High-Bandwidth Clamp Force Control for an Electromechanical Brake

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1 Downloaded from SAE International by University of Melbourne, Tuesday, January 07, :17:13 PM 590 High-Bandwidth Clamp Force Control for an Electromechanical Brake Chih Feng Lee and Chris Manzie The University of Melbourne Published 09/17/2012 Copyright 2012 SAE International doi: / saepcelec.saejournals.org ABSTRACT A controller that fully utilizes the available motor capacity of an electromechanical brake to achieve high closed-loop bandwidth is proposed. The controller is developed based on the time-optimal switching curve derived from Pontryagin's Maximum Principle. The control input is scheduled using a switching surface based on the current motor velocity and position offset. Robustness to modeling errors is achieved by introducing a boundary layer in vicinity of the switching curve, reminiscent of a high gain controller. A flexible tuning procedure is also developed to aid in practical implementation, allowing a balanced choice between tracking speed and energy usage. The controller is implemented on a production-ready prototype EMB, and tested over different braking scenarios to assess the performance and robustness relative to the benchmark controllers. It is demonstrated that significant improvements in step response and dynamic tracking are obtained using the proposed approach. CITATION: Lee, C. and Manzie, C., "High-Bandwidth Clamp Force Control for an Electromechanical Brake," SAE Int. J. Passeng. Cars - Electron. Electr. Syst. 5(2):2012, doi: / INTRODUCTION Compared with traditional hydraulic alternatives, brakeby-wire (BBW) technology offers reductions in components, space requirements, environmental impact, and increased design flexibility through software upgrades. Currently there are two implementations of BBW actuation, differentiated by the availability of self-reinforcement braking mechanisms. Electromechanical brakes (EMBs) are commonly used to identify those designs without self-reinforcement braking, while electronic wedge brakes (EWBs) denote actuators with self-reinforcement braking. Clamping force generation in EMBs and EWBs is facilitated by one or multiple electric motors, connected through a reduction mechanism to the brake pads. The control problem dealt with in this work is the clamp force feedback control in EMB, which directly correlates to the servo control of the brake piston position. It is noted that this problem is closely related to the EWB force control problem that manipulates the wedge position. From a performance perspective, EMBs and EWBs potentially offer finer clamp force control actuation with high closed-loop bandwidth relative to hydraulic systems. This capability is beneficial for maintaining high performance anti-lock braking and electronic stability control. However, the clamp force control performance is dependent on the adopted control architecture and the selected gains. Existing EMB and EWB clamp force controllers borrowed the control architecture from linear motion control applications, where cascaded proportional-integral controllers are implemented [1,2,3]. However, more so than in most motion control applications the EMB and EWB are inherently nonlinear devices, and frequently operate over the full range of operating conditions. As a consequence, high closed-loop bandwidth across the whole clamp force region is not guaranteed using this design with a fixed set of gains. Gain scheduled controllers were consequently proposed in EMB control albeit with tedious tuning procedures [4,5]. As an alternative, this paper proposes a controller design procedure that fully utilizes the available motor capacity to achieve rapid setpoint tracking performance with high closedloop bandwidth, at the same time, offers the flexibility of detuning for energy saving considerations. The controller is developed based on the time-optimal switching curve derived from Pontryagin's Maximum Principle [6]. The control input is scheduled using a switching surface based on the current motor velocity and position offset. Robustness to modeling errors is achieved by introducing a boundary layer in vicinity of the switching curve, reminiscent of a high gain controller. A tuning procedure is developed to aid in practical implementation. The procedure is based on optimizing a function of convergence speed and control effort. A choice of

