# Sliding Mode Control Algorithms for Anti-Lock Braking Systems with Performance Comparisons
## 4 Validation Case Study
Although the control system structure is simple, there are many equations and parameters involved in calculating the redesigned modulating function \(\rho(t)\) from Equation (23) in [38]. The tuning parameters were selected as \(\underline{k}_{p}=0.2\), \(c_{1}=0.5\), the parameters of the reference model transfer function \(1/(0.01s+1)\) were set as \(a_{m}=100\), \(k_{m}=100\), then \(\beta=100\), \(\overline{\theta}=5\), \(\delta=0.1\), \(\Lambda=-100\), \(g=0.1\), \(\phi_{1}=\phi_{2}=0\), \(\overline{d}=2\), \(c_{d}=100\), \(\lambda_{d}=100\). The three most influential parameters of this PSSMC scheme were found \(\lambda_{1}=21.57,\lambda_{2}=9.13\) and \(\varepsilon=0.012\) using a GA to minimize the performance index \(I_{test}\) on the test scenario under control input saturation in [-1;1]. While the others parameters have been determined starting from the values in [38]. The tuned parameters were searched within the domain \([0;100]\times[0;100]\times[0;10]\). Except for \(a_{m}\), \(k_{m}\), there is a number of 15 tuning parameters to search for a large number with respect to the other controllers used for comparisons. The GA solver parameters were set to ensure convergence to approximately the same solution on five runs. The performance index is next measured for all compared controllers, to assess their performance. The set-point tracking results obtained with the controllers LSMC, RSMC, ADC, RL, and MFSMC are shown below in Figure 6 and with the PSSMC in Figure 7. The performance measured with LSMC is \(I_{test}=6.0859\times 10^{-4}\), the one with the RSMC controllers is \(I_{test}=6.0904\times 10^{-4}\), that obtained with ADC is \(I_{test}=7.1224\times 10^{-4}\), that obtained with the RL controller is \(I_{test}=747.75\times 10^{-4}\), for the MFSMC \(I_{test}=7.8029\times 10^{-4}\) and for the PSSMC \(I_{test}=498.42\times 10^{-4}\). Throughout the braking process, the index of the first sample at which the lower wheel speed drops below 10 rad/s are, respectively, \(N_{LSMC}=1272\), \(N_{RSMC}=1272\), \(N_{ADC}=1262\), \(N_{RL}=1330\), \(N_{MFSMC}=1275\), \(N_{PSSMC}=1298\). Revealing that the LSMC and RSMC controllers lead to very similar braking scenarios in terms of the car speed and measured performance. These scenarios are also not very far from the braking scenario with the ADC controller, in the sense that the car speed decreases to 10 rad/s in about the same time (I.262 s vs. 1.272 s with the LSMC and RSMC controllers). In any case, both the LSMC and the RSMC controllers offer superior performance compared to the rest of the ADC, RL, MFSMC and PSSMC controllers in terms of the measured performance index. Although the RL controller displays a poorer performance (of about two orders of magnitude), it is noticeable that the braking learning was performed without knowing the ABS dynamics. The MFSMC controller lies within a similar model-free paradigm, since the simplified model attached to the ABS dynamics makes little use of process knowledge in the control design pipeline. The PSSMC structure reveals a complex design with many tuning parameters and it is better than the RL controller. From a controller complexity viewpoint, the ADC controller requires only two tuning parameters \((k_{0},k_{1})\) which is mostly advantageous with respect to the RSMC controller with three tunable parameters and the LSMC with five tunable parameters. However, the performance of the LSMC and the RSMC is not sensitive but rather robust to various choices of \(\Delta,\xi\) that were introduced for numerical convenience. The remaining number of parameters, one for RSMC and two for LSMC, turn the two controllers into attractive real-time implementations. Figure 6: The slip control results obtained with the Lyapunov-based sliding mode controller (LSMC), RSMC, active dynamic controller (ADC), reinforcement learning (RL) and model-free sliding-mode controller (MFSMC) structures. Figure 7: The slip control result obtained with the periodic switching sliding mode controller (PSSMC) structure.