ROBUST CONTROL OF AN AUTOMOTIVE ELECTROMECHANICAL BRAKE. Chris Line, Chris Manzie and Malcolm Good. The University of Melbourne, Australia

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1 ROBUST CONTROL OF AN AUTOMOTIVE ELECTROMECHANICAL BRAKE Chris Line, Chris Manzie and Malcolm Good The University of Melbourne, Australia Abstract: This paper presents a robust H optimal control design for an automotive electromechanical brake (EMB). The design considers parametric uncertainty, unmodelled dynamics and the need for robust stability and performance. The capacity of a robust EMB control to manage uncertainty and performance is assessed and experimental and simulation results are presented. The tradeoffs involved in meeting a satisfactory performance specification in the presence of a reasonable set of actuator uncertainties are investigated and discussed. The robust control is benchmarked against a standard cascaded PI control for the EMB. Copyright 26 IFAC Keywords: Robust control, Uncertainty, Automotive, Brakes, Force control 1. INTRODUCTION Electromechancial brakes are currently being developed as a potential successor to the hydraulic brakes that are standard on modern automobiles. They offer an attractive alternative with potential advantages including component reduction, system weight reduction, plug, bolt and play modularity, improved brake performance and a brake system that supports a drive-by-wire platform. As shown in Fig. 1, electromechanical brakes use a motor to clamp and release the brake. In the present design a motor torque may be developed between the stator 1 and rotor 2, driving a planetary gear 3 and ball screw 4, to advance the piston 5 and apply the brake pad 6 against the brake rotor. The brake clamp force, F cl, is then reacted over the bridge 7 of the floating caliper by the opposing brake pad. Electromechanical brakes are installed on the vehicle with a mechanical, power and communications interface. Brake commands from the driver or a high level vehicle control are then transmitted to the actuator via a time-triggered communication protocol. Depending on how functionality is arranged, the command may set the brake force or a mode of operation such as standby, off or anti-lock braking (ABS). While brake torque control would be ideal for managing vehicle dynamics, feedback measurements are difficult. For this reason prototype EMBs have typically resorted to controlling brake clamp force, F cl, as the next best, or least removed, alternative. Prior published work on EMB control has mainly considered the issue of brake performance, but there has been little written regarding the handling of uncertainty. A degree of actuator uncertainty may be particularly due to the effects of wear and temperature. For example, stiffness is known to vary with brake pad thickness and temperature (Schwarz, Isermann, Böhm, Nell and Rieth, 1999). Friction in the EMB mechanism may also be liable to significant variation with wear. It is important that such variation and uncertainty is considered in the control design. Fig. 1. EMB from PBR Australia Pty. Ltd. (Wang, Kaganov, Code and Knudtzen, 25)

2 The importance of robust EMB control has motivated the prior work of (Krishnamurthy, Lu, Khorrami and Keyhani, 25) and (Lu, 25). Lu et al. derived a cascaded proportional control and determined conditions for bounded stability by following a method of robust backstepping. However, achieving closed-loop stability is not particularly difficult since the brake apply is open-loop stable from motor torque to clamp force. A considerably more difficult challenge is achieving both robust stability and robust performance on an EMB. This more challenging objective is attempted in this paper. Motivated by the need to ensure stability and performance, this paper presents a robust H control design for an electromechanical brake. In preparation, Section 2 first outlines the EMB model used for simulation and control design. Parametric uncertainty and unmodelled dynamics are then considered in Section 3 with a robust EMB control design. The design includes structured uncertainty to avoid unnecessarily conservative perturbations. A modified control architecture is introduced using feedback linearisation and inverse gain scheduling to facilitate a reduced specification of structured uncertainty. The uncertainty and performance specification is given in Section 4. Section 5 follows with controller synthesis and reduction. The control is then demonstrated with experimental and simulation results in Section 6. A critical discussion ensues and leads to a conclusion on the robust EMB