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Fuel-optimal control of CVT powertrains
Control Engineering Practice 11 (2003) 329–336
Fuel-optimal control of CVT powertrains R. Pfiffner*, L. Guzzella, C.H. Onder Measurement and Control Laboratory, Swiss Federal Institute of Technology (ETH), Zurich, Switzerland . Received 26 October 2001; accepted 18 January 2002
Abstract Continuously variable transmissions (CVTs) offer the potential to substantially improve the part-load fuel efficiency of sparkignited engines. The control of CVTs has traditionally been designed using static arguments, i.e. by identifying the best efficiency points in the engine map for each constant power requirement and by following that curve using some heuristics as much as possible also in transients. In this paper the solution of the fuel-optimal control problem for transient conditions is presented using the numerical optimization package DIRCOL. Based on this optimal solution a simplified but causal control strategy is proposed which offers almost the same benefits. Copyright r 2001 IFAC. The paper was first presented at the third IFAC Workshop on Advances in Automotive Control, April 28–30, 2001, Karlsruhe, Germany r 2003 Elsevier Science Ltd. All rights reserved. Keywords: CVT powertrains; Fuel-optimal control; Transient operation; DIRCOL
1. Introduction Continuously variable transmissions (CVT) have now reached a technological level that permits a large-scale introduction of these devices even in the full-size passenger car class. Maximum torques of approximately 300 Nm can be handled with push belts (cf. Gesenhaus, 2000 or Goppelt, 2000). Other structures (e.g. toroidal drives), which promise to cover even larger torque ranges, have been proposed as well. Although the efficiencies of CVT are inherently lower than those of cog-wheel gear-boxes, a more efficient total-system behavior can be obtained by shifting the engine operating points for a certain demanded power towards higher loads and lower speeds. As shown in Fig. 1, the engine can produce 10 kW of mechanical brake power with efficiencies Z (defined as the ratio between mechanical power output and lower fuel heating value input, Heywood, 1988) anywhere between *Corresponding author. Bystronic Maschinen AG, Entwicklung, Industriestrasse 5, 4922 Bu¨tzberg, Switzerland. Tel.: +41-62-958-7674; fax: +41-62-958-77-33. E-mail addresses: r.pfi[email protected] (R. Pfiffner), guzzella@imrt. mavt.ethz.ch (L. Guzzella), [email protected] (C.H. Onder).
Z ¼ 0:1 and 0:33: Connecting all points at which the engine produces a certain amount of brake power between 0 and Pmax at its best efficiency yields the curve O: The engine is, however, only one part in the picture. Another part is the vehicle which defines at stationary conditions (constant vehicle speed and road level, no wind disturbance, etc.) and constant gear ratio a (parabolic) curve in the engine map. The curve L; shown in Fig. 1, is that vehicle resistance curve which is obtained for maximum gear ratio. The limits in gear ratios impose restrictions on engine efficiency as well. For instance 10 kW of power could be delivered by the engine with an efficiency of Z ¼ 0:33: However, as Fig. 1 shows, with the assumed maximum gear ratio only an efficiency of ZE0:27 can be realized. If this analysis is repeated for every possible demanded power P; the stationary realizable best efficiency curve G will be obtained. As Fig. 1 shows, for low engine speeds G will therefore be equal to L; whereas for higher engine speeds, where points on O can be realized with smaller than maximum gear ratios, G will be equal to O: In transient operating conditions the engine is not limited to operating below or on G but can be operated at any point below the maximum torque curve, e.g. on the ‘‘quasi-static’’ best efficiency curve O: In this case it
0967-0661/03/$ - see front matter r 2003 Elsevier Science Ltd. All rights reserved. PII: S 0 9 6 7 - 0 6 6 1 ( 0 2 ) 0 0 2 1 9 - 8
R. Pfiffner et al. / Control Engineering Practice 11 (2003) 329–336
330
ne O
Nomenclature a1 a2 Z Fr G H J ki L l lp lpWOT mfuel m ’ fuel
rolling friction N aerodynamic friction N=m2 s2 engine efficiency dimensionless multipliers of state constraints dimensionless road load N engine best efficiency curve dimensionless Hamiltonian dimensionless optimization criterion dimensionless torque gains Nm load torque (const. gear ratio and vehicle speeds) dimensionless adjoint variables dimensionless driver’s pedal, input dimensionless lp at wide open throttle dimensionless mass of fuel kg engine’s fuel flow kg/s
is not obvious which trajectory in the engine map will produce the best possible fuel economy for a given vehicle speed trajectory. In the literature several heuristically derived control algorithms have been proposed (see, Pfiffner, 1999 for an overview). In this paper the true optimum under dynamic operating conditions will be analyzed using the following steps. In Section 2 the system models used are introduced. In Section 3 the optimal control problem is defined. In addition a discussion of the optimization criteria that should be chosen and of the boundary conditions that have to be satisfied is included in this part. In Section 4 the theory and the numerical algorithms used to solve the problem are briefly
Λ
P=50 kW
ods oe ow rCVT rf rw s t tf ti Te ye yv ui v
engine speed rpm engine best efficiency curve (no gear limitations) dimensionless CVT speed (wheel side) rad/s engine speed rad/s wheel speed rad/s CVT drive gear ratio dimensionless final drive gear ratio dimensionless wheel radius m distance m time s final time s torque time constants s engine torque Nm engine/flywheel inertia kg m2 vehicle inertia kg m2 input signals dimensionless vehicle speed m/s
reviewed. In Section 5 a solution obtained with the proposed models and algorithms is presented. In Section 6 this solution is analyzed and a simple approximation of the optimal control algorithm is presented. In Section 7 finally, some conclusions are drawn.
2. System modeling The drive train is modeled in its longitudinal behavior only and no drive train elasticities are taken into account. Moreover, only non-zero vehicle velocities are assumed, i.e. the vehicle launch (which needs a clutch or a torque converter) is not analyzed. Fig. 2 shows a sketch of this system. Sections 2.1–2.4 describe the models of all components in detail.
