelerate, whereas the driver clearly wants an acceleration [1], [7], [11], [24]. This inverse behavior can be avoided by a (much) slower downshift of the CVT. Then, it will take more time before the maximum engine power is delivered and before the driver feels any reaction of Fig. 1. Flywheel assisted power train. the vehicle after the pedal kick-down. Fuel-optimal powertrain controllers, therefore, in general result in an unacceptably slow gine speed increases whereas the flywheel speed decreases. The or even inverse response of the vehicle acceleration [4], [11], resulting decrease of the kinetic energy of the flywheel is partly [27], [36], [41]. This inverse response can be explained by the used to accelerate the engine. From a physical point of view occurrence of a nonminimum phase (NMP) zero in the locally it seems that the engine inertia is (partly) cancelled by the fly- linearized transfer function from the CVT ratio to the vehicle wheel inertia. Therefore, the new powertrain is called zero in- speed. This NMP zero imposes considerable limitations on the ertia (ZI), or ZI powertrain [35]. obtainable performance of the closed-loop system [8], [9], [15], The remainder of this paper is organized as follows. In Sec- [16], [29], [42], [50]. Recently, some authors have suggested tion II, a simple model for the powertrain is given. The tradeoff feedback [27] and feedforward control [4], [41] to overcome between driveability and fuel consumption is discussed in Sec- these limitations. They constrain the stationary operating points tion II-C. There also the objectives of the ZI powertrain con- to the OOL or the E-line but allow operating points outside these trollers are considered in more detail. Section IV focuses on lines in transients. However, the small power reserve then still feedback linearization and robust control with the engine speed implies an often unacceptably low driveability. as the output of interest. In Section V,the output of interest is the The driveability can be improved at the expense of increased vehicle speed. The relative degree of the system with this output fuel consumption by increasing the power reserve, i.e., by gen- is not well defined for all CVT ratios, so straightforward feed- erating the required engine power in high-speed low-torque op- back linearization is not always possible. In Section VI, some erating points (far) below the E-line. The driveability can also be methods to overcome this problem are outlined. These methods improved by incorporating a second power source in the pow- include control gain specification and approximate lineariza- ertrain. Modern hybrid electric vehicles combine a combustion tion. Finally, Section VII gives the main conclusion and some engine with a powerful electric motor and a moderate capacity suggestions for future research. battery. Unlike purely electric vehicles with their inherent drawbacks of large weight, small driving range and large recharging II. ZI SOLUTION OF DRIVEABILITY time, the hybrid electric vehicle is a very attractive concept [25], [32]. In stationary situations, the engine can operate in fuel-op- The essential components of the ZI powertrain (see Fig. 1) timal points whereas the extra power, needed to overcome the are a combustion engine, a CVT (torque converter, drive-neu- inverse response in transients, can be delivered by the electric tral-reverse (DNR) set, metal pushbelt variator, oil pump, final motor [22], [28]. The main drawbacks of hybrid electric vehi- reduction, and differential) and a power assist unit, consisting cles are their increased weight, complexity, and price. of a flywheel and a planetary gear set. The sun gear of this set is The power assist can also be delivered by a flywheel. The connected to the flywheel, the annulus gear is connected to the concepts in [12], [23], [34], [39], and [44] require a large high- primary pulley shaft via a gear box with fixed transmission ratio speed flywheel and extra clutches. Appropriate control of these , and the planet carrier is connected to the secondary pulley clutches is difficult. In this paper, the power assist unit consists shaft via a gear box with fixed transmission ratio . The sec- of a fairly small moderate-speed flywheel and a planetary gear ondary pulley is connected to the wheels via the final reduction set in parallel to a standard CVT [36], [41] and without extra and the differential. Numerical values for the powertrain param- clutches. The flywheel speed is constant if the wheel speed and eter are given in the Appendix. the engine speed are constant, meaning that the flywheel will In the next section, a simple model of the ZI powertrain is hardly influence the stationary behavior of the powertrain. If developed. This model is used later to analyze the power flow (for a constant wheel speed) the CVT is shifted down, the en- during fast changes of the CVT ratio and to study the influence SHEN AND VELDPAUS: ANALYSIS CONTROL OF FLYWHEEL HYBRID VEHICULAR POWERTRAIN 647