design in Section ELECTROMECHANICAL BRAKE MODEL The simplified, lumped parameter EMB model used for simulation and control design is shown in Fig. 2 and was based on that in (Line, Manzie and Good, 24). The input is the motor quadrature current, i q, and the output is the brake clamp force, F cl. The halfcaliper model assumes that the brake clamp force developed at one brake pad is reacted over the bridge of the floating caliper by the opposing brake pad. The simplified EMB model considers a torque balance about the motor axis where the net torque of the motor, T m, clamp load, T L, and friction, T F, give the angular acceleration, & θ, of the effective inertia, J. T T T J & θ so that, m L F = i K F N T J & θ q t cl F = (1) where K t is the motor torque constant and N is the total gear reduction. Fig. 2. Simplified electromechanical brake model for simulation and control design. Fig. 3. Friction model describing the load dependent static, Coulomb, and viscous friction The friction, T F (θ &,F cl,t E ), is modelled as a function of the motor velocity, & θ, brake clamp force and external (non-friction) torque, T E =T m -T L. The friction model is depicted in Fig. 3 and may be described as a friction-velocity map with a load dependency factor, G. The friction torque is given by, T F D & θ + ( C + GFcl ) sign( & θ ) = TE ( Ts + GFcl ) sign( TE ) & θ > ε if & θ < ε and TE < otherwise ( T + GF ) s (2) Here, D is the viscous friction coefficient, C is the Coulomb friction, T s is the maximum loadindependent static friction and ε is a small bound for zero velocity detection. The model has three conditions that describe friction in the state of motion, & θ > ε, and near zero velocity, & θ < ε, when the external torque is either fully opposed by the static friction or the maximum static friction is exceeded. The stiffness from to 4 kn was described by the experimentally determined relationship, F cl x x 3.97x x >.125 = (3).1295x otherwise where F cl is given in kilonewtons and the nominal spindle position, x, in millimeters. It should be noted that the simplified stiffness model does not capture the secondary effect of viscoelastic stiffness hysteresis. cl The model parameters were determined from system identification experiments and are summarised in Table 1.

3 Table 1 Parameters of EMB model Parameter Value Units K t.697 N m/a N.263 mm/rad J kg m 2 T s.379 N m D N m s/rad C.34 N m G N m/n The model of the EMB mechanism was augmented with a single-phase motor approximation. The motor circuit comprised a series.5 Ω resistance, R, and 56 µh inductance, L. The electronic transients were described by, diq vq = Riq + L + & θ Kb (4) dt where the back electromotive force (EMF) coefficient was K b =2/3K t and the pulse-width modulated voltage, v q, was limited to 42 V. Finally, a sensor model was included for closed-loop simulations. The controller operated on a merged force-position feedback to handle operation in the clearance region. In this approach the clearance may be represented using negative force values. The sensor model described the transition between force and position (or negative force ) feedback near the zero contact point. 3. ROBUST CONTROL DESIGN The objective of the EMB robust H control design is to handle model uncertainty by optimising the performance under the worst case scenario. The control design is based on a linear system with bounded uncertainty to describe a set of EMB actuators. For actuators within the set of uncertain systems, the robust control design may then guarantee robust stability and a limit on the worst case performance. A careful structuring of model uncertainty is particularly important in the robust control design to avoid unnecessarily conservative perturbations. Approximating the nonlinear plant in Fig. 2 with a linear system and a large degree of uncertainty to allow for unmodelled nonlinearity is not an attractive approach. It may also be difficult to describe discontinuous functions such as friction is this manner. Instead, a reduced specification of uncertainty may be achieved by first applying techniques of feedback linearisation and inverse gain scheduling to partially linearise the apparent plant seen by the controller. A modified control architecture (Line, Manzie and Good, 26) may be introduced to help alleviate the actuator nonlinearty prior to the linear robust control design. The modified control architecture is shown in Fig. 4 and includes direct compensation of the static Fig. 4. Modified control architecture with feedback linearisation and inverse gain scheduling. and Coulomb friction, T s/c, feedback linearisation of