180
2.1. Engine
P=20 kW
150
Ω
Te [Nm]
10 kW
0.35
0.33
0.36
120
Γ
η=0.30
90 5 kW
60
η=0.25
30
η=0.20 η=0.10
1000
2000
3000
4000
5000
6000
ne [rpm]
Fig. 1. Map of a modern 2 l SI engine. Indicated are engine isoefficiency (thin black) and iso-power curves (thin gray), engine best efficiency curve O; stationary driving resistance curve L at maximum gear ratio (thick black) and stationary realizable best efficiency operating curve G:
The ideas presented below are applicable to any kind of IC engine, i.e. to naturally aspirated or supercharged engines. As explained in the introduction, the gains obtained by using a CVT can be attributed to an increased part-load efficiency. In this paper a downsized and supercharged (DSC) engine is taken as an example because such engines are known to have an improved part-load behavior when compared to naturally aspirated engines of the same rated power (Soltic & Guzzella, 2000). Therefore, any benefits obtained by using CVTs in combination with DSC engines can be regarded as the minimum gains to be expected with naturally aspirated engines. For the pressure-wave supercharged engine, several dynamic blocks have been considered. The resulting
R. Pfiffner et al. / Control Engineering Practice 11 (2003) 329–336
Vehicle Inertia
Engine Inertia CVT
ωds
ωe
Te
331
. rCVT, rCVT
Final Drive
ωw
FR. rw
rf
θe
θv Fig. 2. Drive train structure.
ne
k2
+ -
1/τ2
-
1/τ3
-
1/τ4
lp WOT
k3 lp
+ -
+
+ + +
Te
ne
1/τ1
+
+ -
lp WOT
ne
lp WOT
Fig. 3. Engine model.
system model structure is shown in Fig. 3 where the following definitions are used: *
*
* *
lp is the load-pedal position representing the system input u1 lpWOT is the load-pedal position at wide open throttle (WOT), i.e. at the limit of the naturally aspirated part (additional torque is produced by boosting) Te is the engine torque and ne the engine speed ti is the time constants and ki the gains, as identified from measurements with the engine mounted on a dynamometer.
The first low-pass element ðt1 Þ in Fig. 3 represents the electronic throttle. The upper two first-order lags ðt2 ; t3 Þ describe the manifold dynamics for the lower torque range (essentially the naturally aspirated part with the additional receiver of the supercharger) whereas the lower path ðt4 Þ represents the boosting process. The last three low-pass elements mentioned represent essentially the ‘‘empty-and-filling’’ phenomena in the manifolds of the engine (the pressure-wave supercharger itself is very fast, Weber & Guzzella, 2000). The engine’s induction to powerstroke delay (Powell & Cook, 1987) is neglected and the fuel consumption is
calculated by integration of the representative values of a measured stationary engine map m ’ fuel ðoe ; Te Þ: Due to the special DSC engine design the resulting engine map has rather unusual efficiency characteristics, see Fig. 4. The consequences of these special efficiency characteristics will be discussed in Section 7. 2.2. CVT The CVT chosen has the same technical specification as the chain-drive CVT RHVF 154 manufactured by P.I.V. Reimers with a maximum gear ratio difference of 5:5; a maximal power of 55 kW and a maximal input torque of about 100 Nm: The gear ratio can be changed from its maximum to its minimum (and vice versa) in about 2 s: It can be modeled as an input- and stateconstrained integrator d rCVT ðtÞ ¼ u2 ðtÞ; dt
ð1Þ
where u2 is the system input corresponding to the gear ratio change rate. The gear ratio of the CVT is defined by rCVT ðtÞ ¼ ods ðtÞ=oe ðtÞ
ð2Þ
R. Pfiffner et al. / Control Engineering Practice 11 (2003) 329–336
332 70
60
Λ
e
engine torque T [Nm]
50
Γ
40
Ω 254 258
30
262
270 280
290 310 330
20
350 370 400
10
0 1000
1500
2000
2500
3000 3500 4000 engine speed ne [rpm]
4500
5000
5500
6000
Fig. 4. DSC-engine map, fuel consumption in g/kWh. Also indicated are the corresponding L; O and G as defined in Section 1.
with the state constraints rCVTmin prCVT ðtÞprCVTmax
ð3Þ
and the (state-dependent!) input constraints as modeled in (Naunheimer, 1995) rCVT ðtÞ const:pu2 ðtÞprCVT ðtÞ const:
ð4Þ
The CVT efficiency is assumed to be constant and equal to 1: Including non-ideal CVT efficiencies requires additional modeling steps and the solution of the optimization process is more complex as well. Readers interested in these aspects are referred to (Pfiffner & Guzzella, 2001). 2.3. Powertrain In addition to the CVT and the engine two additional components have to be taken into account (cf. Fig. 2). The first one is the final constant gear ratio rf ¼ ow =ods producing an overall gear ratio of r ¼ rCVT rf and the second one is the vehicle resistance force modeled as FR ¼ a1 þ a2 v2 ; where v ¼ ow rw is the vehicle speed and rw is the wheel radius. The resulting driveline dynamics then reads as follows: o ’ e ðtÞ ¼
1 ðTe ðtÞ rðtÞrw FR ðtÞ ye þ yv rðtÞ2 rðtÞrf u2 ðtÞyv oe ðtÞÞ;
ð5Þ
where ye represents the engine inertia including the primary inertia of the CVT and yv the vehicle inertia
including the wheels, the final drive and the secondary inertia of the CVT. 2.4. Combined system The complete system is a 6th-order system of nonlinear ordinary differential equations, where four state variables are associated with the engine model and two state variables with the powertrain and the CVT. The system has two inputs, namely the load pedal signal u1 and the CVT gear ratio change rate u2 :
3. Optimization criterion, boundary conditions and constraints The problem considered in this paper is the minimization of the specific fuel consumption of the vehicle modeled in Section 2 during transient system operation. Of course this objective is not the only optimization goal in a complete drive train design procedure. Other issues like pollutant emission, engine and drive train wear, etc. will have to be addressed also in this context. However, such questions are beyond the scope of this paper and the results presented below will have to be re-examined in a second step if they comply with these other design criteria. The control functions load pedal u1 ¼ lp and gear ratio change u2 ¼ r’CVT ; are to be found such that a prescribed vehicle speed trajectory within a given time range tf is achieved with a minimum amount of fuel consumed per distance traveled. In addition some
R. Pfiffner et al. / Control Engineering Practice 11 (2003) 329–336
