Predictive Brake Control for Electric Vehicles

IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. , NO. , 2017

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Predictive Brake Control for Electric Vehicles Clemens Satzger, Student Member, IEEE, Ricardo de Castro, Member, IEEE

Abstract—This article presents a predictive braking control algorithm for electric vehicles with redundant braking actuators, composed of friction brake actuator(s) and electric motor(s). The proposed algorithm is based on a model predictive control framework and is able to optimally tackle several control goals, such as maximization of energy recuperation and wheel slip regulation, while taking into account actuation dynamics and constraints. The braking control algorithm was simulated and experimentally validated on the ROMO research vehicle. The obtained results demonstrate that, in comparison with state-ofart control techniques, the proposed MPC approach is able to: i) reduce the torque tracking error up to 60%, and ii) improve the deceleration during emergency braking up to 10%. Index Terms—wheel slip control; model predictive control; brake blending; recuperation; hybrid braking; electric vehicles

I. I NTRODUCTION Although the initial uptake of electric drivetrains has been slow - e.g. today’s sales of such powertrains represent a small fraction of the automotive market -, electromobility is still regarded as one of the most promising technologies for sustainable transportation systems [1]. Besides the environmental benefits, such as zero (local) pollutant emissions, electric drivetrains might also have an important impact on the performance of the vehicle traction and braking systems. For instance, investigations carried out by Toyota [2] demonstrated that the use of wheel-individual motors (WIMs) can be instrumental to decrease the braking distances (up to 7%) during emergency decelerations over low-adhesion surfaces. The WIM’s fast and precise torque response [3] represents some of the features that contributed to this increase in braking performance. Another key advantage of WIMs lies in the possibility to individually allocate torques to the left and right wheels [3], [4]; this paves the way for the deployment of torque vectoring strategies controlled by a global chassis controller, enabling active shaping of the vehicle’s handling characteristics [4]. Spurred by these features, WIMs have received an increased attention by the automotive industry, as attested by the several in-wheel (and close-to-wheel) motor concepts recently proposed by Elaphe [5], Brembo [6] and Schaeffler [7]. Due to the limited torque range of WIMs and fail-safe concerns, electric vehicles still require the deployment of friction brakes [8]. This implies that during the vehicle operation, the braking torque needs to be divided between the WIMs and Manuscript received October 30, 2016; revised April 13, 2017 and June 30, 2017; accepted August 21, 2017. Copyright (c) 2017 IEEE. Personal use of this material is permitted. However, permission to use this material for any other purposes must be obtained from the IEEE by sending a request to [email protected]. Clemens Satzger and Ricardo de Castro are with the Institute of System Dynamics and Control, German Aerospace Center (DLR), Münchner Strasse 20, 82234 Wessling, Germany (e-mail: [email protected]).

the friction brakes. This division is often performed by socalled brake blending or hybrid braking algorithms e.g. [9], [10]. These algorithms have two main goals. The first is to maximize regenerative braking, contributing to an increase in the vehicle’s energy efficiency and driving range. The second goal is to prevent excessive wheel slip, i.e. anti-lock braking capabilities. To achieve these goals, several types of strategies have been proposed, where either the tire-road friction force is maximized [11], [12], [13], [14], [15] or the wheel slip is controlled. Ivanov et al. [16] gives an extensive overview on the control approaches applied to the wheel slip control problem; they range from linear parameter varying methods [17], [18], [19], nonlinear control [20], [21], [22], [23], [24], [25], [26] to Model Predictive Control (MPC) approaches [27], [28], [29], [30], [26], [31], [32]. A common element in the majority of these approaches is the adoption of a cascaded control architecture, composed of two loops: 1) an inner loop, named the control allocator (CA) responsible for the splitting of the brake torque (torque blending) among the WIMs and the friction brake; and 2) an outer control loop to handle the wheel slip regulation/ABS tasks. As discussed in [33], these cascaded approaches, despite easier to design and tune, might provide only sub-optimal transient performances for the overall braking control system. Furthermore, the principle of control allocation relies on the assumption that the actuator dynamics are much faster than the plant dynamics [34], [35]. This assumption is not always justified in the case of the wheel slip control. In order to extract the full potential of the hybrid braking system, in the recent past, there has been an increased interest in exploiting centralized control paradigms (i.e., ’single-loop’), where the torque blending and wheel slip regulation tasks are jointly performed in a brake control unit (see [31], [36], [32]). The MPC framework [37] has been the enabling technology for the materialization of this centralized paradigm. With the help of MPC, the control goals, such as maximization of energy recuperation and wheel slip regulation, can be optimally handled, while explicitly considering the actuator dynamics and range limits [31], [32], [36]. The architecture of the electric vehicle’s axle is another important factor in the design of the braking control algorithm. As depicted in Fig. 1, the electric vehicle’s (EV) architectures can generally be divided into three main categories: a) wheelindividual actuation, where each wheel has a dedicated WIM and a friction brake; b) central electric motor with distributed wheel-individual friction brakes; c) central friction brake with distributed WIM. From a control point of view, the topology with wheel-individual actuators [5], [6], [7] is the most attractive and flexible. Besides offering torque vectoring capabilities, it paves the way to the execution of hybrid braking at the wheel-level [22], [19], [31], [26]. The second configuration,


a) Wheel Individual

b) Central Motor

c) Central Brake

ue,l

uf,l Friction Brake

ue,l

WIM uf,l Friction Brake

ue,m Main Motor

uf,r Friction Brake ue,r WIM

WIM

uf,r Friction Brake

diff.

uf,l Friction Brake

ue,r WIM

Fig. 1: Possible drivetrain topologies of electric vehicles.

with a central electric motor, has been the preferable option for the mass-production of electric vehicles (e.g. BMW i3 [38] and Renault Zoe [39]). The third configuration represents an emergent architecture based on centralized friction brakes Fig. 1 c) -, which, so far, has received limited attention in the literature. It represents the main focus of this work. In comparison with wheel-individual configurations, (see Fig. 1 a), the central friction brake configuration offers packaging, modularity and cost benefits, since only one friction brake actuator (per axle) needs to be accommodated and acquired. Another benefit of this configuration consists in the inclusion of two WIMs, which enables the generation of asymmetric braking torque in the left/right wheels, necessary to tackle µsplit braking maneuvers or when the vehicle chassis controller requests corrective yaw-moment [4]. Moreover, in this configuration, the friction brakes can be seen as mechanical backup actuator, which should be deployed when strong or emergency braking is requested. Since the braking demands in the rear axle are normally much inferior than the ones in the front [40], this central brake configuration is particularly well suited for the rear-axle of electric vehicles. From a technical point of view, the development of braking controllers for central friction brake configurations is challenging. Besides requiring the coordination of a higher number of actuators (three), this braking controller also needs to take into account the coupling of the central friction brake actuator in the left/right wheels. This coupling increases the complexity of the control task, e.g., the central friction brake configuration needs a braking controller that can simultaneously control both the left and right wheels (i.e. an ’axle’ braking controller), which is normally not necessary in wheel-individual configurations. While many works can be found on wheel-individual [22], [26], [41], [42] (Fig. 1 a) and central electric motor [43], [44], [28] configurations (Fig. 1 b), very few investigated braking controllers for central friction brake configurations (Fig. 1 c)). To the best of the authors’ knowledge this article represents one of the first contributions to the torque blending and wheel-slip control of such powertrains. In order to tackle this control challenge, an MPC-based framework was developed. The robustness of this control framework against uncertainties in the friction force was

