**3. Results and Discussion**

Figures 8–10 show the results of a 10 kN clamp force step response and the linear and non-linear optimization of brake-by-wire actuators. These simulations are performed using the nonlinear plant of the actuators. The nonlinearities that exist in the EMB and EWB include motor current and voltage saturation, brake caliper saturation, and the Lugre friction model. The nonlinearities in the EHB include the valve nonlinearities and the dead-zone. The "Initial" represents the initial set of parameters of the plant before the optimization. The initial setting for each actuator is compared with a similar setting in the literature to make sure the results are sound and follow other researchers' results. However, for the optimized plants' results, since this is the first study that discusses optimization on these physical parameters, there are no other papers to compare the results with. The "TF-based Opt." represents the linear transfer function optimization, and "Nonlinear Opt." represents the results for the set of plant parameters after the optimization is performed using the nonlinear plants as discussed in Section 2.3.

For the EHB, Figure 8 shows that the transfer function-based and the nonlinear optimization both have reduced the cylinder pressure. It must be noted that the results for the initial set of parameters are consistent with [33]. The cylinder's pressure in the Zhao et al. reaches steady-state around 0.3 s, similar to the results for this study. This shows that robust control along with the linearization is working for this EHB actuator. The readers have to note that the difference between the EHB model studied here is that the valves are considered to change continuously, and therefore, a continuous Youla control scheme is used to control the valves; however, in reality, this needs to be taken care of using a digital controller and pulse width modulation technique. The clamping force response time for the optimized simulation has also decreased from 0.5 s to around 0.3 s and 0.2 s for TF-based and nonlinear optimization, respectively. The power usage plot shows that, in all the cases, the power consumption stops once the actuators reach the steady-state target. This is because of the switching logic that closes both valves once they reach the steady-state value of the clamping force. Since the optimized plants reach the steady-state faster, and they use less actuation to do so, their energy and power usage is reduced (see Table 2).


**Table 2.** Comparison of the amount of energy used in the 10 kN step response for 2 seconds (the amounts are in Joules).

**Figure 8.** Comparison of the initial parameter sets vs. optimized in an EHB for a 10 kN step input. The clamp force plot for the initial parameter setting was consistent with Zhao et al. [33]. The optimized results are the novelty of this paper.

For the EMB, Figure 9 illustrates that the clamping force step response is about the same for all plants. This is because the controllers are set to have the same bandwidth; they all have the same response. In this case, the nonlinearities are mitigated by robust controllers, and the current/voltage saturation is taken care of by the anti-windup compensator. As shown, the current reaches its saturation level for the initial EMB plant, and the gain anti-windup is shown to be working. However, the difference between the plants manifests itself in the power consumption plot. The initial plant uses a lot more power and energy to perform the same task as the optimized plants. Tf-based and nonlinear optimized plants both have a significantly smaller power usage, with the nonlinear optimized plant consuming a slightly lower amount of power. The overall energy consumption for these plants is summarized in Table 2. Comparing to Line et al., for the initial parameter setting, the clamping force also reaches the steady-state around 0.2 s [6]. The current is higher than the results shown in Line et al.; however, the voltage is not plotted for their results. One explanation is that a higher amount of current would result in lower voltage and vice versa.

**Figure 9.** Comparison of the initial parameter sets vs. optimized in an EMB for a 10 kN step input. The clamp force plot for the initial parameter setting was consistent with Line et al. [6]. The optimized results are the novelty of this paper.

For the EWB, Figure 10 shows the step response for all the plants. For the initial parameter setting, the clamping force reaches to steady-state in around 0.5 s. Compared to Che Hasan et al., which uses a PID controller, this is around 0.2 s faster [12]. Additionally, using the Youla parameterization along with cascaded control, the overshoot is also smaller. The voltage peak for both are around the same, although the voltage for this paper is slightly higher. The nonlinear optimized plant is showing a faster response than the transfer function optimized and the initial plant. The overshoot in the nonlinear optimization of the plant has also slightly decreased when compared with the Tf-based optimization plant. As shown, the current once again saturates for the initial plant. For this plant, the current was saturated for around 0.3 s, and the anti-windup compensation has taken care of this; however, this has negatively impacted the closed-loop response and made it slower. Looking at the power consumption, it is clear that the Tf-based opt. has used a slightly lower amount of power, and the nonlinear opt. has used a significantly lesser amount of actuation power. It should be noted that the voltage and current have undershot in all the plants, which comes from the fact that overshoots in the clamping force, as shown in Figure 10, are being compensated by these undershoots.

**Figure 10.** Comparison of the initial parameter sets vs. optimized in an EWB for a 10 kN step input. The clamp force plot for the initial parameter setting was consistent with Che Hassan et al. [12]. The optimized results are the novelty of this paper.

Similar to the step response, we have performed a ramp response of 10 kN/s with the saturation of 10 kN for the given plants. Figures 11–13 show similar results to the ones of the step response as discussed previously. It should be noted that the ramp response is only discussed in this paper, and the cited papers above did not mention performing this test on the actuators.

Table 3 shows the amount of energy usage by each plant with a different set of parameters. Comparing the energy usage of brake-by-wire actuators in Tables 2 and 3, we can conclude that EMB and EWB use significantly lower amounts of energy. However, looking once again at the Figures 8–10, we can see that EHB has at least a 0.1–0.2 s faster response than the dry brake-by-wire actuators such as EMB and EWB.

**Table 3.** Comparison of the amount of energy used in the ramp response for 2 seconds (the amounts are in Joules).


**Figure 11.** Comparison of the initial parameter sets vs. optimized in an EHB for a ramp input.

**Figure 12.** Comparison of the initial parameter sets vs. optimized in an EMB for a ramp input.

**Figure 13.** Comparison of the initial parameter sets vs. optimized in an EWB for a ramp input.

#### **4. Conclusions**

In this paper, we presented the modeling and a new control strategy for three different brake-by-wire actuators. The physical optimization of these plants using linear transfer functions, and nonlinear plants were discussed, and the results were presented. The optimized results show a promising energy reduction when compared with the nominal parameters. EHB's, EMB's, and EWB's energy consumption were reduced to around 10%, 3%, and 20% of their original sets of parameters, respectively. This method can be effectively utilized for other brake-by-wire actuators to reduce their energy consumption while increasing their dynamic response. It should be noted that in practice, other criteria such as structural, electrical, and heat transfer measures should be added to the optimization method. This can result in more constraints for the optimization problem, which in turn will alter the final results. However, the optimization framework and the objectives will

stay the same. Usually, having more constraints will result in a lesser deviation from the initial results. However, as shown in the results, the gains in energy consumption and dynamic responsiveness are high enough to be considered even with added constraints to the optimization problem. This calls for more future studies.

This paper aims to create a framework for optimizing brake-by-wire actuators by considering the problem from the perspective of energy consumption and actuator dynamic response. This framework can be further expanded to add other measures such as cost, weight, and reliability.

**Author Contributions:** Conceptualization, E.A. and F.A.; methodology, E.A. and F.A.; software, E.A.; validation, E.A. and F.A.; formal analysis, E.A. and F.A.; supervision: F.A.; writing—original draft preparation, review and editing: E.A. and F.A.; funding acquisition: F.A. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research received no external funding.

**Institutional Review Board Statement:** Not applicable.

**Informed Consent Statement:** Not applicable.

**Data Availability Statement:** Not applicable.

**Conflicts of Interest:** The authors declare no conflict of interest.

#### **Abbreviations**

The following abbreviations are used in this manuscript:


