*1.2. Objective Metrics for Brake-by-Wire Systems*

Objective metrics are required for performing a comparative analysis of the systems under consideration. These metrics are utilized to measure each system's performance, robustness, and safety correctly. Similar metrics have already been used in other automotive applications to optimize or compare different topologies (different configurations). For example, Shankar et al. use several criteria for optimization and component sizing of plug-in hybrid electric vehicles. The objective functions in their optimization include all-electric range (AER), the CO2 emission from the drive-cycle, and the cost of components [21].

Gombert et al. provide some basic metrics for brake-by-wire actuators and their vehicle configurations [4]. They provide some background for the objective metrics that need to be considered for Brake-By-Wire (BBW) actuators. Yao et al. consider a multiobjective optimization with a few constraints for their combined electromagnetic and electronic wedge brake-by-wire actuator. The objective comprises a time to braking at

an acceptable slew rate, maximum initial braking torque, and electric power of the DC motor. Their constraints include the maximum power of the DC motor, brake slew rate, and maximum braking torque (maximum ground friction coefficient) [22,23]. Kwon et al. use a multi-objective formulation to optimize a caliper for the wedge brake. Their objective function includes the minimization of weight and the maximization of caliper stiffness. They then use the response surface model to optimize and find the best possible set of caliper parameters [24].

Metrics and metric-based optimization have also been used in the control architecture of brake-by-wire systems. Fengjiao et al. use multi-objective optimization for their control strategy of an electro-hydraulic brake system in an EV. Their objectives include 1. braking stability, which can be expressed as a quadratic function of friction adhesion on the rear and front wheels and the brake input, and 2. regenerative energy recovery. The constraints include battery charging power, motor peak torque, and the relationship between vehicle stability while braking and road surface friction [23]. Hielinger et al. used parameter optimization for an autonomous emergency braking system. Their cost function includes safety performance and customer acceptance. Safety performance is measured as the reduction of the impact speed (the speed at which the vehicle might collide to the nearest obstacle; if there is no collision, the cost becomes zero). Customer acceptance includes a sub-cost function for the brake profile (the deceleration of the vehicle summed over time) and braking the distance (minimum distance between the vehicle and the obstacle) [25]. Kelling et al. studied a distributed electronic and control architecture design for brakeby-wire systems and compared a conventional centralized architecture with a proposed fault-tolerant and distributed system in terms of safety and cost advantages [26].

#### *1.3. Control Strategies for Brake-by-Wire Systems*

Many researchers have used the sliding mode method to control the wheel slip for Anti-Lock Braking (ABS). Sliding Mode Controller (SMC) is a nonlinear control technique and an inherently non-continuous control law, which requires additional filtering to suitably smooth out this discontinuous control law, to force the system to operate on a sliding surface which defines the system's closed-loop dynamic. Compared to bang-bang control, SMC has the benefits of smaller actuation and added robustness. Anwar utilized a sliding mode controller to control slip in a hybrid BBW system that resulted in a good slip regulation in low friction surfaces and a smooth operation of the ABS, and reduced noise, vibration, and harshness (NVH) in EHB systems [27]. Tanelli et al. use pseudo-sliding mode control combining slip-deceleration (MSD), which continuously controls slip and deceleration while avoiding chattering and is robust against measurement noise and low sampling frequency [28]. However, SMC is not widely used in the automotive industry due to its design complexity, calibration difficulties, proper consideration of actuator delays, and difficulties with addressing robustness. Actuators have delays that can make the sliding mode lead to chatter, energy loss, and the excitation of unmodeled dynamics. However, this is not as much of a problem in the continuous control design [29]. Soltani et al. use a linearized model of EHB and synthesize closed-loop shaping Youla parameterization for the wheel slip control. The stability and performance of the controller were tested on an HiL (hardware in the loop) setup [30].

#### *1.4. Contribution and Paper Structure*

This paper discusses a novel approach to optimize three different brake-by-wire actuators. The novelty of this paper is as follows:


4. The use of a robust control method (Youla parameterization) to control an EHB brake with build and dump valves (the use of Youla parameterization for EMB and EWB has already been investigated in another paper by the authors [31]);.

The structure of this paper is as follows: In the materials and methods section, the procedures used to achieve the results are discussed. The actuator modeling subsection discusses how each actuator has been mathematically modeled. The model-based control synthesis subsection discusses the robust control design. The optimization section discusses the transfer function and nonlinear plant optimization. In the section results and discussion, the optimization results are presented and discussed. Moreover, in the final section, the conclusions, the final conclusion is drawn, and the benefits and the pitfalls of the optimization framework are discussed.
