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Improving Continuously Variable Transmission Efficiency With Extremum Seeking Control Stan van der Meulen, Bram de Jager, Frans Veldpaus, Erik van der Noll, Francis van der Sluis, and Maarten Steinbuch
Abstract—The design of an efficiency optimal controller for the variator in a vehicular continuously variable transmission is studied. A conventional controller aims at tracking a prescribed reference for the transmission ratio and at preventing damage, but does not address efficiency. Sufficiently accurate models for the efficiency as function of the clamping forces are not available, whereas measurement of the efficiency requires extra sensors. In this brief, a controller is proposed that improves the efficiency without needing extra sensors. The maps between the clamping forces (input) and the efficiency or the speed ratio (output) are studied with test rig experiments. These maps exhibit a maximum, but the location of this maximum is uncertain. So, an extremum seeking controller is developed. This controller can adapt the input to maximize the output, without needing a model. Experiments show that this approach is feasible and that a conventional controller is outperformed. A robustness analysis for disturbances indicates that these are effectively handled. Index Terms—Continuously variable transmissions (CVTs), experiments, extremum seeking control (ESC), nonlinear systems, robustness.
I. INTRODUCTION
C
URRENT automotive research aims at reducing emissions and fuel consumption, e.g., with advanced transmissions, like the continuously variable transmission (CVT). A CVT realizes any transmission ratio within a finite range and changes this ratio without interruption of the power transfer from the engine to the wheels. Furthermore, the overall efficiency of a CVT driveline improves, because the demanded engine power can be generated in fuel optimal operating points [1]. This brief focuses on the efficiency of the CVT itself. The CVT encloses a torque converter, a drive-neutral-reverse set, a variator, an actuation system, and a final drive. The variator consists of a metal V-belt (the pushbelt), a primary pulley, and a secondary pulley. One of the conical sheaves of each pulley is axially fixed, whereas the other can move in the axial
Manuscript received November 10, 2010; revised May 11, 2011; accepted May 20, 2011. S. van der Meulen, B. de Jager, F. Veldpaus, and M. Steinbuch are with the Department of Mechanical Engineering, Control Systems Technology Group, Eindhoven University of Technology, 5600 MB Eindhoven, The Netherlands (e-mail: [email protected]; [email protected]; [email protected]; [email protected]). E. van der Noll and F. van der Sluis are with the Department of Advanced Engineering, Bosch Transmission Technology, 5000 AM Tilburg, The Netherlands (e-mail: [email protected]; [email protected]). Color versions of one or more of the figures in this brief are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TCST.2011.2160980
direction. Changes of the transmission ratio are achieved by simultaneous adjustment of the clamping forces on the movable sheaves. This results in an axial displacement of these sheaves and, due to this displacement, in a change of the running radii of the pushbelt on the pulleys and therefore also of the transmission ratio. Although electromechanical [2] and electrohydraulical [3] actuation systems exist, the majority of CVTs are actuated by a hydraulic system, which is the system considered here. The main sources of power loss in a CVT are found in the variator and its actuation system [4]–[7]. There is a strong relation between these losses and the magnitude of the clamping forces. These forces have to be large enough to transfer a given torque without excessive slip between the pushbelt and the pulleys. When, starting at a high level, the clamping forces are decreased, the slip increases and eventually a transition of the variator from open loop stable (micro-slip) to open loop unstable (macro-slip) will occur. For decreasing clamping forces, the variator efficiency initially increases, subsequently reaches a maximum, and ultimately decreases. The existence of this maximum is shown in [8] with experiments. It occurs within the open loop stable region [9]. Recent research [10] has shown that the variator can withstand substantial levels of slip without an unacceptable reduction of its life span. This leads to the idea to improve the efficiency by reducing the clamping forces to a level for which the given torque can be transferred with an acceptable level of slip. A straightforward approach is to determine a slip reference [11, Sec. 7.2] that corresponds to the maximum efficiency and to control the slip to track this reference [12]. To determine a usable estimate for the slip reference is problematic, because the available variator models are not reliable nor accurate enough. This leads to alternative approaches, which aim at maximizing the efficiency of the variator without slip control. Extremum seeking control (ESC), see [13, Sec. 13.3] and [14], is a control method that exploits the presence of a maximum in an input-output map by adapting the input (here: the clamping forces) to maximize the output (here: the efficiency). Normally, the torques at the pulleys are not measured, so determination of the efficiency and controlled manipulation of the clamping forces to maximize the efficiency is not feasible. From experiments it is known that the input-output map from clamping force to speed ratio (the ratio of the secondary and primary angular velocity) also has a maximum [9] and that the argument of this maximum is close to the argument for which the efficiency is maximal. Hence, near optimal efficiency will be obtained if ESC is used to control the variator to the maximum of the clamping force versus speed ratio map, see [15], [16], [21], and [22]. This approach is detailed, experimentally
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(4a) (4b) where and are the pressure surfaces of the hydraulic cylinders, whereas and are centrifugal coefficients. Based on first principles models and many experiments [8], it is concluded that the torques and are functions of geometric ratio , primary angular velocity , speed ratio (or relative slip ), and clamping forces and , i.e., (5a) (5b) Fig. 1. Schematic illustration of pushbelt variator.
evaluated, and tested for robustness in this brief, which are its main contributions. Definitions and control objectives are given in Section II. The experimental setup is introduced in Section III and the feasibility of the control objectives is shown in Section IV. ESC is described in Section V, together with closed loop experiments. The robustness of ESC against disturbances is analyzed in Section VI. A discussion in Section VII concludes this brief.
Torque cannot be transferred without slip, so, for . For the rate of change of the speed ratio holds
,
(6) Some commonly used models for , i.e., the Carbone Mangialardi Mantriota (CMM) model, the Ide model, and the Shafai model, are described in [11, Sec. 2.5]. It is assumed that for stationary situations, i.e., for , the clamping force can be solved from (6) in the form
II. DEFINITIONS AND CONTROL OBJECTIVES
(7)
This section introduces the notation, provides the physical background, and states the control objectives. Consider Fig. 1, where the primary (input, subscript “ ”) side and the secondary (output, subscript “ ”) side of the variator are depicted. The torques are denoted by and , the angular velocities by and , the clamping forces by and , and the axially movable sheave positions by and . Furthermore, denotes half the pulley wedge angle, which is equal to 11 deg. Deformations of the variator are neglected, so the path of the pushbelt on the pulleys is a circular arc with the running radii and . The geometric ratio , the speed ratio , and the relative slip (see, e.g., [8]), are defined by
where is the clamping force ratio. Using (5b) and (7) the relation for the torque can be written as (8) where
the
secondary traction coefficient satisfies for . It is also assumed that the speed ratio can be solved from (6) with , resulting in (9)
(1)
With this relation and (3), (5a), and (7), it follows that the efficiency will be a function of , , , and , so
(2)
(10)
The variator can realize any transmission ratio in the range from to . The efficiency of the variator is given by (3) with the input power and the output power of the variator. The required power for the hydraulic actuation system is not accounted for in this definition, so is not the efficiency of the complete CVT. The clamping forces and on the movable sheaves scale with the oil pressures and in the hydraulic cylinders of the pulleys. Centrifugal effects and a preloaded spring also contribute
Test rig experiments with quasi-stationary clamping force and a number of fixed values for the ratio , the angular velocity , and the torque , showed that the speed ratio (9) and the efficiency (10) are concave functions of with a maximum for and , respectively. The experiments also showed that with a small difference between these forces. It is assumed that these findings hold for all practically relevant values of , , and . This assumption is crucial for the remainder of this brief, is therefore verified in Section IV for a subset of data, and should be the topic of future analytical and experimental research. Stability analysis of the variator centers around the relative slip (11)
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Based on the same set of test rig experiments mentioned above, it is concluded that is a function of with for small and for larger . The stability of (11) is then determined by the level of and is stable for small and becomes unstable for larger . These situations are referred to as open loop stable and open loop unstable, respectively. The relation between and is normally monotonic, decreasing corresponds to increasing , so decreasing may lead to instability of the relative slip. The transition occurs for , with , see [9]. Current variator controllers employ a relative or absolute safety strategy to determine the clamping forces such that the slip will remain small even in the presence of large torque disturbances. The starting point for these strategies is given by (8). For given , , and this relation determines the clamping force that is required to transfer a given torque . In vehicular applications is unknown, but usually an estimate for the torque at the engine side of the variator is available. Using (2), (3), and (8), it follows that
(12) In the relative safety strategy the unknown power losses and the relative slip are neglected, is replaced by the available estimate , and a safety factor is introduced to account for model uncertainties and torque disturbances. Then, the reference for the secondary clamping force is given by [4]
(13) where is an estimate for the traction coefficient. Often and are used. The relative safety strategy outperforms the absolute safety strategy [4] in terms of efficiency, because the latter results in larger clamping forces. The objectives of variator control are to track a speed ratio reference and to optimize the efficiency, under the condition that variator damage is avoided. The reference for the primary clamping force is chosen to ensure that the speed ratio tracks its reference , which is prescribed by the driveline management system [17]. Here, the focus is on the efficiency. To avoid the tracking problem, the position of the primary axially movable sheave (and hence the geometric ratio) is mechanically constrained. The reaction force exerted by the stop on the sheave is not controllable. So, the variator controller only has to deliver a reference for the secondary clamping force. The objective of ESC is to make equal to , where the input-output map is maximal. Since this clamping force is slightly larger than , for which the efficiency is maximal, it is expected that the obtained efficiency is near optimal. From the earlier mentioned experiments it is known that is always smaller than , so the efficiency, obtained with ESC, will be better than the efficiency with the relative safety strategy. However, torque disturbances may cause problems when using ESC because of the lower clamping forces. This is addressed in Section VI.
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Fig. 2. Experimental setup with electric motors and pushbelt variator.
TABLE I STATIONARY OPERATING POINTS FOR LOW AND HIGH
III. EXPERIMENTAL SETUP The experimental setup in Fig. 2 allows detailed studies of a variator and consists of two identical electric motors (Siemens, type 1PA6184-4NL00-0GA03), a pushbelt variator (Bosch Transmission Technology, type P811), a hydraulic actuation system, and a data acquisition system. The inertia of the electric motors is 0.503 kgm [18, pp. 3/9]. The primary and secondary inertia in a passenger car are approximately 3 smaller and 16 larger, respectively, than the inertia of the electric motors. These large differences should be kept in mind when conclusions, based on experiments with this setup, are to be translated to vehicles. Each electric motor is equipped with a rotary encoder (Heidenhain, type ERN 1387) and a low-level current control system. The reference of this control system is the torque that is to be delivered by the motor. In terms of high-level control, the primary motor is closed loop velocity controlled with PI control, whereas the secondary motor is open loop torque controlled. Each shaft of the variator is connected to one of the electric motors with two elastic couplings with a torque sensor (HBM, type T20WN) in between. The position of the secondary axially movable sheave is measured with an incremental length gauge (Heidenhain, type ST 3078). The hydraulic actuation system consists of several hydraulic pumps for actuation and lubrication. The pressures and are controlled by two servo valves (Mannesmann Rexroth, type 4 WS 2 EE 10), which are fed from a shared accumulator. Each hydraulic cylinder is equipped with a pressure sensor (GE Druck, type PTX 1400). Both hydraulic circuits are closed loop pressure controlled [23]. IV. QUASI-STATIONARY EXPERIMENTS A number of experiments are performed to show that the assumptions of Section II are satisfied. Table I specifies the stationary operating points from which the experiments are started. The geometric ratio in these experiments is either Low or High. For both ratios, the primary axially movable sheave is pressed against a stop. This is enforced by the choice of the primary pressure, 0 bar for Low and 20 bar for High.
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Fig. 5. Clamping force and efficiency for ESC and conventional control, for (black line) and several levels of torque [Left: (grey line); Right: (black line) and (grey line)]. Fig. 3. Velocity and torque for decreasing clamping force and (black line: Measurement; grey line: Reference).
14 Nm
for the development of a controller that looks for the maximum in the input-output map. V. EXTREMUM SEEKING CONTROL
Fig. 4. Ratio and efficiency for decreasing clamping force and 14 Nm ; black dashed line: (black solid line: Measurement; grey solid line: ; grey dashed line: ).
Models to estimate the clamping force in a given stationary operating point are not sufficiently accurate. Since the ESC approach is not model based, ESC will be applied. Consider given, constant, geometric ratio , primary angular velocity , and secondary torque . Then, (9) becomes (15)
In each experiment the primary pressure, primary angular velocity, and secondary torque are kept equal to their stationary values , , and , respectively. The reference for the secondary pressure changes slowly as function of time , i.e., (14) The experiment is stopped when the relative slip becomes larger than 5%. This situation is open loop unstable. To obtain a concise presentation of the results, only the experiment with High and 14 Nm is considered in detail. Results for other torques are given in summary. Those obtained for Low were consistent with these and are omitted. Angular velocities and torques are depicted in Fig. 3. For 48 s the results diverge because the variator becomes open loop unstable and the test rig control system is unable to cope with the drastic change in the system dynamics. The velocity error is very small, since the primary electric motor is closed loop velocity controlled. Although the secondary electric motor is open loop torque controlled the torque error is acceptably small. The secondary angular velocity initially increases, reaches a maximum just before 48 s, and ultimately decreases. The sharp drop signifies the instability. The primary torque decreases, since the power loss in the variator reduces. Results as function of the clamping force are given in Fig. 4. From these results follows that the assumptions of Section II are satisfied, since . From Fig. 5 (left), it is concluded that indeed for all applied secondary torques. The average values of the ratios and are equal to 105% and 143%, respectively. From Fig. 5 (right), it is concluded that for all applied secondary torques the efficiency is higher if the reference for the secondary clamping force is than if this reference is . The average values of the ratios and are equal to 99.7% and 98.7%, respectively. This completes the motivation
In practice, a variator controller is implemented in terms of pressures instead of clamping forces. The relations between these are given by (4). In the sequel, will be used instead of and the relation is replaced by . The value of that maximizes is denoted by . The control objective becomes: Maximize the value of without requiring knowledge of the input-output function nor of the location of the maximum. The output of this controller is used as a reference for the hydraulic actuation system. The feedback mechanism is depicted in Fig. 6. It utilizes a perturbation with amplitude and frequency . This perturbation is added to the output of the controller. When the sinusoidal perturbation is slow in comparison with the variator dynamics there will be no interference between the variator dynamics and the feedback mechanism. The feedback mechanism from to consists of the following operations: (16) (17) (18) (19) (20) where is the measured secondary pressure and the speed ratio derived from the measured angular velocities. The bandpass filter suppresses DC components and noise in and . The product has a DC component. The low-pass filter extracts this component, yielding . Finally, results from integration of with integrator gain and initial condition . The blocks and represent the hydraulic actuation system and the variator, respectively. The feedback mechanism incorporates five design options, being the perturbation amplitude and frequency , the filters and , and the integrator gain . The parameter selection for these options is closely related to the proof of stability
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Fig. 6. ESC feedback mechanism from
to
Fig. 7. Velocity and torque, ESC on at 50 s, for Measurement; grey line: Reference).
