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Vehicle System Dynamics Vol. 44, No. 5, May 2006, 387406 Control of a hydraulically actuated continuously variable transmission MICHIEL PESGENS*, BAS VROEMEN, BART STOUTEN, FRANS VELDPAUS and MAARTEN STEINBUCH Drivetrain Innovations b.v., Horsten 1, 5612 AX, The Netherlands Technische Universiteit Eindhoven, PO Box 513, 5600 MB Eindhoven, The Netherlands Vehicular drivelines with hierarchical powertrain control require good component controller tracking, enabling the main controller to reach the desired goals. This paper focuses on the development of a transmission ratio controller for a hydraulically actuated metal push-belt continuously variable transmission (CVT), using models for the mechanical and the hydraulic part of the CVT. The controller consists of an anti-windup PID feedback part with linearizing weighting and a setpoint feedforward. Physical constraints on the system, especially with respect to the hydraulic pressures, are accounted for using a feedforward part to eliminate their undesired effects on the ratio. The total ratio controller guarantees that one clamping pressure setpoint is minimal, avoiding belt slip, while the other is raised above the minimum level to enable shifting. This approach has potential for improving the efficiency of the CVT, compared to non-model based ratio controllers. Vehicle experiments show that adequate tracking is obtained together with good robustness against actuator saturation. The largest deviations from the ratio setpoint are caused by actuator pressure saturation. It is further revealed that all feedforward and compensator terms in the controller have a beneficial effect on minimizing the tracking error. Keywords: Continuously variable transmission; Feedforward compensation; Feedback linearization; Hydraulic actuators; Constraints 1. Introduction The application of a continuously variable transmission (CVT) instead of a stepped transmis- sion is not new. Already in the 50s Van Doorne introduced a rubber V-belt CVT for vehicular drivelines. Modern, electronically controlled CVTs make it possible for any vehicle speed to operate the combustion engine in a wide range of operating points, for instance in the fuel optimal point. For this reason, CVTs get increasingly important in hybrid vehicles, see for example 13. Accurate control of the CVT transmission ratio is essential to achieve the intended fuel economy and, moreover, ensure good driveability. The ratio setpoint is generated by the hierarchical (coordinated) controller of figure 1. This controller uses the accelerator pedal position as the input and generates setpoints for the local controllers of the throttle and of the CVT. *Corresponding author. Email: pesgensdtinnovations.nl Michiel Pesgens was previously affiliated with Technische Universiteit Eindhoven. Vehicle System Dynamics ISSN 0042-3114 print/ISSN 1744-5159 online 2006 Taylor F shift = F p (r cvt , prime s ) F s (10) An axial force difference F shift , weighted by the thrust ratio results in a ratio change, and is therefore called the shift force. The occurrence of p in the model (10) is plausible because an increasing shift force is needed for decreasing pulley speeds to obtain the same rate of ratio change. The reason is that less V-shaped blocks enter the pulleys per second when the pulley speed decreases. As a result the radial belt travel per revolution of the pulleys must increase and this requires a higher shift force. However, it is far from obvious that the rate of ratio change is proportional to both the shift force and the primary pulley speed. k r is a non-linear function of the ratio r cvt and has been obtained experimentally. Experimental data has been used to obtain a piecewise linear fit, which are depicted in figure 6. The estimation of k r has 392 M. Pesgens et al. Figure 5. Contour plot of (r cvt , prime s ). Figure 6. Fit of k r (r cvt ); greyed-out dots correspond to data with reduced accuracy. Hydraulically actuated CVT 393 Figure 7. Comparison of shifting speed, Ides model vs. measurement. been obtained using the inverse Ide model: k r (r cvt ) = r cvt | p |F shift (11) In the denominator F shift is present, the value of which can become (close to) zero. Obviously, the estimate is very sensitive for errors in F shift when its value is small. The dominant dis- turbances in F shift are caused by high-frequency pump generated pressure oscillations, which do not affect the ratio (due to the low-pass frequency behavior of unmodeled variator pulley inertias). The standard deviation of the pressure oscillations and other high-frequency distur- bances has been determined applying a high-pass Butterworth filter to the data of F shift .To avoid high-frequency disturbances in F shift blurring the estimate of k r , estimates for values of F shift smaller than at least three times the disturbances standard deviation have been ignored (these have been plotted as grey dots in figure 6), whereas