Fig. 2. Scheme of the ZI powertrain. Fig. 3. Engine map. of the flywheel unit on the inverse response. Finally, it will be The engine map of Fig. 3 gives some curves of constant fuel used for controller design. To evaluate the proposed controllers, mass flow per unit engine power [brake specific fuel consump- the far more realistic model from [37] will be used. However, tion (bsfc)]. Also shown is the E-line, i.e., the set of stationary this simulation model with accurate descriptions of the effi- operating points in which the delivered power is generated with ciencies of the powertrain components, flexibilities of the drive minimum bsfc. shafts, etc., will not be described in any detail in this paper. The torque converter is locked and the DNR set is in drive mode, so the primary pulley speed equals the engine speed A. The Controller Design Model . The secondary pulley speed is related to the angular The controller design model is based on simple models for wheel speed by where is the transmission the engine, the CVT, and the flywheel unit. It is assumed that the ratio of the differential plus final reduction. The pulley speeds vehicle moves along a straight line, that the DNR set is in drive are also related by , where the transmission ratio mode, and that the torque converter lock-up clutch is closed. All of the applied variator is lower bounded by and flexibilities (including those in the locked convertor and in the upper bounded by the overdrive ratio . Combination drive shafts) are neglected. A schematic representation of the of the given relations results in powertrain is given in Fig. 2. The angular speed of the engine is bounded by (3) rad/s and rad/s. In stationary situations, the engine torque is a function of and the throttle opening , where the so-called CVT ratio , i.e., the overall ratio of the so2 complete transmission between the engine and the driven wheels, is bounded by and . This (1) ratio is controlled by the clamping forces on the variator pulleys ([47]). The CVT is modeled as a first-order system with input The engine torque at speed is upper bounded by the wide and output ,so open throttle torque , see Fig. 3. At speed , each torque can be realized with (4) an appropriate throttle opening . The torque reserve in operating point is the difference be- The torques and of the push-belt on the primary and tween the maximum torque at speed and the torque secondary pulley are related by , where the in that operating point, i.e., CVT efficiency is assumed to be constant. The speed of the sun gear of the planetary set and of the (2) flywheel is a linear function of the annulus speed and the carrier speed and is given by , where In the controller design model, the time delay between a change the ratio of the annulus radius and the sun gear radius. Besides, in the throttle opening and the corresponding change in the en- is related to the primary pulley speed by gine torque is neglected, so (1) is also used in transient situ- whereas is related to the secondary pulley speed by ations. Hence, with the engine operating in a stationary point . Combination with yields , a stepwise change of the throttle opening to wide open will result in a stepwise increase of the torque from the sta- (5) tionary torque to the maximum torque at speed . with and . Hence, the flywheel The required fuel mass flow to generate a stationary is at rest for any engine speed if the CVT ratio is equal to the engine power is a function of and ,so so-called geared neutral ratio , i.e.,

(6)

2The powertrain is equipped with a drive-by-wire system. The applied ac- tuator and controller guarantee that even highly dynamical excursions of the The kinetic energies and the power losses in the flywheel unit throttle are realized with negligible errors. are small and are neglected. Therefore, the torque in the shaft 648 IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY, VOL. 12, NO. 5, SEPTEMBER 2004 between the annulus gear and the fixed gearing (see Fig. 2) with and use of (7) and (3) yields the stationary power bal- and the torque in the shaft between the planet carrier and the ance equation fixed gearing are related to the torque in the shaft between the flywheel and the sun gear by (12) where is the engine power, required to maintain the given situation, whereas is the required power at the The equations of motion for the engine side of the powertrain wheels (engine, torque converter, DNR set, primary pulley, gearing , and annulus wheel), for the wheel side (planet carrier, gearing , secondary pulley, final reduction, differential, wheels, and Combination of (12) with results in a set of vehicle inertia) and for the flywheel part (flywheel and sun gear) two equations for , and . A solution ( , , ) for are given by a given combination ( , ) is called admissible if the constraints , , and are satisfied. Any combination ( , ) with at least one admissible solution ( , , ) is called an admissible vehicle state. In general, there will be more than one admissible solution for an admissible vehicle state. One trivial is the moment of inertia of the engine side (reduced to the possibility to arrive at a unique solution is to require that the engine shaft), the moment of inertia of the wheel side (re- fuel mass flow in this state is minimal. The problem then is to duced to the drive shaft), and the moment of inertia of the determine , , and , such that for the given stationary, flywheel part. Furthermore, is the torque in the drive shafts admissible vehicle state ( , ) with wheel power to the wheels and is given by the fuel mass flow is minimized under the equality conditions and and the inequality conditions , , and . The obtained fuel-op- Finally, is the external load, consisting of the constant, timal engine speed is a function of the required engine power known rolling resistance torque , the air drag torque ,so . With with known constant , and a disturbance torque due to the optimal throttle opening can be written as a function of road slopes, wind gusts, etc. the engine speed. In summary (7) (13) Elimination of , , , , , , and and use of The optimal operating line is the set of all operating points ( , (1) results in ) for .

(8) C. Behavior After a Pedal Kick Down where , the total moment of inertia, is a function of the CVT Fuel-optimal operating points ( , ) in general combine ratio and is given by a high-torque with a low-speed and a small torque reserve . As a consequence, the behavior (9) of a vehicle with an optimally controlled nonhybrid powertrain after a drive pedal kick-down may be rated unacceptable. with equivalent moments of inertia and for the Suppose that for the state of the vehicle is stationary engine side and the wheel side and characterized by ( , ). Let , , , and be the corresponding fuel-optimal (10) ratio, engine speed, throttle opening, and engine torque. Furthermore, suppose that at time the drive pedal is kicked (11) down completely, meaning that the driver wants the vehicle to accelerate as fast as possible from wheel speed to a With the parameters from the Appendix, it follows that new higher speed. To achieve this, the throttle can be opened and completely as fast as possible, yielding a nearly instantaneous for all ,so is positive but increase from engine torque to the can change sign. maximum torque at speed . The equation of motion directly after opening the throttle can be written as B. Fuel-Optimal Operating Points Suppose that the disturbance torque is constant and equal to (14) , that the vehicle moves with constant wheel speed and that the CVT ratio is constant, so , and Then only the torque reserve or, formulated in terms of . Multiplication of this torque balance equation power, the power reserve is available to accelerate SHEN AND VELDPAUS: ANALYSIS CONTROL OF FLYWHEEL HYBRID VEHICULAR POWERTRAIN 649 the vehicle and the engine. A further fast increase of the power For the conventional vehicle this torque is negative whenever the is possible only if the engine is speeded up quickly by making CVT is shifted down, as will be the case after a pedal kick-down. large negative, i.e., by a fast downshift of the CVT. However, The so-called torque assist , defined by to avoid vehicle decelerations it follows from (14) that has to satisfy