.
of the closed loop system, see [14]. The feedback mechanism in [14] is similar to the feedback mechanism in Fig. 6, but in [14] a high-pass filter instead of a band-pass filter is employed. Here, a band-pass filter is chosen to suppress noise. Furthermore, for signal a filtered version of is used, instead of the excitation signal as in [14], to avoid effects of the (nonlinear) block . When the input-output relation indeed has a maximum in the open loop stable region and the path towards this maximum is also in this region, convergence of the solution towards a certain neighborhood of this maximum is guaranteed for a suitable choice of the design options. On the one hand, upper bounds have to be imposed on and to confine the region to which the solution converges [14, Th. 5.1]. On the other hand, and have to be sufficiently large to excite the variator and to achieve convergence, see [19, Sec. 1.2.3]. The band-pass filter is designed in accordance with the perturbation frequency and such that the DC components and noise in and are suppressed. The cutoff frequency of the low-pass filter is a fraction of the perturbation frequency, see [14]. The integrator gain has to be limited to confine the region to which the solution converges [14, Th. 5.1], but a sufficiently large gain is desired to accelerate convergence. Here, , , , , and are given by bar Hz
(21) (22) (23) (24) (25)
The performance of the feedback mechanism is evaluated with closed loop experiments. For 50 s, the connection between the low-pass filter and the integrator is broken, so the controller output is equal to . For 50 s, the connection is closed. Results for angular velocities and torques are depicted in Fig. 7 and show stable performance. The speed ratio and the efficiency as function of the clamping force are similar to those in Fig. 4, except that the open loop unstable region is not entered. VI. ROBUSTNESS ANALYSIS In the closed loop experiments in the previous section, the torques and are nearly constant. Torque disturbances en-
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14 Nm (black line:
TABLE II STATIONARY OPERATING POINT FOR GEOMETRIC RATIO HIGH AND PRIMARY AND SECONDARY DISTURBANCE
countered by a variator in a passenger car are transient of nature and possibly enforce a transition of the variator from open loop stable to open loop unstable. This may destabilize the ESC feedback loop. So, the robustness for torque disturbances is analyzed. The torque disturbances are induced by the engine (primary side) or the road (secondary side). Primary torque disturbances are typically imposed by the driver who depresses or releases the accelerator pedal. The engine control unit provides an estimate of the primary torque, see [20, Sec. 3.3.2], so a priori information for the primary torque disturbance will be available. Secondary torque disturbances are typically induced by the road. Usually no a priori information for the secondary torque disturbance will be available. A priori information for torque disturbances can be used in a feedforward controller. The robustness for torque disturbances is analyzed in the next subsections. Only ratio High is considered, being critical. The experiments start from the operating point in Table II. The system with ESC converges towards a small neighborhood of the extremum , which is reached for 100 s. At 400 s, the disturbances are applied. To mimic the situation in a passenger car, the test rig control system is modified. The primary motor is open loop torque controlled and the secondary motor is closed loop velocity controlled with PI control. A. Primary Disturbance To the nominal torque a disturbance is added representing a depression of the accelerator pedal with moderate impact. The experiment is performed twice, without and with feedforward of the torque disturbance. In the experiment with feedforward a contribution is added to the estimate of the optimum input. This contribution is determined using (13) (26) Without feedforward, the disturbance (see Fig. 8, top left) mainly accelerates the primary pulley (see Fig. 9, top left), since
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Fig. 8. Torque for a primary disturbance at 400 s (Left: Without feedforward; Right: With feedforward) (black line: Measurement; grey line: Reference).
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Fig. 11. Efficiency and slip for a primary disturbance at 400 s (Left: Without feedforward; Right: With feedforward).
Fig. 9. Velocity for a primary disturbance at 400 s (Left: Without feedforward; Right: With feedforward) (black line: Measurement; grey line: Reference).
Fig. 12. Pressure for a primary disturbance at 400 s (Left: Without feedforward; Right: With feedforward) (black line: Measurement; grey line: Reference).
Fig. 10. Ratio for a primary disturbance at 400 s (Left: Without feedforward; Right: With feedforward).
the variator is unable to transfer the additional torque. The resulting secondary angular velocity and secondary torque are given in Fig. 9 (bottom left) and Fig. 8 (bottom left). The speed ratio decreases (see Fig. 10, top left), whereas the geometric ratio slightly increases, due to deformations of the variator. As a result, the relative slip increases (see Fig. 11, bottom left) and the efficiency decreases (see Fig. 11, top left). The contribution and the estimate of the optimum input are depicted in Fig. 12 (left), together with the measured secondary pressure . The estimate of the optimum input slightly decreases, whereas the disturbance is positive. So, ESC without feedforward is unable to handle the change of the operating condition. With feedforward, the disturbance (see Fig. 8, top right) accelerates both pulleys (see Fig. 9, right), since now the variator transfers the additional torque. The secondary torque is given in Fig. 8 (bottom right) and follows . The speed ratio (see Fig. 10, top right) and the geometric ratio slightly decrease, whereas the relative slip slightly increases (see Fig. 11, bottom right). The variator remains open loop stable. The efficiency slightly increases (see Fig. 11, top right), because it improves for higher loads. The contribution and the estimate of
the optimum input are depicted in Fig. 12 (right), together with the measured secondary pressure . The estimate of the optimum input hardly changes and the effect of the feedforward on is clearly visible. So, ESC with feedforward can handle the change of the primary torque. B. Secondary Disturbance For secondary (torque) disturbances, a disturbance of the secondary angular velocity is added to the nominal value, , to avoid the issue of low inertia for the secondary motor. This disturbance represents the passage of a step bump of moderate impact, lasting 0.6 s. The angular velocities and the torques are depicted in Fig. 13. For 400 s, both and are accurately tracked. For 400 s, the secondary angular velocity clearly lags its reference and there is a large mismatch of the amplitude except for the first peak. After 400.6 s is sub-critically damped and strongly deviates from its reference. This is attributed to the nonlinearity of the power electronics of the secondary electric motor. The main effect visible in is that it follows , just like follows . Only the components of the disturbance at high frequencies are not passed on. The speed ratio shows high frequency oscillations of small amplitude for 400.6 s, whereas the geometric ratio is not or hardly disturbed, see Fig. 14. The efficiency is also depicted here, but the significance of this graph is dubious, since definition (3) makes sense only in stationary situations. The oscillations in the speed ratio are the source of the two peaks in the relative slip in Fig. 14. Despite these peaks the variator remains open loop stable. The relative slip measured at
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Fig. 13. Velocity and torque for a secondary disturbance at 400 s (black line: Measurement; grey line: Reference).
Fig. 14. Ratio, efficiency, and slip for a secondary disturbance at 400 s.