the other points have been plotted as black dots. The white line is the resulting fit of this data. The few points with negative value for k r have been identified as local errors in the map of . To validate the quality of Ides model, the shifting speed r cvt , recorded during a road exper- iment, is compared with the same signal predicted using the model. Model inputs are the hydraulic pulley pressures (p p , p s ) and pulley speeds ( p , s ) together with the estimated primary pulley torque ( T p ). The result is depicted in figure 7. The model describes the shifting speed well, but for some upshifts it predicts too large values. This happens only for high CVT ratios, i.e. r cvt 1.2, where the data of is unreliable due to extrapolation (see figure 5). 3. The hydraulic system The hydraulic part of the CVT (see figure 3) consists of a roller vane pump directly connected to the engine shaft, two solenoid valves and a pressure cylinder on each of the moving pulley 394 M. Pesgens et al. sheaves. The volume between the pump and the two valves including the secondary pulley cylinder is referred to as the secondary circuit, the volume directly connected to and including the primary pulley cylinder is the primary circuit. Excessive flow in the secondary circuit bleeds off toward the accessories, whereas the primary circuit can blow off toward the drain. All pressures are gage pressures, defined relative to the atmospheric pressure. The drain is at atmospheric pressure. The clamping forces F p and F s are realized mainly by the hydraulic cylinders on the move- able sheaves and depend on the pressures p p and p s . As the cylinders are an integral part of the pulleys, they rotate with an often very high speed, so centrifugal effects have to be taken into account and the pressure in the cylinders will not be homogeneous. Therefore, the clamping forces will also depend on the pulley speeds p and s . Furthermore, a preloaded linear elastic spring with stiffness k spr is attached to the moveable secondary sheave. This spring has to guarantee a minimal clamping force when the hydraulic system fails. Together this results in the following relations for the clamping forces: F p = A p p p + c p 2 p (12) F s = A s p s + c s 2 s k spr s s + F 0 (13) where c p and c s are constants, whereas F 0 is the spring force when the secondary moveable sheave is at position s s = 0. Furthermore, A p and A s are the pressurized piston surfaces. In the hydraulic system of figure 3, the primary pressure is smaller than the secondary pressure if there is an oil flow from the secondary to the primary circuit. Therefore, to guarantee that in any case the primary clamping force can be up to twice as large as the secondary clamping force, the primary piston surface A p is approximately twice as large as the secondary surface A s . It is assumed that the primary and the secondary circuit are always filled with oil of constant temperature and a constant air fraction of 1%. The volume of circuit ( = p, s) is given by: V = V ,min + A s (14) V ,min is the volume if s = 0 and A is the pressurized piston surface. The law of mass conservation, applied to the primary circuit, combined with equation (14), results in: oil V p p p = Q sp Q pd Q p,leak Q p,V (15) Q sp is the oil flow from the secondary to the primary circuit, Q pd is the oil flow from the primary circuit to the drain, Q p,leak is the (relatively small) oil flow leaking through narrow gaps from the primary circuit and Q p,V is the oil flow due to a change in the primary pulley cylinder volume. Furthermore, oil is the compressibility of oil. The oil flow Q sp is given by: Q sp = c f A sp (x p ) radicalBigg 2 |p s p p |sign(p s p p ) (16) where c f is a constant flow coefficient and is the oil density. A sp , the equivalent valve opening area for this flow path, depends on the primary valve stem position x p . Flow Q pd follows from: Q pd = c f A pd (x p ) radicalBigg 2 p p (17) Here, A pd is the equivalent opening area of the primary valve for the flow from primary circuit to the drain. The construction of the valve implies that A sp (x p ) A pd (x p ) = 0 for all possible x p . Hydraulically actuated CVT 395 Flow Q p,leak is assumed to be laminar with leak flow coefficient c pl , so: Q p,leak = c pl p p (18) The flow due to a change of the primary pulley cylinder volume is described by: Q p,V = A p s p (19) with s p given by equation (4). Application of the law of mass conservation to the secondary circuit yields oil V s p s = Q pump Q sp Q sa Q s,leak Q s,V (20) The flow Q pump , generated by the roller vane pump, depends on the angular speed e of the engine shaft, on the pump mode m (m = SS for single sided and m = DS for double sided mode), and the pressure p s at the pump outlet, so Q pump = Q pump ( e ,p s ,m). Q sa is the flow from the secondary circuit to the accessories and Q s,leak is the leakage from the secondary circuit. Flow Q sa is modeled as: Q sa = c f A sa (x s ) radicalBigg 2 |p s p a |sign(p s p a ) (21) where A sa , the equivalent valve opening of the secondary valve, depends on the valve stem position x s . The laminar leakage flow Q s,leak is given by (with flow coefficient c sl ): Q s,leak = c sl p s (22) The flow due to a change of the secondary pulley cylinder volume is: Q s,V = A s s s (23) with s s according to equation (3). The accessory circuit contains several passive valves. In practice, the secondary pressure p s will always be larger than the accessory pressure p a , i.e. no backflow occurs. The relation between p a and p s is approximately linear, so p a = c a0 + c a1 p s (24) with constants c a0 0 and c a1 (0, 1). Now that a complete model of the pushbelt CVT and its hydraulics is available, the controller and its operational constraints can be derived. 4. The constraints The CVT ratio controller (in fact) controls the primary and secondary pressures. Several pressure constraints have to be taken into account by this controller: 1. the torque constraints p p ,torque to prevent slip on the pulleys; 2. the lower pressure constraints p p ,low to keep both circuits filled with oil. Here, fairly arbitrary, p p,low = 3 bar is chosen. To enable a sufficient oil flow Q sa to the accessory circuit, and for a proper operation of the passive valves in this circuit it is necessary that 396 M. Pesgens et al. Q sa is greater than a minimum flow Q sa,min . A minimum pressure p s,low of 4 bar turns out to be sufficient; 3. the upper pressure constraints p p ,max , to prevent damage to the hydraulic lines, cylinders and pistons. Hence, p p,max = 25 bar, p s,max = 50 bar; 4. the hydraulic constraints p p ,hyd to guarantee that the primary circuit can bleed off fast enough toward the drain and that the secondary circuit can supply sufficient flow toward the primary circuit. The pressures p p,torque and p s,torque in constraint 1 depend on the critical clamping force F crit , equation (5). The estimated torque T p is calculated using the stationary engine torque map, torque converter characteristics and lock-up clutch mode, together with inertia effects of the engine, flywheel and primary gearbox shaft. A safety factor k s = 0.3 with respect to the estimated maximal primary torque T p,max has been introduced to account for disturbances on the estimated torque T p , such as shock loads at the wheels. Then the pulley clamping force (equal for both pulleys, neglecting the variator efficiency) needed for torque transmission becomes: F torque = cos() (| T p |+k s T p,max ) 2 R p (25) Consequently, the resulting pressures can be easily derived using equations (12) and (13): p p,torque = 1 A p parenleftBig F torque c p 2 p parenrightBig (26) p s,torque = 1 A s parenleftbig F torque c s 2 s k spr s s F 0 parenrightbig (27) Exactly the same clamping strategy has been previously used by ref. 3 during test stand efficiency measurements of this gearbox and test vehicle road tests. No slip has been reported during any of those experiments. As the main goal of this work is to an improved ratio tracking behavior, the clamping strategy has remained unchanged. A further elaboration of constraints 4 is based on the law of mass conservation for the primary circuit. First of all, it is noted that for this elaboration the leakage flow Q p,leak and the compressibility term oil V p p p may be neglected because they are small compared to the other terms. Furthermore, it is mentioned again that the flows Q sp and Q pd can never be unequal to zero at the same time. Finally, it is chosen to replace the rate of ratio change r cvt by the desired rate of ratio shift r cvt,d , that is specified by the hierarchical driveline controller. If r cvt,d 0 and Q sp = 0. Constraint 4 with respect to the primary pulley circuit then results in the following relation for the pressure p p,hyd : p p,hyd = oil 2 parenleftbigg A p p max(0, r cvt,d ) c f A pd,max parenrightbigg 2 (28) where A pd,max is the maximum opening of the primary valve in the flow path from the primary cylinder to the drain. In a similar way, a relation for the secondary pulley circuit pressure p s,hyd in constraint 4 can be derived. This constraint is especially relevant if r cvt 0, i.e. if the flow Q sp from the secondary to the primary circuit has to be positive and, as a consequence, Q pd = 0. This then Hydraulically actuated CVT 397 results in: p s,hyd = p p,d + oil 2 parenleftbigg A p p max(0, r cvt,d ) c f A sp,max parenrightbigg 2 (29) where A sp,max is the maximum opening of the primary valve in the flow path from the secondary to the primary circuit. For the design of the CVT ratio controller it is advantageous to reformulate to constraints in terms of clamping forces instead of pressures. Associating a clamping force F , with the pressure p , and using equations (12) and (13) this results in the requirement: F ,min F F ,max (30) with minimum pulley clamping forces: F ,min = max(F ,low ,F ,torque ,F ,hyd ) (31) 5. Control design It is assumed in this section that at