Robust Extremum Seeking and Speed Ratio Control for High-Performance CVT Operation

Transcription

1 Robust Extremum Seeking and Speed Ratio Control for High-Performance CVT Operation Stan van der Meulen, Bram de Jager, Frans Veldpaus, and Maarten Steinbuch Abstract The variator in a pushbelt continuously variable transmission (CVT) enables a stepless variation of the transmission ratio within a finite range. Nowadays, the variator is electronically controlled and the variator control objectives are twofold: ) tracking a transmission ratio reference; ) optimizing the variator efficiency. Recently, the extremum seeking control (ESC) technique is exploited in view of optimizing the variator efficiency, which only uses measurements from sensors that are standard. However, the operating conditions are fixed and tracking a transmission ratio reference is omitted, for simplicity. In this paper, extensions are proposed that overcome these limitations. This is achieved via the construction of a disturbance feedforward control design and a speed ratio control design. Experiments illustrate the effectiveness of these extensions when the operating conditions are varied and tracking a transmission ratio reference is required. I. INTRODUCTION The pushbelt continuously variable transmission (CVT) incorporates several components, e.g., the variator and the hydraulic actuation system. The variator enables that the transmission ratio is continuously varied in between two bounds, i.e., Low and High. The variator consists of a metal V-belt, i.e., a pushbelt, which is clamped between two pairs of conical sheaves, i.e., two pulleys, see Fig.. A primary (input, subscript p ) pulley and a secondary (output, subscript s ) pulley are distinguished. Each pulley consists of one axially moveable sheave and one axially fixed sheave. Each axially moveable sheave is connected to a hydraulic cylinder, which is pressurized by the hydraulic actuation system. Essentially, the hydraulic actuation system translates a desired pressure p j,ref into a realized pressure p j, where the pressure p j in the hydraulic cylinder is directly related to the clamping force F j on the axially moveable sheave, where j {p, s}. The level of the clamping forces determines the torque capacity, whereas the ratio of the clamping forces determines the transmission ratio. When the level of the clamping forces is too high, variator efficiency is compromized, since the friction loss is increased. When the level of the clamping forces is too low, variator damage is introduced, since the slip is increased. Hence, there exists a choice of the clamping forces that guarantees the functionality of the CVT and optimizes the efficiency of the CVT, which demands a control design in which both items are explicitly addressed. The objective for the variator control system is twofold: ) tracking a speed ratio reference r s,ref, which is prescribed 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, PO Box 3, MB Eindhoven, The Netherlands. S.H.v.d.Meulen@tue.nl, A.G.de.Jager@tue.nl, F.E.Veldpaus@tue.nl, M.Steinbuch@tue.nl This research is partially funded by Bosch Transmission Technology, Tilburg, The Netherlands. Fig.. a R s T p F s ω p x s x p ω s F p β T s R p Schematic illustration of pushbelt variator. by the driveline control system; ) optimizing the variator efficiency η. The transmission ratio is represented by the speed ratio r s, which is easily computed from the ratio of the measurements of the angular velocities. The variator efficiency η is defined by the ratio of the powers, which are not measured. Traditionally, the majority of the approaches control the speed ratio via the primary pulley with the primary hydraulic circuit and the torque capacity via the secondary pulley with the secondary hydraulic circuit, see, e.g., []. The primary pressure that is required in order to achieve the speed ratio is computed by means of a feedback controller (closed loop). Several feedback control designs are encountered, e.g., PI(D) control [], fuzzy control [], robust control [3]. The secondary pressure that is required in order to transfer the torque is computed by means of a variator model (open loop). Since the variator model is uncertain, a safety strategy is employed, which utilizes a safety factor. Generally, the safety factor ranges from. [-] to.3 [-], which implies that the variator efficiency is seriously compromized. Recently, the variator efficiency is explicitly addressed in the control design that is proposed in []. The existence of a certain optimum for the variator efficiency as a function of the slip is shown by means of experiments. As a result, a straightforward approach is to control the slip in such a way that a certain slip reference is tracked, which corresponds to the optimum variator efficiency []. However, this approach involves two issues. First, the determination of the slip reference [, Section 7.]. Since the optimum variator efficiency depends on, e.g., the transmission ratio, the variator load, and the variator wear, the determination of the slip reference is not straightforward and often time-consuming, which is typically caused by the unreliability of the available variator

2 models. Second, the reconstruction of the slip in the variator. This typically requires a dedicated sensor, e.g., measurement of the pushbelt running radius [] or measurement of the axially moveable sheave position [], which increases both the complexity and the costs. In addition, the reconstruction of the slip in the variator on the basis of one of these measurements is extremely sensitive to deformations in the variator, which are unknown. These drawbacks are avoided in the control design that is proposed in [7], [], which effectively improves the variator efficiency and only uses the measurements of the angular velocities and the secondary pressure, which are standard. The control design exploits the observation that the maximum of the (ps, rs ) equilibrium map and the maximum of the (ps, η) equilibrium map are achieved for values of ps that nearly coincide. This motivates the consideration of the input-output map in which the secondary pressure ps is the input and the speed ratio rs is the output, although the location of the maximum is unknown. For this reason, the maximum of the input-output map is found by means of extremum seeking control (ESC) [9], which aims to adapt the input in order to maximize the output. In [], however, two simplifications are adopted. First, the torques that are exerted on the variator, which are considered in terms of disturbances, are stationary of nature. Obviously, the torque disturbances that are encountered when the variator is installed in a vehicle are transient of nature. These torque disturbances possibly enforce a transition of the variator behavior from open loop stable to open loop unstable, since the torque capacity is nearly consumed. This possibly destabilizes the ESC feedback mechanism, since the ESC feedback mechanism is not robustified. Second, the control problem for optimizing the variator efficiency is isolated from the control problem for tracking the speed ratio reference. With this simplification, a single-input singleoutput (SISO) control problem is obtained (input: ps, output: rs ). Without this simplification, a multi-input single-output (MISO) control problem is obtained (inputs: pp and ps, output: rs ), which is not treated. The main contribution of this paper is twofold. The first contribution concerns a solution for the robustness problem of the ESC feedback mechanism in view of a primary side disturbance that resembles a depression of the accelerator pedal. This is achieved via the addition of a disturbance feedforward control design. The second contribution concerns a solution for the MISO control problem that simultaneously satisfies both variator control objectives. This is achieved via the integration of the ESC design with a speed ratio control (SRC) design. The remainder of this paper is organized as follows. The preliminaries are addressed in Section II, which includes the definitions and the experimental setup. In Section III, the ESC design is described and the disturbance feedforward control design is introduced. In Section IV, the ESC design is optimized and the SRC design is introduced. Finally, the paper concludes with a discussion in Section V. Notation: Consider the variator in Fig.. The torques that are exerted on the variator are denoted by Tp and Ts. Furthermore, the angular velocities are denoted by ωp and ωs, the clamping forces by Fp and Fs, the axially moveable sheave positions by xp and xs, and the running radii by Rp and Rs. Finally, a denotes the variator centre distance and β denotes half the pulley wedge angle, i.e., β = [deg]. II. P RELIMINARIES A. Definitions The geometric ratio rg and the speed ratio rs of the variator are defined by: rg = rs = Rp Rs ωs. ωp () () The relative slip ν is defined by: ν= ωp Rp ωs Rs rs =, ωp Rp rg (3) see []. The variator efficiency η is defined by: η= Pout Ts ωs =, Pin Tp ωp () where Pin and Pout denote the input power and the output power, respectively. B. Experimental Setup The experimental setup is depicted in Fig. and consists of five main components. These are given by two identical electric motors, a pushbelt variator, a hydraulic actuation system, and a data acquisition system. The experimental setup incorporates additional sensors, which are primarily used for analysis purposes. Æ Å Ç Ã Á À Â Ä Fig.. Experimental setup with pushbelt variator (À: Pushbelt variator; Á: Primary torque sensor; Â: Secondary torque sensor; Ã: Primary electric motor; Ä: Secondary electric motor; Å: Hydraulic actuation system; Æ: Accumulator; Ç: Data acquisition system). III. ROBUST E XTREMUM S EEKING C ONTROL The control configuration, which incorporates the ESC design, is introduced in Section III-A. The ESC design is subsequently described in Section III-B. The ESC design options are addressed in Section III-C. A detailed analysis of stability and performance of the ESC design is presented in []. Then, the background of the torque disturbances is

3 highlighted in Section III-D. Finally, the robustness of the ESC design with respect to a primary side disturbance is analyzed in Section III-E. A. Control Configuration Consider the control configuration that is depicted in Fig. 3. Here, G V denotes the relation between p s and r s and T Hp and T Hs denote the relations between p p,ref and p p and p s,ref and p s, respectively. Furthermore, T p and T s denote the deviations that are possibly superposed to the nominal values T p and T s. gradient update law, which enables the adaptation of ˆp s,ref towards the optimum input p s,ref. p s,ref T Hs α m sin(πf m t) ˆp s,ref s I p s H b ξ ξ G V H l ξ 3 r s ξ H b T p T p p s,ref [ ]. ESC Fig. 3. p p,ref T Hp p p [ ] r s. GV p s,ref p s T Hs T s Control configuration. T p T s T s B. Extremum Seeking Control Design The feedback mechanism is depicted in Fig.. Obviously, the feedback mechanism utilizes a sinusoidal perturbation α m sin(πf m t), which is added to ˆp s,ref, i.e., the estimate of the optimum input p s,ref. As a result, the input of the hydraulic actuation system p s,ref is defined by: p s,ref (t) = ˆp s,ref (t) + α m sin(πf m t), () where α m denotes the perturbation amplitude and f m denotes the perturbation frequency. When ˆp s,ref is on either side of p s,ref, the periodic perturbation enforces a periodic response of the output of the variator r s, which is either in phase or out of phase with the periodic perturbation. With this information, the feedback mechanism from r s to ˆp s,ref is designed, which consists of the following operations: ξ = H b (s)r s () ξ = H b (s)p s (7) ξ 3 = ξ ξ () ξ = H l (s)ξ 3 (9) ˆp s,ref = s Iξ. () Here, H b (s) denotes a band-pass filter, H l (s) denotes a lowpass filter, and I denotes the integrator gain. The band-pass filter H b (s) enforces the suppression of DC components and noise for r s and p s, which results in ξ and ξ, respectively. As a result, ξ and ξ are approximately two sinusoids, which are out of phase for ˆp s,ref > p s,ref and in phase for ˆp s,ref < p s,ref. In either case, the product of both sinusoids ξ 3 has a DC component. The low-pass filter H l (s) extracts the DC component of ξ 3, which results in ξ. Finally, ˆp s,ref results from integration of ξ, with integrator gain I. The initial condition for the integrator is equal to p s,ref, which corresponds to a stationary operating point. Observe that (9) contains the gradient information and () represents the Fig.. Feedback mechanism from r s to p s,ref for ESC. Obviously, the feedback mechanism incorporates five design options. These are the perturbation amplitude α m, the perturbation frequency f m, the band-pass filter H b (s), the low-pass filter H l (s), and the integrator gain I. The selection of these design options is closely related to the proof of stability for the closed loop system, which is addressed in [9]. The feedback