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shown using Lyapunov theory, which is of great relevance for the regulation of the wheel slip. Previous works on MPCbased hybrid braking control either ignored this analysis [26] or investigated it in an ad-hoc manner via numerical simulations [31], [30], [28]. Another important contribution of this work is related with the implementation of the proposed MPC approach in a real-time control platform and its experimental validation in the research prototype ROboMObil (ROMO). This contrasts with previous works on MPC-based brake controllers, which only offered validations via numerical simulations [31], [29], [26] or hardware-in-the-loop tests [27], [30], [28]. The remaining of the paper is organized as follows. Chapter II introduces the experimental vehicle, the ROMO, as well as the actuators and wheel-slip models. The formulation of the MPC problem for the powertrain with central friction brake actuators, as well as its stability properties, is presented in Chapter III. Chapter IV validates the proposed MPC strategy, including simulative results and experimental tests on the ROMO. Finally, the conclusions and outlook for future work are presented in Chapter V. II. T HE ROMO AND ITS M ODEL A. The ROMO The braking controller proposed in this work is designed having the ROMO as target vehicle (see Fig. 2a). The ROMO is a space-inspired highly actuated drive-by-wire electric vehicle developed by the DLR for demonstrating the benefits of transferring advanced space and robotics technologies to road vehicles (see [4] for details). The wheels of this vehicle are equipped with the so-called Wheel Robots, each consisting of an in-wheel electric motor and steer-by-wire actuator, plus (axle-based) brake by-wire units (see Fig. 2b). The in-wheel electric motors are based on 16 kW permanent magnet synchronous machines and are able to provide a torque of up to 160 Nm/wheel. The brake-by-wire units rely on an electro-hydraulic disk brake (EHB) operating at the axle level. Fig. 3 depicts the block diagram of this EHB actuator. Its principle of operation goes as follows: a small electric motor generates rotational motion, which is then converted in linear displacement by a ball-screw device, building up the braking pressure in the master cylinder. Afterward, the hydraulic pressure is conducted through two pipelines to the slave cylinders, which finally press the braking disk to generate the clamping force. Notice that in this configuration, the EHB applies the same brake pressure to both left and right wheels, while the differential braking capabilities are provided by the WIMs. This configuration corresponds to the single brake topology shown in Fig. 1. Each EHB actuator is capable of generating a maximum braking torque of about 534 Nm per wheel. B. Control Architecture As depicted in Fig. 4, the ROMO’s motion controller is based on an hierarchical control structure composed of four main layers: the vehicle level application (VLA), the vehicle dynamics controller (VDC), the braking control unit (BCU)


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Vehicle level application (VLA) Chassis‘ side-slip (b*)

Yaw-rate (y *)

Velocity or acceleration

Vehicle dynamics controller (VDC) Wheel torques (Tw*) Wheel steer (dk*)

Brake control unit (BCU)

kÎ{1,2,3,4}

IWM’s torques

(a) ROMO - the robotic electric vehicle.

Braking pressure

Wheel robots actuators

In-Wheel Motor

Fig. 4: Central control structure of the ROMO. lower levels, the BCU is responsible for performing the torque blending tasks (i.e. divide the wheel torque requested by the VDC into the WIM and friction brake demands) and avoid excessive wheel-slip (i.e. ABS functions). In the remaining of this article we discuss in detail the design, analysis and validation of the BCU.

Disc Brake

(b) In-wheel motor and disc brake of the ROMO.

Fig. 2: The ROMO and its actuators.

and the wheel robot actuator control. The VLA is responsible for performing the high-level motion tasks, such as interfacing human-controlled driving [45], executing autonomous path planning [46] and path following [47]. On the middle layer, the VDC distributes the control effort among wheel-steering and wheel-torques Tw∗ such that kinematic motion demand composed of yaw-rate, chassis side-slip angle and longitudinal acceleration setpoint generated by the VLA - is accurately and efficiently tracked (see [48], [49] for details). On the

C. Actuator Models In order to represent the dynamics of these ROMO’s braking actuators (i.e., WIM and EHB), discrete first-order models with delay were adopted in this work. Besides being able to capture the fundamental dynamics of the actuator’s response, these control-oriented models are numerically attractive for the incorporation into the MPC formulation. The mathematical representation of these models is described as follows: xk+1 = Aa xk + Ba uk min

max

(1a)

≤ uk ≤ u (1b) T Ce Cf 0 Tw,k = wl,k = Ca xk ; Ca = (1c) Twr,k 0 Cf Ce     Ae 0 0 Be 0 0 Aa =  0 Af 0  ; Ba =  0 Bf 0  (1d) 0 0 Ae 0 0 Be Slave T where uk = uel,k uf,k uer,k represents the desired left cylinder Disk left WIM’s electric torque, EHB friction brake torque and right WIM’s electric torque. The triple (Af , Bf , Cf ) characterizes Master cylinder the EHB’s dynamics, (Ae , Be , Ce ) the WIM’s response and Motor pf min Gap u , umax the minimum and maximum torque provided T Inverter Pipeline Slave by the actuators. Additionally, xk = xel,k xf,k xer,k Disk right cylinder Ball-screw represent the torque of the motor left (xel,k ), the friction brake states xf,k = [pf xs ], consisting of brake pressure pf and a shift state xs , and the torque of the motor right (xer,k ). Finally T Tw,k = Twl,k Twr,k represents the vector containing the Gap left and right wheel torques. Fig. 3: Schematic diagram of the EHB employed in the Field-oriented control methods [50] are employed to track ROMO. the electric torque generated by the WIM. According to [50], u


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Equation (4a) describes the torque equilibrium at the wheel, where J is the wheel inertia, ωi the wheel speed of the left or right wheel i ∈ {l, r}, re the effective tire radius, µi (λi ) the wheel friction coefficient, Fzi the vertical force on the wheel and Twi the wheel torque provided by the braking actuators. Equation (4b) states the force equilibrium of an axle, where m is the half vehicle mass, a the vehicle acceleration and v the vehicle speed. The wheel slip λi is expressed by equation (4c).

60 uf

pf(model)

pf(exp)

50

Pressure [bar]

40

30

20

Differentiating (4c) and inserting (4a) and (4b) allows expressing the wheel slip dynamics as:

10

0

0

0.2

0.4

0.6

0.8

1 1.2 Time [s]

1.4

1.6

1.8

2

Fig. 5: Experimental data of the master cylinder pressure pf (exp) versus the first order model pf (model) when the EHB input uf is excited with pseudo random binary signal steps of 50 bar. this torque response is mainly dependent on dynamics of the inner current-loop. Since the dominant time constant of this inner-loop (τe = 0.8 ms) is much faster than the BCU’s sample time Ts , (4 ms), the WIM’s dynamics are neglected in the MPC’s prediction model. Only the communication delay between the BCU and the actuators (equal to one sample time) is considered in the prediction model. This leads to the following parametrization: Ae = 0; Be = 1; Ce = 1

(2)