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motivates the use of the input-output map instead of the input-output map in the design of a controller that optimizes the efficiency. Since the location of the maximum of the input-output map is not known with sufficient accuracy, ESC is used. Experiments show that the efficiency is improved in comparison with a conventional controller. The robustness of the closed loop system with ESC for some disturbances is analyzed. The primary disturbance, representing a depression of the accelerator pedal, is mainly nullified by feedforward, whereas the secondary disturbance, representing a step bump, is blissfully ignored by ESC. The ESC design requires further research to improve convergence and to handle possible transitions of the variator from open loop stable to open loop unstable. An instrumented test vehicle would be useful for this, as well as to see if the excitation signal of ESC does not interfere with other components of the driveline. The first objective for the variator controller in a vehicle is to track the speed ratio reference that is prescribed by the driveline management system. The second objective is to improve the variator efficiency. This first objective is not included here, since only fixed transmission ratios are considered. The extension of ESC, as described here, towards a feedback controller that satisfies both control objectives will be subject of another contribution. REFERENCES
Fig. 15. Pressure for a secondary disturbance at 400 s (black line: Measurement; grey line: Reference).
the test rig is large in comparison with the slip in a passenger car. This is caused by the primary inertia of the test rig. This inertia is larger than the one in a passenger car and, as a consequence, the peaks in the relative slip are larger. The peaks in the relative slip appear and disappear within the interval 400.5 s when the fast changes in occur. The secondary disturbance is not noticed in the ESC output. This is deduced from Fig. 15. The visible effects of the disturbance in and are filtered by the ESC filters and are not visible in . In this case the variator operates as a torque fuse for high frequency components in the disturbance, the relative slip is shortly increased, and those components are not suppressed by the variator control system. VII. DISCUSSION From a series of test rig experiments with a pushbelt variator it is confirmed that the variator efficiency and the speed ratio as function of the secondary clamping force have a global maximum and that the clamping force for which is maximal is only slightly different from the clamping force for which is maximal. These findings need to be confirmed for other conditions. In automotive applications the variator efficiency is not measured, whereas the speed ratio follows from the standard measurements of the primary and secondary angular velocity. This
[1] R. Pfiffner and L. Guzzella, “Optimal operation of CVT-based powertrains,” Int. J. Robust Nonlinear Control, vol. 11, no. 11, pp. 1003–1021, 2001. [2] K. G. O. van de Meerakker, P. C. J. N. Rosielle, B. Bonsen, T. W. G. L. Klaassen, and N. J. J. Liebrand, “Mechanism proposed for ratio and clamping force control in a CVT,” presented at the FISITA World Autom. Congr., Barcelona, Spain, 2004, Art. No. F2004F108. [3] S. Shastri and A. A. Frank, “Comparison of energy consumption and power losses of a conventionally controlled CVT with a servo-hydraulic controlled CVT and with a belt and chain as the torque transmitting element,” presented at the Proc. Int. Continuously Variable Hybrid Transmission Congr., Davis, CA, 2004, Art. No. 04CVT-55. [4] F. van der Sluis, T. van Dongen, G.-J. van Spijk, A. van der Velde, and A. van Heeswijk, “Fuel consumption potential of the pushbelt CVT,” presented at the FISITA World Autom. Congr., Yokohama, Japan, 2006, Art. No. F2006P218. [5] S. Akehurst, N. D. Vaughan, D. A. Parker, and D. Simner, “Modelling of loss mechanisms in a pushing metal V-belt continuously variable transmission. Part 1: Torque losses due to band friction,” in Proc. Inst. Mechan. Eng., Pt. D: J. Auto. Eng., 2004, vol. 218, no. 11, pp. 1269–1281. [6] S. Akehurst, N. D. Vaughan, D. A. Parker, and D. Simner, “Modelling of loss mechanisms in a pushing metal V-belt continuously variable transmission. Part 2: Pulley deflection losses and total torque loss validation,” in Proc. Inst. Mechan. Eng., Pt. D: J. Auto. Eng., 2004, vol. 218, no. 11, pp. 1283–1293. [7] S. Akehurst, N. D. Vaughan, D. A. Parker, and D. Simner, “Modelling of loss mechanisms in a pushing metal V-belt continuously variable transmission. Part 3: Belt slip losses,” in Proc. Inst. Mechan. Eng., Pt. D: J. Auto. Eng., 2004, vol. 218, no. 11, pp. 1295–1306. [8] B. Bonsen, T. W. G. L. Klaassen, K. G. O. van de Meerakker, M. Steinbuch, and P. A. Veenhuizen, “Analysis of slip in a continuously variable transmission,” presented at the ASME Int. Mechan. Eng. Congr., Washington, DC, 2003, Art. No. IMECE2003-41360. [9] K. Sakagami, T. Fujii, H. Yoshida, and T. Yagasaki, “Study on belt slip behavior in metal V-belt type CVT,” in Proc. Int. Congr. Continuously Variable Hybrid Transmissions, 2007, pp. 135–139. [10] M. van Drogen and M. van der Laan, “Determination of variator robustness under macro slip conditions for a push belt CVT,” presented at the SAE World Congr., Detroit, MI, 2004, Art. No. 2004-01-0480.
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[11] T. W. G. L. Klaassen, “The Empact CVT: Dynamics and control of an electromechanically actuated CVT,” Ph.D. dissertation, Dept. Mechan. Eng., Eindhoven Univ. Technol., Eindhoven, The Netherlands, 2007. [12] B. Bonsen, T. W. G. L. Klaassen, R. J. Pulles, S. W. H. Simons, M. Steinbuch, and P. A. Veenhuizen, “Performance optimisation of the push-belt CVT by variator slip control,” Int. J. Veh. Design, vol. 39, no. 3, pp. 232–256, 2005. [13] K. J. Åström and B. Wittenmark, Adaptive Control, ser. Addison-Wesley Series in Electrical Engineering: Control Engineering, 2nd ed. Reading, MA: Addison-Wesley, 1995. [14] M. Krstić and H.-H. Wang, “Stability of extremum seeking feedback for general nonlinear dynamic systems,” Automatica, vol. 36, no. 4, pp. 595–601, 2000. [15] S. van der Meulen, B. de Jager, E. van der Noll, F. Veldpaus, F. van der Sluis, and M. Steinbuch, “Improving pushbelt continuously variable transmission efficiency via extremum seeking control,” in Proc. 3rd IEEE Multi-Conf. Syst. Control, 2009, pp. 357–362. [16] E. van der Noll, F. van der Sluis, T. van Dongen, and A. van der Velde, “Innovative self-optimising clamping force strategy for the pushbelt CVT,” presented at the SAE World Congr., Detroit, MI, 2009, Art. No. 2009-01-1537. [17] M. H. Smith, E. J. Barth, N. Sadegh, and G. J. Vachtsevanos, “The horsepower reserve formulation of driveability for a vehicle fitted with a continuously variable transmission,” Veh. Syst. Dynam., vol. 41, no. 3, pp. 157–180, 2004.
[18] Siemens AG, Munich, Germany, “ROTEC—Low-voltage motors for variable-speed drives,” Advance Catalog DA 65.3, 1997. [19] K. B. Ariyur and M. Krstić, Real-Time Optimization by ExtremumSeeking Control. Hoboken, NJ: Wiley, 2003. [20] G. Lechner and H. Naunheimer, Automotive Transmissions: Fundamentals, Selection, Design and Application. Berlin, Germany: Springer-Verlag, 1999. [21] S. van der Meulen, B. de Jager, F. Veldpaus, and M. Steinbuch, “Robust extremum seeking and speed ratio control for high-performance CVT operation,” in Proc. 6th Int. Conf. Continuously Variable Hybrid Transmissions, 2010, pp. 110–115. [22] S. H. van der Meulen, “High-performance control of continuously variable transmissions,” Ph.D. dissertation, Eindhoven Univ. Technol., Eindhoven, The Netherlands, 2010. [23] T. Oomen, S. van der Meulen, O. Bosgra, M. Steinbuch, and J. Elfring, “A robust-control-relevant model validation approach for continuously variable transmission control,” in Proc. Amer. Control Conf., 2010, pp. 3518–3523.