vehicle speed constraints have to be met during this maneuver. The transient vehicle trajectory was chosen to be an acceleration part of the FTP75 driving cycle depicted in Fig. 6. The vehicle starts at a speed of 49:1 km=h and is supposed to accelerate within 10 s to a final speed of 75:3 km=h: The FTP75 driving cycle requires the vehicle speed to remain between certain upper and lower speed limits, also indicated in Fig. 6. Since the whole transient operation has to be finished in exactly 10 s; the optimal control problem investigated herein is one with a fixed final time tf ¼ 10 s: The optimization criterion was chosen to be the amount of fuel used during this transient operation relative to the distance traveled: mfuel ðtf Þ Jðu1 ; u2 ; tf Þ ¼ : sðtf Þ Thereby the total amount of fuel used is given by Z tf mfuel ðtf Þ ¼ m ’ fuel ðoe ðtÞ; Te ðtÞÞ dt
ð6Þ
ð7Þ
0
and the distance traveled can be calculated as Z tf vðoe ðtÞ; rCVT ðtÞÞ dt: sðtf Þ ¼
ð8Þ
0
This special choice of the optimization criterion J corresponds to the reference used for fuel consumption measurements during a driving cycle, where the final value is not related to the prescribed distance, but to the one actually traveled. Remark 1. Since the optimization criterion consists of two additional integrators (Eqs. (7) and (8)), the system order is increased by two. Before and after the transient operation considered, the system is assumed to operate on the best efficiency curve for stationary operation G pictured in Fig. 4, because this operation represents the fuel-optimal operation in stationary conditions, see Section 1. Thus the system states have to fulfill several boundary conditions at the beginning as well as at the end of the trajectory. The engine speed oe ; the gear ratio of the CVT rCVT and the engine torque Te are fixed at t ¼ 0 and t ¼ tf to the corresponding values which allow to operate the engine on G: The boundary conditions have to be fulfilled at the beginning ðt ¼ 0Þ and at the end ðt ¼ tf Þ: But there are also constraints which always have to be met, i.e. the state and control constraints. These constraints are physically motivated as for example the gear ratio range of the CVT, see Eq. (3), or the gear ratio change limitation of Eq. (4). Additional constraints stem from the specification of the vehicle speed trajectory as for example the condition that the actual vehicle speed vðtÞ
333
has to be in the allowed range shown in Fig. 6. Of course all these constraints have to be taken into account when solving the optimal control problem.
4. Numerical optimization approach The optimal control problem just described was solved using the numerical optimization package DIRCOL (von Stryk, 1999). This package is a collection of Fortran 77 subroutines which can be used to solve optimal control problems numerically. By discretizing the continuous time interval 0ptptf into an N-dimensional time interval 0 ¼ t1 ot2 o?otN ¼ tf
ð9Þ
and with an appropriate interpolation of the control and state variables between these time steps, the constrained infinite dimensional control problem is transformed into a sequence of finite dimensional nonlinearly constrained optimization problems (NLPs). These NLPs are solved either by the dense Sequential Quadratic Programming (SQP) method NPSOL (Gill, Murray, Saunders, & Wright, 1998b) or by the sparse SQP method SNOPT (Gill, Murray, & Saunders, 1998a). Although DIRCOL does not use the optimality conditions proposed in optimal control theory to solve the problem, it evaluates and checks it. Because relatively accurate estimates of the adjoint variables lðtÞ and the multipliers of state constraints ZðtÞ can be obtained from DIRCOL, the computation of the Hamiltonian H and the inspection of the first-order optimality conditions can be performed (von Stryk, 1993). Remark 2. The numerically obtained optimal solution fulfills the first-order necessary optimality conditions (Bryson & Ho, 1975). Especially the Hamiltonian H is negative and constant.
5. Optimal solution The solution of the optimal control problem defined in Section 3 was solved with DIRCOL. The resulting engine operation trajectory is pictured in Fig. 5. As described earlier, the system starts on the stationary best efficiency curve G: Then the engine speed is quickly raised to approximately 3500 rpm by changing the gear ratio of the CVT as fast as possible, before the engine gets back on the best efficiency curve for stationary operation G at 1 s: Actually, it gets on the ‘‘quasi-static’’ best efficiency curve O which coincides with G for engine speed values greater than 2800 rpm: After the system has reached O; it stays on this ‘‘quasi-static’’ best
R. Pfiffner et al. / Control Engineering Practice 11 (2003) 329–336
334 70
60
e
engine torque T [Nm]
50
Γ
40
1.0 s
9.6 s
254 258
Ω 30
262
270 280
290 310 330
20
10.0 s
350 370 400
10
0 1000
0.0 s
1500
2000
2500
3000
3500
4000
4500
5000
5500
6000
engine speed n [rpm] e
Fig. 5. Engine map trajectory of the optimal solution.
4000
80
3000 upper limit of v
e
prescribed v
engine speed n [rpm]
vehicle speed [km/h]
70
60
2000 lower limit of v
50
1000
40
0 0
1
2
3
4
5
6
7
8
9
10
time [s]
Fig. 6. Vehicle and engine speed traces of the optimal solution: vehicle speed (solid black) and engine speed (solid gray).
efficiency curve O for a relatively long time ð8:6 sÞ before it leaves it again to return to the final stationary operating point on G: The resulting time traces of engine and vehicle speed are depicted in Fig. 6. The optimal solution therefore consists of bringing the system to the O curve as fast as possible, and to keep the system on this curve as long as possible. During the periods when the system moves towards or away from this curve the gear ratio has to be changed with maximal possible speed while the optimal engine power trajectory is more or less constant over time.
Remark 3. It can be shown that the system is on a singular arc if the engine operates at the O curve, i.e. the Hamiltonian H becomes independent of the gear ratio change rate u2 : Thus the optimal solution above is on a singular arc between t ¼ 1:0 s and 9:6 s: The behavior of the optimal solution described above does not correspond to any classical control strategy known in the literature, cf. Pfiffner, 1999. Consequently controlling the engine at the ‘‘quasi-static’’ best efficiency curve O during transient vehicle operation cannot
R. Pfiffner et al. / Control Engineering Practice 11 (2003) 329–336
be fuel-optimal, although this has been assumed so far. Comparing the optimal solution with the classical approaches, fuel economy improvements of approximately 2.5% to 5% can be achieved, depending on the classical approach used.
335
(3) derive the engine speed and engine torque of O that corresponds to this wheel power (4) bring the engine speed and engine torque to these desired values as fast as possible (5) when the driver command is decreased, do the same but hold the engine torque at the present value as long as possible before changing it—also as fast as possible—towards the new desired value.
6. Approximation of the optimal solution The computation requirements to solve the optimal control problem on-line are too high, even with modern state-of-the-art processors. Therefore optimal trajectories for all possible acceleration maneuvers have to be computed off-line as described above and saved as lookup tables in the controller. Another possibility is to derive an approximation of the optimal solution that is simple enough to be realizable on-line. Fortunately, it is possible to derive such an approximation which shows only a degradation of approximately 0:7% from the fuel economy promised by the optimal solution. This suboptimal strategy takes advantage of the stated features of the optimal solution outlined in Section 5. As a starting point it is assumed that a suitable driver input interpretation algorithm is active which ‘‘pre-filters’’ the accelerator pedal position such that the desired vehicle acceleration is obtained. This approach comprises the following five steps: (1) filter the accelerator signal such that it ‘‘inverts’’ the power dynamics of the system (2) calculate the total wheel power that will accelerate the vehicle with this filtered accelerator signal
This suboptimal strategy needs an accelerator pedal or a sophisticated driver interpretation. Fig. 7 shows the resulting trajectory in the engine map for the acceleration part of the FTP75 driving cycle defined in Section 3 for an accelerator step. Notice that the driver command was set back to zero at approximately t ¼ 9:6 s; so that the boundary conditions at the final time tf could be fulfilled. Fortunately this approximation still fulfills all the constraints defined in Section 3. Surprisingly this approximation strongly resembles a known classical approach, the so-called single track modified control strategy (Pfiffner, 1999), but only if the load pedal is interpreted as described above and the gear ratio is always changed by its maximum rate (Pfiffner, 2001).
7. Conclusions Since classical control approaches are based on heuristic deliberations only, there exists indeed a need to investigate what the fuel-optimal control solution looks like and how much fuel consumption can be
70
60
e
engine torque T [Nm]
50
Γ
40 254 258
Ω 30
262
270 280
290 310 330
20
10.0 s
350 370 400
10
0 1000
0.0 s
1500
2000
2500
3000
3500
4000
4500
5000
5500
6000
engine speed ne [rpm]
Fig. 7. Comparison of the optimal solution (dashed) and its approximation (solid).
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improved. Thus an appropriate system model of the whole powertrain was introduced. Then the optimal control problem for CVT-based powertrains in transient conditions was defined and solved using state-of-the-art numerical optimization software. Compared to the best of the current approaches, the expected fuel economy improvements are in the range of 2.5% to 5% for the downsized and supercharged engine analyzed in this paper. Based on this optimal solution, a new simplified control strategy was derived which can realize almost the same fuel economy benefits, but with a substantially lower control complexity. It is shown that the well-known single track modified classical strategy can be adjusted such that it coincides with the simplified optimal control strategy. The improvements in fuel consumption are obtained by changing the controller software. Since no new powertrain components are needed, the costs associated with these changes are expected to be reasonably small. References Bryson, A. E., & Ho, Y.-C. (1975). Applied optimal control. Revised printing, Washington, DC: Hemisphere Publishing Corporation. Gesenhaus, R. (2000). Audi multitronic: The new generation of automatic transmission. In Proceedings of the 1st ricardo international conference, vehicle systems integration—the way ahead. Brighton, U.K. Gill, P. E., Murray, W., & Saunders, M. A. (1998a). User’s guide for SNOPT 5.3: A fortran package for large-scale nonlinear programming. Technical Report. Department of Operations Research, Stanford University, Stanford, CA. Gill, P. E., Murray, W., Saunders, M. A., & Wright, M. H. (1998b). User’s guide for NPSOL 5.0: A fortran package for nonlinear
programming. Technical Report SOL 86-1. Department of Operations Research, Stanford University, Stanford, CA. Goppelt, G. (2000). Stufenloses Automatikgetriebe Multitronic von Audi. Automobiltechnische Zeitschrift (ATZ), 102(2), 110–111. Heywood, J. B. (1988). Internal combustion engine fundamentals. New York, NY: McGraw-Hill Book Company. Naunheimer, H. (1995). Beitrag zur Entwicklung von Stufenlosgetrieben mittels Fahrsimulation. Dissertation. Universit.at Stuttgart, Germany. . Pfiffner, R. (1999). Steuerstrategien von CVT-Getrieben: Ubersicht und Ausblick. In UMSCHLINGUNGSGETRIEBE: Systemelemente der modernen Antriebstechnik (pp. 313–331). VDI-Berichte 1467. Dusseldorf, . Germany: VDI-Verlag GmbH. Pfiffner, R. (2001). Optimal operation of CVT-based powertrains. Dissertation no. 14136. Swiss Federal Institute of Technology, Zurich, . Switzerland. Pfiffner, R., & Guzzella, L. (2001). Optimal operation of CVT-based powertrains. International Journal of Robust and Nonlinear Control, 11(11), 1003–1021. Powell, B. K., & Cook, J. A. (1987). Nonlinear low frequency phenomenological engine modeling and analysis. In Proceedings of the American control conference, Vol. 1 (pp. 332–340). Minneapolis, MN. Soltic, P., & Guzzella, L. (2000). Optimum SI engine based powertrain systems for lightweight passenger cars. SAE technical paper 200001-0827, Detroit, MI. von Stryk, O. (1993). Numerical solution of optimal control problems by direct collocation. In R. Bulirsch, A. Miele, J. Stoer, & K.-H. Well (Eds.), Optimal control—calculus of variations, optimal control theory and numerical methods, (pp. 129–143). Number 111 In International series of numerical mathematics. Basel, Switzerland; Birkh.auser. von Stryk, O. (1999). User’s guide for DIRCOL. A direct collocation method for the numerical solution of optimal control problems. (Version 2.0). Technische Universit.at Munchen, . Germany, Lehr. stuhl M2 Hohere Mathematik und Numerische Mathematik. Weber, F., & Guzzella, L. (2000). Control oriented modeling of a pressure wave supercharger. SAE technical paper 2000-01-0567, Detroit, MI.