The EHB’s master cylinder pressure is controlled by a cascaded linear control law, as presented in [9]. Its dominant dynamics are approximated as a first order system with a dominant time constant τf and a delay of one sample time Ts (see 3). Details on the identification of the dominant time constant can be found in [9]. a 1 0 Af = 0 ; Bf = ; Cf = c0 0 (3a) 0 0 b0 Ts τf − Ts ; b0 = ; c0 = kp (3b) a0 = τf τf kp Fig. 5 shows the comparison of the master cylinder and the EHB model pressure dynamics during pseudo random excitation. These results demonstrate the validity of the simplified EHB model in capturing the main dynamics of the actuator (see [19] for details). D. Wheel Slip Dynamics The wheel slip dynamics employed in the design of the controller is based on a combination of two single corner models [51]: J ω˙ i = −re µ(λi )Fz + Tw,i X ma = mv˙ = µi (λi )Fz,i

(4a) (4b)

i∈{l,r}

ωi re − v λi = = h(ωi , v), i ∈ {l, r} v

(4c)

re ∂h(ωi , v) = (Γi (λi ) − Twi ) (5) λ˙ i = ∂t Jv which uses the auxiliary function X J Γi (λi ) = −re µi (λi )Fzi − (1 + λi ) 2 re µj (λj )Fzj mre j∈{l,r}

(6) Note that the coupling between left and right wheel has a J modest effect on Γi (λi ) as mr 2 1. e

Linearizing the wheel slip around the equilibrium point (Twi,eq , λi,eq ) and replacing the variables by the deviations from the equilibrium for the wheel slip difference δλi and the wheel torque difference δTwi leads to: δλi = λi − λi,eq ; δTwi = Twi − Twi,eq

(7)

This results in a first order differential equation that describes the dynamics of the wheel slip deviation from the equilibrium point ˙ i = re ∂Γi δλi + δTwi = re (θi δλi + δTwi ) (8) δλ Jv ∂λi Jv ∂Γi θi = (9) ∂λi λi =λi,eq Considering now the vehicle speed as a slowly varying bounded parameter v ∈ V compared to the wheel slip dynamics, the main source of uncertainty in equation (8) is the parameter θi , which is assumed to be bounded by [θmin , θmax ] = Θ ⊂ R. Note that, from a practical perspective, θi represents the sensitivity of the function Γi to the tire slip. Furthermore, since the controller operates in discrete time, it is necessary to obtain a discrete representation for the wheel slip dynamics. Toward that goal, (8) was discretized with the forward integration method (Euler method [52]), leading to: Ts re Ts re θi δλi,k + δTwi,k (10a) δλi,k+1 = 1 + Jv Jv = Aλ,i (v, θi )δλi,k + Bλ,i (v)δTwi,k (10b) where Ts is the discrete sample time. Finally, for control design purposes, it is convenient to aggregate the left and right wheel slips into the vector δλk = δλl,k δλr,k , which leads to the following state-


Parameter J re τf

Value 0.75 kg.m2 0.27 m 54.3 ms

τe

800 µs

kp

4.45 Nm/bar

Ts Te Tf m

4 ms ∈ [−160, 160] Nm ∈ [−534, 0] Nm 540 kg

Description wheel inertia effective tire radius dominant time constant of EHB dominant time constant of WIM pressure to torque coefficient controller sample time motor torque limits EHB torque limits half vehicle mass

BCU

Tˆw

λ Supervisor WS TT

RE , R ∆

MM left right QT , Qλ

Tw∗ MPC

λ∗

uel uf

Central Brake Topology Tel Motor Twl Tf Twr

Friction Brake

uer v λ

TABLE I: Parameters of the ROMO

Motor

Ter

x

Static Control p Allocation u

space representation: δλk+1 = Aλ (v, θ)δλk + Bλ (v)δTw,k Aλ,i (v, θl ) 0 Aλ (v, θ) = 0 Aλ,i (v, θr ) Bλ,i (v) 0 Bλ (v) = 0 Bλ,i (v)

5

(11a)

Fig. 6: Brake control unit for powertrain configurations with central brake configuration.

(11b) (11c)

where Aλ (v, θ), Bλ (v) are the state space matrices for the T axle wheel slip model with the uncertainties θ = θl θr . Table I presents the vehicle parameters, which are further used in this paper. III. B RAKE C ONTROL U NIT The braking control developed in this work (see Fig. 6) must provide two important functionalities: torque tracking, and wheel slip regulation (i.e., ABS). In both modes, torque blending is necessary in order to divide the wheel-torque demand Tw∗ between the WIMs (Tel , Ter ) and friction brake actuator (Tf ). This allocation process aims at maximizing the energy recuperation of the WIM, while offering a high bandwidth torque control. The second main functionality, wheelslip regulation, limits the wheel torque whenever excessive values of tire-slip are detected. The main goal of this section is to present an MPC formulation that simultaneously addresses the ABS and torque blending functions, i.e., to provide a framework to handle both problems. In the sequel this will be called brake control unit (BCU). Fig. 6 depicts the three main components of the BCU (delimited by a dashed red box): i) the MPC controller that combines wheel slip and torque tracking; ii) the supervisor that modifies the weights of the MPC and iii) the static control allocator, which computes the preferred setpoint upk .

Introducing these variables in the actuator and wheel slip dynamics (1) and (11), and stacking all the states into zk = T [∆xTk Tw,k ∆λTk λTk ]T ∈ Rn enables to compactly represent the system model as: zk+1 = A(v, θ)zk + B∆uk yk = Czk 

(12b) 

Aa 0 0 0  Ca Aa I 0 0   (12c) A(v, θ) =  Bλ (v)Ca 0 Aλ (v, θ) 0 Bλ (v)Ca 0 Aλ (v, θ) I   Ba Ca Ba  0 I 0 0   ; C= B= (12d) 0  0 0 0 I 0 T T λTk where yk = Tw,k is the output vector consisting of the wheel torques and wheel slips. B. Cost Function The cost function of the MPC formulation is composed of primary and secondary optimization goals. The primary ∗ objective is to track either the wheel torque set points Tw,j ∗ or the wheel slip references λj at the prediction time instant j depending on the operation mode. These two goals can be combined in the following cost function:

∗ 2

λj − λ∗j 2 Jj = Tw,j − Tw,j + (13) Q Q T

A. System Model with Embedded Integrator In order to obtain good disturbance rejection at steady-state conditions, the BCU controller must contain integral action. Toward this goal, the system model is augmented with integral action using the differential formulation proposed in [37]. The difference of the axle actuator states is defined as ∆xk = xk − xk−1 , the difference of the axle wheel slip is ∆λk = δλk − δλk−1 and the difference of the axle control variables is ∆uk = uk − uk−1 .

(12a)

2 where k.kW diag QT l

λ

is the quadratic norm with basis W , QT = QT r , Qλ = diag Qλl Qλr are weights on the torque tracking and the wheel slip control. The design of the secondary goal was driven by energy concerns. Note that, during braking maneuvers, the WIM can be used as generator to convert the vehicle’s kinetic energy into electric energy and charge the vehicle’s energy storage system (e.g. batteries). Consequently, by designing the secondary goal such that the regenerative braking energy is


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maximized, we are contributing to decrease the overall energy consumption of the vehicle [53], [54]. This is achieved by extending the MPC’s cost function with an additional term that promotes the convergence of the actuator set points towards a T p p preferred input setpoint up = uel uf uper , whose value is chosen to maximize the recuperation of the electric motors. Mathematically, this means: JE,j = kuj −

2 upk kRE

(14)

where RE = diag([RE,e , RE,f , RE,e ]) is a tuning weight and upk the preferred input values obtained from a static control allocation as explained in Appendix A. It is worth mentioning that the attraction of uj towards upk represents a widely used technique in the control of over actuated aeronautic systems (e.g., to minimize the drag of aircraft [55]). Our motivation here is to extend this technique to the torque blending problem.