(15) represents the influence of the flywheel. This positive torque For the conventional vehicle (no extra flywheel, so and assist partly compensates or even overcompensates the negative ) the condition reduces to conventional torque whenever and . D. Nonminimum Phase Zero From a control point of view, the initial inverse response can Clearly, this condition is not satisfied for large negative values be explained by the occurrence of a nonminimum phase (NMP) of , meaning that the desired fast downshift will result in a zero in the linearized transfer function from the transmission highly undesirable inverse response of the conventional vehicle input to the wheel acceleration . Linearization of system because this vehicle will decelerate initially whereas the driver (8) around a stationary ratio , engine speed , and throttle clearly wants an acceleration. opening , followed by Laplace transformation yields the To solve this problem, the torque reserve can be increased by transfer functions from perturbations of the throttle moving the stationary engine operating point from the OOL to a opening to perturbations of the wheel acceleration, point (far) below this line with higher speed smaller torque and from perturbations of the CVT input to and larger torque reserve but also with a (strongly) increased fuel from perturbations of the disturbance to . The function of consumption. From a fuel economy point of view, it is more interest here is . A straightforward calculation results in attractive to integrate a torque assist unit in the powertrain. In the ZI vehicle, this is materialized by the flywheel unit. For this (18) vehicle, the condition to avoid the inverse response is given by (15). For a further investigation, (11) for the moment of inertia where , the zero of the linearized conventional system, is rewritten as and the pole are given by (16) where the ratio is given by with partial derivative , respectively, , of the engine torque , respectively, the engine power (17) , with respect to . For all realistic engine operating points, is positive, meaning that the It is seen that if , meaning that the engine conventional system is nonminimum phase. For the ZI vehicle, inertia is compensated by the flywheel if . Therefore, this situation only occurs if since only then is is called the zero inertia ratio. The engine inertia is more negative. There is no zero if .For , the zero is than compensated if whereas it is negative, i.e., minimum phase. Hence, if no problems partly compensated if . Fi- are to be expected for the ZI vehicle, even not if the CVT is nally, if .Anad hoc optimization shifted down as fast as possible. of the ZI parameters [6], [37], [48] resulted in a geared neutral ratio and a zero inertia ratio with III. NONLINEAR CONTROL PHILOSOPHY and with and close to . In all states with moderate to large vehicle speeds the fuel-optimal CVT ratio The driveline management system (DMS) for the ZI power- is close to . Starting in such a state, after a pedal kick-down train has to determine setpoints for throttle and CVT ratio such initially is negative and must be smaller than a posi- that the fuel consumption is minimized without compromising tive number to avoid an inverse response. This is not a restriction driveability. The DMS also has to specify the desired state of since a large negative value for is wanted to obtain the desired the lockup clutch in the torque convertor and of the drive clutch large positive engine acceleration. in the DNR set. Here only the setpoints for the throttle and the For a more physical interpretation of the effect of the fly- CVT ratio are considered. The design of the DMS is based on wheel, (14) is rewritten as the nonlinear model, given by (4) and (8). Fig. 4 gives a skeleton of the powertrain controller. It consists of two layers. The first layer comprises the DMS with supervisor, pedal interpreter, and setpoint generator. The pedal interpreter translates the drive pedal position into a desired power where the torque is given by or desired torque at the wheels whereas the supervisor speci- fies, amongst others, the desired state of the clutches. The output of the interpreter and of the supervisor is used by the setpoint 650 IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY, VOL. 12, NO. 5, SEPTEMBER 2004

If (an estimate for) the disturbance is given, the desired wheel speed follows from

(22)

if or, if , from

(23)