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Improving Continuously Variable Transmission Efficiency With Extremum Seeking Control Stan van der Meulen, Bram de Jager, Frans Veldpaus, Erik van der Noll, Francis van der Sluis, and Maarten Steinbuch
Abstract—The design of an efficiency optimal controller for the variator in a vehicular continuously variable transmission is studied. A conventional controller aims at tracking a prescribed reference for the transmission ratio and at preventing damage, but does not address efficiency. Sufficiently accurate models for the efficiency as function of the clamping forces are not available, whereas measurement of the efficiency requires extra sensors. In this brief, a controller is proposed that improves the efficiency without needing extra sensors. The maps between the clamping forces (input) and the efficiency or the speed ratio (output) are studied with test rig experiments. These maps exhibit a maximum, but the location of this maximum is uncertain. So, an extremum seeking controller is developed. This controller can adapt the input to maximize the output, without needing a model. Experiments show that this approach is feasible and that a conventional controller is outperformed. A robustness analysis for disturbances indicates that these are effectively handled. Index Terms—Continuously variable transmissions (CVTs), experiments, extremum seeking control (ESC), nonlinear systems, robustness.
I. INTRODUCTION
C
URRENT automotive research aims at reducing emissions and fuel consumption, e.g., with advanced transmissions, like the continuously variable transmission (CVT). A CVT realizes any transmission ratio within a finite range and changes this ratio without interruption of the power transfer from the engine to the wheels. Furthermore, the overall efficiency of a CVT driveline improves, because the demanded engine power can be generated in fuel optimal operating points [1]. This brief focuses on the efficiency of the CVT itself. The CVT encloses a torque converter, a drive-neutral-reverse set, a variator, an actuation system, and a final drive. The variator consists of a metal V-belt (the pushbelt), a primary pulley, and a secondary pulley. One of the conical sheaves of each pulley is axially fixed, whereas the other can move in the axial
Manuscript received November 10, 2010; revised May 11, 2011; accepted May 20, 2011. S. van der Meulen, B. de Jager, F. Veldpaus, and M. Steinbuch are with the Department of Mechanical Engineering, Control Systems Technology Group, Eindhoven University of Technology, 5600 MB Eindhoven, The Netherlands (e-mail: [email protected]; [email protected]; [email protected]; [email protected]). E. van der Noll and F. van der Sluis are with the Department of Advanced Engineering, Bosch Transmission Technology, 5000 AM Tilburg, The Netherlands (e-mail: [email protected]; [email protected]). Color versions of one or more of the figures in this brief are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TCST.2011.2160980
direction. Changes of the transmission ratio are achieved by simultaneous adjustment of the clamping forces on the movable sheaves. This results in an axial displacement of these sheaves and, due to this displacement, in a change of the running radii of the pushbelt on the pulleys and therefore also of the transmission ratio. Although electromechanical [2] and electrohydraulical [3] actuation systems exist, the majority of CVTs are actuated by a hydraulic system, which is the system considered here. The main sources of power loss in a CVT are found in the variator and its actuation system [4]–[7]. There is a strong relation between these losses and the magnitude of the clamping forces. These forces have to be large enough to transfer a given torque without excessive slip between the pushbelt and the pulleys. When, starting at a high level, the clamping forces are decreased, the slip increases and eventually a transition of the variator from open loop stable (micro-slip) to open loop unstable (macro-slip) will occur. For decreasing clamping forces, the variator efficiency initially increases, subsequently reaches a maximum, and ultimately decreases. The existence of this maximum is shown in [8] with experiments. It occurs within the open loop stable region [9]. Recent research [10] has shown that the variator can withstand substantial levels of slip without an unacceptable reduction of its life span. This leads to the idea to improve the efficiency by reducing the clamping forces to a level for which the given torque can be transferred with an acceptable level of slip. A straightforward approach is to determine a slip reference [11, Sec. 7.2] that corresponds to the maximum efficiency and to control the slip to track this reference [12]. To determine a usable estimate for the slip reference is problematic, because the available variator models are not reliable nor accurate enough. This leads to alternative approaches, which aim at maximizing the efficiency of the variator without slip control. Extremum seeking control (ESC), see [13, Sec. 13.3] and [14], is a control method that exploits the presence of a maximum in an input-output map by adapting the input (here: the clamping forces) to maximize the output (here: the efficiency). Normally, the torques at the pulleys are not measured, so determination of the efficiency and controlled manipulation of the clamping forces to maximize the efficiency is not feasible. From experiments it is known that the input-output map from clamping force to speed ratio (the ratio of the secondary and primary angular velocity) also has a maximum [9] and that the argument of this maximum is close to the argument for which the efficiency is maximal. Hence, near optimal efficiency will be obtained if ESC is used to control the variator to the maximum of the clamping force versus speed ratio map, see [15], [16], [21], and [22]. This approach is detailed, experimentally
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(4a) (4b) where and are the pressure surfaces of the hydraulic cylinders, whereas and are centrifugal coefficients. Based on first principles models and many experiments [8], it is concluded that the torques and are functions of geometric ratio , primary angular velocity , speed ratio (or relative slip ), and clamping forces and , i.e., (5a) (5b) Fig. 1. Schematic illustration of pushbelt variator.
evaluated, and tested for robustness in this brief, which are its main contributions. Definitions and control objectives are given in Section II. The experimental setup is introduced in Section III and the feasibility of the control objectives is shown in Section IV. ESC is described in Section V, together with closed loop experiments. The robustness of ESC against disturbances is analyzed in Section VI. A discussion in Section VII concludes this brief.