the nonlinear stiffness load, T mc2, and an inverse gain schedule from F cl to ν. This inverse function produces a linear compound gain from the position, x, to the control variable ν. As a consequence of the modified control architecture the controller K c is largely isolated from actuator nonlinearity except for the unavoidable actuator saturation. For brevity some detail is not shown in Fig. 4. For example, the plant from Fig. 2 is redrawn using a mechanism block. Also, the power electronics that execute the current command, i q *, are represented by the current control block. In reality the three phase motor commutation is managed by the switching of power transistors in a dc-link converter. The motor operation was approximated using the motor circuit in (4) with a discrete proportional-integral current control. This regulated the current by adjusting the pulse-width modulation of the 42V supply. Using the modified control architecture in Fig. 4 it is possible to specify a linear model with a reduced level of uncertainty. A large uncertainty is no longer necessary to describe unmodelled nonlinearity. Instead, a reduced uncertainty is only specified to tolerate unmodelled dynamics arising from imperfect compensation. The introduction of uncertainty for imperfect compensation is graphically depicted in Fig. 5. Firstly, the block diagram of Fig. 2 may be expanded in (5a) under the condition of motion. Error terms are then introduced in (5b) to represent the imperfect feedback linearisation. The errors are replaced with bounded uncertain gains in (5c). In this case, W G and W F describe uncertainties associated with imperfect compensation of the load dependent friction and the nonlinear stiffness load. For example, W G =.2 would allow for a 2% error in the friction compensation of GF cl. K w represents a stiff linearisation of the stiffness curve such that the uncertain loads remain conservatively overestimated. Meanwhile, the uncertain load TC Tˆ C due to imperfect compensation of the Coulomb friction is rearranged near the input and described with an input uncertainty, W C. For brevity the feedback current control from iq * to i q is not fully shown in (5c) and omitted entirely in (5d). Finally, the linear compound gain from x to ν may be represented with the uncertain parameter K v.

4 (5a) (5b) (5c) (5d) Fig. 5. Development of a framework for plant uncertainty in a linear design model Appropriate parametric uncertainties may then be introduced for the motor torque constant, K t, inertia, J, viscous friction coefficient, D, and the gear ratio, N. The uncertain system in (5d) may then be described with bounded uncertainties, δ 1 and x 1, as shown in Fig. 6. For example, the uncertain torque constant in Fig. 6 is given as K t =K to +δ mkt δ Kt where δ mkt describes the magnitude of uncertainty and the bounded uncertainty is δ Kt 1. An offset term is sometimes included to re-centre the uncertainty bound. For example, the uncertainty for D in Table 2 is non-symmetric and the D offset term in Fig. 6 re-centers the nominal gain within the uncertain bound. x Fig. 7. General framework for uncertainty The designer may then pull out the deltas to achieve the general framework for robust control design shown in Fig. 7. In this diagram K indicates the controller, P is the extended plant and represents the set of bounded uncertainty. Fig. 7 may be related to Fig.6 by the signals z p and w p. For example, when the uncertainty δ Kt is pulled out in Fig. 7 the same signals z p1 and w p1 describe the input and output to the uncertainty block as in Fig. 6. In Fig. 7, K is specified as a two-degree-of-freedom controller for improved performance. The control feedback, y= [ r, y m] = [ F *, cl F ] cl, contains both the reference and the measurement. The controlled input is u=i q. w= [ n, r] comprises a noise and clamp force reference signal. A mixed sensitivity S-KS design is followed with the output, z= [ z 1, z 2 ] = [ W 1e, W2u ], including a weighted tracking error, e = F * cl Fcl, and input. If all the uncertainties were set to zero then the problem would reduce to an H optimal control design for the weighted plant shown in Fig. 8. This shows the S-KS mixed sensitivity weighting of the motor current input, u, and the clamp force tracking error, e. With zero uncertainty the H optimisation would seek the controller K that minimises the H norm of the transfer matrix from inputs [ n, r] to outputs W e, W ]. [ 1 2u Fig. 6. Bounded uncertainty in robust design Fig. 8. Two-degree-of-freedom control configuration with mixed sensitivity S-KS weighting on the input and tracking error.