The remainder of this paper concentrates on the behavior of the vehicle after pedal motions, starting at wheel speeds higher than the switching speed. For a given pedal position the desired stationary engine power then follows from . To minimize fuel consumption it is required that this power is delivered in a fuel-optimal operating point. Hence, according to (13), the desired stationary engine Fig. 4. Scheme of the powertrain controller. speed is given by , meaning that the pedal position can be translated into a desired stationary engine speed. generator to produce setpoints for the local controllers of the For the ZI vehicle, the relation between and turns out to throttle, the CVT and the clutches in the second layer. After a be approximately linear, specially for large pedal positions. short discussion on the pedal interpreter, two strategies for the DMS are suggested. These strategies are elaborated and evalu- B. Control Strategies ated in the next sections. The objective of the DMS is to determine setpoints for the CVT ratio and the throttle opening to bring the system from A. Pedal Interpretation the actual state in the desired stationary state. In literature [31], The vehicle is equipped with a drive-by-wire system, so there various laws for the throttle opening are suggested. Only three is no mechanical connection between the drive pedal and the of them will be used here. throttle and it is necessary to interpret the pedal position ( • Power law with such that if the pedal is released and if it is completely depressed) in terms of a desired powertrain quantity. An easy and intuitive (24) way is to translate into a desired stationary power at the wheels, using a relation of the form where • Torque law with such that is a strictly increasing function with and . Furthermore, is the maximum power at the wheels, so with maximum engine power . This (25) interpretation is problematical for low-wheel speeds where it is more appropriate to translate the pedal position into a de- • Fuel-optimal law with such that sired stationary wheel torque , using a relation of the form with a strictly increasing function with (26) and . The maximum torque in the drive shafts, , is limited amongst others by the maximum en- With this choice, the engine operates always in points on the op- gine torque and the maximum force that can be transmitted be- timal operating line, even in transient situations. Two strategies tween the tires and the road. For high-wheel speeds this torque for the determination of setpoints for the CVT ratio are distin- interpretation results in unrealistic large values for the desired guished. The first strategy, discussed in the next section, controls wheel power. Therefore, the torque interpretation is used if the ratio to obtain and maintain the desired engine speed with is lower than some switching speed whereas the power in- a strictly increasing wheel speed. The second strategy, aiming terpretation is used if . The transition must be con- at a smooth control of the wheel speed to the desired speed, is tinuous with respect to the wheel torque, so considered in Sections V and VI. must hold. Numerical experiments showed that the choices and result in an acceptable in- IV. ENGINE ORIENTED CONTROLLER terpretation. In summary The engine oriented CVT controller has to bring the engine if (19) speed to the desired value . The adopted controller is based on input–output linearization [19], [43] of the nonlinear model, if (20) given by (4) and (8). The output of interest is the engine speed. where the switching speed is given by Differentiating the output equation and using (4) and (8) results in (21) (27) SHEN AND VELDPAUS: ANALYSIS CONTROL OF FLYWHEEL HYBRID VEHICULAR POWERTRAIN 651

Fig. 5. Powertrain control results. (a) Engine map. (b) Engine speed. (c) Vehicle acceleration. (d) Powertrain ratio.

Because is strictly positive it makes sense to in- For each of the control laws (24), (25), and (26) the transmission troduce a new input , such that input can be determined from the combination of (28) and (30). (28) The dynamics of the considered second-order system with relative degree 1 is split in an external part, given by (27), and an where is an estimate for the external torque . The internal part, given by . To prove stability of the simple estimator from [41], based on measurements of the closed-loop system, it suffices to show that the zero dynamics wheel speed and the engine speed and on the engine torque is stable [19]. With , it follows that if the estimate from the engine management system, can be used to output tracks the desired value , so the zero dynamics is estimate . With the new input it is readily seen that

(29) The equilibrium point , therefore, satisfies

There exists a variety of control laws for , such that the output will approach the desired value , even in the presence of system uncertainties and disturbances. Here, a simple law with and the zero dynamics can be rewritten as a feedforward term and a proportional feedback term law with gain is adopted, so The gain , given by (30)

The equation for the output error then becomes3

(31) is strictly positive, so the equilibrium point is asymptotically stable. 3The earlier outlined pedal interpretation results in a desired future stationary The simulation results in the rest of this section are obtained value for the engine speed. Therefore, 3 is supposed to be zero in the sequel. with the earlier mentioned control laws applied to the advanced 652 IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY, VOL. 12, NO. 5, SEPTEMBER 2004

Fig. 6. Results for engine oriented control. (a) Engine speed. (b) Ratio. simulation model of the ZI vehicle. The disturbance torque is neglected. For , the pedal position is , corresponding to an engine speed of 125.5 rad/s and an engine power of 7.7 kW. At time s the pedal is moved to position , corresponding to a desired speed of 191.6 rad/s and a desired power of 20.25 kW. Fig. 5 gives some results for the ZI powertrain. The marks 1–3 indicate the results of, respectively, the power, torque, and fuel-optimal throttle control law. Initially the engine operating point is below the E-line because of the limitations on the CVT ratio [see Fig. 5(a)]. According to Fig. 5(b), there is a small steady-state error in the engine speed. This and the other small differences between the realized and the desired engine speed are caused by the differences between the control design model and the advanced simulation model, especially with respect to the modeling of the efficiency of the powertrain components. From Fig. 5(b), it also follows that for s the engine is at its final speed and no power is needed anymore to accelerate Fig. 7. Vehicle acceleration. the engine. A very small part of the engine power is used then to accelerate the ZI flywheel whereas the rest is available for trol law (30). Two values of are chosen, i.e., s for the vehicle. From the same figure it is seen that the different the fast case and s for the slow case. From Fig. 6(b), throttle control laws result in almost the same course of the en- it is seen that the initial down shift of the CVT ratio for the con- gine speeds. The reason is that the engine speed, commanded ventional powertrain (CCDL in the plots) is marginally faster by (29) and (30), does not depend on the applied throttle con- than for the ZI powertrain (ZIPT in the plots). The reason is trol law. However, as can be seen from Fig. 5(c), the different that the equivalent moment of inertia of the wheel side throttle control laws yield quite different vehicle accelerations. of the ZI powertrain is somewhat larger than the corresponding The power law produces the largest accelerations whereas the moment of the conventional powertrain if . fuel-optimal law results in the smallest ones. This can also be The results in Fig. 7 clearly demonstrates the initial inverse concluded from Fig. 5(a), where it is seen that the power law response of the conventional vehicle in the fast case: the acceler- uses the most of the torque reserve. As a consequence, the CVT ation is negative for s until s. Furthermore, it is ratio shift with the power law is a little bit less than with the seen that in the slow and also in the fast case, it takes 0.5 s before torque and the fuel-optimal law [see Fig. 5(d)]. the conventional vehicle accelerates in the desired direction. For To get an idea of the effect of the extra flywheel, some simu- the ZI powertrain the engine and the vehicle are boosted by the lations are performed with the fuel-optimal throttle control law, power from the flywheel unit. For a faster downshift (increasing applied to the model for the conventional vehicle. This model ), the power flow is larger but can be delivered only during a originates from the ZI simulation model after substitution of shorter time interval since the kinetic energy of the flywheel is . The realized engine speeds for the ZI powertrain (solid fairly limited. After reaching the desired engine speed, the con- lines in Fig. 6) are very similar to those of the conventional ventional vehicle accelerates somewhat faster than the ZI ve- powertrain (dotted lines). Again, this is not surprising since the hicle because then some engine power is needed to accelerate course of the engine speed is governed by the gain in the con- the flywheel. SHEN AND VELDPAUS: ANALYSIS CONTROL OF FLYWHEEL HYBRID VEHICULAR POWERTRAIN 653

