# Sliding Mode Control Algorithms for Anti-Lock Braking Systems with Performance Comparisons
## 4 Validation Case Study  
The sliding-mode control algorithms LSMC and RSMC (from Algorithms 1 and 2, respectively) are next validated on the slip control for the ABS process that was introduced in the second Section. First, the controllers' parameters are experimentally tuned to their best values who attain the smallest performance index on a given test scenario. Towards the section's end, a systematic optimization-based tuning is employed, to challenge the best achievable control performance from the LSMC and the RSMC. The feedback slip control system is presented in Figure 5.  
Figure 4: The family of first order differential equations for the RSMC stability analysis case. For subsequent comparisons, the following test scenario for all the controllers uses a desired slip \(\lambda_{d}\) calculated as a step input with amplitude 0.15 that is filtered through the first order lag transfer function \(1/(0.01s+1)\). This way, the controlled slip should track the output of a linear output reference model. The angular speed of the wheels at the start of the braking cycle is 180 rad/s. This angular speed is usually obtained after accelerating the two wheels in contact through the lower wheel. The continuous-time controller equations are numerically integrated using a fifth order solver with the fixed step size of \(10^{-3}\) seconds (s), which is consistent with the real-world execution times. The sampling time corresponding to the fixed step solver allows for data acquisition in order to collect the discrete-time values of \(\lambda(t),\lambda_{d}(t)\) as \(\lambda(k),\lambda_{d}(k)\), respectively. Which in turn allows the computation of the following performance index  
\[I_{test}=\frac{1}{N}\sum_{k=k_{i}}^{k_{i}+N-1}(\lambda(k)-\lambda_{d}(k))^{2}, \tag{15}\]  
where \(k_{i}\) is the sampling instant index corresponding to the start of the braking process, and \(N\) samples of the controlled slip and desired slip are measured, until the angular speed \(x_{2}\) of the lower wheel (the \"car\") drops below 10 rad/s towards the end of the braking cycle. The experimental parameter values for the LSMC control law (8) are selected as \(\Delta=10^{-3},\xi=10^{-3},\delta=0.1,v_{\text{max}}=1\). And the parameters of the RSMC law (13) are selected as \(\Delta=10^{-3},\xi=10^{-3},k=3\). Notice that \(U_{\text{F}}\) for the RSMC needs not to be estimated since it can be accounted for, by the selection of a large enough \(k\). Several additional controllers are next used to benchmark the performance of the LSMC and RSMC structures. We choose the first adaptive active dynamic controller (ADC) since it is a recent one implemented on the same ABS system, obeying the same working principles as our system. In fact, the ABS system is from the same manufacturer. The second reinforcement learning (RL) model-free controller stems from a different paradigm that does not use the system dynamics, therefore it is designed to cope with significant amount of uncertainty, as it is the case with RSMC and LSMC. This controller was originally proposed, developed and validated on the same ABS system. The third model-free sliding-mode controller combines the feature of the second reinforcement learning controller (in the sense that it is model-free and it does not use the system dynamics knowledge) and also the working principle of sliding-mode control (SMC). Therefore, it fits the comparisons goal. The fourth periodic switching sliding mode controller (PSSMC) is added for comparisons since it is designed to deal with uncertainties under output feedback only with no full state measurement and has been applied to a simplified ABS model before. (1) _An adaptive active dynamic controller_ (ADC) is also implemented according to the laws presented in [36], that is  
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