Fuel-optimal control of CVT powertrains R. Pfiffner*, L. Guzzella, C.H. Onder Measurement and Control Laboratory, Swiss Federal Institute of Technology (ETH), Zurich, Switzerland . Received 26 October 2001; accepted 18 January 2002
Abstract Continuously variable transmissions (CVTs) offer the potential to substantially improve the part-load fuel efficiency of sparkignited engines. The control of CVTs has traditionally been designed using static arguments, i.e. by identifying the best efficiency points in the engine map for each constant power requirement and by following that curve using some heuristics as much as possible also in transients. In this paper the solution of the fuel-optimal control problem for transient conditions is presented using the numerical optimization package DIRCOL. Based on this optimal solution a simplified but causal control strategy is proposed which offers almost the same benefits. Copyright r 2001 IFAC. The paper was first presented at the third IFAC Workshop on Advances in Automotive Control, April 28–30, 2001, Karlsruhe, Germany r 2003 Elsevier Science Ltd. All rights reserved. Keywords: CVT powertrains; Fuel-optimal control; Transient operation; DIRCOL
1. Introduction Continuously variable transmissions (CVT) have now reached a technological level that permits a large-scale introduction of these devices even in the full-size passenger car class. Maximum torques of approximately 300 Nm can be handled with push belts (cf. Gesenhaus, 2000 or Goppelt, 2000). Other structures (e.g. toroidal drives), which promise to cover even larger torque ranges, have been proposed as well. Although the efficiencies of CVT are inherently lower than those of cog-wheel gear-boxes, a more efficient total-system behavior can be obtained by shifting the engine operating points for a certain demanded power towards higher loads and lower speeds. As shown in Fig. 1, the engine can produce 10 kW of mechanical brake power with efficiencies Z (defined as the ratio between mechanical power output and lower fuel heating value input, Heywood, 1988) anywhere between *Corresponding author. Bystronic Maschinen AG, Entwicklung, Industriestrasse 5, 4922 Bu¨tzberg, Switzerland. Tel.: +41-62-958-7674; fax: +41-62-958-77-33. E-mail addresses: r.pfi[email protected] (R. Pfiffner), guzzella@imrt. mavt.ethz.ch (L. Guzzella), [email protected] (C.H. Onder).
Z ¼ 0:1 and 0:33: Connecting all points at which the engine produces a certain amount of brake power between 0 and Pmax at its best efficiency yields the curve O: The engine is, however, only one part in the picture. Another part is the vehicle which defines at stationary conditions (constant vehicle speed and road level, no wind disturbance, etc.) and constant gear ratio a (parabolic) curve in the engine map. The curve L; shown in Fig. 1, is that vehicle resistance curve which is obtained for maximum gear ratio. The limits in gear ratios impose restrictions on engine efficiency as well. For instance 10 kW of power could be delivered by the engine with an efficiency of Z ¼ 0:33: However, as Fig. 1 shows, with the assumed maximum gear ratio only an efficiency of ZE0:27 can be realized. If this analysis is repeated for every possible demanded power P; the stationary realizable best efficiency curve G will be obtained. As Fig. 1 shows, for low engine speeds G will therefore be equal to L; whereas for higher engine speeds, where points on O can be realized with smaller than maximum gear ratios, G will be equal to O: In transient operating conditions the engine is not limited to operating below or on G but can be operated at any point below the maximum torque curve, e.g. on the ‘‘quasi-static’’ best efficiency curve O: In this case it
0967-0661/03/$ - see front matter r 2003 Elsevier Science Ltd. All rights reserved. PII: S 0 9 6 7 - 0 6 6 1 ( 0 2 ) 0 0 2 1 9 - 8
R. Pfiffner et al. / Control Engineering Practice 11 (2003) 329–336
330
ne O
Nomenclature a1 a2 Z Fr G H J ki L l lp lpWOT mfuel m ’ fuel
rolling friction N aerodynamic friction N=m2 s2 engine efficiency dimensionless multipliers of state constraints dimensionless road load N engine best efficiency curve dimensionless Hamiltonian dimensionless optimization criterion dimensionless torque gains Nm load torque (const. gear ratio and vehicle speeds) dimensionless adjoint variables dimensionless driver’s pedal, input dimensionless lp at wide open throttle dimensionless mass of fuel kg engine’s fuel flow kg/s
is not obvious which trajectory in the engine map will produce the best possible fuel economy for a given vehicle speed trajectory. In the literature several heuristically derived control algorithms have been proposed (see, Pfiffner, 1999 for an overview). In this paper the true optimum under dynamic operating conditions will be analyzed using the following steps. In Section 2 the system models used are introduced. In Section 3 the optimal control problem is defined. In addition a discussion of the optimization criteria that should be chosen and of the boundary conditions that have to be satisfied is included in this part. In Section 4 the theory and the numerical algorithms used to solve the problem are briefly
Λ
P=50 kW
ods oe ow rCVT rf rw s t tf ti Te ye yv ui v
engine speed rpm engine best efficiency curve (no gear limitations) dimensionless CVT speed (wheel side) rad/s engine speed rad/s wheel speed rad/s CVT drive gear ratio dimensionless final drive gear ratio dimensionless wheel radius m distance m time s final time s torque time constants s engine torque Nm engine/flywheel inertia kg m2 vehicle inertia kg m2 input signals dimensionless vehicle speed m/s
reviewed. In Section 5 a solution obtained with the proposed models and algorithms is presented. In Section 6 this solution is analyzed and a simple approximation of the optimal control algorithm is presented. In Section 7 finally, some conclusions are drawn.
2. System modeling The drive train is modeled in its longitudinal behavior only and no drive train elasticities are taken into account. Moreover, only non-zero vehicle velocities are assumed, i.e. the vehicle launch (which needs a clutch or a torque converter) is not analyzed. Fig. 2 shows a sketch of this system. Sections 2.1–2.4 describe the models of all components in detail.