C. MPC Formulation Taking into account the goals (13) and (14), the unified MPC formulation for the wheel slip and torque blending problem can be posed as:   Np Nc X X min  Jj + JE,j + ∆uTj R∆,j ∆uj  (15a) ∆uj

s.t.

j=0

(15b)

zj+1 = A(v, θ0 )zj + B∆uj , j = [0, ..., Np ] (15c) uj+1 = uj + ∆uj , j = [0, ..., Nc − 1] u

Torque Tracking (TT) Mode: operates with QT 6= 0, Qλ = 0 and it is normally appropriate for braking maneuvers when both left and right wheels have low values of wheel-slip. Wheel Slip (WS) Mode: operates with QT = 0, Qλ 6= 0. This mode is normally suitable for strong braking maneuvers, where the torque demand Tw∗ exceeds the tire-road friction limits, and both left and right wheel-slip needs to be regulated in order to maximize the braking force. Mixed Mode (MM): operates with either QT l = 0, QT r 6= 0, Qλl 6= 0, Qλr = 0 or QT l 6= 0, QT r = 0, Qλl = 0, Qλr 6= 0. In this case one wheel operates in the TT mode and the other in the WS mode, which might occur when braking on µ-split roads. Remark 1. Although focusing mainly on the central-brake configuration (Fig. 1 c), the BCU formalized in (15) can be straightforwardly modified to tackle other powertrain configurations, e.g. with central-motor (Fig. 1 b) or with wheelindividual actuators as shown in Fig. 1 a). Only small modifications are required in the actuators models (1) and cost function to take into account specifications associated with each powertrain configuration (see [36] for details).

j=0

z0 = zk

min

operates with one of the Q parameters equal to zero, which renders four main operation modes for the controller:

≤ uk ≤ u

max

, j = [1, ..., Nconst ]

(15d) (15e)

where (15c) represents the combined model of actuator and wheel slip dynamics, (15d) is related with the controller integral formulation and (15e) imposes the actuator’s range limits. The parameter θ0 represents the nominal value for θ. This MPC formulation contains three main sets of tuning parameters. The first set is composed by the prediction horizon Np , the control horizon Nc and the constraint horizon Nconst . The selection of these parameters was performed using the recommendations proposed by [37], [56]. As a general rule of thumb, Np has to be long enough to capture the fundamental transient response of the plant [37]; in our case, this response is dominated by the EHB dynamics (3) and wheel-slip dynamics (8). The control horizon Nc is normally upper bounded by computational constraints (i.e., increasing Nc raises the number of independent decision variables in the optimization problem and its computational load [56]), and lower bounded by the number of samples necessary to cover the system’s dead time [57]. Finally, the constraint horizon Nconst can be set at the same length as the control horizon, but for real time MPC applications Nconst is normally reduced to further decrease the computational load [56]. The second set of tuning parameters is composed by the matrices RE , which penalizes deviations from the preferred input, and R∆ , which penalizes actuator control variations. And finally, the last set of tuning parameters is represented by the error tracking weights QT and Qλ . In practice, the MPC

In order to manage the switching between the above mentioned modes and MPC’s weights a supervisor was developed. The supervisor’s logic is based on the concepts described in [51] and extended here for the braking control of the vehicle’s axle.

D. Stability Analysis Since the MPC formulation does not have a final weight and no final state constraint (they were omitted to simplify the real-time implementation of the MPC), it is not possible to guarantee nominal stability at first glance [21]. However, the unconstrained solution of the problem (15) can be rendered into a linear closed loop form for a fixed theta θ0 (with a nominal value of [−600, −600]T in this study) and used for analysis of the control properties. The state feedback gain κ(vk , θ0 ) associated with the unconstrained MPC operation can be computed by inverting the Hessian matrix [22]. This leads to the closed loop dynamics: Acl (vk , θk , θ0 ) = A(vk , θk ) − Bκ(vk , θ0 )

(16)

Based on (16) a sensitivity analysis on the weights of the MPC cost function and the prediction and control horizon can be drawn out. Because (16) represents an LPV system, stability for the unconstrained MPC operation can be proven with standard Lyapunov theory. To that end, a parameter dependent positive definite Lyapunov matrix function (PDLF) P (vk ) is introduced and the Lyapunov function V (zk , vk ) = zkT P (vk )zk is defined. Acl (vk , θk , θ0 ) is stable, if the two


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Params Np Nc Nconst θ0 W1 W2

0 -1

Vehicle acceleration a [m/s2 ]

-2 -3 -4

Value 20 5 1 [−600, −600]T diag 100 100 diag 0 0.01 0

Description Prediction horizon Control horizon Constraint horizon Nominal uncertainty Tracking weight Penalization weight

(a) Parameters (Params) of the unified brake controller. Weights WS TT 1.2 QT diag 1.2 diag 0 0 diag 0 0 diag 1.3 1.3 Qλ 0 0 0 RE 10−3 diag 2 2 2 diag R∆ 10−5 diag 1 1 1 diag 400 1335 400

-5 -6 -7 -8 -9 -10 -11 400 200 0 Un cer

-200 tain ty θ

-400 -600 -800

5

10

15 le Vehic

20 speed

v

25 [m/s]

30

Fig. 7: Operational area of the BCU. Note that each grid point fulfills the stability conditions (20) and (21).

(b) Weights of TT and WS mode. Weights Mixed mode (MM) WS left, TT right WS right, left TT QT diag 0 3 diag 3 0 Qλ diag 1.3 0 diag 0 1.3 0 0 0 RE diag R∆ diag 400 1335 400 (c) Weights of mixed mode.

TABLE II: Parameters of the BCU. following Lyapunov conditions are fulfilled. V (zk , vk ) > 0

(17)

V (zk , vk ) − V (zk+1 , vk + ∆vk ) > 0; ∀(vk , ∆vk , θk ) ∈ V × ∆V × Θ

(18)

where V, ∆V, Θ are the bounded sets of vk , ∆vk , θk . In order to derive a numerical tractable LMI expression from these conditions, several steps are required as presented by Knoblach [58], [59]. First, the infinite dimensional parameter space (vk , ∆vk , θk ) ∈ [V ×∆V ×Θ] is approximated by a three dimensional grid. Second, a basis function for PDLF must be defined. Inspired by the work of Petersen [60], the following basis is adopted: P (vk ) = P0 + P1 vk + P2 vk2 + P3 vk3

(19)