Fig. 8. Results with fuel-optimal law for the throttle. (a) Torque at wheels. (b) Speeds.

From the given results, it may be concluded that the drive- a proportional feedback term with gain , and a differential ability (seen as a measure for the vehicle reaction on drive pedal feedback term with gain is used, so motions) of the ZI vehicle is much better than that of the conventional vehicle. Besides, it turns out that large values of the (35) gain are undesirable for the ZI vehicle to avoid large jerks and for the conventional vehicle to avoid serious inverse responses. The output error then follows from:

V. V EHICLE ORIENTED CONTROLLER (36) In the vehicle oriented CVT controller, a reference for the wheel speed is determined from the desired wheel power, using the first-order filter The differential feedback term in (35) is not necessary to guarantee stability. However, as can be seen from the error equation, this term is helpful in reducing the effect of, e.g., model errors (32) and external disturbances. The control law for the transmission input follows from with the initial condition . The controller (33) and (35). This law requires an estimate for the external aims to make the actual wheel speed equal to the reference speed torque and, if , also for the wheel acceleration. The ear- in order to realize a smooth transition from the initial speed lier mentioned estimator from [41] can be used for this purpose. to the desired final speed . With the proposed filter To evaluate the proposed control law for , simulations are for the reference speed, the actual wheel power will con- performed in which the pedal position and the desired power at verge to the desired value if converges to . In the final the wheels change from and kW for state the engine has to operate in a fuel-optimal operating point. sto and for s. The Like the engine oriented controller, the vehicle oriented final value of the desired power is fairly low to guarantee that CVT controller is based on input–output linearization, but now the CVT ratio will remain larger than for s. with the wheel speed as the output of interest. Thus, with The results in Fig. 8(a) for the driving torque at the wheels as the input, the system (8) is already in the desired form for are presented to emphasize that the value of the gains and input–output linearization. It follows that the relative degree is in (35) is very important. The solid line, marked “With ,” 1if , i.e., if the CVT ratio differs from the zero is determined with the s and whereas inertia ratio . This will be assumed in the rest of this section. both gains are zero for the line, marked “Without .” The given The case where is investigated in Section VI. values for the gains are fairly arbitrary. Fine tuning is desired but If , it makes sense to introduce a new input , such is not a subject of this paper. The results for the wheel speed in that Fig. 8(b) are obtained with the gains s and . This figure shows that goes ahead of . The reason is that (33) the adopted tire model in the simulation model requires a certain amount of slip between the tire and the road to produce the force and to rewrite the input–output (8) as to propel the vehicle. (34) The results in Fig. 9 show that the power law (24), marked with 1, the torque law (25), marked with 2, and the fuel-optimal The objective is to find a law for , such that will track the law (26), marked with 3, yield practically the same results for reference . Here a simple law with a feedforward term , the torque and the power at the wheels. However, the power law 654 IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY, VOL. 12, NO. 5, SEPTEMBER 2004

Fig. 9. Results for different throttle control laws. (a) Engine map. (b) Torque at wheels. (c) Power at wheels. (d) Powertrain ratio. obviously cannot ensure that in the final stationary state the en- where is a small positive number. Otherwise, the power gine operates in a point on the optimal operating line. This also law is replaced by follows from a closer examination of the zero dynamics. The internal dynamics of the controlled system is described by (4) and (32) after substitution of the control law for . Substitution of (perfect tracking, ) and of the assumption The parameter controls the speed of convergence to the in the internal dynamics relations results in the zero optimal operating line. The results of the modified strategy in dynamics, i.e., Fig. 10 show that the operating point now indeed converges to this line. The relation for the desired engine power neglects the power required to accelerate the engine and the flywheel. To compensate for this inertia effect, the relation for is modified into With (23) for and (7) for , it follows that the equilibrium point of the zero dynamics system is given by and a solution for of This modification is meaningful if the power law or the torque law are used to control, but not if the fuel-optimal law is used. The results in Fig. 11 are obtained with the torque law. With If the torque law (25) or the fuel-optimal law (26) are used to this inertia compensation, the engine delivers a somewhat larger control the throttle, then the solution for is unique. In both torque whereas the smallest CVT ratio is somewhat larger. The laws it is guaranteed that, if the desired power is delivered, it modification has hardly any influence on the power and torque is delivered in an operating point on the optimal operating line. at the wheels nor on the wheel speeds. This is not the case for the power law because then any value Several strategies to control the throttle and the CVT are dis- of , such that represents an operating point on the cussed in this section. The behavior of the engine and of the isopower line , is a solution. To overcome transmission is quite different for the various strategies but the this problem, the power law (24) is used only if torque and the power at the driven wheels and the speed of these SHEN AND VELDPAUS: ANALYSIS CONTROL OF FLYWHEEL HYBRID VEHICULAR POWERTRAIN 655

Fig. 10. Modified control results. (a) Engine map. (b) Ratio.