2 Downloaded from SAE International by University of Melbourne, Tuesday, January 07, :17:13 PM Lee et al / SAE Int. J. Passeng. Cars - Electron. Electr. Syst. / Volume 5, Issue 2(September 2012) 591 appropriate weightings on the objective function allows tradeoffs between these two conflicting metrics to be made. The proposed controller is implemented on a productionready prototype EMB, and tested over different braking scenarios to assess the performance and robustness relative to the benchmark controllers. It is demonstrated that significant improvements in step response and dynamic tracking are obtained using the proposed approach. Therefore, it is suggested as a candidate for future production EMB controller design. The organization of the paper is as follows. The second section describes the control-oriented model of an EMB that serves as the basis for the proceeding controller designs. An ideal time-optimal control for an EMB is first proposed, followed by the robust near time-optimal control (NTOC). A gain tuning procedure for the NTOC will be presented, proceeded by discussions on obtaining robust performance against modeling errors and noise. Then, experimental results will be presented and discussed. The conclusion encompasses the main results and establishes further work required to take advantage of the achieved high closed-loop bandwidth. CONTROL-ORIENTED EMB MODEL A production-ready EMB prototype shown in Figure 1 is chosen in this work to illustrate the proposed high-bandwidth clamp force control design procedure. The EMB consists of a three phase, permanent magnet synchronous motor, a gear reduction stage, a ball screw, a floating caliper and control electronics. A low order, lumped parameter model from [4] is chosen as the basis for the proposed controller design. While these simplifications facilitate the construction of a low order controller, simulations in this paper are performed using higher-order models presented in [7]. velocity respectively. The motor model is approximated using an equivalent DC motor, where the torque is given by where i q and K t are the motor quadrature current and motor constant respectively. The lumped stiffness is presented as the relationship between the clamp force and motor position. It is experimentally found to be sufficiently well approximated using a third-order polynomial. Additionally, it is modeled using a first-order continuously differentiable function with respect to the motor position, a requirement for the derivation of the ideal time-optimal solution. With x 1 = 0 defined at the contact position between the brake pad and disk, the clamp force is represented by By considering Coulomb, viscous, and load-dependent sources of friction, the lumped non-linear friction torque about the motor's rotational axis is modeled as where the friction model parameters D, C, and G are the viscous friction coefficient, load-independent Coulomb's friction torque, and load-dependent Coulomb's friction torque coefficient respectively. The tanh function is used to approximate the Coulomb's friction in order to ensure the first-order continuous differentiability of friction torque with respect to the states, a requirement for the derivation of the ideal time-optimal solution. A sufficiently large ε is chosen such that the friction model is close to the experimentally obtained measurements. Additionally, motor torque and velocity are limited to Tm T mmax and x 2 θ max respectively, while the clamp force region considered in this work is between 0 to 30 kn. The experimentally identified EMB parameters are listed in Table 1. (2) (3) (4) Table 1. EMB Parameters Figure 1. The production-ready prototype electromechanical brake used in this work. The torque equation about the motor's rotational axis is given by where F cl, J, N, T f, T m, x 1, and x 2 represent brake clamp force, effective moment of inertia, gear ratio, friction torque, motor torque, motor angular position, and motor angular (1) A wide range of BBW actuators can be approximated using the lumped parameter control-oriented EMB model (1),

3 Downloaded from SAE International by University of Melbourne, Tuesday, January 07, :17:13 PM 592 Lee et al / SAE Int. J. Passeng. Cars - Electron. Electr. Syst. / Volume 5, Issue 2(September 2012) which will be utilized for the controller design presented in the following section. CONTROLLER DESIGN In this section, an idealized time-optimal control design is first illustrated, which provides the theoretical upper-bound limit for rapid clamp force reference tracking. However, this controller is impractical for implementation, and leads to chatter during operation due to modeling errors and measurement noise. The robust near-time-optimal control (NTOC) structure is then proposed to eliminate chatter while facilitating fast tracking response. Finally, tuning guidelines are provided to assist the design, followed by a discussion on the choice of tuning parameters. on these factors. During experimentation, it was observed that chatter led to overheating of the power electronics. Furthermore, it is also expected that chatter could lead to excessive mechanical wear and component fatigue failures, therefore should be avoided. Idealized Time-Optimal Control High performance BBW actuators are capable of tracking the clamp force reference in minimum time, given the power rating and velocity limits of the drive. A fast clamp force tracking response is important, especially during safetycritical operations, such as ABS. Henceforth, time-optimality is a natural control performance metric. The idealized time-optimal clamp force tracking solution can be derived from Pontryagin's Maximum Principle [6] and