As a result, conditions (17) can be re-written as: P (vk ) > 0

(20) T

Acl (vk , θk , θ0 ) P (vk + ∆vk )Acl (vk , θk , θ0 ) − P (vk ) < 0 (21) The gridding approach, together with the LMI conditions (20) and (21), enables to determine the stable parameter range. For example, Fig. 7 illustrates the parameter range where the PDLF fulfills the Lyapunov stability conditions considered in (20), (21). Each red cross in Fig. 7 represents a stable grid point. The calculations considered a speed range of V = [2, 30.5] m/s, an incremental speed range of ∆V ∈ [−0.051, −4 · 10−4 ], which corresponds to an acceleration range of ak ∈ [−12.75, −0.1] m/s2 , and an uncertainty range of Θ = [−800, 400]. Inspecting these results, it can be observed that the unconstrained MPC is able to robustly stabilize the wheel slip for velocities as low as 2.5 m/s and with a very significant level of uncertainty in the friction conditions (captured by the parameter θk ). Of course, this stability analysis does not consider actuator

constraints; therefore it does not explicitly prove stability for the entire operation domain of the proposed MPC controller. Nevertheless, it gives some insight in the stable operational range of the controller. IV. VALIDATION The proposed BCU was validated by simulations and experimental tests on the ROMO. In both cases the performance was compared against the nominal cascade-based braking controller, based on a combination of a static allocation (SA) torque blending and a proportional and integral (PI) control law for the wheel-slip regulation (see [19] for details). This later braking controller is referred in the sequel as PI+SA. The simulations were carried out based on a co-simulation environment between Matlab/Simulink and Dymola. The former is responsible for the implementation of the BCU, while the latter contains the model of the braking actuators. These models are based on the DLR’s Modelica Powertrain library, [61], [9], parametrized for the ROMO prototype. To model the wheel slip dynamics, nonlinear tire force models (MFSwift 6.2.0.3 [62]) were used. Furthermore, in order to provide a realistic measurement noise, the vehicle speed signal was corrupted with noise having a power spectral density (PSD) of P SDv = 10−6 m/s, while the wheel-speed noise has P SDω = 10−5 rad/s. The parameters employed in the BCU are presented in Tables II. A. Simulative Evaluation of the Torque Tracking The first set of simulation tests intends to evaluate the performance of the torque-tracking mode. Toward that goal, several types of torque profiles Tw∗ were tested. The first profile consists in a rectangular shape - with an amplitude of ± 70 Nm, a frequency of 1 Hz and an offset of -200 Nm -, which is equally applied to the left and right wheels (see Fig. 8a).


8

IAE(λ) = Ts E = Ts

N sim X i=1 N sim X i=1 N sim X

∗

Tw,i − Tw,i kλi − λ∗i k T Te,i ωi

(22)

Wheel Torque right [Nm]

IAE(T ) = Ts

Wheel Torque left [Nm]

The energy E, the integral absolute tracking error IAE of either the wheel torque IAE(T ) or the wheel slip IAE(λ), similar to the ones defined in [31], were considered as performance metrics for the evaluation of the braking controllers.

(23)

(24)

0

*

Tw,l SA

Tw,l

Tw,l BCU

-100 -200 -300

0

0.5

1

1.5

0 T*

T

w,r

w,r

SA

T

w,r

BCU

-100 -200 -300

0

0.5

1

1.5

It is also worth pointing out that, although this work only considers a single TT mode with a constant RE , the curves shown in Fig. 8b pave the way to the introduction of additional Pareto-optimal tunings in the MPC formulation. For example, the BCU could be extended with an ecologically TT sub-mode (high RE ) and a ’sporty’ TT variant (low RE ). The former promotes the maximization of the energy recuperation (but with a penalization in torque tracking), while the latter enables the driver to access the best torque tracking performance (but with an energy penalization). The time-domain operation of the BCU with RE = 0.002I3 (the default BCU’s parametrization in the TT mode) is analyzed in more detail inFig. 8a. Inspecting the wheel torque response one can verify that the BCU and the SA have similar steady-state performance, with zero steady-state torque tracking error. However, during transients (e.g., see the step at 0.5s) the BCU is able to offer a faster wheel torque

-100 -150 -200

EHB Torque [Nm]

Te,l SA 0

0.5

0

0.5

Te,r SA

Te,l BCU

Te,r BCU

1

1.5

0 -200 -400 u SA

T SA

f

-600

u BCU

f

T BCU

f

f

1

1.5

Time [s]

(a) Torque tracking of the SA (dash dotted) and the BCU in TT mode with RE = 0.002I3 . RE

−0.94

−0.95

−0.96

−6

−1

10

SA BCU

RE=10 I3 RE=0.002 I3

−2

10

Normalized energy E/|ESA|

For instance, Fig. 8b shows that by selecting RE = 0.002I3 the BCU produces a normalized torque’s IAE that is 42% of the IAE obtained with the SA - thus providing a 58% reduction in this metric - but introduces a 4% penalization in the energy consumption. Note that the SA has a normalized energy consumption of −1, while the BCU (RE = 0.002I3 ) requires a normalized energy of −0.96, i.e. a 4% increase in the energy consumption. These results put in evidence the trade-offs in energy consumption vs torque tracking that the designer will have to handle in the commissioning of the BCU.

0 -50

-250

+4% Energy consumption

where Nsim is the number of time steps over the duration of the simulation, ω = [ωl , ωr ]T are the wheel speeds of the left and right wheel. Both metrics were normalized with respect to the SA solution. In order to investigate trade-offs between torque tracking and energy recuperation the BCU’s energy weight RE was varied in the range of [10−6 , 0.15], leading to a quasi-Pareto optimal curve (see Fig. 8b and Fig. 9b). Inspecting the obtained results one can find that with an increase of RE the energy recuperation of the BCU approaches the SA energy performance (notice that a negative energy means that the motor is recuperating energy). Qualitatively speaking, these results are in-line with our expectations: the increase of RE attracts the BCU solution toward the preferable input, which is generated by the SA. On the other hand, as RE is decreased one can verify an improvement in the BCU’s torque-tracking performance, with a small penalization in the energy recuperation.

Motor Torque [Nm]

i=1

−3

10

−0.97

−0.98

−4

10

RE=0.15I3

−0.99

−5

10 −1

−1.01 0.4

−6

0.5

0.6

0.7

0.8

0.9

1

10

IAE(T)/IAE(T)SA 58% IAE(T) reduction

(b) Comparison between BCU and SA in terms of IAE(T ) and the energy, both normalized with respect to the SA.

Fig. 8: Comparison between BCU and SA for symmetric reference torques.

response than the SA. The main reason for these differences lies in the predictive nature of the MPC formulation. More specifically, since the MPC incorporates the models of the braking actuators, the algorithm is able to pre-act on the control signals and compensate for the actuator dynamics. Another factor worth highlighting in these results is the interaction between primary and secondary goals of the MPC, i.e. between torque tracking and energy recuperation. For example, inspecting the torque allocation results in the time range [0.4, 0.6] s, one can observe that immediately after the


1.0027 1.0017 1.0013

TABLE III: Torque tracking results running the Artemis rural, the Artemis URM150 and the Inrets Route2 driving cycles [63] with RE = 0.002I2 . application of the step in the wheel torque set point, the electric torque compensates for the slow friction brake torque change. Posterior, as the torque tracking error decreases, the cost term associated with the torque tracking also shrinks, which then allows the BCU to refocus on the secondary performance metric, i.e., energy recuperation. In order to evaluate the performance of the braking control algorithms when asymmetric torque demands are requested to the left/right wheel, a second torque profile was prepared. This second profile consists of constant braking torque demands (−347 Nm) for the right wheel, while the left torque setpoint alternates between −100 and −390 Nm (see Fig. 9a). This type of torque profile might occur when the vehicle dynamics controller requests differential wheel-torques to the braking system, e.g., for stabilizing the vehicle motion. Similarly to the previous simulation, the BCU is able to improve the braking performance. Compared to the SA, the BCU significantly reduces the tracking error (51% with respect to SA), while the energy recuperation is only slightly reduced (1% with respect to SA). The transient response depicted in Fig. 9a also demonstrates the superior coordination of the braking actuator during transient maneuvers, which is particularly visible in the torque step performed at t = 0.8 s.