Fig. 11. Inertia compensation. (a) Engine map. (b) Ratio. wheels show nearly the same response for each of these strate- (38) gies. It is emphasized that the results in this section are valid only if at each moment the CVT ratio is greater than the zero inertia ratio . (39) VI. BIFURCATION AND ITS CONTROL Straightforward input–output linearization of the first-order An obvious choice for the function is to replace in single-input–single-output (SISO) system the neighborhood of by a linear function of , such that with input and output is not applicable if changes sign in the control interval. Then a bifurcation can occur. This is the if (40) case for the system in Section V, since for . if One possible approach to solve this problem is to replace the term in the ratio control law (33) by a function , where is a small positive-constant and , that is defined and continuous for all . Then, so . assuming , using control law (35) for the input There are two equilibrium points for the controlled system. and (32) for the reference speed (but now with instead In the first point , and therefore, . Furthermore, of ) and again denoting the output error by , the it follows from (37) that controlled system is described by

whereas (38) results in

(37) 656 IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY, VOL. 12, NO. 5, SEPTEMBER 2004

Fig. 12. Bifurcation around r`r . (a) Engine map. (b) Power at wheels. (c) Powertrain ratio.

From these relations, the error and the reference speed 0 and remain equal for some time. Obviously, the CVT cannot in this first equilibrium point can be determined as soon as the shift down to ratios below . The reason is that a bifurcation (modified) control law for the throttle is specified. [38] takes place for . For a further investigation, (39) In the second equilibrium point , meaning that is linearized around . Substitution of ,of and . Furthermore, it follows that: according to (40), and of results in

(41) and, finally, that It turns out that the gain is positive until s, meaning that and that remains equal to at least until s. These results, combined with any of the earlier given (modified) In the period form supto s, the engine speed, the laws for the throttle, imply that in the second equilibrium point engine power, and the power at the wheels increase but it is not and . possible to influence the power flow in the powertrain by the An analysis of the stability of these equilibrium points learns CVT. At s the gain becomes negative, meaning that the that the second equilibrium point is locally asymptotically stable is no longer a stable solution of (41). As a consequence, whereas the first point is unstable. after s the CVT ratio can increase to the desired level. Fig. 12 gives some results, obtained with the given function The main reason for the observed behavior of the CVT ratio for a desired stepwise change of the power at the wheels is that the function changes sign for . This problem from 6.4 kW for s to 53.5 kW for s. Two values of can be shunned by choosing a function that is strictly neg- are considered, being (corresponding to ) ative or strictly positive for all . This choice and (corresponding to ). The gain in strongly influences the rate of ratio change and the accelera- the control law (35) is equal to 0. The torque law (25) is used tion of the driven wheels. Amongst others, it has to be guaran- to control the throttle. Initially, (because ) teed that this acceleration is nonnegative after a kick down of and , so the CVT will shift down. At s the ratio the drive pedal. It is noted that may be quite large if is much becomes equal to and and become equal to larger than and that should be low if is close to or smaller SHEN AND VELDPAUS: ANALYSIS CONTROL OF FLYWHEEL HYBRID VEHICULAR POWERTRAIN 657

Fig. 13. New control gain to overcome bifurcation. (a) Engine speed. (b) Power at wheels. (c) Powertrain ratio. than . In a first attempt, is chosen as a continuous func- decreases but also results in more oscillations and a larger over- tion, given by shoot. To improve the performance, the gain is decreased from to and the ratio control law (33) is modified (42) in two ways. First, the term is replaced not by ,but by with according to (42), (43), (44), and de- if ,by fined by (43) if and by a linear function of if , i.e., where is a positive constant, and . The result of this modification is that the CVT will shift speed (44) is decreased for . Second, to amplify the influence of the engine torque, the term is multiplied by a factor . The resulting control law for the Both and are positive, so is negative for all CVT ratio then follows from: . Some results for the closed-loop system, using this with and using the torque law (25) for the throttle are given in Fig. 13. Now the CVT can shift to lower ratios than be- Because is positive, is strictly positive and is strictly cause the bifurcation is removed. The stationary error in the en- negative, it is easy to prove the stability of the equilibrium point gine speed in Fig. 6(a) is caused by the differences between the of the closed-loop system. Fig. 14 presents some results for the control design model and the simulation model. From Fig. 6(b), modified control law with and . Strategy 1 it is seen that the response of the power at the wheels now has means and , strategy 2 refers to a peak at s and oscillates with a large overshoot in the and , whereas strategy 3 means and . period from sto s. Increasing the proportional Comparing these results with those in Fig. 6(b), it can be con- gain in the control law (35) results in a decrease of the peak cluded that the improvement is significant. Small values of 658 IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY, VOL. 12, NO. 5, SEPTEMBER 2004