Torque cannot be transferred without slip, so, for . For the rate of change of the speed ratio holds
,
(6) Some commonly used models for , i.e., the Carbone Mangialardi Mantriota (CMM) model, the Ide model, and the Shafai model, are described in [11, Sec. 2.5]. It is assumed that for stationary situations, i.e., for , the clamping force can be solved from (6) in the form
II. DEFINITIONS AND CONTROL OBJECTIVES
(7)
This section introduces the notation, provides the physical background, and states the control objectives. Consider Fig. 1, where the primary (input, subscript “ ”) side and the secondary (output, subscript “ ”) side of the variator are depicted. The torques are denoted by and , the angular velocities by and , the clamping forces by and , and the axially movable sheave positions by and . Furthermore, denotes half the pulley wedge angle, which is equal to 11 deg. Deformations of the variator are neglected, so the path of the pushbelt on the pulleys is a circular arc with the running radii and . The geometric ratio , the speed ratio , and the relative slip (see, e.g., [8]), are defined by
where is the clamping force ratio. Using (5b) and (7) the relation for the torque can be written as (8) where
the
secondary traction coefficient satisfies for . It is also assumed that the speed ratio can be solved from (6) with , resulting in (9)
(1)
With this relation and (3), (5a), and (7), it follows that the efficiency will be a function of , , , and , so
(2)
(10)
The variator can realize any transmission ratio in the range from to . The efficiency of the variator is given by (3) with the input power and the output power of the variator. The required power for the hydraulic actuation system is not accounted for in this definition, so is not the efficiency of the complete CVT. The clamping forces and on the movable sheaves scale with the oil pressures and in the hydraulic cylinders of the pulleys. Centrifugal effects and a preloaded spring also contribute
Test rig experiments with quasi-stationary clamping force and a number of fixed values for the ratio , the angular velocity , and the torque , showed that the speed ratio (9) and the efficiency (10) are concave functions of with a maximum for and , respectively. The experiments also showed that with a small difference between these forces. It is assumed that these findings hold for all practically relevant values of , , and . This assumption is crucial for the remainder of this brief, is therefore verified in Section IV for a subset of data, and should be the topic of future analytical and experimental research. Stability analysis of the variator centers around the relative slip (11)
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Based on the same set of test rig experiments mentioned above, it is concluded that is a function of with for small and for larger . The stability of (11) is then determined by the level of and is stable for small and becomes unstable for larger . These situations are referred to as open loop stable and open loop unstable, respectively. The relation between and is normally monotonic, decreasing corresponds to increasing , so decreasing may lead to instability of the relative slip. The transition occurs for , with , see [9]. Current variator controllers employ a relative or absolute safety strategy to determine the clamping forces such that the slip will remain small even in the presence of large torque disturbances. The starting point for these strategies is given by (8). For given , , and this relation determines the clamping force that is required to transfer a given torque . In vehicular applications is unknown, but usually an estimate for the torque at the engine side of the variator is available. Using (2), (3), and (8), it follows that
(12) In the relative safety strategy the unknown power losses and the relative slip are neglected, is replaced by the available estimate , and a safety factor is introduced to account for model uncertainties and torque disturbances. Then, the reference for the secondary clamping force is given by [4]
(13) where is an estimate for the traction coefficient. Often and are used. The relative safety strategy outperforms the absolute safety strategy [4] in terms of efficiency, because the latter results in larger clamping forces. The objectives of variator control are to track a speed ratio reference and to optimize the efficiency, under the condition that variator damage is avoided. The reference for the primary clamping force is chosen to ensure that the speed ratio tracks its reference , which is prescribed by the driveline management system [17]. Here, the focus is on the efficiency. To avoid the tracking problem, the position of the primary axially movable sheave (and hence the geometric ratio) is mechanically constrained. The reaction force exerted by the stop on the sheave is not controllable. So, the variator controller only has to deliver a reference for the secondary clamping force. The objective of ESC is to make equal to , where the input-output map is maximal. Since this clamping force is slightly larger than , for which the efficiency is maximal, it is expected that the obtained efficiency is near optimal. From the earlier mentioned experiments it is known that is always smaller than , so the efficiency, obtained with ESC, will be better than the efficiency with the relative safety strategy. However, torque disturbances may cause problems when using ESC because of the lower clamping forces. This is addressed in Section VI.
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Fig. 2. Experimental setup with electric motors and pushbelt variator.
TABLE I STATIONARY OPERATING POINTS FOR LOW AND HIGH
III. EXPERIMENTAL SETUP The experimental setup in Fig. 2 allows detailed studies of a variator and consists of two identical electric motors (Siemens, type 1PA6184-4NL00-0GA03), a pushbelt variator (Bosch Transmission Technology, type P811), a hydraulic actuation system, and a data acquisition system. The inertia of the electric motors is 0.503 kgm [18, pp. 3/9]. The primary and secondary inertia in a passenger car are approximately 3 smaller and 16 larger, respectively, than the inertia of the electric motors. These large differences should be kept in mind when conclusions, based on experiments with this setup, are to be translated to vehicles. Each electric motor is equipped with a rotary encoder (Heidenhain, type ERN 1387) and a low-level current control system. The reference of this control system is the torque that is to be delivered by the motor. In terms of high-level control, the primary motor is closed loop velocity controlled with PI control, whereas the secondary motor is open loop torque controlled. Each shaft of the variator is connected to one of the electric motors with two elastic couplings with a torque sensor (HBM, type T20WN) in between. The position of the secondary axially movable sheave is measured with an incremental length gauge (Heidenhain, type ST 3078). The hydraulic actuation system consists of several hydraulic pumps for actuation and lubrication. The pressures and are controlled by two servo valves (Mannesmann Rexroth, type 4 WS 2 EE 10), which are fed from a shared accumulator. Each hydraulic cylinder is equipped with a pressure sensor (GE Druck, type PTX 1400). Both hydraulic circuits are closed loop pressure controlled [23]. IV. QUASI-STATIONARY EXPERIMENTS A number of experiments are performed to show that the assumptions of Section II are satisfied. Table I specifies the stationary operating points from which the experiments are started. The geometric ratio in these experiments is either Low or High. For both ratios, the primary axially movable sheave is pressed against a stop. This is enforced by the choice of the primary pressure, 0 bar for Low and 20 bar for High.
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Fig. 5. Clamping force and efficiency for ESC and conventional control, for (black line) and several levels of torque [Left: (grey line); Right: (black line) and (grey line)]. Fig. 3. Velocity and torque for decreasing clamping force and (black line: Measurement; grey line: Reference).
14 Nm
for the development of a controller that looks for the maximum in the input-output map. V. EXTREMUM SEEKING CONTROL
Fig. 4. Ratio and efficiency for decreasing clamping force and 14 Nm ; black dashed line: (black solid line: Measurement; grey solid line: ; grey dashed line: ).
Models to estimate the clamping force in a given stationary operating point are not sufficiently accurate. Since the ESC approach is not model based, ESC will be applied. Consider given, constant, geometric ratio , primary angular velocity , and secondary torque . Then, (9) becomes (15)
In each experiment the primary pressure, primary angular velocity, and secondary torque are kept equal to their stationary values , , and , respectively. The reference for the secondary pressure changes slowly as function of time , i.e., (14) The experiment is stopped when the relative slip becomes larger than 5%. This situation is open loop unstable. To obtain a concise presentation of the results, only the experiment with High and 14 Nm is considered in detail. Results for other torques are given in summary. Those obtained for Low were consistent with these and are omitted. Angular velocities and torques are depicted in Fig. 3. For 48 s the results diverge because the variator becomes open loop unstable and the test rig control system is unable to cope with the drastic change in the system dynamics. The velocity error is very small, since the primary electric motor is closed loop velocity controlled. Although the secondary electric motor is open loop torque controlled the torque error is acceptably small. The secondary angular velocity initially increases, reaches a maximum just before 48 s, and ultimately decreases. The sharp drop signifies the instability. The primary torque decreases, since the power loss in the variator reduces. Results as function of the clamping force are given in Fig. 4. From these results follows that the assumptions of Section II are satisfied, since . From Fig. 5 (left), it is concluded that indeed for all applied secondary torques. The average values of the ratios and are equal to 105% and 143%, respectively. From Fig. 5 (right), it is concluded that for all applied secondary torques the efficiency is higher if the reference for the secondary clamping force is than if this reference is . The average values of the ratios and are equal to 99.7% and 98.7%, respectively. This completes the motivation
In practice, a variator controller is implemented in terms of pressures instead of clamping forces. The relations between these are given by (4). In the sequel, will be used instead of and the relation is replaced by . The value of that maximizes is denoted by . The control objective becomes: Maximize the value of without requiring knowledge of the input-output function nor of the location of the maximum. The output of this controller is used as a reference for the hydraulic actuation system. The feedback mechanism is depicted in Fig. 6. It utilizes a perturbation with amplitude and frequency . This perturbation is added to the output of the controller. When the sinusoidal perturbation is slow in comparison with the variator dynamics there will be no interference between the variator dynamics and the feedback mechanism. The feedback mechanism from to consists of the following operations: (16) (17) (18) (19) (20) where is the measured secondary pressure and the speed ratio derived from the measured angular velocities. The bandpass filter suppresses DC components and noise in and . The product has a DC component. The low-pass filter extracts this component, yielding . Finally, results from integration of with integrator gain and initial condition . The blocks and represent the hydraulic actuation system and the variator, respectively. The feedback mechanism incorporates five design options, being the perturbation amplitude and frequency , the filters and , and the integrator gain . The parameter selection for these options is closely related to the proof of stability
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Fig. 6. ESC feedback mechanism from
to
Fig. 7. Velocity and torque, ESC on at 50 s, for Measurement; grey line: Reference).