Wheel Torque right [Nm]

0.8418 0.8901 0.9539

Motor Torque [Nm]

E ESA

Wheel Torque left [Nm]

BCU IAE(T ) IAE(T )SA

T*

-100

Tw,l SA

w,l

-300 -400

0

0.2

0.4

0.6

0.8

1

1.2

0.8

1

1.2

-100 -200 -300 *

T

Tw,r -400

0

w,r

0.2

SA

T

w,r

0.4

BCU

0.6

0 -50 -100 -150 -200

Te,l SA 0

0.2

0.4

0

0.2

0.4

Te,r SA 0.6

Te,l BCU 0.8

1

The next set of simulation tests intends to evaluate the performance of the BCU’s Mixed Mode (MM). Toward that goal, a braking maneuver, with asymmetric torque demands in the left/right wheels is carried out over a surface with a maximum friction coefficient of µ = 0.5. Fig. 10a shows the mixed tracking modes that the braking controller must handle during this maneuver: wheel-slip tracking on the left wheel and torque tracking on the right wheel.

1.2

-200 -400

uf SA

Tf SA 0.6 Time [s]

uf BCU 0.8

Tf BCU 1

1.2

(a) Torque vectoring of the SA (dash dotted) and the BCU in TT mode with RE = 0.002I3 . -0.97

RE=10 I3

-0.98

-1

RE

10

SA BCU

-6

C. Simulative Evaluation of the Mixed Mode

Te,r BCU

0

B. Drive Cycle Evaluation of the Torque Tracking Mode

-2

10 -0.99

-1

-1.01

RE=0.002 I3

-3

10

+1% Energy consumption

Normalized energy E/|ESA|

The performance of the SA and the BCU on the Inrets Route2, Artemis rural and the Artemis URM150 driving cycles [63] is compared in Table III. The BCU was parametrized with a RE = 0.002I2 , which according to the previous simulation tests provides a good trade-off between energy and tracking performance. The obtained results, shown in Table III, demonstrate that the BCU is able to improve the torque tracking IAE by 5-16%, while the overall energy recuperation is slightly penalized (0.1-0.3%). These results put in evidence the potential benefits of the BCU under representative driving cycles.

Tw,l BCU

-200

-250

EHB Torque [Nm]

Drive Cycle Metric Inrets Route2 Artemis rural Artemis URM150

9

-4

10

RE=0.15I3

-5

10

-1.02

-1.03 0.4

-6

0.5

0.6

0.7

0.8

0.9

1

10

IAE(T)/IAE(T)SA 51% IAE(T) reduction

(b) Comparison between BCU and SA in terms of integral absolute error and energy for asymmetric reference torques both normalized with respect to the SA.

Fig. 9: Comparison between BCU and SA for asymmetric reference torques.


0

Normalized speed v/v

0.6 0.4 0.2 0.6

0.8

1

1.4

1.6

1.8

2

−0.2

λ*

−0.4

−0.4

−0.6

−0.6

0.5

1

1.5

2

0 0

[Nm]

0 0

*

0.5

1

1.5

2

0.5

1

1.5

2

0.5

1

1.5

2

w,r

Tw −200

Wheel torque right T

Wheel torque left Tw,l [Nm]

1.2

0

Wheel slip right λr

0.4

−400 −600 0

0.5

1

1.5

−200 −400 −600 0

2 Motor torque right Te,r [Nm]

Wheel slip left λl

0.2

−0.2

Motor torque left Te,l [Nm]

Torque Limits 100 0

100 0

−100

−100 0

0.5

1

1.5

2

0

0 −100 −200 −300 −400 −500 0

0.2

0.4

0.6

0.8

1 Time [s]

1.2

1.4

1.6

1.8

2

(a) Time series plot of the mixed mode operation. 6

10

1 0.95

Q

Tr

PI+SA BCU 4

PI+SA

0.9 IAE(T)/IAE(T)

The experimental validation of the BCU was performed on the ROMO prototype, introduced in Chapter II. On the realtime platform, the optimization problem of the BCU (15) can be solved within less than 1 ms using the Hildreth quadratic problem (QP) solver [64]. Therefore, the BCU was able to fulfill with the 4 ms cycle time, the nominal sample time of ROMO’s realtime platform. The main feedback signals for the braking controller were the wheel-rotational velocities, obtained via the WIM’s encoders, and the vehicle speed, which is measured by an advanced inertial measurement unit (OXTS RT4003). The parameters of the actuator models for the ROMO prototype are listed in Table I. It is worth stressing that during the WS mode - normally active during emergency braking - the vehicle safety and the wheel-slip tracking performance (weighted by parameter Qλ ) have priority over the of the energy recuperation (weighted by RE ). To enforce this design policy, the WS mode was parametrized with Qλ RE = 0 (see Table IIb). Since the TT mode was already extensively investigated in simulations and for the sake of brevity, the experimental tests presented here focus mainly on the validation of the WS, which represents the most challenging control task. In these experimental braking tests, two different types of surfaces were used: i) dry asphalt and ii) low µ. In all cases, the torque demands Tw∗ were generated with a ramp-like-setpoint (gradient of 750 Nm/s). Additionally, a target slip of λ∗ = 0.05 was considered, which enabled the tyres to operate close to its friction peak (see Fig. 11) during both experimental maneuvers. 1) Braking on Dry Asphalt: The results for braking on dry asphalt can be seen in Fig. 12. At around t = 0.8 s, excessive wheel slip is detected, and the BCU switches to the WS mode. Immediately after the activation of this mode, the electric torques Te,l , Te,r are rapidly modulated in order to decrease the excessive wheel slip, while the slower EHB setpoint remains almost unchanged. This allocation strategy contributes to a very good transient and steady-state response of the BCU, which is underlined by the low average wheel slip tracking errors presented in Fig. 14b. The PI+SA controller, albeit offering satisfactory steady-state behavior, does not perform so well during the transient. The wheel-slips reaches a peak of almost 0.6 and its response is particularly oscillatory during the time range t ∈ [0.8, 1.2] s. The main reason for this poor performance lies in the SA’s torque allocation strategy: since

PI+SA BCU

0 0 0

EHB torque Tf [Nm]

D. Experimental Validation

1 0.8

+41% IAE(T) reduction

These results also reveal that the BCU, due its intelligent coordination of the braking allocation effort, outperforms the PI+SA, particularly during the transient response. See, e.g., the PI+SA’s large wheel slip overshoot in the left wheel or the oscillatory response in the right wheel’s motor torque. To further gain insight into the tuning of the MPC, Fig. 10b shows the effect of the torque tracking weight QT r in the IAE(T ) of the torque and IAE(λ) of the wheel slip tracking. The results highlight that QT r = 3 produces a reduced wheelslip error with minimum penalization in the torque tracking performance.

10

0.85

10

QTr=103

QTr=10−3

2

10

0.8 0.75

0

10

0.7 0.65

QTr=3

0.6 0.55 0.75

0.8

0.85 0.9 IAE(λ)/IAE(λ)PI+SA

−2

10 0.95

1

23% IAE(λ) reduction

(b) Pareto-curve between IAE(T ) and IAE(λ) both normalized with respect to PI+SA.