mechanism in [9] is similar to the feedback mechanism in Fig.. However, a high-pass filter is employed instead of a band-pass filter. The main reason for the application of a band-pass filter concerns the suppression of noise. Three assumptions are required, see [9], which are satisfied for the operating conditions that normally occur, see []. Then, convergence of the solution (ˆp s,ref (t), ξ (t), r s (t)) towards a certain neighborhood of the point (p s,ref,, r s,max) is guaranteed by [9, Theorem.] for a suitable choice of the design options. A suitable choice of the design options is made in Section III-C. C. ESC Design Options The perturbation amplitude α m, the perturbation frequency f m, the band-pass filter H b (s), the low-pass filter H l (s), and the integrator gain I are given by: α m =.7 () f m = 3 () H b (s) = πf m.s (πf m) s + πf m.s + (3) H l (s) = π s + () I =. () D. Background of Torque Disturbances The torque disturbances are induced by the internal combustion engine (ICE) (primary side) or the road (secondary side). The primary torque disturbance T p is typically imposed by the driver who depresses or releases the accelerator pedal. The secondary torque disturbance T s is typically induced by obstacles or unevennesses. Regarding the primary side, the engine control unit (ECU) of the ICE measures several variables, e.g., the accelerator pedal position or the throttle valve angle and the crankshaft angular velocity. These variables in combination with the performance map of the ICE enable the estimation of the torque. Subsequently, this estimate of the torque is transferred to the transmission

4 control unit (TCU). Hence, a priori information with respect to the primary torque disturbance T p is typically available. Regarding the secondary side, a priori information with respect to the secondary torque disturbance T s is typically unavailable. E. Primary Disturbance A specific experiment is performed in order to analyze the robustness of the ESC feedback mechanism with respect to a primary torque disturbance T p,ref. The experiment is started from a certain stationary operating point. The ESC feedback mechanism converges towards a small neighborhood of the extremum r s = r s,max, which is approximately reached for t [s]. Finally, the deviation T p,ref = [Nm] is superposed to the nominal value T p,ref = 7 [Nm] for t. [s]. The chosen deviation T p,ref is representative for a depression of the accelerator pedal with moderate impact. The experiment is performed twice. First, without feedforward of the primary torque disturbance T p,ref. Second, with feedforward of the primary torque disturbance T p,ref. A contribution p s,ref is added to the estimate of the optimum input ˆp s,ref, which is defined by: p s,ref = (a) p s,ref = A s T p,ref cos(β) µr p, (b) for the case without feedforward and the case with feedforward, respectively. Here, A s denotes the secondary hydraulic cylinder pressure surface and µ denotes the traction coefficient, i.e., µ =.9 [-]. The presentation of the experimental results shows the case without feedforward on the left and the case with feedforward on the right. Tp [Nm] η [%] ν [%] Fig.. Experimental results for High and primary disturbance (Left: Without feedforward; Right: With feedforward). ps,ref [bar] ˆps,ref [bar] ps,ref [bar] ps [bar] Fig. 7. Experimental results for High and primary disturbance (Left: Without feedforward; Right: With feedforward). Ts [Nm] Fig.. Experimental results for High and primary disturbance (Left: Without feedforward; Right: With feedforward) (black: Measurement; grey: Reference). The torques are depicted in Fig.. For the case without feedforward, the variator is unable to transfer the additional torque, which is given by the deviation T p (Fig. (top left)). For the case with feedforward, the variator is able to transfer the additional torque, which is given by the deviation T p (Fig. (top right)). For the case without feedforward, the relative slip ν sharply rises (Fig. (bottom left)). The experiment is stopped when the relative slip exceeds the predefined value ν = [%]. Obviously, the variator enters in the macro-slip region, i.e., the variator behavior is open loop unstable. Observe that the variator efficiency η sharply drops (Fig. (top left)). The contribution p s,ref and the estimate of the optimum input