Fig. 14. New control gain with better performance. (a) Engine speed. (b) Power at wheels. reduce the peak in the response of the power at the wheels but with and given by also produce more overshoot. Large values of not only improve the peak response but also reduce the overshoot, however at the expense of a slower system response. It must be noted that the given values for , and are the result of some trial and error, not of a more or less systematic optimization. The second method to overcome the problems, caused by the where in all relevant situations. Again in agreement fact that the relative degree is not well defined for , with [14], the control law is chosen as is based on approximate linearization, e.g., [10] and [14]. It is assumed that . For simplicity, only the torque law (47) (25) is used to control the throttle and it is assumed that at each moment with and With the choice for the third new variable the . The error between the reference wheel speed closed-loop system is described by and the actual wheel speed is seen as the output of interest. According to [14], a coordinate transformation from , and to , , and is introduced with the output of interest, so The factor in the control law (47) is introduced here in con- (45) travention of [14] to get more freedom in controlling the shift speed of the CVT. Fig. 15 gives some the results, obtained with Differentiation of this relation with respect to time results in this control law with and . Furthermore, four values of , being , , , and , are used. The initial conditions of the system and the desired final state are the same as in the earlier simulations. The realized power at the wheels in Fig. 15(b) is quite similar to that in Fig. 14. where the functions and are given by It turns out that the peak in the response of the power at the wheels can be decreased by increasing the control gain at the expense of a slower response and an increasing the stationary error. Fine tuning of the control parameters , , and to significantly improve the performance has not been a topic of The relative degree is not well defined in because then research and will be very time consuming. Nevertheless, it may . In agreement with [14], the new variable is defined be concluded that the approximate linearization method is very by suitable to control the ZI powertrain without bifurcation problems. (46) VII. CONCLUSION Differentiation of this relation with respect to time results in A new concept for a CVT powertrain, called the ZI powertrain, is presented and analyzed. The extra flywheel in this powertrain can resolve the driveability problems of conventional powertrains that are controlled to maximize fuel economy. It is SHEN AND VELDPAUS: ANALYSIS CONTROL OF FLYWHEEL HYBRID VEHICULAR POWERTRAIN 659

Fig. 15. Results for approximated linearization. (a) Ratio. (b) Power at wheels. shown that the inverse behavior of the acceleration of a conven- ACKNOWLEDGMENT tional vehicle after a pedal kick down, due to the occurrence of This study is part of EcoDrive, a joint project of Van Doorne’s the nonminimum phase zero in the linearized transfer function Transmissie B.V., The Netherlands Organization for Applied of the rate of ratio change to this acceleration, is eliminated in Scientific Research, and the Technische Universiteit Eindhoven. the ZI vehicle. To realize the desired power at the wheels, specified by the pedal interpreter, two types of controllers are designed. The first type focuses at the control of the engine speed. REFERENCES Feedback linearization combined with simple techniques from [1] C. Chan, T. Volz, D. Breitweiser, A. Frank, F. S. Jamzadah, and T. linear control theory suffice to obtain the desired behavior of Omitsu, “System Design and Control Considerations of Automotive Continuously Variable Transmissions,” in Proc. Int. Congr. Exposition, this speed. The controllers of the second type concentrate on Detroit, MI, 1984, SAE-Paper 840048. the wheel speed. Feedback linearization is more problematic be- [2] V. H. L. Cheng and C. A. Desoer, “Limitations on the closed-loop cause the relative degree is not well defined then. This results transfer function due to right-half plane transmission zeros of the plant,” IEEE Trans. Automat. Contr., vol. TAC–25, pp. 1218–1220, June 1980. in a bifurcation if the transmission ratio becomes equal to the [3] D. Cho and J. K. Hedrick, “Automotive powertrain modeling for con- so-called zero inertia ratio. Parameter specification and approx- trol,” ASME J. Dyn. Sys. Meas. Contr., vol. 111, pp. 568–576, 1989. imate linearization are conducted to overcome the problems, as- [4] M. Deacon, C. J. Brace, N. D. Vaughan, C. R. Burrows, and R. W. Horrocks, “Impact of alternative controller strategies on exhaust emis- sociated with the bifurcation. sions from an integrated diesel/continuously variable transmission pow- The proposed controllers force the engine to work in fuel-op- ertrain,” in Proc Inst. Mech. Eng. D, Transp. Eng., vol. 213, 1999, pp. timal operating points in stationary situations. In transient situa- 95–107. [5] R. M. van Druten, B. G. Vroemen, P. C. J. N. Rosielle, F. E. Veldpaus, tions they use the kinetic energy of the extra flywheel to greatly and W. J. M. Schouten, “Dynamic modeling for dimensioning and con- improve the behavior of the ZI vehicle compared to an otherwise trol of a flywheel assisted driveline,” in Proc. CVT Conf., Eindhoven, identical vehicle without this flywheel. Although the controllers The Netherlands, 1999, pp. 231–237. [6] R. M. van Druten, “Transmission design of the zero inertia powertrain,” are designed for the ZI powertrain, they can also be applied to Ph.D. dissertation, Tech. Univ. Eindhoven, Eindhoven, The Netherlands, conventional powertrains. 2001. [7] F. A. Frank and P.B. Pires, “A high torque, high efficiency cvt for electric vehicles,” in Elect. Veh. Design Development, 1991, SAE-Paper 910251. [8] J. S. Freudenberg and D. P. Looze, “Right half poles and zeros and de- APPENDIX sign tradeoffs in feedback systems,” IEEE Trans. Automat. Contr., vol. NOMINAL PARAMETERS TAC–30, pp. 555–565, June 1985. [9] J. S. Freudenberg, R. H. Middleton, and A. Stefanopoulou, “A survey 0.0113 Nms . of inherent design limitation,” in American Contr. Conf. Workshop Tu- 0.224 kgm . torial, vol. 5, Chicago, IL, 2000, pp. 2987–3001. [10] R. Ghanadan and G. L. Blankenship, “Adaptive control of nonlinear sys- 135.43 kgm . tems via approximate linearization,” IEEE Trans. Automat. Contr., vol. 0.4 kgm . 41, pp. 618–625, Apr. 1996. 10 s . [11] L. Guzzella and A. M. Schmid, “Feeback linearization of spark-ignition engines with continuously variable transmission,” IEEE Trans. Contr. 0.0915. Syst. Technol., vol. 3, pp. 54–60, Jan. 1995. 0.4998. [12] L. Guzzella, C. Wittmer, and M. Ender, “Optimal operation of drive 0.1227. trains with SI-engines and flywheels,” in IFAC 13th World Congr., San Francisco, CA, 1996, pp. 237–242. 0.1976. [13] I.-J. Ha, A. K. Tugcu,˘ and N. M. Boustany, “Feeback linearization con- 59.93 Nm. trol of vehicle longitudinal acceleration,” IEEE Trans. Automat. Contr., 0.9578. vol. 34, pp. 689–698, July 1989. [14] J. Hauser, S. Sastry, and P. Kokotovic, “Nonlinear control via approx- 7.8. imate input–output linearization: the ball and beam example,” IEEE 0.87. Trans. Automat. Contr., vol. 37, pp. 392–398, Mar. 1992. 660 IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY, VOL. 12, NO. 5, SEPTEMBER 2004