determined numerically, which consists of two intervals of maximum acceleration and deceleration, and possibly an intermediate interval with constant maximum velocity. Figure 2 illustrates the two different step responses for 0.8 kn and 4 kn. Beginning at t = 0 when x1 = x2 = 0, a clamp force setpoint reference is commanded, demanding the clamp force and motor velocity to reach the target states, which are the commanded clamp force reference and zero velocity respectively in minimum time. The time-optimal transient consists of maximum initial acceleration, operation at maximum speed and controlled deceleration to the prescribed target point, as illustrated in the 4 kn step response. The motor torque trajectory demonstrates the control switching to initiate the maximum acceleration, to maintain constant speed (from 0.03 to 0.05 seconds), and to start the maximum deceleration. Control chatter is observable during the constant speed region, as maintaining velocity at perfectly constant is not possible due to modeling and numerical errors during simulation, and discrepancies arise from modeling, measurement and discretization in experiments. In the state plane, the intervals of maximum acceleration and deceleration are characterized by arcs, while the interval at constant velocity is portrayed by a straight line parallel to the horizontal axis. For a small step response, as depicted by the 0.8 kn case, the maximum velocity may not be reached. The acceleration is immediately followed by deceleration with the switch-over taking place at fractions of the distance traveled. Notably, control chatter occurs for both the large and small step commands when the references are reached. This is due to modeling, numerical and measurement errors, which also signifies the sensitivity of the idealized time-optimal control Figure 2. Time-optimal clamp force setpoint tracking for 800 N (dashed) and 4000 N (solid) with constrained motor torque and velocity: (a) Response vs. time; (b) State plane, where the dots mark the 0.01 seconds intermissions. To aid in real-time implementation, it is beneficial to compute the time-optimal solution offline and store the control switching instants in a lookup table referred to as a switching curve. Construction of Switching Curve As eluded before, there can be at most one segment of maximum acceleration or deceleration that leads to the final state. If the final state is given, the terminal segment can be calculated by solving the equation of motion (1) backwards in time, which is attainable by solving the following negated

4 Downloaded from SAE International by University of Melbourne, Tuesday, January 07, :17:13 PM Lee et al / SAE Int. J. Passeng. Cars - Electron. Electr. Syst. / Volume 5, Issue 2(September 2012) 593 equation of motion (1) from t = [ t a,0], t a > 0 using a standard software package such as MATLAB: where T m,cv represents the motor torque that maintains constant velocity at the velocity limit, governed by The final state is employed as the initial condition and torque with the maximum magnitude, T m = ±T mmax taken as input. A few samples of terminal segment are plotted in Figure 3. It is noted that the control magnitude switching takes place when the time-optimal trajectory coincides to the terminal segment, also called the switching curve. The control magnitude on area on the left of the switching curve is T mmax, and T mmax on the right. (5) The idealized time-optimal solution represents the theoretical upper-bound performance subjected to the power rating and velocity of the motor. However, this control law is impractical, as the smallest modeling error and measurement noise will cause the control to chatter between the maximum and the minimum values across the switching curve in vicinity of the zero state errors. In the context of EMB application, control chatter will lead to vibration and overheating of the electric motor. Nevertheless, the idealized control will be used as the basis for further development of a practical high-bandwidth control robust to modeling error and noises. Robust Near-Time-Optimal Control To alleviate control chatter, a near-time-optimal clamp force control robust to modeling and measurement errors is presented. Simple implementation with rapid tracking performance is emphasized during the design. To this end, the storage requirement for the exact switching curve (8) is first reduced by considering an approximated switching curve : Figure 3. Time-optimal switching curves are plotted for various x 1 * and x 2 *. Cases for x 1 *=0,10,,50 rad are shown, with state constraints indicated by dashed lines. Idealized Time-Optimal Control Law Based on the current motor position and velocity with respect to the switching curve, a time-optimal control law can be devised. To this end, the motor position offset before control switching is defined as where denotes the motor position where control switching should occur. The idealized time-optimal control law can be stated as (6) The motor position offset before control switching using the approximated switching curve is where the tracking error is designated by (9) (10) (11) Figure 4 shows the exact switching curves overlaid after the change of variable. It is noted that the shape of the exact switching curves differs due to the plant nonlinearity. The family of curves is approximated using a single switching curve that corresponds to. (7)