Fig. 10: Comparison between BCU and PI+SA with the left wheel in WS and the right wheel in TT (Mixed mode).


11

1 BCU PI+SA

0.8 Dry asphalt MF−Tire Low mu

0.6

0.8 0.6

0.4 0.2

0.5 0 0 0

0.4 0.3

1

1.5

0.05

0.1

0.15

3

−0.3 −0.4

−0.2 −0.3 −0.4 −0.5

λ* 0

Fig. 11: Identified tire characteristics of the ROMO tires for braking on dry asphalt and the MF-tire used in simulation.

0.5

1

1.5

2

2.5

3

150 Motor Torque left Te,l [Nm]

100 50 0 −50 −100

0.5

1

1.5

2

2.5

3

0

0.5

1

1.5

2

2.5

3

100 50 0 −50 −100 −150

−150 0

0 150

Torque limits

0.5

1

1.5

2

2.5

3

0 −50 EHB Torque Tf [Nm]

the traction motor is saturated during the WS mode, the SA is employing the slower actuator (EHB) for the wheel-torque modulation, which significantly compromises the controller response. 2) Braking on Low-µ Road: The second braking scenario was performed on a watered surface with a special polymer cover (µ ∼ = 0.25). In this case, the BCU and the PI+SA achieve similar results during the initial transient, as shown in Fig. 13. Note that, in these low-µ conditions both control strategies (PI+SA and the BCU) are able to perform the wheel-torque modulation using the fast actuator (the electric motor), which contributes to a similar response. However, when the velocities are lower, the performance of PI+SA degrades. The key reason for this degradation lies in the effect of the vehicle velocity in the plant’s dynamics. In particular, inspecting (8) shows that the wheel-slip dynamics have a e θi dominant pole at s = rJv , dependent on the vehicle velocity v. As v → 0 one can verify that these dynamics become infinitely fast (please see [65] and [51] for a detailed discussion of the effect of the velocity in the plant’s model). This fact contributes to the excitation of the plant’s unmodeled dynamics, making the control task more challenging at lower velocity. 3) Statistical Summary: A statistical summary of the experimental validation is given in Fig. 14. It includes the mean and standard deviation of the acceleration (Fig. 14a) and wheel-slip tracking error (Fig. 14b). In order to facilitate the comparison between the control strategies, all the results are normalized with respect to the uncontrolled case. Inspecting the obtained results, one can find that the BCU provides the highest acceleration on both surfaces (5 to 10% higher than the PI+SA and 11 to 30% higher than the uncontrolled case). As a result, the vehicle equipped with BCU is able to achieve shorter braking distances, representing an important contribution to the vehicle’s safety during emergency braking. Additionally, one can also observe that the BCU generates the smallest standard deviation in the acceleration (particularly on

−0.2

−0.5

Wheel Slip |λ|

2.5

−0.1 Wheel Slip right λr

0.1

2 0

Motor Torque right Te,r [Nm]

0.2

0 0

0.5

−0.1 Wheel Slip left λl

Friction Coefficient |µ|

0.7

Normalized Speed v/v0

0.9

−100 −150 −200 −250 −300 −350 −400 0

0.5

1

1.5 Time [s]

2

2.5

3

Fig. 12: Braking maneuver on dry asphalt with a starting speed of 11.8 m/s. dry asphalt), which delivers a smoother and more comfortable braking. Fig. 14b demonstrates the superior tracking performance of the BCU, yielding a mean tracking error that is 75% inferior to the PI+SA on dry (25% inferior on wet). It also shows that the BCU offers the smallest standard deviation in the tracking error, which confirms its good disturbance rejection capabilities. V. C ONCLUSION A predictive braking controller for electric vehicles equipped with WIMs and axle-based friction brakes was presented. This controller relies on MPC framework and enables to simultaneously and optimally tackle several control goals, such as torque tracking, energy recuperation and wheel slip regulation, as well as actuator dynamics and constraints. This multi-objective nature of the MPC framework provides a widerange of tuning possibilities, enabling the designer to pursue


12

1,60

Normalized Speed v/v0

v BCU v PI+SA 0.8 0.6 0.4 0.2

Normalized acceleration a/aNC

1

1,50 1,40 1,30 NC

1,20

PI+SA

1,10 1,00

1,30

0,90 1,00

1,07

BCU

1,17

1,11 1,00

0,80 0.5

1

1.5

2 0

−0.1

−0.1

−0.2 −0.3 −0.4 λ*

0 150

1

2

3

3.5

4

0,70 dry

50 0 −50 −100 −150

−0.3 −0.4

0

1

2

3

4

2

3

and its standard deviation.

1,20 1,00 0,80 NC 0,60 1,00

PI+SA

1,00

0,40

BCU

0,04 0,04

0,20

0,03 0,01

0,00

100

-0,20 50

dry

0

low-µ

(b) Mean normalized wheel slip tracking error standard deviation.

−50

λ−λ∗ λN C −λ∗

and its

−100 −150

1

a aN C

1,40

150

Torque limits

low-µ

(a) Mean normalized acceleration

−0.2

4

100

0

3

−0.5

Motor Torque right Te,r [Nm]

Wheel Slip left λ

l

Wheel Slip right λr

0

−0.5

Motor Torque left Te,l [Nm]

2.5

Normalized wheel slip tracking error (ߣ−ߣ*)/(ߣNC−ߣ*)

0 0

4

0

1

2

3

4

0

Fig. 14: Statistical analysis normalized with respect to case with no control (NC) of the braking maneuvers on dry asphalt and a low-µ.

EHB Torque Tf [Nm]

−20

robust MPC formulations is needed, which will be the focus of future publications. Additionally, the integration of the BCU with online algorithms for estimating the tyre-road friction peak and optimal wheel-slip setpoint λ∗ (such as [66]) is planned for future endeavors.

−40

−60

−80

−100 0

0.5

1

1.5 Time [s]

2

2.5

3

Fig. 13: Braking maneuver on low-µ with a starting speed of 10.7 m/s.

different trade-offs between energy recuperation vs torque/slip tracking performance. The BCU was first evaluated in simulations and then experimentally validated on the ROMO prototype vehicle. It was shown that, in comparison with a cascade-based braking strategy - relying on static torque allocation and linear wheel slip regulation law - , the BCU is able to increase the mean deceleration by 5 to 10% on dry asphalt and low-µ roads. The BCU was also able to reduce the torque tracking error up to 60%. These gains were obtained due to the intelligent actuation coordination offered by the MPC-based BCU. In particular, the proposed BCU demonstrates how the fast dynamics of the WIMs can be explored to improve the wheel slip regulation (during an emergency braking) and the torque tracking performance (during normal braking). Although the BCU was proven to be robust in the experiments, robustness for the constrained MPC case was not theoretically proven. In this latter case, further analysis using

ACKNOWLEDGMENT The authors would like to thank the ROMO team for the support on the experiments and to Franciscus van der Linden for his valuable review. A PPENDIX A S TATIC T ORQUE A LLOCATION The goal of the static torque allocation employed in this work consists in finding a (steady-state) preferable actuation setpoint up that: i) produces the wheel-torque setpoint Tw∗ ; ii) maximizes the use of the regenerative braking and iii) fulfills the actuation constraints (umin ≤ up ≤ umax ). This task is mathematically formulated as: min p u

s.t.