.
of the closed loop system, see [14]. The feedback mechanism in [14] is similar to the feedback mechanism in Fig. 6, but in [14] a high-pass filter instead of a band-pass filter is employed. Here, a band-pass filter is chosen to suppress noise. Furthermore, for signal a filtered version of is used, instead of the excitation signal as in [14], to avoid effects of the (nonlinear) block . When the input-output relation indeed has a maximum in the open loop stable region and the path towards this maximum is also in this region, convergence of the solution towards a certain neighborhood of this maximum is guaranteed for a suitable choice of the design options. On the one hand, upper bounds have to be imposed on and to confine the region to which the solution converges [14, Th. 5.1]. On the other hand, and have to be sufficiently large to excite the variator and to achieve convergence, see [19, Sec. 1.2.3]. The band-pass filter is designed in accordance with the perturbation frequency and such that the DC components and noise in and are suppressed. The cutoff frequency of the low-pass filter is a fraction of the perturbation frequency, see [14]. The integrator gain has to be limited to confine the region to which the solution converges [14, Th. 5.1], but a sufficiently large gain is desired to accelerate convergence. Here, , , , , and are given by bar Hz
(21) (22) (23) (24) (25)
The performance of the feedback mechanism is evaluated with closed loop experiments. For 50 s, the connection between the low-pass filter and the integrator is broken, so the controller output is equal to . For 50 s, the connection is closed. Results for angular velocities and torques are depicted in Fig. 7 and show stable performance. The speed ratio and the efficiency as function of the clamping force are similar to those in Fig. 4, except that the open loop unstable region is not entered. VI. ROBUSTNESS ANALYSIS In the closed loop experiments in the previous section, the torques and are nearly constant. Torque disturbances en-
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14 Nm (black line:
TABLE II STATIONARY OPERATING POINT FOR GEOMETRIC RATIO HIGH AND PRIMARY AND SECONDARY DISTURBANCE
countered by a variator in a passenger car are transient of nature and possibly enforce a transition of the variator from open loop stable to open loop unstable. This may destabilize the ESC feedback loop. So, the robustness for torque disturbances is analyzed. The torque disturbances are induced by the engine (primary side) or the road (secondary side). Primary torque disturbances are typically imposed by the driver who depresses or releases the accelerator pedal. The engine control unit provides an estimate of the primary torque, see [20, Sec. 3.3.2], so a priori information for the primary torque disturbance will be available. Secondary torque disturbances are typically induced by the road. Usually no a priori information for the secondary torque disturbance will be available. A priori information for torque disturbances can be used in a feedforward controller. The robustness for torque disturbances is analyzed in the next subsections. Only ratio High is considered, being critical. The experiments start from the operating point in Table II. The system with ESC converges towards a small neighborhood of the extremum , which is reached for 100 s. At 400 s, the disturbances are applied. To mimic the situation in a passenger car, the test rig control system is modified. The primary motor is open loop torque controlled and the secondary motor is closed loop velocity controlled with PI control. A. Primary Disturbance To the nominal torque a disturbance is added representing a depression of the accelerator pedal with moderate impact. The experiment is performed twice, without and with feedforward of the torque disturbance. In the experiment with feedforward a contribution is added to the estimate of the optimum input. This contribution is determined using (13) (26) Without feedforward, the disturbance (see Fig. 8, top left) mainly accelerates the primary pulley (see Fig. 9, top left), since
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Fig. 8. Torque for a primary disturbance at 400 s (Left: Without feedforward; Right: With feedforward) (black line: Measurement; grey line: Reference).
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Fig. 11. Efficiency and slip for a primary disturbance at 400 s (Left: Without feedforward; Right: With feedforward).
Fig. 9. Velocity for a primary disturbance at 400 s (Left: Without feedforward; Right: With feedforward) (black line: Measurement; grey line: Reference).
Fig. 12. Pressure for a primary disturbance at 400 s (Left: Without feedforward; Right: With feedforward) (black line: Measurement; grey line: Reference).
Fig. 10. Ratio for a primary disturbance at 400 s (Left: Without feedforward; Right: With feedforward).
the variator is unable to transfer the additional torque. The resulting secondary angular velocity and secondary torque are given in Fig. 9 (bottom left) and Fig. 8 (bottom left). The speed ratio decreases (see Fig. 10, top left), whereas the geometric ratio slightly increases, due to deformations of the variator. As a result, the relative slip increases (see Fig. 11, bottom left) and the efficiency decreases (see Fig. 11, top left). The contribution and the estimate of the optimum input are depicted in Fig. 12 (left), together with the measured secondary pressure . The estimate of the optimum input slightly decreases, whereas the disturbance is positive. So, ESC without feedforward is unable to handle the change of the operating condition. With feedforward, the disturbance (see Fig. 8, top right) accelerates both pulleys (see Fig. 9, right), since now the variator transfers the additional torque. The secondary torque is given in Fig. 8 (bottom right) and follows . The speed ratio (see Fig. 10, top right) and the geometric ratio slightly decrease, whereas the relative slip slightly increases (see Fig. 11, bottom right). The variator remains open loop stable. The efficiency slightly increases (see Fig. 11, top right), because it improves for higher loads. The contribution and the estimate of
the optimum input are depicted in Fig. 12 (right), together with the measured secondary pressure . The estimate of the optimum input hardly changes and the effect of the feedforward on is clearly visible. So, ESC with feedforward can handle the change of the primary torque. B. Secondary Disturbance For secondary (torque) disturbances, a disturbance of the secondary angular velocity is added to the nominal value, , to avoid the issue of low inertia for the secondary motor. This disturbance represents the passage of a step bump of moderate impact, lasting 0.6 s. The angular velocities and the torques are depicted in Fig. 13. For 400 s, both and are accurately tracked. For 400 s, the secondary angular velocity clearly lags its reference and there is a large mismatch of the amplitude except for the first peak. After 400.6 s is sub-critically damped and strongly deviates from its reference. This is attributed to the nonlinearity of the power electronics of the secondary electric motor. The main effect visible in is that it follows , just like follows . Only the components of the disturbance at high frequencies are not passed on. The speed ratio shows high frequency oscillations of small amplitude for 400.6 s, whereas the geometric ratio is not or hardly disturbed, see Fig. 14. The efficiency is also depicted here, but the significance of this graph is dubious, since definition (3) makes sense only in stationary situations. The oscillations in the speed ratio are the source of the two peaks in the relative slip in Fig. 14. Despite these peaks the variator remains open loop stable. The relative slip measured at
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Fig. 13. Velocity and torque for a secondary disturbance at 400 s (black line: Measurement; grey line: Reference).
Fig. 14. Ratio, efficiency, and slip for a secondary disturbance at 400 s.