ˆp s,ref are depicted in Fig. 7 (left), together with the secondary pressure reference p s,ref and the secondary pressure p s. Apart from the sinusoidal perturbation, p s,ref is only determined by ˆp s,ref, since the case without feedforward is considered, i.e., p s,ref is equal to (a). It is observed that the estimate of the optimum input ˆp s,ref slightly decreases, which is undesired, since the deviation T p is obviously positive. This behavior is also observed from the signals in the feedback loop (Fig. (left)), especially from ξ, which contains the gradient information. Hence, the ESC feedback mechanism without feedforward is unable to adapt to the 3

5 x 3 x 3 ω p ξ [-] r s,ref [ ] SRC ESC p p,ref p s,ref [ ] THpp T Hps T Hsp T Hss p p p s [ GVp G Vs ] r s ξ [-] T s ξ3 [-] x x Fig. 9. Control configuration. ξ [-] x x Fig.. Experimental results for High and primary disturbance (Left: Without feedforward; Right: With feedforward). change of the operating condition. For the case with feedforward, the relative slip ν slightly increases (Fig. (bottom right)). Obviously, the variator operates in the micro-slip region, i.e., the variator behavior is open loop stable. Observe that the variator efficiency η slightly increases (Fig. (top right)). The contribution p s,ref and the estimate of the optimum input ˆp s,ref are depicted in Fig. 7 (right), together with the secondary pressure reference p s,ref and the secondary pressure p s. Apart from the sinusoidal perturbation, p s,ref is mainly determined by p s,ref, since the case with feedforward is considered, i.e., p s,ref is equal to (b). It is observed that the estimate of the optimum input ˆp s,ref slightly changes, since the contribution p s,ref is apparently inaccurate. This behavior is also observed from the signals in the feedback loop (Fig. (right)), especially from ξ, which contains the gradient information. Hence, the ESC feedback mechanism with feedforward is able to adapt to the change of the operating condition. IV. INTEGRATION WITH SPEED RATIO CONTROL The control configuration, which incorporates the ESC design, is introduced in Section IV-A. The SRC design is subsequently described in Section IV-B. A detailed analysis of stability and performance of the SRC design is presented in []. The ESC design options are addressed in Section IV- C. In Section IV-D, a closed loop experiment is performed, which shows that the variator control objectives are simultaneously satisfied. A. Control Configuration Consider the control configuration that is depicted in Fig. 9. A cascade control design is employed. The inner loop includes the hydraulic actuation system, which is closed loop controlled []. The desired pressures p p,ref and p s,ref are the inputs and the realized pressures p p and p s are the outputs. The outer loop includes the variator, which is controlled by the combination of ESC and SRC. B. Speed Ratio Control Design The SRC design K(s) computes the input p p,ref on the basis of the tracking error r s,ref r s. The SRC design K(s) is given by the product of the following parts: K gain = 3 (7a) K int (s) = (7b) s (. π K lead (s) = s + ) (7c) (πf K notch (s) = m) s + πf m.s + (πf m) s + πf m.s + (7d) K roll-off (s) = (s + π). (7e) The notch filter (7d) is implemented in order to reduce the suppression of the periodic response that is enforced by the sinusoidal perturbation of the ESC design. C. ESC Design Options The perturbation amplitude α m, the perturbation frequency f m, the band-pass filter H b (s), the low-pass filter H l (s), and the integrator gain I are given by: α m =.7 () f m = (9) H b (s) = πf m.3s (πf m) s + πf m.3s + () H l (s) = π s + () I =. () The design options are changed in comparison with the design options in Section III-C. Both the perturbation frequency f m and the integrator gain I are increased in order to improve the convergence speed, see []. The increase of the perturbation frequency f m enforces a change of the bandpass filter H b (s). Finally, the damping ratio of the band-pass filter H b (s) is modified. D. Closed Loop Experiment The operation of the combination of the ESC design and the SRC design is evaluated by means of a closed loop experiment, see Fig. 9. The operating conditions are given by ω p = [rpm] and T s = [Nm], whereas the speed ratio reference is equal to r s,ref =. [-]. The closed loop experiment is started from a stationary