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He re- etrain With a CVT,” SAE, Tech. Paper 960233, 1996. ceived the B.S., M.S., and Ph.D. degrees in mechan- [29] R. H. Middleton, “Trade-offs in linear control system design,” Auto- ical engineering from the Jilin University of Tech- matica, vol. 27, no. 2, pp. 281–292, 1991. nology, Jilin, China, in 1990, 1992, and 1996, respec- [30] A. Numazawa et al., “Automatic transmission optimization for better tively. fuel economy,” in Proc. FISITA Conf., 1978. He worked at the Shanghai Jiaotong University, [31] R. A. J. Pfiffner, “Optimal operation of CVT-based powertrains,” Ph.D. China as a Post-Doctororal Fellow from 1996 to dissertation, Swiss Federal Inst. Technol., Zürich, Switzerland, 2001. 1998, as a Guest Lecturer at Yokohama National [32] B. K. Powell, K. E. Bailey, and S. R. Cikanek, “Dynamic modeling and University, Japan, from 1998 to 1999, and as a control of hybrid electric vehicle powertrain systems,” IEEE Control Research Fellow at the Eindhoven University of Syst. Mag., vol. 18, no. 5, pp. 17–23, 1998. Technology, The Netherlands, from 1999 to 2001. [33] S. Sakaguchi, E. Kimura, and K. Yamamoto, “Development of an En- He currently works as a Research Fellow at the University of Leeds, Leeds, gine-CVT Integrated Control System,” in Proc. Int. Congr. Exposition, U.K. His research interests include modeling and control of advanced and Detroit, MI, Mar. 1999, SAE-Paper 1999-01-0754. hybrid vehicular power trains, steering control, vehicular handling and, more [34] A. Schmid, P. Dietrich, S. Ginsburg, and H. P. Geering, “Controlling a general, robust and nonlinear control and their applications to complex CVT-Equipped Hybrid Car,” SAE, Tech. Paper 950 492, 1995. mechanical systems. [35] A. F. A. Serrarens and F. E. Veldpaus, “Dynamic modeling for dimensioning and control of a flywheel assisted driveline,” in Proc. EAEC Conf., Barcelona, Spain, 1999, pp. 261–268. [36] , “New concepts for power transients in flywheel assisted drivelines with a CVT,” presented at the FISITA World Automotive Conf., Seoul, Korea, 2000, [CD-ROM] Paper nr F2000A129. Frans E. Veldpaus was born on July 1, 1942. He [37] A. F. A. Serrarens, “Coordinated control of the zero inertia powertrain,” received the M.S. and Ph.D. degrees from the Tech- Ph.D. dissertation, Tech. Univ. Eindhoven, Eindhoven, The Netherlands, nische Universiteit Eindhoven (TU/e), Eindhoven, 2001. The Netherlands, in 1966 and 1973, respectively. [38] R. Seydel, Practical Bifurcation and Stability Analysis: From Equilib- Since 1996, he has worked in the Department of rium to Chaos. New York: Springer-Verlag, 1994. Mechanical Engineering, TU/e, first in the area of [39] E. Shafai, A. Schmid, and H. P. Geering, “Torque Pedel for a Car With structural dynamics and solid mechanics and then in a Continuously Variable Transmission,” Transmission Drivelines, ser. the field of control of mechanical systems. His main SAE Special Publication, vol. 1032, 1994. research activities concern modeling and control of [40] S. Shen, H. Tanaka, and Y. Sato, “Simulating a CVT vehicle economy automotive systems, i.e., of active wheel suspensions with the method of an engine universal performance map,” in Proc. 15th and power trains in general and of automotive com- Int. Combustion Engine Symp., Seoul, Korea, 1999. ponents like clutches and continuously variable transmissions in particular.