# Sliding Mode Control Algorithms for Anti-Lock Braking Systems with Performance Comparisons
## 4 Validation Case Study
After each episode, the Q-function estimate is improved (in fact, a Q-NN was trained towards the optimal Q-function) and the current episode controller is improved towards the optimal controller. The objective was to minimize an infinite-horizon but discounted version of the performance index \(l_{test}\), which makes the tuning objective approximately minimize \(l_{test}\). For additional details please consult [33]. (3) _A model-free sliding-mode controller_ (MSSMC). In the work of [37], two model-free sliding mode control system structures were proposed, the second one proposes the control law:  
\[ \begin{array}{l}u(t)=\frac{1}{\alpha}\left[-\hat{F}(t)+\dot{\lambda}_{d}(t) -K_{P}e(t)-K_{l}\int\limits_{0}^{t}e(\tau)d\tau-\right.\\ \left.\frac{1}{T}\times\left(\delta\alpha T\right.\left.+\psi+T\right.e_{\text{ test max}}+\left|K_{l}T\int\limits_{0}^{t}e(\tau)d\tau+(K_{P}T-1)e(t)\right|)\text{sgn}( \sigma(t))\right],\end{array}  \tag{17}\]  
where \(e(t)=\lambda(t)-\lambda_{d}(t)\) is the tracking error with respect to the target slip and \(\hat{F}(t)=\hat{\lambda}(t)-\alpha\ u(t)\) is the uncertainty estimate stemming from a first order phenomenological model simplification of the ABS system that reads \(\dot{\lambda}(t)=F(t)+\alpha\ u(t)\). Here, \(\alpha\) is a user-selectable constant parameter seen as a minimal process knowledge, \(K_{P},K_{I}\) are the user-selectable parameters of the intelligent PI (iPI) controller part, \(\sigma(t)=\Sigma_{1}(t)+T\ \Sigma_{2}(t)\) is a switching variable defined in terms of \(\Sigma_{1}(t)=\ \int_{0}^{t}e(\tau)d\tau,\ \Sigma_{2}(t)=e(t)\), the user-selectable \(T>0\) imposes the desired behavior on the sliding surface \(\sigma(t)=0\). Additionally, \(e_{est\ max}\) acts as another design parameter as the upper bound of the uncertain tracking error dynamics and \(\psi>0\) is again user-selectable to ensure a conservative stability condition. Subjected to a tuning aimed at minimizing the performance index \(I_{test}\) on the test scenario using genetic algorithm (GA), the parameters values for the MFSMC controller were found as \(\alpha=2.02\), \(K_{P}=15.01\), \(K_{I}=0.05\), \(\Psi=0.05\), \(T=100.09\), in their respective search domain \([0;10]\times[0;100]\times[0;100]\times[0;10]\times[0;200]\), with \(e_{est\max}=10^{-3}\) and \(\delta=10^{-3}\) being fixed. The GA solver parameters were set to ensure convergence to approximately the same solution on five runs. (4) _A periodic switching sliding-mode controller_ (PSSMC) from [38]. The nonlinear ABS model is rewritten as a linear model in the state with disturbance term complemented with a nonlinear output equation depending on a linear state combination, to comply with the model (1) and (2) from [38], as:  
\[\left( \begin{array}{c}\dot{x}_{1}\\ \dot{x}_{2}\end{array} \right)=\left( \begin{array}{cc}-1&0\\ 0&-1\end{array} \right)\left( \begin{array}{c}x_{1}\\ x_{2}\end{array} \right)+\left( \begin{array}{c}1\\ 0.5\end{array} \right)u+\left( \begin{array}{c}f_{1}+g_{1}u+x_{1}-u+v_{1}\\ f_{2}+g_{2}u+x_{2}-0.5u+v_{2}\end{array} \right), \tag{18}\] \[ \begin{array}{c}\mathbf{A}\\ \mathbf{B}\end{array}\begin{array}{c}d(t)\\ d(t)\end{array} \] \[y(t)=\lambda=\frac{\mathbb{C}[x_{1}\ x_{2}]}{r_{2}x_{2}}=h(\mathbf{C }\mathbf{x},t),\]  
where \(\mathbf{C}=[-0.01\ 0.01]\) was selected, as well as the matrices \(\mathbf{A}\) and \(\mathbf{B}\), ensuring that \(\mathbf{C}\mathbf{B}\
eq 0\). Possible exogeneous disturbances, unmodeled dynamic as well as parametric uncertainties are modeled under terms \(v_{1}(t),v_{2}(t)\) who enter the larger disturbance term \(d(t)\) for which a simplifying upper bound \(\overline{d}\) is found. In the following, the control law (17) from [38] is implemented based on Equations (5), (14), (16), (18) and (23) from the same work. This control scheme is designed for output model reference tracking control, it includes dynamical filters for the underlying uncertain controlled system missing full state measurement. The proposed PSSMC structure was tested in [38] on a linear, simplified, first order ABS model.