180
2.1. Engine
P=20 kW
150
Ω
Te [Nm]
10 kW
0.35
0.33
0.36
120
Γ
η=0.30
90 5 kW
60
η=0.25
30
η=0.20 η=0.10
1000
2000
3000
4000
5000
6000
ne [rpm]
Fig. 1. Map of a modern 2 l SI engine. Indicated are engine isoefficiency (thin black) and iso-power curves (thin gray), engine best efficiency curve O; stationary driving resistance curve L at maximum gear ratio (thick black) and stationary realizable best efficiency operating curve G:
The ideas presented below are applicable to any kind of IC engine, i.e. to naturally aspirated or supercharged engines. As explained in the introduction, the gains obtained by using a CVT can be attributed to an increased part-load efficiency. In this paper a downsized and supercharged (DSC) engine is taken as an example because such engines are known to have an improved part-load behavior when compared to naturally aspirated engines of the same rated power (Soltic & Guzzella, 2000). Therefore, any benefits obtained by using CVTs in combination with DSC engines can be regarded as the minimum gains to be expected with naturally aspirated engines. For the pressure-wave supercharged engine, several dynamic blocks have been considered. The resulting
R. Pfiffner et al. / Control Engineering Practice 11 (2003) 329–336
Vehicle Inertia
Engine Inertia CVT
ωds
ωe
Te
331
. rCVT, rCVT
Final Drive
ωw
FR. rw
rf
θe
θv Fig. 2. Drive train structure.
ne
k2
+ -
1/τ2
-
1/τ3
-
1/τ4
lp WOT
k3 lp
+ -
+
+ + +
Te
ne
1/τ1
+
+ -
lp WOT
ne
lp WOT
Fig. 3. Engine model.
system model structure is shown in Fig. 3 where the following definitions are used: *
*
* *
lp is the load-pedal position representing the system input u1 lpWOT is the load-pedal position at wide open throttle (WOT), i.e. at the limit of the naturally aspirated part (additional torque is produced by boosting) Te is the engine torque and ne the engine speed ti is the time constants and ki the gains, as identified from measurements with the engine mounted on a dynamometer.
The first low-pass element ðt1 Þ in Fig. 3 represents the electronic throttle. The upper two first-order lags ðt2 ; t3 Þ describe the manifold dynamics for the lower torque range (essentially the naturally aspirated part with the additional receiver of the supercharger) whereas the lower path ðt4 Þ represents the boosting process. The last three low-pass elements mentioned represent essentially the ‘‘empty-and-filling’’ phenomena in the manifolds of the engine (the pressure-wave supercharger itself is very fast, Weber & Guzzella, 2000). The engine’s induction to powerstroke delay (Powell & Cook, 1987) is neglected and the fuel consumption is
calculated by integration of the representative values of a measured stationary engine map m ’ fuel ðoe ; Te Þ: Due to the special DSC engine design the resulting engine map has rather unusual efficiency characteristics, see Fig. 4. The consequences of these special efficiency characteristics will be discussed in Section 7. 2.2. CVT The CVT chosen has the same technical specification as the chain-drive CVT RHVF 154 manufactured by P.I.V. Reimers with a maximum gear ratio difference of 5:5; a maximal power of 55 kW and a maximal input torque of about 100 Nm: The gear ratio can be changed from its maximum to its minimum (and vice versa) in about 2 s: It can be modeled as an input- and stateconstrained integrator d rCVT ðtÞ ¼ u2 ðtÞ; dt
ð1Þ
where u2 is the system input corresponding to the gear ratio change rate. The gear ratio of the CVT is defined by rCVT ðtÞ ¼ ods ðtÞ=oe ðtÞ
ð2Þ
R. Pfiffner et al. / Control Engineering Practice 11 (2003) 329–336
332 70
60
Λ
e
engine torque T [Nm]
50
Γ
40
Ω 254 258
30
262
270 280
290 310 330
20
350 370 400
10
0 1000
1500
2000
2500
3000 3500 4000 engine speed ne [rpm]
4500
5000
5500
6000
Fig. 4. DSC-engine map, fuel consumption in g/kWh. Also indicated are the corresponding L; O and G as defined in Section 1.
with the state constraints rCVTmin prCVT ðtÞprCVTmax
ð3Þ
and the (state-dependent!) input constraints as modeled in (Naunheimer, 1995) rCVT ðtÞ const:pu2 ðtÞprCVT ðtÞ const:
ð4Þ
The CVT efficiency is assumed to be constant and equal to 1: Including non-ideal CVT efficiencies requires additional modeling steps and the solution of the optimization process is more complex as well. Readers interested in these aspects are referred to (Pfiffner & Guzzella, 2001). 2.3. Powertrain In addition to the CVT and the engine two additional components have to be taken into account (cf. Fig. 2). The first one is the final constant gear ratio rf ¼ ow =ods producing an overall gear ratio of r ¼ rCVT rf and the second one is the vehicle resistance force modeled as FR ¼ a1 þ a2 v2 ; where v ¼ ow rw is the vehicle speed and rw is the wheel radius. The resulting driveline dynamics then reads as follows: o ’ e ðtÞ ¼
1 ðTe ðtÞ rðtÞrw FR ðtÞ ye þ yv rðtÞ2 rðtÞrf u2 ðtÞyv oe ðtÞÞ;
ð5Þ
where ye represents the engine inertia including the primary inertia of the CVT and yv the vehicle inertia
including the wheels, the final drive and the secondary inertia of the CVT. 2.4. Combined system The complete system is a 6th-order system of nonlinear ordinary differential equations, where four state variables are associated with the engine model and two state variables with the powertrain and the CVT. The system has two inputs, namely the load pedal signal u1 and the CVT gear ratio change rate u2 :
3. Optimization criterion, boundary conditions and constraints The problem considered in this paper is the minimization of the specific fuel consumption of the vehicle modeled in Section 2 during transient system operation. Of course this objective is not the only optimization goal in a complete drive train design procedure. Other issues like pollutant emission, engine and drive train wear, etc. will have to be addressed also in this context. However, such questions are beyond the scope of this paper and the results presented below will have to be re-examined in a second step if they comply with these other design criteria. The control functions load pedal u1 ¼ lp and gear ratio change u2 ¼ r’CVT ; are to be found such that a prescribed vehicle speed trajectory within a given time range tf is achieved with a minimum amount of fuel consumed per distance traveled. In addition some
R. Pfiffner et al. / Control Engineering Practice 11 (2003) 329–336