5 Downloaded from SAE International by University of Melbourne, Tuesday, January 07, :17:13 PM 594 Lee et al / SAE Int. J. Passeng. Cars - Electron. Electr. Syst. / Volume 5, Issue 2(September 2012) where x2sat > 0. The overall controller can be summarized as (15) The control magnitude over different region in the state plane is shown in Figure 5 and the controller implementation is shown in Figure 6. In order to aid implementation, tuning guidelines for x2sat and ssat will be provided in the following section. Figure 4. Overlaid exact switching curve plotted in error state plane. The family of curves will be approximated using the thick curve. As a remedy to alleviate chatter, the sgn function that is reminiscent of an infinite gain operator and governs the abrupt control switching at the switching curve is replaced by a saturation function. It provides a finite slope approximation and gives the system a finite bandwidth, therefore is much more practical for the EMB applications. This leads to a graduate transition of the control magnitude across the error state plane and in vicinity of the origin. The resultant finite bandwidth control law was proposed in Lee et. al. [8], which was optimized for ramp tracking performance, and is represented by the following: (12) where ks 2, ssat > 0, and e2sat > 0 are tunable gains. The sat function is defined as: Figure 5. Control magnitude in different regions of the e1 - x2 state plane. Figure 6. Block diagram of the EMB clamp force tracking control structure. Gain Tuning Guidelines (13) However, in clamp force control applications, and are not available online and have to be numerically approximated from. To avoid numerical differentiation of the reference, which is prone to noise generation, a simpler control law that considers reference tracking is proposed: The tuning parameters ks, ssat and x2sat represent the size of the boundary layer of the NTOC. However, it is unclear how these parameters affect the performance and robustness of the controller. Hence, a systematic procedure is proposed to address these problems. Noted that the control chatter occurs in vicinity of zero tracking error, therefore the dynamics of the EMB system around this region is first studied. To this end, the switching curve is linearized at the origin, and is given by and is optimized for step (16) where the first order linear approximation is (14)

6 Downloaded from SAE International by University of Melbourne, Tuesday, January 07, :17:13 PM Lee et al / SAE Int. J. Passeng. Cars - Electron. Electr. Syst. / Volume 5, Issue 2(September 2012) 595 (17) This approximation is shown in Figure 7, indicating that ζ (x 2 ), which represents the higher order terms is small near the origin, and increases significantly when x2 > 200. (21) Therefore, the relationship between the new gains K 1, K 2 and the original tuning parameters are (22) The LQR design problem is characterized by the following objective function consists of penalties on tracking speed z(t) and control effort, weighted by ρ: (23) The 2 represents the square of the Euclidean norm of a vector. The penalty on tracking speed is a weighted function of the motor position and velocity tracking, and is given by Figure 7. Comparison of switching curve with its first order linear approximation. When the state is in vicinity of zero tracking error and the control magnitude is unsaturated, the closed-loop dynamics can be calculated from (1), (10), (14), (16) and the maximum angular acceleration, to be (18) (24) Since the tuning procedure now translates to the selection of weightings directly related to tracking error and control effort, it is more transparent than the direct alteration of the boundary layer size. Figure 8 illustrates the trend of s sat and x 2sat with respect to ρ and γ. It is noticed that increasing the penalty on control effort, ρ leads to a larger boundary layer. Furthermore, increasing the penalty on motor velocity tracking resulted in the stretching of the boundary layer vertically. (19) Noted that (17) can be approximated using a linear equation, whereby suitable tools for controller gain tuning are readily available. One of these choices is the linear quadratic regulator (LQR) design, based upon optimizing an objective function with respect to tracking speed and control effort. To this end, the closed-loop dynamics are re-written as (20) where the last term is treated as a small perturbation to the system. The input of the new system (18), u equates to Figure 8. Controller parameters s sat and x 2sat are plotted against ρ and γ, when k s =2. The robustness of the controller is studied using the openloop transfer function; from the plant input to the controller's