2

2

kCax up − Tw∗ kW1 + kup kW2 u

min

p

≤u ≤u

max

(25a) (25b)

where the weights W1 , W2 represent tuning factors that the designer can use to penalize tracking errors and promote the use of the most energy-efficient actuators, respectively. Note that (25) is a lightweight optimization problem: besides being a QP problem (which can be efficiently solved online), it only has three decision variables. The solution of this QP problem was calculated in real-time using the Hildreth’s QP solver [64].


A PPENDIX B ACRONYMS Acronym ABS BCU CA EHB EV IAE LMI LPV MPC PI PI+SA PDLF QP ROMO SA TT VDC VLA WIM WS

Description Anti-lock braking systems Brake control unit Control allocator Electro-hydraulic disk brake Electric Vehicle Integral absolute error Linear matrix inequality Linear parameter varying Model predictive control Proportional+Integral Cascade of PI and SA Positive definite Lyapunov matrix function Quadratic problem ROboMobil Static allocation Torque tracking Vehicle dynamics controller Vehicle level application Wheel independent motor Wheel Slip

Clemens Satzger (S’14) was born in Munich, Germany, in 1983. He received the bachelor and diploma in electrical and computer engineering from the University of Munich, Germany, in 2006 and 2010. He also received a second diploma as ´ Ingénieur Généraliste from the Ecole Centrale de Lyon, France, in 2010. Since 2010 he is researcher at the German Aerospace Center (DLR) and in 2015 he started his Ph.D. studies at the Friedrich-AlexanderUniversity in Erlangen-Nürnberg, Germany. His research interests include robust model predictive control, embedded control, vehicle dynamics control and brake control of electric vehicles.

Ricardo de Castro (S’09, M’13) was born in Porto, Portugal, in 1983. He received the Licenciatura and Ph.D. degrees in electrical and computer engineering from the Faculdade de Engenharia da Universidade do Porto, Portugal, in 2006 and 2013, respectively. During 2007-08, he was an entrepreneur with the WeMoveU project, targeting the development of powertrain control solutions for light electric vehicles. Since 2013 he is researcher at the German Aerospace Center (DLR). His research interests include platooning, vehicle dynamics, tire-road friction estimation, and batteries-supercapacitors hybridization.

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R EFERENCES [1] Amsterdam Roundtable Foundation McKinsey & Company. Electric vehicles in Europe: gearing up for a new phase? Technical report, 2014. [2] Satoshi Murata. Innovation by in-wheel-motor drive unit. Vehicle System Dynamics, 50(6):807–830, 2012. [3] Y Hori. Future vehicle driven by electricity and control-research on fourwheel-motored. IEEE Transactions on Industrial Electronics, 51(5):954– 962, 2004. [4] Tilman Bünte, Stephan Kaspar, Sören Hohmann, and Jonathan Brembeck. Inverse model based torque vectoring control for a rear wheel driven battery electric vehicle. In World Congress, pages 12016–12022. IFAC, 2014. [5] Matej Biˇcek, Gorazd Lampiˇc, Samo Zupan, and Borut Obrul. High Torque ’In-Wheel’ Motors for Rescue Vehicles. In Innovative Automobile Technology, pages 1–14, Dolenjske Toplice, 2012. Fisita. [6] Brembo. Brembo präsentiert auf der iaa sein erstes in-wheel projekt. http://www.brembo.com/en/ComunicatiStampa/2015/ BrembopresentsitsfirstIn-Wheelapplication EN.pdf, 2015. [7] Schaeffler. Schaeffler and ford unveil concept fiesta with new spacesaving electric wheel hub drives. http://www.schaeffler.com/ remotedocuments/en 2/ global/download pdf rtf/pdf/pressrelease 27225472.pdf, 2013. [8] Alexander Fraser. In-Wheel Electric Motors. In 10th International CTI Symposium, pages 1–15, Berlin, 2011. Protean Electric Ltd. [9] Clemens Satzger, Jonathan Brembeck, and Martin Otter. Framework for the Evaluation of Wheel Torque Blending Algorithms. In Advances in Automotive Control, volume 46, pages 347–352, Tokyo, 2013. IFAC. [10] Jiweon Ko, Sungyeon Ko, Hanho Son, Byoungsoo Yoo, Jaeseung Cheon, and Hyunsoo Kim. Development of Brake System and Regenerative Braking Cooperative Control Algorithm Hybrid Electric Vehicles. IEEE Transactions on Vehicular Technology, 64(2):431–440, 2015. [11] Yoichi Hori. A Novel Traction Control for EV Based on Maximum Transmissible Torque Estimation. IEEE Transactions on Industrial Electronics, 56(6):2086–2094, 2009. [12] Reza Hoseinnezhad and Alireza Bab-hadiashar Senior Member. Efficient Antilock Braking by Direct Maximization of Tire - Road Frictions. IEEE Transactions on Industrial Electronics, 58(8):3593–3600, 2011. [13] Sajad Sadr, Davood Arab Khaburi, and Jose Rodr´ıguez. Predictive Slip Control for Electrical Trains. IEEE Transactions on Industrial Electronics, 63(6):3446–3457, 2016. [14] Guoqing Xu, Kun Xu, Chunhua Zheng, Xinye Zhang, and Taimoor Zahid. Fully Electrified Regenerative Braking Control for Deep Energy Recovery and Maintaining Safety of Electric Vehicles. IEEE Transactions on Vehicular Technology, 65(3):1186–1198, 2016. [15] Tesheng Hsiao. Robust estimation and control of tire traction forces. IEEE Transactions on Vehicular Technology, 62(3):1378–1383, 2013. [16] Valentin Ivanov, Dzmitry Savitski, and Barys Shyrokau. A Survey of Traction Control and Anti-lock Braking Systems of Full Electric Vehicles with Individually-Controlled Electric Motors. IEEE Transactions on Vehicular Technology, 64(9):3878–3898, 2014. [17] Matteo Corno, Sergio M. Savaresi, and Gary J. Balas. On linear parameter varying (LPV) slip controller design for two wheeled vehicles. International Journal of Robust and Nonlinear Control, 19(12):1313– 1336, 2009. [18] Tohru Kawabe. Model Predictive PID Traction Control Systems for Electric Vehicles. In International Conference on Control Applications (CCA), pages 112 – 117, Dubrovnik, 2012. IEEE. [19] Clemens Satzger, Ricardo de Castro, and Tilman Bünte. A model predictive control allocation approach to hybrid braking of electric vehicles. In Intelligent Vehicles Symposium, pages 286–292, Dearborn, 2014. IEEE. [20] Ricardo De Castro, Rui E Araújo, Mara Tanelli, and Sergio M Savaresi. Torque blending and wheel slip control in EVs with in-wheel motors with in-wheel motors. Vehicle System Dynamics, 50(Supplement 1):71– 94, 2012. [21] Ricardo De Castro, Fabio Todeschini, Rui Esteves Ara, Sergio M Savaresi, Matteo Corno, and Diamantino Freitas. Adaptive-Robust Friction Compensation in a Hybrid Brake-by-Wire Actuator. Journal of Systems and Control Engineering, 228(10):1–18, 2013. [22] Martin Ringdorfer and Martin Horn. A robust wheel slip controller for a hybrid braking system. International Journal of Computer, Electrical, Automation, Control and Information Engineering, 5(11):2269–2273, 2011. [23] S Kanarachos and M Alirezaei. Control allocation for regenerative braking of electric vehicles with an electric motor at the front axle


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