vehicle speed constraints have to be met during this maneuver. The transient vehicle trajectory was chosen to be an acceleration part of the FTP75 driving cycle depicted in Fig. 6. The vehicle starts at a speed of 49:1 km=h and is supposed to accelerate within 10 s to a final speed of 75:3 km=h: The FTP75 driving cycle requires the vehicle speed to remain between certain upper and lower speed limits, also indicated in Fig. 6. Since the whole transient operation has to be finished in exactly 10 s; the optimal control problem investigated herein is one with a fixed final time tf ¼ 10 s: The optimization criterion was chosen to be the amount of fuel used during this transient operation relative to the distance traveled: mfuel ðtf Þ Jðu1 ; u2 ; tf Þ ¼ : sðtf Þ Thereby the total amount of fuel used is given by Z tf mfuel ðtf Þ ¼ m ’ fuel ðoe ðtÞ; Te ðtÞÞ dt
ð6Þ
ð7Þ
0
and the distance traveled can be calculated as Z tf vðoe ðtÞ; rCVT ðtÞÞ dt: sðtf Þ ¼
ð8Þ
0
This special choice of the optimization criterion J corresponds to the reference used for fuel consumption measurements during a driving cycle, where the final value is not related to the prescribed distance, but to the one actually traveled. Remark 1. Since the optimization criterion consists of two additional integrators (Eqs. (7) and (8)), the system order is increased by two. Before and after the transient operation considered, the system is assumed to operate on the best efficiency curve for stationary operation G pictured in Fig. 4, because this operation represents the fuel-optimal operation in stationary conditions, see Section 1. Thus the system states have to fulfill several boundary conditions at the beginning as well as at the end of the trajectory. The engine speed oe ; the gear ratio of the CVT rCVT and the engine torque Te are fixed at t ¼ 0 and t ¼ tf to the corresponding values which allow to operate the engine on G: The boundary conditions have to be fulfilled at the beginning ðt ¼ 0Þ and at the end ðt ¼ tf Þ: But there are also constraints which always have to be met, i.e. the state and control constraints. These constraints are physically motivated as for example the gear ratio range of the CVT, see Eq. (3), or the gear ratio change limitation of Eq. (4). Additional constraints stem from the specification of the vehicle speed trajectory as for example the condition that the actual vehicle speed vðtÞ
333
has to be in the allowed range shown in Fig. 6. Of course all these constraints have to be taken into account when solving the optimal control problem.
4. Numerical optimization approach The optimal control problem just described was solved using the numerical optimization package DIRCOL (von Stryk, 1999). This package is a collection of Fortran 77 subroutines which can be used to solve optimal control problems numerically. By discretizing the continuous time interval 0ptptf into an N-dimensional time interval 0 ¼ t1 ot2 o?otN ¼ tf
ð9Þ
and with an appropriate interpolation of the control and state variables between these time steps, the constrained infinite dimensional control problem is transformed into a sequence of finite dimensional nonlinearly constrained optimization problems (NLPs). These NLPs are solved either by the dense Sequential Quadratic Programming (SQP) method NPSOL (Gill, Murray, Saunders, & Wright, 1998b) or by the sparse SQP method SNOPT (Gill, Murray, & Saunders, 1998a). Although DIRCOL does not use the optimality conditions proposed in optimal control theory to solve the problem, it evaluates and checks it. Because relatively accurate estimates of the adjoint variables lðtÞ and the multipliers of state constraints ZðtÞ can be obtained from DIRCOL, the computation of the Hamiltonian H and the inspection of the first-order optimality conditions can be performed (von Stryk, 1993). Remark 2. The numerically obtained optimal solution fulfills the first-order necessary optimality conditions (Bryson & Ho, 1975). Especially the Hamiltonian H is negative and constant.
5. Optimal solution The solution of the optimal control problem defined in Section 3 was solved with DIRCOL. The resulting engine operation trajectory is pictured in Fig. 5. As described earlier, the system starts on the stationary best efficiency curve G: Then the engine speed is quickly raised to approximately 3500 rpm by changing the gear ratio of the CVT as fast as possible, before the engine gets back on the best efficiency curve for stationary operation G at 1 s: Actually, it gets on the ‘‘quasi-static’’ best efficiency curve O which coincides with G for engine speed values greater than 2800 rpm: After the system has reached O; it stays on this ‘‘quasi-static’’ best
R. Pfiffner et al. / Control Engineering Practice 11 (2003) 329–336
334 70
60
e
engine torque T [Nm]
50
Γ
40
1.0 s
9.6 s
254 258
Ω 30
262
270 280
290 310 330
20
10.0 s
350 370 400
10
0 1000
0.0 s
1500
2000
2500
3000
3500
4000
4500
5000
5500
6000
engine speed n [rpm] e
Fig. 5. Engine map trajectory of the optimal solution.
4000
80
3000 upper limit of v
e
prescribed v
engine speed n [rpm]
vehicle speed [km/h]
70
60
2000 lower limit of v
50
1000
40
0 0
1
2
3
4
5
6
7
8
9
10
time [s]
Fig. 6. Vehicle and engine speed traces of the optimal solution: vehicle speed (solid black) and engine speed (solid gray).
efficiency curve O for a relatively long time ð8:6 sÞ before it leaves it again to return to the final stationary operating point on G: The resulting time traces of engine and vehicle speed are depicted in Fig. 6. The optimal solution therefore consists of bringing the system to the O curve as fast as possible, and to keep the system on this curve as long as possible. During the periods when the system moves towards or away from this curve the gear ratio has to be changed with maximal possible speed while the optimal engine power trajectory is more or less constant over time.
Remark 3. It can be shown that the system is on a singular arc if the engine operates at the O curve, i.e. the Hamiltonian H becomes independent of the gear ratio change rate u2 : Thus the optimal solution above is on a singular arc between t ¼ 1:0 s and 9:6 s: The behavior of the optimal solution described above does not correspond to any classical control strategy known in the literature, cf. Pfiffner, 1999. Consequently controlling the engine at the ‘‘quasi-static’’ best efficiency curve O during transient vehicle operation cannot
R. Pfiffner et al. / Control Engineering Practice 11 (2003) 329–336
be fuel-optimal, although this has been assumed so far. Comparing the optimal solution with the classical approaches, fuel economy improvements of approximately 2.5% to 5% can be achieved, depending on the classical approach used.