6 operating point, which is defined by the initial condition for the integrator of the ESC feedback mechanism. The initial condition for the integrator is equal to p s,ref = 7. [bar]. Actually, this stationary operating point corresponds to the secondary pressure reference p s,ref that is achieved by the absolute safety strategy, where the absolute safety factor is equal to.3 [-]. This absolute safety strategy is commonly used by the automotive industry [3]. For this reason, this absolute safety strategy is adopted for comparison purposes. The experimental results for the proposed strategy and the absolute safety strategy are depicted in Figs.,, and. From Fig., it follows that the speed ratio reference r s,ref is accurately tracked. The variator efficiency η for both the proposed strategy and the absolute safety strategy is depicted in Fig.. Obviously, when the pressure references for the proposed strategy decrease in comparison with the absolute safety strategy, see Fig., the variator efficiency increases. The gain with respect to the variator efficiency is approximately equal to. [%]. pp [bar] ps [bar] Fig.. Experimental results for pressures (Left: Proposed strategy; Right: Absolute safety strategy) (black: Measurement; grey: Reference). rs [-] Fig.. Experimental results for speed ratio (Left: Proposed strategy; Right: Absolute safety strategy) (black: Measurement; grey: Reference). V. DISCUSSION In this paper, a variator control system for a pushbelt CVT is proposed on the basis of the extremum seeking control (ESC) technique, which effectively optimizes the variator efficiency. A disturbance feedforward control design is proposed in order to deal with a primary torque disturbance η [%] Fig.. strategy; 3 3 Experimental results for variator efficiency (black: Proposed grey: Absolute safety strategy). that is typically imposed by the driver. A speed ratio control design is proposed in order to deal with a speed ratio reference that is typically imposed by the driveline control system. The effectiveness of both extensions is successfully demonstrated by means of experiments. Two opportunities for future research are highlighted. First, the acceleration of the convergence of the ESC design, via optimization or extension of the ESC design. Second, the integration of the separate control designs and the evaluation for a driving cycle. REFERENCES [] K. Sato, R. Sakakiyama, and H. Nakamura, Development of electronically controlled CVT system equipped with CVTip, in Proc. 99 Int. Conf. Continuously Variable Power Transmissions, no., Yokohama, Japan, 99, pp. 3. [] W. Kim and G. Vachtsevanos, Fuzzy Logic Ratio Control for a CVT Hydraulic Module, in Proc. IEEE Int. Symp. Intell. Contr., vol., Rio, Patras, Greece,, pp.. [3] K. Adachi, T. Wakahara, S. Shimanaka, M. Yamamoto, and T. Oshidari, Robust control system for continuously variable belt transmission, JSAE Rev., vol., no., pp. 9, 999. [] 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. Des., vol. 39, no. 3, pp. 3,. [] T. W. G. L. Klaassen, The Empact CVT: Dynamics and Control of an Electromechanically Actuated CVT, Ph.D. Thesis, Eindhoven University of Technology, Eindhoven, The Netherlands, 7. [] H. Nishizawa, H. Yamaguchi, and H. Suzuki, Friction Characteristics Analysis for Clamping Force Setup in Metal V-belt Type CVTs, R&D Rev. Toyota CRDL, vol., no. 3, pp.,. [7] 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, in Proc. SAE 9 World Congr., no , Detroit, MI, 9, CD-ROM. [] 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. Contr., Saint Petersburg, Russia, 9, pp [9] M. Krstić and H.-H. Wang, Stability of extremum seeking feedback for general nonlinear dynamic systems, Automatica, vol. 3, no., pp. 9,. [] S. van der Meulen, B. de Jager, F. Veldpaus, and M. Steinbuch, Combining Extremum Seeking Control and Tracking Control for High- Performance CVT Operation, in Proc. 9th IEEE Conf. Decision Contr., Atlanta, GA,. [] 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. Contr. Conf., Baltimore, MD,, pp [] Y. Tan, D. Nešić, and I. Mareels, On the choice of dither in extremum seeking systems: A case study, Automatica, vol., no., pp.,. [3] 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, in Proc. FISITA World Automotive Congr., no. FP, Yokohama, Japan,, CD-ROM.