7 Downloaded from SAE International by University of Melbourne, Tuesday, January 07, :17:13 PM 596 Lee et al / SAE Int. J. Passeng. Cars - Electron. Electr. Syst. / Volume 5, Issue 2(September 2012) output. Figure 9 and Figure 10 show the effect of ρ and γ on the open-loop gain. At low frequencies, the open-loop gain magnitude matches that of the EMB transfer function from u to x1. Additionally, the transfer function has an infinite positive gain margin, where the penalty on control effort ρ varies the magnitude of the Bode plot vertically. Furthermore, larger γ results in a larger phase margin, which facilitates controller implementation with larger delays. 2. Controller synthesis using the structure (8), (13) and (14). 3. Select the controller gains using the guidelines provided in this subsection. RESULTS AND DISCUSSIONS Experimental investigation of the proposed controller is conducted on the static test rig of a production-ready prototype EMB shown in Figure 11. The test bench consists of a PC laptop, a data acquisition system, a 42V power supply, and an EMB. The proposed controller is embedded on the EMB onboard controller, and measurements are taken using the onboard sensors, both with sampling periods of 250 Hz. The PC sends the clamp force reference and logs the measured signals in real-time. Communications between the PC and the EMB are performed via a CAN bus. Figure 9. Open-loop gain for several values of ρ (fixed γ =0.04). This parameter moves the whole magnitude Bode plot vertically. Figure 11. Bench top experimental setup for a production-ready prototype EMB. Figure 10. Open-loop gain for several values of γ (fixed ρ=4). Larger values for this parameter result in a larger phase margin. The performance of an EMB is in part determined by how well it responds to the clamp force reference. To facilitate repeatability of experiments, the controller performance is systematically studied using a set of standard waveforms as the reference command. Tracking characteristics, such as short rise time and high bandwidth are significant for the assessment of EMB performance. The former is important for providing a rapid brake response during critical braking scenario, while the latter is vital for ABS operations. The tracking error is assessed using the mean absolute percentage error, defined as To summarize, the implementation procedure for the proposed method can be broken up into three steps: 1. Determine the switching curve: Firstly, a switching curve is produced by integrating (5) between t = [ ta,0], ta > 0, setting the input Tm = ±Tmmax and the initial condition x* = (0,0). Secondly, change the coordinate of x-axis from x1 to e1 and approximate the curve using a polynomial fit, as shown in Figure 4. For the final release of the control software during production, it is expected that the mean system parameters for a particular EMB design will be used to calculate the switching curve. (25) where e, Fcl,r(mean) and T respectively represent the clamp force tracking error, the mean and the period of the clamp force reference trajectories. The proposed controller is benchmarked against the baseline PI controller proposed in [4], which represents the standard clamp force controller design commonly adopted in the industry. In order to perform fair tests, the PI gains are tuned such that the available motor torque is fully utilized

8 Downloaded from SAE International by University of Melbourne, Tuesday, January 07, :17:13 PM Lee et al / SAE Int. J. Passeng. Cars - Electron. Electr. Syst. / Volume 5, Issue 2(September 2012) while avoiding integral windup due to input and state constraints over the whole operating range. Figure 12 and Figure 13 respectively show the large and small step transients. It is noticed that the baseline PI and the NTOC performed similarly for the large step maneuver, indicating that the baseline PI is well-tuned for large step references. However, for smaller references, the baseline PI shows sluggish performance because the full motor torque range is not utilized. On the other hand, NTOC fully utilizes the motor torque range and resulted in rapid setpoint tracking, while alleviating the control chatter in the ideal time-optimal control. It is noted that the NTOC tracking responses are uniform across the whole operating range. In contrast, the baseline PI which is tuned for large setpoint tracking maneuver performed poorly for small reference levels. Conversely, if the baseline PI is tuned for small maneuvers, then it may cause significant overshoot and integral windup during large setpoint tracking. While gain scheduled PI was proposed in [4,5] to improve the uniformity of performance across operating range, the tuning procedure is tedious. Consequently, the relatively simple control structure and tuning procedure offered by the NTOC warrants itself to be an appealing candidate for future EMB control design. Figure 12. Comparison of baseline PI (dashed) and near time-optimal (solid) controller for 15kN step tracking. The characteristic of PI and NTOC is also investigated using a phase portrait, where the motor velocity is plotted against the position in Figure 14. The NTOC demonstrates longer acceleration period and achieves higher velocity, while maintaining the trajectories at high velocity range for longer. There is a slight velocity constraint violation for NTOC 15 kn setpoint transient, due to modeling error and noise. However, the observed level of constraint violation is insignificant from an operational perspective. 