335
(3) derive the engine speed and engine torque of O that corresponds to this wheel power (4) bring the engine speed and engine torque to these desired values as fast as possible (5) when the driver command is decreased, do the same but hold the engine torque at the present value as long as possible before changing it—also as fast as possible—towards the new desired value.
6. Approximation of the optimal solution The computation requirements to solve the optimal control problem on-line are too high, even with modern state-of-the-art processors. Therefore optimal trajectories for all possible acceleration maneuvers have to be computed off-line as described above and saved as lookup tables in the controller. Another possibility is to derive an approximation of the optimal solution that is simple enough to be realizable on-line. Fortunately, it is possible to derive such an approximation which shows only a degradation of approximately 0:7% from the fuel economy promised by the optimal solution. This suboptimal strategy takes advantage of the stated features of the optimal solution outlined in Section 5. As a starting point it is assumed that a suitable driver input interpretation algorithm is active which ‘‘pre-filters’’ the accelerator pedal position such that the desired vehicle acceleration is obtained. This approach comprises the following five steps: (1) filter the accelerator signal such that it ‘‘inverts’’ the power dynamics of the system (2) calculate the total wheel power that will accelerate the vehicle with this filtered accelerator signal
This suboptimal strategy needs an accelerator pedal or a sophisticated driver interpretation. Fig. 7 shows the resulting trajectory in the engine map for the acceleration part of the FTP75 driving cycle defined in Section 3 for an accelerator step. Notice that the driver command was set back to zero at approximately t ¼ 9:6 s; so that the boundary conditions at the final time tf could be fulfilled. Fortunately this approximation still fulfills all the constraints defined in Section 3. Surprisingly this approximation strongly resembles a known classical approach, the so-called single track modified control strategy (Pfiffner, 1999), but only if the load pedal is interpreted as described above and the gear ratio is always changed by its maximum rate (Pfiffner, 2001).
7. Conclusions Since classical control approaches are based on heuristic deliberations only, there exists indeed a need to investigate what the fuel-optimal control solution looks like and how much fuel consumption can be
70
60
e
engine torque T [Nm]
50
Γ
40 254 258
Ω 30
262
270 280
290 310 330
20
10.0 s
350 370 400
10
0 1000
0.0 s
1500
2000
2500
3000
3500
4000
4500
5000
5500
6000
engine speed ne [rpm]
Fig. 7. Comparison of the optimal solution (dashed) and its approximation (solid).
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improved. Thus an appropriate system model of the whole powertrain was introduced. Then the optimal control problem for CVT-based powertrains in transient conditions was defined and solved using state-of-the-art numerical optimization software. Compared to the best of the current approaches, the expected fuel economy improvements are in the range of 2.5% to 5% for the downsized and supercharged engine analyzed in this paper. Based on this optimal solution, a new simplified control strategy was derived which can realize almost the same fuel economy benefits, but with a substantially lower control complexity. It is shown that the well-known single track modified classical strategy can be adjusted such that it coincides with the simplified optimal control strategy. The improvements in fuel consumption are obtained by changing the controller software. Since no new powertrain components are needed, the costs associated with these changes are expected to be reasonably small. References Bryson, A. E., & Ho, Y.-C. (1975). Applied optimal control. Revised printing, Washington, DC: Hemisphere Publishing Corporation. Gesenhaus, R. (2000). Audi multitronic: The new generation of automatic transmission. In Proceedings of the 1st ricardo international conference, vehicle systems integration—the way ahead. Brighton, U.K. Gill, P. E., Murray, W., & Saunders, M. A. (1998a). User’s guide for SNOPT 5.3: A fortran package for large-scale nonlinear programming. Technical Report. Department of Operations Research, Stanford University, Stanford, CA. Gill, P. E., Murray, W., Saunders, M. A., & Wright, M. H. (1998b). User’s guide for NPSOL 5.0: A fortran package for nonlinear
programming. Technical Report SOL 86-1. Department of Operations Research, Stanford University, Stanford, CA. Goppelt, G. (2000). Stufenloses Automatikgetriebe Multitronic von Audi. Automobiltechnische Zeitschrift (ATZ), 102(2), 110–111. Heywood, J. B. (1988). Internal combustion engine fundamentals. New York, NY: McGraw-Hill Book Company. Naunheimer, H. (1995). Beitrag zur Entwicklung von Stufenlosgetrieben mittels Fahrsimulation. Dissertation. Universit.at Stuttgart, Germany. . Pfiffner, R. (1999). Steuerstrategien von CVT-Getrieben: Ubersicht und Ausblick. In UMSCHLINGUNGSGETRIEBE: Systemelemente der modernen Antriebstechnik (pp. 313–331). VDI-Berichte 1467. Dusseldorf, . Germany: VDI-Verlag GmbH. Pfiffner, R. (2001). Optimal operation of CVT-based powertrains. Dissertation no. 14136. Swiss Federal Institute of Technology, Zurich, . Switzerland. Pfiffner, R., & Guzzella, L. (2001). Optimal operation of CVT-based powertrains. International Journal of Robust and Nonlinear Control, 11(11), 1003–1021. Powell, B. K., & Cook, J. A. (1987). Nonlinear low frequency phenomenological engine modeling and analysis. In Proceedings of the American control conference, Vol. 1 (pp. 332–340). Minneapolis, MN. Soltic, P., & Guzzella, L. (2000). Optimum SI engine based powertrain systems for lightweight passenger cars. SAE technical paper 200001-0827, Detroit, MI. von Stryk, O. (1993). Numerical solution of optimal control problems by direct collocation. In R. Bulirsch, A. Miele, J. Stoer, & K.-H. Well (Eds.), Optimal control—calculus of variations, optimal control theory and numerical methods, (pp. 129–143). Number 111 In International series of numerical mathematics. Basel, Switzerland; Birkh.auser. von Stryk, O. (1999). User’s guide for DIRCOL. A direct collocation method for the numerical solution of optimal control problems. (Version 2.0). Technische Universit.at Munchen, . Germany, Lehr. stuhl M2 Hohere Mathematik und Numerische Mathematik. Weber, F., & Guzzella, L. (2000). Control oriented modeling of a pressure wave supercharger. SAE technical paper 2000-01-0567, Detroit, MI.