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motivates the use of the input-output map instead of the input-output map in the design of a controller that optimizes the efficiency. Since the location of the maximum of the input-output map is not known with sufficient accuracy, ESC is used. Experiments show that the efficiency is improved in comparison with a conventional controller. The robustness of the closed loop system with ESC for some disturbances is analyzed. The primary disturbance, representing a depression of the accelerator pedal, is mainly nullified by feedforward, whereas the secondary disturbance, representing a step bump, is blissfully ignored by ESC. The ESC design requires further research to improve convergence and to handle possible transitions of the variator from open loop stable to open loop unstable. An instrumented test vehicle would be useful for this, as well as to see if the excitation signal of ESC does not interfere with other components of the driveline. The first objective for the variator controller in a vehicle is to track the speed ratio reference that is prescribed by the driveline management system. The second objective is to improve the variator efficiency. This first objective is not included here, since only fixed transmission ratios are considered. The extension of ESC, as described here, towards a feedback controller that satisfies both control objectives will be subject of another contribution. REFERENCES
Fig. 15. Pressure for a secondary disturbance at 400 s (black line: Measurement; grey line: Reference).
the test rig is large in comparison with the slip in a passenger car. This is caused by the primary inertia of the test rig. This inertia is larger than the one in a passenger car and, as a consequence, the peaks in the relative slip are larger. The peaks in the relative slip appear and disappear within the interval 400.5 s when the fast changes in occur. The secondary disturbance is not noticed in the ESC output. This is deduced from Fig. 15. The visible effects of the disturbance in and are filtered by the ESC filters and are not visible in . In this case the variator operates as a torque fuse for high frequency components in the disturbance, the relative slip is shortly increased, and those components are not suppressed by the variator control system. VII. DISCUSSION From a series of test rig experiments with a pushbelt variator it is confirmed that the variator efficiency and the speed ratio as function of the secondary clamping force have a global maximum and that the clamping force for which is maximal is only slightly different from the clamping force for which is maximal. These findings need to be confirmed for other conditions. In automotive applications the variator efficiency is not measured, whereas the speed ratio follows from the standard measurements of the primary and secondary angular velocity. This
[1] R. Pfiffner and L. Guzzella, “Optimal operation of CVT-based powertrains,” Int. J. Robust Nonlinear Control, vol. 11, no. 11, pp. 1003–1021, 2001. [2] K. G. O. van de Meerakker, P. C. J. N. Rosielle, B. Bonsen, T. W. G. L. Klaassen, and N. J. J. Liebrand, “Mechanism proposed for ratio and clamping force control in a CVT,” presented at the FISITA World Autom. Congr., Barcelona, Spain, 2004, Art. No. F2004F108. [3] S. Shastri and A. A. Frank, “Comparison of energy consumption and power losses of a conventionally controlled CVT with a servo-hydraulic controlled CVT and with a belt and chain as the torque transmitting element,” presented at the Proc. Int. Continuously Variable Hybrid Transmission Congr., Davis, CA, 2004, Art. No. 04CVT-55. [4] F. van der Sluis, T. van Dongen, G.-J. van Spijk, A. van der Velde, and A. van Heeswijk, “Fuel consumption potential of the pushbelt CVT,” presented at the FISITA World Autom. Congr., Yokohama, Japan, 2006, Art. No. F2006P218. [5] S. Akehurst, N. D. Vaughan, D. A. Parker, and D. Simner, “Modelling of loss mechanisms in a pushing metal V-belt continuously variable transmission. Part 1: Torque losses due to band friction,” in Proc. Inst. Mechan. Eng., Pt. D: J. Auto. Eng., 2004, vol. 218, no. 11, pp. 1269–1281. [6] S. Akehurst, N. D. Vaughan, D. A. Parker, and D. Simner, “Modelling of loss mechanisms in a pushing metal V-belt continuously variable transmission. Part 2: Pulley deflection losses and total torque loss validation,” in Proc. Inst. Mechan. Eng., Pt. D: J. Auto. Eng., 2004, vol. 218, no. 11, pp. 1283–1293. [7] S. Akehurst, N. D. Vaughan, D. A. Parker, and D. Simner, “Modelling of loss mechanisms in a pushing metal V-belt continuously variable transmission. Part 3: Belt slip losses,” in Proc. Inst. Mechan. Eng., Pt. D: J. Auto. Eng., 2004, vol. 218, no. 11, pp. 1295–1306. [8] B. Bonsen, T. W. G. L. Klaassen, K. G. O. van de Meerakker, M. Steinbuch, and P. A. Veenhuizen, “Analysis of slip in a continuously variable transmission,” presented at the ASME Int. Mechan. Eng. Congr., Washington, DC, 2003, Art. No. IMECE2003-41360. [9] K. Sakagami, T. Fujii, H. Yoshida, and T. Yagasaki, “Study on belt slip behavior in metal V-belt type CVT,” in Proc. Int. Congr. Continuously Variable Hybrid Transmissions, 2007, pp. 135–139. [10] M. van Drogen and M. van der Laan, “Determination of variator robustness under macro slip conditions for a push belt CVT,” presented at the SAE World Congr., Detroit, MI, 2004, Art. No. 2004-01-0480.
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[11] T. W. G. L. Klaassen, “The Empact CVT: Dynamics and control of an electromechanically actuated CVT,” Ph.D. dissertation, Dept. Mechan. Eng., Eindhoven Univ. Technol., Eindhoven, The Netherlands, 2007. [12] B. Bonsen, T. W. G. L. Klaassen, R. J. Pulles, S. W. H. Simons, M. Steinbuch, and P. A. Veenhuizen, “Performance optimisation of the push-belt CVT by variator slip control,” Int. J. Veh. Design, vol. 39, no. 3, pp. 232–256, 2005. [13] K. J. Åström and B. Wittenmark, Adaptive Control, ser. Addison-Wesley Series in Electrical Engineering: Control Engineering, 2nd ed. Reading, MA: Addison-Wesley, 1995. [14] M. Krstić and H.-H. Wang, “Stability of extremum seeking feedback for general nonlinear dynamic systems,” Automatica, vol. 36, no. 4, pp. 595–601, 2000. [15] S. van der Meulen, B. de Jager, E. van der Noll, F. Veldpaus, F. van der Sluis, and M. Steinbuch, “Improving pushbelt continuously variable transmission efficiency via extremum seeking control,” in Proc. 3rd IEEE Multi-Conf. Syst. Control, 2009, pp. 357–362. [16] E. van der Noll, F. van der Sluis, T. van Dongen, and A. van der Velde, “Innovative self-optimising clamping force strategy for the pushbelt CVT,” presented at the SAE World Congr., Detroit, MI, 2009, Art. No. 2009-01-1537. [17] M. H. Smith, E. J. Barth, N. Sadegh, and G. J. Vachtsevanos, “The horsepower reserve formulation of driveability for a vehicle fitted with a continuously variable transmission,” Veh. Syst. Dynam., vol. 41, no. 3, pp. 157–180, 2004.
[18] Siemens AG, Munich, Germany, “ROTEC—Low-voltage motors for variable-speed drives,” Advance Catalog DA 65.3, 1997. [19] K. B. Ariyur and M. Krstić, Real-Time Optimization by ExtremumSeeking Control. Hoboken, NJ: Wiley, 2003. [20] G. Lechner and H. Naunheimer, Automotive Transmissions: Fundamentals, Selection, Design and Application. Berlin, Germany: Springer-Verlag, 1999. [21] S. van der Meulen, B. de Jager, F. Veldpaus, and M. Steinbuch, “Robust extremum seeking and speed ratio control for high-performance CVT operation,” in Proc. 6th Int. Conf. Continuously Variable Hybrid Transmissions, 2010, pp. 110–115. [22] S. H. van der Meulen, “High-performance control of continuously variable transmissions,” Ph.D. dissertation, Eindhoven Univ. Technol., Eindhoven, The Netherlands, 2010. [23] T. Oomen, S. van der Meulen, O. Bosgra, M. Steinbuch, and J. Elfring, “A robust-control-relevant model validation approach for continuously variable transmission control,” in Proc. Amer. Control Conf., 2010, pp. 3518–3523.