597 Figure 13. Comparison of baseline PI (dashed) and near time-optimal (solid) controller for 2kN step tracking. Figure 14. Trajectories in error state plane for 2 kn and 15 kn setpoint tracking. The switching curve (solid) and its associated boundary layer (dash-dot) are shown. The reduction of tracking error using NTOC is demonstrated using ramp and sinusoidal waveforms. The ramp tracking profile between 0 to 20 kn with 1 s period is shown in Figure 15, while the 1 Hz sinusoidal tracking is shown in Figure 16. The NTOC tracks the ramp reference very well, where the experimentally obtained tracking response visually appears to overlap the reference (see Figure 15). Furthermore, reduction in phase shift is observed in the sinusoidal tracking (see Figure 16). Overall, the NTOC

9 Downloaded from SAE International by University of Melbourne, Tuesday, January 07, :17:13 PM 598 Lee et al / SAE Int. J. Passeng. Cars - Electron. Electr. Syst. / Volume 5, Issue 2(September 2012) demonstrates improvements in tracking error, henceforth higher bandwidth. Figure 17. Energy used for step responses initiated from 0 kn. Figure 15. Comparison of baseline PI (dashed) and near time-optimal (solid) controller for ramp responses from 0 to 20 kn. Figure 18. Rise time for step responses initiated from 0 kn. CONCLUSIONS Figure 16. Comparison of baseline PI (dashed) and near time-optimal (solid) controller for 1 Hz sinusoidal responses. The energy usage and rise-time for step transients with NTOC and baseline PI is shown in Figure 17 and Figure 18. In order to achieve shorter rise time, NTOC uses more energy to achieve higher acceleration. Nevertheless, the proposed design method is flexible enough to penalize energy usage, in the face of losing tracking speed. A high-bandwidth clamp force control for a BBW actuator is presented. The proposed controller achieves neartime-optimal performance by making full use of the available motor capacity, while taking into account of motor torque and velocity constraints. A controller parameters tuning method that balances between tracking speed and energy usage is also given to aid in practical implementation. Experimental results demonstrate lower tracking error and higher bandwidth are achieved using the proposed method, suggesting it to be a candidate for future EMB controller design. Although this work considers an EMB with force sensor, it can be adapted for EMBs without a force sensor if the mapping between clamp force and motor position is available. This can be achieved by implementing a clamp force estimator.

10 Downloaded from SAE International by University of Melbourne, Tuesday, January 07, :17:13 PM Lee et al / SAE Int. J. Passeng. Cars - Electron. Electr. Syst. / Volume 5, Issue 2(September 2012) 599 Further research directions include taking advantage of the achieved high-bandwidth for additional functionality of the BBW actuators, as well as validating the braking performance with the higher-level controller such as ABS and electronic stability control (ESC). REFERENCES 1. Maron, C., Dieckmann, T., Hauck, S., and Prinzler, H., Electromechanical Brake System: Actuator Control Development System, SAE Technical Paper , 1997, doi: / Schwarz, R., Isermann, R., Böhm, J., Nell, J. et al., Modeling and Control of an Electromechanical Disk Brake, SAE Technical Paper , 1998, doi: / Fox, J., Roberts, R., Baier-Welt, C., Ho, L. et al., Modeling and Control of a Single Motor Electronic Wedge Brake, SAE Technical Paper , 2007, doi: / Line, C., Manzie, C., and Good, M., Electromechanical brake modelling and control: from PI to MPC, IEEE Transaction Control System Technology, 16(3): , Jo, C., Hwang, S., and Kim, H., Clamping-force control for electromechanical brake, IEEE Transaction Vehicular Technology, 59(7): , Pontryagin, L. S., Boltyanskii, V. G., Gamkrelidze, R. V., and Mishchenko, E. F., The mathematical theory of optimal processes, Wiley, New York, Line, C., Manzie, C., and Good, M., Control of an Electromechanical Brake for Automotive Brake-By-Wire Systems with an Adapted Motion Control Architecture, SAE Technical Paper , 2004, doi: / Lee, C.F. and Manzie, C., Near-time-optimal tracking controller design for an automotive electromechanical brake, Proceedings of the Institution of Mechanical Engineers, Part I: Journal of Systems and Control Engineering, 226(4): , CONTACT INFORMATION Chih Feng Lee Department of Mechanical Engineering The University of Melbourne Parkville, Victoria 3010 Australia. Tel.: +61(3) c.lee33@pgrad.unimelb.edu.au ACKNOWLEDGMENTS This research was in part supported by the Australian Research Council through the grant FT