5 During controller synthesis the mixed sensitivity weights to shape the transfer functions S and KS were chosen as, 1 s + ω11 M s ω 22 s + ω12 W1 = and W2 = (5) s + ω M ω s + ω 21 W 1 and W 2 are frequency domain specifications of the desired performance. W 1 shapes the desired sensitivity, S, from the reference, r, to the error, e. Meanwhile, W 2 shapes the desired transfer function KS from the reference, r, to the input, u, and may be used to include actuator saturation in the design. Writing the plant transfer matrix from T [ x &, z, y] as, u T [ x, w, u] to A B1 B2 P = C1 D11 D12 (6) C2 D21 D22 a suboptimal H controller may be characterised by means of Riccati equations, as in (Mackenroth, 24) pp , if i) (A,B 2 ) is stabilisable and (C 2, A) is detectable A iωi B2 ii) has full column rank for every ω C1 D12 A iωi B1 iii) has full row rank for every ω C2 D21 iv) D 12 has full column rank and D 21 has full row rank To ensure these conditions are satisfied a small weighting, W n, is applied to the noise signal, n. A nominal small load offset, W offset, is also included in Fig. 6. The satisfaction of conditions and the mixed sensitivity weighting are similar to that in (Mackenroth, 24), p.431. The performance weights and bounded uncertainties need to be chosen for robust performance over a reasonable set of actuator uncertainty. Design specifications are subsequently given in Section DESIGN SPECIFICATION Two design specifications are considered in the robust control design. The first specification is given in Table 2 and contains a practical level of uncertainty with desirable performance. Table 3 then contains a relaxed specification with a reduced uncertainty and performance. The uncertainty in Table 2 was specified to offer a reasonable tolerance to imperfect compensation and inaccurate modelling of friction and stiffness. The performance weights were shaped to provide reasonable step and frequency responses, attenuate high frequency noise and avoid actuator saturation. Nominal values for the gear ratio and friction load dependency are denoted by N o and G o respectively. Table 2 Design Specification 1 Compensation uncertainty Lower bound Upper bound W C W G -1.15G o 1.15G o W F -.15N o.15n o Parameter Lower bound Upper bound uncertainty - % + % _ D -5 1 K v /J -2 2 K t -1 1 N -1 1 Performance 1 st corner 2 nd corner weights Peak frequency frequency _ W 1 M s =1.2 ω 11 = 25 ω 21 =.1 ω 11 W 2 4 M u = 3 ω 12 = 5 ω 22 =1 ω 11 Table 3 Design Specification 2 Compensation uncertainty Lower bound Upper bound _ W C W G -.1G o.1g o W F -.5N o.5n o Parameter Lower bound Upper bound uncertainty - % + % _ D -5 5 K v /J K t N Performance 1 st corner 2 nd corner weights Peak frequency frequency _ W 1 M s =1.2 ω 11 = 15 ω 21 =.1 ω W 2 M u = ω 12 = 5 ω 22 =1 ω CONTROLLER SYNTHESIS AND REDUCTION With the control design in the general form of Fig. 7, controller synthesis may be performed using commercial software such as MATLAB. After defining the problem framework, uncertainty, and performance weights, the MATLAB function dksyn() was used to synthesis a robust controller via D-K iteration. The controllers produced by µ synthesis are typically of high order. Hence, a controller reduction is applied for a more practical realisation. A reduction of the state-space controller is possible by methods of

6 1 From: r Bode Diagram From: y m case the 4ms period is more than an order of magnitude below the.1-.2 s step apply time in Fig. 1 and Fig. 11. Magnitude (abs) ; Phase (deg) To: u To: u Frequency (Hz) Fig. 9. Reduction of the controller order K Kreduced Hankel singular values. The Hankel singular values may be used to preserve the most energetic states during reduction. Controller reduction was performed using the MATLAB function reduce(). As an example, the controller from the 2nd design specification was reduced from 8 states to 5 and the corresponding Bode diagram is shown in Fig. 9. The plot shows a close agreement between the high order and reduced controller. Following synthesis and reduction, the state-space controller was converted to a discrete time system and balanced for implementation on a fixed-point, discrete time embedded controller. The discretised controller, K, resulting from the 2nd design specification in Table 3 was ultimately given by, Ka Kb K = with a period of T = 4 ms. (7) Kc Kd where, K a = K b = K c = [ ] K d = [ ] The robust control design is a continuous time procedure and it is recognised that the discretisation of the synthesised control is an approximation. This may be considered reasonable when the sampling period is fast relative to the system dynamics. In this 6. RESULTS AND DISCUSSION The success of the robust EMB control design can be partly assessed in terms of the γ value achieved. γ indicates the magnitude of the specified uncertainty for which the control offers robust stability and performance and γ =1 would be ideal. However, with the design specification in Table 2 the robust control synthesis produced a large value of γ 2.7. This indicated that the robust design was not particularly successful. In other words, the control design did not guarantee robust performance over the specified actuator uncertainty. Since the robust EMB control design was not successful for the desired level of performance and uncertainty in Table 2, it is appropriate to either stop at this conclusion or otherwise relax the design specification. For the purpose of continuing the investigation, the second approach of reducing the design specification was followed. To determine a more achievable level of robustness the specification of actuator uncertainty and performance was reduced until an acceptable γ value was achieved. This lead to the design specification in Table 3 and a smaller value of γ Following reduction, discretisation and balancing the state-space controller in (7) was produced. While it was possible to reduce the specification of EMB uncertainty and performance to obtain a more successful control design (with a low γ value), this somewhat defeats the initial objective of robust performance over a reasonable set of actuators. It may be seen that the design specification in Table 3 only offers a limited tolerance to uncertainty. Further, with the magnitude of M u increased, the controller no longer avoids the 4 A current saturation for force commands approaching a magnitude of 3 kn. This further reduces the operational envelope for which robustness is guaranteed. Another criticism of the robust EMB control design may be that it considers a continuous time system and only approximates the discrete nature of the fixedpoint embedded control. Also, the approach of mainly avoiding actuator saturation inevitably produces a conservative response. This compounds the conservative nature of the robust control design. Aware of the limitations of the robust EMB control design, it is still useful to confirm the anticipated performance by experiment. To this end a large 25 kn brake apply was measured and simulated on the prototype actuator in Fig. 1 using the robust controller given in (7). A comparison of the results is shown in Fig. 1. Generally, the results indicate a reasonable

7 Force (kn) Velocity (rad/s) Current (A) Simulated Measured Fig. 1. Measured and simulated 25 kn brake apply with the robust EMB controller in (7) agreement between the simulated and measured response. The discrepancies are likely to arise from simulation inaccuracy. Also, it was difficult to ensure the controller states had identical initial conditions. The simulated controller could be initialised, but the experimental controller had an operational history and had to be settled at an initial state. The performance of the robust EMB control was benchmarked against a standard cascaded PI control. This had feedback loops to regulate the clamp force, motor velocity and current. Conditional integration was included for integral anti-windup. The PI gains for the force and velocity control were optimally tuned by minimising a penalty on the squared force tracking error and the mechanical power demand. The cost function had a dominant weighting on the tracking error. This lead to gains that best suited large manoeuvres such as a full brake apply. The experimental results in Fig. 11 compare the cascaded PI control and robust control performance for a light 5 kn brake apply. As it might be expected, the robust control has a comparatively conservative performance. However, the benefit of the robust design is that the performance may be guaranteed over a set of uncertain actuators. This may be contrasted with the situation for the cascaded PI control where the handling of uncertainty would need to be checked by an exhaustive simulation of whatif scenarios. To complement the performance results on the nominal actuator, the EMB control is also simulated on a perturbed system. The simulation offers a controlled environment for adjusting the model parameters and investigating the response. Two perturbed systems were constructed by selecting actuator uncertainty within the Table 2 specification to approach best and worst case scenarios. For example, a degraded actuator had increased friction, greater inertia, decreased stiffness, a decreased motor torque constant and so on. Velocity (rad/s) Current (A) Force (kn) Cascaded PI control Robust control Fig. 11. Measured 5 kn brake apply comparing the EMB robust control performance against that of a cascaded PI control As a simple test of open-loop sensitivity a 5 A step response was simulated on