each point of time t, the primary speed p (t), the ratio r cvt (t), the primary pressure p p (t) and the secondary pressure p s (t) are known from measurements, filtering and/or reconstruction. Furthermore, it is assumed that the CVT is mounted in a vehicular driveline and that the desired CVT ratio r cvt,d (t) and the desired rate of ratio change r cvt,d (t) are specified by the overall hierarchical driveline controller. This implies, for instance, that at each point of time the constraint forces can be determined. The main goal of the local CVT controller is to achieve fast and accurate tracking of the desired ratio trajectory. Furthermore, the controller should also be robust for disturbances. An important subgoal is to maximize the efficiency. It is quite plausible (and otherwise supported by experiments, 3) that to realize this sub-goal the clamping forces F p and F s have to be as small as possible, taking the requirements in equation (30) into account. The output of the ratio controller is subject to the constraints of equation (31). The constraints F F ,min can effectively raise the clamping force setpoint of one pulley, resulting in an undesirable ratio change. This can be counteracted by raising the opposite pulleys clamping force as well, using model-based compensator terms in the ratio controller. Using Ides model, i.e. using equation (10), expressions for the ratio change forces F p,ratio and F s,ratio (figure 8) can be easily derived: F p,ratio = F shift,d + F s,min (32) F s,ratio = F shift,d + F p,min (33) where F shift,d is the desired shifting force, basically a weighted force difference between both pulleys. As explained earlier, depends on prime s , which in turn depends on F s . This is an implicit relation (F s,ratio depends on F s ), which has been tackled by calculating from pressure measurements. It will now be shown that at each time, one of the two clamping forces is equal to F ,min , whereas the other determines the ratio. Using equations (30), (32) and (33) the desired primary 398 M. Pesgens et al. Figure 8. Ratio controller with constraints compensation and secondary clamping forces F p,d and F s,d are given by: F p,d = F p,ratio F s,d = F s,min bracerightBigg if F shift,d + F s,min F p,min (34) F p,d = F p,min F s,d = F s,ratio bracerightBigg if F shift,d + F s,min F p,min F p,min F s,ratio = F shift,d if F shift,d + F s,min F p,max F s,ratio F s,max (40) If either pressure saturates (p p = p p,max or p s = p s,max ), the shifting speed error inevitably becomes large. The anti-windup algorithm ensures stability, but the tracking behavior will deteriorate. This is a hardware limitation which can only be tackled by enhancing the variator and hydraulics hardware. The advantage of a conditional anti-windup vs. a standard (linear) algorithm is that the linear approach requires tuning for good performance, whereas the con- ditional approach does not. Furthermore, the performance of the conditional algorithm closely resembles that of a well-tuned linear mechanism. 6. Experimental results As the CVT is already implemented in a test vehicle, in-vehicle experiments on a roller bench have been performed to tune and validate the new ratio controller. To prevent a non- synchronized operation of throttle and CVT ratio, the accelerator pedal signal (see figure 1) has been used as the input for the validation experiments. The coordinated controller will track the maximum engine efficiency operating points. A semi kick-down action at a cruise-controlled speed of 50 km/h followed by a pedal back out has been performed in a single reference exper- iment. The recorded pedal angle (see figure 9) has been applied to the coordinated controller. This approach cancels the limited human drivers repeatability. The upper plot of figure 10 shows the CVT ratio response calculated from speed measure- ments using equation (1), the plot depicts the tracking error. As this is a quite demanding experiment, the tracking is still adequate. Much better tracking performance can be obtained with more smooth setpoints, but the characteristics of the responses will become less distinct as well. Figure 11 shows the primary and secondary pulley pressures. The initial main peak in the error signal (around t = 1.5 s) is caused by saturation of the secondary pressure (lower plot of figure 11), due to a pump flow limitation. If a faster initial response were required, adaptation of the hydraulics hardware would be necessary. After the initial fast downshift, the ratio reaches its setpoint (around t = 7 s) before downshifting again. All changes in shifting 400 M. Pesgens et al. Figure 9. Pedal input for the CVT powertrain. direction (t = 1.3, t = 1.6 and t = 7.5 s) occur with a relatively small amount of overshoot, which shows that the integrator anti-windup algorithm performs well. Looking at the primary pressure in the vicinity of t = 1.5 s, it can be observed that this pressure peaks repeatedly above its setpoint. This behavior is caused