the perturbed actuators. The responses are shown in Fig. 12 and compared with that of the nominal plant. It may be observed that the practical level of uncertainty contained in Table 2 is sufficient to produce a reasonable variation in the open-loop EMB response. Following different rates of apply, the enhanced actuator with low friction approaches 25 kn before undergoing reversal. Meanwhile, the other actuators lock-up without reversal due to static friction. To investigate how the perturbed actuators were managed a series of closed-loop step responses was simulated. The robust controller given in (7) was again compared with the cascaded PI control. While it is appreciated that the large perturbations exceed the guaranteed level of robustness in Table 3, an analysis of the best and worst case scenarios may better expose the behaviour of the control. The simulated responses for a 1 kn brake apply are shown in Fig. 13. Two curves are apparent for each controller and correspond to the best and worst case actuator perturbations. As was the case in Fig. 11, it is seen that the robust control offers a conservative response. To an extent both controllers reject the disturbances observed in the open-loop responses of Fig. 12 by the mechanism of feedback. However, it is seen that brake apply profile with the robust control is Force (kn) Nominal plant Perturbed plants Fig. 12. Simulated open-loop step response to a 5A motor current for the nominal EMB and two perturbed systems within actuator uncertainty of Table 2

8 Force (kn) 1 5 Cascaded PI control Robust control Fig. 13. Simulated 1 kn brake apply comparing the EMB robust control performance against that of a cascaded PI control for the two perturbed plants less sensitive to perturbation and shows less deviation than that with the cascaded PI control. When compared with a cascaded PI control, the robust EMB control appears be less sensitive to actuator perturbations. Further, the robust control design can explicitly guarantee stability and a level of performance over a set of uncertain actuators. On the other hand, the robust EMB control has the disadvantage that it tends to be conservative. Successfully designing a robust EMB control is challenging due to the conflicting requirements for high performance and tolerance to uncertainty. To achieve a successful design it was necessary to reduce the design specification. As a consequence the guaranteed tolerance to perturbations was rather limited. During emergency or anti-lock braking the electromechanical brake may be actuated to its limits. Given the need for a rapid brake response the conservative robust EMB control may not be the most suitable approach for handling uncertainty. REFERENCES Krishnamurthy, P., W. Lu, F. Khorrami and A. Keyhani (25). A robust force controller for an SRM based electromechanical brake system. 44th IEEE Conference on Decision and Control and 25 European Control Conference, Seville, Spain. Line, C., C. Manzie and M. Good (24). "Control of an electromechanical brake for automotive brakeby-wire systems with an adapted motion control architecture." SAE Technical Paper Line, C., C. Manzie and M. Good (26). Electromechanical brake control: limitations of, and improvements to, a cascaded PI control architecture. Mechatronics 26: The 1th Mechatronics Forum Biennial International Conference, Penn State Great Valley. Lu, W. (25). Modeling and control of switched reluctance machines for electromechanical brake systems, PhD dissertation. Electrical and computer engineering, The Ohio State University. Mackenroth, U. (24). Robust control systems. Berlin, Springer. Schwarz, R., R. Isermann, J. Böhm, J. Nell and P. Rieth (1999). "Clamping force estimation for a brake-by-wire actuator." SAE Technical Paper Wang, N., A. Kaganov, S. Code and A. Knudtzen (25). Actuating Mechanism and Brake Assemby. WO 25/12418 A1, PBR Australia Pty. Ltd. 7. CONCLUSION Despite a careful incorporation of structured uncertainty it was difficult to achieve an electromechanical brake control design with robust stability and performance over a reasonable set of actuator uncertainty. When compared with a benchmark cascaded PI control the robust control performance was overly conservative. Part of the problem was that the need for a high brake performance conflicted with a conservative handling of uncertainty. To approach a reasonable brake performance it was necessary to relax the specification on model uncertainty. However, this would somewhat defeat the original intent of maintaining robustness over a reasonable set of actuator uncertainty. Consequently, future work should concentrate on improving the performance of the actuator and investigating different approaches to address actuator uncertainty in the controller design.